### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# BOUNDARY LAYER MET ATM S 547

UW

GPA 4.0

### View Full Document

## 36

## 0

## Popular in Course

## Popular in Atmospheric Science

This 97 page Class Notes was uploaded by Ms. Imani Mante on Wednesday September 9, 2015. The Class Notes belongs to ATM S 547 at University of Washington taught by Bretherton in Fall. Since its upload, it has received 36 views. For similar materials see /class/192396/atm-s-547-university-of-washington in Atmospheric Science at University of Washington.

## Similar to ATM S 547 at UW

## Reviews for BOUNDARY LAYER MET

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/09/15

Atm S 547 Boundary Layer Meteorology BretheIton Lecture 12 The diurnal cycle and the nocturnal BL Over at land under clear skies and with weak thermal advection the atmospheric boundary layer undergoes a pronounced diurnal cycle A schematic and an example from the Wangara ex periment are shown on the next page This archetypical diurnal cycle is muted by clouds and can be entirely obscured by rapid changes in the free atmospheric conditions due for instance to the passage of a midlatitude cyclone or front It is also highly modi ed by terrain or nearby landsea contrasts Despite these caveats it is illuminating to study the archetypical case in more depth During the night the BL is stable due to surface longwave cooling and a shallow temperature inversion of typically 100500 m builds up After dawn surface heating builds up a shallow con vective mixed layer which deepens slowly and rapidly warms until it fully erodes the nocturnal stable layer At this point the top of the new mixed layer starts to penetrate into the residual layer the remnants of the previous day s aftemoon mixed layer This layer is very weakly stratified so the new mixed layer rapidly deepens into it until it encounters the top of the previous day s mixed layer which tends to be marked by a weak inversion At this point further BL warming occurs much more slowly as a much deeper layer must be warmed than in the early morning In the late afternoon the solar heating is no longer sufficient to maintain upward surface buoyancy uxes Within an hour a few eddy turnover times turbulence collapses through most of the boundary layer and becomes restricted to a shallow layer typically 100 m deep driven by surface drag Dur ing the night clearair radiative cooling is most intense near the cold surface enhancing the static stability of the lowest couple of hundred meters of air Much of the nocturnal inversion can be at tributed to this cooling rather than downward turbulent heat uxes However downward heat uxes of up to 50 W m392 can occur near the surface at night under moderately strong geostrophic winds Morning growth of the boundary layer Garratt 61 The rate of growth of the convective mixed layer is dictated primarily by energy balance though entrainment dynamics also play a significant role As a simple example consider the growth of a mixed layer driven by a surface buoyancy ux Bo into an atmosphere of constant buoy ancy frequency N2 The mean buoyancy profile in the free troposphere is bz N22 g9vz GVRGVR where we have chosen GVR as the initial 9 at z 0 We assume i that the buoyancy ux is turned on at time t 0 and ii that it leads to a convective mixed layer of depth ht governed by the entrainment closure w39b39h weAb BBO B 02 empirically 1 It is interesting to compare the solution with a realistic Bto the case B 0 In the latter limit called encroachment convection is assumed not to be penetrative and the mixed layer entrains air only when its buoyancy is no larger than that of the mixed layer air Lastly iii we neglect any mean vertical motion within the atmosphere so we dhdt The buoyancy bz t obeys Bbdt BBzw39b39 121 Atm S 547 Boundary Layer Meteorology Bretherton k2 km I I d 1 l39 Residual layer I Mixe ayer I I l I I Top of nocturnal inversion I 39 100 300m I I NBL z y NBL Surfacelayer H H Rise A B Set Fig 61 Schematic representation of ABL evolution throughout the diurnal period over land under clear skies 20 Arya 039 II LllIJl LJ zlkm 0 5 ILIII 3 hay39139 L 1 o 5 10 5 O N O hlk lo m 05 B uluunl o 4 van N 18 0 LOCAL TIME h Fig 52 Diurnal variation of potential temperature pro les and the PBL height during a day 33 and b days 3334 of the Wangara Experiment c Curve A convective Curve B stable After Deardorff 1978 0quot 122 Atm S 547 Boundary Layer Meteorology Bretherton Integrating from the surface up to a fixed height H above the mixed layer top we see that 3 H H 0bdziiwb 0 730 Graphically letA be the net area added to the buoyancy pro le by the heating of the BL Then A Bot 2 We can now compare the cases of encroachment and an entraining boundary layer The encroach ing BL has depth given by hN2h2 A 30 gt hm ZBotN212 As expected h deepens more slowly as it gets larger since more heat must be imparted to a deeper boundary layer to raise its buoyancy by a given amount For the entraining BL there is a similarity solution in which the buoyancy profile retains the same shape as it grows so that Abt cN2ht c is an as yet unknown constant Consistency of l and 2 determines c From 1 BBQ weAb dhdUcNZh Integrating this equation from time 0 to t starting with h0 0 we get BBQ cN2h22 3 Turning now to 2 we write A as the difference of the positive area prhere the mixed layer buoyancy bmt exceeds the original environmental buoyancy and the negative area A Nwhere pen Encroachment A Entrainment Z b 2 ha hencrt A Bot AP gt V DJ gt BO m b Convective mixed layer evolution illustrating more rapid deepening if entrainment is assumed to be I quot B 02 J to 39 B 0 r 123 Atm S 547 Boundary Layer Meteorology Bretherton etrative convection has reduced the buoyancy From the figure above we see that bm Ab bt Nzh so bm lcN2h The heights ofthe triangles making up AP and AN are N392 times as long as their bases so the area ofAP is bmbmN22 and similarly for AN Hence 2 can be written Bot A AP A N mezv2 Ab22N2 1 c2 c2N2h22 1 2cN2h22 4 Dividing 3 by 4 we see that B cl 2c or that c Bl 2B It follows from 4 that hm zBorN20 2a 2 230ml ZBN2 2 1 BMW We conclude that entrainment contributes about 5 20 to the boundary layer deepening For a 1 km deep BL and N2 10394 s39l the inversion strength would be Ab l4N2h H AGV cs 04 K re gardless of the surface buoyancy ux Entrainment hardly changes the boundary layer temperature The nocturnal jet As turbulence dies down in the residual layer in late aftemoon it decouples from the BL The momentum ux convergence that was helping to reduce and turn the wind during the day suddenly disappears leaving a wind profile in which there is an imbalance between the two main horizontal 124 Atm S 547 Boundary Layer Meteorology Bretherton forces Coriolis force and pressure gradient force The figure below shows the resulting evolution of the wind during one night of the Wangara experiment which took place over at ground Dur ing the night a strong jet develops above the nocturnal BL In the bottom panel is another example in which the geostrophic wind is also plotted During the daytime the wind component along the geostrophic wind direction is subgeostrophic but at night it is supergeostrophic This is one of the cleanest atmospheric examples of an inertial oscillation The pressure gradi ent is horizontally uniform so the ageostrophic wind ua u ug rotates clockwise with the Coriolis period 27tf which at midlatitudes is somewhat less than a day Supergeostrophic winds ensue dur z km Garratt a 05 O 0000 0400 0800 l 200 1600 Local time I 200 1 600 2000 c 6 8 10 ums Fig618 Observations illustrating the formation of the nocturnal jet a Windspeed pro les on day 13 of WANGARA local times indicated b Height time crosssection of Wind speed in m S on days 1314 at WANGARA Isopleths of wind speed are drawn at 15 rns 1 intervals c Profiles of the ucomponent of the wind velocity with the xaxis along the geostrophic wind direction for midafternoon 1330 UT 6 August 1974 and early morning 0514 7 August 1974 near Ascot England After Thorpe and Guymer 1977 Quarterly Journal of the Royal Meteorological Society 125 Atm S 547 Boundary Layer Meteorology Bretherton t nIfl Garratt zm2m tznIfl b t712f Fig 619 Illustrated solutions of the unbalanced momentum equation Eq 677 for a a lowroughness surface and b a high roughness surface undamped inertial oscillations are shown for the southern hemisphere in the form of anticlockwise rotation of the wind vector V about the geostrophic wind vector Vg ing the night as shown in the gure above In the morning the convective mixed layer deepens into the residual layer so the wind pro le becomes frictionally coupled again The Great Plains nocturnal southerly jet prominent during the springtime when it can achieve speeds of 30 m s391 less than 1 km above the surface partially owes its origin to this mechanism In this region cli matological southerly geostrophic ow occurs due to a thermal low over the elevated terrain to the west i e the Rockies The strong enhancement of lowlevel southerlies during the night help pump humid air northward where it can help fuel severe thunderstorms and mesoscale convective systems through the night 126 Atm S 547 Boundary Layer Meteorology Bretherton Lecture 8 Pararnete zation of BL Turbulence I In the next two lectures we will summarize several approaches to parameterization of BL ver tical turbulent transports that are commonly used in largescale forecast and climate models In such models the horizontal grid resolution is insufficient to resolve the most energetic BL turbu lent eddies which might be tens of meters to 12 km across Furthermore while the lowest one or two model levels are usually taken to be 100 m or less from the ground to resolve stable BLs the vertical grid spacing at a height of 1 km is typically 100500 In so the vertical structure of the BL can be at best coarsely resolved Table 1 shows the distribution of thermodynamic gridpoints in the lowest 20 of the atmosphere for three representative models the NCAR Community Climate Model version 3 CCM3 18 levels overall the ECMWF operational forecast model 60 levels overall and the MM5 mesoscale model as used for realtime forecasting in the Pacific Northwest 37 levels overall Three parameterization approaches are popular In order of simplicity they are 1 Mixed layer models 2 Local closures based on eddy diffusivity 3 Nonlocal closures Horizontal turbulent uxes are invariably neglected as they are very small compared to advection by the mean wind We will reserve discussion of parameterization of cloudy boundary layers for later Model Levels in the Lower Troposphere CCM3 ECMWF MM5 000 mb 850 mb 900 mb 950 b m i I 1000 mb 8l Atm S 547 Boundary Layer Meteorology Bretherton Mixed Layer Models Mixed layer models MLMs assume that E 3 and g in the BL are uniform wellmixed They are most applicable to convective BLs and represent stable BLs rather poorly However they are relatively simple to add moist physics too and do not require a ne vertical grid to work They are mainly used by researchers and teachers as a conceptual tool for understanding the impacts of different physical processes on BL turbulence However at least one GCM the CSUUCLA GCM uses a mixed layer model to describe the properties and depth of its lowest grid layer For simplicity we will consider a case with no horizontal advection or mean vertical motion no ther mal wind and no diabatic effects above the surface We will assume that the surface momentum and buoyancy uxes are given in general these will depend upon the mixed layer variables but we needn t explicitly worry about this now We let h be the mixed layer top at which there may be jumps in the winds and potential temperature denoted by A Turbulence in the mixed layer en trains freetropospheric air from just above the mixed layer causing h to rise at the entrainment rate we mvivggt 1 g fmiug 2 3 we 4 Since the left hand sides ofl3 are heightindependent the right hand sides must be too so the uxes of u v and 9 are linear with 2 note that this would no longer be the case for a baroclinic BL in which 11g varied with height or in the presence of internal sources or sinks of 9 The uxes are given at the surface The entrainment deepening of the BL in which freetropospheric air with value 5 Ad of some property a u v 9 is replaced by BL air with a 5 at the rate we requires a ux w39a39h weAa Thus the right hand side of the mixed layer equation for Z is just a iweAaiw aKO w 61 7 32 h This closes the set of equations l4 except for a specification of we called the entrainment clo sure This is the big assumption in any MLM For cloudfree unstable to nearly neutral mixed lay ers formulas such as that from last time Moeng and Sullivan 1994 are commonly used w b h wAb 02w 3 u3h where Ab gAGvGvo Recall that w and m are determined by the surface buoyancy and momentum uxes respectively so this closure determines we in terms of known variables enabling l4 to be integrated forward 82 Atm S 547 Boundary Layer Meteorology Bretherton 1 05 0 05 10 w 0V w 0v 0 Fig 62 Experimental data on the vertical variation of the virtual heat ux normalized by its surface value h is the depth of the mixed layer Data are for three days from the 1983 ABL experiment see Stull 1988 Figs 31 32 and 33 See also Fig 623 of this volume In a convective BL entrainment buoyancy ux is OZBO in time This type of entrainment closure is wellsupported by observational evidence and LES simulations especially in the purely convective BL in which the above relation reduces to w b 39m O2w3 My 02BO An observational veri cation of this from data taken in a daytime convective BL over land is shown above In fact a classic application of a MLM is to the deepening of a convective boundary layer due to surface heating we ll look at this when we discuss the diurnal cycle of BLs over land Local Eddy diffusivity parameterizations Garratt 87 In eddydiffusivity often called Ktheory models the turbulent ux of an adiabatically con served quantity a such as G in the absence of saturation but not temperature T which decreases when an air parcel is adiabatically lifted is related to its gradient dc z I I K w a ad 5 The key question is how to specify Ka in terms of known quantities Three approaches are com monly used in mesoscale and global models i Firstorder closure in which Ka is specified from the vertical shear and static stability or by prescribing a ii 15order closure or TKE closure in which TKE is predicted with a prognostic energy equa tion and Ka is specified using the TKE and some lengthscale iiiKpro les in which a specified profile of Ka is applied over a diagnosed turbulent layer depth From here on we will drop overbars except on uxes so az will refer to an ensemble or horizontal average at level 2 The following discussion of theseapproaches is necessarily oversimpli ed a lot of work was done in the 1970 s on optimal ways to use them An excellent review of first 15 and secondorder closure is in Mellor and Yamada 1982 Rev Geophys Space Phys 20 851875 Firstorder closure We postulate that Ka depends on the vertical shear s dudz the buoyancy frequency N2 and an eddy mixing lengthscale Z In most models saturation or cloud fraction is accounted for in the 83 Atm S 547 Boundary Layer Meteorology Bretherton computation of N2 From the shear and stability one de nes a Richardson number Ri Nzsz Di mensionally K length2time 1st4110 6 One could take the stability dependence in F Ri the same as found for the surface layer in Mo ninObhukov theory e g FRi Ia m 391 where C depends on Ri as in the surface layer and Pa 1quot ifa is momentum or 1 ifa is a scalar This is fine in the stable BL In the convec tive BL it gives see notes p 62 FmRi 1 16Ri 2 andFmRi 1 16Ri34 However in nearly unsheared convective ows one would like to obtain a finite K independent of s in the limit of small 3 This requires Fa cc Ri12 so Kg cc 12s Ri12 12N21 This is consistent with the M0 form for Km but not for Kh Thus we just choose Kh Km to obtain 1 7 16Ri12 unstable Fh mRi 17 mi2 stable No turbulent mixing is diagnosed unless Ri lt 02 Every model has its own form of F Ri but most are qualitatively similar to this Usually if this form is used within the stable BL the F s are en hanced near the surface no Ri 02 cutoff to account for unresolved ows and waves driven for instance by landsea or hillvalley circulations that can result in spatially and temporally intermit tent turbulent mixing Many prescriptions for I exist The only definite constraint is that l gt kz near the surface to match 6 to the eddy diffusivity in neutral conditions to that observed in a log layer K a kmz One commonly used form for I suggested by Blackadar 1962 is 70 WMk2 where the asymptotic lengthscale 7 is chosen by the user A typical choice is 7 50 100 m or roughly 10 of the boundary layer depth The exact form of l is less important than it might ap pear since typically there will be i layers with large K a and small gradients i e fairly well mixed layers in which those small gradients will just double but still be small if K a is halved to maintain the same uxes ii layers with small K a where physical processes other than turbulence will tend to dictate the vertical profiles of velocity and temperature and iii a surface layer in which the form of K a is always chosen to match MoninObuhkov theory and so is on solid obser vational ground l5order closure Now we prognose the TKE e q2 2 based on the shear and stability profiles Using the same eddy mixing lengthscale as above dimensionally K a lengthvelocity lq S aGM GH a M momentum orH heat GM 123292 GH 12N2q2 Closure assumptions and measurements discussed in Mellor and Yarnada dictate the form of SM and SH in terms of the nondimensional shear and stratification GM and GH These are complicated algebraic expressions but are shown in the figure on the next page As in first order closure the 84 Atm S 547 Boundary Layer Meteorology Bretherton stability functions are larger in unstable stratif1cation GHgt 0 than in stable strati cation GH lt 0 To determine the evolution of q we use the TKE equation a q2 B T 7 322 5 8 0 We model the shear and buoyancy production terms using eddy diffusion to find the uxes z l I 37 Z 2 S u w a Z v w E qumdudz Mellor and Yamada 1982 m 30 4 Fig 3 The stability functions SMGH GM and 5M0 GM The heavy solid lines are contours of SM whereas the dashed lines are contours of S The lighter solid lines are contours of P PbE One could also draw lines of constant R GHGM which are radial lines on this diagram The shaded portion is where w2q2 S 012 Stability functions for TKE closure B W quth We model the transport term by neglecting pressure correlations and using eddy diffusion to model the ux of TKE wpzo 2 2 Z a q 2 a q w e KqE 2 quqE 2 Sq 1s often taken to be 02 2 a f 1 a a q T 3le e pow p l azllqsqazl 2 Lastly the dissipation term is modelled in terms of characteristic turbulent velocity and length scales While the lengthscale in 8 is related to the master lengthscale it is necessary to intoduce a scaling factor to get the TKE to have the right magnitude 8 elf131 Blz15 85 Atm S 547 Boundary Layer Meteorology I n p mm m rum a lip I mann4 I A a pad t n i umrvwmalu1v 4 mm n m M mu nmmm39 quot 1 Virtual temperature evolution observed during two days of the Wangara eXpt top and modelled With a TKE closure bottom 39quotIxquotli l quot39 11quot quot 391 r r I t It I 1 u I t I lll 1 It lll l I III l i E I llll j 39 II 1 39 l v 39 NJ I 39 139 y r r r u I39I Kl39l I ll all I 39 ling 39 I I E i n II I hrw1 bit39l rnmuomdugll Warnmuses M u n u u n R u milHm Emaimmpn pmyj39 Same for u velocity 86 Bretherton Atm S 547 Boundary Layer Meteorology Bretherton nFl SH39 MM l V U rquot Bu 1 C a LE Irg N Tall39r xnll 1quot L AILIHV H vl urllquotu m w 39 ilwr tm39mrnl humIx gnu5w mm 1 Mn WEIquot1 urr 39 Ind lhr 39IIIFpIL II Juan man4U Innum Murry ll 1 III 39 n39 Modelled TKE pro le for simulation on previous page With these forms for all the terms in the TKE equation it can now be integrated forward in time The basic improvement in using TKE vs 1st order closure is that there is TKE transport through eddy diffusion and storage In the surface layer where storage and transport are negligible com pared to local shear and buoyancy production of TKE the latter must balance dissipation and one findsthate S B so 413031 lq Smldwdzl2 lq ShN2 q2 BllzSmldudzl2 ShNZ Hence q can be eliminated in place of the local shear and stratification and we recover the first order closure method In the S H SM figure the thin line P SPB8 1 corresponds to this case K profile methods For specific types of boundary layer one can use measurements and numerical experiments to specify a profile of eddy diffusivity which matches the observed uxes and gradients This can be particularly useful in situations such as stable nocturnal BLs which can be difficult for other meth ods Such methods require a diagnosis of BL height h then specify a profile of K For instance Brost and Wyngaard 1978 combined theoretical ideas and observational analysis to proposed the following profile for stable BLs Km kuhPzl SZL 132 zh1 zh32 This method is designed to approach the correct form ku mzL in the surface layer where zh ltlt 1 Similar approaches are have been used for convective BLs Advantages of the K profile method are that it is computationally simple and works well even with a coarsely resolved BL as long as the BL height h can be diagnosed fairly well On the other hand it is tuned to specific types of BLand may work poorly if applied more generally than the situations for which it was tuned 87 Atm S 547 Boundary Layer Meteorology Bretherton Comments on local closure schemes Firstorder closure is most appropriate for neutral to weakly stable BLs in which little transport of TKE is occuring and the size of the most energetic eddies is a small fraction of the BL depth In this case it is reasonable to hope that the local TKE will be dependent on the local shear and sta bility and that since the eddies are small they can be well repressented as a form of diffusion However it works tolerably well in convective boundary layers as well except near the entrain ment zone In an entrainment zone Transport of TKE into the entrainment zone is required to sus tain any turbulence there Since this is ignored in 1st order closure there is no way for such a model to deepen by entrainment through an overlying stable layer as is observed BL layer growth must instead be by encroachment i e the incorporation of air above the BL which has a buoyancy lower than that within the BL This does allow a surfaceheated convective boundary layer to deepen in a not too unreasonable manner but creates severe problems for cloudtopped boundary layer mod eling Almost all largescale models e g CCM3 ECMWF and MMS include a firstorder clo sure scheme to handle turbulence that develops above the BL due to KelvinHelmholtz instability or elevated convection for instance l5order closure is also widely used especially in mesoscale models where the timestep is short enough not to present numerical stability issues for the prognostic TKE equation The Mel lorYamada and GaynoSeaman PBL schemes for MMS are 15 order schemes that include the ef fect of saturation on N2 The BurkThompson scheme for MMS is a 15 order scheme with additional prognostic variables for scalar variances MellorYamada Level 3 The TKE equa tion in l5order closure allows for some diffusive transport of TKE This creates a more uniform diffusivity throughout the convective layer and does permit some entrainment to occurQuite re alistic simulations of the observed diurnal variation of boundary layer temperature and winds have been obtained using this method see figures on next page However getting realistic entrainment rates for clear and cloudtopped convective BLs with this approach requires considerable witch craft The BL top tends to get locked to a fixed grid level if there is a significant capping inversion and vertical grid spacing of more than 100 m or so TKE closure has also proved successful for cloudtopped boundary layers but again only with grid spacings smaller than is currently feasible for GCMs Grenier and Bretherton 2001 M WR 129 357377 showed that this method works well for convective BLs even at coarse resolution when combined with an explicit entrainment pa rameterization at the BL top implemented as an effective diffusivity K profile methods are widely used in GCM BL parameterizations e g CCM3 For convective boundary layers a nonlocal contribution is usually also added to the uxes see below Nonlocal closure schemes Any eddy diffusivity approach will not be entirely accurate if most of the turbulent uxes are carried by organized eddies filling the entire boundary layer such as boundary layer rolls or con vection Consequently a variety of nonlocal schemes which explicitly model the effects of these boundary layer filling eddies in some way have been proposed A difficulty with this approach is that the structure of the turbulence depends on the BL stability baroclinicity history moist pro cesses etc and no nonlocal pararneterization proposed to date has comprehensively addressed the effects of all these processes on the largeeddy structure Nonlocal schemes are most attractive when the vertical structure and turbulent transports in a specific type of boundary layer i e neutral or convective must be known to high accuracy For instance successful applications include the detailedthermal structure i e deviation from neutral static stability within a convective boundary layer or the velocity structure and relation of nearsurface wind to geostrophic wind within a 88 Atm S 547 Boundary Layer Meteorology Bretherton nearneutral boundary layer this is the motivation for the PBL model developed here at UW by Bob Brown s group 89 Atm S 547 Boundary Layer Meteorology Bretherton Lecture 2 Turbulent Flow 165 Turbulem water m annndurml numum The spinal re nluuun is admumu m resaIVe um mm mm he mmmmon uiju flmd m the plane of ym gomv was in the downstream w of the photograph 7month mm mm The Reynolds number n appmmmsu39ly 1300 Note the diverse scales of eddy motion and selfsimilar appearanc at different lengthscales ofthe turbulence in this Waterjet Only eddies of size 001L or smaller are subject to substantial Viscous dissipation 2l Atm S 547 Boundary Layer Meteorology Bretherton Description of Turbulence Turbulence is characterized by disordered eddying uid motions over a wide range of length scales While turbulent ows still obey the deterministic equations of uid motion a small initial perturbation to a turbulent ow rapidly grows to affect the entire ow loss of predictability even if the external boundary conditions such as pressure gradients or surface uxes are unchanged We can imagine an in nite family or ensemble of turbulent ows all forced by the same boundary con ditions but starting from a random set of initial ows One way to create such an ensemble is by adding random small perturbations to the same initial ow then looking at the resulting ows at a much later time when they have become decorrelated with each other Turbulent ows are best characterized statistically through ensemble averaging i e averag ing some quantity of interest across the entire ensemble of ows By definition we cannot actually measure an ensemble average but turbulent ows vary randomly in time and along directions of symmetry in space so a sufficiently long time or space average is usually a good approximation to the ensemble average Any quantity a which may depend on location or time can be parti tioned a Z a39 where Z is the ensemble mean of a and a39 is the fluctuating part or perturbation of a The ensem ble mean of a39 is zero by definition a39 can be characterized by a probability distribution whose spread is characterized by the variance a a One commonly referred to measure of this type is the turbulent kinetic energy TKE per unit mass TKE 017 v v w w This is proportional to the variance of the magnitude of the velocity perturbation TKE 517 9739 u392 v392 w 212 We may also be interested in covariances between two quantities a and b These might be the same field measured at different locations or times i e the spatial or temporal autocorrelation or dif ferent fields measured at the same place and time e g the upward eddy heat ux is proportional to the covariance w T between vertical velocity w and temperature T Variances and covari ances are called second order moments of the turbulent ow These take a longer set of measure ments to determine reliably than ensemble means The temporal autocorrelation of a perturbation quantity of measured at a fixed position a ta t T a ta t Rm can be used to define an integral time scale 17a j RTdT which characterizes the timescale over which perturbations of a are correlated One may similarly 22 Atm S 547 Boundary Layer Meteorology Bretherton 1 gtl Tritton Figure 205 Typical correlation curves de ne an integral length scale One commonly referredto statistic for turbulence in which buoyancy forces are important in volves thirdorder moments The vertical velocity skewness is de ned w w w 32 w Iw S The skewness is positive where perturbation updrafts tend to be more intense and narrower than perturbation downdrafts e g in cumulus convection and is negative where the downdrafts are more intense and narrower e g at the top of a stratocumulus cloud Skewness larger than 1 indi cates quite noticeable asymmetry between perturbation up and downdrafts Fourier spectra in space or time of perturbations are commonly used to help characterize the distribution of the uctuations over different length and time scales For example given a long time series of a quantity al we can take the Fourier transform of its autocovariance to get its tem poral power spectrumvs frequency 0 3510 a la l Texp imTdT this is real and positive for all 0 Given the power spectrum one can recover the autocovariance by an inverse Fourier transform and in particular the variance is the integral of the power spectrum over all frequencies a la l 13mwm so we can think of the power spectrum as a partitioning of the variance of a between frequencies For spatially homogeneous turbulence one can do a 3D Fourier transform of the spatial auto covariance function to obtain the spatial power spectrum vs wavevector k A 1 Sak 3 a ra r ReXp lk Rdk 2n again the variance a is the integral of the power spectrum over all wavenumbers a ra r otsamwk 23 Atm S 547 Boundary Layer Meteorology Bretherton If the turbulence is also isotropic i e looks the same from all orientations then the power spec trum depends only on the magnitude k of the wavenumber and we can partition the variance into different wavenumber bands a ra r jsak4nk2dk In particular for homogeneous isotropic turbulence we can partition TKE into contributions from all wavenumbers this is called the energy spectrum Ek 1f TKE Eq rq r J OEkdk Roughly speaking the energy spectrum at a particular wavenumber k can be visualized as being due to eddies whose characteristic size diameter is 27ck Turbulent Energy Cascade Ultimately boundary layer turbulence is due to continuous forcing of the mean ow toward a state in which shear or convective instabilities grow These instabilities typically feed energy most ly into eddies whose characteristic size is comparable to the boundary layer depth When these ed dies become turbulent considerable variability is also seen on much smaller scales This is often described as an energy cascade from larger to smaller scales through the interaction of eddies It is called a cascade because eddies are deformed and folded most ef ciently by other eddies of com parable scales and this squeezing and stretching transfers energy between nearby lengthscales Thus the large eddies feed energy into smaller ones and so on until the eddies become so small as to be viscously dissipated There is typically a range of eddy scales larger than this at which buoy ancy or shear of the mean ow are insignificant to the eddy statistics compared to the effects of other turbulent eddies in this inertial subrange of scales the turbulent motions are roughly homo geneous isotropic and inviscid and if fact from a photograph one could not tell at what length scale one is looking i e the turbulence is selfsimilar Dimensional arguments have always played a central role in our understanding of turbulence due to the complexity and selfsimilarity of turbulent ow Kolmogorov 1941 postulated that for large Reynolds number the statistical properties of turbulence above the viscous dissipation scale are independent of viscosity and depend only on the rate at which energy produced at the largest scale L is cascaded down to smaller eddies and ultimately dissipated by viscosity This is measured Garratt 1 Equilibrium range Energy input 1 Inertial subrange r Viscous dissipation l Energy Ll Frequency or wavenumber gt Fig 21 Schematic representation of the energy spectrum of turbulence 24 Atm S 547 Boundary Layer Meteorology Bretherton by the average energy dissipation rate 8 per unit mass units of energy per unit mass per unit time or m2 s393 If the largest scale eddies have characteristic eddy velocity V dimensional analysis im plies eoc V3L and the dissipation timescale is the eddy turnover timescale LVwhich is typically 01000 ml m s39l 1000 s in the ABL This means that if its largescale energy source is cut off turbulence decays within a few turnover times The viscous dissipation lengthscale or Kolmogorov scale T depends on 8 m2s393 and V m2s391 so dimensionally n v3e14 z 1 mm for the ABL Re3934L Kolmogorov argued that the energy spectrum E k within the inertial subrange can depend only on the lengthscale measured by wavenumber k and 8 Noting that Ek has units of TKEwave number m2s392m391 m s392 dimensional analysis implies the famous 53 power law 0C 823k53 Ll ltlt k ltlt nil Similarly the spatial power spectra of velocity components and scalars a also follow 339 ak cc k3953 in the inertial range The spatial power spectrum can be measured in one direction by a sensor moving with respect to the boundary layer at a speed U comparable to or larger than V i e if the wind is blowing dif ferent turbulent eddies past a sensor on the ground or if we take measurements from an aircraftWe must invoke Taylor s frozen turbulence hypothesis that the statistics of the turbulent field are similar to what we would measure if the turbulent field remained unchanged and just advected by with the mean speed U In general empirically this appears to be a good assumption Temporal power spectra S a D gathered in this way can be converted to spatial power spectral by substitut ing 0 Uk ak U aUk for turbulence moving by with mean speed U Thus we expect an 03958 temporal power spectrum for scalars and velocity components in the in ertial subrange The figures below show measurements from a tethered balloon stationed in a convecting cloudtopped boundary layer at 85 of the inversion height The mean wind of U 7 m s391 is con siderable larger than the characteristic largeeddy velocity of V l m s39l so Taylor s hypothesis is safe The time series shows up and downdrafts associated with large eddies with width and height comparable to the BL depth of 1 km with turbulent uctuations associated with smaller ed dies The corresponding temporal power spectrum triangles is plotted as DSaD as expected this has a 039 dependence in the inertial range and decays at low frequencies that correspond to lengthscales larger than L The second spectrum circles is in the entrainment zone which is in a very sharp and strong inversion stable layer at the BL top Here large scale strong vertical motions are suppressed and the turbulence is highly anisotropic at these scales but at small scales a few meters or less an inertial range is still observed 25 Ams 547 BanndaryLayzxMz eamlngy Brahman n 39 1W 2 l W H quotH M A1 I if IPA 4 252 L 39 i W Vemcalvelncx tmemxal mm tz micams hdhgman mm W F P an distance mm m m u km dzep camcnng banndarylayzx 1 W mT mu m quotw Tm quothm mmrmmm T 1mm mm W shamb andcmles m m Kmmn m m 331 mp m m39l Intuesmwgly 20 Il mllznce39 dnesn39t pmdvae an emxgycascadz a mu scalzs mm m 21m gmulmzd an m campm emxgytznds a be transferred m w my scale mmmns pap mmzdbythz bawdsz andbmadngmns afnnaathlywrgnng nwappar mmmynzdbyshzu ms and Intense Inngrlmdwmces Atm S 547 Boundary Layer Meteorology Bretherton Lecture 7 More on BL wind pro les Stability Above the surface layer the wind pro le is also affected by stability As we mentioned previ ously unstable BLs tend to have much more wellmixed wind pro les than stable BLs The gures below show observations from the Wangara experiment on how the velocity defect laws and tem perature pro le are altered by BLstability as measured by hL Within stability classes the veloc ity pro les collapse when scaled with a velocity scale m and the observed BL depth h but there is a large difference between the stability classes Baroclinicity We would expect baroclinicity vertical shear of geostrophic wind to also affect the observed wind pro le This is most easily seen for an Ekman layer in a geostrophic wind with constant ver tical shear ugz G Mz NZ where M gfT0BTBy N gfTO BTdx fv Nz v 51215122 u G Mz v d2vd22 u00u gtGMz asz gt00 v0 0 v gtNz asz gt oo Resultant BL velocity pro le just has thermal wind added onto it uz G1 e39 cos C Mz vz G e39 sin C Nz C 25 5 2Vf12 This can considerably alter the BL wind pro le The largest crossing angle of the surface wind di rection across the isobars is seen if Mlt 0 N gt 0 corresponding to surface cold advection This effect is clearly seen in the gure below of crossing angle vs thermal wind orientation in 23000 wind pro les over land Hoxit 1974 On weather maps one can see much larger crossing angles behind cold fronts than ahead of them On the other hand the wind turns less with height if N gt 0 surface cold advection Turbulence Pro les Garratt 33 For applications such as the dispersion of pollutants it is important to understand the charac teristics of turbulence in different types of BL LES simulations illustrate some of these charac teristics Most of the gures below are from Moeng and Sullivan 1994 JAS 51 9991022 Neutral BLS Moeng and Sullivan simulated a neutral BL capped by a strong 8 K inversion at a height of 21 500 m The geostrophic wind is 15 m s39 in the x direction and m 05 m s39l The figures on 414 show x y slices ofu39 at various heights and the wind hodograph Because of the capping inversion the wind shear within the bulk of the BL is fairly small nearly a mixed layer with strong wind shear across the inversion 71 Atm S 547 Boundary Layer Meteorology Bretherton a b 05 u c 05 I I l 40 720 0 20 40 u gu0 Fig 313 Profiles of the normalized velocity defect for the ucomponent as a function of normalized height zh based on Eq 382 and an analysis of Wangara observations Three stability regimes are presented a 150 lt hL lt 120 b 0 lt h lt 30 c 180lt hL lt210 Curves are drawn by eye After Yamada 1976 Journal of Atmos pheric Sciences American Meteorological Society I a z 05 I I I 40 20 0 20 40 7 40 r 7 05 39 I I I 40 20 J 20 40 ea90 A t ivyH Same for v Same for 6 72 Atm S 547 Boundary Layer Meteorology Bretherton Sorb39an VA 04 J 02 00 02 393 Vquot o Vbmz uh I I Vb mz 04 I I I I 06 08 10 12 UB 00 02 04 Figure 611 Ekman spirals obtained for the baroclinic correction of the V component of the wind velocity 1 0001 m 00001 Points are plotted every 100 m starting on the surface Uh Vhbarotropic components of the wind vectors Dotted line shows directions of the thermal wind vectors Ekman spirals for thermal wind with M O and N gt O N 0 no thermal wind N lt O Nearsurface wind is oriented more in y direction larger crossing angle for Ngt O WEMgt 0 A L E 51 N 0 quotg ml g ELF d 3 an a 239 1 339 an H E 39uql lFI E ur Warm Advactinn Cold Adventinm at t s 7 aw If m w 39iff 215i z IWJ39 my a Isobaric crossing angle of surface wind vs angle of thermal wind Afternoon 00 Z soundings show stronger effect due to stronger vertical mixing in a more convective BL Hoxit 1974 73 Atm S 547 Boundary Layer Meteorology Bretherton r I 1 m Ulmm 1 H la n H v 15 Fin 3 Cantur ul m In he 4 mllmr 1 Eva height haul fur maulHug 5 ml Ia md m cannun 34 1 I5 L U KJ D I 05 I IS 2 51 dirk minimum 111w llr l llllu 39 mullJ 1 415 74 Ams 547 BanndaryLayzxMz eamlngy Brahman Vemcal secnan dqmth a mum EL Nam sfmng anommlaonnbemen u and w We cause um m m tapaf m surface Mina1 n 1 11 axgamud msmaks camspundmg m Inngcylmdncaleddusax m s39 ammedahantZWm m lz a hz geasfmphmwmd Th2 wmd pnm ua ans wukzn Amman less Imzadyaxgamud wnhhzlght Th2 gure belawshnws an H cmss secnanaf u w andu w acmss m camerath damnmy Hex an cause m sfmng mganve cammmnbemenu andw n 115 have asm 11 than dnwndn s espcu y fanzlt n 5 mm m sandmancaef cumbethenu w 15 in Ambean mm mm b afwhlch accnrs m m mm 2w fth EL Wm 5m and mamznmm uxes are bath largest and mfmdzm disgpunan m 1nd camx mman mm Mm12m rmspu Althnughthzxe 15m 5 bunynncy nx m Mulznce dnesemdz m capping Invexsmn unsung a small dawnwuxd enmmmzmbunymcy nx In rm W cm mm m 4m 1m assumz 1mm my baundaxylayzx is wannzd equ ybyenmmmzm afwann m mm 75 Atrn S 547 Boundary Layer Meteorology Bretherton Moeng and Sullivan 1994 E i V f V V u2 u2 u2 2 WE w2 0 1 2 3 4 5 6 0 02 04 06 08 1 b Convective PBL I I o 05 10 W W W m2 5392 3 v7 w2 m2 3392 FIG 9 Vcrlicxl diatrlhunnns of he velocity variances of almulaunns S and B above the inversion we can associate a buoyancy ux pro le with the entrainment which varies linearly from 0 at z 0 to W at the inversion The consumption rate of TKE by this buoyancy ux vertically averaged over the BL is W10 05u3zl If we compare this to the overall dissipation rate of TKE we nd that the TKE dissipation rate is much larger than this at the surface but about 2m3zl in the upper part of the BL39 i e entrainment is consuming around 25 of the TKE generated in this region Weakly Unstable BLs Moeng and Sullivan also simulated a weakly unstable boundary layer also under a capping inversion This was similar to their neutral case but with a surface hear ux of 50 W m 39 giving an Obuhkov lengthL 300 m comparable to 21 In this case page 417 the streaky structure is still apparent at the lowest levels but large convective rolls dominate the turbulence higher in the BL and help keep it wellmixed The buoyant and shear contributions to TKE are comparable in this case A velocity scale based on surface buoyancy ux can be derived from the TKE equation w 0902 Note that zlL kw 31 3 for this case wit 09 m s39l For the buoyancy and shear driven BL a combined velocity scale wm 5m W 3 seems to work best In particular with any combina tion of surface buoyancy ux and shear Moeng and Sullivan found that the entrainment buoyancy ux is roughly w39E39l 02 wm3zl 76 Ams 547 BanndaryLayzxMz eamlngy Brahman A 3 yimrmr ilymurn m mxsu M m y M tum Md m m Mama 4 1 mm mmmmummnmrn 5 m mmwmmmm mummnmkw M Atm S 547 Boundary Layer Meteorology Bretherton 5 j I my V 7 a v gt Ewemnm H Vernal mm mmquot mm m unwind A B C ml autumn In g 9 F iMmFmLummfm lm a 78 Atm S 547 Boundary Layer Meteorology Bretherton Convective BLS Lastly let s look at a purely buoyancydriven or convective BL The simulations shown Moeng and Rotunno 1990 JAS are below a rigid boundary and do not include entrainment but do show the overall structure well At the bottom there is avery good correlation between w39 and 939 with polygonal regions of updraft separating circular patches of downdraft As we move close to the BL top the updrafts accelerate and combine to become circular and the temperature uctu ations become much less well correlated with the updrafts For penetrative convection in fact the updrafts would be a bit cooler than the surrounding air at the highest level shown The velocity variances previous page show a very different structure than for a sheardriven BL They are dominated by the large eddies which have updrafts in the middle of the BL and pre dominantly lateral motions at its top and bottom There is much more velocity variance in the up per part of the BL so the TKE and TKE dissipation rate are almost uniform with height and equal to 04w 3z As in the upper part of a sheardriven BL about 25 of the TKE generated is going into consumption by entrainment which averaged over the BL is w39b39i 2 0 lw z Below are shown LES simulations Sullivan et al 1998 JAS of the top of a convective BL penetrating a moderate inversion of 4 K grid resolution at top right of each plot White indicates 9 lt 304 K other shades increasing 9 up to 308 K Arrows indicate velocity in the xz plane Plots show a sequence of times 10 s apart Note the undulations in the BL top with downward moving air on the edge of hummocks where updraft air has partly mixed with freetropospheric air These motions produce the negative buoyancy flux in the entrainment zone which for a pure convective BL reaches 0ZBO Also note in panels eh the formation of an entrainment tongue at x 1750 m of partly mixed buoyant air that is getting sucked into the BL H rrvv E n Iml 79 Atm S 547 Boundary Layer Meteorology Bretherton Lecture 5 Surface roughness and the logarithmic sublayer Garratt Ch 3 similarity theory Ch 4 surface characteristics Near a solid boundary in the surface layer vertical uxes are transported primarily by eddies with a lengthscale much smaller than in the center of the BL A very successful similarity theory is based on dimensional reasoning Monin and Obuhkov 1954 It postulates that near any given surface the wind and thermodynamic profiles should be determined purely by the height 2 above the surface which scales the eddy size and the surface uxes which drive turbulence 1 Surface mom ux W0 often expressed as friction velocity m W012 2 Surface buoyancy ux Bo W0 One can construct from these uxes the Obuhkov length L u3kBO positive for stable negative for unstable BLs Here k 04 is the von Karman constant whose physical significance we ll discuss shortly In the ABL a typical m might be 03 m s391 and a typical range of buoyancy ux would be 3gtlt10394 m2s39 nighttime to l5gtlt10392 m2s393midday i e a virtual heat ux of 10 W m392 at night 500 W m39 at midday giving L 200 m nighttime and 5 m midday The logarithmic sublayer Garratt p 41 At height 2 the characteristic eddy size velocity and buoyancy scale with z m and B m If the buoyant acceleration acts over the eddy height it would make a vertical velocity 25b 2 zBou12 If 2 lt lLl this buoyancy driven contribution to the vertical velocity is much smaller than the sheardriven inertial velocity scale m so buoyancy will not significantly affect the eddies In this case the mean wind shear will depend only on m and 2 so dimensionally dZdz mkz z lt lLl 1 This can also be viewed in terms of mixing length theory with eddy diffusion Km cc velocitylength ukz W0 Kmd dz gt m2 kmz dZdz equivalent to l The von Karman constant k is the empirically determined constant of proportionality in l Inte grating we get the logarithmic velocity pro le law mm k1 lnzzo 2 ltlt lLl 2 The constant of integration zo depends on the surface and is called the roughness length It is loosely related to the typical height of closely spaced surface obstacles often called roughness el ements e g water waves trees buildings blades of grass It depends on the distribution as well as the height ho of roughness elements see figure below but as a rule of thumb zo0lhc 5l Atm S 547 Boundary Layer Meteorology Bretherton DAY 43 TIME ums39 I39Ya A 0702 0364 I 0802 0380 O 090 0370 O 1000 0309 0276 80 2 GEOMETRIC MEAN 22 000l5m 410 N 39 5 c N 20 0 Io o5 1 I I I I l o I 2 3 4 5 6 7 8 9 IO U ms39 Fig 104 Comparison of the observed wind pro les in the neutral surface layer of day 43 of the Wangara Experiment with the log law Eq 106 solid lines Data from Clarke et a1 1971 Example of logarithmic velocity pro le in a neutral surface layer Garratt ZOhc Observed 0 II G l A U D C il E F 00 0001 I I I l l I 1 l I 001 01 1 11 Fig 41 Variation of zohC with element density based on the results of Kutzbach I961 Lettau 1969 and Wooding et al 1973 represented by the shaded area and solid curve Some specific atmospheric data are also shown as follows A and B trees C and D Wheat E pine forest F parallel flow in a vineyard G normal ow in a Vineyard Analogous windtunnel data are described in Seginer 1974 From Garratt 1977b Dependence of roughness length on density 7L of roughness elements 52 Atm S 547 Boundary Layer Meteorology Bretherton 1 UV 3 Rocky Mountains Stu 11 W Virginia 180m mtns 1o Appalacian Mountains 3 E Tenn 100150 m mts Centers 0 cities with very tal buildings Very hilly or moderate mountainous areas 11 Centers of large towns and cities Itow quotlifts 8 American average r ores I Centers of small towns quot 5 A93quot average Average U plains Outskirts of towns D quots 95 Fa zotr zly fmded Many trees hedges and few burldings S Africa average N America average 10 1 USSFt average Europe average Many hedges Australia average E Few trees summer time Farmland Long grass Pacerquot crops 0 N 39W39aied quot995 Airpons runway area Uncut grass 2 c3 N Asia average 0 I Few Wag winter Ema Fairly level grass plains Cut grass 3cm N Alrica average Natural snow surface farmland 3 10 2 Ol39lsea wind in coastal areas Desert at Large expanses of water 4 Calmopensea Snowcovered at or rolling ground 05 Ice mud ats Fig 96 Aerodynamic roughness lengths for typical terrain es After Garratt 1977 SmedmanHbgstrbm amp HegstrOm 19 8 Kondo amp Yamazawa 1986 Thompson 1978 Napo 1977 and Hicks 1975 Z0 varies greatly depending on the surface but a typical overall value for land surfaces is Z0 01 m see table on next page In the rare circumstance that the surface is so smooth that the Viscous sublayer is deeper than roughness elements ZO01vu 0015 mm foru0l ms391 53 Atm S 547 Boundary Layer Meteorology Bretherton Near the surface the log pro le ts best if z is offset by a zero plane displacement do which lies between 0 and the height ho of roughness elements and is typically roughly 07 uzu k l lnz al0zo 2 ltlt lLl 3 Roughness of Water Surfaces Garratt p 97100 The roughness of a water surface depends on wind speed and the spectrum of waves A strong wind blowing from S to N across the SR 520 bridge shows the importance of fetch on wave spec trum On the south side large waves will be crashing onto the bridge deck On the N side the water surface will be nearly smooth except for short wavelength ripples cats paws associated with wind gusts As one looks further N from the bridge one sees chop then further downwind longer waves begin to build It can take a fetch of 100 km for the wave spectrum to reach the steady state or fully developed sea assumed by most formulas for surface roughness It is thought that much of the wind stress is associated with boundary layer separation at sharp wave crests of breaking waves or whitecaps which start forming at wind speeds of 5 m s391 and cover most of the ocean surface at wind speeds of 15 m s391 or more For wind speeds below 25 m s39l the water surface is approximately aerodynamically smooth and the viscous formula for 20 applies For intermediate wind speeds the ow is aerodynamically smooth over some parts of the water surface but rough around and in the lee of the breaking white caps and for wind speeds above 10 m s391 it is fully rough For rough ow Charnock 1955 sug gested that 20 should depend only on the surface stress on the ocean and the gravitational restoring force i e u and g leading to Chamock s formula 20 occmZg OLC 0016 20 from empirical measurements This formula appears reasonably accurate for 10 m wind speeds of 450 m s39l For 10 m wind speeds of 510 m s39 this gives roughness lengths of 01 1 mm much less than almost any land surface Even the heavy seas under in a tropical storm have a roughness length less than mown grass This is because a the large waves move along with the wind and b drag seems to mainly be due to the vertical displacements involved directly in breaking rather than by the much larger amplitude long swell The result is that nearsurface wind speeds tend to be much higher over the ocean while surface drag tends to be smaller over the ocean than over land surfaces Snow and Sand Surfaces Garratt p 8788 The roughness of sand or snow surfaces also increases of wind speed apparently due to sus pension of increasing numbers of particles Charnock s dimensional argument again applies and remarkably the same 0L0 appears to work well though now the minimum 20 is larger typically at least 005 mm associated with the roughness of the underlying solid surface BulkAerodynamic Drag Formula Garratt p 100101 Suppose that a wind measurement is taken at a standard reference level ZR within the log layer A typical shipboard height of ZR 10 m is often used for ocean measurements Then ignoring zeroplane displacement for simplicity 52 uvk391 lnzRzo The bulk aerodynamic formula re lates the surface stress pOWto the reference wind speed in terms of a drag coefficient C DNwhich depends on surface roughness Pou39W39 Pom2 POCDN772ZR 54 Atm S 547 Boundary Layer Meteorology Bretherton 50 I I I I I I I I I I a o L 20 11H quotl 390 9 o x r jH Z O 0 LO 16 217433l63562795106 I l 3 2 2 l8 3 56 34 7 5 ll 34 IB l4 9 6 4 l O l I I I l I I I 1 o 2 4 e a 10 I2 l4 l6 IS 20 22 UIo msl I I I I l 0 4 b 3 j no I 1 z 2 quotquot 0 0 I I O I 3 6 6 5 2 6 5 1 lo I 5 2 5 3 3 2 0 I l l l 1 IO 20 30 4o 50 Ulo ms Fig 134 Neutral drag coef cient as a function of wind speed at a lOm height compared with Charnock s formula Eq 135 indicated by the arrows in a and b with a 00144 Blockaveraged values are shown for a l m sec I intervals based on eddy correlation and pro le methods and b S m seC I intervals based on geostrophic departure method and wind ume simulation experiments After Garratt 1977 4 CDN x 103 CHN X 103 CEN x 103 25 Garratt 0 l 2 3 5 10 15 20 O m wind speed m srl Fig 49 Drag coefficient CDN heat transfer coefficient CHN and water vapour transfer coefficient CEN as functions of the 10 m wind speed Curves A are for smooth ow solid curve CDN Eq 422 peeked curve CHN Eqs410 and 426a dotted curve CEN Eqs 411 and 426b Curves B are for rough ow solid curve CDN Eq 423 pecked curve CHN Eqs 410 and 427 dotted curve CEN Eqs 411 and 428 Observational data are from Large and Pond 1982 55 Atm S 547 Boundary Layer Meteorology Bretherton CDN k2lnzRzo2 5 The N for neutral in the suffix is to remind us that this formula only applies if when ZR ltlt lLl which for typical reference heights 2 m or 10 m requires fairly neutrally stratified conditions as are often observed over the oceans but less often over land For ZR 10 m wind speed and 20 01 m C DN 8 X 1039 Over the water C DN is a function of surface roughness m and hence implicitly of wind speed While Chamock s formula gives an awkward transcendental equation to solve for C DN in terms of 5ZR a good approximation using mean 10 m wind speed ulo is C DN 075 0067u10 X 10393 water neutrally stratified BL Heat anal Moisture Transfer in Neutral Conditions Let a be a scalar 9 q etc transported by the turbulence In the loglayer we again might hope for a fluxgradient relation of the form w39a39 Kadadz Kg kazm The nondimensional constant kg need not equal the von Karman constant since momentum per turbations of uid parcels are affected by eddyinduced pressure gradients while scalars are not However empirical measurements do suggest that kg k in a neutral BL A scale for turbulent per turbations a39 in the log layer is a WlOm Since the flux is approximately equal to its surface value throughout the surface layer daalz Wl0 kzm akz 52 50 ak lnzzoa This has the same logarithmic form as the velocity profile but the scaling length 20a need not be and usually isn t the same as 20 In fact it is often much smaller because pressure form drag on roughness elements helps transfer momentum between the interfacial viscous sublayer around roughness elements to the inertial sublayer No corresponding nonadvective transfer mechanism eXists for scalars so they will be transferred less efficiently out of the interfacial layer 2a lt 20 un less their molecular diffusivity is much larger than that of heat This can be converted into a bulk aerodynamic formula like 5 but the transfer coefficient may be different p07 poCaN5ZR50 5ZR CaN k21nzRzolnzR20a For most land surfaces the heat and moisture scaling lengths 20H and zoq are 1030 as large as 20 resulting in typical CHNof07095 CDN For water surfaces the heat and moisture coefficients are comparable to CDNfor 10 m winds of7 m s391 or less but remain around 1 15 gtlt10393 rather than increasing as wind speed increases This corresponds to heat and moisture scaling lengths appro priate for laminar ow even at high wind speeds For instance ECMWF uses 20H zoq 04 062Vu following Brutsaert 1982 Bulk aerodynamic formulas are quite accurate as long as i an appropriate transfer coefficient 56 Atm S 547 Boundary Layer Meteorology Bretherton 0I8 Arya 0J5 o POND SAN DIEGO 97 u PONDBOMEX 197l X DREYEROWAX I974 i FRIEHE AND SCHMITT NORPAX 974 44 quotA oilz DUNCKEL etg ATEX 1974 H E 3 I 009 E it 006 mquot o N X 9 3 003 0 x I O I l I l l I I l 0 IO 20 30 40 so so 70 so 90 UlQo39Qms grna Fig 136 Observed moisture ux at the sea surface as a function of UQ0 Q compared with Eq 138 with Cw 2 132 X 10 indicated by the arrow After Friehe and Schmitt 1976 is used for the advected quantity the reference height and the BL stability and ii Temporal vari ability of the mean wind speed or airsea differences are adequately sampled The gure below shows comparisons between direct eddycorrelation measurements of moisture ux in nearly neutrally strati ed BLs over ocean surfaces compared with a bulk formula with constant Cq l32gtlt10393 In individual cases discrepancies of up to 50 are seen which are as likely due to sampling scatter in the measured uxes as to actual problems with the bulk formula but the overall trend is well captured However due to this type of scatter no two books or papers seem to exactly agree on the appropriate formulas to use though all agree within about 20 57 Atm S 547 Boundary Layer Meteorology Bretherton Lecture 51 Nonlocal Parameterizations for Unsaturated BLs In this lecture we describe three nonlocal parameterizations for unsaturated BLs 1 HoltslagBoville scheme used in CCM3 2 Blackadar scheme MMS 3 UW PBL scheme used by Bob Brown s group for using satellite microwave scatterometer measurements of surface wind to determine geostrophic wind We describe 1 and 2 in the notes 3 will be discussed by guestlecturer Dr Ralph Foster HoltslagBoville Scheme References Troen I and L Mahrt 1986 A simple model of the atmospheric boundary layer Sensitivity to surface evaporation Bound Layer Meleor 37 129148 Holtslag A A M and CH Moeng 1991 Eddy diffusivity and countergradient transport in the convective atmospheric boundary layer J Almos Sci 48 16901698 Holtslag A A M and B A Boville 1993 Local versus nonlocal boundary layer diffusion in a global climate model J Climate 6 18251842 Holtslag and Moeng 1991 JAS examined the prognostic equation for an advected scalar a in a surfaceheated convective BL By modeling the individual terms they concluded that w39a39Kal ya Oltzlth The second term on the right which can be interpreted as a nonlocal ux of a is due to bound arylayer lling convective eddies which transport the surface ux of a upward regardless of the local gradient of a Assuming the surface ux of a is positive the result of the nonlocal term is to produce a BL with in which a decreases less with height than if pure rstorder closure were used Kg kwtz1 zh2 k 04 is von Karman constant Holtslag an Boville 1993 FIG 2 Typical vertical pro les for virtual potential temperature 0 and speci c humidity q for a dry convective boundary layer mod i ed after Stull 1991 The arrows to the left illustrate thespeci c humidity ux Wt and the arrows to the right the heat ux w Also an uprising parcel is indicated up to its intersection height he The three regions are discussed in the text 5ll Atm S 547 Boundary Layer Meteorology Bretherton WW0 39Ya AT A wth wt Pru 3 clw cl 06 Pr l momenta 06 l scalars Ricruh2 vlthgt21 h g Ric 05 optimal value depends on model AZ 9 S9Vh 93 The nonlocal ux is largest near the center of the boundary layer with a maximum value w39a39nonlocal max KLLmaXya 043w wtw39a390 atz h3 Since the nonlocal ux is proportional to w wz it is only active in unstable boundary layers where the convective velocity w is signi cant In stable or neutral BLs the parameterization reduces to aKprofile eddy diffusivity The surface uxes are computed using an approximation to MoninObuhkov theory In a coarsely resolved model the actual gradient of a as a function of z is not explicitly computed so bulk aerodynamic formulas due to Louis 1979 Bound Layer Meteor which are based only on the difference between the surface value a0 and its value do at the lowest gridpoint at height 21 are used The transfer coefficient for a scalar a given roughness length 20 is of the standard form C a C NFRi0 where C N k2ln2zlZO2 is standard neutral transfer coeff Rio 21b1 b0lu1l2 where b g9w 9R9R is mean buoyancy at level 139 15Ri0 l unstable Ri0 lt 0 T 12 75CNR1021zo 4 110Ri01 8Ri0 FRi0 stable RiO gt 0 Note that F Rio is always positive regardless of how large Rio is This is because even if Rio is too large to support steady turbulence at height 21 there will be turbulence and turbulent uxes closer to the ground which should modify the lowest model layer The nonlocal closure tends to produce a warmer deeper convective BL than firstorder closure This is often a step in the right direction but can be misleading for cloudtopped boundary layers where the estimated BL depth can be too deep 512 Atm S 547 Boundary Layer Meteorology Holtslag and Boville 1993 IrIIlIIItlIIIIIIIIIIIItT Ill lIIiIIlllllllllll 650 650 Local K Nonlocal ABL 700 700 3 750 750 E V 800 800 0 L 3 850 850 I a L 0 900 900 950 Local K 950 Nonlocal ABL 1000 1000 e I I I I I l I l I I I I I I I I I I I I I 1 I I I I I I l I I I I I I I I I I I I I I I I I I 260 270 280 290 300 310 0 5 10 15 20 San Juan Puer ro Rico Bretherton Temperature K FIG 3 As Fig 1 but for San Juan Puerto Rico 183 N 66 W q gKg 25 Comparison of CCM3 with local solid and nonlocal dashed closures with July climatology for San Juan Puerto Rico a tradecumulus regime Blackac ar convective BL from Grell et al 1994 II I I 2h Ir Ox 2m P 21 23quot3k 0 a FIG 2 Schematic diagram illustrating free convective mad Plumes originating at level 2 rise and mix at various level a changing heat moisture and momentum with air at these Some thermals overshoot the level 2 of zero buoyancy The min of negative area N on the thermodynamic diagram to the posit area P is the entrainment rate see text 5l3 Atm S 547 Boundary Layer Meteorology Bretherton Blockadar high resolution PBL scheme References Blackadar A K 197939Advances in Environmental Science and Engineering 1 No 1 Pfaf in and Ziegler Eds Gordon and Breach Publishers 5085 Zhang DL and R A Anthes 1982 A highresolution motdel of the planetary boundary lay er sensitivity tests and comparisons with SESAME79 data J Appl Meteor 21 1594 1609 Grell G A J Dudhia and D R Stauffer 1994 A Description of the Fifth Generation Penn StateNCAR Mesoscale Model MM5 NCAR Tech Note NCAIUTN398 pp 9197 Like the HoltslagBoville scheme the Blackadar scheme distinguishes between stable and un stable BLs For stable BLs conventional firstorder closure is used Turbulence is reduced to weak background values if Rio gt 02 Numerically efficient approximations to the M0 relations are used in the stable to neutral regime in which hL gt 15 where h is a diagnosed BL height For unstable BLs a nonlocal scheme is used It is based on conceptual models and observa tions of BL convection The lowest model thermodynamic level is assumed to represent the surface layer and is labeled by subscript a Vertical exchange is visualized as the result of plumes origi nating in the surface layer mixing with air at each level below h The BL depth h is taken to be the maximum penetration height of undilute plumes They are assumed to accelerate due to their buoy ancy until they reach their level of neutral buoyancy znb At this point their upward kinetic energy wp22 cc P where P is their vertically integrated buoyancy perturbation Due to their inertia the plumes overshoot topping out at a level n at which their vertically integrated buoyancy deficitN 02P see figure above This defines the BL top 7 417 bpzdz Ebbpawz The unstable stratification of the lowest model layer above the surface layer is assumed to be related to the sensible heat ux through this layer following observations of Priestley 1956 NoIZ 02 atz h where bpz g9va 9Vz9R 7 32 w 9v1 7 Bevaiev32 where B is a coefficient that depends only on the heights of the first two model levels In the surface layer 3901 7 w GV 1 7 w GV O 3t 21 These equations will reach an equilibrium in which 9m is larger than am by an amount sufficient to carry the surface heat ux out of the surface layer into the rest of the BL The scheme now postulates a mass exchange m between the surface layer and each other layer below 2 h 514 Atm S 547 Boundary Layer Meteorology Bretherton r71 w b 1064P where the buoyancy ux w b l eiw ev l R The value of a scalar such as 9 in the BL is now assumed to change due to turbulent exchange with the surface layer according to E 9 E m 9a 7 9 For momenta is multiplied by a factor 1 zh to account for the fact the momentum mixing is somewhat less efficient than mixing of scalars in a convective BL Comparisons of this pararneterization with LES results have not been presented and two case studies presented by Zhang and Anthes 1982 show fair but not excellent agreement with ob served BL evolution overland Thus the convective nonlocal part of this scheme should probably be regarded as being on a shakier footing than the HoltslagBoville scheme It is not entirely clear that either of these schemes is superior to rst order closure in practice 5l5 Atm S 547 Boundary Layer Meteorology Bretherton 516 Atm S 547 Boundary Layer Meteorology Bretherton 517 Atm S 547 Boundary Layer Meteorology Bretherton Lecture 4 Boundary Layer Turbulence and Mean Wind Pro les Turbulence Closure Models G 24 The equations for ensemble averaged quantities involve the divergence of the eddy correla tions which arise from averaging the nonlinear advection terms Similarly prognostic equations for the ensemble averaged secondorder correlations include averages of triple correlations etcso this approach does not lead to a closed set of equations In a turbulence closure model TCM higherorder correlations are parameterized in terms of lowerorder correlations to close the system In a firstorder TCM all secondorder correlations are parameterized in terms of the mean elds In a secondorder TCM 1st and second order moments are prognosed but thirdorder correlations are parameterized in terms of them TCMs of up through third order have been used Third order TCMs can do a fairly realistic job of predicting the profiles of mean elds and even secondorder moments but are quite complicated and computationally intensive F irst order turbulence closure mixing length theory and eddy di usivity For now we will just introduce firstorder turbulence closure which is the most common pa rameterization of turbulent mixing currently used in largescale numerical models such as GCMs The usual approach is inspired by mixing length theory Prandtl 1925 We idealize eddies as tak ing random uid parcels from some level and advecting them up or down over some characteristic height or mixing length 52 at some characteristic speed V where the uid parcel gets homogenized with the other air at that level Except near the surface the transport is primarily by eddies whose scale is the boundary layer depth so we think of Vas the largeeddy velocity and 52 as proportional to the boundary layer height scale H Near the surface a different scaling applies which we discuss later At any location half the time there is an updraft with wu39 Vcarrying uid upward from an average height 2 522 and the other half of the time there is a downdraft with wd39 Vcarrying uid downward from an average height 2 522 Consider the corresponding vertical ux of some advected quantity a In updrafts au39 52 522 52 If we assume that d varies roughly linearly between 2 522 and 2 then Similarly in downdrafts ad39 52 522 52 cs 532 Hence taking the ensemble average l w a wu39au wd39ad39 rs Kad where Kg V522 E d Thus the eddy ux of a is always down the mean gradient and acts just like diffusion with an eddy diffusivity K a For typical ABL scales V l m s39 52 1 km and mixing length theory would predict K a 500 m s39l Most first order turbulence closure models assume that turbulence acts as an eddy diffusivity and try to relate Vand 52 to the profiles of velocity and static stability more on how this is done later when we talk about parameterization 41 Atm S 547 Boundary Layer Meteorology Bretherton Observing the BL The turbulent nature of BL ow presents special challenges for observations and modeling On the other hand its nearness to the surface makes surfacebased observing systems particularly use ful Chapter 10 of Stull s book handout is an excellent summary of sensors and the principles by which they work types of measurement and analysis methods for ABL observations It also has a list of major BL eld experiments through early 1987 and describes numerical modelling of boundary layer turbulence Fast response sensors capable of insitu measurements of turbulent per turbations in velocity components temperature pressure humidity and some trace gases such as C02 from different platforms e g an airplane balloon mast or surface site are now widely avail able and can be used to calculate vertical turbulent uxes and moments Due to the sensitivity of the instruments and their high data rate these measurements are restricted to dedicated field exper iments Remote sensors measure waves generated or modified by the atmosphere at locations dis tant from the sensor Active remote sensorsgenerate sound sodar light lidar or other EM waves e g radar Passive remote sensors rely on electromagnetic waves generated by the earth infra red microwave the atmosphere infrared or the sun visible Remote sensors can often scan over a large volume and are invaluable in characterizing aspects of the vertical structure of the BL but typically provide poor time and space resolution However Doppler lidar in clear air with some scatterers and mmwave radar in cloud have proved capable of resolving larger turbulent eddies and characterizing some of the turbulent statistics of the ow and are particularly useful for characterizing the structure of the entrainment zone at the top of the boundary layer Large eddy simulation Numerical modeling in particular large eddy simulation LES has also become a formida ble tool for understanding BL turbulence Atwo or preferably threedimensional numerical domain somewhat deeper than the anticipated boundary layer depth H and at least 23H wide is covered by a grid of points Atypical domain size for an ABL simulation might be 5X5gtlt2 km The grid spac ing must be small enough to accurately resolve the larger eddies which are most energetic and transport most of the uxes Grid spacings of 100 m in the horizontal and 50 m in the vertical are adequate for a convective boundary layer without a strong capping inversion Such a simulation might run nearly in real time on a fast workstation Higher resolution 1020 m is required near strong inversions and for stable sheardriven BLs putting such simulations at the edge of what can currently be done on a workstation The Boussinesq equations or some other approximation to the dynamical equations are discretized on the grid A subgridscale model is used to parameterize the effects of unresolved eddies on the resolved scale There is no consensus on the ideal subgrid scale model Luckily as long as the gridspacing is fine enough LES simulations have been found to be relatively insensitive to this One can understand this as a consequence of the turbulent en ergy cascade in which energy uxes down to small scales in a manner relatively independent of the details of the viscous drain In an LES the energy cascade must be terminated at the grid scale but as long as the gridscale is in the inertial range and the gridscale eddies are efficiently damped this should not affect the statistics of the large eddies The simulation is started from an idealized usually nonturbulent initial profileand forced with realistic surface uxes geostrophic winds etc Small random perturbations are added to some field such as temperature these seed shear or convective instability which develops into a qua sisteady turbulent ow typically within an hour or two of simulated time for ABL simulations The simulation is run for a few more hours and ow statistics and structures from the quasisteady period are analyzed For cloudtopped boundary layers radiative uxes and a model of cloud mi 42 Atm S 547 Boundary Layer Meteorology Bretherton crophysics are also part of the LES lntercomparisons between different LES codes and comparisons with data show that for a con vective boundary layer without a strong capping inversion the simulation statistics are largely in dependent of the LES code used building confidence in the approach For cloudtopped boundary layers different codes agree on the vertical structure of the large eddies within the BL but predict considerably different rates of entrainment or freetropospheric air for the same forcing This is not surprising as most current LES models are run with 2550 m resolution at the inversion which is often insufficient As soon as other physical parameterizations such as cloud microphysics radi ation or landsurface models are coupled into the LES the results are only as good as the weakest parameterization Thus LES models of most realistic BLs are illuminating but are no substitute for observations Laboratory Experiments Turbulence is important in many contexts outside atmospheric science such as aerodynamics hydraulics oceanography astrophysics etc Most of our fundamental understanding of turbulence derives from laboratory experiments with these contexts in mind Convection has been studied mainly in liquids in tanks a few cm to a few m in size Shear flows have been studied in water tunnels or rotating tanks Salt can be used to produce stratification Turbulence can be created by stirring or passing moving uid through a grid Many sophisticated visualization techniques using dye insitu sensors laser velocimetry etc are used Many simple models of atmospheric turbu lence are tuned based on laboratory results Typical boundary layer pro les Mixing length theory predicts that vigorous turbulence should strongly diffuse vertical gradi ents of mean quantities in the BL resulting in a wellmixed BL with only slight residual vertical gradients How well does turbulence mix up observed boundary layers For clear unstable con vective BLs mixed layer structure is observed in 6 usually in a and often in E 5 with slight veer ing of the wind with height Jllllll o Zkm 2 3 285 Q g kg 9K Fig 65 Measured wind potential temperature and speci c humidity pro les in the PBL under convective conditions on day 33 of the Wangara Experiment From Deardorff 1978 Typical mixed layer structure of a convective boundary layer visible even in a v 43 Atm S 547 Boundary Layer Meteorology Bretherton Arya sve 290 o to 20 9K wmo COMPONENTS ms Fig 67 Observed vertical pro les of mean wind components and potential temperature and the calculated Ri pro le in the nocturnal PBL under moderately stable conditions From Deardorff 1978 after Izumi and Barad 1963 n I l I 285 O OSRIIO LS 20 I For moderately stable BLs in which turbulence is largely continuous in space and time the BL is far from wellmixed but the Richardson number Ri remains less than 14 see gure above In extremely stable boundary layers the turbulence is sporadic and the mean Ri can be 1 or more see below The lowlevel veering of the wind with height is much larger in very stable boundary lay Arya D 5 IO WIND COMPONENTS mseC Fig 68 Observed wind and potential temperature pro les under very stable sporadic turbulence conditions at night during the Wangara Experiment From Deardorff 1978 ers where most of the surface stress is distributed as momentum ux convergence near to the bot tom of the BL see below 0 I g a 4 5 6 Aryafig610 I 39 T 39 rT 3910 Wind hodographs at South Pole Station Categories 18 correspond to increasingly stable BLs dots are composites of measurements at 05 1 2 4 8 12 16 20 24 32 rn yaXis is in surface wind direction Note large turning of wind with height in stable BLs 44 Atm S 547 Boundary Layer Meteorology Bretherton Lecture 6 Monin Obuhkov similarity theory Garratt 33 Because so many BL measurements are made within the surface layer i e where wind veering with height is insigni cant but strati cation effects can be important at standard measurement heights of 2 m for temperature and moisture and 10 m for winds it is desirable to correct the loglayer pro les for strati cation effects Based on the scaling arguments of last lecture Monin and Obuhkov 1954 suggested that the vertical variation of mean ow and turbulence characteristics in the surface layer should depend only on the surface momentum ux as measured by friction velocity 11 the buoyancy uxBo and the height 2 One can form a single nondimensional combination of these which is traditionally chosen as the stability parameter CzL The logarithmic scaling regime of last time corresponds to Cltlt 1 Thus within the surface layer we must have kzmxamz MC 1 Ia609 MC 2 where MG and 19 are universal similarity functions which relate the uxes of momentum and 9 i e sensible heat to their mean gradients Other adiabatically conserved scalars should be have similarly to 9 since the transport is associated with eddies which are too large to be affected by molecular diffusion or viscosity To agree with the log layer scaling mQ and PAC should approach 1 for small We can express 1 and 2 in other equivalent forms First we can regard them as de ning sur face layer eddy Viscosities Km W amz ur2 mg mkz kmz Mg Kh W 32 mar149 9kz kmz Mg By analogy to the molecular Prandtl number the turbulent Prandtl number is their ratio Pr KmKh MC mC Another commonly used form of the similarity functions is to measure stability with gradient Ri chardson number Ri instead of Recalling that N2 dEdZ and again noting that the surface layer is thin so vertical uxes do not vary signi cantly with height within it Ri is related to C as follows Ri dFdzdZdz2 V170 KhW0Km2 BOW New u cvmmcmz emf Given expressions for mQ and PAC we can write C and hence the similarity functions and eddy diffusivities in terms of Ri The corresponding formulas for dependence of eddy diffusivity on Ri stability are often used by modellers even outside the surface layer with the neutral Km and Km 6l Atm S 547 Boundary Layer Meteorology Bretherton estimated as the product of an appropriate velocity scale and lengthscale Field Experiments The universal functions must be determined empirically In the 195060s several eld exper iments were conducted for this purpose over regions of at homogeneous ground with low ho mogeneous roughness elements culminating in the 1968 Kansas experiment This used a 32 m instrumented tower in the middle of a 1 mi2 eld of wheat stubble Businger et al 1971 JAS 28 181189 documented the relations below which are still accepted D 17719714 for 72 lt C lt 0 unstable quot1 1 BC for 0 S C lt1 stable PrIN1 7 YZCflZ for 72 lt C lt 0 unstable PrIN BC for 0 S C lt 1 stable Pk The values of the constants determined by the Kansas experiment were PrtN 074 B 47 71 15 72 9 The quality of the ts to observations are shown on the next page Other experiments have yielded somewhat different values of the constants Garratt Appendix 4 Table A5 so we will follow Gar ratt p 52 and Dyer 1974 Bound Layer Meteor 7 363372 and assume P14th 1 BT59 Y1 12516 In neutral or stable strati cation this implies 1quot M i e pressure perturbations do not affect the eddy transport of momentum relative to scalars such as heat and the turbulent Prandtl number is 1 In unstable strati cation the eddy diffusivity for scalars is more than for momentum Solving these relations for Ri Ri for 72 lt Ri lt 0 unstable C R lt 39 1 7 SRi for 0 R1 lt 02 stable Limiting cases Garratt p 50 i Neutral limit 1quot 1 gt 1 as C gt 0 as expected recovering loglayer scaling forz ltlt 1L1 ii Stable limit Expect eddy size to depend on L rather than 2 2 less scaling since our scaling analysis of last time suggests that stable buoyancy forces tend to suppress eddies with a scale larger than L This implies that the eddy diffusivity Km kuzm cc velocitylength cc ML gt 1quot N zL C and similarly for K h The empirical formulas imply Km N for large C which is consistent with this limit Hence they are usually assumed to apply for all positive 62 Bretherton l Atm S 547 Boundary Layer Meteorology L I y I H I Aryall23 I g g l r l t a l i 39 I 1 1 39 if g N h l M II n IFl 39 M iii I Elias 20quot g quotEquot In inquot v I i ll 439 3 cm 39I F quotf u L A if I g quot5 36 F r v n A 5 if 39239 5 n W I at a V wquot auri l mg 2 4 quot quotquot39quot U 5 L D n I h i 3 391 quot5 II a i a z is 5 39m 433 a 35 5 Empirical determination of similarity functions from Kansas experiment I A1yall5 3 quotLLquot Km kzu quot s ltkzut 2 l 0398 3906 3904 O392 E 02 Ri Eddy Viscosity and diffusivity as functions of stability measured by Ri 63 Atm S 547 Boundary Layer Meteorology Bretherton iii Unstable limit Convection replaces shear as the main source of eddy energy so we expect the eddy velocity to scale with the buoyancy ux Bo and not the friction velocity We still assume that the eddy size is limited by the distance 2 to the boundary In this free convective scaling the eddy velocity scale is uf 1302 3 and the eddy viscosity should go as Km kuzIm cc ufz gt bmocmuf cc zL3913 C3913 A similar argument applies to eddy diffusivity for scalars K h The empirical relations go as C 2 for scalars and C 4 for momenta but reliable measurements only extend out to C 2 Free convective scaling may be physically realized but only at higher Wind and thermodynamic pro les The similarity relations can be integrated with respect to height to get m k1 lnzzo VmzL 90 g 9 kquot1 lnzzm whzL and similarly for other scalars where ifx l Y1C14 wma 17 mltCHdCC 2 2 J 2tan71x g for 72 lt C lt 0 unstable 45C for 0 SC stable WAQEUMVC 2 21n1 2x for 72 lt C lt 0 unstable 45C for 0 SC stable Wind profiles in stable neutral and unstable conditions are shown in the gure below Lowlevel wind and shear are reduced compared to the log profile in unstable conditions when Km is larger From thesewe derive bulk aerodynamic coefficients which apply in nonneutral conditions 2 2 k k C 3 D a H 1nZZO 7 WmZL2 111ZZT0 WmZLlnZZT0 WhZLl These decrease considerably in stable conditions see gure on next page In observational anal yses and numerical models 3 and the formula forL are solved simultaneously to find surface heat and momentum uxes from the values of u and 90 9 at the measurement or lowest gridlevel z 64 Atm S 547 Boundary Layer Meteorology Bretherton Garratt 1n zzo 061 00 8 z m 32 0 10 20 30 141444 Fig 35 Three wind profiles from the Kansas field data Izumi 1971 plotted in normalized form at three values of the gradient Ri z 566m Both normalized and absolute heights are shown whilst the magnitude of the horizontal arrows indicates the effect of buoyancy on the wind relative to the neutral profile see Eq 334 Garratt a CllCHN 1 7 05 0 05 1 zL Fig 37 Values of a CDCDN and b CHCHN as functions of zL for two values of zzo as indicated In b the solid curves have zo zr and the peeked curves have zozT 74 see Chapter 4 65 Atm S 547 Boundary Layer Meteorology Bretherton Scaling for the entire boundary layer Garratt 32 In general the BL depth h and turbulence pro le depend on many factors including history stability baroclinicity clouds presence of a capping inversion etc Hence universal formulas for the velocity and thermodynamic pro les above the surface layer i e where transports are prima rily by the large BLfllling eddies are rarely applicable However a couple of special cases are illuminating to consider The first is a well mixed BL homework in which the uxes adjust to ensure that the tendency of 0 q and velocity remain the same at all levels Well mixed BLs are usually either strongly convective or strongly driven stable BLs capped by a strong inversion Mixed layer models incorporating an entrainment closure for determining the rate at which BL turbulence incorporates aboveBL air into the mixed layer are widely used The other interesting though rarely observable case is a steadystate neutral barotropic BL This is the turbulent analogue to a laminar Ekman layer Here the fundamental scaling parameters are G lugl f and 20 Out of these one can form one independent nondimensional parameter the surface Rossby number Ros Gfzo which is typically 104 108 The friction velocity which measures surface stress must have the form mG FRoS Hence one can also regard m G as a proxy nondimensional control parameter in place of Ros The steadystate BL momentum equations are 23 fu7ug 7 72va 23 fv7vg 7 a Zu w On the next page are velocity and momentum ux profiles from a direct numerical simulation 384x384x85 gridpoints in which mG 0053 Coleman 1999 J Atmos Sci 56 891900 The geostrophic wind is oriented in the x direction and is independent of height the barotropic as sumption Height is nondimensionalized by 5 mf In the thin surface layer extending up to z 0025 the wind increases logarithmically with height without appreciable turning this is most clearly seen on the wind hodograph and is turned at 200 from geostrophic this angle is an increas ing function of mG The neutral BL depth defined as the top of the region of significantly ageo strophic mean wind is hN For m 03 m s391 andf 10394 s39l hN 24 km Real ABLs are rarely this deep because of strati fication aloft but fair approximations to the idealized turbulent Ekman layer can occur in strong winds over the midlatitude oceans The wind profile qualitatively resembles an Ekman layer of with an Ekman thickness 2Vf12 0 l2uf except much more of the wind shear is compressed into the surface layer 66 Atm S 547 Boundary Layer Meteorology Bretherton 20 r v I I I 39 I 15 O o k 1 05 00 39 39 39 A I o2 00 02 04 06 00 10 12 ltugtc ltugtG Wind pro les in a neutral barotropic BL with uG 0053 Coleman 1999 06 39 I I 39 I 39 I 39 I 04 G K E v I 0392 I at Lo ay 1 I I I I I I L 1 00 02 04 00 08 10 12 ltuGgt Wind hodograph dashed Ekman layer Log surface layer is part of pro le to right of dashes 20 26 3 00 39 A 10 05 05 10 00 Total Stress 102 Stress pro les in geostrophic coordinate system Solid in direction of ug dashed transverse dir 67 Atm S 547 Boundary Layer Meteorology Bretherton The wind and momentum ux pro les depend weakly on Ros but we will describe below a scaling that collapses these into a single universal pro le independent of Ros above the surface layer As we go up through the boundary layer the magnitude of the momentum ux will decrease from m2 at the surface to near zero at the BL top so throughout the BL the momentum ux will be Ou2Hence throughout the BL the turbulent velocity perturbations u39 w39 should scale with m to be consistent with this momentum ux We assume that the BL depth scales with uf These scalings suggest a nondimensionalization of the steady state BL momentum equations u ug 3v w ui u 3zfu v vg 3u w ui 15 3Zfu If we adopt a coordinate system in which the x aXis is in the direction of the surfacelayer wind the boundary conditions on the momentum ux are u w ui l v w ui O as z gt O at surface layer top 2 2 u w u O v w u O asz gtoo This momentum balance and boundary conditions are consistent with a universal velocity defect law of the form u ugu Fxzfu v vgu Fyzfu where Fx and Fy are universal functions which must be determined empirically or via LES simu lation that apply for any Ros In the surface layer these universal functions cease to apply and the logarithmic wind profile ukm lIlZZO v 0 matches onto the defect laws The figure below shows that Coleman s simulations and laboratory experiments with different parameters are con sistent with the same Fx and Fy supporting their universality O lllllll I IIIIIII I l39lllIllI Illllll 996VvaV VVN I 7 log layer VVI V 039 O l vaV V V I velocity defect scaling ltugt Gu ltvgt Gyu lllllll I llllllll 10quot 10 20 lJlllllll 1 llllllll 1 104 10 3 102 26 I 68 Atm S 547 Boundary Layer Meteorology Bretherton Lecture 14 Marine and cloud topped boundary layers Marine Boundary Layers Garratt 63 Marine boundary layers typically differ from BLs over land surfaces in the following ways a Near surface air is moist with a typical RH of 75100 b The diurnal cycle tends to be weak though not negligible since surface energy uxes get distributed over a considerable depth 10100 of water which has a heat capacity as much as hundreds of times as large as the atmospheric BL c Airsea temperature differences tend to be small except near coasts The air tends to be 02 K cooler than the water This is because the BL air is usually radiatively cooling and some of this heat is supplied by sensible heat uxes off the ocean surface However if the air tem perature is much lower than the SST vigorous convection will reduce the temperature differ ence and except where there are large horizontal gradients in SST horizontal advection cannot maintain the imbalance Hence the surface layer is nearly neutral over almost all of the oceans d Due to the small airsea temperature difference the Bowen ratio tends to be small typically 01 in the tropical oceans and more variable in midlatitudes latent heat uxes are 50200 W m392 while except in cold air outbreaks off cold landmasses sensible heat uxes are 030 W m392 e Over 95 of marine boundary layers contain cloud The only exceptions are near coasts where warm dry continental air is advected over a colder ocean and in some regions such as the eastern equatorial Pacific cold tongue and some western parts of the major subtropical oceans in which air is advecting from warmer to colder SST tending to produce a more sta ble sheardriven BL which does not deepen to the LCL of surface air Cloud profoundly af fects the BL dynamics as we discuss below Many large field experiments have studied marine BLs Particular focus areas and particularly seminal field experiments have included i Tropical BLs associated with deep convection GATE 1973 tropical E Atlantic TO GACOARE 1992 tropical W Paci c ii Trade cumulus boundary layers BOMEX 1969 Caribbean ATEX 1973 tropical E Atlantic iii Subtropical stratocumuluscapped BLs FIREMSC 1987 California ASTEX 1992 NE At lantic DYCOMSH 2001 NE Pacific EPIC2001 SE Pacific iv Midlatitude summertime BLs JASIN 1978 NE Atlantic v Midlatitude wintertime BLs AMTEX 1974 S China Sea MASEX 1983 Atlantic Coast vi Arctic stratus Arctic Stratus Experiment 1980 BASE 1994 SHEBNFIREACE 19978 While there are interesting issues associated with the formation of stable cloudless marine boundary layers due to advection of air off a continent see Garratt 64 the study of marine bound ary layers largely comprises an important subset of the study of cloudtopped boundary layers to which we turn now 141 Am 5 547 BunndaryLayzx MMwmlngy Brahman chumps huumhly layzn Gama Ch 7 agima and gaoyaphmzl dumbutmn Ovexmvcha hz glnbe mmcmmxmsmn means lawrlymgclaudplaysakz m m ymm afpxecxp man mm layzxs hemath m claud and Lhm afbaththz ELnselfandthz mum m 5m Pmcesses a ecnng camc m clnudrmppdELs Th2 mast cammmdyseen clmldyEL malqu 1 Shallnwcnmnhs Cnbmmdarylayus mqnnmls mmcunlc m seen mmmmmm means as m humus afcaldmm 1 wmdxegxmeshma en b cleurmrndunvecaal thanks Thzse m dnv en pnman y y 1 Stnmcumulus ScrcappdELs typica yfannd mamwyclnnlc nwvvenhz swamp and mum means and a m seen am m can seamn mm masher landmasses Thzse aLs maymc a Cube wax mm mm m Sc mam dnvenmhrge Mbymm mg nth taps afthz clmlds andsecandanlybysnrface cald mam 1Thz pmgxes an m a mum said My andnxeak mm sha awclnud sums m hm pu chzs and hmsafSc and m ypulygaml Amy afCu Th2 By s drwenbysrmng surface several hundred w m 2 can hm uxes afnpt w swarm shallnw slums layus amnseenmmldlamudzs m warm adveconn Hue m dynamical and mum Tee s am am an pmbablysecandaxy sum mm are a zn mdymg clauds mxeduce m mm mm afthz la clnud and ennth funk Imam hzanng m m clauds m be Impunam 2 maybe law clmld and mm 25 mt have a clan asmcmmg wnh EL pmcesses asmcmed wnh sympmscus m 142 Atm S 547 Boundary Layer Meteorology Bretherton V Summertime arctic stratus under a weak anticyclone in which there may be multiple cloud layers driven by surface chilling of and cloud top radiative cooling of moist warm air advect ed over cold pack ice The global distribution of low cloud at heights of 2 km or less above the surface is best document ed in routine synoptic observations of cloud type and cover by untrained surface observers using a simple classification scheme from WMO These have been archived over the past 50 years and were compiled by Warren et al 1988 Below are shown the annually averaged cloud cover fre quency of occurrence multiplied by fractional sky cover when cloud type is present for low lying stratus stratusstratocumulusfog which encompasses the most radiatively important cloud types and for cumulus cloud These cloud layers are typically 100500 m thick with a cloud base anywhere from the surface to 1500 m and tend to be nonprecipitating Over much of the midlati tude oceans and parts of the eastern subtropical oceans stratus cloud cover exceeds 50 Annual Stratus Cloud Amount Klein and Hartmann 1993 from surface observations Cumulus Cloud Amount cumann2 143 Atm S 547 Boundary Layer Meteorology Bretherton Klein and Hartmann 1993 showed that the cloud cover in these regions is highest when the seasurface is coldest compared to the air above the boundary layer which tends to occur in the summertime In some parts of the Aleutian Islands the average stratus cloud cover in June July and August is 90a dreary sky indeed Over land there is much less stratus cloud due to the lesser availability of surface water In most of the tropical and subtropical oceans stratus clouds are rare There is a very strong correlation between TOA cloud radiative forcing and stratus cloud amount due to the high albedo of these clouds coupled with the smallness of their greenhouse ef fect since being low clouds they are at a similar temperature to the underlying surface There is an obvious correlation between cumulus cloud and a relatively warm surface Note that cumulus cloud amount is everywhere low even though over much of the trade wind belts the frequency of occurrence is 7090 More than 100200 km offshore a complete lack of BL cloud is rare oc curing 12 of the time in most ocean locations BL structure of subtropical convective C T BLS The figure above shows composite soundings from four eld experiments that studied marine subtropical and tropical CTBLs Albrecht et al 1995 The experiments were conducted over lo cations with very different seasurface temperature SSTThe typical observed boundary layer cloud structure and circulations are sketched The experiments are FIRE SNI July 1987 33 N 120 W SST 289 K Cloud Fraction 083 ASTEX June 1992 SM 37 N 25 w SST 291 K CF 067 VALD 28 N 24 w SST 294 K CF 040 and TIWE December 1991 O N 140 W SST 300 K CF 026 The deeper BLs tend to have less cloud cover a weaker inversion and a less wellmixed struc ture in the total water mixing ratio q qv c which is conserved following uid motions in the absence of mixing The stratification of 6 is roughly dryadiabatic below cloud base In the cloud layer it is moistadiabatic within the shallow FIRE stratocumulus cloud layer and conditionally unstable in the other cases In general one can identify three types of BL structure i wellmixed e g FIRESNI A speci c example is shown on the next page ii diurnally decoupled some daytime shallow Sc layers in which there are wellmixed surface and cloud layers separated by a stable layer across which there is no turbulent transport An exam ple is shown on next page iii conditionally unstable in which a wellmixed subcloud layer is topped by cumulus clouds and 3 1 1 V l 1 l 4 I 3 I r 39I a 5 Tl 2 SNI b I 3 ASTEX SM g 2 ASTEX VALD Vl 1 TIWE 2 2 1 N N l FIRE SNI l 39 39 TIWE 0 1 l I I 0 1 139 1 1 L 285 290 295 300 305 310 315 320 0 2 4 6 8 1o 12 14 16 18 20 6vK W9kg Composite 6 and q from four CTBL experiments Albrecht et al 1995 144 Atm S 547 Boundary Layer Meteorology Bretherton there may or may not be a thin stratocumulus layer below the capping inversion formed by detrain ment from the cumuli There is a very weak lt 1 K inversion at the base of the cumuli called the transition layer that separates the region of subcloud convection below cloud base from the drier cumulus layer above Essentially the transition layer acts as a valve to allow only the strongest sub cloud updrafts to form cumulus clouds The capping inversion tends to be sharp if there is more Sc cloud see figure on next page and extends over 100500 m if only Cu is present In the deep convective regions of the tropics conditionally unstable cumulus boundary layers are also often seen extending up to around 800 mb when deep convection is suppressed A capping inversion is not evident around deep convection here the BL is complicated by internal BLs as shallow as 100 m due to cold dry out ow from deep convective systems Even over a uniform seasurface mesoscale temperature variationsof 35 K are common in this situation Surface ux es restore the out ow air to a typical nonout ow thermodynamic state in 624 hrs Over midlatitudes when stratocumulus or cumulus cloud is observed the soundings again fall into the above categories Norris 1998 However the RH of surface air may be lower and hence the depth of the subcloud layer may be as much as 1500 m especially over land t Garratt H a p hPa a m i 39 020 960 39 900 1000 quot 1000 l 3 I I I l 1 1 l l 1 275 295 8 295 315 quot0 390 3 9 K 4 g kg 96 K Wellmixed Sc layer TT TTT39IFT39T39TWFIITIIIIjIIIIIIIHIGarratt Z 39139 v 1400 1200 1000 800 600 400 200 0 300 305 310 288 2 4 6 8 r6 3 0 3 K K g kg m squot Fig 75 Observed mean profiles of thermodynamic variables and wind components made in the CTBL over the ocean during JASIN for a decoupled stratocumulus layer The peeked horizontal lines delineate layer boundaries as follows 1 cloud top 2 cloud base 3 bottom of subcloud layer 4 top of the surfacerelated Ekman layer After Nicholls and Leighton 1986 Quarterly Journal of the Royal Meteorological Society Decoupled Sc layer 145 Atm S 547 Boundary Layer Meteorology Bretherton PORTO SANTO ASTEX June 1 1992 33 N 16 W 30 20 10 0 10 20 30 3000 2500 2000 1500 1000 on layer 0 Conditionally unstable sounding with shallow Cu rising into an Sc layer Stable C T BLS Some cloud types such as stratus and fog are associated with stable BLs Norris 1998 has used soundings from ocean weather ships taken during the 1970s to form composites for different cloud types In these cases the sounding is absolutely stable and the presence of cloud just reduces the effective stability We will not discuss these BLs more as the impact of cloud on convective BLs is more profound especially when surface sensible heat uxes are weak as they usually are over the ocean OWS C JJA Norris 1998 650 39 650 r 700 4 700 3 750 3 750 E E e 800 e 800 a 850 4 a 850 u m 9 900 9 900 LL 3 qS o 950 950 1000 1000 o 2 4 6 8 10 280 290 300 310 320 Mixing Ratio gkg Potential Temperature K Composite profiles for stratus St and fractostratus Fscapped stable CTBLsat Ocean Weathership C in the N Atlantic Ocean 146 Atm S 547 Boundary Layer Meteorology Bretherton Lecture 3 Turbulent uxes and TKE budgets Garratt Ch 2 The ABL though turbulent is not homogeneous and a critical role of turbulence is transport and mixing of air properties especially in the vertical This process is quanti ed using ensemble averaging often called Reynolds 39 of the 39 J J 39 eqnatirms Boussinesq Equations G 22 For simplicity we will use the Boussinesq approximation to the NavierStokes equations to de scribe boundarylayer ows This is quite accurate for the ABL and ocean BLs as well since 1 The ABL depth Ol km is much less than the density scale height 010 km 2 Typical uid velocities are 01 10 m s39l much less than the sound speed The Boussinesq equations of motion are 1 fk X 11 7 Vpp bk VVzu where buoyancy b gGVVGO 0 V r u 0 De 7 2 1 RN E 7 Se KV 9 Se 7m 1n the absence of clouds Dq 7 2 E 7 Sq KqV q Sq 0 1n the absence ofprecrprtatron Here p39 is a pressure perturbation 9 is potential temperature q qv q is mixing ratio including water vapor qv and liquid water q if present and 9V 91 608qv ql is virtual potential tem perature including liquid water loading S denotes source sink terms and p0 and 90 are character istic ABL density and potential temperature K and Kq are the diffusivities of heat and water vapor The most important source term for 9 is divergence of the net radiative ux RN usually treated as horizontally uniform on the scale of the boundary layer though this needn t be exactly true espe cially when clouds are present For noprecipitating BLs S q 0 For cloudtopped boundary lay J quot 39 quot quot and quot can also be important as r r 139 Using mass continuity the substantial derivative of any quantity a can be written in ux form DaDt Bel3t Vua Ensemble Averaging G 23 The ensemble average of DaDt is E 371 Dr 3 vua Zz 67 7m u Zz a 2370 v Zz a w w ZI a iamdeJWdm Bx B 3t 32 31 Atm S 547 Boundary Layer Meteorology BretheIton The three eddy correlation terms at the end of the equation express the net effect of the turbulence Consider a BL of characteristic depthH over a nearly horizontally homogeneous surface The most energetic turbulent eddies in the boundary layer have horizontal and vertical lengthscale H and by mass continuity the same scale U for turbulent velocity perturbations in both the horizontal and vertical The boundary layer structure and hence the eddy correlations will vary horizontally on characteristic scales L S gtgt H due to the impacf on the BL of mesoscale and synopticscale variabil ity in the free troposphere If we let denote the scale ofquot and assume a39 A we see that the vertical ux divergence is dominant iu a U Altlt iw a U A Bx LS 232 H Thus noting also that V gt 1 1 0 to undo the ux form of the advection of the mean E a a a r V Dr Bra u a Bzw a If we apply this to the ensembleaveraged heat equation and throw out horizontal derivatives of 9 in the diffusion term using the same lengthscale argument LS gtgt H as above we find 2 E 7 E E Eemrve 7 73w 9 SeKa 29 2 Thus the effect of turbulence on 6 is felt through the convergence of the vertical eddy correlation or turbulent ux of 9 The turbulent sensible and latent heat uxes are the turbulent uxes of 9 and q in energy units of W m39 Turbulent sensible heat ux pOpr 9 Turbulent latent heat ux pOLw q Except in the interfacial layer within mm of the surface the diffusion term is negligible so we ve written it in square brackets If geostrophic wind ugis defined in the standard way the ensembleaveraged momentum equa tions are 82 at aquot 7 3 Euer 7 7fu7ug7vw Hiya fv7vg7u w Often but not always the tendency and advection terms are much smaller than the two terms on the right hand side and there is an approximate threeway force balance see figure below between momentum ux convergence Coriolis force and pressure gradient force in the ABL such that the mean wind has a component down the pressure gradient The cross isobar ow angle 0c is the an gle between the actual surface wind and the geostrophic wind If the mean profiles of actual and geostrophic velocity can be accurately measured the momen 32 Atm S 547 Boundary Layer Meteorology Bretherton Reynolds stress friction Coriolis Surface layer force balance in a steady state BL f gt 0 Above the surface layer the force balance is similar but the Reynolds stress need not be along V tum ux convergence can be calculated as a residual in the above equations and vertically inte grated to deduce momentum ux This technique was commonly applied early in this century before fastresponse high data rate measurements of turbulent velocity components were perfect ed It was not very accurate because small measurement errors in either u or ug can lead to large relative errors in momentum ux In most BLs the vertical uxes of heat moisture and momentum are primarily carried bylarge eddies with lengthscale comparable to the boundary layer depth except near the surface where smaller eddies become important The gure below shows the cospectrum of w39 and T 39 which is the Fourier transform of WT 39 from tethered balloon measurements at two heights in the cloudtopped boundary layer we plotted in the previous lecture The cospectrum is positive i e positive correlation between w39 and T 39 at all frequencies typical of a convective boundary layer Most ofthe covariance between w39 and T39 is at the same low frequencies n 021 N 10392 Hz that had the maximum energy Since the BL is blowing by the tethered balloon at the meanwind speed 7 7 m s39l this frequency corresponds to large eddies ofwavelength 9 Un 700 m which is comparable to the BL depth of 1 km Cospectrum of w39 and T 39 at cloud base triangles top circles in convective BL 33 Atm S 547 Boundary Layer Meteorology Bretherton Turbulent Energy Equation G 256 To form an equation for TKE E u r u 2 we dot 11 into the momentum equation and take the ensemble average After considerable manipulation we nd that for the nearly horizontally ho mogeneous BL H ltlt LS rV SBTD where S 7 u w il 7 v w i shear production 32 32 B w b buoyancy ux T iw p transport and pressure work 32 po 2 D VIVXuI d1ss1patron always negatrve 81n Garratt Shear production of TKE occurs when the momentum ux is downgradient i e has a compo nent opposite or down the mean vertical shear To do this the eddies must tilt into the shear Kinetic energy of the mean ow is transferred into TKE Buoyancy production of TKE occurs where relatively buoyant air is moving upward and less buoyant air is moving downward Gravi tational potential energy of the mean state is converted to TKE Both S and B can be negative at some or all levels in the BL but together they are the main source of TKE so the vertical integral of S B over the BL is always positiveThe transport term mainly uxes TKE between different levels but a small fraction of TKE can be lost to upwardpropagating internal gravity waves excit ed by turbulence perturbing the BL top The dissipation term is the primary sink of TKE and for mally is related to enstrophy In turbulent ows the enstrophy is dominated by the smallest dissipation scales so D can be considerable despite the smallness of V Usually the left hand side the storage term is smaller than the dominant terms on the right hand side The figure on the next page shows typical pro les of these terms for a daytime convectively driven boundary layer and a nighttime sheardriven boundary layer In the convective boundary layer transport is considerable Its main effect is to homogenizing TKE in the vertical With ver tically fairly uniform TKE dissipation is also uniform except near the ground where it is enhanced by the surface drag Shear production is important only near the ground and sometimes at the boundary layer top In the sheardriven boundary layer transport and buoyancy uxes are small everywhere and there is an approximate balance between shear production and dissipation The ux Richardson number Rif BS characterizes whether the ow is stable Rifgt 0 neutral Rif cs 0 or unstable Riflt 0 34 Atm S 547 Boundary Layer Meteorology Bretherton Garratt Normalized TKE budget terms b 10 8 7 o 4 2 o 2 4 6 8 10 TKE budget terms x 10 m2 s 3 Fig24 Terms in the TKE equation 274b as a function of height normalized in the case of the clear daytime ABL a through division by w3h actual terms are shown in b for the clear nighttime ABL Profiles in a are based on observations and model simulations as described in Stull 1988 Figure 54 and in b are from Lenschow et a 1988 based on one aircraft ight In both B is the buoyancy term D is dissipation S is shear generation and T is the transport term Reprinted by permission of Kluwer Academic Publishers 35 Atm S 547 Boundary Layer Meteorology BretheIton Lecture 11 Surface Evaporation Garratt 53 The partitioning of the surface turbulent energy ux into sensible vs latent heat ux is very important to the boundary layer development Over ocean SST varies relatively slowly and bulk formulas are useful but over land the surface temperature and humidity depend on interactions of the BL and the surface How then can the partitioning be predicted For saturated ideal surfaces such as saturated soil or wet vegetation this is relatively straight forward Suppose that the surface temperature is To Then the surface mixing ratio is its saturation value qT0 Let 21 denote a measurement height within the surface layer e g 2 m or 10 m at which the temperature and humidity are T1 and q1 The stability is characterized by an Obhukov lengthL The roughness length and thermal roughness lengths are 20 and 2T Then MoninObuhkov theory implies that the sensible and latent heat uxes are HS PCpCHV1To 39 T1 HL pLCHV1q0 ql where CH fnV1 21 20 ZT L We can eliminate To using a linearized version of the ClausiusClapeyron equations qo qT1 dqdTRT0 T1 R indicates a value at a reference temperature that ideally should be close to To T12 HL 3HS pLCHV1qT1 ql 3 LcpdqdTR 07 at 273 K 33 at 300 K 1 This equation expresses latent heat ux in terms of sensible heat ux and the saturation de cit at the measurement level It is immediately apparent that the Bowen ratio H SH L must be at most s391 over a saturated surface and that it drops as the relative humidity of the overlying air decreas es At higher temperatures latent heat uxes tend to become more dominant For an ideal surface 1 together with energy balance RN HG HS HL can be solved for HL HL LEP FRN39HG 1 FPLCHV19T1 91 2 F 3 s l 04 at 273 K 077 at 300 K The corresponding evaporation rate E P is called the potential evaporation and is the maximum possible evaporation rate given the surface characteristics and the atmospheric state at the measure ment height If the surface is not saturated the evaporation rate will be less than EP The gure on the next page shows HL vs the net surface energy in ux RN HG for T1 293 K and RHI 57 at a height ole 10 m with a geostrophic wind speed of 10 m s39l assuming a range of surface roughness Especially over rough surfaces forest H L often exceeds RN H 6 so the sensible heat ux must be negative by up to 100 W m392 The Bowen ratio is quite small 02 or less for all the saturated surfaces shown in this gure lll Atm S 547 Boundary Layer Meteorology Bretherton J 1 I I I 200 400 600 RNO 00 W miz Fig 56 Potential evaporation for different wet neutr ildconditions have been assumed and in b the full stability correction in r V is lIlC u 396 see Eqs 347 and 357 Note how the effects of thermal stability tend to reduce the direct in uence of aerodynamic roughness Values 0f 20 are as follows 0 001 001 m rass 01111 b m lake Webb 13975 SCH 1 In forest Further detalls of the calculations can be found in surfaces calculated from Eq 526 In a Evaporation from dry vegetation We consider a fully vegetated surface with a single effective surface temperature and humidity a singlelayer canopy The sensible heat ux is originates at the leaf surfaces Whose tempera ture is T 0 The latent heat ux is driven by evaporation of liquid water out of the intercellular spaces within the leaves through the stomata which are channels from the leaf interior to its surface The evaporation is proportional to the humidity difference between the saturated inside of the stomata and the ambient air next to the leaves The constant of proportionality is called the stomatal resis tance units of inverse velocity rs pqTo qoE 3 Plants regulates transport of water vapor and other gasses through the stomata to maintain an op timal internal environment largely shutting down the stomata when moisturestressed Hence rsZ depends not only on the vegetation type but also soil moisture temperature etc Table 51 of Gar ratt shows measured rst which varies form 30 300 s m39l By analogy we can define an aerodynamic resistance r0 CHV1391 pltqo qoE 4 Typical values of rd are 100 s m39l decreasing in high wind or highly convective conditions This is comparable to the stomatal resistance Working in terms of aerodynamic resistance in place of CH is convenient in this context as we shall see next because these resistances add rst Fa PqTo qoE Pqo CIDE PqTo tinE 5 i e E is identical to the evaporation rate over an equivalent saturated surface with aerodynamic resistance rsZ rd The same manipulations that led to l and 2 now lead to H5 pcpwo am Hi LE pltqltTogt q1gtrsz r0 sHS pLqT1q1gtraltrsz r0 HL 2 1TKU QN HG 1 rPLCIT1 610 rs Fa 6 112 Atm S 547 Boundary Layer Meteorology Bretherton where 17 s s l rSra This is the PenmanMonteith relationship Comparing 6 to 2 we find that 17 lt F so the heat ux will be partitioned more into sensible heating especially if stomatal resistance is high winds are high or the BL is unstable The effect is magni ed at cold temperatures where s is small The ratio of H L to the saturated latent heat ux 2 given the same energy in ux RN HG is 1 HLHLa W 1 1 FrsZra Calculations of this ratio for neutral conditions a 10 m s391 geostrophic wind speed and various surface roughnesses are shown in the figure below For short grass the surface transfer coefficient is low so the aerodynamic resistance is high and stomatal resistance does not play a crucial role at high temperatures though at low temperatures it cuts off a larger fraction of the latent heat ux For forests stomatal resistance is very important due to the high surface roughness low aerody namic resistance Garratt EKJEL Forest 02 l l l l 500 250 100 50 25 10 5 25 rss mil Fig 58 Variations of EoEL Eq 537 with surface resistance Values of ray have been calculated for neutral conditions with zq 2074 For short grass zo 00025 m curve 1 T 303 K curve 2 T 278 K For forest zo 075 m curve 3 T 303 K curve 4 T 278 K Soil moisture If the surface is partly or wholly unvegetated the evaporation rate depends on the available soil moisture Soil moisture is also important because it modulates the thermal conductivity and hence the ground heat ux and affects the surface albedo as well as transpiration by surface vegetation For instance Idso et al 1975 found that for a given soil albedo varied from 014 when the soil was moist to 031 when it was completely dry at the surface If the soilsurface relative humidity RHO is known then the evaporation is E Z PRHOCIT0 39 CIDra Note that net evaporation ceases when the mixing ratio at the surface drops below the mixing ratio at the measurement height which does not require the soil to be completely dry Soil moisture can be expressed as a volumetric moisture content 11 unitless which does not exceed a saturated value 11 S usually around 04 When the soil is saturated moisture can easily ow through it but not all pore spaces are waterfilled As the soil becomes less saturated water is increasingly bound to the ll3 Atm S 547 Boundary Layer Meteorology Bretherton soil by adsorption chemicals and surface tension The movement of water through the soil is down the gradient of a combined gravitational po tential gz here we take 2 as depth below the surface plus a moisture potential gwm The moisture potential is always negative and becomes much more so as the soil dries out and its remaining wa ter is tightly bound Note 11 has units of height The downward ux of water is FW pWKnaw zBz Darcy s law where Km is a hydraulic conductivity units of m s39l which is a very rapidly increasing function of soil moisture Conservation of soil moisture requires pwanaz BFwBz The surface relative humidity is 1HO 2 130le ORVTO i e the more tightly bound the surface moisture is to the soil the less it is free to evaporate Em pirical forms for 111 and K as functions of n have been tted to eld data for various soils 111 wsmn s39b K Ksmn 921 where 111s and K s are saturation values depending on the soil and the exponent b is 412 For b 5 halving the soil moisture increases the moisture potential by a factor of 32 and decreases the hy draulic conductivity by a factor of 4000 Because these quantities are so strongly dependent on 1 one can de ne a critical surface soil moisture the wilting point nw above which the surface rel ative humidity RHO is larger than 99 and below which it rapidly drops The wilting point can be calculated as the 1 below which the hydraulic suction 411 exceeds 150 m Garratt Table A9 Soil moisture quantities for a range of soil types based on Clapp and Homberger 1978 Quantities shown are as follows 115 is the saturation moisture content volume per Volume 11w is the wilting value of the moisture constant which assumes 150 m suction 16 the value of 17 when 1p 150 m 105 is the saturation moisture potential and K S is the saturation hydraulic conductivity b is an index parameter see Eqs 5 46 548 n so type 775 WS Kns b 7w m3 m 3 m 10 6 m squot m3 m 3 1 sand 0395 0121 176 405 00677 2 loamy sand 0410 0090 1563 438 0075 3 sandy loam 0435 0218 341 490 01142 4 silt loam 0485 0786 72 530 01794 5 loam 0451 0478 70 539 01547 6 sandy clay loam 0420 0299 63 712 01749 7 silty clay loam 0477 0356 17 775 02181 8 clay loam 0476 0630 25 852 02498 9 sandy clay 0426 0153 22 1040 02193 10 silty clay 0492 0490 10 1040 02832 11 clay 0482 0405 13 1140 02864 ll4 Atm S 547 Boundary Layer Meteorology Bretherton rh 100 l l l Garratt a 60 20 0 01 02 03 04 05 06 07 08 EUIEL 06 04 02 0 1 1 r l l 1 1 J 0 01 02 03 0 4 05 06 07 08 Fig59 a Relative humidity rh as a function of relative soil moisture content 17715 based on Eq 549 and data in Table A9 for soil types 1 sand 6 loam and 11 clay Calculations are for a temperature T0 of 303 K The vertical arrows indicate the wilting points Note that combining Eqs 546 and 549 allows rh to be calculated from In rh gRvTowsnns b b EOEL as a function of the relative soil moisture content based on numerical simulations in an atmospheric model for a range of climate conditions midlatitude summer represented by the shaded regions the temperature range is 283 303 K and q 0005 ll5 Atm S 547 Boundary Layer Meteorology Bretherton Landuse Moisture Emissivity Roughness Thermal Inertia Integer Landuse Albedo Avail at 9 pm Length cm cal cm39z K39l 512 ldenu cauon Description Sum Win Sum Win Sum Win Sum Win Sum Win 1 Urban land 18 18 S 10 88 88 50 50 003 003 2 Agriculture 17 23 3O 60 92 92 15 5 004 004 3 Rangegrassland 19 23 15 30 92 92 12 10 0 03 004 4 Deciduous forest 16 17 30 60 93 93 50 50 004 005 5 Coniferous forest 12 12 30 6O 95 95 50 50 0 04 005 6 Mixed forest and 14 14 35 70 95 95 4O 40 O 05 006 wet Ian 7 Water 8 8 100 100 98 98 0001 0001 006 006 8 Marsh or wet land 14 14 50 75 95 95 20 20 006 006 9 Desert 25 25 2 5 85 85 10 10 0 02 002 10 Tundra 15 70 50 9O 92 92 10 10 0 05 005 1 1 Permanent ice 55 70 95 95 95 95 5 5 0 05 005 12 Tropical or sub 12 12 50 50 95 95 50 50 0 05 005 tropical forest 13 Savannah 20 20 15 15 92 92 15 15 003 003 MMS surface types and their characteristics Appendix 4 of MMS manual Parameterizatz39on of surface evaporation in largescale models In practice simplified formulations of soil moisture and transpiration are used in most models We will defer most of these until later However the MMS formulation of surface evaporation is particularly simplified It is E PLCHV1MqTo 611 i e the standard formula for evaporation off a saturated surface at the ground temperature T 0 cal culated by the model multiplied by a moisture availability factor M between 0 and 1 that is as sumed to depend only on the surface type This formulation avoids the need to initialize soil moisture but is tantamount to assuming a surface resistance that is proportional to the aerodynamic resistance with rAra l MM While this type of formulation can be tuned to give reasonable results on an annually averaged ba sis it is likely to be in error by a factor of two or more in individual situations because rs and rd are both subject to large and independent uctuations More sophisticated schemes explicitly prog nose soil moisture often using relaxation to specified values deep within the soil to control uc tuations and vegetation characteristics and determine the evaporation from these 116 Atm S 547 Boundary Layer Meteorology Bretherton Lecture 10 Surface Energy Balance Garratt 5152 The balance of energy at the earth s surface is inextricably linked to the overlying atmospheric boundary layer In this lecture we consider the energy budget of different kinds of surfaces Con sider rst an ideal surface which is a very thin interface between the air and an underlying solid or liquid medium that is opaque to radiation Because it is thin this surface has negligible heat capacity and conservation of energy at the surface requires that RN HS HL HG where note sign conventions H S often just called H is the upward surface sensible heat ux H L LE is the upward surface latent heat ux due to evaporation at rate E HG is the downward ground heat ux into the subsurface medium RN is the net downward radiative ux longwave shortwave The ratio B H LH S is called the Bowen ratio L 25gtlt106 J kg391 is the latent heat of vaporization Over land there is a large diurnal variation in the surface energy budget see schematic below Over large bodies of water the large heat capacity of the medium and the absorption of solar radi ation over a large depth combine to reduce the near surface diurnal temperature variability so H S and H L vary much less However the surface skin temperature of a tropical ocean can vary diur nally by up to 3 K in sunny lightwind conditions An ideal surface is not usually encountered Real surfaces may include a plant canopy or other features such as buildings not opaque to radiation and with a significant heat capacity In this case it is more appropriate to de ne an interfacial layer which includes such features We let WU be the energy stored within this layer per unit horizontal area The revised layer energy budget is RN H5HL HGdWdl We could also consider the energy budget of control volumes with nite horizontal extent e g a parking lot city or larger geographic region In this case horizontal transfer of energy may also be important we won t consider this complication here RN 0 r LAND SURFACE I G b RN H HL LAND SURFACE He Fig 21 Schematic representation of typical surface energy budgets during a daytime and b nighttime lOl Atm S 547 Boundary Layer Meteorology Bretherton N O O 398 OO ENERGY FLUXWm Z I 8 80 I I I I l 24 04 oh 2 lie 20 54 0394 0 8 l 2 re 20 TIMEh Fig 23 Observed diurnal energy balance over a dry lake bed at El Mirage California on June 10 and 11 1950 After Vehrencamp 1953 Examples The energy budget measured over a dry desert lake bed is shown above In this case latent heat uxes are negligible During the day copious solar radiation is absorbed at the surface and the ground heats up rapidly Initially most of the heat is conducted down into the soil but as the layer of warmed soil thickens H S dominates the heat is primarily transferred to the air This is promot ed by extreme differences up to 28 K between the ground temperature and the 2 m air tempera ture At night surface radiative cooling is balanced by an upward ground heat ux Since the nocturnal boundary layer is very stable the turbulent heat ux H S is negligible The energy budget of a barley eld is shown below During the daytime radiative heating of the surface is balanced mainly by latent heat ux due to evapotranspiration i e evaporation from the soil surface and transpiration by the plant leaves In the lingo the Bowen ratio is small O3 to 03 H L can be so large that the surface gets cooler than the air during early morning and late af ternoon and the heat ux is downward For a eld heat storage is usually negligible At night all terms become much smaller as before radiative cooling is mainly balanced by ground heat ux r l RN l l I 1 I Arya 200 r s oo ol4lo39e39 2 16 20 24 TIME h Fig 24 Observed diurnal energy budget of a barley eld at Rothamsted England on July 23 1963 From Oke 1987 after Long 61 a1 1964 ENERGY FLUX Wm E The last example is a Douglas r forest next page Here latent and sensible heat uxes are comparable during the day The storage and ground heat ux are lumped in the curves but for deep forest the storage term dominates At night release of heat from the tree canopy and condensation dew balance radiative energy loss 102 Atm S 547 Boundary Layer Meteorology Bretherton Arya 0quot O O p O O O ENERGY FLUX Wm zl N O O 398 0 IS 20 24 8 08 l2 TIME h Fig 25 Observed energy budget ofa Douglas r canopy at Haney British Columbia on luly 23 1970 From Oke 1987 after McNaughton and Black 1973 Net radiation at the surface The net radiation RN is due to the difference between downwelling and upwelling shortwave plus longwave radiative uxes The net shortwave ux depends on the incident solar radiation Rsy and on surface albedo as The net longwave ux depends upon the downwelling longwave radia tion R L y the surface emissivity es and the radiating temperature T S RN Rssl 39 RsT RLJ 39 RLT 1 39 asRslz RLJ 39 139 8sRL lx l 8SGTS4 Thus the surface characteristics critically in uence RN A table of typical surface radiative char acteristics is given below Albedos are quite diverse while emissivities are usually near but not equal to l Table 31 Alya Radiative Properties of Natural Surfaces Albedo Emissivity Surface type Other speci cations a a Water Small zenith angle 003010 092 097 Large zenith angle 010 050 092 097 Snow Old 040 070 082 089 Fresh 045 095 090 099 Ice Sea 030 040 092 097 Glacier 020 040 Bare sand Dry 035 045 084 090 Wet 020 030 091 095 Bare soil Dry clay 020 035 095 Moist clay 010 020 097 Wet fallow eld 005 007 Paved Concrete 017 027 071 088 Black gravel road 005 010 088 095 Grass Long 1 m Short 002 m 016 026 090 095 Agricultural Wheat rice etc 010 025 090 099 Orchards 015 020 090 095 Forests Deciduous 010 020 097 098 Coniferous 005 015 097 099 Compiled from Sellers 1965 Kondratyev 1969 and Oke 1978 103 Atm S 547 Boundary Layer Meteorology Bretherton 800 D O O b O O 200 ENEIRGY FLUX Wm2 O 200 400 RU 600 l l I l l 0 4 8 l2 IS 24 TIME h Fig 34 Observed radiation budget over a 02m tall stand of native grass at Matador Saskatchewan on July 30 197 From Oke 1987 after Ripley and Redmann 1976 An example of the surface radiation components is shown above Soil temperatures and heat ux The surface or skin temperature is important for the radiative balance of the surface and for pre dicting frost and dew It can be quite different than the surface air temperature which is conven tionally measured at 152 rn In fact it can be dif cult to even measure in silu because it is dif cult to shield and ventilate a sensor placed at the surface Furthermore if there is a plant canopy or sur face inhomogeneity there is no single uniquely de nable surface temperature Radiatively an ap parent surface temperature can be determined from the upward longwave energy ux if the emissivity is known Large diurnal variations in skin temperature are achieved for bare dry sur faces in clear calm conditions Under such conditions midday skin temperature may reach 5060 C while early morning skin temperatures can drop to 1020 C The surface temperature is related to the pro le of temperature in the subsurface medium as illustrated in the gures on the next page In a solid medium the subsurface temperature pro le is governed by heat conduction Deeper in the soil the diurnal temperature cycle decreases and lags the cycle of skin temperature Over an annual cycle similar waves penetrate further into the soil If 2 is depth into the soil and T z I is soil temperature Fourier s law of heat conduction states HG k3Taz k thermal conductivity Therrnal energy conservation implies that car 2 3H G p at 32 Combining these two equations and assuming that the subsurface medium is homogeneous so that material constants do not depend on 2 we obtain the diffusion equation p density 0 heat capacity 104 Atm S 547 Boundary Layer Meteorology Bretherton Observed diurnal subsurface soil temperature variability Arya I l 8 0 TEMPERATURE C 8 20 l l l l l I l I 24 06 I2 l8 24 06 l2 IS 24 06 I2 IS 24 TlMElh Fig 41 Observed diurnal course of subsurface soil temperatures at various depths in a sandy loam with bare surface 25 cm 15 cm 30 cm From Deacon 1969 after West 1952 Observed annual subsurface soil temperature variability Arya I2 x 1 I I I I l l T A X 8390 s 8 x 39 o 6 x lt E 4 x 5 2 x X 0 52 x 34 as 55quotquot Depth x25cm 1640 Xx Depth0243m 42 x E i3 i 1 l L i l l 1 I 0 IO 20 3O 40 50 TIMElWEEKS Fig 42 Annual temperature waves in the weekly averaged subsurface soil temperatures at two depths in a sandy loam soil X 25 cm O 243 m Fitted solid curves are sine waves From Deacon 1969 after West l952 105 Atm S 547 Boundary Layer Meteorology Bretherton Table A7 Representative values of the thermal conductivity ks specific heat cs density pS and thermal diffusivity KS for various types of surface based mainly on Table 113 in Pielke 1984 Data for clay and sand are approximately consistent with Eq A24 in which Csi is equal to 27 X 106 and 22 X 106 Jm 3 K 1 for clay and sand respectively CW is equal to pwcl with pw 2 1000 kgm 3 and C 4186 Jkg 1 K l and 175 is taken from Table A9 The reader should also consult eg Geiger 1965 Table 10 Hillel 1982 Table 93 and Oke 1987 Table 21 Surface kS cS pS KS W m lK39l J kg lK l kg m 3 10quot m2 3quot Sand soil dry 03 800 1600 023 r 02 19 1260 1800 084 77 04 22 1480 2000 074 Clay soil dry 025 890 1600 018 1 02 11 1170 1800 052 71 04 16 1550 2000 052 rock 29 750 2700 14 ice 25 2100 910 13 snow old 10 2090 640 07 new 01 2090 150 03 water 06 4186 1000 014 a BZT T a K 2 K kpc thermal d1ffus1v1ty 1 Z A table of material properties is given below the thermal conductivity varies over almost two or ders of magnitude from new snow low to rock high Wet soils have conductivities about ve times as large as dry soils The thermal diffusivity shows similar trends but less variation Sur prisingly K is smallest for water due to its large heat capacity It is illuminating to look at a soil temperature wave forced by a sinusoidal variation in surface temperature We assume a deep soil temperature T z gt oo Tand take T O T A cos mt We look for a solution to 1 that is also sinusoidal in time with the same frequency 0 T z t T Reazexpi0t Here 61Z is a complexvalued function of 2 To satisfy 1 ima 4c dZadz2 2 To satisfy the boundary conditions a0 A az gt oo O The solution of 2 that satis es the BCs is 612 A eXpl izD D 2K012 106 Atm S 547 Boundary Layer Meteorology Bretherton 25 Ts C 20 I 1 10 39 39 4 12 24 36 t hr Fig 51 Idealized variation of soil temperature through a diurnal cycle for several depths in the soil indicated in metres The curves represent the solutions to Eq57 for sinusoidal forcing these are given by Eq 58 A uniform soil is assumed with KS 08 x 10 6 m2 s 1 and kS 168 Wm 1 K l T z t T eXpzDcosot zD 3 This solution is shown above The temperature wave damps exponentially with depth 2 and lags the surface temperature wave by a phase zD which increases with depth see observations at bot tom of page The damping depth D to which the temperature wave penetrates increases as the oscillation frequency slows and is larger if the thermal diffusivity is larger For moist soil K O8gtltlO396 mzs39l D 014 m for the diurnal cycle and 28 m for the annual cycle The ground heat ux at the surface is HG k aTaz0 kAD Rel z39 eXpz39ot pcKo12 cosot n4 It leads the surface temperature wave by l 8 cycle Hence the ground heat ux is largest three hours ahead of the surface temperature for a diurnally varying surface temperature cycle In practice the diurnal cycle of surface temperature is not sinusoidal Furthermore the surface temperature interacts with the sensible and latent heat uxes so that the surface boundary condition is really the energy balance of the surface which is coupled to the atmosphere Lastly testing of these formulas is complicated by the fact the temperature within 1 cm of the ground can be non uniform so the surface temperature and ground heat ux must be inferred from measurements 60 12 5 l or O o AMPLITUDE TIME LAG as l l 4 o 398 AMPLITUDE C 039 1 0 x TIME LAG d I N O 4 I I 5 O 0 l l I 0 5 I I5 2 25 DEPTH IN SOlL m Fig 44 Variations of amplitude and time lag of the annual soil temperature waves with depth in the soil From Deacon 1969 107 Atm S 547 Boundary Layer Meteorology Bretherton across a buried uX plate a thin plate buried within the soil that measures heat ux based on the temperature difference across it typically at a depth of 12 cm 108 Atm S 547 Boundary Layer Meteorology Bretherton Lecture 1 Scope of Boundary Layer BL Meteorology In classical uid dynamics a boundary layer is the layer in a nearly inviscid uid next to a sur face in which frictional drag associated with that surface is signi cant term introduced by Prandtl 1905 Such boundary layers can be laminar or turbulent and are often only mm thick In atmospheric science a similar definition is useful The atmospheric boundary layer ABL sometimes called Planetary BL is the layer of uid directly above the Earth s surface in which significant uxes of momentum heat and or moisture are carried by turbulent motions whose hor izontal and vertical scales are on the order of the boundary layer depth and whose circulation ti mescale is a few hours or less Garratt p l A similar definition works for the ocean The complexity of this definition is due to several complications compared to classical aerody namics i Surface heat exchange can lead to thermal convection ii Moisture and effects on convection iii Earth s rotation iv Complex surface characteristics and topography BL is assumed to encompass surfacedriven dry convection Most workers but not all include shallow cumulus in BL but deep precipitating cumuli are usually excluded from scope of BLM due to longer time for most air to recirculate back from clouds into contact with surface Air surface exchange BLM also traditionally includes the study of uxes of heat moisture and momentum between the atmosphere and the underlying surface and how to characterize surfaces so as to predict these uxes roughness thermal and moisture uxes radiative characteristics Includes plant canopies as well as water ice snow bare ground etc Characteristics ofABL The boundary layer itself exhibits dynamically distinct sublayers Interfacial sublayer in which molecular viscositydiffusivity dominate vertical uxes V Inertial layer in which turbulent uid motions dominate the vertical uxes but the dominant scales of motion are still much less than the boundary layer depth This is the layer in which most surface wind measurements are made V Layers i ii comprise the surface layer Coriolis turning of the wind with height is not evident within the surface layer iii Outer layer turbulent uid motions with scales of motion comparable to the boundary layer depth large eddies At the top of the outer layer the BL is often capped by an entrainmentzone in which turbulent BL eddies are entraining nonturbulent freeatmospheric air This entrainment zone is often associated with a stable layer or inversion For boundary layers topped by shallow cumulus the outer layer is subdivided further into ll Atm S 547 Boundary Layer Meteorology Bretherton subcloud transition cumulus and inversion layer Garratt fig 11 Outer Ekman layer 1 lt h z E 01 ll Inncr surface Inertial layer sublayer 5 gt 51gt Interfacial roughness j H r H l sublayer Boundary layers are classi ed as unstable if the air moving upward in the turbulent motions tends to be buoyant less dense than in the downdrafts and stable if the reverse is true If there is negligible buoyancy transport within the BL it is called neutral On a hot sunny morning surface heating causes the boundary layer to become strongly unstable and convect vigorously with outer layer updrafts of 13 m s391 which are a few tenths of a K warmer than the downdrafts transporting several hundred W m392 of heat upward In desert regions such BLs can grow to a depth of 5 km or more by afternoon though typical summer early afternoon BL depths over Midwest Seattle etc are l2 km At night the surface cools by radiation The BL depth can become as little as 50 m on a clear calm night and the BL tends to be stable with weak downward buoyancy uxes Rarely is an ideal neutral ABL observed but with strong winds buoyancy effects can become relatively unimportant especially for winds over the oceans blowing along contours of constant SST Typical ABLs over the ocean tend to be slightly unstable with little diurnal cycle due to the nearconstancy of SST BL depths vary from a few hundred m in regions of warm advection to 153 km where cold advection has led to shallow cumuli subtropical trade wind belts cold air outbreaks In regions of deep convection a BL top can be dif cult to de ne Within the ocean there is also an oceanic BL driven by surface wind stress and sometimes con vection and considerably affected by the absorption of radiation in the upper ocean It is usually but not always stable The oceanic BL can vary from a few m deep to a few km deep in isolated locations e g Labrador Sea and times where oceanic deep convection is driven by intense cold air advection overhead Applications and Relevance ofBLM The boundary layer is the part of the atmosphere in which we live and carry out most human activities Furthermore almost all exchange of heat moisture momentum naturally occurring par ticles aerosols and gasses and pollutants occurs through the BL Speci c applications i Climate simulation and NWP parameterization of surface characteristics airsurface ex change BL thermodynamics uxes and friction and cloud No climate model can succeed without some consideration of the boundary layer In NWP models a good boundary layer is critical to proper prediction of the diurnal cycle of lowlevel winds and convergence of l2 Atm S 547 Boundary Layer Meteorology Bretherton effects of complexterrain and of timing and location of convection Coupling of atmospheric models to ocean ice landsurface models occurs through BL processes iiAirPollution and Urban Meteorology Pollutant dispersal interaction of BL with mesoscale circulations Urban heat island effects iii Agricultural meteorology Prediction of frost dew evapotranspiration ivAviation Prediction of fog formation and dissipation dangerous windshear conditions V Remote Sensing Satellitebased measurements of surface winds skin temperature etc in volve the interaction of BL and surface and must often be interpreted in light of a BL model to be useful for NWP 1900 1910 1910 1940 1940 1950 1950 1960 1960 1970 1970 1980 1980 1990 1990 History of BLM Development of laminar boundary layer theory for aerodynamics starting with a seminal paper of Prandtl 1904 Ekman 19051906 develops his theory of laminar Ekman layer Taylor develops basic methods for examining and understanding turbulent mixing Mixing length theory eddy diffusivity von Karman Prandtl Lettau Kolmogorov 1941 similarity theory of turbulence Buoyancy effects on surface layer Monin and Obuhkov 1954 Early field experiments e g Great Plains Expt of 1953 capable of accurate direct turbulent ux measurements The Golden Age of BLM Accurate observations of a variety of boundary layer types including convective stable and tradecumulus Verificationcalibration of surface similarity theory Introduction of resolved 3D computer modelling of BL turbulence largeeddy simulation or LES Application of higherorder turbulence closure theory Major field efforts in stratocumulustopped boundary layers FIRE 1987 and landsurface J 39 modeling The Age of Technology New surface remote sensing tools lidar cloud radar and extensive spacebased coverage of surface characteristics LES as a tool for improving pararneterizations and bridging to observations Coupled oceanatmosphereicebiosphere and mediumrange forecast models create stringent accuracy requirements for BL pararneterizations Accurate routine mesoscale modelling for urban air ow coupling to air pollution Boundary layer deep convection interactions e g TOGACOARE 1992 u C 1 411011 13 Atm S 547 Boundary Layer Meteorology Bretherton Why is the boundary layer turbulent We characterize the BL by turbulent motions but we could imagine a laminar BL in which there is a smooth transition from the freetropospheric wind speed to a noslip condition against a surface e g a laminar Ekman layer Such a BL would have radically different characteristics than are observed Steady Ekman BL equations 2 height surface at z 0 free troposphere is z fv v dZudzz fu G v ciZVciz2 u0 O uoo G v0 O voo 0 Solution C 28 for BL velocity pro le uZ 01 e39C cos C vz G e39C sin C vug uug Fig 54 Hodograph of the wind components in the Ekman spiral solution The arrows show the velocity vectors for several levels in the Ekman layer while the spiral curve trace out the velocity variation as a function of height Points labeled on the spiral shovl the values of yz which is a nondimensional measure of height Flow adjusts nearly to geostrophic within Ekman layer depth 5 2vf 2 of the surfaceWith a free tropospheric geostrophic velocity of G in the x direction the kinematic molecular viscosity of air v l4gtltlO395 m2 s391 and a Coriolis parameterf 10394 s391 5 05 m which is far thinner than observed Hydrodynamic Instability Laminar BLs like the Ekman layer are not observed in the atmosphere because they are hydro dynamically unstable so even if we could artif1cially set such a BL up perturbations would rapidly grow upon it and modify it toward a more realistic BL structure Three forms of hydrodynamic in stability are particularly relevant to BLs i Shear instability ii KelvinHelmholtz instability iii Convective RayleighBenard instability By examining these types of instability we can not only understand why laminar boundary layers are not observed but also gain insight into some of the turbulent ow structures that are observed The Shear Instabilit Instability of an unstratif1ed shear ow Uz occuring at high Reynolds numbers Re VLN 14 Atm S 547 Boundary Layer Meteorology Bretherton E a b c d e Figure 1713 To illustrate that the velocity profiles of a pipe flow b a boundary layer c a wake d a jet and e a free convection boundary layer are all shear ows Tritton Some shear ows Dots indicate in ection points where Vis a characteristic variation in the velocity across the shear layer which has a characteristic height L Here high means at least 103 an ABL with a shear V 10 m s391 through a boundary layer of depth 1 km would have Re 10 m s3911000 m 10395 m2 s391109 which is plenty high Inviscid shear ows can be linearly unstable only if they have an in ection point where a 2 U dz2 O Rayleigh s criterion 1880 and are de nitely unstable if the vorticity dUdz has an extre mum somewhere inside the shear layer not on a boundary Fjortoft s criterion 1950 This ex cludes pro les such as linear shear ows or pipe ows between boundaries but some such pro les are in fact unstable at small but nonzero viscosity and may still break down into turbulence The Ekman layer pro le has an in ection point so is subject to shear instability as well as a second class of instability at moderately large Re of a few hundred In shear instability a layer of high vorticity rolls up into isolated vortices A good example is the von Karman vortex street that forms the the wake behind a moving obstacle MIN 39I M i l Inffl I39i39l39liIWJELIIIUJIFL THIHL YMI at I L u Hr mi 533 n h 5 m E H39quot39 jquotquotquotquot E Il 39r39quotifi 39 39F IU39J i SHINE1quot M lfluSlm nanrt L r r39m Inrm Nah cm l Jul39zir mm ll39lEHTIllril iIrz alaiuuti um I39 an i 17 ml 3 quot213 1 i i39mlrnmhrn III a armquot 39 J I hl an 23 mun9M1 van Dyke p 56 l5 Atrn S 547 Boundary Layer Meteorology Bretherton KelvinHelmholtz Instabili cram mmquotan down ll rl slept while h water hhm 145 KalvmHclmhnlrz inslabilrw hr slrall cd shnr n u R my a mh Watcr him hohmu bum The rhmh are allowed to he xmtrhu urcurs aim a N suonds and has her ThmPc l 77l van Dyke p 85 wx mama setting the fluid mm nohuh The bnnc mu rur 39 39 39 39 m er 39 39 in tabilitv ofthe shear layermay still occur if the strati cation is suf ciently weak Shear instability at the interface between two of different densities was rst investigated by Helmholtz 1868 Miles 1960 showed that for a continuously Varying system insta ility cannot occur if the static stability as measured by buoy cy requency Nis large en Ri NZdUdz2gt 14 throughout the shear layer For lesser Values ofRi instability usually does occur The general form ofthis criterion can be rationalized by considering the mixing oftwo parcels of uid ofvolume Vat different heights In a ow relative coordinate system Lowerparcel has height 62 initial density p 7 5p Velocity BU Upper parcel has height 62 initial density p 5p Velocity BU Here BU dUdzBz and 5p dpdzBz where N2 gp dpdz For simplicity we consider an incompressible uid and assume each parcel has Volume V at heighm The total initial energy of the parcels is E KE PE 05V P i 5P 5UZ P 5P 5UZ VP 5Pg 52 P 5Pg52 Vltplt est2 2 gsp z If the parcels are homogenized in density and momentum Lowerparcel has height 62 final density p Velocity 0 Upper parcel has height 62 final density p Velocity 0 The total final energy is Ef KEer PEf 0 VPg 52 Pg52 0 so the change in total energy is E 3 16 Atrn S 547 Boundary Layer Meteorology Bretherton AE EfE Vp amz 2 ngBz 1435Z dUdzZ ZNZ energy reduction occurs ifdUdzZ gt 2N i e ifRi lt 12 In this case residual energy is avail able to stir up eddy circulations The reason this argument gives a less restrictive criterion for in stability than 39 39 39 quot i tahilitie ofa shear layer Convection herrnal convection occurs ifthe potential density decreases with height in some layer Clas sically this instability quot hv 39 39 39 between in an 39 39 39 39 39 larger than that of the upper plate In the absence ofconvection the temperature pro le within the uid would vary linearlywith 39 W4 A 39 Ifm I 4 L 1 A difference AT and if 39 39 39 iscosity v 439 39 39 x 2x105 m2 s391 for air then convective instability occurs when the Rayleigh number Ra thBVK gt 1700 Here AB is the buoyancy change gApp associated with a temperature increase ofAT at a given pressure for air and other ideal gasses AB gAT TI The instability is a circulation with cells with L A hr er Rolls and hexagonal patterns are equally unstable L r L the shear vector as seen in the cloud Fig7 Commu u t n5mblcnylrahtnmlm surnrs nlnng meanecmmmm mh hg 4 up x wluvlr 31w rim unwind by l mm a nu e gt r minew am can shut m convection l humnuwh u 17 Atrn S 547 Boundary Layer Meteorology Bretherton F r ART i n 39 39 Luau Lu l pica ary layer air temperature Even with a small AT1 K we can estimate AB 10 m s39Z1 K300 K 003 m 211 1000 m and Ra 003 m s39Z1000 m3 14 x105 mZ s3912X10395 mZ s39l1017 z m m small viscosities TVansi on to turbulence Each of these instabilities initially has a simple regular circulation panem However ifthe uid is suf ciently inviscid threedimensional secondary instabilities grow on the initial circulation 39 in nlarintime A 39 39 quot 39 on a variety of scales This is a transition into turbulent motion We don t generally see this tran quot39 39t D39 39 quot 39 quot quotquot39 quotquot growsisrarelyrealized mz hunbrliw at an nxisymmuric jet A lzmmar cdgrullhewrdzvelnpsaxisvmmuncoscillalians rullaup 5mm or in ow from a circular rub m Raymlds mm Vnna rings and the abruptly becomes turbulent Th 18 Ams 547 BanndaryLayzxMz eamlngy Brahman Lemuel m mm m Gama 6 2 H mm mchmlELNBLhas pmvedam fth mm m w tiypsafELm undzxstnnd andmadzl mbamwuwmm mm anysnrznn m dzep Tm mllznce tends mm mmnm zm gummy m nuns a mmnmngledwht Ilzncees c1 pun lay m mmm m mmm Itself nachmg 1 K ham axmme mm unwesamnn mbycam an a JUszamraNaLh1 Elmwuuldcaahuumz aqua ZKhr Evemhzhxgestm emeddusdnmtspmduenm EL sa mm xs 5 tendzncym 1mm afchzmmals and nemsals wnhm m EL espcm ymthz pp pun afthz EL Wm mmm 15 mm Wmd mm a much lass wz rmxxed m mgm than mu m daynmz canvashve EL a 5 H 2 4 M 3m W uuz v 2 wavingquot 39 y u 39 nun mum mmmwammmsz mm mm mmmm n 21 my A an NHL Mulznce dzcnases sharplywnh mm 131 Ams 547 BanndaryLayzxMz eamlngy Brahman y R1 112 Mmdmauwm amusmw m m mammmmmdwww W 539 my wnh gmiywm mum um can mndulm Inca 51W smm39xca nn and 1mm mmm H m Neaxacam sunset rame swung mm snrpnsmgkyxasum hzlps mmmnm a 5mm smu cman b typqu sfmngrwmd NEL a a wukwmd may nndzx clzu sky 132 Atm S 547 Boundary Layer Meteorology Bretherton An idealized NBL model One illuminating theoretical idealization is a NBL of constant depth driven by surface cooling only Nieuwstadt 1984 J Atmos Sci 41 22022216 In practice this is most realistic when winds are strong producing sufficient turbulence to make substantial downward buoyancy uxes that are much larger than the radiative ux divergence across the NBL which is typically less than 10 W m392 We take the friction velocity 11 the geostrophic wind U g taken to be in the x direc tion and the Coriolis parameter f as given In a practical application we would likely know the surface roughness length 20 not 11 but we could use the solution below to relate these two param eters We assume i The entire BL extending up to a fixed but unknown height h is cooling at the same rate and maintains fixed vertical profiles of stratification and wind ii No turbulence at the top of the BL iii Within the bulk of the BL above the surface layer the sink of TKE due to buoyancy uxes is assumed to be a fixed fraction Rfrs 02 of the shear production of TKE The remaining fraction 08 of the shearproduced TKE goes to turbulent dissipation as transport is observed to be neg ligible This is the same as saying that the ux Richardson number Rf 02 iv No radiative cooling within the BL v The unknown Obuhkov length L is assumed much smaller than the boundary layer depth Hence the largest eddies have a depth which is order of L since deeper eddies do not have enough TKE to overcome the stratification by the scaling arguments we made in discussing the z less scal ing at zgtgtL when we discussed MoninObhukov theory vi The eddies act as an unknown heightdependent eddy viscosity and diffusivitme K h as sug gested by MoninObuhkov theory Hence the gradient Richardson number Ri Rf so is also 02 throughout the BL vii The BL is barotropic Scaling Note that one could also use firstorder closure on this problem instead of invoking assump tions iii v and vi about the eddies and their transports This would give a largely similar an swer as long as the lengthscale in the firstorder closure was on the order of L through most of the boundary layer depth and could also be used to relax the assumptions of steadiness uniform cool ing rate no radiative cooling and no thermal wind However the equations would not permit a closedform solution which displays the parametric dependences clearly We first scale the steadystate momentum equations then use a clever approach to solve them Assumptions i and ii imply that if the unknown surface buoyancy ux is Bo lt 0 then Bz W Bol zh l The steadystate BL momentum equations are fv vg WWW 2 u ug BBZW 3 If indicates scale of the above assumptions imply u39 V39 W39 W Km eddy velocity scaleeddylengthscale ML 133 Atm S 547 Boundary Layer Meteorology Bretherton gt Bu32 WKm mzmL mL similarly for v 332 h391 To apply this scaling to 2 3 we differentiate them with respect to z noting that the geostrophic wind is constant with respect to height by assumption vii favaz BZBz2 W 4 fauaz BZBz2 7W 5 Scaling the two sides of 4 we nd favaz fmL Bl822W ui2h2 The same scaling holds for 5 This implies a scaling for BL depth h h YcuLf12 6 where yc is an as yet unknown proportionality constant Solution Now we have understood the scaling of the equations we solve them in nondimensional form This is a bit technical so feel free to skip to the results It is mathematically advantageous to com bine 4 and 5 into one nondimensional complexvalued equation Let the nondimensional height shear momentum ux and eddy viscosity be E zh sv Lm 3u ivBz and 039 W iwymz KmE KWmL Then 4 and 5 can be written s WG2 aleBE 7 The boundary conditions come from the de nition of friction velocity and assumption ii that stress vanish at the BL top The surface momentum ux m is in the direction opposite the wind If the unknown surface crossisobaric wind turning angle is 0L then the two BCs are 50 equot 1 0139 01 0 0139 The eddy viscosity assumption vi implies that 039 mEsv Since the nondimensional eddy vis cosity is real this is equivalent to requiring that the complex numbers 039 and sv have opposite phase at all nondimensional heights sv The last condition we must enforce is iii that buoyant consumption of TKE is 02 of shear production 02 Rf BzSz Bol zhu39w39BuBz v39w39BvBz Bol E u3LRe039sv denotes complex conjugate Substituting 7 in for Sv noting that by definition of Obuhkov length BO u3kL and that the eddy viscosity assumption implies that 5 sv is guaranteed to be real we obtain the nonlinear ODE 134 Atm S 547 Boundary Layer Meteorology Bretherton 6BZGBE2 iMl E where 70 Yczka is unknown 8 This equation can be solved systematically by substituting 039 reie and obtaining a pair of ODEs for rE and 9E However an easier approach is to look for a trial solution in the form 6a equot 1 a This solution automatically obeys the boundary conditions and has the right form to match the RHS of 10 Substituting into 10 we nd that this trial solution works if aa2 1 6161 1l7 Setting a ar 10 the rst of these equations implies that ar 32 From the second we deduce that 0 Reaa 1 Re32z39a12z39a 34 a2 gt ai3122 s 6a equot 1 96 W3 1 Imaa 1 1m32z39a12z39a 2a 312 yfka gt 7 312kR 12 037 so h037uLf12 Z SV WG 2 326352 iaa1Yc 2eid1 1N32 M39c39zeia E391 T NW2 is nondim shear gt KmE 53 1 Ef m 0081 E2 is nondim eddy viscosity Hence remarkably we have been able to deduce the BL depth There is one shortcoming which is thatL must still be deduced The deduced eddy viscosity decreases with height to zero at the BL top as we d expect since turbulence is concentrated at the surface The shear pro le can be inte grated from E 1 z h and the resulting velocity pro le redimensionalized to obtain uiv2 uhL2Mc3921Jen3eio 1 1N32 At the BL top the velocity is Ug At the surface the velocity is zero Hence setting E 0 on the RHS and noting that l i32 eXpz39TE3 and that we have u ml Ug uhL70YC392eXpiX m3 9 For consistency the RHS must be real and have the same magnitude as the LHS Thus 0L TE3 surface isobaric wind turning angle of 60 degrees 10 Ug uhLMc392 uhLlka 11 Summary of Results and Comparison to Observations PBL depth h ycuLf12 where y 312kRnl2 037 135 Atm S 547 Boundary Layer Meteorology Bretherton Wind pro le u ivUg 1 1 zhlt1 t W Note that h can be expressed in terms of the given parameters as h 037uLf12 037u4kBOf12 037u40 12kf2Ug212 17urzng A larger friction velocity smaller geostrophic wind or lower latitude will increase h Also note that the wind pro le is independent of the surface roughness except in the surface layer 2 ltlt L where the assumed eddy scale of L is no longer applicable and 12 is invalid The surface isobaric turning angle is 60 degrees and the wind turns to geostrophic at the PBL top We can solve l l for the Obhukov lengthL h uUglk Rf l25huUg Substituting for h this can also be written as L 70uLf12 uUglk M or L uPngzmk R32 This can be used to deduce the surface buoyancy ux which by definition of L is Surface buoyancy ux BO u3kL 012ng2 The constant is Rf312 Remarkably the downward surface buoyancy ux is depends only on the geostrophic wind and is independent of surface roughness The NBL structure obtained from this approach is fairly realistic For reasonable values of m 03 m s39l Ug 10 m s39l and f 10394 s39l we nd that h 037uLf12 150 m close to ob served NBL depths ofO100 m The Obhukov lengthL 56 m and Lh rs 038 ltlt 1 consistent with our original assumption that the vertical eddy mixing scale is much less than the PBL depth The the downward surface buoyancy ux BO l2gtlt10393 m2s393 i e a virtual heat ux chGRgBO cs 40 W m392 For Ug 5 ms 391 the downward buoyancy ux would be only 25 as large as this These are not a large heat ux atmospheric turbulence cannot keep the ground from cooling rap idly at night under clear skies unless the geostrophic wind is large Instead ground heat ux is the major counterbalance to nocturnal radiative cooling The surface energy budgets e g over a dry lake bed nicely showed the fairly small role of surface heat uxes in the nocturnal boundary layer The NBL strati cation can also be deduced dbdz N2 Rildudzlz RiuL2lsvl2 Rik2Rf2uL21 zh 1 Since Ri Rf 02 the constant is lk2Rf 31 Integrating with respect to z we obtain 132 130 3lhuL21nl a This has a singularity at the BL top which is a bit disturbing but relates to the assumption that there must be uniform cooling all the way to the BL top even though there is very little turbulence near the BL top The small turbulent diffusivity then requires a large gradient there For our ex ample values N2 9gtlt10394 s392 26 K per 100 m at the surface rising with height To get a more stable BL than this we must have diabatic e g radiative cooling within the BL A comparison of this theory to observations is shown in the figure on the next page It should 136 Atm S 547 Boundary Layer Meteorology Bretherton 1 Garratt 212 0 WI 05 W 1 1 I I I 1 l l I I 05 IVVIV I l 20 60 I 1 1 g 9 76 0W 0 20 40 a 0 0 z m 150 100 10 72 0 Vm s4 6OC Fig 615 a Predicted values of crossisobar ow and normalized wind speed Eq 668 and of normalized temperature difference Eq 669 as functions of normalized height b Observations from Cabaow of cross isobar ow angle wind speed and temperature as functions of height in the NBL From Nieuwstadt 1985 by permission of the Oxford University Press Comparison of steady NBL theory top with tower observations bottom in a case of strong geostrophic wind be noted that this case has a high geostrophic wind speed so that the surface buoyancy ux is large and the relative importance of radiative cooling in the NBL dynamics is smaller than usual The comparison is quite good under these conditions The predicted linear increase of wind with height in the BL and the concentration of the wind turning at the BL top are both observed The observed wind turning of 30 is less than predicted however As predicted the strongest stratif1ca tion is near the BL top Normally however the NBL is most strongly stratif1ed near the ground where clearair radiative cooling is strongest as seen in other soundings in these notes The one step in applying this approach that we have not discussed is how to relate m to Ug and the surface roughness Z0 The velocity pro le deduced above linearly approaches zero at the sur face rather than the loglinear behavior of MO theory Empirical formulas given on pp 6364 137 Atm S 547 Boundary Layer Meteorology Bretherton of Garratt can be used to relate m to U They are given in terms of two functions A204 and Bzu of LL hL and are typically expressed in coordinates parallel to the surface wind Translating these formulas into our notation we find Cg u Ug2 kzlnhZO A22 322 12 where for moderately stable conditions 0 lt LL lt 35 Garratt s eqn 389 implies that A2 1 03811 32 45 0311 For our example Cg u1Ug2 00009 and u 27 so A2 00 32 53 The surface roughness that could give this NBL is found by solving l3 anal20 12 322 k2Cg gt lnhZOA2kzCgBZZ12 122 2020001 m typical of ow over a smooth land surface such as sand A change in Z0 of several orders of mag nitude is necessary to move m up or down by 50 for a given geostrophic wind speed Kalaball39cF lows Sloping terrain has a large in uence on stable boundary layers The cold dense air near the surface is now accelerated by the downslope component b sin 01 of its buoyant acceleration 01 is the slope angle and b lt 0 is the buoyancy of air within the BL relative to aboveBL air at the same height Viewed in terrainparallel coordinates b sin 01 is like an effective pressure gradient force which is strongly heightdependent since b depends on 2 In this sense the slope acts similar to a thermal wind which would also be associated with a height dependent PGF Slopes of as little as 2 in 1000 can have an impact on the BL scaling As the slope increases or BL stability increases the velocity pro le is increasingly determined by drag created by turbulent mixing with air above rather than surface drag As for the NBL the BL is typically 10s to 100s of m thick Over glaciers katabatic winds often occur during the day as well as during the night since the net radiation balance of a hi ghalbedo surface is negative even during much of the day and evaporative cooling due to surface snowmelt can also stabilize the air near the surface On the coast of Antarctica persistent katabatic ows down from the interior ice caps can produce surface winds in excess of 50 m s39l Qjmb z Garratt Fig 622 Schematic representation of the downslope ow typical of nighttime ow under light wind clear sky conditions Here a is the slope angle and d is the 6 deficit of the ow relative to the ambient field 138

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I used the money I made selling my notes & study guides to pay for spring break in Olympia, Washington...which was Sweet!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.