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# Atmospheric Convection MEA 714

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This 8 page Class Notes was uploaded by Ian Davis on Thursday October 15, 2015. The Class Notes belongs to MEA 714 at North Carolina State University taught by Matthew Parker in Fall. Since its upload, it has received 20 views. For similar materials see /class/223853/mea-714-north-carolina-state-university in Marine Science at North Carolina State University.

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Date Created: 10/15/15

MEA714 Review of cloudmeso scale governing equations Dr Matthew D Parker The purpose of this document is to show the details necessary to derive some key results that appear in Houze s Cloud Dynamics textbook You have been assigned to read Houze s Chapter 7 We also recommend that you review Houze s Chapter 2 download it from the from course website which is a review of many of the basic equations that are used in later chapters of the teXt The primary variables ofinterest are u 1 u p 0 Oz lp qv and qn Where n l ofhydrometeor species Vector equation for motion 7anigFr72 xui x xrianigFri2 xu 1 This is equivalent to Houze s 21 wherein vectors are denoted by bold type 11 is the Wind vector with components u 1 and u g is Newton s gravitational acceleration g g Q x Q x r is the appar ent gravitational acceleration including Newton s gravity and the centrifugal acceleration aassociated with planetary rotation is the planetary rotation vector Fr is the frictional acceleration vector and D Dt is the material parcelifollowing derivative operator D Dt 8825 11 V The components of this equation are Du 8p 5770472w cos 2st1n gtFTh 2 Du 3 E 7aaiii2u s1n gtFMg 3 Du 3 affaiig2u cos Frzg 4 wherein j is latitude Scale analysis shows that for miaLlatitude cloudiscale and mesoscale analysis of moist convection we may simplify to Du 7 8p E 7 043 5 5 Du 8p E 7 Otaiy fuy 6 Du 8p Dt 7 704 7 97 7 wherein f 2Q sin 1 is the Coriolis parameter We wish to apply our equations to convective clouds and an obvious characteristic of cloudy air is that hydrometeors liquid and solid water are present The mass of hydrometeors adds to the weight of an air parcel To include this effect consider 7 in terms of forces per unit volume rather than in terms of accelerations Du 8p 7 if 7 8 0 Dt 82 pg Review of cloudmesmscale governing equations M D Parker 2 If q H is the mixing ratio of all hydrometeors in the air the mass per unit volume of all the hydrometeors is qu Thus the force associated with the weight of the hydrometeors is included via Du 7 8p 9 P Dt 82 P9 Pquy or in terms of accelerations 8 Du p 7 7 i 7 1 10 D t 0 82 9 l qH Conservation ofmass D p 7 7 V 11 Dt 0 u which is equivalent to Houze s 220 Or alternatively D 17 av u 12 Consider splitting 04 into mean and perturbation parts 04 040 0 for most reasonable ie none nuclearvolcanic convective clouds 0 ltlt 040 In other words 040 o z 040 Therefore u V040 040V u 13 or alternatively V pan 0 14 This is the anelastic sounaLproo continuity equation which is equivalent to Houze s 254 Because the mean p0 and 040 are only functions of height if desired 13 and 14 can also be written dao dPo w iaovuwdz poV u 0 15 For some applications 14 can be further simpli ed via the Boussinesq approximation in which p0 is treated as a constant In this case the continuity equation expresses threeidimensional incompressibility V u O which is equivalent to Houze s 255 Treating density as invariant in the vertical the Boussinesq approximation can be acceptable when studying shallow motions However it is a poor assumption when studying deep motions eg cumulonimbi Thermodynamic equation D0 7 S 16 Dt 97 where 59 is a source or sink term eg diabatic heating This is equivalent to Houze s 29 Conservation ofwater substance 13 i Dt 75 17 13 S 18 D luv for n 1 of hydrometeor species Again SW is a source or sink term related to conversions among species This is equivalent to Houze s 221 Review of cloudmesmscale governing equations M D Parker 3 Equation of state pa RT 19 wherein R is the gas constant for the moist air mixture and 04 is understood to represent the parcel s speci c volume considering all gaseous species including water vapor For simplicity we commonly use the virtual temperature 1904 RdTM 20 which is equivalent to Houze s 24 and wherein 7 lqvE N p RCp FFltW 0W 1061q 21 Splitting 20 into mean and perturbation parts p00pvma41avaRdLUIWZQ an Removing the base state ie dividing by poao RdTvo and taking the logarithm of 22 yields 1n1p po 1n1a a0 1n1T TO 23 Provided that the perturbations are reasonably small a Taylor series approximation 1n 1 z z mfg2 3 3 7 m44 can be used to rewrite 23 to a very good approximation as T iiw po 040 Two which is equivalent to Houze s 274 This form often appears in cloudiscale analyses The equation of state is also occasionally written using the virtual potential temeprature 100000 P RCp 0 T z 0 1 061 25 P Substituting and following through the preceding analysis then yields R 0 lt177gt33i 26 CI P0 0 0 0710 This form is also used in cloudiscale analyses eg the buoyancy term in Houze s 252 Anelastic equations The above represent a closed set with the exception of the Sr terms which are speci ed from thermody namicmicrophysical equations in the variables u v w p 0 a qv and q Because acoustic waves are generally thought to be meteorologically irrelevant it is often preferable to use 13 or 14 For consistency with the anelastic continuity equation the assumption that 040 o z 040 should also be canied through in in the equations of motion For 5 and 6 this is simply D a 8 iaoa p0p fvziaoaiifm 27 D07 8 N 829 E7 04004P0P fu aoaifu 28 Note again that the horizontal gradients of all mean atmospheric variables eg p0 are zero Review of cloudmesmscale governing equations M D Parker 4 More care is required in the case of w so that small but potentially important buoyant accelerations are retained Rewriting 10 Du a Eaoa jpop 91qH 29 8 The base state is hydrostatic and free of hydrometeors 040 819082 7 g removing this hydrostatic mean state yields Dw 7 829 3190 E aotaa a 82 qu 3 8 7aoa 87ia ai7gq 31 819 0 aota 59lt qHgt 32 819 0 3 0405 9 070 qHgt 33 This formulation at last explicitly reveals the concept of buoyancy in the vertical accelerations Using 24 or 26 buoyancy can be de ned as 0 T p 0 R p B E 7 7 7 7 7 7 7 7 71 7 7 34 g 0 0 qH 9 TWO P0 qH g 0110 lt61 gt190 qu7 so that 33 is more compactly written Du 819 7 7 0 7 Dt 0 82 Collecting 27 28 and 35 the vector equation of motion that is compatible with the anelastic continuity equation is B 35 7040Vp 7 fk x u Bk 36 This is equivalent to Houze s 72 except that Houze has ignored the Coriolis term If we make the Boussi nesq shallow motion approximation 36 continues to apply to the horizontal wind components u and 12 However the Boussinesq continuity equation V 11 O constrains the relationship among u v and w so that we can only predict two of the three Therefore in the Boussinesq system w is not predicted from 36 Instead it is determined diagnostically from u and 1 using the continuity equation Diagnostic pressure equation The anelastic set has the bene t that sound waves are excluded But it is no longer possible to predict the pressure eld because the total density is never predicted when using 13 or 14 However the pressure eld must be obtained in order to evaluate the pressure gradient acceleration in 36 For this reason a diagnostic pressure equation is needed Moreover the diagnostic pressure equation proves quite useful for cloud7scale analysis Combining ul4 with p36 converts the vector equation for motion to ux form 8 a Poll V pouu 7W 7 pofk x u poBk 37 The components of this vector equation are 3 819 i p a 1001 l V Pouu l a Pofv 07 38 Review of cloudmesmscale governing equations M D Parker 5 8 8 a pow V pow 8 pofu 0 39 8 8 a pow V pawn 8 poB 0 40 The diagnostic pressure equation is derived by taking V 37 in other words by combining 88m38 88y39 and 88240 The resultant component equations are 0 0 0 0219 0 a pawl a uy pan pou w 82 p0a m 0 lt41 0 0 0 0219 0 a 87 pawl 87 W pan pou w 872 pea m 0 lt42 0 0 0 0 0219 0 a pow E w V pou p0u Vw 822 93013 0 43 0 or in vector form 6 2 a A a Vpou VpouVuV 75P0Bp0V ltkaugt 0 44 0 This is equivalent to Houze s 73 except that again Houze has neglected the Coriolis term Regrouping yields the traditional form of the diagnostic pressure equation 8 A vzp V p0 u V u 5 p013 pov M x u 45 At any time this can be used to infer the perturbation pressure eld from other known variables Interpretation Unfortunately 45 is not easy to interpret straightaway For this reason the perturbation pressure eld is often broken into three parts a part that is in geostrophic balance with the wind eld a part that is due to local buoyancy and a a part that is due to dynamic effects ie p p G p B p33 and therefore 2 7 2 2 2 V V p G V pB V pp Separat1ng 45 1nto 1ts geostrophic buoyant and dynamic parts y1elds vng pov 1 x u 46 8 2 V p a p03 47 which is equivalent to Houze s 75 and W 7V mu V u 48 which is equivalent to Houze s 76 The geostrophic part in 46 is treated at length in dynamic and synoptic meteorology courses The meaning of 47 is that the pressure is minimized where buoyancy increases with height and maXi mized where buoyancy decreases with height Review of cloudmesmscale governing equations M D Parker 6 In order to gain insight from 48 however some additional work is needed Applying the vector identity V1Ja aV1J 1 V a to 48 yields 7Vng 11 V u We poV 11 V u 49 or as p0 po 2 8 was 11 v w 8 pov 11 v u so It proves useful to eXpand all of the terms in 50 so that they can be regrouped in a more ef cacious way 7vng 20 118710 087w 0 3m 3m 3x2 3m 3y 3m3y 3m 32 3m32 314 31 32v 3U 31 3 3w 31 32v wwawaaaa aaw 3u 3w 32w 3U 3w 32w 3w 3w 32w 5 Wag 87 My 5 10 51 Vzp vaiw 108710 222 00 u3m 3m 3y 32 U3y 3m 3y 32 w32 3m 3y 32 3m 3y 32 3m 3y 32 3m 32 3y 7 or 30 3w 3w w ivzp 8 ltuv87ywggtpouVVu Polltgt2ltgt2ltgtz 2 en 00 3m3y323z323y 39 The second term on the rhs of 53 can now be reiwritten using 15 poluVVul ipo luv 54 7 300 3w 300 3w 2 32 300 3w u 32 3m U 32 3y pow 322 11100 w 32 3239 55 Substituting 55 into 53 and cancelling corresponding terms then leaves 314 2 au 2 3w 2 82 2 7 7 7 7 7 7 2 MD poilt8zgt lt8ygt lt8z w 8220 uid extension terms 2 31 314 3u3w 3U3w P0 V uid shear terms 56 Review of cloudmesmscale governing equations M D Parker 7 This is the form of the diagnostic equation for dynamic pressure pertubations that is most often given in the literature and in educational textbooks The meaning of 56 is that the pressure is minimized in regions of vortical ow ie the shear term is gt 0 and is maximized in regions of convergence ie the extension term is lt 0 or deformation ie the shear term is lt 0 For some analyses the diagnostic pressure equation is divided into linear and nonlinear parts Dividing the components of the velocity vector into mean and perturbation parts u 1402 u v 1202 1 w w substituting into 56 and then removing all nonilinear terms ie products of primes yields du 3w d1 3w 2 77 07 07 VPDWT 23900lt2l23x01231 The other remaining terms are therefore the nonlinear part 81 2 ea 2 3w 2 82 2 7 7 7 7 7 7 27 VPDMquot 7 p0 lt3ygt lt82gt w 322011100 2 8718710 81 00 32 3y 32 32 32 3y 39 57 58 This can be a useful separation because p33 in in 57 can be thought of as the pressure perturbation that is due to the presence of an updraft in the base state environmental shear Vorticity The prognostic equation for vertical vorticity C k V x u is obtained by taking k Vx 36 or equivalently by combining 3 3228 733y27 2 81 59 Dt 3273y if 3273y 3z3y 3y327332 if 3z3y ivdy7 or 314 31 3w 3u 3w 31 g 60 3y 32 32 32 D 7 577ltf87y ivdy often neglected This equation is equivalent to Houze s 259 if we make the Boussinesq approximation and treat 1 as a constant as Houze does The vorticity equation should be familiar from dynamic and synoptic meteorology As indicated the effect of 1 s change with latitude is usually neglected as small in cloud and mesoscale studies The prognostic equation for horizontal vorticity about the yiaxis E j V x u is often of interest in quasi72D squall line studies in which 2 is usually taken to be the acrossiline dimension and y is taken to be the alongiline dimension It is obtained by taking j V x 36 or equivalently by combining 3 3227 7339529 2883 310 8163 daogp 6261 Dt 3273x if 3273z am82 323y7323y 7d2 3zfd273x7 01 lt1 lt2 lt3 lt4 lt5 A g77873 Qquot mew 3U3ugtda037p7 at 3w Dt 7 32 32 d2 32 32 32 62 often neglected eg Houze s 261 Review of cloudmesmscale governing equations M D Parker 8 This is equivalent to Houze s 258 if we make the Boussinesq approximation as Houze does Term 1 on the rhs shows that horizontal gradients in buoyancy baroclinically generate horizontal vorticity This term is often thought to predominate in cloud and mesoscale studies and as indicated the other terms on the rhs have frequently been ignored Term 2 is the twisting of planetary vorticity into the horizontal it is often neglected either because it is much smaller than the twisting of the relative vertical vorticity in term 3 or in squall line studies because the vertical shear of the alongiline wind ie 191282 is small or is set to zero Term 3 is the twisting of offiaXis vorticity into the yidimension it has often been ignored because in 2D squall line studies there are no yivariations ie 881 0 Term 4 is the solenoidal generation of horizontal vorticity owing to the vertical gradient in 040 it is often assumed to be small Term 5 is the convergence of horizontal vorticity this term could be signi cant in some cases picture a gust front but it has been neglected by many 2D squall line investigators who assume that the ow is incompressible if V 11 0 and 881 0 then auaz 81082 0

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