Dynamics of the Atmosphere
Dynamics of the Atmosphere ATOC 5060
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This 10 page Class Notes was uploaded by Jon Johns on Thursday October 29, 2015. The Class Notes belongs to ATOC 5060 at University of Colorado at Boulder taught by David Noone in Fall. Since its upload, it has received 18 views. For similar materials see /class/232050/atoc-5060-university-of-colorado-at-boulder in Marine Science at University of Colorado at Boulder.
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Date Created: 10/29/15
Boundary layers Equations for the boundary layer Bu 7814 au 7817i l3i73 at 3xv3 Wag pdx 31W 817 8 8 8 187113 at udxv8ywdz pay azW Simplify for balanced ow making use of geostrophic definition Am BB ZW fi ig av w States that turbulent transfer is responsible for a geostrophic ow FluxGradient Theory and the Mixing Length Hypothesis 39 A very simple form of turbulence closure can be obtained by assuming that a vertically displaced parcel of uid carries its mean horizontal velocity to its new level where it completely mixes with its new environment and causes a turbulent velocity uctuation 16 W 2 Cd My Another way a single parameter representing the typical scale of an eddy is required This parameter is known as the mixing length Mixing length hypothesis A vertically displaced parcel of uid originally at level z carries its mean horizontal velocity z to another level zl The mean velocity at zl is z 1 Assume that the parcel completely mixes with its environment at the new level causing a turbulent uctuation LL 1 7 7 7 Z u71uzl uz ilaiu l l 32 Z l rnixinglength 32 Thus quantities in eddy stress tenns such as pu w can be written as 7 pm a Z If we further assume that turbulence is about the same in all directions isotropic then a dy viscosi y 7 coefficien 7 87 If we de nekgad2 82 g Then we can write the eddy stress tenns as 71714 me 1 and 7pv w me a Z Z 7 7 37 37 w 1 vlawdaai Thuspuwplza7u Z L4 Ekman layer We can move toward a solution to these equations by using the definition of the geostrophic wind to substitute for the pressure gradient term Wag Bi 462 p Bx m Bzz g pf Bx u riiir rial p 3y Bz2 g pf 3y After some simple manipulation we obtain the classical Ekman layer equations named for Swedish oceanographer V W Ekman 2 K 3 144 fv V 0 These equations can be solved to m Bzz g determine the departure of the wind az from geostrophic balance as a Km V f u u 0 function of height in the boundary 312 g layer Flow in the Ekman layer Ekman layer equations are second order differential equations To solve establish some boundary conditions 140 v0 at z0 M Hlg v Wg as z cgto With these boundary conditions a can be obtained Pugh 2 sz where 7f2Km12 e72 v uge smyz This solution which is valid in the Northern Hemisphere yields the Ekman spiral The Ekman spiral describes the turning of the winds with height in the boundary layer as the effects of friction diminish with height Ekman spiral The Ekman spiral depicts the turning of the wind with height PZ7 low 1 dimens39onless height M M p If the wind was 02 7 p2 purely geostrophic P 4 2W3 3 W 039 o 2 o 4 06 o B l 0 P4 P5lthighgt The wind approaches the geostrophic wind at the top of the PBL Within the PBL the wind blow across the isobars toward lower pressure Cross isobaric ow produces boundary layer convergence in cyclones and boundary layer divergence in anticyclones Boundary layer closure Mixing length hypothesis Theoretical basis for selection K K smaller with stronger stability larger with stronger wind shear In a uniform vertical gradient a parcel lifted a height I differs from the background by V 7 7 7 a 7 a n n2 n1 n li n i dz dz 3V Assuming spherical edd1es wlai z BV 81 31 3V 7lziii 7 27 W Bz Bz Bz K 1 az Includes information of eddy generation Requires selection of I 30 meters Surface layer Small enough that we can neglect Coriolis and pressure gradients 1030 m Dependant only turbulent character shear independent of height uf K aa Define a friction velocity Z Eddy size is limited distance to surface 2 2 57 So I kz dz k is the von Karman constant 0304 M Z L7 ln k Z0 Integrating from u0 at zz0 zo is the roughness length depends on surface Wind speed increases logarithmically with height Note m C dure so we can estimate Cd by 3 measuring the wind profile i we Ekman layer 32 Using Ktheory fV Vg K 3Z2 fu ug 2 Kg Ageostrophic acceleration balanced by turbulent momentum ux Turbulence slows ow reducing Coriolis closer to the surface allowing greater cross isobaric ow Vertical integration shows net ageostrophic ow High As such convergence to cyclones K pressure which must be matched with accent Di ers samewhatfmm Ekman layer in Ocean Low pressure y Ekman layer in laboratory ocean Rotating tank Coriolis with applied surface stress Side View Top view L lWind driven W CllI I thS P man eptb I mixedlryer l H V V 7 7 1 Deep water does not feel surface stress Seen in ocean Ekman drift of sea ice Ekman s observation and upwelling where Ekman transport causes divergence Ekman layer in laboratory atmosphere Rotating tank Coriolis with applied surface stress quotSidexiew Top View l surface Requires stable boundary layer 7 rarely seen easily in atmosphere However effects of turning are evident Today s sea surface temperature Boundary layer mass uX Decompose ow into pressure driven and turbulent stress driven components uzupu vvpv Vfiaj 32 K7 p 3x dz t azz v 1 37 3 32 77 K7 7 p 3y m 3z2 At steady state assume geostrophic balance and ageostrophic mass ow due to turbulence Integrating over boundary layer depth to obtain cross isobaric ageostrophic mass ux Mx I Oo rdZ My 2 I 00 7rdZ Boundary layer divergence 39 From continuity horizontal mass ux divergence is associated with vertical motions 1 1 a BM wd 0 i wiy 0 00 ex By Thus the Ekman pumping wd Alternatively for balanced ow can write 1 3 3217 3 32V Wd EK azz j axEK azz Ekman pumping and vorticity Making use of the Ekman solution uugl e zcosz 722K 772 v uge s1nz After some algebra write Ekman pumping in terms of geostrophic vorticity 1 1 Bug avg 3 Wd 27la7 l lzfl Turbulent drag on rotating uid causes vertical motion Barotropic vorticity equation L537 31 ELWNL dt 7 fEafayjifaHWW Since the thermal wind vanishes for barotropic conditions integrate as before E fK dt 2H2 l 2 41 Differential has exponential solution gt g0expit1 1 5 Where the characteristic e folding time is T H i typically about 4 days fK So in the absence of vorticity sources weather features will almost completely dissipate in about 10 days 3 e folds What does this tell us about weather prediction with barotropic vorticity equation Example cup of tea 39 A cup of tea is stirred and spoon is removed ie put into cyclostrophic motion The bottom friction slows down the spinning uid in the cup near the bottom surface This breaks the balance between the PGF and centrifugal force The stronger PGF causes the secondary circulation ow towards the center of the cup near the bottom pushing tea leaves to the center of the cup Compensating ow above this layer away from the central axis over the remaining depth of the cup To conserve angular momentum the spin has to slow down This is analogous to the spin down of a cyclone Predictions based on tea leaves would best use vorticity equation Circulation view of spin down In the region of a cyclone convergence in PBL causes vertical motion and divergence aloft in the free troposphere Consider a ring of parcels that divergence ie the area eXpands Circulation must be conserved tangential velocity must continuall decrease y fl C 1dp 2o dim sin 2 T 2 That is out ow allows a Coriolis force which acts against the primary rotation Spin down experiment Rotating tank Geometry not quite the same as atmosphere Secondary circulation radial direction Q ConverM Divergence Ekman layer Primary circulation 011 bottom about rotation axis lO
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