Quasi-geostrophic motion

Transcription

1 Capter 8 Quasi-eostropic motion Scale analysis for synoptic-scale motions Simplification of te basic equations can be obtained for synoptic scale motions. Consider te Boussinesq system ρ is assumed to be constant in as muc as it affects te fluid inertia and continuity. Introduce nondimensional variables, ( ), and typical scales (in capitals) as follows: (x, y) (x', y') z Hz' t (/U)t' (u,v) U(u', v') w Ww' p Pp'; b Σb '; and f f f', f is a typical middle latitude value of f.

2 Te orizontal component of te momentum equation takes te nondimensional form ' W Ro t U H w ' ' P ' u ' u + f' k u p' ' ' ρuf + + N M were ' denotes te operator ( / x', / y', ) and Ro is te nondimensional parameter U/(f ), te Rossby number. Definition of scales > all ( )-quantities ave manitude ~ (). Typical values of te scales for middle latitude synoptic systems are: l 6 m, H l 4 m, U ms, P 3 Pa ( mb), b δt/t *3/3 ms, ρ k m 3 and f ~ 4 s. Clearly, we can take P Uf. Ten, assumin tat (W/UH) ~ (), te key parameter is te Rossby number. ' W Ro t U H w ' ' P ' u ' u + f' k u p' ' ' ρuf + + N M For synoptic scale motions at middle latitudes, Ro ~.l so tat, to a first approximation, te D'u' /Dt' can be nelected and te equation reduces to one of eostropic balance. In dimensional form it becomes fk u p ρ We solve it by takin k ^ of bot sides. u + k p Tis equation defines te eostropic wind. ur scalin sows to be a ood approximation to te total orizontal wind u.

3 As noted earlier, it is a dianostic equation from wic te wind can be inferred at a particular time wen te pressure radient is known. In oter words, te limit of ' W Ro t U H w ' ' P ' u ' u + f' k u p' ' ' ρuf + + N M as Ro is deenerate in te sense tat time derivatives drop out. We cannot use te eostropic equation to predict te evolution of te wind field. If f is constant te eostropic wind is orizontally nondiverent; i.e., u. Te difference between te orizontal wind and te eostropic wind is called te aeostropic wind: ua u u ' W Now Ro t U H w ' ' P ' u ' u + f' k u p' ' ' ρuf + + N M for Ro <<, u ~ u wile u a is of order Ro. A suitable scale for u a is URo. Because u, te continuity equation reduces to te nondimensional form (assumin tat f is constant).

4 Te second term of is important if Ro ' u' a+ N M W H U W H U w' ' Ro. a typical scale for w is U(H/)Ro ms. te operator u ' ' in ' W Ro t U H w ' ' P ' u ' u + f' k u p' ' ' ρuf + + N M is muc larer tan w'/. To a first approximation, advection by te vertical velocity can be nelected, bot in te momentum and termodynamic equations. Two important results To a first approximation, advection by te vertical velocity can be nelected, bot in te momentum and termodynamic equations. Also te dominant contribution to u ' ' is u' ' in quasi-eostropic motion, advection is by te eostropic wind.

5 Vertical momentum equation In nondimensional form, te vertical momentum equation is W D'w' p' ΣH Ro + H U Dt' ' Uf b' It is easy to ceck tat ΣH/(Uf ) Ro(WH/U) Ro (H/) 6. Synoptic scale perturbations are in a very close state of ydrostatic balance. Te overnin equations for quasi-eostropic motion in dimensional form + u u + fk ua p b ρ w ua + + u + u b Nw + k p an approximated form of te termodynamic equation were at present f is assumed to be a constant.

6 A prediction equation for te flow at small Ro First derive te vorticity equation: Use: u u ( u ) u ( u ) u Takin k ^ ives a + ( u ) u ( u ) + fk u + u ( ζ + f) -f u a were ζ k ^ u is te vertical component of relative vorticity computed usin te eostropic wind. If f is constant, + u f. Assume N is constant Take of + u + b Nw b u w + u + b+n t of u + k p u + k p Usin p b ρ u + k b f b u w + u + b+n t w u a

7 b + u N u a + u ( ζ + f) -f u a f b + ζ + f + u N Assumes tat f is a constant (ten we can omit te sinle f in te middle bracket). If te meridional displacement of air parcels is not too lare, we can allow for meridional variations in f witin te small Rossby number approximation - see exercise 8.. Te quasi-eostropic potential vorticity equation Suppose tat f f + βy, ten + u q f b were q ζ + f + N Tis is an equation of fundamental importance in dynamical meteoroloy; it is te quasi-eostropic potential vorticity equation It states tat te quasi-eostropic potential vorticity q is conserved alon eostropically computed trajectories. It is te pronostic equation wic enables us to calculate te time evolution of te eostropic wind and pressure fields.

8 Expression of q in terms of pressure ζ k u p p b ρ z f q ζ + f + N b f q p+ f + ρ N p Solution procedure Write + u q in te form q u q u k p

9 Suppose tat we make an initial measurement of te pressure field p(x,y,z,) at time t. Calculate q(x,y,z,) usin f q p+ f + ρ Calculated u (x,y,z,) usin u p Predict te distribution of q(x,y,z, Δt) usin N k p q u q Dianose p(x,y,z,δt) by solvin te elliptic partial differential equation for p: Dianose u (x,y,z,δt) usin p f p + ρ f q f ( ) N u k p Repeat te process...

10 Boundary conditions In order to carry out te interations, appropriate boundary conditions must be prescribed. For example, for flow over level terrain, w at z. Use + u + b Nw and p b ρ z p + u at z More on te approximated termodynamic equation Wen N is a constant, te nondimensional form of tis equation is Db + w Dt B were te Burer number f B and NH R te Rossby radius of deformation NH f R An important feature of quasi-eostropic motion is te assumption tat ~ R, or equivalently tat B ~.

11 Wen B ~, Db + w Dt B Te rate-of-cane of buoyancy (and temperature) experienced by fluid parcels is associated wit vertical motion in te presence of a stable stratification. Since in quasi-eostropic teory te total derivative D/Dt is approximated by / +u, te rate-of-cane of buoyancy is computed followin te (orizontal) eostropic velocity u. Te vertical advection of buoyancy w / is neliible. Tus quasi-eostropic flows "see" only te stratification of te basic state caracterized by N (/θ)dθ /dz ---- tis is independent of time; suc flows cannot cane te effective static stability caracterized locally by N + b/. Te derivation of te potential vorticity equation for a compressible atmospere is similar to tat for a Boussinesq fluid. s c p ln θ Te equation for te conservation of entropy, or equivalently, for potential temperature θ, replaces te equation for te conservation of density: Dρ Dt ter canes are Te quasi-eostropic equation for a compressible atmospere ρ ρ b ρ* o θ θ b θ* Dθ Dt o dθ N o θ dz *

12 Te teory applies to small departures from an adiabatic atmospere in wic θ (z) is approximately constant, equal to θ*. For a deep atmosperic layer, te continuity equation must include te vertical density variation ρ (z): u a + ( ρw) ρ Te vorticity equation is D f ( ζ + f) ( ρw) Dt ρ Te potential vorticity equation is + f ρ( z) ψ ζ + f + ρ z N M ( ) N u Quasi-eostropic flow over a bell-saped mountain For steady flow ( / ) te quasi-eostropic potential vorticity equation takes te form u q. Assume tat f is a constant, u q is satisfied (e..) by solutions of te form q f. For tese solutions, p satisfies p f p + ρ f q f ( ) or N ψ ψ ψ x + f y + N ψ is te eostropic streamfunction ( p/). Tese solutions ave zero perturbation potential vorticity

13 mit te zero subscript on f, and assume tat N is a constant. ψ x + ψ f ψ y + N Put z ( N / f) z aplace s equation ψ ψ ψ + + x y Two particular solutions are: ψ Uy u ψ y U a uniform flow ψ S /4πr a source solution, strent S at z r x + y + (z + z ) * z * Streamfunction equation is linear ψ Uy S /4πr is a solution. In q-flow p b f ρ ψ because ψ p/ Te vertical displacement of a fluid parcel, η is related to σ by b η N Since b is a constant on isentropic surfaces, te displacement of te isentropic surface from z constant for te flow defined by ψ Uy S/4πr is iven (in dimensional form) by 3 / + S N η x + y + ( z+ z ) ( z z ) 4πf f

14 3 / + S N η x + y + ( z+ z ) ( z z ) 4πf f Te displacement of fluid parcels wic, in te absence of motion would occupy te plane at z is xy (, ) Sz N x + y + 4πf f m ( R / R ) + z 3 / 3 / m S/(4πNR * ) R ( x + y ) R * Nz * /f xy (, ) ( R / R ) m 3 / + is an isentropic surface of te quasi-eostropic flow defined by ψ Uy S/4πr. Wen S 4πNR * m and z f R * /N, ψ Uy S/4πr represents te flow in te semi-infinite reion z > of a uniform current U past te bell-saped mountain wit circular contours iven by (x,y). Te mountain eit is m and its caracteristic widt is R *. In terms of m etc., te displacement of an isentropic surface in tis flow is m( z/ z + ) η( xyz,, ) / ( R / R ) + ( z/ z + ) 3

15 Te vorticity canes in stratified quasieostropic flow over an isolated mountain U A B R * x Te streamline pattern in quasi-eostropic stratified flow over an isolated mountain NH case Te incident flow is distorted by te mountain anticyclone, but te perturbation velocity and pressure field decay away from te mountain (after R. B. Smit, 979).

16 η - η 5 A B R * x 5 Heit of te lowest isentrope above te toporapy in as a function of x. Unit scale equals te lent of te four vertical lines in Fi. 8.. End of Capter 8