E 3 LECRE 3A 3FEB8 Objective: o analyze baroclinic instability of normal-mode Rossby waves in a vertically sheared zonal flow. Reading: CH 7, pp 3-3 Problems: 7. and 7.3 on p. 6 --------------------------------------- We will use a two-layer quasigeostrophic model where the upper layer extends -5 mb and the lower layer -5 mb. We know the geostrophic wind and vorticity at levels and 3 We know the pressure-coordinate vertical velocity at level he pressure coordinate vertical velocity is zero at levels (top of the atmosphere) and (surface). Define the QG streamfunctionψ φ / f ˆ ˆ vψ k ψ ( k/ f) φ. se this definition to write the QG vorticity and thermodynamic energy equations: ψ ω ψ + vψ ( ψ) + β f t x p ψ ψ σ + vψ ω t p p f Apply the vorticity equation at levels and 3, and apply the thermodynamic energy equation at level. se finite differences to represent the vertical derivatives of ω. ω ω( p+ δp) ω( p) p δ p ω ω ω ω ω p δp δp δp ω ω ω ω ω p δp δp δp 3 With this substitution, the vorticity equations at levels and 3 become: ψ fω ψ + v ( ψ) + β t x δ p ψ3 fω ψ + 3 v 3 ( ψ3) + β t x δ p
E 3 LECRE 3A 3FEB8 Applying the same reasoning to the thermodynamic equation, we write ψ / p ( ψ3 ψ)/ δpand ψ ( ψ + ψ ). 3 ψ 3 ψ 3 v ψ ψ + σ ω t δp δp f Or σδ p ( ψ ψ3) + v ( ψ ψ3) ω t f Redefine the streamfunctions and pressure-coordinate vertical velocities: ψ y+ ψ (,) xt ψ y+ ψ (,) xt 3 3 3 ω ω (,) xt u ψ / y, u ψ / y ψ ψ ( ) y+ ψ ψ 3 3 3 hen, just when you started to get used to the prime streamfunctions, we define mean and thermalwind mean flows: ( + ), ( ) 3 3 And vertical mean and thermal-wind perturbation stream functions. ψ ( ψ + ψ ), ψ ( ψ ψ ) Note that 3 3 +,, ψ ψ + ψ, ψ ψ ψ 3 he vorticity equations become: fω + ( + ) ( ) ( ) ψ + ψ + β ψ + ψ x x δ p fω + ( ) ( ψ ) ( ) ψ β ψ ψ + x x δ p Which transform to:
E 3 LECRE 3A 3FEB8 ψ ψ fω + ( ) ( ) ψ + ψ + + + β ψ + ψ x x x x x x δ p ψ ψ fω + ( ψ ) ( ) ψ β ψ ψ + + x x x x x x δ p Adding the vorticity equations to get the vertical-mean vorticity equation: ψ ψ ψ + + + x x x x β Subtracting the vorticity equations to get the thermal-wind vorticity equation: ψ ψ ψ fω + β + + x x x x δ p In this notation [e.g., ψ ψ3 ψ, ( 3) y y ], the thermodynamic equation becomes: ψ ψ ψ ( y) σδ p + + ω t x x y f ψ ψ ψ σδ σ δ f ω p ( p) + ω t x x f f δ p Or f ψ ψ f ψ f ω σδ ( p) + t x σδ ( p) x δp Subtracting the manipulated thermodynamic-energy equation from the thermal-wind vorticity equation and writing f / σδ ( p) : ψ ψ ψ ψ + ψ β + + + x x x x What about? R θ g R θ gh θ σ P θ p P g θ p P θ p f f f f σδ ( p) gh θ gh δp θ gh δp δθ ( δp) δp P θ p P θ p P θ So is the inverse Rossby radius squared for this system. As the pressure interval or the stability increase the Rossby radius also increases (inverse decreases) because the shallow-water gravity wave speed increases. he thermal-wind vorticity equation thus becomes a potential vorticity equation. 3
E 3 LECRE 3A 3FEB8 We assume wavelike solutions, where in this case we use the phase speed ( ω/k) instead of the frequency: ψ Aexp{ ik( x ct)}, ψ Bexp{ ik( x ct)} Substituting into the vertical-mean vorticity equation: ik c k A ik k B ik A ( )( ) + ( ) + β [( c ) k + β ] A kb Substituting into the thermal-wind potential vorticity equation: ik c k B ik k A ik B ( )( ) + ( + ) + β [( c )( k + ) + β] B ( k ) A Rewriting these equations in matrix form: ( c ) k + β k A ( ) ( )( ) B k c k + + β For nontrivial solutions, the determinant must be zero: [( c ) k + β][( c )( k + + β)] ( k )[ ( k )] ultiplying: c k k + + β c k + c k + + β k k, ( ) ( ) [( ) ( )( )] ( ) which simplifies to: ( c ) k ( k + ) + β( c )( k + )] + β k ( k ) Dividing through by k (k + ) and rearranging a bit, β( k + ) β k ( c ) + ( c ) + k ( k + ) k ( k + ) k + Solving the quadratic equation for c, β( k + ) β ( k + ) β k c ± k ( k + ) k ( k + ) k ( k + ) k + Simplifying:
E 3 LECRE 3A 3FEB8 β( k + ) β ( k + ) β k ( k + ) k ( k )( k + ) c ± k ( k + ) k ( k + ) k ( k + ) k ( k + ) And still more: β + ± β + + β + + c k ( k + ) ( k ) ( k k ) ( k k ) k ( k ) o get βk + ± β k k c k ( k + ) ( ) ( ) his looks like a mess, but let s try to get some insight by looking at two special cases: and β. First, If : c β β( k + ) ± β k + k ( k + ) β k (minus root) (plus root) hese are divergent and nondivergent neutral Rossby waves. No surprises here, but must be nonzero for instability. Second, If β. ± k ( k) k c ± i k ( k + ) + k When k < (i.e., when the waves are longer than the Rossby radius divided by ), they are unstable. hey also move exactly with the mean wind with no wind-relative propagation, For these unstable waves on an f plane, the complex frequency (of which the imaginary part is the e-folding time) is: ω k ± ik k + k Shorter, stable, waves can propagate relative to the mean flow as a sort of inertia-gravity wave. For the general case, with both β and nonzero, instability happens when the radical is <, or when: β k( k) β ( k k) < 8 Divide through by 8 and reverse order of the terms to get: 5
E 3 LECRE 3A 3FEB8 8 k k β + < 8 Solving for (k/) : k ( β ) ± 6 β / ± / Or k ( β ) k ± / As before, if β is small or is large (k / ) is zero (minus root) or one (plus root), so these values define the limiting for instability in the wavenumber direction. he value where the two roots join is β/, or equivalently β/. his value defines the smallest thermal wind that can cause instability for given Coriolis Parameter and Rossby Radius. he wavenumber of the first wave to become unstable is (k / ) ½, or k / ½. Key points to remember: Waves shorter than k / are always stable. his is the famous Short-Wave cutoff. When < β/, waves are stable aximum instability is at k / ½.77, sort of Longer waves are generally stable unless is really large because the β-effect stabilizes them. 6
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