GENERALIZED STABILITY OF THE TWO-LAYER MODEL

Transcription

1 GENERALIZED STABILITY OF THE TWO-LAYER MODEL The simplest mid-latitude jet model supporting the baroclinic growth mechanism is the two-layer model The equations for the barotropic and baroclinic geostrophic streamfunction perturbations in this model assuming meridional wavenumber zero are: t + U ψ m m x x + β ψ m x + U ψ T T 0 x x t + U ψ T m x x λ ψ T + β ψ T x + U ψ m T + λ ψ x x m 0 Assuming the general temporal dependence: ψ m x, t ˆψ m te ikx ψ T x, t ˆψ T te ikx the resulting dynamical system can be written in standard form: d ˆψm dt ˆψ T βi U k Mik U T ik λ k λ +k U T ik βik U λ +k M ik ˆψm ˆψ T The essential baroclinic dynamics is retained with the simplifying assumption U m β 0 The additional assumptions U T and λ 0 correspond to choosing space scale /λ 0 and time scale /λ 0 U T /λ 0 is taken to have the typical value of 000 km and a typical value for U T is 5 m/s so the time scale is 077 day In order to see the effect of static stability on baroclinic growth the nondimensional Rossby radius /λ is retained as a variable The system under these assumptions is: d ˆψm dt ˆψ T 0 ik ik λ k λ +k 0 ˆψm ˆψ T

2 Writing the above system as dψ Aψ first we solve the eigenproblems for A and A+A A is dt the Hermitian transpose of A: A 0 ik ik λ k λ +k 0 A + A 0 ikλ λ +k ikλ λ +k 0 Recall that the maximum real part of the eigenvalue of A is the growth rate of the perturbation of modal form For a general perturbation to a system dx Ax the instantaneous perturbation dt growth rate can be found by differentiating x x x: d ln x dt xa + A x xx The rhs of the above expression is maximized by the eigenfunction of A + A associated with its necessarily real maximum eigenvalue This eigenvalue is called the numerical range The eigenvalues of A are: λ k σ A ±k λ + k, and the maximum real part of the eigenvalues is shown as a function of k and λ in Fig The eigenvalues of A+A are: σ A+A ±k λ k λ + k

3 5 5 λ k Figure : The modal growth rate as a function of wavenumber k and inverse Rossby radius λ The maximum of these necessarily real eigenvalues is the numerical range and it is shown as a function of k and λ in Fig The eigenvectors of A are: ˆψm ˆψ T ±i λ k λ+k The upper and lower eigenvector streamfunctions are: ˆψ ψm ˆ + ˆ ψ T ˆψ ψˆ M ψ ˆ T Normalizing so that ˆψ : k i +k ˆψ ˆψ ±i k +k

4 5 5 λ k Figure : The maximum instantaneous growth rate the numerical range as a function of wavenumber k and inverse Rossby radius λ From which we see that the phase shift between the upper and lower layer eigenfunction streamfunctions vanishes for k > which is the stable region In the unstable region k < the modes have phase shift θ arctan k +k k +k with negative phase for the growing mode This phase shift between the upper and lower mode streamfunction is shown as a function of k in Fig Turning to the numerical range the eigenfunctions of A + A are: ˆψ ˆψ i ±i The maximum instantaneous growth occurs for upper layer phase π/ relative to the lower

5 θ k Figure : Phase shift between the upper and lower layers for the growing mode as a function of k for λ 5

6 numerical range λ mode growth rate k Figure : Growth rate of the mode compared to the maximum instantaneous growth rate for λ layer; and maximum decay for phase difference π/ The numerical range is greater or equal to the maximum mode growth rate as can be seen in Fig in which these quantities are shown as a function of k for γ The standard solution form Ψτ R τ Ψ0 maps the initial state at t0 to the final state at t τ in terms of the propagator R τ e Aτ written as a matrix exponential expm Aτ in Matlab we can evaluate the e Aτ by noticing that e Aτ Ee Λτ E in which E is the matrix with columns the eigenvectors of A and Λ is the diagonal matrix of the eigenvalues: e Aτ i γ / i γ / e kγ / τ 0 0 e kγ/ τ +iγ/ / iγ / / 6

7 This results in the propagator: e Aτ coshkγ / τ i γ / sinhkγ/ τ iγ / sinhkγ / τ coshkγ / τ Taking our example values k, λ so that γ / the propagator becomes: e Aτ cos τ i sin τ i sin τ cos τ We can find the optimal growth as a function of τ using only eigenanalysis by noticing that R τ is the square root of the first eigenvalue of e A τ e Aτ and the associated eigenvector is the optimal perturbation use a Rayleigh quotient argument to verify this e A τ e Aτ cos τ + sin τ i cos τ sin τ i cos τ sin τ cos τ + sin τ The leading eigenvalue of this matrix is: R τ + sin τ + + sin τ The optimal growth as a function of τ for k and γ is shown in Fig 5 The first maximum occurs at τ max π with growth R τ This maximum is called the global optimal because there is no damping in this problem the upper and lower layers come 7

8 R τ τ Figure 5: Growth of the optimal perturbation for k and λ in and out of phase and this interference results in multiple maxima; there is typically only one global maximum when damping is allowed for The eigenvector at τ max π is: ˆψ ˆψ This corresponds to phase shift of π between the layers with the upper level lagging by twice as much phase as in the case of the numerical range This norm of R τ can be found in general using normr τ in Matlab The norm is the first singular value of the propagator and this growth is attained by the initial perturbation in the form of the right singular vector of the propagator these are found in general using the singular value decomposition command in Matlab: [U,S,V] svdr τ 8