Vertical Structure and the WKB Approximation

Vertical structure and the WKB approximation Content: In the presence of a basic state shear flow, (U(y,\xi),) our QG equations for the waves become [(U-c)(\nabla^{2}\psi+\frac{1}{\overline{\rho}}\frac{\partial}{\partial\xi}\frac{\overline{\rho}f^{2}}{N^{2}}\frac{\partial}{\partial\xi}\psi)+\frac{\partial\overline{q}}{\partial y}\psi=0] (the Rayleigh equation as extended) with [\frac{\partial\overline{q}}{\partial y}=\beta-\frac{\partial^{2}U}{\partial y^{3}}-\frac{1}{\overline{p}}\frac{\partial}{\partial\xi}\frac{\overline{<0>rho}f^{2}}{N^{2}}\frac{\partial U}{\partial\xi}] The boundary equations are tricky, however. If U is not zero at the bottom, then we will have pressure gradients along the surface so the the surface coordinate, (\overline{\xi}{s}(y)) will be a function of y. The condition that the bottom be at geopotential 0 becomes [\overline{\phi}(y,\overline{\xi}{s})=0] Taking a y derivative and using the geostrophic relation gives [-fU_{a}+\overline{b}{a}\frac{\partial\overline{\xi}{s}}{\partial y}=0] where (U_{s}) and (b_{s}) are the surface values of velocity and buoyancy. The linearized boundary conditions become [\overline{b}{s}\xi{s}^{\prime}+\phi^{\prime}(x,y,\overline{\xi}{s},t)=0] [(\frac{\partial}{\partial t}+U{s}\frac{\partial}{\partial x})\xi_{s}^{\prime}+v^{\prime}\frac{\partial\overline{\xi}_{s}}{\partial y}=\omega^{\prime}] Multiplying the second equation by (-\overline{b}{s}) and substituting from the first (and the equation for the basic gradient of the surface coord.) gives [(\frac{\partial}{\partial t}+U{s}\frac{\partial}{\partial x})\phi^{\prime}-v^{\prime}fU_{s}=-\overline{b}_{s}\omega^{\prime}] We combine this with the buoyancy equation [(\frac{\partial}{\partial t}+U\frac{\partial}{\partial x})\frac{\partial\phi^{\prime}}{\partial\xi}-v^{\prime}f\frac{\partial U}{\partial\xi}=-N^{2}\omega^{\prime}] To get ((\frac{\partial}{\partial t}+U_{s}\frac{\partial}{\partial x})(\frac{\partial}{\partial\xi}-\frac{N^{2}}{\overline{b}{s}})\phi^{\prime}-v^{\prime}f(\frac{\partial}{\partial\xi}-\frac{N^{2}}{\overline{b}{s}})U=0) at [\xi=\overline{\xi},] Simple modes Content: For the purpose of examining simple modes, we shall assume the surface velocity (U_s) vanishes. Then we can take (\bar{\xi}_s = 0). We also take the vertical shear to be zero at the ground so that the surface density ((\bar{\rho})) is constant and (\bar{b}_s = g). Our boundary condition is now [\frac{\partial \theta}{\partial \xi} = \frac{N^2}{g}\theta] at (\xi = 0)   We shall take the isothermal basic state so that [\frac{1}{\bar{\rho}}\frac{\partial}{\partial \xi}\frac{\bar{\rho}f^2}{N^2}\frac{\partial}{\partial \xi} \longrightarrow \frac{f^2}{N^2} \left(\frac{\partial^2}{\partial \xi^2} - \frac{1}{H_} \frac{\partial}{\partial \xi}\right)] and look for solutions with (\psi = e^{\xi/2H_} \Psi) which implies [\frac{\partial^2 \Psi}{\partial \xi^2} = \left(\frac{1}{4H_^2} - \frac{1}{H_} \frac{\partial}{\partial \xi}\right) \Psi]   Our interior equation simplifies to [\frac{\partial^2 \Psi}{\partial \xi^2} = -\Gamma(\xi) \Psi] (1) under the assumption that c is less than the minimum of U (or greater than the maximum) with [\Gamma(\xi) = \frac{N^2}{f^2} \frac{\frac{\partial \bar{q}}{\partial y}}{U - c} - \frac{N^2}{f^2} k^2 - \frac{1}{4H_*^2}]   The lower boundary condition is [\frac{\partial \Psi}{\partial \xi} = -\kappa \Psi] (2) with [\kappa = \frac{1}{2H_} - \frac{N^2}{g} = \frac{1}{H_} \left(\frac{1}{2} - \frac{R}{c_p} \right)] No flow: In the absence of flow, we have (\Psi = exp(-\kappa \xi)) and [c = -\frac{\beta}{k^2 + \frac{f^2}{N^2 H_^2} \frac{R}{c_p} \left(1 - \frac{R}{c_p} \right)}] or [c = -\frac{\beta}{k^2 + \frac{f^2}{g H_e}}] with [H_e = H_ \frac{c_p}{c_v}] In the presence of flow can we have more than one mode? Given a U(ξ) profile, we can integrate (1) starting from the initial conditions Ψ(0) = 1, Ψ' = -κ and vary c until the solution decays at ∞. With [U = \frac{s}{2}\xi^{2}] we have [\Gamma(\xi) = \frac{\frac{N^{2}}{f^{2}}\beta-\frac{s}{H_s} + \frac{s\xi}{H_s}}{\frac{1}{2}s\xi^{2}-c} - \frac{N^{2}}{f^{2}}k^{2} - \frac{1}{4H_*^{2}}] For s < N²β/f², we can take c < 0 so that the first term is positive but becomes small in the upper atmosphere, while the second and third terms are negative. Thus we have sinusoidal solutions in the lower atmosphere where 1 Γ(ξ) is positive and exponentially decays solutions at height where Γ becomes negative. As long as the sinusoidal range is big enough, we should be able to find several modes.   WKB We can look first for trapped mode solutions to (1) with Γ > 0 everywhere using a WKB form. We let [\Psi = A(\xi)exp(-i\theta(\xi))] giving [A'' + A\theta'^{2} - 2A'\theta' - A\theta'' = |\Gamma|A] Taking the presumed largest terms to balance gives [\theta' = \sqrt{|\Gamma|}] [\theta = \int_{0}^{\xi} \sqrt{|\Gamma|} d\xi^{'}] The next order terms give us the structure of A, but that isn't really necessary here. The lower boundary condition gives, to the first approximation, [-i\theta'(0)A = -\kappa A \implies |\Gamma(0)|^{1/2} = \kappa \text{ or } \Gamma(0) = -\kappa^{2}] so that [c = -\frac{\beta - \frac{f^{2}}{N^{2}}s}{k^{2} + \frac{f^{2}}{4N^{2}H_*^{2}} - \frac{f^{2}}{N^{2}}\kappa^{2}}] To solve (1) approximately using WKB for the other modes, we must deal with the turning point at ξ = ξ₀ where Γ(ξ₀) = 0 (although we don't know where exactly that is until we determine c). Below ξ₀, we can take [\Psi = A(\xi) cos~\theta(\xi)]