Rossby Waves on Non-zonal Currents: Structural Stability of Critical Layer Effects

Abstract

The problem of the propagation of linear Rossby waves in horizontally inhomogeneous non-zonal flows is studied. The explicit solution within the geometric optics (WKBJ) approximation is found to be identical to the exact Cauchy problem solution for the case of a constant horizontal velocity shear.The effect of the short-wave transformation of Rossby waves near the so-called critical layer is detailed for the arbitrary direction of non-zonal flow. In the general case, this transformation can occur in two ways: (1) as an adhering, a monotonic approaching of wave packets to the critical layer for an infinitely long time. The sign of the intrinsic frequency of the packet remains the same all the time; (2) as an adhering with overshooting when the wave packet, first, crosses its critical layer at finite wavenumber. The wave changes the sign of the intrinsic frequency when overshooting the critical layer and then keeps the sign when it is adhering to this layer asymptotically similarly to the previous scenario. The latter regime does not exist for zonal flows, that degenerates the short-wave dynamics of Rossby waves in this special case. On the contrary, the anisotropy of the dispersion relation permits both positive and negative frequencies in a non-zonal flow. It allows for the effective use of the concept of waves of negative energy for the analysis of the stability of large-scale currents.

Appendix: Wave Guide Trajectories Near the Critical Layer Within the WKBJ-Approximation

Integration of the ray equations (38) for the wave vector component l with the initial condition $l=l_0$ at $t=0$ gives a linear dependence on time

$$\begin{aligned} l=l_0-kU_y t. \end{aligned}$$

(40)

Introduce the notation

$$\begin{aligned} \kappa ^2=k^2+a^2. \end{aligned}$$

(41)

Substituting (40) into ray equations for coordinates with initial condition $x=0,\,y=0$ at $t=0$ one gets

$$\begin{aligned} y=\frac{\beta }{kU_y}\left( \frac{k\cos \alpha - l \sin \alpha }{\kappa ^2+l^2 } - \frac{k\cos \alpha - l_0 \sin \alpha }{\kappa ^2+l_0^2 }\right) . \end{aligned}$$

(42)

$$\begin{aligned} \begin{aligned} x&= \frac{\beta }{k U_y} \frac{k^2\cos \alpha }{\kappa ^{3}} \left( \arctan \left( \frac{l_0}{\kappa } \right) - \arctan \left( \frac{l}{\kappa }\right) \right) \\&\quad - \frac{\beta }{kU_y}\dfrac{\sin \alpha }{2k} \ln \left( \frac{l^2+\kappa ^2}{l_0^2+\kappa ^2}\right) \\&\quad - \frac{\beta }{U_y} \left( \dfrac{k l \cos {\alpha } +a^2 \sin \alpha }{\kappa ^2\left( l^2+\kappa ^2\right) } - \dfrac{k l_0 \cos {\alpha } +a^2 \sin \alpha }{\kappa ^2 \left( l_0^2+\kappa ^2\right) } \right) +U_y y t. \end{aligned} \end{aligned}$$

(43)

The coordinate y tends to its limit at $t=0$

$$\begin{aligned} \lim _{t\rightarrow \infty } y = -\frac{1}{kU_y}\frac{\beta \left( k\cos \alpha -l_0 \sin \alpha \right) }{\kappa ^2+l_0^2} \end{aligned}$$

(44)

that is exactly the condition of the critical layer (22) with the dispersion relation (17). The asymptotic expression for the along-current coordinate in the general case of non-zonal current is

$$\begin{aligned} \lim _{t \rightarrow \infty }x=U_y y t+ O(\ln t). \end{aligned}$$

(45)

The case of zonal current, again, shows its degeneracy

$$\begin{aligned} \lim _{t \rightarrow \infty }x =U_y y t+ O(t^{-1}). \end{aligned}$$

(46)