Explosive Growth of Unsymmetric Perturbations in a Flow with a Vertical Shear

Abstract

The classical problem of geophysical hydrodynamics is the problem of instability of a zonal geostrophic flow with a vertical velocity shear. At present, the instability with respect to symmetric perturbations that do not depend on the coordinate along the flow has been most thoroughly studied. For a symmetric instability to arise, the two-dimensional perturbation wave vector must lie inside a certain sector in the vertical plane of the wave numbers. In this paper, we study the instability with respect to unsymmetric perturbations oriented at an angle to the flow. Fundamentally new features of the temporal dynamics of the amplitudes of such perturbations are found. The main feature is associated with the existence of a stage of exponential explosive growth of finite duration. A kinematic interpretation of this stage is given that is related to the passage of the projection of the three-dimensional wave vector onto the plane transversely to the flow through the sector of symmetric instability.

APPENDIX.

1.1 CONSERVATION OF POTENTIAL VORTICITY FOR A LINEARIZED SYSTEM OF EQUATIONS

The fully nonlinear Boussinesq system of equations implies the potential vorticity conservation law (Pedlosky, 1987): $\partial q{\text{/}}\partial t + {\mathbf{u}}\nabla q = 0$, where

$$q = (\operatorname{curl} {\mathbf{u}} + f{\mathbf{k}},\nabla \sigma )$$

(A.1)

(the round brackets denote the scalar product). Potential vorticity of a horizontal flow with a vertical shear ${\mathbf{U}} = U(z){\mathbf{i}} = \Lambda z{\mathbf{i}}$ and buoyancy distribution $\bar {\sigma } = \Sigma (y,z) = {{N}^{2}}z - f\Lambda y$

$$\bar {q} = (\operatorname{curl} {\mathbf{U}} + f{\mathbf{k}},\nabla \bar {\sigma }) = f({{N}^{2}} - {{\Lambda }^{2}}) = {\text{const}}{\text{.}}$$

(A.2)

It follows that if $q = \bar {q} + q{\kern 1pt} '$, then, within the linearized system of equations, perturbation $q{\kern 1pt} '$ satisfies the conservation law ${{Dq{\kern 1pt} '} \mathord{\left/ {\vphantom {{Dq{\kern 1pt} '} {dt}}} \right. \kern-0em} {dt}} = 0$ or $q_{t}^{'} + \Lambda zq_{x}^{'} = 0$. Setting ${\mathbf{u}} = {\mathbf{U}} + {\mathbf{u}}{\kern 1pt} '$, $\sigma = \bar {\sigma } + \sigma {\kern 1pt} '$, for $q{\kern 1pt} '$ from (A.1), we obtain the expression

$$q{\kern 1pt} ' = (\operatorname{curl} {\mathbf{U}} + f{\mathbf{k}},\nabla q{\kern 1pt} ') + (\operatorname{curl} {\text{t}}{\mathbf{u}}{\kern 1pt} ',\nabla \bar {\sigma }),$$

(A.3)

or, in more detail (we omit the primes at the variables),

$$q = {{N}^{2}}({{{\text{v}}}_{x}} - {{u}_{y}}) - f\Lambda ({{u}_{z}} - {{w}_{x}}) + \Lambda {{\sigma }_{y}} + f{{\sigma }_{z}}.$$

(A.4)

This law is the conservation law (3). Note that this law can be also obtained directly from the linearized system (2); however, this derivation is a rather cumbersome procedure. It is interesting to obtain the conservation law directly from first principles.