Baroclinic instability of spatially-periodic flows in a discrete surface quasi-geostrophic model with two levels

Abstract

Within the framework of a discrete surface quasi-geostrophic (SQG) model with two levels, the hydrodynamic stability of a spatially periodic flow with sinusoidal velocity profile (Kolmogorov flow) is investigated. Such spatially quasi-periodic flows are often observed in the atmospheres of the giant planets and in the Southern Ocean. To describe the dynamics of perturbations, the Galerkin method with three basis trigonometric functions was used. It provides a tool to study both the linear and nonlinear perturbation dynamics. Analytical expressions are obtained for the growth rates of perturbations in the linear theory of stability, and it is shown that the fastest growing perturbations have a wavelength on the order of the spatial period of the main flow. Using the Galerkin method, it is also shown that the exponential growth of perturbations at the linear stage of development is replaced by the stage of stable nonlinear oscillations. Such oscillations are analogous to the nonlinear fluctuations, or vacillations, found in R. Hide's laboratory experiments. In terms of elliptic integrals, an analytical expression for the period of nonlinear oscillations is obtained, and it is shown that for typical values of the parameters relevant to the Earth’s atmosphere, the period of oscillations is about one month. The paper also proposes a long-wavelength version of the discrete SQG model, which describes the dynamics of perturbations with a horizontal scale much larger than the baroclinic Rossby radius of deformation. It is shown that the use of this version simplifies the study of stability and leads to asymptotically correct results.