Mechanism of baroclinic instability based on an idealized equation in a simplest situation

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Baroclinic instability is investigated with emphasis on its mechanism. First a symmetric form of idealized evolution equation is derived for a two-layer flat ocean on an f-plane, which model is purified and simplified as much as possible without loss of the essence of baroclinic instability. Detailed explanation is provided to each term of the equations. The model captures well the features of baroclinic instability in a simple systematic picture, so that the equation may be called the canonical equation for baroclinic instability. The interpretation is based on the evolution of barotropic and baroclinic modes. In particular, the equation is reduced to the Laplace equation for two independent variables of time t and the zonal coordinate x, in the limit of long wavelengths compared with the baroclinic radius of deformation; it is related with the Cauchy-Riemann condition of complex functions. Therefore the instability mechanism becomes almost trivial and the spatial features of growing and decaying modes are interpreted by the property of complex functions (their real and imaginary parts are harmonic functions). On the other hand for short-wave disturbances, the equation is reduced to the one-dimensional wave equation. In both limits, methaphorical systems with the same equations but different physics are presented to explain why the system is unstable (growing/decaying modes) or stable (neutral waves). It also allows simple forms of analytic solutions for the growth rate and stream functions, which make it easy to understand various aspects and roles of baroclinic instability. Also simple representations are given to the meridional transport of buoyancy and the meridional circulation as for Eady's model, within the framework of a two-layer model. In addition, the role of baroclinic instability is argued in relation to Gent-McWilliams parameterization and diffusive stretching.

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