Evolution of disturbances and singular vectors in the shallow-water semi-geostrophic model

Abstract

The first part of this work is devoted to exploring the instability mechanism of the shallow-water semi-geostrophic model by investigating the evolution of disturbances under constant-shear and cosine-type basic flows. For the constant-shear basic-flow case, the wave-packet analysis method is applied to study the evolution of potential and total energy and the meridional structure of disturbances under me assumption of a spatially slowly varying basic state. The results are found to be in good agreement with numerical calculations even when the assumption is slightly relaxed. For the cosine-type basic-flow case, both unstable normal modes and continuous spectra exist if the stability criterion is violated. Numerical results show that the relative importance of regular normal-mode and continuous-spectrum disturbances in the evolution process depends on the configuration of initial disturbances and their projections onto the normal modes and continuous spectra. In the second part of this work, in order to search for initial disturbances based on which the growth of certain norms of disturbances can be optimized, singular vectors of different norms (metrics) are computed. It is found that of all the norms, the structures of singular vectors are mainly determined by the configuration of the zonal basic flow and less affected by model parameters such as the Rossby number, rotating Froude number, and amplitude of topography. However, when the shear of the basic flow is strong, the optimal singular value can be sensitive to the model parameters. When the stability criterion is violated, the optimal singular values of all the norms have clear wave-number preferences. The peaks of spectra of singular values of the energy-type norms, including potential, kinetic and total-energy norms, correspond to very weak normal-mode instability. However, the optimal singular value of the enstrophy norm has almost the same wave-number preference as unstable normal modes.