Stability of a coastal upwelling front: 1. Model development and a stability theorem

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Abstract
A two‐layer shallow water equation model is used to investigate the linear stability of a coastal upwelling front. The model features a surface front parallel to a coastal boundary and bottom topography which is an arbitrary function of the cross‐shelf coordinate. Global conservation statements for energy, momentum and potential vorticity are examined to help elucidate the instability mechanism. By combining these conservation statements, a general stability theorem is established which allows the a priori determination of the stability of a coastal upwelling front. The necessary conditions for instability derived for these ageostrophic flows differ from traditional quasi‐geostrophic criteria. The stability theorem suggests that a coastal upwelling front may be unstable no matter what the basic state flow configuration is. Confirmation of the existence of these unstable waves, a description of their characteristics and a comparison of their properties to observations are presented in a companion paper.