Frontal Wave Stability during Moist Deformation Frontogenesis. Part II: The Suppression of Nonlinear Wave Development

Abstract

In this paper, the role of horizontal deformation and the associated frontogenetic ageostrophic circulation in suppressing the development of nonlinear waves is assessed. Unless linear barotropic frontal waves can become nonlinear, the associated horizontal transports of momentum will not be sufficient to halt frontogenesis or to create nonlinear mixing processes such as vortex roll-up. The analysis of Dritschel et al. suggests that such nonlinear phenomena will not occur if the wave slope remains small. For the linear model described in Part I, a simple relationship between optimal wave slope amplification over a specified time period and the amplification of an initially isolated edge wave is found. Using this relationship, the mechanisms by which strain affects the dependence of optimal wave slope amplification on wavelength and the time of entry of disturbances to the front are investigated. It is found that waves entering the frontal zone when it is intense can experience greater steepening than those appearing earlier in the development of the front. The most rapidly growing waves enter the front with a wavelength about three times the width of the front. As the front collapses, the ratio of wavelength to frontal width rapidly increases. For strain rates greater than 0.6 × 10⁻⁵ s⁻¹, the model predicts that wave slope amplification greater than a factor of e is impossible. The variation of optimal growth with wavenumber and the time of entry of disturbances to the front is explained using diagnostics based on a mathematical model of Bretherton's qualitative description of wave growth in terms of the interaction of counterpropagating edge waves. These diagnostics yield a simple formula for the frontogenesis rate required to completely eliminate wave steepening. For the front considered in Part I, the formula predicts that no amplification is possible for strain rates greater than one-quarter of the Coriolis parameter. Diagnostics of this sort may aid attempts to predict, from the large-scale forcing, the minimum attainable cross-frontal scale of a front.