Chapter 5 Nonlinear Wave Phenomena in Rotating Shallow Water with Applications to Geostrophic Adjustment

Author(s):  
V. Zeitlin
2001 ◽  
Vol 445 ◽  
pp. 93-120 ◽  
Author(s):  
G. M. REZNIK ◽  
V. ZEITLIN ◽  
M. BEN JELLOUL

We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallow-water dynamics. The only assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-time-scale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f−10 and (f0Ro)−1 respectively, where f0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction.The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ΔH/H0, where ΔH and H0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (ΔH/H0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasi-geostrophic potential vorticity equation for times t [les ] (f0Ro)−1. We find modifications to this equation for longer times t [les ] (f0Ro2)−1. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance.For large relative elevations (ΔH/H0 [Gt ] Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrödinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.


2011 ◽  
Vol 68 (6) ◽  
pp. 1234-1252 ◽  
Author(s):  
Julien Lambaerts ◽  
Guillaume Lapeyre ◽  
Vladimir Zeitlin

Abstract Dynamical influence of moist convection upon development of the barotropic instability is studied in the rotating shallow-water model. First, an exhaustive linear “dry” stability analysis of the Bickley jet is performed, and the most unstable mode identified in this way is used to initialize simulations to compare the development and the saturation of the instability in dry and moist configurations. High-resolution numerical simulations with a well-balanced finite-volume scheme reveal substantial qualitative and quantitative differences in the evolution of dry and moist-convective instabilities. The moist effects affect both balanced and unbalanced components of the flow. The most important differences between dry and moist evolution are 1) the enhanced efficiency of the moist-convective instability, which manifests itself by the increase of the growth rate at the onset of precipitation, and by a stronger deviation of the end state from the initial one, measured with a number of different norms; 2) a pronounced cyclone–anticyclone asymmetry during the nonlinear evolution of the moist-convective instability, which leads to an additional, with respect to the dry case, geostrophic adjustment, and the modification of the end state; and 3) an enhanced ageostrophic activity in the precipitation zones but also in the nonprecipitating areas because of the secondary geostrophic adjustment.


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