The fully nonlinear stratified geostrophic adjustment problem

Abstract

Abstract. The study of the adjustment to equilibrium by a stratified fluid in a rotating reference frame is a classical problem in geophysical fluid dynamics. We consider the fully nonlinear, stratified adjustment problem from a numerical point of view. We present results of smoothed dam break simulations based on experiments in the published literature, with a focus on both the wave trains that propagate away from the nascent geostrophic state and the geostrophic state itself. We demonstrate that for Rossby numbers in excess of roughly 2 the wave train cannot be interpreted in terms of linear theory. This wave train consists of a leading solitarylike packet and a trailing tail of dispersive waves. However, it is found that the leading wave packet never completely separates from the trailing tail. Somewhat surprisingly, the inertial oscillations associated with the geostrophic state exhibit evidence of nonlinearity even when the Rossby number falls below 1. We vary the width of the initial disturbance and the rotation rate so as to keep the Rossby number fixed, and find that while the qualitative response remains consistent, the Froude number varies, and these variations are manifested in the form of the emanating wave train. For wider initial disturbances we find clear evidence of a wave train that initially propagates toward the near wall, reflects, and propagates away from the geostrophic state behind the leading wave train. We compare kinetic energy inside and outside of the geostrophic state, finding that for long times a Rossby number of around one-quarter yields an equal split between the two, with lower (higher) Rossby numbers yielding more energy in the geostrophic state (wave train). Finally we compare the energetics of the geostrophic state as the Rossby number varies, finding long-lived inertial oscillations in the majority of the cases and a general agreement with the past literature that employed either hydrostatic, shallow-water equation-based theory or stratified Navier-Stokes equations with a linear stratification.

Strengthening of the Kuroshio current by intensifying tropical cyclones

“…If this balance is disturbed through processes such as the shoaling of isopycnals by strong storms, the flow will adjust itself back to a state of geostrophic balance. This process, known as geostrophic adjustment, was first considered by Rossby in 1938 (19) and has since been investigated in a variety of contexts both linear and nonlinear (20)(21)(22)(23). Common to all those studies is that the flows in question hold less energy at their end states than they do initially, and energy is dispersed in the form of inertial gravity waves (19,24).…”

Internal Bore Evolution across the Shelf near Pt. Sal, California, Interpreted as a Gravity Current

“…The evolution of ISWs are modified by rotation depending on Rossby number, amplitude, and non-dimensional dispersion parameter (e.g., Helfrich and Grimshaw 2008;Coutino and Stastna 2017). In contrast, internal bores on the shelf have large amplitude (isopycnal displacements a 10.1175/JPO-D-21-0095.1.…”

Instabilities within Rotating mode-2 Internal Waves

“…As described in Coutino and Stastna (2017), without the walls the Poincaré waves steepen and eventually breakdown into a non-linear wave packet as dispersive effects take over. The addition of side walls appears to slow the large-scale breakdown of the leading wave.…”

“…In the simpler case of a rotating fluid adjusting without the presence of side walls, it has been observed that if dispersive effects are accounted for, a leading solitary wave is created which then breaks down into a nonlinear wave packet as it propagates (Coutino and Stastna (2017)). Additionally, a geostrophic state is created at the site of the initial condition which oscillates at a near inertial frequency and radiates waves.…”

“…To account for these effects the authors used model equations based on the rotation modified Korteweg-de Vries (KdV) or Ostrovsky equation, which gave qualitatively similar results to the observations. However, these equations do not account for the full nonlinearity which accounts for the differences to Coutino and Stastna (2017) where the full Euler equations were used.…”