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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 24, issue 1
Nonlin. Processes Geophys., 24, 61-75, 2017
https://doi.org/10.5194/npg-24-61-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.
Nonlin. Processes Geophys., 24, 61-75, 2017
https://doi.org/10.5194/npg-24-61-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.

Research article 30 Jan 2017

Research article | 30 Jan 2017

The fully nonlinear stratified geostrophic adjustment problem

Aaron Coutino and Marek Stastna Aaron Coutino and Marek Stastna
  • Department of Applied Mathematics, University of Waterloo, Waterloo, Canada

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 solitary-like 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.

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We have re-examined the classical geostrophic adjustment problem, where a disturbance of a density stratification is released from rest in a rotating frame of reference, from a numerical point of view. This has enabled us to consider the governing equations without approximations. We show that both the waves generated and the remaining state exhibit nonlinear effects. Due to advances in available computational power, we can now revisit classical problems and solve them completely.
We have re-examined the classical geostrophic adjustment problem, where a disturbance of a...
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