scholarly journals Fast and slow resonant triads in the two-layer rotating shallow water equations

2018 ◽  
Vol 850 ◽  
pp. 18-45
Author(s):  
Alex Owen ◽  
Roger Grimshaw ◽  
Beth Wingate
Keyword(s):  
Shallow Water ◽  
Water System ◽  
Layer System ◽  
Interaction Coefficients ◽  
Shallow Water System ◽  
Rotating Shallow Water ◽  
Resonant Triads ◽  
The One ◽  
Baroclinic Modes ◽  
Fast Waves

In this paper, we examine triad resonances in a rotating shallow water system when there are two free interfaces. This allows for an examination in a relatively simple model of the interplay between baroclinic and barotropic dynamics in a context where there is also a geostrophic mode. In contrast to the much-studied one-layer rotating shallow water system, we find that as well as the usual slow geostrophic mode, there are now two fast waves, a barotropic mode and a baroclinic mode. This feature permits triad resonances to occur between three fast waves, with a mixture of barotropic and baroclinic modes, an aspect that cannot occur in the one-layer system. There are now also two branches of the slow geostrophic mode, with a repeated branch of the dispersion relation. The consequences are explored in a derivation of the full set of triad interaction equations, using a multiscale asymptotic expansion based on a small-amplitude parameter. The derived nonlinear interaction coefficients are confirmed using energy and enstrophy conservation. These triad interaction equations are explored, with an emphasis on the parameter regime with small Rossby and Froude numbers.

scholarly journals A New Well-Balanced Reconstruction Technique for the Numerical Simulation of Shallow Water Flows with Wet/Dry Fronts and Complex Topography

Water ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 1661 ◽  
Author(s):  
Zhengtao Zhu ◽  
Zhonghua Yang ◽  
Fengpeng Bai ◽  
Ruidong An
Keyword(s):  
Shallow Water ◽  
Water System ◽  
Reconstruction Technique ◽  
Test Cases ◽  
Bed Topography ◽  
Surface Gradient ◽  
Shallow Water Flows ◽  
Shallow Water System ◽  
The One ◽  
Flow Variables

This study develops a new well-balanced scheme for the one-dimensional shallow water system over irregular bed topographies with wet/dry fronts, in a Godunov-type finite volume framework. A new reconstruction technique that includes flooded cells and partially flooded cells and preserves the non-negative values of water depth is proposed. For the wet cell, a modified revised surface gradient method is presented assuming that the bed topography is irregular in the cell. For the case that the cell is partially flooded, this paper proposes a special reconstruction of flow variables that assumes that the bottom function is linear in the cell. The Harten–Lax–van Leer approximate Riemann solver is applied to evaluate the flux at cell faces. The numerical results show good agreement with analytical solutions to a set of test cases and experimental results.

Ageostrophic instability in rotating shallow water

2012 ◽  
Vol 712 ◽  
pp. 327-353 ◽  
Author(s):  
Peng Wang ◽  
James C. McWilliams ◽  
Ziv Kizner
Keyword(s):  
Shallow Water ◽  
Gravity Wave ◽  
Shear Wave ◽  
Water System ◽  
Wave Mode ◽  
Shear Instability ◽  
Mean Flow ◽  
Shallow Water System ◽  
Rotating Shallow Water ◽  
Wave Modes

AbstractLinear instabilities, both momentum-balanced and unbalanced, in several different $ \overline{u} (y)$ shear profiles are investigated in the rotating shallow water equations. The unbalanced instabilities are strongly ageostrophic and involve inertia–gravity wave motions, occurring only for finite Rossby ($\mathit{Ro}$) and Froude ($\mathit{Fr}$) numbers. They serve as a possible route for the breakdown of balance in a rotating shallow water system, which leads the energy to cascade towards small scales. Unlike previous work, this paper focuses on general shear flows with non-uniform potential vorticity, and without side or equatorial boundaries or vanishing layer depth (frontal outcropping). As well as classical shear instability among balanced shear wave modes (i.e. B–B type), two types of ageostrophic instability (B–G and G–G) are found. The B–G instability has attributes of both a balanced shear wave mode and an inertia–gravity wave mode. The G–G instability occurs as a sharp resonance between two inertia–gravity wave modes. The criterion for the occurrence of the ageostrophic instability is associated with the second stability condition of Ripa (1983), which requires a sufficiently large local Froude number. When $\mathit{Ro}$ and especially $\mathit{Fr}$ increase, the balanced instability is suppressed, while the ageostrophic instabilities are enhanced. The profile of the mean flow also affects the strength of the balanced and ageostrophic instabilities.

scholarly journals Could waves mix the ocean?

2009 ◽  
Vol 638 ◽  
pp. 1-4 ◽  
Author(s):  
G. FALKOVICH
Keyword(s):  
Shallow Water ◽  
Internal Waves ◽  
Particle Velocity ◽  
Small Amplitude ◽  
Water System ◽  
Effective Diffusivity ◽  
Finite Amplitude ◽  
Random Set ◽  
Shallow Water System ◽  
Rotating Shallow Water

A finite-amplitude propagating wave induces a drift in fluids. Understanding how drifts produced by many waves disperse pollutants has broad implications for geophysics and engineering. Previously, the effective diffusivity was calculated for a random set of small-amplitude surface and internal waves. Now, this is extended by Bühler & Holmes-Cerfon (J. Fluid Mech., 2009, this issue, vol. 638, pp. 5–26) to waves in a rotating shallow-water system in which the Coriolis force is accounted for, a necessary step towards oceanographic applications. It is shown that interactions of finite-amplitude waves affect particle velocity in subtle ways. An expression describing the particle diffusivity as a function of scale is derived, showing that the diffusivity can be substantially reduced by rotation.

WELL-BALANCED NUMERICAL SCHEMES BASED ON A GENERALIZED HYDROSTATIC RECONSTRUCTION TECHNIQUE

2007 ◽  
Vol 17 (12) ◽  
pp. 2055-2113 ◽  
Author(s):  
MANUEL J. CASTRO ◽  
ALBERTO PARDO MILANÉS ◽  
CARLOS PARÉS
Keyword(s):  
Shallow Water ◽  
Numerical Scheme ◽  
Source Term ◽  
Hyperbolic Systems ◽  
Water System ◽  
The Other ◽  
Reconstruction Technique ◽  
Numerical Schemes ◽  
Shallow Water System ◽  
The One

The goal of this paper is to generalize the hydrostatic reconstruction technique introduced in Ref. 2 for the shallow water system to more general hyperbolic systems with source term. The key idea is to interpret the numerical scheme obtained with this technique as a path-conservative method, as defined in Ref. 35. This generalization allows us, on the one hand, to construct well-balanced numerical schemes for new problems, as the two-layer shallow water system. On the other hand, we construct numerical schemes for the shallow water system with better well-balanced properties. In particular we obtain a Roe method which solves exactly every stationary solution, and not only those corresponding to water at rest.

Resonant fast–slow interactions and breakdown of quasi-geostrophy in rotating shallow water

2016 ◽  
Vol 788 ◽  
pp. 492-520 ◽  
Author(s):  
Jim Thomas
Keyword(s):  
Evolution Equation ◽  
Shallow Water ◽  
Water System ◽  
Wave Interaction ◽  
Multiple Time ◽  
Slow Dynamics ◽  
Balanced Flow ◽  
Rotating Shallow Water ◽  
Fast Waves ◽  
Wave Modes

In this paper we investigate the possibility of fast waves affecting the evolution of slow balanced dynamics in the regime $Ro\sim Fr\ll 1$ of a rotating shallow water system, where $Ro$ and $Fr$ are the Rossby and Froude numbers respectively. The problem is set up as an initial value problem with unbalanced initial data. The method of multiple time scale asymptotic analysis is used to derive an evolution equation for the slow dynamics that holds for $t\lesssim 1/(fRo^{2})$, $f$ being the inertial frequency. This slow evolution equation is affected by the fast waves and thus does not form a closed system. Furthermore, it is shown that energy and enstrophy exchange can take place between the slow and fast dynamics. As a consequence, the quasi-geostrophic ideology of describing the slow dynamics of the balanced flow without any information on the fast modes breaks down. Further analysis is carried out in a doubly periodic domain for a few geostrophic and wave modes. A simple set of slowly evolving amplitude equations is then derived using resonant wave interaction theory to demonstrate that significant wave-balanced flow interactions can take place in the long-time limit. In this reduced system consisting of two geostrophic modes and two wave modes, the presence of waves considerably affects the interactions between the geostrophic modes, the waves acting as a catalyst in promoting energetic interactions among geostrophic modes.

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