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.

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 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.

Particle dispersion by random waves in rotating shallow water

2009 ◽  
Vol 638 ◽  
pp. 5-26 ◽  
Author(s):  
OLIVER BÜHLER ◽  
MIRANDA HOLMES-CERFON
Keyword(s):  
Shallow Water ◽  
Numerical Study ◽  
Gaussian Random Field ◽  
Water System ◽  
Effective Diffusivity ◽  
Particle Dispersion ◽  
Second Order ◽  
Random Waves ◽  
Effective Particle ◽  
Rotating Shallow Water

We present a theoretical and numerical study of wave-induced particle dispersion due to random waves in the rotating shallow-water system, as part of an ongoing study of particle dispersion in the ocean. Specifically, the effective particle diffusivities in the sense of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, p. 196) are computed for a small-amplitude wave field modelled as a stationary homogeneous isotropic Gaussian random field whose frequency spectrum is bounded away from zero. In this case, the leading-order diffusivity depends crucially on the nonlinear, second-order corrections to the linear velocity field, which can be computed using the methods of wave–mean interaction theory. A closed-form analytic expression for the effective diffusivity is derived and carefully tested against numerical Monte Carlo simulations. The main conclusions are that Coriolis forces in shallow water invariably decrease the effective particle diffusivity and that there is a peculiar choking effect for the second-order particle flow in the limit of strong rotation.

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