The validity of two-dimensional models of a rotating shallow fluid layer

Dritschel, David G.; Jalali, Mohammad Reza

doi:10.1017/jfm.2020.487

Cited by 6 publications

(40 citation statements)

References 36 publications

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“…Pearce and Esler (2010) 69 derived the VA equations in their vorticity-divergence form, widely used in SW studies of atmosphere/ocean dynamics and convenient for a pseudo-spectral numerical treatment. They verified their numerical model for an analytic solution of a propagating uni-directional Cnoidal wave, and studied the PV evolution in an unstable jet (we have also done similar tests for the method used here 49 ). Recently, Alemi Ardakani (2021) 70 derived generalised variational SW and VA models for a shallow fluid sloshing inside a container subject to arbitrary (threedimensional) translations and rotations.…”

Section: Introductionmentioning

confidence: 90%

“…These are valid so long as horizontal scales L are large compared to the mean fluid depth H. In a rotating flow with Coriolis frequency f , there is an additional length scale L D = c/ f called the 'Rossby deformation length' where c = √ gH is the short-scale gravity wave speed and g is the acceleration due to gravity (or reduced gravity). 18 The tacit assumption is that L D H, or at least that L H even when L D ∼ H. However, commonly, rotating SW flows develop small scales, especially in PV, as a result of nonlinear flow interactions 48,49 . Horizontal scales with L < L D inevitably form, and so the validity of the rotating SW model requires H L D , i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, one can show that the non-hydrostatic SW model is simply the vertical average of the three-dimensional Euler equations. 49 All conserved quantities, including PV, are just the vertical averages of their three-dimensional counterparts. For this reason, we call this model the 'Vertically-Averaged (VA) model' below.…”

Section: Introductionmentioning

confidence: 99%

“…Most of these works have been theoretical. Prior to our own work 49,71,72 , only Pearce and Esler 69 studied the evolution of nonlinear rotating flows (and they restricted attention to the PV dynamics in a single example). Little, therefore, is known about the actual differences between the SW and VA models, or about the general properties of rotating VA flows.…”

Section: Introductionmentioning

confidence: 99%

“…Significantly, we have shown that the VA model is substantially more accurate than the SW model by comparing simulations of horizontal shear instability directly with simulations of the three-dimensional Euler equations with a free surface. 49 The greater accuracy of the VA model is well known in studies of unidirectional flows without rotation, and this is due in part to the better representation of wave dispersion, a feature entirely absent in the SW model (without rotation).…”

Section: Introductionmentioning

confidence: 99%

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Balance in non-hydrostatic rotating shallow-water flows

Jalali

Dritschel

2021

Physics of Fluids

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Unsteady nonlinear shallow-water flows typically emit inertia-gravity waves through a process called 'spontaneous adjustment-emission'. This process has been studied extensively within the rotating shallow-water model, the simplest geophysical model having the required capability. Here we consider what happens when the hydrostatic assumption underpinning the shallow-water model is dropped. This assumption is in fact not necessary for the derivation of a two-dimensional or single-layer flow model. All one needs is that the horizontal flow field be independent of height in the fluid layer. Then, vertical averaging yields a single-layer flow model, with the full range of expected conservation laws, similar to the shallow-water model yet allowing for non-hydrostatic effects. These effects become important for horizontal scales comparable to or less than the depth of the fluid layer. In a rotating flow, such scales may be activated if the Rossby deformation length (the ratio of the characteristic gravity-wave speed to the Coriolis frequency) is comparable to the the depth of the fluid layer. Then, the range of frequencies supporting inertia-gravity waves is compressed, and the group velocity of these waves is reduced. We find that this change in wave properties has the effect of strongly suppressing spontaneous adjustment-emission and trapping inertia-gravity waves near regions of relatively strong circulation.

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Balance in non-hydrostatic rotating shallow-water flows

Jalali

Dritschel

2021

Physics of Fluids

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The magnetic non-hydrostatic shallow-water model

Dritschel,

Tobias

2023

J. Fluid Mech.

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We consider the dynamics of a set of reduced equations describing the evolution of a magnetised, rotating stably stratified fluid layer, atop a stagnant dense, perfectly conducting layer. We consider two closely related models. In the first, the layer has, above it, relatively light fluid where the magnetic pressure is much larger than the gas pressure, and the magnetic field is largely force-free. In the second model, the magnetic field is constrained to lie within the dynamical layer by the implementation of a model diffusion operator for the magnetic field. The model derivation proceeds by assuming that the horizontal velocity and the horizontal magnetic field are independent of the vertical coordinate, whilst the vertical components in the layer have a linear dependence on height. The full system comprises evolution equations for the magnetic field, horizontal velocity and height field together with a linear elliptic equation for the vertically integrated non-hydrostatic pressure. In the magneto-hydrostatic limit, these equations simplify to equations of shallow-water type. Numerical solutions for both models are provided for the fiducial case of a Gaussian vortex interacting with a magnetic field. The solutions are shown to differ negligibly. We investigate how the interaction of the vortex changes in response to the magnetic Reynolds number ${Rm}$ , the Rossby deformation radius $L_D$ , and a Coriolis buoyancy frequency ratio $f/N$ measuring the significance of non-hydrostatic effects. The magneto-hydrostatic limit corresponds to $f/N\to 0$ .

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A Lagrangian theory of geostrophic adjustment for zonally invariant flows on a rotating spherical earth

Paldor

Dritschel

2021

Physics of Fluids

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We examine the late-time evolution of an inviscid zonally symmetric shallow-water flow on the surface of a rotating spherical earth. An arbitrary initial condition radiates inertia-gravity waves that disperse across the spherical surface. The simpler problem of a uniformly rotating (f-plane) shallow-water flow on the plane radiates these waves to infinity, leaving behind a nontrivial steady flow in geostrophic balance (in which the Coriolis acceleration balances the horizontal hydrostatic pressure gradient). This is called "geostrophic adjustment." On a sphere, the waves cannot propagate to infinity, and the flow can never become steady due to energy conservation (at least in the absence of shocks). Nonetheless, when energy is conserved a form of adjustment still takes place, in a time-averaged sense, and this flow satisfies an extended form of geostrophic balance dependent only on the conserved mass and angular momentum distributions of fluid particles, just as in the planar case. This study employs a conservative numerical scheme based on a Lagrangian form of the rotating shallow-water equations to substantiate the applicability of these general considerations on an idealized aqua-planet for an initial "dam" along the equator in a motionless ocean.

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The validity of two-dimensional models of a rotating shallow fluid layer

Abstract: Abstract

Cited by 6 publications

References 36 publications

Balance in non-hydrostatic rotating shallow-water flows

Balance in non-hydrostatic rotating shallow-water flows

The magnetic non-hydrostatic shallow-water model

A Lagrangian theory of geostrophic adjustment for zonally invariant flows on a rotating spherical earth

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