scholarly journals Super- and sub-rotating equatorial jets in shallow water models of Jovian atmospheres: Newtonian cooling versus Rayleigh friction

2017 ◽  
Vol 822 ◽  
pp. 484-511 ◽  
Author(s):  
Emma S. Warneford ◽  
Paul J. Dellar
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Rossby Waves ◽  
Giant Planets ◽  
Mean Flow ◽  
Standard Result ◽  
Zonal Jets ◽  
Rayleigh Friction ◽  
Shallow Water Models ◽  
Zonal Momentum

Numerical simulations of the shallow water equations on rotating spheres produce mixtures of robust vortices and alternating zonal jets, as seen in the atmospheres of the gas giant planets. However, simulations that include Rayleigh friction invariably produce a sub-rotating (retrograde) equatorial jet for Jovian parameter regimes, whilst observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted over several decades. Super-rotating equatorial jets have recently been obtained in shallow water simulations that include a Newtonian relaxation of perturbations to the layer thickness to model radiative cooling to space, and in simulations of the thermal shallow water equations that include a similar relaxation term in their temperature equation. Simulations of global quasigeostrophic forms of these different models produce equatorial jets in the same directions as the parent models, suggesting that the mechanism responsible for setting the direction lies within quasigeostrophic theory. We provide such a mechanism by calculating the effective force acting on the thickness-weighted zonal mean flow due to the decay of an equatorially trapped Rossby wave. Decay due to Newtonian cooling creates an eastward zonal mean flow at the equator, consistent with the formation of a super-rotating equatorial jet, while decay due to Rayleigh friction leads to a westward zonal mean flow at the equator, consistent with the formation of a sub-rotating equatorial jet. In both cases the meridionally integrated zonal mean of the absolute zonal momentum is westward, consistent with the standard result that Rossby waves carry westward pseudomomentum, but this does not preclude the zonal mean flow being eastward on and close to the equator.

scholarly journals Modelling Weirs in Two-Dimensional Shallow Water Models

Water ◽  
2021 ◽  
Vol 13 (16) ◽  
pp. 2152
Author(s):  
Gonzalo García-Alén ◽  
Olalla García-Fonte ◽  
Luis Cea ◽  
Luís Pena ◽  
Jerónimo Puertas
Keyword(s):  
Experimental Data ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Model ◽  
Two Dimensional ◽  
Shallow Water Model ◽  
Experimental Dataset ◽  
River Hydraulics ◽  
Shallow Water Models ◽  
Water Models

2D models based on the shallow water equations are widely used in river hydraulics. However, these models can present deficiencies in those cases in which their intrinsic hypotheses are not fulfilled. One of these cases is in the presence of weirs. In this work we present an experimental dataset including 194 experiments in nine different weirs. The experimental data are compared to the numerical results obtained with a 2D shallow water model in order to quantify the discrepancies that exist due to the non-fulfillment of the hydrostatic pressure hypotheses. The experimental dataset presented can be used for the validation of other modelling approaches.

Waves, jets and vortices: Regime transitions in shallow water turbulence

2021 ◽  
Author(s):  
Nikos Bakas
Keyword(s):  
Shallow Water ◽  
Large Scale ◽  
Rossby Waves ◽  
Phase Speed ◽  
Critical Value ◽  
Turbulent Regime ◽  
Wave Regime ◽  
Frequency Spectra ◽  
Spectra Analysis ◽  
Zonal Jets

<p>Forced-dissipative beta-plane turbulence in a single-layer shallow-water fluid has been widely considered as a simplified model of planetary turbulence as it exhibits turbulence self-organization into large-scale structures such as robust zonal jets and strong vortices. In this study we perform a series of numerical simulations to analyze the characteristics of the emerging structures as a function of the planetary vorticity gradient and the deformation radius. We report four regimes that appear as the energy input rate ε of the random stirring that supports turbulence in the flow increases. A homogeneous turbulent regime for low values of ε, a regime in which large scale Rossby waves form abruptly when ε passes a critical value, a regime in which robust zonal jets coexist with weaker Rossby waves when ε passes a second critical value and a regime of strong materially coherent propagating vortices for large values of ε. The wave regime which is not predicted by standard cascade theories of turbulence anisotropization and the vortex regime are studied thoroughly. Wavenumber-frequency spectra analysis shows that the Rossby waves in the second regime remain phase coherent over long times. The coherent vortices are identified using the Lagrangian Averaged Deviation (LAVD) method. The statistics of the vortices (lifetime, radius, strength and speed) are reported as a function of the large scale parameters. We find that the strong vortices propagate zonally with a phase speed that is equal or larger than the long Rossby wave speed and advect the background turbulence leading to a non-dispersive line in the wavenumber-frequency spectra.</p>

A Balanced Approximation of the One-Layer Shallow-Water Equations on a Sphere

2009 ◽  
Vol 66 (6) ◽  
pp. 1735-1748 ◽  
Author(s):  
W. T. M. Verkley
Keyword(s):  
Shallow Water ◽  
Potential Vorticity ◽  
Shallow Water Equations ◽  
Rossby Waves ◽  
Coriolis Parameter ◽  
Beta Plane ◽  
Propagating Waves ◽  
Barotropic Vorticity Equation ◽  
Barotropic Vorticity ◽  
The One

Abstract A global version of the equivalent barotropic vorticity equation is derived for the one-layer shallow-water equations on a sphere. The equation has the same form as the corresponding beta plane version, but with one important difference: the stretching (Cressman) term in the expression of the potential vorticity retains its full dependence on f 2, where f is the Coriolis parameter. As a check of the resulting system, the dynamics of linear Rossby waves are considered. It is shown that these waves are rather accurate approximations of the westward-propagating waves of the second class of the original shallow-water equations. It is also concluded that for Rossby waves with short meridional wavelengths the factor f 2 in the stretching term can be replaced by the constant value f02, where f0 is the Coriolis parameter at ±45° latitude.

scholarly journals Numerical Study of Spurious Inertial Modes in Shallow Water Models for a Variable Bathymetry

2019 ◽  
Vol 11 (6) ◽  
pp. 58
Author(s):  
Gossouhon SITIONON ◽  
Adama COULIBALY ◽  
Jérome Kablan ADOU
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Shallow Water Equations ◽  
Numerical Study ◽  
Ocean Modelling ◽  
Galerkin Approximations ◽  
Shallow Water Models ◽  
Water Models ◽  
Spurious Modes

In this study we perform a modal analysis of the linear inviscid shallow water equations using a non constant bathymetry, continuous and discontinuous Galerkin approximations. By extracting the discrete eigenvalues of the resulting algebraic linear system written on the form of a generalized eigenvalue / eigenvector problem we first show that the regular variation of the bathymetry does not prevent the presence of spurious inertial modes when centered finite element pairs are used. Secondly, we show that such spurious modes are not present in discontinuous Galerkin discretizations when all variables are approximated in the same descrete space. Such spurious inertial modes have been found very damageable for the quality of inertia-gravity and Rossby modes in ocean modelling.

Numerical Simulations of Forced Shallow-Water Turbulence: Effects of Moist Convection on the Large-Scale Circulation of Jupiter and Saturn

2007 ◽  
Vol 64 (9) ◽  
pp. 3132-3157 ◽  
Author(s):  
Adam P. Showman
Keyword(s):  
Numerical Simulations ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Large Scale ◽  
Spherical Geometry ◽  
Small Deformation ◽  
Moist Convection ◽  
Zonal Jets ◽  
Strong Control ◽  
Water Turbulence

Abstract To test the hypothesis that the zonal jets on Jupiter and Saturn result from energy injected by thunderstorms into the cloud layer, forced-dissipative numerical simulations of the shallow-water equations in spherical geometry are presented. The forcing consists of sporadic, isolated circular mass pulses intended to represent thunderstorms; the damping, representing radiation, removes mass evenly from the layer. These results show that the deformation radius provides strong control over the behavior. At deformation radii <2000 km (0.03 Jupiter radii), the simulations produce broad jets near the equator, but regions poleward of 15°–30° latitude instead become dominated by vortices. However, simulations at deformation radii >4000 km (0.06 Jupiter radii) become dominated by barotropically stable zonal jets with only weak vortices. The lack of midlatitude jets at a small deformation radii results from the suppression of the beta effect by column stretching; this effect has been previously documented in the quasigeostrophic system but never before in the full shallow-water system. In agreement with decaying shallow-water turbulence simulations, but in disagreement with Jupiter and Saturn, the equatorial flows in these forced simulations are always westward. In analogy with purely two-dimensional turbulence, the size of the coherent structures (jets and vortices) depends on the relative strengths of forcing and damping; stronger damping removes energy faster as it cascades upscale, leading to smaller vortices and more closely spaced jets in the equilibrated state. Forcing and damping parameters relevant to Jupiter produce flows with speeds up to 50–200 m s−1 and a predominance of anticyclones over cyclones, both in agreement with observations. However, the dominance of vortices over jets at deformation radii thought to be relevant to Jupiter (1000–3000 km) suggests that either the actual deformation radius is larger than previously believed or that three-dimensional effects, not included in the shallow-water equations, alter the dynamics in a fundamental manner.

scholarly journals Modeling convection and zonal winds in giant planets

2006 ◽  
Vol 2 (S239) ◽  
pp. 177-187
Author(s):  
Martha Evonuk ◽  
Gary A. Glatzmaier
Keyword(s):  
Shallow Water ◽  
Equatorial Region ◽  
Giant Planets ◽  
Density Stratification ◽  
Fluid Turbulence ◽  
Zonal Winds ◽  
Shallow Water Models ◽  
Water Models ◽  
Boussinesq Models ◽  
Deep Shell

AbstractThree basic modeling approaches have been used to numerically simulate fluid turbulence and the banded zonal winds in the interiors and atmospheres of giant planets: shallow-water models, deep-shell Boussinesq models and deep-shell anelastic models. We review these models and discuss the approximations and assumptions upon which they are based. All three can produce banded zonal wind patterns at the surface. However, shallow-water models produce a retrograde (i.e., westward) zonal jet in the equatorial region, whereas strong prograde (i.e., eastward) equatorial jets exist on Jupiter and Saturn. Deep-shell Boussinesq models maintain prograde equatorial jets by the classic method of vortex stretching of convective columnar flows; however, they neglect the effects of the large density stratification in these giant planets. Deep-shell anelastic models account for density stratification and maintain prograde equatorial jets by generating vorticity as rising fluid expands and sinking fluid contracts, without the constraint of long thin convective columns.

scholarly journals Forced-Dissipative Shallow-Water Turbulence on the Sphere and the Atmospheric Circulation of the Giant Planets

2007 ◽  
Vol 64 (9) ◽  
pp. 3158-3176 ◽  
Author(s):  
R. K. Scott ◽  
L. M. Polvani
Keyword(s):  
Shallow Water ◽  
Dissipative System ◽  
Water System ◽  
Equilibrium States ◽  
Giant Planets ◽  
Zonal Jets ◽  
Random Forcing ◽  
Wide Range ◽  
Shallow Water System ◽  
Decaying Turbulence

Abstract Although possibly the simplest model for the atmospheres of the giant planets, the turbulent forced-dissipative shallow-water system in spherical geometry has not, to date, been investigated; the present study aims to fill this gap. Unlike the freely decaying shallow-water system described by Cho and Polvani, equilibrium states in the forced-dissipative system are highly dependent on details of the forcing and the dissipation. For instance, it is found that for a given equilibrated energy level, the steadiness of zonal jets depends crucially on the balance between forcing and dissipation. With long (up to 100 000 days) high-resolution (T170) calculations, the dependence of the equilibrium states on Rossby number Ro and Rossby deformation radius LD is explored, for the case when the dissipation takes the form of hypodiffusion (acting predominantly at large scales) and the random forcing at small scales is δ correlated in time. When LD is large compared to the planetary radius, zonal jets are verified to scale closely with the Rhines scale over a wide range of Ro; furthermore, the jets at the equator are found to be both prograde and retrograde with approximately equal likelihood. As LD is decreased, the equatorial jets become increasingly and consistently retrograde, in agreement with the freely decaying turbulence results. Also, the regime recently discussed by Theiss, where zonal jets are confined to low latitudes, is illustrated to emerge robustly in the limit of small LD. Finally, specific calculations with parameter values typical of the giant planets are presented, confirming many of the earlier results obtained in the freely decaying case.

Rotating Shallow-Water Models with Full Coriolis Force

2018 ◽  
Author(s):  
Vladimir Zeitlin
Keyword(s):  
Shallow Water ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Vertical Component ◽  
Tangent Plane ◽  
Water Model ◽  
Shallow Water Models ◽  
Water Models ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations

The derivation of the rotating shallow-water model by vertical averaging is carried on in the tangent plane approximation without neglecting the vertical component of the Coriolis force, and contributions of the vertical component of velocity in its horizontal component (‘non-traditional’ terms), leading to one- and two-layer ‘non-traditional’ rotating shallow-water models. A similar approach on the whole sphere encounters difficulties with conservation of angular momentum. Consistent ‘non-traditional’ rotating shallow-water equations in this case are obtained from the variational principle, which is first formulated for full primitive equations. It is shown that columnar motion hypothesis should be replaced by solid-angle motion one on the sphere. Two-layer non-traditional rotating shallow-water equations are used to analyse inertial instability of jets and compare the results with Chapter 10. It is shown that non-traditional terms can increase the growth rates up to 30% in some configurations and can also change the structure of the unstable modes.

Sign in / Sign up

Export Citation Format

Share Document