The shallow-water equations in non-spherical geometry with latitudinal variation of gravity

2014 ◽  
Vol 141 (687) ◽  
pp. 655-662 ◽  
Author(s):  
Andrew Staniforth ◽  
Andy White
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Spherical Geometry ◽  
Latitudinal Variation

A standard test set for numerical approximations to the shallow water equations in spherical geometry

1992 ◽  
Vol 101 (1) ◽  
pp. 227-228 ◽  
Author(s):  
David L Williamson ◽  
John B Drake ◽  
James J Hack ◽  
Rüdiger Jakob ◽  
Paul N Swarztrauber
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Spherical Geometry ◽  
Standard Test ◽  
Numerical Approximations ◽  
Test Set

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.

A standard test set for numerical approximations to the shallow water equations in spherical geometry

1992 ◽  
Vol 102 (1) ◽  
pp. 211-224 ◽  
Author(s):  
David L. Williamson ◽  
John B. Drake ◽  
James J. Hack ◽  
Rüdiger Jakob ◽  
Paul N. Swarztrauber
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Spherical Geometry ◽  
Standard Test ◽  
Numerical Approximations ◽  
Test Set

MAPPING TIDAL CURRENTS AND RESIDUAL CURRENTS BY USE OF COASTAL ACOUSTIC TOMOGRAPHY

2017 ◽  
Author(s):  
Xiao-Hua Zhu ◽  
Xiao-Hua Zhu ◽  
Ze-Nan Zhu ◽  
Ze-Nan Zhu ◽  
Xinyu Guo ◽  
...  
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Mean Amplitude ◽  
Momentum Equation ◽  
Tidal Currents ◽  
Acoustic Tomography ◽  
Mean Values ◽  
Residual Currents ◽  
Water Depths ◽  
Deep Area

A coastal acoustic tomography (CAT) experiment for mapping the tidal currents in the Zhitouyang Bay was successfully carried out with seven acoustic stations during July 12 to 13, 2009. The horizontal distributions of tidal current in the tomography domain are calculated by the inverse analysis in which the travel time differences for sound traveling reciprocally are used as data. Spatial mean amplitude ratios M2 : M4 : M6 are 1.00 : 0.15 : 0.11. The shallow-water equations are used to analyze the generation mechanisms of M4 and M6. In the deep area, velocity amplitudes of M4 measured by CAT agree well with those of M4 predicted by the advection terms in the shallow water equations, indicating that M4 in the deep area where water depths are larger than 60 m is predominantly generated by the advection terms. M6 measured by CAT and M6 predicted by the nonlinear quadratic bottom friction terms agree well in the area where water depths are less than 20 m, indicating that friction mechanisms are predominant for generating M6 in the shallow area. Dynamic analysis of the residual currents using the tidally averaged momentum equation shows that spatial mean values of the horizontal pressure gradient due to residual sea level and of the advection of residual currents together contribute about 75% of the spatial mean values of the advection by the tidal currents, indicating that residual currents in this bay are induced mainly by the nonlinear effects of tidal currents.

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