Fourier analysis of a class of upwind schemes in shallow water systems for gravity and Rossby waves

2008 ◽  
Vol 57 (4) ◽  
pp. 389-416 ◽  
Author(s):  
A. M. Mohammadian ◽  
D. Y. Le Roux
Keyword(s):  
Fourier Analysis ◽  
Shallow Water ◽  
Rossby Waves ◽  
Water Systems ◽  
Upwind Schemes

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>

Modeling ship-induced waves in shallow water systems: The Venice experiment

2018 ◽  
Vol 155 ◽  
pp. 227-239 ◽  
Author(s):  
D. Bellafiore ◽  
L. Zaggia ◽  
R. Broglia ◽  
C. Ferrarin ◽  
F. Barbariol ◽  
...  
Keyword(s):  
Shallow Water ◽  
Water Systems

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.

Nonlinear Fourier Analysis for Shallow Water Waves

2021 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Shallow Water ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Wave Equations ◽  
Superposition Principle ◽  
Broad Band ◽  
Kp Equation ◽  
Linear Superposition

Abstract I consider nonlinear wave motion in shallow water as governed by the KP equation plus perturbations. I have previously shown that broad band, multiply periodic solutions of the KP equation are governed by quasiperiodic Fourier series [Osborne, OMAE 2020]. In the present paper I give a new procedure for extending this analysis to the KP equation plus shallow water Hamiltonian perturbations. We therefore have the remarkable result that a complex class of nonlinear shallow water wave equations has solutions governed by quasiperiodic Fourier series that are a linear superposition of sine waves. Such a formulation is important because it was previously thought that solving nonlinear wave equations by a linear superposition principle was impossible. The construction of these linear superpositions in shallow water in an engineering context is the goal of this paper. Furthermore, I address the nonlinear Fourier analysis of experimental data described by shallow water physics. The wave fields dealt with here are fully two-dimensional and essentially consist of the linear superposition of generalized cnoidal waves, which nonlinearly interact with one another. This includes the class of soliton solutions and their associated Mach stems, both of which are important for engineering applications. The newly discovered phenomenon of “fossil breathers” is also characterized in the formulation. I also discuss the exact construction of Morison equation forces on cylindrical piles in terms of quasiperiodic Fourier series.

scholarly journals Propagation properties of Rossby waves for latitudinal β-plane variations of <I>f</I> and zonal variations of the shallow water speed

2012 ◽  
Vol 30 (5) ◽  
pp. 849-855 ◽  
Author(s):  
C. T. Duba ◽  
J. F. McKenzie
Keyword(s):  
Shallow Water ◽  
Group Velocity ◽  
Rossby Wave ◽  
Rossby Waves ◽  
Low Frequency ◽  
Wave Number ◽  
Coriolis Parameter ◽  
Propagation Properties ◽  
Wave Normal ◽  
Water Speed

Abstract. Using the shallow water equations for a rotating layer of fluid, the wave and dispersion equations for Rossby waves are developed for the cases of both the standard β-plane approximation for the latitudinal variation of the Coriolis parameter f and a zonal variation of the shallow water speed. It is well known that the wave normal diagram for the standard (mid-latitude) Rossby wave on a β-plane is a circle in wave number (ky,kx) space, whose centre is displaced −β/2 ω units along the negative kx axis, and whose radius is less than this displacement, which means that phase propagation is entirely westward. This form of anisotropy (arising from the latitudinal y variation of f), combined with the highly dispersive nature of the wave, gives rise to a group velocity diagram which permits eastward as well as westward propagation. It is shown that the group velocity diagram is an ellipse, whose centre is displaced westward, and whose major and minor axes give the maximum westward, eastward and northward (southward) group speeds as functions of the frequency and a parameter m which measures the ratio of the low frequency-long wavelength Rossby wave speed to the shallow water speed. We believe these properties of group velocity diagram have not been elucidated in this way before. We present a similar derivation of the wave normal diagram and its associated group velocity curve for the case of a zonal (x) variation of the shallow water speed, which may arise when the depth of an ocean varies zonally from a continental shelf.

Simulation of shallow-water systems using graphics processing units

2009 ◽  
Vol 80 (3) ◽  
pp. 598-618 ◽  
Author(s):  
Miguel Lastra ◽  
José M. Mantas ◽  
Carlos Ureña ◽  
Manuel J. Castro ◽  
José A. García-Rodríguez
Keyword(s):  
Shallow Water ◽  
Graphics Processing Units ◽  
Water Systems ◽  
Graphics Processing
Sign in / Sign up

Export Citation Format

Share Document