scholarly journals Propagation of a finite-amplitude potential vorticity front along the wall of a stratified fluid

2002 ◽  
Vol 468 ◽  
pp. 179-204 ◽  
Author(s):  
MELVIN E. STERN ◽  
KARL R. HELFRICH
Keyword(s):  
Laboratory Experiment ◽  
Shallow Water ◽  
Potential Vorticity ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Similarity Theory ◽  
Wave Structure ◽  
Small Scale ◽  
Reduced Gravity ◽  
Coriolis Parameter

A similarity solution to the long-wave shallow-water equations is obtained for a density current (reduced gravity = g′, Coriolis parameter = f) propagating alongshore (y = 0). The potential vorticity q = f/H1 is uniform in −∞ < x [les ] xnose(t), 0 < y [les ] L(x, t), and the nose of this advancing potential vorticity front displaces fluid of greater q = f/H0, which is located at L < y < ∞. If L0 = L(−∞, t), the nose point with L(xnose(t), t) = 0 moves with velocity Unose = √g′H0 φ, where φ is a function of H1/H0, f2L20/g′H0. The assumptions made in the similarity theory are verified by an initial value solution of the complete reduced-gravity shallow-water equations. The latter also reveal the new effect of a Kelvin shock wave colliding with a potential vorticity front, as is confirmed by a laboratory experiment. Also confirmed is the expansion wave structure of the intrusion, but the observed values of Unose are only in qualitative agreement; the difference is attributed to the presence of small-scale (non-hydrostatic) turbulence in the laboratory experiment but not in the numerical solutions.

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.

On the Spectral Convergence of the Supercompact Finite-Difference Schemes for the f-Plane Shallow-Water Equations

2009 ◽  
Vol 137 (7) ◽  
pp. 2393-2406 ◽  
Author(s):  
S. Ghader ◽  
A. R. Mohebalhojeh ◽  
V. Esfahanian
Keyword(s):  
Finite Difference ◽  
Shallow Water ◽  
Potential Vorticity ◽  
Shallow Water Equations ◽  
Small Scale ◽  
Finite Difference Schemes ◽  
Vorticity Field ◽  
Eighth Order ◽  
Spectral Convergence ◽  
Nonlinear Flows

For the f-plane shallow-water equations, the convergence properties of the supercompact finite-difference method (SCFDM) are examined during the evolution of complex, nonlinear flows spawned by an unstable jet. The second-, fourth-, sixth-, and eighth-order SCFDMs are compared with a standard pseudospectral (PS) method. To control the buildup of small-scale activity and thus the potential for numerical instability, the vorticity field is damped explicitly by the application of a triharmonic hyperdiffusion operator acting on the vorticity field. The global distribution of mass between isolevels of potential vorticity, called mass error, and the representation of the balance and imbalance are used to assess numerical accuracy. In each of the quantitative measures, a clear convergence of the SCFDM to the PS method is observed. There is no saturation in accuracy up to the eighth order examined. Taking the PS solution as the reference, for the fundamental quantity of potential vorticity the rate of convergence to PS turns out to be algebraic and near-quadratic.

Models of non-Boussinesq lock-exchange flow

2011 ◽  
Vol 675 ◽  
pp. 1-26 ◽  
Author(s):  
R. ROTUNNO ◽  
J. B. KLEMP ◽  
G. H. BRYAN ◽  
D. J. MURAKI
Keyword(s):  
Free Boundary ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Water System ◽  
Analytic Solutions ◽  
Analytical Models ◽  
Navier Stokes ◽  
Exchange Flow

Nearly all analytical models of lock-exchange flow are based on the shallow-water approximation. Since the latter approximation fails at the leading edges of the mutually intruding fluids of lock-exchange flow, solutions to the shallow-water equations can be obtained only through the specification of front conditions. In the present paper, analytic solutions to the shallow-water equations for non-Boussinesq lock-exchange flow are given for front conditions deriving from free-boundary arguments. Analytic solutions are also derived for other proposed front conditions – conditions which appear to the shallow-water system as forced boundary conditions. Both solutions to the shallow-water equations are compared with the numerical solutions of the Navier–Stokes equations and a mixture of successes and failures is recorded. The apparent success of some aspects of the forced solutions of the shallow-water equations, together with the fact that in a real fluid the density interface is a free boundary, shows the need for an improved theory of lock-exchange flow taking into account non-hydrostatic effects for density interfaces intersecting rigid boundaries.

scholarly journals Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 345
Author(s):  
Sudi Mungkasi ◽  
Stephen Gwyn Roberts
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Adaptive Mesh ◽  
Finite Volume Methods ◽  
Water Type ◽  
Two Dimensional ◽  
Water Flows ◽  
Shallow Water Flows ◽  
Smoothness Indicators

This paper proposes some formulations of weak local residuals of shallow-water-type equations, namely, one-, one-and-a-half-, and two-dimensional shallow water equations. Smooth parts of numerical solutions have small absolute values of weak local residuals. Rougher parts of numerical solutions have larger absolute values of weak local residuals. This behaviour enables the weak local residuals to detect parts of numerical solutions which are smooth and rough (non-smooth). Weak local residuals that we formulate are implemented successfully as refinement or coarsening indicators for adaptive mesh finite volume methods used to solve shallow water equations.

Introduction to Geophysical Fluid Dynamics

1998 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Fluid Dynamics ◽  
Shallow Water ◽  
Potential Vorticity ◽  
Shallow Water Equations ◽  
Coordinate Frame ◽  
Geophysical Fluid Dynamics ◽  
Homogeneous Fluid ◽  
The One ◽  
Quasigeostrophic Equations ◽  
Inertia Gravity Waves

This second chapter offers a brief introduction to geophysical fluid dynamics—the dynamics of rotating, stratified flows. We start with the shallow water equations, which govern columnar motion in a thin layer of homogeneous fluid. Roughly speaking, the solutions of the shallow-water equations comprise two types of motion: ageostrophic motions, including inertia-gravity waves, on the one hand, and nearly geostrophic motions on the other. In rapidly rotating flow, these two types of motion may, in some sense, decouple. We seek simpler equations that describe only the nearly geostrophic motion. The simplest such equations are the quasigeostrophic equations. In the quasigcostrophic equations, potential vorticity plays the key role: The potential vorticity completely determines the velocity field that transports it, thereby controlling the whole dynamics. We begin by generalizing our previously derived fluid equations to a rotating coordinate frame.

Maximum power from a turbine farm in shallow water

2013 ◽  
Vol 714 ◽  
pp. 634-643 ◽  
Author(s):  
Chris Garrett ◽  
Patrick Cummins
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Momentum Equation ◽  
Basic Flow ◽  
Maximum Power ◽  
Two Dimensional ◽  
Dominant Term ◽  
Tidal Flows ◽  
Linear Friction

AbstractThe maximum power that can be obtained from a confined array of turbines in steady or tidal flows is considered using the two-dimensional shallow-water equations and representing the turbine farm by a uniform local increase in friction within a circle. Analytical results supported by dimensional reasoning and numerical solutions show that the maximum power depends on the dominant term in the momentum equation for flows perturbed on the scale of the farm. If friction dominates in the basic flow, the maximum power is a fraction (half for linear friction and 0.75 for quadratic friction) of the dissipation within the circle in the undisturbed state; if the advective terms dominate, the maximum power is a fraction of the undisturbed kinetic energy flux into the front of the turbine farm; if the acceleration dominates, the maximum power is similar to that for the linear frictional case, but with the friction coefficient replaced by twice the tidal frequency.

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