A quantitative test case for global-scale dynamical cores based on analytic wave solutions of the shallow-water equations

2016 ◽  
Vol 142 (700) ◽  
pp. 2705-2714 ◽  
Author(s):  
Ofer Shamir ◽  
Nathan Paldor
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Global Scale ◽  
Test Case ◽  
Quantitative Test ◽  
Wave Solutions ◽  
Analytic Wave

scholarly journals The Matsuno baroclinic wave test case

2019 ◽  
Vol 12 (6) ◽  
pp. 2181-2193
Author(s):  
Ofer Shamir ◽  
Itamar Yacoby ◽  
Shlomi Ziskin Ziv ◽  
Nathan Paldor
Keyword(s):  
Shallow Water ◽  
Gravity Wave ◽  
Global Scale ◽  
Wave Mode ◽  
Wave Number ◽  
Scale Model ◽  
Water Model ◽  
Test Case ◽  
Wave Speeds ◽  
Wave Solutions

Abstract. The analytic wave solutions obtained by Matsuno (1966) in his seminal work on equatorial waves provide a simple and informative way of assessing the performance of atmospheric models by measuring the accuracy with which they simulate these waves. These solutions approximate the solutions of the shallow-water equations on the sphere for low gravity-wave speeds such as those of the baroclinic modes in the atmosphere. This is in contrast to the solutions of the non-divergent barotropic vorticity equation, used in the Rossby–Haurwitz test case, which are only accurate for high gravity-wave speeds such as those of the barotropic mode. The proposed test case assigns specific values to the wave parameters (gravity-wave speed, zonal wave number, meridional wave mode and wave amplitude) for both planetary and inertia-gravity waves, and suggests simple assessment criteria suitable for zonally propagating wave solutions. The test is successfully applied to a spherical shallow-water model in an equatorial channel and to a global-scale model. By adding a small perturbation to the initial fields it is demonstrated that the chosen initial waves remain stable for at least 100 wave periods. The proposed test case can also be used as a resolution convergence test.

scholarly journals A test case for the inviscid shallow-water equations on the sphere

2015 ◽  
Vol 142 (694) ◽  
pp. 488-495 ◽  
Author(s):  
R. K. Scott ◽  
L. M. Harris ◽  
L. M. Polvani
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Test Case

Predicting mesh density for adaptive modelling of the global atmosphere

2009 ◽  
Vol 367 (1907) ◽  
pp. 4523-4542 ◽  
Author(s):  
Hilary Weller
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Large Scale ◽  
Mesh Adaptation ◽  
Similar Proportion ◽  
Test Case ◽  
Uniform Mesh ◽  
Coarse Mesh ◽  
Mesh Density ◽  
The Cost

The shallow water equations are solved using a mesh of polygons on the sphere, which adapts infrequently to the predicted future solution. Infrequent mesh adaptation reduces the cost of adaptation and load-balancing and will thus allow for more accurate mapping on adaptation. We simulate the growth of a barotropically unstable jet adapting the mesh every 12 h. Using an adaptation criterion based largely on the gradient of the vorticity leads to a mesh with around 20 per cent of the cells of a uniform mesh that gives equivalent results. This is a similar proportion to previous studies of the same test case with mesh adaptation every 1–20 min. The prediction of the mesh density involves solving the shallow water equations on a coarse mesh in advance of the locally refined mesh in order to estimate where features requiring higher resolution will grow, decay or move to. The adaptation criterion consists of two parts: that resolved on the coarse mesh, and that which is not resolved and so is passively advected on the coarse mesh. This combination leads to a balance between resolving features controlled by the large-scale dynamics and maintaining fine-scale features.

scholarly journals Approximate Traveling Wave Solutions of Coupled Whitham-Broer-Kaup Shallow Water Equations by Homotopy Analysis Method

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
M. M. Rashidi ◽  
D. D. Ganji ◽  
S. Dinarvand
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Homotopy Analysis Method ◽  
Shallow Water Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Analysis Method ◽  
Wave Solutions ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is applied to obtain the approximate traveling wave solutions of the coupled Whitham-Broer-Kaup (WBK) equations in shallow water. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the systems of nonlinear partial differential equations.

The Diabatic Contour-Advective Semi-Lagrangian Algorithms for the Spherical Shallow Water Equations

2009 ◽  
Vol 137 (9) ◽  
pp. 2979-2994 ◽  
Author(s):  
Ali R. Mohebalhojeh ◽  
David G. Dritschel
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Relative Strength ◽  
Test Case ◽  
Diagnostic Measures ◽  
Contour Advection ◽  
Rapid Generation ◽  
Quasigeostrophic Model ◽  
Conventional Algorithm ◽  
Grid Based

Abstract The diabatic contour-advective semi-Lagrangian (DCASL) algorithm is extended to the thermally forced shallow water equations on the sphere. DCASL rests on the partitioning of potential vorticity (PV) to adiabatic and diabatic parts solved, respectively, by contour advection and a grid-based conventional algorithm. The presence of PV in the source term for diabatic PV makes the shallow water equations distinct from the quasigeostrophic model previously studied. To address the more rapid generation of finescale structures in diabatic PV, two new features are added to DCASL: (i) the use of multiple sets of contours with successively finer contour intervals and (ii) the application of the underlying method of DCASL at a higher level to diabatic PV. That is, the diabatic PV is allowed to have both contour and grid parts. The added features make it possible to make the grid part of diabatic PV arbitrarily small and thus pave the way for a fully Lagrangian DCASL in the presence of forcing. The DCASL algorithms are constructed using a standard semi-Lagrangian (SL) algorithm to solve for the grid-based part of diabatic PV. The 25-day time evolution of an unstable midlatitude jet triggered by the action of thermal forcing is used as a test case to examine and compare the properties of the DCASL algorithms with a pure SL algorithm for PV. Diagnostic measures of vortical and unbalanced activity as well as of the relative strength of the grid and contour parts of the solution for PV indicate that the superiority of contour advection can be maintained even in the presence of strong, nonsmooth forcing.

Edge waves over a shelf: full linear theory

1984 ◽  
Vol 142 ◽  
pp. 79-95 ◽  
Author(s):  
D. V. Evans ◽  
P. Mciver
Keyword(s):  
Shallow Water ◽  
Wide Class ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Explicit Solution ◽  
Mode Shapes ◽  
Edge Waves ◽  
Wave Solutions ◽  
Linearized Theory ◽  
Sloping Beach

Edge-wave solutions to the linearized shallow-water equations for water waves are well known for a variety of bottom topographies. The only explicit solution using the full linearized theory describes edge waves over a uniformly sloping beach, although the existence of such waves has been established for a wide class of bottom geometries. In this paper the full linearized theory is used to derive the properties of edge waves over a shelf. In particular, curves are presented showing the variation of frequency with wavenumber along the shelf, together with some mode shapes for a particular shelf geometry.

scholarly journals On a Coupled System of Shallow Water Equations Admitting Travelling Wave Solutions

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
D. Burini ◽  
S. De Lillo ◽  
D. Skouteris
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Boussinesq Equations ◽  
Coupled System ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Characteristic Parameters ◽  
Wave Solutions ◽  
Free Interfaces ◽  
Rigid Walls

We consider three inviscid, incompressible, irrotational fluids that are contained between the rigid wallsy=−h1andy=h+Hand that are separated by two free interfacesη1andη2. A generalized nonlocal spectral (NSP) formulation is developed, from which asymptotic reductions of stratified fluids are obtained, including coupled nonlinear generalized Boussinesq equations and(1+1)-dimensional shallow water equations. A numerical investigation of the(1+1)-dimensional case shows the existence of solitary wave solutions which have been investigated for different values of the characteristic parameters.

scholarly journals The Matsuno baroclinic wave test case

2018 ◽  
Author(s):  
Ofer Shamir ◽  
Itamar Yacoby ◽  
Nathan Paldor
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Analytic Solutions ◽  
Wave Mode ◽  
Wave Number ◽  
Water Model ◽  
Test Case ◽  
Wave Solutions ◽  
Barotropic Vorticity ◽  
Spatial Spectra

Abstract. The analytic wave-solutions obtained by Matsuno (1966) in his seminal work on equatorial waves provide a simple and informative way of assessing atmospheric and oceanic models by measuring the accuracy with which they simulate these waves. These solutions approximate the solutions of the shallow water equations on the sphere for small speeds of gravity waves such as those of the baroclinic modes in the atmosphere and ocean. This is in contrast to the solutions of the non-divergent barotropic vorticity equation, used in the Rossby-Haurwitz test case, which are only accurate for large speeds of gravity waves such as those of the barotropic mode. The proposed test case assigns specific values to the wave-parameters (gravity wave speed, zonal wave-number, meridional wave-mode and amplitude) for both planetary and inertia gravity waves, and confirms the accuracy of the simulation by employing Hovmöller diagrams and temporal and spatial spectra. The proposed test case is successfully applied to a standard finite-difference, equatorial, non-linear, shallow water model in spherical coordinates, which demonstrates that Matsuno’s wave-solutions can be accurately simulated for at least 10 wave-periods, which for oceanic planetary waves is nearly 1300 days. In order to facilitate the use of the proposed test case, we provide Matlab, Python and Fortran codes for computing the analytic solutions at any time on arbitrary latitude-longitude grids.

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