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At least 19 records

Multigrid-Reduction-in-Time for the Rotating Shallow Water Equations

We consider multilevel time-parallel methods for the numerical solution of the rotating shallow water equations. In particular, the multigrid-reduction-in-time (MGRIT) algorithm is used for the parallel time integration. An asymptotic model is used at the coarse levels while the full model is employed at the finer levels. The asymptotic model is well-suited for highly oscillatory partial differential equations like the rotating shallow water equations because it can accurately and stably take the required large time-steps on coarse levels. Our work exploits the flexibility of the MGRIT algorithm in terms of the number of levels and relaxation schemes to show some computational benefits, especially with respect to FCF-relaxation and data reuse.

97 MATHEMATICS AND COMPUTING↗

On the transition towards slow manifold in shallow-water and 3D Euler equations in a rotating frame

The long-time, asymptotic state of rotating homogeneous shallow-water equations is investigated. Our analysis is based on long-time averaged rotating shallow-water equations describing interactions of large-scale, horizontal, two-dimensional motions with surface inertial-gravity waves field for a shallow, uniformly rotating fluid layer. These equations are obtained in two steps: first by introducing a Poincare/Kelvin linear propagator directly into classical shallow-water equations, then by averaging. The averaged equations describe interaction of wave fields with large-scale motions on time scales long compared to the time scale 1/f(sub o) introduced by rotation (f(sub o)/2-angular velocity of background rotation). The present analysis is similar to the one presented by Waleffe (1991) for 3D Euler equations in a rotating frame. However, since three-wave interactions in rotating shallow-water equations are forbidden, the final equations describing the asymptotic state are simplified considerably. Special emphasis is given to a new conservation law found in the asymptotic state and decoupling of the dynamics of the divergence free part of the velocity field. The possible rising of a decoupled dynamics in the asymptotic state is also investigated for homogeneous turbulence subjected to a background rotation. In our analysis we use long-time expansion, where the velocity field is decomposed into the 'slow manifold' part (the manifold which is unaffected by the linear 'rapid' effects of rotation or the inertial waves) and a formal 3D disturbance. We derive the physical space version of the long-time averaged equations and consider an invariant, basis-free derivation. This formulation can be used to generalize Waleffe's (1991) helical decomposition to viscous inhomogeneous flows (e.g. problems in cylindrical geometry with no-slip boundary conditions on the cylinder surface and homogeneous in the vertical direction).

Mahalov, A.↗

Localized Exponential Time Differencing Method for Shallow Water Equations: Algorithms and Numerical Study

Here, we explore the performance of the exponential time differencing (ETD) method applied to the rotating shallow water equations. Comparing with explicit time stepping of the same order accuracy in time, the ETD algorithms could reduce the computational time in many cases by allowing the use of large time step sizes while still maintaining numerical stability. To accelerate the ETD simulations, we propose a localized approach that synthesizes the ETD method and overlapping domain decomposition. By dividing the original problem into many subdomain problems of smaller sizes and solving them locally, the proposed approach could speed up the calculation of matrix exponential vector products. Several standard test cases for shallow water equations of one or multiple layers are considered. The results show great potential of the localized ETD method for high-performance computing because each subdomain problem can be naturally solved in parallel at every time step.

58 GEOSCIENCES↗

A Lagrangian method for the shallow water equations based on a Voronoi mesh - Flows on a rotating sphere

A Lagrangian scheme using the Voronoi mesh is applied to study shallow water flow on a sphere. Discrete approximations to the shallow water equations are obtained for the surfaces of a nonrotating and a rotating sphere, and discrete differential operators are defined for the gradient and the divergence on the sphere. Dissipation is put into the model, when needed, by merging fluid points when they get too close to each other. The full numerical scheme is described and results of numerical computations on various test cases are given, including zonal flow and the Riemann problem.

Augenbaum, J. M.↗

A conservative discontinuous-Galerkin-in-time (DGiT) multirate time integration framework for interface-coupled problems with applications to solid–solid interaction and air–sea models

In this paper we extend the DGiT multirate framework, developed in Connors and Sockwell (2022) for scalar transmission problems, to a solid–solid interaction (SSI) problem involving two coupled elastic solids and a coupled air–sea model with the rotating, thermal shallow water equations. In so doing we aim to demonstrate the broad applicability of the mathematical theory and governing principles established in Connors and Sockwell (2022) to coupled problems characterized by subproblems evolving at different temporal scales. Further, multirate time integration algorithms employing different time steps, optimized for the dynamics of each subproblem, can significantly improve simulation efficiency for such coupled problems. However, development of multirate algorithms is a highly non-trivial task due to the coupling, which can impact accuracy, stability or other desired properties such as preservation of system invariants. DGiT provides a general template for multirate time integration that can achieve these properties. To elucidate the manner in which DGiT accomplishes this task, we fully detail each step in the application of the framework to the SSI and air–sea coupled problems. Numerical examples illustrate key properties of the resulting multirate schemes for both problems.

42 ENGINEERING↗

Topological Signature of Stratospheric Poincaré-Gravity Waves

The rotation of Earth breaks time-reversal and reflection symmetries in an opposite sense north and south of the equator, leading to a topological origin for certain atmospheric and oceanic equatorial waves. Away from the equator, the rotating shallow-water and stably stratified primitive equations exhibit Poincaré inertia–gravity waves that have nontrivial topology as evidenced by their strict superinertial time scale and a phase singularity in frequency–wavevector space. This nontrivial topology then predicts, via the principle of bulk-interface correspondence, the existence of two equatorial waves along the equatorial interface, the Kelvin and Yanai waves. To directly test the nontrivial topology of Poincaré-gravity waves in observations, we examine ERA5 data and study cross correlations between the wind velocity and geopotential height of the midlatitude stratosphere at the 50 hPa height. We find the predicted vortex and antivortex in the relative phase of the geopotential height and velocity at the high frequencies of the waves. By contrast, lower-frequency planetary waves are found to have trivial topology also as expected from theory. These results demonstrate a new way to understand stratospheric waves and provide a new qualitative tool to investigate waves in other components of the climate system.

54 ENVIRONMENTAL SCIENCES↗

Semi-Lagrangian integration of a gridpoint shallow water model on the sphere

A stable, semi-Lagrangian, semi-implicit, two-time-level, gridpoint integration scheme for the shallow water equations on the sphere is presented. A rotated spherical coordinate system is used to integrate the equations of motion at each gridpoint poleward of a certain latitude, thus overcoming problems associated with the polar singularity. The results of medium term integrations of large scale test patterns using a long time step are presented.

Mcdonald, A.↗

The effects of the Asselin time filter on numerical solutions to the linearized shallow-water wave equations

In the present investigation, a one-dimensional linearized analysis is used to determine the effect of Asselin's (1972) time filter on both the computational stability and phase error of numerical solutions for the shallow water wave equations, in cases with diffusion but without rotation. An attempt has been made to establish the approximate optimal values of the filtering parameter nu for each of the 'lagged', Dufort-Frankel, and Crank-Nicholson diffusion schemes, suppressing the computational wave mode without materially altering the physical wave mode. It is determined that in the presence of diffusion, the optimum filter length depends on whether waves are undergoing significant propagation. When moderate propagation is present, with or without diffusion, the Asselin filter has little effect on the spatial phase lag of the physical mode for the leapfrog advection scheme of the three diffusion schemes considered.

Schlesinger, R. E.↗

A fully implicit scheme for global numerical weather prediction

A fast-slow factored scheme is presented for use with shallow-water primitive equation numerical weather prediction models. The technique was developed to reduce the rotational mode errors which arise when the fast and slow terms of the governing differential equations are treated simultaneously. The method factors out the fast and slow terms along the coordinate directions by means of a modified Crank-Nicolson scheme. A finite-difference spatial discretization is carried out in the zonal and meridional directions to reduce the factorization error to near-zero, and that time steps of 60-90 min can be used to obtain acceptably accurate results, even in the presence of fine spatial structures in the flow.

Augenbaum, J. M.↗

Semi-Lagrangian integration of a grid-point shallow water model on the sphere

This paper describes a semi-Lagrangian technique for integrating the equations of motion on the global domain. The technique uses an auxiliary spherical coordinate system at each near-polar gridpoint of the latitude-longitude grid; the auxiliary system is obtained by a rotation such that the new equator passes through the gridpoint in question and the new coordinate directions coincide with those of the original system at that point. The technique was applied to the shallow water equations, incorporating a semiimplicit treatment of the adjustment terms on a C-grid, with two-time levels. A five day integration was successfully carried out for a situation involving strong cross-polar flow. No filtering or diffusion was required to maintain stability over a five day period.

Mcdonald, A.↗

Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements

Within finite element models of fluids, vector-valued fields such as velocity or momentum variables are commonly discretised using the Raviart-Thomas elements. However, when using the lowest-order quadrilateral Raviart-Thomas elements, standard finite element discretisations of the vector transport equation typically have a low order of spatial accuracy. This paper describes two schemes that improve the accuracy of transporting such vector-valued fields on two-dimensional curved manifolds. The first scheme that is presented reconstructs the transported field in a higher-order function space, where the transport equation is then solved. The second scheme applies a mixed finite element formulation to the vector transport equation, simultaneously solving for the transported field and its vorticity. In this work, an approach to stabilising this mixed vector-vorticity formulation is presented that uses a Streamline Upwind Petrov-Galerkin (SUPG) method. These schemes are then demonstrated, along with their accuracy properties, through some numerical tests. Two new test cases are used to assess the transport of vector-valued fields on curved manifolds, solving the vector transport equation in isolation. The improvement of the schemes is also shown through two standard test cases for rotating shallow-water models.

97 MATHEMATICS AND COMPUTING↗

Amplification and decay of long nonlinear waves.

The interaction of weakly nonlinear waves with slowly varying boundaries is considered. Special emphasis is given to rotating fluids, but the analysis applies with minor modifications to waves in stratified fluids and shallow-water waves. An asymptotic solution of a variant of the Korteweg-de Vries equation with variable coefficients is developed that produces a 'Green's law' for the amplification of waves of finite amplitude. For shallow-water waves in water of variable depth, the result predicts wave growth proportional to the -1/3 power of the depth.

Leibovich, S.↗

Stability of propagating modons for small-amplitude perturbations

A stability analysis based on Arnol'd-Liapunov arguments in a gauge-variable formalism is applied to a generalized form of propagating modons. The method was previously applied to the special case of Stern's modon in a quiescent background flow. The analysis presented here shows that propagating-modon solutions in the shallow-water equations are stable to small-amplitude perturbations, regardless of the sign of their translational speed, as long as this is much smaller in magnitude than the solid-rotation speed of the earth. Sufficient conditions for stability are that the given flow be quasigeostrophic, i.e., that the Rossby number be small, and that the associated Froude number be one order of magnitude smaller than the Rossby number.

Sakuma, H.↗

Monsoon Depression Amplification by Horizontal Shear and Humidity Gradients: A Shallow Water Perspective

Transient, synoptic-scale vortices produce a large fraction of total rainfall in most monsoon regions and are often associated with extreme precipitation. However, the mechanism of their amplification remains a topic of active research. For monsoon depressions, which are the most prominent synoptic-scale vortex in the Asian–Australian monsoon, recent work has suggested that meridional gradients in zonal wind in the vortex environment may produce growth through barotropic instability, while meridional gradients in environmental humidity have also been proposed to cause amplification through coupling with precipitating convection. Here, a two-dimensional shallow water model on a sphere with parameterized precipitation is used to examine the relative role played by these two environmental gradients. By systematically varying the meridional moisture gradient and meridional wind shear for both weak, quasi-linear waves and finite-amplitude isolated vortices, we show that rotational winds in the initial vortex are amplified most strongly by meridional shear of the environmental zonal wind, while vortex precipitation rates are most sensitive to environmental moisture gradients. The growth rate in the presence of both gradients is less than the sum of growth rates in the presence of isolated gradients, as the phase relation between moisture and vorticity anomalies becomes distorted with increasing shear. In conclusion, these results suggest that background meridional gradients in both zonal wind and environmental humidity can contribute to the amplification of vortices to monsoon depression strength, but with some degree of decoupling of the dry rotational flow and the moist convection.

54 ENVIRONMENTAL SCIENCES↗

Symmetric Equations on the Surface of a Sphere as Used by Model GISS:IB

Standard vector calculus formulas of Cartesian three space are projected onto the surface of a sphere. This produces symmetric equations with three nonindependent horizontal velocity components. Each orthogonal axis has a velocity component that rotates around its axis (eastward velocity rotates around the north–south axis) and a specific angular momentum component that is the product of the velocity component multiplied by the cosine of axis’ latitude. Angular momentum components align with the fixed axes and simplify several formulas, whereas the rotating velocity components are not orthogonal and vary with location. Three symmetric coordinates allow vector resolution and calculus operations continuously over the whole spherical surface, which is not possible with only two coordinates. The symmetric equations are applied to one-layer shallow water models on cubed-sphere and icosahedral grids, the latter being computationally simple and applicable to an ocean domain. Model results are presented for three different initial conditions and five different resolutions.

Icosahedral grid↗

Orbital, Rotational and Climatic Interactions: Energy Dissipation and Angular Momentum Exchange in the Earth-Moon System

A numerical ocean tide model has been developed and tested using highly accurate TOPEX/Poseidon (T/P) tidal solutions. The hydrodynamic model is based on time stepping a finite difference approximation to the non-linear shallow water equations. Two novel features of our implementation are a rigorous treatment of self attraction and loading (SAL), and a physically based parameterization for internal tide (IT) radiation drag. The model was run for a range of grid resolutions, and with variations in model parameters and bathymetry. For a rational treatment of SAL and IT drag, the model run at high resolution (1/12 degree) fits the T/P solutions to within 5 cm RMS in the open ocean. Both the rigorous SAL treatment and the IT drag parameterization are required to obtain solutions of this quality. The sensitivity of the solution to perturbations in bathymetry suggest that the fit to T/P is probably now limited by errors in this critical input. Since the model is not constrained by any data, we can test the effect of dropping sea-level to match estimated bathymetry from the last glacial maximum (LGM). Our results suggest that the 100 m drop in sea-level in the LGM would have significantly increased tidal amplitudes in the North Atlantic, and increased overall tidal dissipation by about 40%. However, details in tidal solutions for the past 20 ka are sensitive to the assumed stratification. IT drag accounts for a significant fraction of dissipation, especially in the LGM when large areas of present day shallow sea were exposed, and this parameter is poorly constrained at present.

Egbert, Gary D.↗

Integration of the shallow water equations on the sphere using a vector semi-Lagrangian scheme with a multigrid solver

A vector semi-Lagrangian semi-implicit two-time-level finite-difference integration scheme for the shallow water equations on the sphere is presented. A C-grid is used for the spatial differencing. The trajectory-centered discretization of the momentum equation in vector form eliminates pole problems and, at comparable cost, gives greater accuracy than a previous semi-Lagrangian finite-difference scheme which used a rotated spherical coordinate system. In terms of the insensitivity of the results to increasing timestep, the new scheme is as successful as recent spectral semi-Lagrangian schemes. In addition, the use of a multigrid method for solving the elliptic equation for the geopotential allows efficient integration with an operation count which, at high resolution, is of lower order than in the case of the spectral models. The properties of the new scheme should allow finite-difference models to compete with spectral models more effectively than has previously been possible.

Bates, J. R.↗