Thermal versus isothermal rotating shallow water equations: comparison of dynamical processes by simulations with a novel well-balanced central-upwind scheme
04 Mar 2021-Geophysical and Astrophysical Fluid Dynamics (Taylor & Francis)-Vol. 115, Iss: 2, pp 125-154
TL;DR: In this paper, a well-balanced central-upwind scheme for two-dimensional rotating shallow water equations with horizontal temperature/density gradients is proposed, based on thermal rotating shallow.
Abstract: We introduce a new high-resolution well-balanced central-upwind scheme for two-dimensional rotating shallow water equations with horizontal temperature/density gradients – thermal rotating shallow
Citations
More filters
TL;DR: Fluid dynamics is a broad field within fluid mechanics that deals with the flow of fluids such as liquids and gases under various conditions, mostly with the dynamics of flows on macroscopic scales as mentioned in this paper.
Abstract: Fluid dynamics is a broad field within fluid mechanics that deals with the flow of fluids such as liquids and gases under various conditions, mostly with the dynamics of flows on macroscopic scales...
37 citations
TL;DR: In this article, a numerical dissipation switch is proposed to control the amount of dissipation present in central-upwind schemes without risking oscillations, which is achieved with the help of a more accurate estimate of the local propagation speeds in the parts of the computational domain.
Abstract: We propose a numerical dissipation switch, which helps to control the amount of numerical dissipation present in central-upwind schemes. Our main goal is to reduce the numerical dissipation without risking oscillations. This goal is achieved with the help of a more accurate estimate of the local propagation speeds in the parts of the computational domain, which are near contact discontinuities and shears. To this end, we introduce a switch parameter, which depends on the distributions of energy in the x - and y -directions. The resulting new central-upwind is tested on a number of numerical examples, which demonstrate the superiority of the proposed method over the original central-upwind scheme.
12 citations
TL;DR: In this article, Cheng et al. study the flux globalization based central-upwind scheme for the Saint-Venant system of shallow water equations and develop a well-balanced scheme, which can accurately handle both still and moving-water equilibria.
Abstract: We study the flux globalization based central-upwind scheme from Cheng et al. (J Sci Comput 80:538–554, 2019) for the Saint-Venant system of shallow water equations. We first show that while the scheme is capable of preserving moving-water equilibria, it fails to preserve much simpler “lake-at-rest” steady states. We then modify the computation of the global flux variable and develop a well-balanced scheme, which can accurately handle both still- and moving-water equilibria. In addition, we extend the flux globalization based central-upwind scheme to the case when “dry” and/or “almost dry” areas are present. To this end, we introduce a hybrid approach: we use the flux globalization based scheme inside the “wet” areas only, while elsewhere we apply the central-upwind scheme from Bollermann et al. (J Sci Comput 56:267–290, 2013), which is designed to accurately capture wet/dry fronts. We illustrate the performance of the proposed schemes on a number of numerical examples.
8 citations
TL;DR: In this paper, the authors show how the thermal effects affect trajectories, intensity, and formation of secondary structures during the passages of strong tropical cyclone-like vortices over oceanic warm and cold pools as well as over an island-type topography.
Abstract: In this paper, we show how the thermal effects affect trajectories, intensity, and formation of secondary structures during the passages of strong tropical cyclone-like vortices over oceanic warm and cold pools as well as over an island-type topography. Our results are obtained using the moist-convective thermal rotating shallow-water atmospheric model recently developed in [A. Kurganov et al., “Moist-convective thermal rotating shallow-water model,” Phys. Fluids 32, 066601 (2020)]. This model introduces thermodynamics of the moist air and moist convection in the standard rotating shallow-water models and allows to include in the latter atmosphere–ocean interactions in an elementary way.
6 citations
TL;DR: In this paper , a reduced-order model was developed for the rotating thermal shallow water equation (RTSWE) in the non-canonical Hamiltonian form with state-dependent Poisson matrix.
Abstract: In this paper, reduced-order models (ROMs) are developed for the rotating thermal shallow water equation (RTSWE) in the non-canonical Hamiltonian form with state-dependent Poisson matrix. The high fidelity full solutions are obtained by discretizing the RTSWE in space with skew-symmetric finite-differences, while preserving the Hamiltonian structure. The resulting skew-gradient system is integrated in time by the energy preserving average vector field (AVF) method. The ROM is constructed by applying proper orthogonal decomposition (POD) with the Galerkin projection, preserving the reduced skew-gradient structure, and integrating in time with the AVF method. The nonlinear terms of the Poisson matrix and Hamiltonian are approximated with the discrete empirical interpolation method (DEIM) to reduce the computational cost. The solutions of the resulting linear-quadratic reduced system is accelerated by the use of tensor techniques. The accuracy and computational efficiency of the ROMs are demonstrated for a numerical test problem. Preservation of the energy (Hamiltonian), and other conserved quantities, i.e., mass, buoyancy, and total vorticity show that the reduced-order solutions ensure the long-term stability of the solutions while exhibiting several orders of magnitude computational speedup over the full-order model (FOM). Furthermore, we show that the ROMs are able to accurately predict the test and training data, and capture the system behavior in the prediction phase.
5 citations
References
More filters
TL;DR: In this article, a second-order extension of the Lagrangean method is proposed to integrate the equations of ideal compressible flow, which is based on the integral conservation laws and is dissipative, so that it can be used across shocks.
6,630 citations
TL;DR: In this article, a simple analytic model is constructed to elucidate some basic features of the response of the tropical atmosphere to diabatic heating, showing that there is considerable east-west asymmetry which can be illustrated by solutions for heating concentrated in an area of finite extent.
Abstract: A simple analytic model is constructed to elucidate some basic features of the response of the tropical atmosphere to diabatic heating. In particular, there is considerable east-west asymmetry which can be illustrated by solutions for heating concentrated in an area of finite extent. This is of more than academic interest because heating in practice tends to be concentrated in specific areas. For instance, a model with heating symmetric about the equator at Indonesian longitudes produces low-level easterly flow over the Pacific through propagation of Kelvin waves into the region. It also produces low-level westerly inflow over the Indian Ocean (but in a smaller region) because planetary waves propagate there. In the heating region itself the low-level flow is away from the equator as required by the vorticity equation. The return flow toward the equator is farther west because of planetary wave propagation, and so cyclonic flow is obtained around lows which form on the western margins of the heating zone. Another model solution with the heating displaced north of the equator provides a flow similar to the monsoon circulation of July and a simple model solution can also be found for heating concentrated along an inter-tropical convergence line.
3,799 citations
"Thermal versus isothermal rotating ..." refers background or methods in this paper
...In order to emphasise the importance of this phenomenon, let us recall that our understanding of the circulation patterns in the tropical atmosphere is largely based on the classical paper of Gill (1980), where the RSWmodel on the equatorial β-plane was used to study the effects of large-scale heating by adding a source to the mass conservation equation (1b)....
[...]
...(Gill 1980), the buoyancy, that is, the potential temperature in the atmospheric context, acts mainly as a catalyzer of pressure evolution not much changing itself in the absence of diabatic effects in the TRSWmodel....
[...]
...2) problemwhich was solved inGill (1980), the results can be understood by considering an initial value problem with a localised initial perturbation of h; the equatorial adjustment problem....
[...]
...…phenomenon, let us recall that our understanding of the circulation patterns in the tropical atmosphere is largely based on the classical paper of Gill (1980), where the RSWmodel on the equatorial β-plane was used to study the effects of large-scale heating by adding a source to the mass…...
[...]
...…is that in the long-wave approximation, which is traditionally used in the analysis of tropical circulation starting from the pioneering work (Gill 1980), the buoyancy, that is, the potential temperature in the atmospheric context, acts mainly as a catalyzer of pressure evolution not much…...
[...]
TL;DR: The technique of obtaining high resolution, second order, oscillation free (TVD), explicit scalar difference schemes, by the addition of a limited antidiffusive flux to a first order scheme is described in this article.
Abstract: The technique of obtaining high resolution, second order, oscillation free (TVD), explicit scalar difference schemes, by the addition of a limited antidiffusive flux to a first order scheme is expl...
2,490 citations
TL;DR: This paper reviews and further develops a class of strong stability-preserving high-order time discretizations for semidiscrete method of lines approximations of partial differential equations, and builds on the study of the SSP property of implicit Runge--Kutta and multistep methods.
Abstract: In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.
2,199 citations
"Thermal versus isothermal rotating ..." refers methods in this paper
...In our simulations, we have used the threestage third-order strong stability preserving (SSP) Runge-Kutta method (see, e.g. Gottlieb et al. 2011, 2001) with the time steps restricted using the following CFL condition with the CFL number 1/2: t = 1 2 min { x axmax , y aymax } , where axmax := maxj,k…...
[...]
TL;DR: It is proved that a scalar version of the high-resolution central scheme is nonoscillatory in the sense of satisfying the total-variation diminishing property in the one-dimensional case and the maximum principle in two-space dimensions.
1,685 citations
"Thermal versus isothermal rotating ..." refers methods in this paper
...…solved using the Riemann-problem-solver-free central-upwind scheme, which was derived for general multidimensional hyperbolic systems of conservation laws in a series of works (see, e.g. Kurganov and Tadmor 2000, Kurganov et al. 2001, Kurganov and Tadmor 2002, Kurganov and Lin 2007, Kurganov 2016)....
[...]