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conservative and is able to capture physically relevant shocks and discontinuities. DG is a scalable method because numerical information of each element is only passed locally through numerical fluxes to the nearest neighbors. In a body of work, the DG method was successfully implemented on the sphere for advection models ( Nair et al. 2005b ) and the shallow-water equation ( Nair et al. 2005a ). Multiwavelets are a discontinuous, orthogonal, compactly supported, multiscale set of functions with
conservative and is able to capture physically relevant shocks and discontinuities. DG is a scalable method because numerical information of each element is only passed locally through numerical fluxes to the nearest neighbors. In a body of work, the DG method was successfully implemented on the sphere for advection models ( Nair et al. 2005b ) and the shallow-water equation ( Nair et al. 2005a ). Multiwavelets are a discontinuous, orthogonal, compactly supported, multiscale set of functions with
over an isolated mountain, and a barotropically unstable jet are presented in section 4 and conclusions are drawn on the suitability of different meshes and the current modeling framework in section 5 . 2. Discretization of the shallow-water equations on polygons The shallow-water equations on a rotating sphere have been implemented in the open-source C++ library OpenFOAM (Open Field Operation and Manipulation; information online at www.opencfd.co.uk ), to create AtmosFOAM [which has in part
over an isolated mountain, and a barotropically unstable jet are presented in section 4 and conclusions are drawn on the suitability of different meshes and the current modeling framework in section 5 . 2. Discretization of the shallow-water equations on polygons The shallow-water equations on a rotating sphere have been implemented in the open-source C++ library OpenFOAM (Open Field Operation and Manipulation; information online at www.opencfd.co.uk ), to create AtmosFOAM [which has in part
1. Introduction In most textbooks on geophysical fluid dynamics ( Cushman-Roisin 1994 ; Holton 1992 ; Gill 1982 ; Pedlosky 1987 ; Salmon 1998 ) the shallow-water equations (SWE) are studied in their linearized form to derive the dispersion relations and the velocity and free-surface height profiles for Kelvin, Poincaré (inertial–gravity), and Rossby waves; however, the waves are derived in two slightly different contexts. The high-frequency Kelvin and Poincaré waves are derived on an f
1. Introduction In most textbooks on geophysical fluid dynamics ( Cushman-Roisin 1994 ; Holton 1992 ; Gill 1982 ; Pedlosky 1987 ; Salmon 1998 ) the shallow-water equations (SWE) are studied in their linearized form to derive the dispersion relations and the velocity and free-surface height profiles for Kelvin, Poincaré (inertial–gravity), and Rossby waves; however, the waves are derived in two slightly different contexts. The high-frequency Kelvin and Poincaré waves are derived on an f
use in theoretical studies. In the present article we will combine Daley’s (1983) simplest form of the geostrophic relationship with a linearized expression of the potential vorticity to formulate a much simpler balanced model. We will do this in the context of a global one-layer shallow-water model of the atmosphere, introduced in section 2 . In section 3 we show how our simplifying procedure leads to an equation that is identical in form to the equivalent barotropic vorticity equation
use in theoretical studies. In the present article we will combine Daley’s (1983) simplest form of the geostrophic relationship with a linearized expression of the potential vorticity to formulate a much simpler balanced model. We will do this in the context of a global one-layer shallow-water model of the atmosphere, introduced in section 2 . In section 3 we show how our simplifying procedure leads to an equation that is identical in form to the equivalent barotropic vorticity equation
scheme ( Arakawa 1966 ) for two-dimensional incompressible flow is volume conserving. Here we show results of a more general physical and numerical setting. Section 2 gives a definition of phase space volume conservation for numerical schemes in terms of the Liouville equation and introduces the concept of time reversibility. In section 3 , three different spatial discretization schemes of the shallow-water equations on the sphere are compared regarding their phase space volume conservation
scheme ( Arakawa 1966 ) for two-dimensional incompressible flow is volume conserving. Here we show results of a more general physical and numerical setting. Section 2 gives a definition of phase space volume conservation for numerical schemes in terms of the Liouville equation and introduces the concept of time reversibility. In section 3 , three different spatial discretization schemes of the shallow-water equations on the sphere are compared regarding their phase space volume conservation
-dimensional nonlinear shallow-water equations in flux convergence form. As noted by a reviewer, there is a growing body of implicit and semi-implicit Runge–Kutta schemes available in the literature. For example, Ascher et al. (1997) has proposed a family of semi-implicit Runge–Kutta schemes applied to the advection–diffusion equation; Butcher (1964) has developed fully implicit Runge–Kutta schemes that require a simultaneous solution of equations at each time step. We have not explored such alternative
-dimensional nonlinear shallow-water equations in flux convergence form. As noted by a reviewer, there is a growing body of implicit and semi-implicit Runge–Kutta schemes available in the literature. For example, Ascher et al. (1997) has proposed a family of semi-implicit Runge–Kutta schemes applied to the advection–diffusion equation; Butcher (1964) has developed fully implicit Runge–Kutta schemes that require a simultaneous solution of equations at each time step. We have not explored such alternative
advection in solutions for diabatic PV is avoided in the DCASL algorithm of Dritschel and Ambaum (2006) by periodically transferring the diabatic PV into adiabatic PV. This is made possible by merging adiabatic and diabatic solutions periodically and “recontouring” the merged solution to reconstruct the contour representation for PV. In realistic circumstances, however, diabatic effects may lead to nonsmooth forcing, as in the case of the thermally forced spherical shallow water equations studied in
advection in solutions for diabatic PV is avoided in the DCASL algorithm of Dritschel and Ambaum (2006) by periodically transferring the diabatic PV into adiabatic PV. This is made possible by merging adiabatic and diabatic solutions periodically and “recontouring” the merged solution to reconstruct the contour representation for PV. In realistic circumstances, however, diabatic effects may lead to nonsmooth forcing, as in the case of the thermally forced spherical shallow water equations studied in
between the large-scale jet structure and the field of eddy turbulence is nonlinear and results in a nonlinear trajectory for the jet and the ensemble-mean eddy field associated with it, which for parameter values in the shallow-water equations (SWEs) on an equatorial beta plane typical of Jovian conditions asymptotically approaches a stable equilibrium with multiple jets. This equilibrium arises from a linear instability of the coupled turbulence–mean flow system. This jet-forming instability is
between the large-scale jet structure and the field of eddy turbulence is nonlinear and results in a nonlinear trajectory for the jet and the ensemble-mean eddy field associated with it, which for parameter values in the shallow-water equations (SWEs) on an equatorial beta plane typical of Jovian conditions asymptotically approaches a stable equilibrium with multiple jets. This equilibrium arises from a linear instability of the coupled turbulence–mean flow system. This jet-forming instability is
(SCFDM), which offers a general formula for approximating even and odd derivatives with a three-point stencil size for an arbitrary order of accuracy. At second and fourth orders, SCFDM becomes the same as the familiar centered and the Padé schemes, respectively. Esfahanian et al. (2005 , hereafter EGM05) have examined the numerical accuracy of the second-, fourth-, and sixth-order SCFDM in prototype problems of (i) linear inertia–gravity wave propagation in f -plane shallow-water (SW) equations
(SCFDM), which offers a general formula for approximating even and odd derivatives with a three-point stencil size for an arbitrary order of accuracy. At second and fourth orders, SCFDM becomes the same as the familiar centered and the Padé schemes, respectively. Esfahanian et al. (2005 , hereafter EGM05) have examined the numerical accuracy of the second-, fourth-, and sixth-order SCFDM in prototype problems of (i) linear inertia–gravity wave propagation in f -plane shallow-water (SW) equations
1. Introduction Arakawa and Lamb (1981 , hereafter AL81 ) developed a finite-difference scheme for the shallow-water equations that simultaneously conserves energy and potential enstrophy. As pointed out in subsequent work (e.g., Salmon 2004 ), this was a significant achievement because neither kinetic energy nor potential enstrophy are simple quadratic quantities due to the divergent nature of shallow-water motion. Conserving analogs of these two global invariants in numerical models is
1. Introduction Arakawa and Lamb (1981 , hereafter AL81 ) developed a finite-difference scheme for the shallow-water equations that simultaneously conserves energy and potential enstrophy. As pointed out in subsequent work (e.g., Salmon 2004 ), this was a significant achievement because neither kinetic energy nor potential enstrophy are simple quadratic quantities due to the divergent nature of shallow-water motion. Conserving analogs of these two global invariants in numerical models is
