Equations In Spherical Geometry Research Articles

Some Results Concerning the Singular Solution of the Conservative Transport Equation in Spherical Geometry

We examine in this work one of the exact solutions of the conservative transport equation for isotropic scattering in spherical geometry, specifically the solution that is singular at the origin and vanishes at infinity. Two representations are known for that solution: one expressed as an infinite divergent series that is derived from the spherical harmonics method and another given by an integral that results from the technique of integration along the particle path and is confirmed here by the method of characteristics. We establish a connection between these representations by showing that the Borel sum of the first reproduces the latter. We also examine computational aspects of the solution expressed in various forms and discuss some standing issues related to it.

Domain-specific implementation of high-order Discontinuous Galerkin methods in spherical geometry

In recent years, domain-specific languages (DSLs) have achieved significant success in large-scale efforts to reimplement existing meteorological models in a performance portable manner. The dynamical cores of these models are based on finite difference and finite volume schemes, and existing DSLs are generally limited to supporting only these numerical methods. In the meantime, there have been numerous attempts to use high-order Discontinuous Galerkin (DG) methods for atmospheric dynamics, which are currently largely unsupported in main-stream DSLs. In order to link these developments, we present two domain-specific languages which extend the existing GridTools (GT) ecosystem to high-order DG discretization. The first is a C++-based DSL called G4GT, which, despite being no longer supported, gave us the impetus to implement extensions to the subsequent Python-based production DSL called GT4Py to support the operations needed for DG solvers. As a proof of concept, the shallow water equations in spherical geometry are implemented in both DSLs, thus providing a blueprint for the application of domain-specific languages to the development of global atmospheric models. We believe this is the first GPU-capable DSL implementation of DG in spherical geometry. The results demonstrate that a DSL designed for finite difference/volume methods can be successfully extended to implement a DG solver, while preserving the performance-portability of the DSL.

Reviewing and clarifying the derivation of the hydrostatic primitive equations

AbstractHaving adopted a spherical geopotential approximation, textbooks generally derive the hydrostatic primitive equations from the fully compressible Euler equations by making three further approximations—usually termed “hydrostatic”, “shallow”, and “traditional”. However, the derivation given by Phillips in 1966 indicates that the traditional approximation is one of the consequences of the consistently applied shallow approximation. This is demonstrated here without recourse to the “vector‐invariant” form of the momentum equation used by him. This new derivation is not only simpler, but also in accord with the known existence of a quartet of dynamically consistent models, depending on whether approximations of shallow and/or hydrostatic type are or are not made in the fully compressible Euler equations in spherical geometry. A brief survey of other derivation strategies is given. To provide further insight, this includes the approximation of Lagrangian density—a scalar quantity—followed by application of either Hamilton's principle or the Euler–Lagrange equations. Relatedly, the consistent specific forms of the spherical geopotential approximation for the (deep) Euler equations and for the (shallow) hydrostatic primitive equations are obtained and discussed. The analysis herein can be generalised to spheroidal geometry. This is of interest, since the Earth is more realistically represented as an oblate spheroid than as a sphere, and opens the way to the development of more accurate dynamical cores for global atmospheric and oceanic modelling.

An efficient covariant frame for the spherical shallow water equations: Well balanced DG approximation and application to tsunami and storm surge

In this work we consider an efficient discretization of the Shallow Water Equations in spherical geometry for oceanographic applications. Instead of the classical 2d-covariant or 3d-Cartesian approaches, we focus on the mixed 3d/2d form of Bernard et al. (2009) which evolves the 2d momentum tangential to the sphere by projecting the 3d-Cartesian right-hand side on the sphere surface. Differently from the last reference we consider the exact representation of the sphere instead of the finite element one, mixed with a covariant basis projection of the momentum. This leads to several simplifications of the Discontinuous Galerkin scheme: the local mass matrix goes back to the standard block-diagonal form; the Riemann Problem does not require any tensor or vector rotations to align the bases on the two sides of an edge. Second we consider well balancing corrections related to relevant equilibrium states for tsunami and storm surge simulations. These corrections allow to compensate for the inherent non-exactness of the quadrature induced by the non-polynomial nature of both the geometrical mapping and of the covariant basis. In other words, these corrections are the order of the cubature error. We show that their addition makes the scheme exactly well balanced, and is equivalent to recasting the integral of the hydrostatic pressure term in strong form. The proposed method is validated on academic benchmarks involving both smooth and discontinuous solutions, and applied to realistic tsunami and a historical storm surge simulation.

Radiative–conductive transfer equation in spherical geometry: arithmetic stability for decomposition using the condition number criterion

The radiative–conductive transfer equation in the $$S_N$$ approximation for spherical geometry is solved using a modified decomposition method. The focus of this work is to show how to distribute the source terms in the recursive equation system in order to guarantee arithmetic stability and thus numerical convergence of the obtained solution, guided by a condition number criterion. Some examples are compared with results from literature and parameter combinations are analyzed, for which the condition number analysis indicates convergence of the solutions obtained by the recursive scheme.

An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables

AbstractWe derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal discretization. The formulation is a version of Dubos and Tort (2014, https://doi.org/10.1175/MWR-D-14-00069.1) rewritten in terms of primitive variables. It is valid for terrain following mass or height coordinates and for both Eulerian or vertically Lagrangian discretizations. The discretization relies on an extension to Simmons and Burridge (1981, https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2) vertical differencing, which we show obeys a discrete derivative product rule. This product rule allows us to simplify the treatment of the vertical transport terms. Energy conservation is obtained via a term‐by‐term balance in the kinetic, internal, and potential energy budgets, ensuring an energy‐consistent discretization up to time truncation error with no spurious sources of energy. We demonstrate convergence with respect to time truncation error in a spectral element code with a horizontal explicit vertically implicit implicit‐explicit time stepping algorithm.

A numerically stable spherical harmonics solution for the neutron transport equation in a spherical shell

A numerically stable version of the spherical harmonics (PN) method for solving the one-speed neutron transport equation with Lth order anisotropic scattering in a spherical shell is developed. Implementing a stable PN solution for this problem is a challenging task for which no satisfactory answer has been given in the literature. The approach used in this work follows and generalizes a previous work on a problem whose domain is defined by the exterior of a sphere. First, a transformation is used to reduce the original transport equation in spherical geometry to a plane-geometry-like transport equation, where the angular redistribution term in spherical geometry is treated as a source. Then, a PN solution in plane geometry given by a combination of the solution of the associated homogeneous equation and a particular solution is developed. This is followed by a post-processing step which is very effective in improving the PN solution. An additional amount of work with respect to that required for solving problems in plane geometry occurs in the form of a system of N+1 Volterra integral equations of the second kind that has to be solved for the coefficients of the particular solution. The proposed approach has, in any case, the merit of avoiding the ill-conditioning caused by the presence of modified spherical Bessel functions in the standard PN solution, as demonstrated by numerical results tabulated for some test cases.

Finite Speed of Propagation for the Thin-Film Equation in Spherical Geometry

We show that a doubly degenerate thin-film equation obtained in modeling the flows of viscous coatings on spherical surfaces has a finite speed of propagation for nonnegative strong solutions and, hence, there exists an interface or a free boundary separating the regions, where the solution u > 0 and u = 0. By using local entropy estimates, we also establish the upper bound for the rate of propagation of the interface.

Variance reduction method for particle transport equation in spherical geometry

This article is devoted to the design of importance sampling method for the Monte Carlo simulation of a linear transport equation. This model is of great importance in the simulation of inertial confinement fusion experiments. Our method is restricted to a spherically symmetric idealized design: an outer sphere emitting radiation towards an inner sphere, which in practice should be thought of as the hohlraum and the fusion capsule, respectively. We compute the importance function as the solution of the corresponding stationary adjoint problem. Doing so, we have an important reduction of the variance (by a factor 50 to 100), with a moderate increase of computational cost (by a factor 2 to 8).

Mathematical Development and Computational Analysis of Harmonic Phase-Magnetic Resonance Imaging (HARP-MRI) Based on Bloch Nuclear Magnetic Resonance (NMR) Diffusion Model for Myocardial Motion.

Harmonic Phase-Magnetic Resonance Imaging (HARP-MRI) is a tagged image analysis method that can measure myocardial motion and strain in near real-time and is considered a potential candidate to make magnetic resonance tagging clinically viable. However, analytical expressions of radially tagged transverse magnetization in polar coordinates (which is required to appropriately describe the shape of the heart) have not been explored because the physics required to directly connect myocardial deformation of tagged Nuclear Magnetic Resonance (NMR) transverse magnetization in polar geometry and the appropriate harmonic phase parameters are not yet available. The analytical solution of Bloch NMR diffusion equation in spherical geometry with appropriate spherical wave tagging function is important for proper analysis and monitoring of heart systolic and diastolic deformation with relevant boundary conditions. In this study, we applied Harmonic Phase MRI method to compute the difference between tagged and untagged NMR transverse magnetization based on the Bloch NMR diffusion equation and obtained radial wave tagging function for analysis of myocardial motion. The analytical solution of the Bloch NMR equations and the computational simulation of myocardial motion as developed in this study are intended to significantly improve healthcare for accurate diagnosis, prognosis and treatment of cardiovascular related deceases at the lowest cost because MRI scan is still one of the most expensive anywhere. The analysis is fundamental and significant because all Magnetic Resonance Imaging techniques are based on the Bloch NMR flow equations.

A PN particular solution for the radiative transfer equation in spherical geometry

A PN particular solution for the radiative transfer equation in spherical geometry is reported. A general source that varies with position and direction is considered.

Exact solution of the neutron transport equation in spherical geometry

Abstract Solution of the neutron transport equation in one dimensional slab geometry construct a basis for the solution of neutron transport equation in a curvilinear geometry. Therefore, in this work, we attempt to derive an exact analytical benchmark solution for both neutron transport equations in slab and spherical medium by using PN approximation which is widely used in neutron transport theory.

Criticality calculations with PN approximation for certain scattering parameters of Anlı-Güngör and Henyey-Greenstein phase functions in spherical geometry

Abstract PN approximation is applied to neutron transport equation in spherical geometry using Anlı-Güngör (AG) and Henyey-Greenstein (HG) phase functions. These phase functions are used for calculating critical radius and results of PN approximation obtained from AG and HG phase functions are compared with each other for different collision scattering function. The numerical results obtained from AG and HG phase functions are presented in the tables and these results are compared.

Dynamically consistent shallow‐water equation sets in non‐spherical geometry with latitudinal variation of gravity

The shallow‐water equations in spherical geometry have proven to be an invaluable prototypical tool to advance geophysical fluid dynamics. Many of the fundamental terms and properties, including dynamical consistency, needed to model Earth's atmosphere and oceans are embodied within them. Limitations to their wider use include the following: representation of the Earth as a sphere, rather than as an ellipsoid; gravity not varying latitudinally, but being held constant; the Coriolis force being incompletely represented; and no representation of vertical acceleration.Recent work has addressed the first three limitations, but not simultaneously. The present work addresses all four simultaneously by endowing the equations selectively with supplementary terms, whilst respecting dynamical consistency. This is accomplished using Hamilton's principle of least action, but could be done otherwise. It leads to a switch‐controlled quartet of shallow‐water equation sets, all of which include topography.Potential future applications include the following: further sensitivity tests to assess the possible impact of latitudinal variation of gravity; development of improved numerical methods with good conservation properties; examination of the stability of discretizations of the Coriolis terms; and sensitivity tests for the possible impact of vertical acceleration as a function of decreasing horizontal scale. Sensitivity experimentation would, however, necessitate scale analysis of flow regimes to ensure correct interpretation of results.

Least-Squares Finite Element Discretization of the Neutron Transport Equation in Spherical Geometry

The main focus of this paper is the numerical solution of the steady-state, mono-energetic Boltzmann transport equation for neutral particles through mixed material media in a spherically symmetric geometry. Standard solution strategies, like the discrete ordinates method, may lead to nonphysical approximate solutions. In particular, a point source at the center of the sphere yields undesirable ray effects. Posing the problem in spherical coordinates avoids ray effects and other nonphysical numerical artifacts in the simulation process, at the cost of coupling all angles in the PDE setting. In addition, traditional finite element or finite difference techniques for spherical coordinates often yield incorrect scalar flux at the center of the sphere, known as flux dip, and oscillations near steep gradients. In this paper, a least-squares finite element method with adaptive mesh refinement is used to approximate solutions to the nonscattering, one-dimensional neutron transport equation in spherically symmetric geometry. It is shown that the resulting numerical approximations avoid flux dip and oscillations. The least-squares discretization yields a symmetric, positive definite, linear system which shares many characteristics with systems obtained from Galerkin finite element discretization of totally anisotropic elliptic PDEs. In general, standard algebraic multigrid techniques fail to scale on non-grid-aligned anisotropies. In this paper, a new variation of smoothed aggregation is employed and shown to be essentially scalable. The effectiveness of the method is demonstrated on several mixed-media model problems.

The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity

Abstract. We evaluate the performance of the Community Atmosphere Model's (CAM) spectral element method on variable-resolution grids using the shallow-water equations in spherical geometry. We configure the method as it is used in CAM, with dissipation of grid scale variance, implemented using hyperviscosity. Hyperviscosity is highly scale selective and grid independent, but does require a resolution-dependent coefficient. For the spectral element method with variable-resolution grids and highly distorted elements, we obtain the best results if we introduce a tensor-based hyperviscosity with tensor coefficients tied to the eigenvalues of the local element metric tensor. The tensor hyperviscosity is constructed so that, for regions of uniform resolution, it matches the traditional constant-coefficient hyperviscosity. With the tensor hyperviscosity, the large-scale solution is almost completely unaffected by the presence of grid refinement. This later point is important for climate applications in which long term climatological averages can be imprinted by stationary inhomogeneities in the truncation error. We also evaluate the robustness of the approach with respect to grid quality by considering unstructured conforming quadrilateral grids generated with a well-known grid-generating toolkit and grids generated by SQuadGen, a new open source alternative which produces lower valence nodes.

The shallow‐water equations in non‐spherical geometry with latitudinal variation of gravity

AbstractThe shallow‐water equations in spherical geometry are well known. They are derived as a constant‐density, constant‐gravity specialization of the hydrostatic primitive equations for a thin layer of fluid, bounded below by topography and above by a free surface. It is shown herein that it is possible to derive an analogous set of shallow‐water equations in non‐spherical (but zonally symmetric) geometry using orthogonal curvilinear coordinates. This equation set is dynamically consistent, possessing conservation principles for mass, axial angular momentum, energy and potential vorticity. Furthermore, gravity is allowed to vary, as it does physically, as a function of latitude. This prepares the way for performing sensitivity tests, in an idealized framework, to assess the possible impact of latitudinal variation of gravity. Illustrative examples of models of gravity and specific non‐spherical coordinate systems are given.

A Multiscale Model for the Planetary and Synoptic Motions in the Atmosphere

Abstract A reduced asymptotic model valid for the planetary and synoptic scales in the atmosphere is presented. The model is derived by applying a systematic multiple-scales asymptotic method to the full compressible-flow equations in spherical geometry. The synoptic-scale dynamics in the model is governed by modified quasigeostrophic equations, which take into account planetary-scale variations of the background stratification and of the Coriolis parameter. The planetary-scale background is described by the planetary geostrophic equations and a new closure condition in the form of a two-scale evolution equation for the barotropic component of the background flow. This closure equation provides a model revealing an interaction mechanism from the synoptic scale to the planetary scale. To obtain a quantitative assessment of the validity of the asymptotics, the balances on the planetary and synoptic scales are studied by utilizing a primitive equations model. For that purpose, spatial and temporal variations of different terms in the vorticity equation are analyzed. It is found that, for planetary-scale modes, the horizontal fluxes of relative and planetary vorticity are nearly divergence free. It is shown that the results are consistent with the asymptotic model.

Spherical electron acoustic solitary waves in plasma with suprathermal electrons

A reductive perturbation technique is employed to solve the fluid-Poisson equations in spherical geometry describing a weakly nonlinear electron–acoustic (EA) waves in unmagnetized plasma consisting of stationary ions, cold electrons and kappa distributed hot electrons. It is shown that a variable coefficient Kadomtsev–Petviashvili (KP) equation governs the evolution of scalar potential describing propagation of EA waves. The influence of suprathermality and geometry effects on propagation of EA solitary waves is investigated. We found that when electrons evolve toward their thermodynamic equilibrium, EA solitons are generated with large amplitudes. Also it is shown that EA solitary structures can be significantly modified by transverse perturbations.

Numerical simulations of highly non-linear coupled full MHD equations in spherical geometry

Numerical simulations have been performed to solve highly non-linear coupled full MHD equations in spherical polar coordinates. The control of flow separation behind a sphere using Lorentz forces is investigated at moderate magnetic Reynolds numbers. An external magnetic field is applied in the direction of the steady, viscous and electrically conducting flow such that it is aligned at large distances from the sphere. The governing equations are coupled non-linear Navier–Stokes and non-linear Maxwell's equations. The parameters that governs the flow are Reynolds number Re, magnetic Reynolds number Rm and Alfvén number β. The finite difference method combined with multigrid technique is used to solve the full MHD equations which are expressed in vorticity, stream function and magnetic stream function form. All the non-linearities in the momentum equation due to Lorentz force are handled effectively. It is found that the separation for highly conducting fluids can be suppressed with low magnetic fields. The drag coefficient is found to decrease for β≤1 and then increase. The results agree with experimental results.