Conservation Of Potential Enstrophy Research Articles

Discrete conservation properties for shallow water flows using mixed mimetic spectral elements

A mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as higher moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.

A Potential Enstrophy and Energy Conserving Scheme for the Shallow-Water Equations Extended to Generalized Curvilinear Coordinates

An energy and potential enstrophy conserving finite-difference scheme for the shallow-water equations is derived in generalized curvilinear coordinates. This is an extension of a scheme formulated by Arakawa and Lamb for orthogonal coordinate systems. The starting point for the present scheme is the shallow-water equations cast in generalized curvilinear coordinates, and tensor analysis is used to derive the invariant conservation properties. Preliminary tests on a flat plane with doubly periodic boundary conditions are presented. The scheme is shown to possess similar order-of-convergence error characteristics using a nonorthogonal coordinate compared to Cartesian coordinates for a nonlinear test of flow over an isolated mountain. A linear normal mode analysis shows that the discrete form of the Coriolis term provides stationary geostrophically balanced modes for the nonorthogonal coordinate and no unphysical computational modes are introduced. The scheme uses centered differences and averages, which are formally second-order accurate. An empirical test with a steady geostrophically balanced flow shows that the convergence rate of the truncation errors of the discrete operators is second order. The next step will be to adapt the scheme for use on the cubed sphere, which will involve modification at the lateral boundaries of the cube faces.

Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods – Part 1: Derivation and properties

Abstract. The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertia-gravity and Rossby waves, and a (quasi-) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Z-grid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids, and the S07 scheme is restricted to uniform square grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows for both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos, and others) and discrete exterior calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp, and others). Detailed results of the schemes applied to standard test cases are deferred to part 2 of this series of papers.

A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: comparison of hexagonal–icosahedral and cubed-sphere grids

Abstract. A new algorithm is presented for the solution of the shallow water equations on quasi-uniform spherical grids. It combines a mimetic finite volume spatial discretization with a Crank–Nicolson time discretization of fast waves and an accurate and conservative forward-in-time advection scheme for mass and potential vorticity (PV). The algorithm is implemented and tested on two families of grids: hexagonal–icosahedral Voronoi grids, and modified equiangular cubed-sphere grids. Results of a variety of tests are presented, including convergence of the discrete scalar Laplacian and Coriolis operators, advection, solid body rotation, flow over an isolated mountain, and a barotropically unstable jet. The results confirm a number of desirable properties for which the scheme was designed: exact mass conservation, very good available energy and potential enstrophy conservation, consistent mass, PV and tracer transport, and good preservation of balance including vanishing ∇ × ∇, steady geostrophic modes, and accurate PV advection. The scheme is stable for large wave Courant numbers and advective Courant numbers up to about 1. In the most idealized tests the overall accuracy of the scheme appears to be limited by the accuracy of the Coriolis and other mimetic spatial operators, particularly on the cubed-sphere grid. On the hexagonal grid there is no evidence for damaging effects of computational Rossby modes, despite attempts to force them explicitly.

A mass, energy, vorticity, and potential enstrophy conserving lateral fluid–land boundary scheme for the shallow water equations

A numerical scheme for treating fluid–land boundaries in inviscid shallow water flows is derived that conserves the domain-summed mass, energy, vorticity, and potential enstrophy in domains with arbitrarily shaped boundaries. The boundary scheme is derived from a previous scheme that conserves all four domain-summed quantities only in periodic domains without boundaries. It consists of a method for including land in the model along with evolution equations for the vorticity and extrapolation formulas for the depth at fluid–land boundaries. Proofs of mass, energy, vorticity, and potential enstrophy conservation are given. Numerical simulations are carried out demonstrating the conservation properties and accuracy of the boundary scheme for inviscid flows and comparing its performance with that of four alternative boundary schemes. The first of these alternatives extrapolates or finite-differences the velocity to obtain the vorticity at boundaries; the second enforces the free-slip boundary condition; the third enforces the super-slip condition; and the fourth enforces the no-slip condition. Comparisons of the conservation properties demonstrate that the new scheme is the only one of the five that conserves all four domain-summed quantities, and it is the only one that both prevents a spurious energy cascade to the smallest resolved scales and maintains the correct flow orientation with respect to an external forcing. Comparisons of the accuracy demonstrate that the new scheme generates vorticity fields that have smaller errors than those generated by any of the alternative schemes, and it generates depth and velocity fields that have errors about equal to those in the fields generated by the most accurate alternative scheme.

Nonextensive statistical mechanics for rotating quasi-two-dimensional turbulence

We have conducted experiments on an asymmetrically forced quasi-two-dimensional turbulent flow in a rapidly rotating annulus. Assuming conservation of potential enstrophy and energy, we maximize a nonextensive entropy function to obtain the azimuthally averaged vorticity as a function of radial position. The predicted vorticity profile is in good accord with the observations. A nonextensive formalism is appropriate because long-range correlations between small-scale vortices give rise to large coherent structures in the turbulence. We also derive probability distribution functions for the vorticity from both extensive and nonextensive entropies, and we find that the prediction from nonextensive theory is in better accord with experiment, especially in the tails of the distribution function. The nonextensive parameter q has the value 1.9±0.2.

Nonlinear Saturation of Topographically Forced Rossby Waves in a Barotropic Model

A quasigeostrophic barotropic model is used to examine the nonlinear saturation of forced Rossby waves and the role of wave‐wave interactions in limiting the wave growth. A simple mechanism, based on wave interference, is used to produce strong transient eddy growth and an analytical linear solution for the flow evolution is used as a starting point. Given the rigid upper bound on wave growth, set by the potential enstrophy conservation principle, the linear solution is bound to break down at high forcing amplitudes. An analytical quasi-linear solution, which guarantees potential enstrophy conservation, is formulated and its domain of validity is examined with a numerical nonlinear model. The nonlinear flow evolution is shown to bear strong similarity to the analytical quasi-linear solution and wave‐mean flow interactions are found to be always sufficient to limit wave growth. The saturation of the forced disturbances is shown to come through the deceleration of the mean flow and the modification of the topographic vorticity forcing. Overall, wave‐wave interactions prove not to be important in the saturation process in the examples considered. While the authors consider the implications of this result for the observationally more relevant case of vertically propagating Rossby waves, explicit calculations are clearly called for.

Geostrophic Adjustment and Inverse Cascades in Rotating Stratified Turbulence

Abstract Rotating stratified turbulence is examined both numerically and analytically, guided by energy and potential enstrophy conservation as well as resonant interaction theory, in order to investigate the cascade properties of rotational and wave modes at Froude numbers of order one or below, over a range of Rossby numbers. As Ro → 0, rotational modes are only weakly coupled to wave modes, and there are only weak rotational wave energy exchanges when initial conditions are random. A catalytic interaction involving two waves and a rotational mode, leaving the rotational mode unchanged, then provides the mechanism for geostrophic adjustment via a downscale cascade of wave energy. When simulations are initially balanced, gravity modes act to damp large-scale rotational modes through a transfer into intermediate-scale gravity modes and a subsequent downscale wave cascade involving the catalytic interaction. At larger Ro transfer from rotational to wave modes is important at any Froude number, and geostrop...

On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields. Part II: Comparison of Numerical Model Solutions in Doubly Periodic Domains

As part of a program to improve understanding of the dynamics of the complicated, vigorous eddy and jet flow fields recently observed over the continental shelf and slope, we investigate the potential of intermediate models for use in both process and data assimilation studies of these flows. Intermediate models incorporate physics simpler than that contained in the full primitive equations yet more complete than in the quasi-geostrophic equations, and are capable of representing subinertial flows over O(1) bottom topographic variations and/or with O(1) isopycnal slopes. In addition, intermediate models dynamically filter out high-frequency gravity- inertial motions leading, potentially, to higher computational efficiency and well-posed limited area forecast/hindcast models. Initial studies focus on single layer flows on an f-plane with a free surface, governed by the shallow-water equations. In Part I, various intermediate models are formulated and their accuracy assessed by comparing some exact nonlinear analytical solutions that exist for the shallow-water equations with corresponding analytical solutions of the intermediate models. Here in Part II, an extensive set of numerical finite-difference solutions to initial-value problems in doubly periodic domains (to isolate model differences from the influence of boundary condition implementation on solid walls) is used to determine the accuracy of various intermediate models by comparing their predictions with those of a shallow-water equation model that uses a potential enstrophy and energy conserving numerical scheme (SWE). Intermediate model results are also contrasted with those from a quasi-geostrophic (QG) model. The intermediate models considered are based on the geostrophic momentum (GM) approximation, the derivation of Salmon utilizing Hamilton's principle (HP), a geostrophic vorticity (GV) approximation, a combination of the quasi-geostrophic momentum and full continuity equations (IM), the linear balance equations (LBE), the balance equations (BE), the related balance-type (HBE, BEM, NBE) and modified linear balance equations (LQBE), and on Lynch's slow equations in their original form (SE) and in a modified form (MSE). In addition, a semi-implicit version (SEMI) of the shallow-water equations, which numerically filters high-frequency motions, is included in the study. The basic initial-value problem used to test the various intermediate models involves sinusoidal flow over a symmetric Gaussian-shaped bottom topographic feature. Comparisons are made for a range of the relevant dimensionless model parameters including the strength of the flow (as measured by the Rossby number), the square of the ratio of a characteristic horizontal length scale to the Rossby radius of deformation, and the height of the topographic feature. A second initial-value problem involves the evolution of a rotating elliptical vortex over both flat and variable bottom topography. Results show that in cases with low local Rossby number flow, but with large topographic height, most of the intermediate models are substantially more accurate than QG. Even for flows with O(1) local Rossby numbers, some of the intermediate models continue to give excellent results. Specifically, BE, BEM and SEMI consistently do the best in the comparisons. Although the relative ordering of the remaining models is somewhat parameter-dependent, in general the next most accurate models are MSE and LQBE followed by HP and NBE. These are followed in quality by GV and GM while the remaining models, HBE, IM and LBE, perform least accurately for the range of parameters studied. Generally, intermediate models that have an integral invariant corresponding to conservation of potential enstrophy do better than those without, with the best results coming from models which have this property and a nonlinear balance equation.

Generalized Energy and Potential Enstrophy Conserving Finite Difference Schemes for the Shallow Water Equations

The conditions under which finite difference schemes for the shallow water equations can conserve both total energy and potential enstrophy are considered. A method of deriving such schemes using operator formalism is developed. Several such schemes are derived for the A-, B- and C-grids. The derived schemes include second-order schemes and pseudo-fourth order schemes. The simplest B-grid pseudo-fourth-order schemes are presented.

Implementation of APosterioriMethods for Enforcing Conservation of Potential Enstrophy and Mass in Discretized Shallow-Water Equations Models

Abstract A Sasaki variational approach is for the first time applied to enforce a posteriori conservation of potential enstrophy and total mass in long-term integrations of two ADI finite-difference approximations of the nonlinear shallow-water equations on a limited-area domain. The performance and accuracy of the variational approach is compared with that of a modified Bayliss-Isaacson a posteriori technique, also designed to enforce conservation of potential enstrophy and total mass, and with that of a periodic application of a two-dimensional high-order Shapiro filter. While both the variational and the Bayliss-Isaacson a posteriori techniques yielded very satisfactory results after 20 days of numerical integration with regard to conservation of the integral constraints of the shallow-water equations and the accuracy of the solution, the high-order filtering approach performed in a less satisfactory way. This is attributed to the effects of the boundary conditions in the limited-area shallow-water equ...

Compressible Model Flows on the Sphere

Abstract The effect of potential enstrophy conservation in finite-difference model flows on the stability of large- scale Rossby waves is investigated. It is also shown that cylindrical coordinates can be used successfully in global numerical experiments. Coherent boundary conditions at the poles are derived and tested on Rossby waves allowing cross-polar flow.