A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids
Introduction
In a previous and related effort, Thuburn et al. [35], hereafter T09, consider the formulation of a discrete method for robust simulation of geostrophic adjustment. When the geostrophic adjustment problem is considered with a constant Coriolis parameter, the continuous linear system supports a stationary mode in geostrophic balance that is characterized by purely rotational flow; divergence is everywhere identical to zero for all time and the time tendency of vorticity is also zero for all time. In the limit of geostrophic balance, the steady-state flow is along lines of constant pressure with an exact cross-flow balance between the horizontal pressure gradient force and the Coriolis force.
The discrete method considered in T09 is a finite-volume approach based on a C-grid staggering. This approach retains prognostic equations for mass at the center of finite-volume cells and for the normal component of velocity at the faces (or edges in 2D) of finite-volume cells. While C-grid methods lead to an excellent representation of gravity waves in relation to other finite-volume grid staggerings [4], the method’s ability to reproduce geostrophic balance is sometimes problematic [20]. The root cause of this problem is in the discrete formulation of the Coriolis force; the Coriolis force is proportional to the tangential component of velocity, which is not known. Since the tangential velocities are not known, these velocities must be reconstructed based on neighboring values of normal velocity. The primary result of T09 is the derivation of a robust method for the reconstruction of the tangential velocities on arbitrarily-structured C-grids such that the discrete system allows exact geostrophic balance to be maintained when appropriate.
T09 limited the scope of analysis to the linearized shallow-water equations. The primary purpose of this present effort is to extend the previous analysis to the nonlinear shallow-water equations with focus on the analysis of potential vorticity dynamics and system energetics within the discrete system. Even though the analysis of T09 is carried out for the linearized shallow-water equations with a constant Coriolis parameter, their results are essential for the extension to the full nonlinear shallow-water equations solved on the sphere.
With the growing interest in the use of unstructured meshes for the simulation of ocean and atmosphere flows, a significant amount of effort has been devoted to the treatment of the Coriolis force in C-grid methods. While the reconstruction of the tangential velocity required for computing acceleration due to the Coriolis force is straightforward on square meshes, the task becomes significantly more difficult on other meshes, such as Voronoi diagrams or Delaunay triangulations. While in the continuous system the Coriolis force is orthogonal to the velocity vector and, thus, energetically-neutral, this property is not trivial to satisfy for a C-grid staggering on arbitrarily-structured meshes. Since the Coriolis force can serve as an infinite source of kinetic energy in discrete systems where it is not energetically-neutral, efforts in reconstructing the discrete Coriolis force have sometimes focused exclusively on system energetics (e.g. [30], [12], [15]). Significantly fewer efforts have recognized that the Coriolis force is not only central to the energetics of the discrete system, but also plays a fundamental role in the discrete vorticity budget (e.g. [23], [6]). While conservation of energy and absolute vorticity are critically important aspects of discrete models that are intended to simulate geophysical flows, these properties are not sufficient for the robust simulation of geophysical flows. The critical aspect of the discrete system that has to be included is that of potential vorticity dynamics, as was address by [26], [2]. The primary purpose of this contribution is to derive a discrete method on arbitrarily-structured C-grids that allows for the conservation of total energy and a robust simulation of potential vorticity dynamics. As shown below, a single term in the momentum equation, the nonlinear Coriolis force, plays a central role in both the energy and potential vorticity dynamics. As a result, energy and potential vorticity cannot be adequately treated in isolation, but require a unified approach.
The results of T09 are particularly relevant to the construction of the discrete potential vorticity (PV) equation. Potential vorticity has proven to be a key quantity in the theoretical and observational interpretation of atmosphere and ocean dynamics (e.g. [7], [18]). As discussed in Hoskins et al. [14], the broad utility of PV stems both from its ability to inform local processes by acting as a Lagrangian tracer and from its ability to inform large-scale, balance-dominated processes through the principle of invertibility. Given the fundamental importance of PV in geophysical flows, numerical models are sometimes constructed to faithfully represent some aspects of the PV dynamics within the discrete system (e.g. [2], [26], [33], [16]). We carry forward that idea here, but have to overcome two significant hurdles to be successful.
The first hurdle is one of compatibility with the momentum equation. In a C-grid method, the discrete PV equation is obtained by applying the discrete curl operator to the velocity equation, then combining that result with a discrete equation for layer thickness. Compatibility with the momentum equation is the ability to derive a flux-form expression of the PV equation that is a direct analog of its continuous counterpart in the sense that the tendency of thickness-weighted PV is due solely to one term and that term is the divergence of a PV flux. Compatibility in the sense defined here is sufficient to guarantee local and global conservation since the sole forcing term in the discrete PV equation is the divergence of a flux.
The second hurdle is one of consistency with the Lagrangian behavior of PV. In the frictionless and adiabatic limit, PV evolves aswhere q is PV and is the material derivative. For the numerical scheme to be considered consistent with respect to the Lagrangian behavior of PV, we require the discrete system to possess an analog to (1) in the sense that the discrete PV field evolves with the same material derivative as the underlying continuity equation from which the PV is derived.
In a finite-volume formulation, a consistent representation of the PV Lagrangian property requires that the flux-form PV equation be consistent with an underlying flux-form thickness equation such that if is uniform (i.e. independent of ) at some time t, then is uniform for all time. We consider consistency with the Lagrangian properties of PV to include the preservation of a constant PV field for all time. It is important to note that within a finite-volume formulation, compatibility with the momentum equation is a necessary prerequisite for consistency with the Lagrangian property of PV.
Finite-volume methods designed for simulation of geophysical flows are often constructed to mimic integral constraints found in the continuous system, such as conservation of thickness-weighted PV in the shallow-water system that follows from compatibility. However, conservation of PV in the discrete system is not sufficient to guarantee that the discrete system mimics the Lagrangian property of PV defined in (1). In particular, conserving PV in the shallow-water system does not guarantee consistency. The lack of control that conservation exerts on the Lagrangian property of PV is frequently overlooked, e.g. [6], and results in less robust simulations.
The aim of this contribution is to construct the discrete principles of PV compatibility and consistency, along with the principle of total energy conservation, within the very general framework of arbitrarily-structured meshes. These discrete principles have been demonstrated before, but only for regular grids, such as square meshes (e.g. [26], [2]). The results produced below hold for a broad class of meshes that utilize the C-grid staggering, including arbitrary Voronoi diagrams [11], arbitrary Delaunay triangulations [6], stretched-poles grids [29] and conformally-mapped cubed-sphere meshes [1]. The requirement for the method derived below to hold is that the mesh be locally orthogonal in the sense that the edges that define mass cells and the edges that define vorticity cells are perpendicular at their intersection. The findings of Sadourny [26] on a square mesh are recovered as a special case of this general method.
While the method holds for a broad class of meshes, the example simulations used to confirm the analytic results are constructed using the Voronoi diagram and Delaunay triangulations. Voronoi diagrams and Delaunay triangulation are deeply connected. A specification of either uniquely determines the other. For this reason, the two meshes are often referred to as duals. The strong relationship proves extremely useful when developing staggered-grid methods that require extensive use of both a prime mesh and a dual mesh. In addition, the Voronoi–Delaunay combination along with the C-grid staggering combine to form a discrete analog of the Helmoltz decomposition [21]. Thus, the vorticity and divergence fields contain equivalent information to the velocity field in the discrete system, as they do in the continuous system. This relationship proves useful when attempting to connect the evolution of velocity to the evolution of vorticity.
The paper is presented as follows: Section 2 discusses principles of PV dynamics and energy conservation within the context of the continuous system. Special attention in Section 2 is given to the relationship between the PV flux and the nonlinear Coriolis force. Section 3 develops the discrete system such that the principles of PV consistency, compatibility and conservation and total energy conservation hold. Section 4 presents numerical results intended primarily to confirm the analytical findings. Section 5 concludes with a discussion of possible extensions of the proposed method and the types of problems that become tractable using these techniques.
Section snippets
Nonlinear shallow-water equations
The nonlinear shallow-water equations can be expressed aswhere prognostic equations are written for the evolution of the fluid thickness, h, and the fluid vector velocity, . The unit vector, , points in the local vertical direction. We consider the velocity field to exist in and the Lagrange multiplier, , is formally included following [8] to constrain to the surface of the sphere. We assume throughout that . The three parameters in the
Notation
The discrete system requires the definition of seven elements. These seven elements are composed of two types of cells, two types of lines, and three types of points. These elements are depicted in Fig. 1 and defined in Table 1. Let the space (either the plane or the surface the sphere) be tessellated by two meshes, a primal mesh composed of cells and a dual mesh composed of cells. Each corner of a primal mesh cell is uniquely associated with the “center” of a dual mesh cell and vice
Definition of error norms
In order to facilitate comparison to previously published error norms of shallow-water test case simulations, we specify the and norms aswhereThe function is the numerical solution defined at the positions on the numerical mesh. The index j can represent the cells on the primal or dual mesh. The function is the reference solution that has been calculated at or interpolated to the same
Discussion
The primary complication with the C-grid method arises during the consideration of the discrete system’s PV dynamics. By construction, the C-grid method staggers the mass and vorticity fields, with mass defined at the centers of the primal mesh and vorticity defined at the centers of the dual mesh. Since shallow-water PV is defined as the ratio of vorticity to fluid thickness, it is not immediately obvious how, or even where, to define PV when using a C-grid staggering.
By extending the results
Acknowledgments
The careful review provided by three anonymous reviewers led to significant improvements to this contribution. This work was supported by the DOE Office of Science’s Climate Change Prediction Program DOE 07SCPF152. NCAR is supported by the National Science Foundation. We would like to thank Lili Ju for generating the meshes used for the simulations presented in Section 4.
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