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)

Consider the motion of a (multi-component) fluid on a rotating spheroid under
the influence of gravity and radiation. This is the fundamental subject of
inquiry for geophysical fluid dynamics, covering fields such as weather
prediction, climate dynamics, and planetary atmospheres. Central to our
current understanding of these subjects is the use of numerical models to
solve the otherwise intractable equations (such as the fully compressible
Euler equations) that result. As a first step towards developing a numerical
model for simulating geophysical fluid dynamics, schemes are usually
developed for the rotating shallow water equations (RSWs). The RSWs provide a
useful analogue of the fully compressible Euler equations since they have
similar conservation laws, many of the same types of waves, and a similar
(quasi-) balanced state. It is desirable that a numerical model possesses at
least some these same properties (see Fig.

In fact, there exists some evidence

A pioneering scheme developed over 30 years ago possesses many of these
properties (including both total energy and potential enstrophy
conservation): the 1981 Arakawa and Lamb scheme (AL81,

Recently, there has been an effort to extend the AL81 scheme to more general
grids, using tools from discrete exterior calculus (commonly referred to as
the TRiSK scheme;

Rather than using finite differences, it is also possible to extend the AL81
scheme to arbitrary grids by using compatible finite elements (see

As an alternative to the AL81 scheme, which preserves many of its valuable
mimetic properties, but has good wave dispersion properties independent of
Rossby radius,

This work combines the discrete exterior calculus approach from

The remainder of this paper is structured as follows: Sect.

A diagram of some desirable model properties for the shallow water equations, organized thematically into groups. Similar considerations apply for the Euler, hydrostatic primitive, and other equation sets used in atmospheric models. There is vigorous discussion in the literature and between model designers about the importance of various properties for different applications (such as weather forecasting or long-term climate prediction). The schemes presented here satisfy all of these properties, with the exception of accuracy. There are additional desirable model properties, such as consistent physics–dynamics coupling, compatible and accurate tracer advection, and tractable treatment of acoustic waves that are not presented.

The RSWs for both planar and spherical domains are presented below in several forms: the vector-invariant formulation, the vorticity–divergence formulation, the symplectic Hamiltonian formulation based on the vector-invariant form and both Poisson bracket, and Nambu bracket formulations based on the vorticity–divergence formulations. Although all of these formulations are equivalent in the continuous case, they lead to very different discretizations.

The mass continuity equation for the RSWs is expressed in vector-invariant
form as

As discussed in

Since the rotating shallow water equations form a (non-canonical) Hamiltonian system, we know from Noether's theorem and other considerations (such as the singular nature of the symplectic operator) that there are at least two categories of conserved quantities: Hamiltonian and Casimirs.

The first is simply the Hamiltonian itself. In this case, the Hamiltonian is
the total energy of the system. Conservation of the Hamiltonian arises due to
the skew-symmetric nature of the Poisson bracket. In particular, using
Eq. (4)
the evolution of

The second category of conserved quantities consists of Casimir invariants.
Since the rotating shallow water equations are a non-canonical Hamiltonian
system, the Poisson bracket

By taking the divergence (

As shown in

The use of the Poisson (and Nambu) bracket formulation of the shallow water
equations is motivated by the intimate connection between these formulations
and the conserved quantities. As is well known, the conservation of energy

Fortunately, there is a closely related formulation of the shallow water
equations in terms of Nambu brackets (see

These brackets are useful because they are triply anti-symmetric (which
ensures the conservation of

Following

Specifically, the brackets in Eqs. (

In fact, by making alternative choices for

As discussed in

As is well known, the linearized version of a Hamiltonian system about a
steady state can be found by evaluating the brackets at that state and using
the quadratic approximation to the associated pseudo-energy as the
Hamiltonian

A subset of discrete variables and their staggering on the
computational grid for the C-grid scheme. A subscript

Summary of required operator properties for obtaining the desirable
mimetic properties along with total energy and potential enstrophy
conservation. A example of operators that satisfy these properties can be
found in Appendix

This scheme has many important properties, including the following:

Mass and potential vorticity conservation: both mass

No spurious vorticity production: by construction,

Linear stability (pressure gradient force and Coriolis force conserve energy):
this is due to the fact that

Steady geostrophic modes: by construction,

PV Compatibility: again by construction

Other conservation properties: see below for a discussion on total energy and potential enstrophy conservation.

Table

Following S04, total energy will be conserved for any choice of

Following S04, potential enstrophy is a Casimir and therefore will be
conserved when

In the case of a uniform square grid, the C-scheme grid above reduces to the
well-known Arakawa and Lamb 1981 total energy and potential enstrophy scheme
(modified to prognose

Since this is an extension of Arakawa and Lamb 1981 scheme, it seems
extremely likely that the proposed scheme will suffer from the Hollingsworth
instability, especially if applied in a height coordinate framework using a
Lorenz staggering in the vertical (as discussed in

The principal novelty of the new C-grid scheme is the specification of a

Loosely following S04,

A diagram of the stencil of

It remains to determine the coefficients

Following S04, in order for

From Eq. (

A diagram of the stencil

Specifically, for each grid cell

The solution procedure outlined above gives a large matrix system

A diagram of the stencil EVE

Instead, following

The astute reader will note that nothing has been said yet about enforcing PV
compatibility (

Unlike the C-grid scheme, the

The functional derivative of a general functional

Following

Loosely following S07, the

The mixed bracket is trickier since it contains an apparent singularity

Since the

The Hamiltonian

By taking variations of

After a lot of algebra, these can be grouped (half of each term involving

A natural definition of the discrete potential enstrophy is

By plugging these back into the

As noted before, the mimetic and conservation properties of the discrete
scheme are completely independent of the choice of discrete Hamiltonian

By setting

The Laplacian and flux-divergence operators (which come from the mixed
bracket) can be written as

The Jacobian operators (which come from the Jacobian brackets) can be written
as

Note that on a polygonal grid with a purely triangular dual (including the
important case of an icosahedral grid),

Under the assumption of linear variations around a state of rest (

For the cases of a uniform planar square grid and a general orthogonal planar
polygonal grid with triangular dual, the general discretization scheme
presented above reduces to the schemes given in S07. However, this
discretization scheme is more general, and it also makes specific choices for
the total energy

The discrete scheme as outlined above possesses the following (among others) key
properties:

Linear stability (Coriolis and pressure gradient forces conserve energy):
provided that

No spurious vorticity production: by construction, the pressure gradient term does not produce spurious vorticity since the curl is taken in the continuous system, prior to discretization.

Conservation: by construction, this scheme conserves mass, potential vorticity, total energy, and potential enstrophy in both a local (flux-form) sense and global (integral) sense.

PV compatibility and consistency: by inspection, the mass-weighted potential
vorticity equation is a flux-form equation that ensures both local and global conservation
of mass-weighted potential vorticity. In addition, an initially uniform potential
vorticity field will remain uniform. This rests on the fact that

Steady geostrophic modes: since the same divergence

Linear properties (dispersion relations, computational modes): as expected,
the scheme possesses the same linear mode properties on uniform planar grids as
those presented in

Accuracy: unfortunately, as shown in

To test the utility of the C- and

As a short preview of the more detailed results in

A plot of the absolute vorticity from the

This paper presents an extension of AL81 to arbitrary non-orthogonal
(spherical) polygonal grids in a manner that preserves almost all of the
desirable properties of that scheme (including both total energy and
potential enstrophy conservation) through a new

This work has also presented an extension of the total energy and potential
enstrophy conserving

A detailed comparison of the two schemes, including an analysis of the
accuracy of the operators used and results from a variety of test cases, can
be found in second part of this series

The schemes described in this manuscript have been implemented in a
Python/Fortran mixed language code, and are freely available at

The schemes described above are designed to work on arbitrary (spherical)
polygonal grids along with an associated dual grid. In the case of the
C- grid scheme, the grid can be either orthogonal or non-orthogonal, while
the

Consider a (primal) conformal grid constructed of polygons (or spherical
polygons). A dual grid is constructed such that there is a unique one to one
relationship between elements of the primal grid and element of the dual
grid: primal grid cells are associated with dual grid vertices, primal grid
edges are associated with dual grid edges, and primal grid vertices are
associated with dual grid cells. This grid configuration covers the majority
of grids that are used in current and upcoming atmospheric dynamical cores,
including cubed sphere and icosahedral grids (both hexagonal–pentagonal and
triangular variants). Once the dual grid vertices have been placed, there are
several important geometric quantities that are needed in order to construct
the discrete operators (shown graphically in Fig.

The geometric quantities on a planar grid. Primal grid edge lengths
are denoted as “de”, dual grid edge lengths are denoted as “le”, the area
associated with an edge by

Following

In order to close the C-grid scheme presented in Sect.

For the

List of discrete C- grid variables and their diagnostic equations.

Table

Table

List of discrete

The authors declare that they have no conflict of interest.

The authors would like to thank Colin Cotter and an anonymous reviewer for their helpful and thorough reviews, which greatly improved the clarity, presentation, and content of this manuscript. The authors would also like to thank Pedro Peixoto for his comments and suggestions on an earlier draft of this manuscript. This work has been supported by the National Science Foundation Science and Technology Center for Multi-Scale Modeling of Atmospheric Processes, managed by Colorado State University under cooperative agreement no. ATM-0425247. Christopher Eldred was also supported by the Department of Energy under grant DE-FG02-97ER25308 (as part of the DOE Computational Science Graduate Fellowship administered by the Krell Institute). Edited by: S. Marras Reviewed by: C. Cotter and one anonymous referee