Standard vector calculus formulas of Cartesian three space are projected onto the surface of a sphere. This produces symmetric equations with three nonindependent horizontal velocity components. Each orthogonal axis has a velocity component that rotates around its axis (eastward velocity rotates around the north–south axis) and a specific angular momentum component that is the product of the velocity component multiplied by the cosine of axis' latitude. Angular momentum components align with the fixed axes and simplify several formulas, whereas the rotating velocity components are not orthogonal and vary with location. Three symmetric coordinates allow vector resolution and calculus operations continuously over the whole spherical surface, which is not possible with only two coordinates. The symmetric equations are applied to one-layer shallow water models on cubed-sphere and icosahedral grids, the latter being computationally simple and applicable to an ocean domain. Model results are presented for three different initial conditions and five different resolutions.

According to the “hairy ball theorem” of Poincaré (proved by Brouwer), every
continuous horizontal vector field on the surface of a sphere must have a

Many numerical schemes for solving the fluid dynamic equations on the surface
of a sphere use two independent horizontal velocity components aligned with
underlying coordinates. On a lat–long grid, polar singularities occur as well
as other problems discussed in the introduction of

To use the lat–long grid, but to avoid its polar deficiencies, the HYbrid
Coordinate Ocean Model (HYCOM)

Recent research on one-layer models has been directed at icosahedral grids:

If two-component velocity is the prognostic transport variable that is
advected in flux form, then spatial derivatives of vector components will
cause discontinuities to occur. This is a principal reason why researchers
developed forms of the shallow water equations wherein scalar quantities such
as potential vorticity, specific kinetic energy, and divergence are
continuous everywhere. Computations are performed on local spherical
coordinates or on the local tangent plane after which the horizontal velocity
components are resurrected or time integrated by manipulating spatial
derivatives of scalar quantities; spatial derivatives of vector components
are not needed. Such forms include vector-invariant

The approach here uses three symmetric coordinates on the surface of a sphere to represent two-dimensional flow. When one coordinate reaches a singularity, it is ignored and the other two coordinates become perpendicular on the spherical surface. Symmetric equations are used to develop one-layer shallow water equation models: one for a gnomonic cubed-sphere grid (CSK), one for an icosahedral B grid with momentum defined at the primary grid cell corners (IB), and one for an icosahedral grid with a Voronoi tessellation (IK).

Several formulas of the symmetric equations are simplified by using relative
specific angular momentum on the unit sphere,

The shallow water equations based on

Computers require grid representations of differential equations; this causes
numerical errors that relate to grid imprinting, mass variations, time
integration, etc. Grid imprinting is easily recognized when integrating the
solid body rotation Test Case 2 of

Mass variations cause advection errors in numerical models and cause
grid-matched alternating patterns. Mass is usually conserved by programming
advection to use flux form, but when mass is needed at different locations,
it is specific mass or concentration that is interpolated. Tracers that
follow mass advection include linear momentum and velocity, angular momentum
and specific angular momentum, kinetic energy and specific kinetic energy, or
absolute vorticity and potential vorticity.

If tracer concentration is a linear function of mass over several grid cells
in one dimension, then “the linear upstream scheme” of

To be applied to the Earth, models should be tested with mass variations
comparable to those on Earth. In some mountainous regions, surface pressure
gradients and mass variations increase with finer resolution and so do their
errors. Test Case 5 of

Section 2 explains symmetric mathematics, including coordinates and variables, symmetric calculus operators, and the differential solution to the shallow water equations in terms of symmetric coordinates. Section 3 presents the discrete implementation of alignment, the pressure gradient force, advection, the Coriolis force, and other aspects of icosahedral models including grid arrangements and time steps. Section 4 applies three test cases to lat–long, cubed-sphere, and icosahedral one-layer models with various resolutions. Section 5 contains discussion and conclusions.

Unlike computer languages, division has lower precedence than does
multiplication in this paper. Vector quantities are indicated by bold capital
letters; when displayed by three coordinates, e.g.,

A sphere of radius 1 is centered at the origin in three-dimensional Cartesian
space with axis unit vectors

Cartesian coordinates on the surface of the
unit sphere are labeled

Three horizontal velocity components,

Relative specific angular momentum on the unit sphere,

Spherical angular rotation coordinates, measured in radians, that rotate
around the

The new symmetric equations to be presented here are applicable to many grid
arrangements; one is the gnomonic cubed-sphere grid. A symmetric tessellation
of the surface of a cube [

Upper diagram shows arrangement of velocity components on Face 1 of
cubed-sphere grid centered around the North Pole. Lower diagram shows arrangement
of velocity components at the intersection of Faces 1, 2, and 3. Although

Two new shallow water equation models were developed on icosahedral grids.
The centers of primary grid cells are the vertices of a triangular lattice
covering the sphere; the cells themselves are pentagons and irregular hexagons.
The B-grid icosahedral model, labeled IB, is discussed in detail in this paper.
Its momentum cells (described as the dual mesh by

Four triangles of an icosahedron are surrounded by thick, bold lines. Vertices of the bold triangles are centers of incomplete pentagonal primary cells. Edges of primary cells, for grid level 2, are indicated by thin solid lines. Dotted lines, between primary cell centers, indicate edges of triangular momentum cells. Primary cell corners are momentum cell centers.

The symmetric del operator on the surface of a sphere

The gradient of a scalar

The divergence of a horizontal vector

Noting that

The curl of a horizontal vector

The upward vertical component of relative vorticity is

If

Many of the new symmetric forms have no varying quantities outside their
derivatives and can be integrated using Green's theorem. Proofs of several of
the equivalences used above are available at

The differential form for conservation of mass, using Eq. (2.18), is applied to mass per unit
area:

The three-component advective form for specific angular momentum is

Replacing

Symmetric versions of the shallow water equations using vorticity and
divergence or vector-invariant form are shown at

Alignment of the three momentum components at

Given unaligned velocity components

Performing this analysis with specific angular momentum components yields

The benefit of using three aligned components instead of two for advection on
a particular cubed-sphere grid is shown at

Change of velocity by the pressure gradient force is proportional to the
gradient of the field top geopotential

Application of the pressure gradient force to velocity averaged over an arc
usually involves interpolating

Acceleration of velocity averaged over a primary cell of a cubed-sphere model
starts by knowing

Relationships like Eqs. (3.10) or (3.11) apply to any great circle arc, not
just cubed-sphere edges. To check this, any arc from

Renaming the vertices of a spherical polygon with

Appendix B shows that if

Change in primary cell mass by advection, in flux form, is derived from Green's theorem and Eq. (2.27).

The change in grid cell mean relative angular momentum by advection, in flux
form, is equal to the summation of the angular momentum fluxes

For stability purposes, if

Primary and momentum cells are not identical for model IB; triangular
momentum cells are centered at the corners of primary cells. The solution for
momentum cell mass presented here is different from that of

C points are primary cell centers or triangular momentum cell corners.
A and B points are primary cell corners or momentum cell centers.
Quadrilaterals are the intersection area between primary and momentum
cells. Arc intersection D points are overwritten by

For computer implementations of symmetric equations created so far, momentum
components have not been defined on staggered locations; all three components
reside at the same locations. For each velocity component, the other two
components determine the velocity that is perpendicular to the first
component. Thus,

Starting with 12 vertices and 20 triangles of the icosahedron, new vertices
for the raw grid are formed from the center points of triangular edges. After

Properties of the raw, tweaked, and centroid grids; numerical
columns are for grid levels 4 through 8. ArcA is the arc between adjacent primary
cell corners or momentum cell centers. ArcC is the arc between nearby primary
cell centers or momentum cell corners. Point D is intersection of ArcA and
ArcC. The ideal number is 1 for first three properties and 0 for last two
properties. Radius is the square root of cell area divided by

Starting from the raw grid,

Table 1 shows various properties of the three grids for grid levels 4 through
8. A significant property is “smallest arc length from primary cell corner to
D divided by half of ArcA length”; ArcA is between two primary cell corners
and D is the intersection of ArcA and the arc between the primary centers.
The raw grid value in Table 1 stays at 81%, while the tweaked grid converges
to unity with increasing resolution. For this reason, IKT is superior to IKR.
IB performs edge computations using only values of the two primary cells,
ignoring the fact that D is not the center of ArcA. IK, CSK, and the
icosahedral model of

Considering the most extreme momentum cells in Table 1, the raw grid
triangular momentum cells of model IB are more equilateral than are those of
the tweaked grid. Momentum cells are more important for model IB, and
consequently IBR is as good as IBT. An expanded version of Table 1 with
additional parameters is available at

The time scheme is leap-frog initialized every 8 to 10 time steps by
forward–backward steps. Alignment of

For model IB, the unadjusted

Model IK uses

The following two lat–long climate models, reduced to one layer for the
shallow water equations, were applied to the test cases below. Arakawa's
second-order B-grid (velocity components defined at primary cell corners)
lat–long model with the NASA Goddard Institute for Space Studies (GISS) ModelE's pole modifications and filters on mass and
velocity

Symmetric equation models IBR, IBT, IBC, IKT, and CSK are represented; IKR
and IKC are not, being worse than IKT. Each model was tested with different
initial conditions and different horizontal resolutions that approximate

Abbreviations, number of primary cells at

Using the parameters of Test Case 2 of

Mass field error

Figure 5 shows the

In general, the mass field error of IBR is less than that of IBT, often
exceeding more than 10 % for the coarser resolutions, but closer to 1 %
for

Mass field errors

Horizontal plots of

For cubed-sphere models that initialize from repetitive wave 4 initial
conditions, each of the four wavelengths lies above the same grid arrangement
and copies remain identical when the wave's axis passes through the center of
a cube face as is the case for CSK. The same statement applies to icosahedral
models with wave 5 initial conditions when the axis passes through a vertex
of the icosahedron as is the case for IB and IK. The initial conditions used
here are Rossby–Haurwitz wave 3, 3 being relatively prime to both 4 and 5.
With wave 3, separate wavelengths diverge among themselves for models CSK,
IB, and IK, and there is more variety in the errors that occur. Plots (not
displayed) show that CSK for Rossby–Haurwitz wave 4 and IB or IK for
Rossby–Haurwitz wave 5 maintain their wavelength shape, but other situations
do not. Some results of CSK and IBR for Rossby–Haurwitz waves 3, 4, and 5 are
shown at

Figure 6 shows horizontal plots of

Relative change (%) of specific kinetic energy for wave numbers 0 and 3 and total energy for various resolutions and models IBR, IBT, IBC, IKT, CSK, LLB, LLC, and National Center for Atmospheric Research spectral transform model at T42 resolution (Hack and Jakob, 1992) after 40 days of integration. Initial conditions are Rossby–Haurwitz wave 3. “Diverge” means the simulation diverged.

Table 3 shows the change in spectral specific kinetic energy for wave numbers
0 and 3 and the change in total energy for models IBR, IBT, IBC, IKT, CSK,
LLB, LLC, and National Center for Atmospheric Research spectral transform
model (STSWM) at T42 resolution

Except for LLC, which maintains its pattern well for 100 days, the other

The initial velocity is eastward, 50 m s

Although the lat–long models use Fourier polar filters on east–west mass flux and pressure gradient force and other filters on prognostic variables, the symmetric equation models do not. The stability of IB, IK, and CSK is maintained by using the proper amount of linear upstream advection of momentum.

For the videos discussed later, the leap-frog time step of each model is
interrupted every eight time steps, and

Table 5 shows the change in kinetic energy after 50 days of integration for
different models and resolutions. For resolutions

For each model and approximate horizontal resolution, the largest dynamical time step, half of leap-frog time step, in seconds for which the model survives for 50 days without major instabilities, or after what day the model diverged (in parentheses), for solid body rotation initial conditions with Earth's bottom topography.

Percentage change in kinetic energy for different models and resolutions after 50 days of integration from solid body rotation initial conditions with Earth's bottom topography. “Diverge” means the simulation diverged.

Figure 7 shows horizontal plots of

Horizontal plots of

Videos of 50 simulated days for models IBR, IKT, CSK, LLB, and LLC are
displayable of limited quality on YouTube for all resolutions
(

This paper presents symmetric calculus operators on the surface of a sphere
that are projected from three-dimensional Cartesian formulas. Symmetric
equations are simplified by using specific angular momentum on the unit
sphere,

Summarizing the results from the test cases, no one model is clearly superior
in all tests for all resolutions and times of integration.

The SBRZ test is the most difficult but the most realistic test. Given that
these one-layer models will be the basis for multi-layer climate models with
realistic topography, IB models must be considered the best overall.
Parallelogram-shaped grid cells in CSK with non-perpendicular grid line edges
lasting over large swaths of the globe cause a systematic error in numerical
flow, most noticeably evident in the RH3 test case.

The symmetric equation models presented here use the smallest sensible grid
cell stencil needed for a computation. Enlarging the stencil, as used by

As noted in Sect. 3.5, IB uses only two adjacent primary cell centers when performing computations on their common edge, even though point D is not the center of the primary cell edge. Except for the stability requirement of mass fluxes entering momentum cells (Sect. 3.3 and Fig. 3), the computational subroutines of IB are extremely simple. There are no separate lines of code for angular momentum; all components use the same lines. IK is slightly more complicated, but CSK is worse, requiring frequent interpolation to position variables.

Flux form velocity, represented by two horizontal components, has significant problems where the coordinates become discontinuous. An improvement has been to use forms of the shallow water equations where scalar quantities such as potential vorticity, specific kinetic energy, divergence, stream function, and velocity potential are continuous over the whole sphere and from which the local horizontal velocity can be resurrected, or integrated using the manipulated scalar quantities. Deficiencies of these methods are complexity of understanding and computer coding. This paper presents another method: vector angular momentum is continuous over the whole sphere and its application via the symmetric equations is simpler than using velocity. Each component of relative angular momentum is conserved by flux form advection without discontinuities. Further work is needed to determine the practical advantages that one scheme may have over others.

Fortran source code for model GISS:IB is available at

Using the notation for

Using Eq. (B3) instead of Eq. (3.13), the formula of Eq. (3.14) is replaced with

Most of the formulas and equations of Sect. 2 have probably been known for at least two centuries, but were rederived from scratch by the authors. Use of three horizontal components, vector angular momentum instead of velocity, and the combination of both is new. The Fortran programs used for this paper are original and were written from scratch.

The authors declare that they have no conflict of interest.

The origins of this paper came from discussions with Maxwell Kelley in 2012 and 2013. He has made contributions to the present paper, and models IK and CSK are based on his ideas that may be published in the future. The authors thank Ross Heikes of Colorado State University for the tweaked grid locations. Rainer Bleck was helpful with knowledge of other icosahedral models. Robert Schmunk assisted in creating the YouTube video files. The paper was significantly improved by the editorial process of Monthly Weather Review.

Climate modeling at GISS is supported by the NASA Modeling, Analysis, and Prediction program, and resources supporting this work were provided by the NASA High-End Computing Program through the NASA Center for Climate Simulation at Goddard Space Flight Center. Edited by: Paul Ullrich Reviewed by: two anonymous referees