In this article we propose two grid generation methods for global ocean general circulation models. Contrary to conventional dipolar or tripolar grids, the proposed methods are based on Schwarz–Christoffel conformal mappings that map areas with user-prescribed, irregular boundaries to those with regular boundaries (i.e., disks, slits, etc.). The first method aims at improving existing dipolar grids. Compared with existing grids, the sample grid achieves a better trade-off between the enlargement of the latitudinal–longitudinal portion and the overall smooth grid cell size transition. The second method addresses more modern and advanced grid design requirements arising from high-resolution and multi-scale ocean modeling. The generated grids could potentially achieve the alignment of grid lines to the large-scale coastlines, enhanced spatial resolution in coastal regions, and easier computational load balance. Since the grids are orthogonal curvilinear, they can be easily utilized by the majority of ocean general circulation models that are based on finite difference and require grid orthogonality. The proposed grid generation algorithms can also be applied to the grid generation for regional ocean modeling where complex land–sea distribution is present.

The generation of the model grid preludes the simulation with ocean general
circulation models (OGCMs) and sea ice models. The majority of OGCMs use
orthogonal curvilinear dipolar or tripolar grids with the North Pole
relocated to continental areas, including models in the Coupled Model
Intercomparison Project, fifth phase (CMIP5;

Contrary to OGCMs, regional ocean models

As an important trend for global ocean modeling, high-resolution simulation
has been applied in oceanic forecast

In this paper, we propose two new grid generation algorithms that improve
existing grids in various aspects. They are based on Schwarz–Christoffel
conformal mappings. The first algorithm improves dipolar grids, with an
enlarged latitudinal–longitudinal (lat–lon) portion of the grid and smooth
transition of grid sizes and scaling factors. The second algorithm aims at
supporting high-resolution and multi-scale modeling, including (1) the
removal of a major continental area from the grid, (2) enhanced spatial
resolution in coastal regions, and (3) the alignment of grid lines to
large-scale coastlines. In the following part of this section, we first
review the design of orthogonal curvilinear grids for OGCMs in
Sect.

The practice of modeling the ocean involves (1) the selection of an equation set to characterize the thermodynamic and dynamic evolution of the ocean, and (2) numerical treatments of the equation set, including spatial discretization, numerical approximation and time-domain integration. The majority of OGCMs utilize a mixture of finite difference and finite volume spatial and temporal discretization, and assume local orthogonality of the underlying grid. In the traditional latitude–longitude grid, which is a special case for general orthogonal curvilinear grids, meridians converge at the North Pole. This brings several challenges to the ocean modeling: (1) the small grid step sizes in the zonal direction near the North Pole impose a strict limitation on the maximum allowed time step sizes and computational efficiency, especially for high-resolution modeling; and (2) there exists large grid step size anisotropy near the North Pole, which negatively affects the effective spatial resolution. To overcome these shortcomings, current OGCMs usually utilize angle-preserving mappings that relocate the grid's pole from the North Pole to one or several continental locations.

Figure

Orthogonal curvilinear grids for OGCMs.

To summarize, we outline the design requirements for OGCM grids as follows. They are loosely sorted according to relative importance, starting from more important or classic ones to less important or more modern ones.

Grid orthogonality.

Relocation of the grid pole to continental locations. The farther the grid poles from the ocean, the better.

The scaling factor is close to 1 for the whole grid; i.e., there is no jump of local grid cell sizes.

The major part of the grid is still lat–lon.

Grid cell size anisotropy should be low.

The grid does not induce very small time steps.

The grid is indexable as a Cartesian grid.

The grid can reduce as many unused grid points (i.e., grid points on land) as possible.

The grid can support high-resolution and multi-scale modeling.

This list is arguably more comprehensive than that in

It is worth noting that finite-element methods-based OGCMs utilize irregular
meshes

Conformal mappings

Northern patch constructed from a Moebius transformation.

For a single-connected region

For an area with a connectivity of

Schwarz–Christoffel mapping for a single-connected polygon.

Multiple-connected regions and canonical forms.

Despite the fact that the canonical forms of multiple-connected regions have
long been recognized, the construction theory for such a Schwarz–Christoffel
mapping between a user-specified polygonal region and its canonical forms is
a recent discovery since

Hereby we denote SCSC as the Schwarz–Christoffel mapping for
single-connected regions, and MCSC the Schwarz–Christoffel mapping for
multiple-connected regions. In Sect.

In this section we apply the Schwarz–Christoffel mapping for
single-connected regions (SCSC) to the generation of dipolar grids for OGCMs.
We mainly address the following grid design objectives: (1) the enlargement
of the lat–lon portion of the grid; (2) the mitigation of scale changes
across patch boundaries; and (3) reduced grid cell size anisotropy in polar
regions. Traditional dipolar grids

Patching scheme for the global grid.

Longitudinal segment across the Atlantic Ocean at a certain latitude
(at 47

Longitudinal segment across the Pacific Ocean at a certain latitude (across the Bering Strait), which extends into the Eurasian and North American continents on both ends;

Smooth linkage for the first two segments in Eurasia;

Smooth linkage for the first two segments in North America.

Statistics of the sample orthogonal curvilinear grids.

The last two segments were constructed to ensure that the overall boundary is smooth. Due to the irregular boundary of NP, an SCSC mapping is constructed to map (1) a unit disk to the stereographic projection of NP and (2) the origin of the unit disk to the prescribed continental position in Greenland (i.e., the location for the grid's pole).

The remaining patches are (1) the southern patch (SP), to cover the middle and high latitudes in the Southern Hemisphere; (2) the equatorial band patch (EBP), to cover low-latitude areas (with meridional refinement); (3) the North Pacific patch (NPP), covering the area between EBP and NP on the Pacific Ocean and Indian Ocean; and (4) the North Atlantic patch (NAP), covering the leftover area between EBP and NP (which mainly corresponds to the North Atlantic Ocean). The boundaries between NPP and NAP are in the meridional direction and in continental areas, i.e., Eurasia and North America.

Furthermore, we construct the oceanic part of NPP and NAP to be lat–lon (and orthogonal). The non-oceanic areas in NPP and NAP are filled with non-orthogonal grid cells. Since the grid points in these areas are not active in the simulation, it is guaranteed that the loss of orthogonality of these points will not affect simulation results. During the construction of each patch, meridional refinements are introduced where the grid size anisotropy is large.

The grid generation algorithm is outlined as follows.

Generation of the NP boundary.

Grid generation for NP, by constructing an SCSC mapping from a unit disk to the stereographic projection of NP. An orthogonal polar coordinate is generated for the unit disk and mapped back by the SCSC mapping and backward stereographic projection. The meridional grid cell sizes along the boundary of NP are computed.

Generation of the Pacific and North Atlantic basin patches, i.e., NAP and NPP. Linkage between (1) NAP and NPP on the eastern and western boundaries; (2) NAP (or NPP) and NP are also constructed.

Generation of the equatorial and southern grid patches, i.e., EBP and SP, according to grid cell anisotropy requirements.

Assembly of the patches into a global grid.

Generation of land and depth masks.

As a consequence of the irregular boundary of NP, the grid cell sizes in the
meridional direction along its boundary are not uniform. Because the Atlantic
Ocean and the Pacific Ocean are not directly connected between 40 and
70

Before introducing the detailed design, we show a sample grid with nominal
1

The choice for the boundary of NP is a trade-off between several factors: the
reduction of the length of the boundary on the Pacific and Atlantic oceans,
the smoothness of the scaling factor in both meridional and zonal directions,
etc. In the sample grid, we choose (1) on the Pacific side, the longitudinal
segment crossing the Bering Strait, i.e, at about 66

Smooth linkage of NP boundary and meridional step size mitigation.

We use a discretized boundary (nominal resolution in the zonal direction) as
the polygonal boundary (shown in Fig.

For dipolar grids, grid cells in the polar regions tend to feature very large
anisotropy in cell sizes. Meanwhile, equatorial regions are often modeled
with higher meridional resolution for purposes such as higher accuracy in the
simulation of tropical waves and ENSO

Sample grid with nominal 1

In the proposed grid generation method, we introduce a bespoken threshold value to limit the maximum anisotropy in polar regions. This value is also used for the meridional refinement in EBP. Hence, the maximum anisotropy of the whole grid is kept below this value. For SP, due to the fact that it is purely lat–lon, the latitudes and longitudes of the grid points could be computed as one-dimensional arrays. We start from the lowest latitude (the southern boundary of EBP) and numerically integrate to higher latitudes by gradually decreasing the sizes of latitudinal steps. The gradual decrease in meridional step sizes is designed to ensure that the maximum anisotropy does not increase beyond the predefined threshold. For NP, a similar strategy to gradually reduce the latitudinal steps is used, except that due to the uneven edge sizes for any circle in the zonal direction, the anisotropy of a certain zonal circle of the grid is computed as the average meridional edge size divided by the average zonal edge size on the circle.

One important property of the anisotropy control scheme is that although it
increases the number of unknowns, it has a limited effect on the largest
allowed time step size

In order to maintain the accuracy of finite difference operators, we should
keep the local scaling factors close to 1. To ensure that there are no abrupt
scale changes across the boundary of NP, after the grid generation for NP, we
compute the cell edge sizes along the two oceanic segments of the NP boundary
as the average meridional cell edge size of all the cells in each of the
segments. Then, NAP and NPP are treated separately to ensure a smooth
transition of meridional edge sizes, starting from the uniform meridional
edge size in the south (i.e., their boundaries with EBP), and gradually
changing to the meridional edge sizes on its northern boundary with NP. We
use a cosine function to construct the meridional edge sizes: (1) on both the
southern and northern ends the transition of cell sizes is smooth, and
(2) the numerical integration (i.e., the sum) of all the meridional edges
equals the latitude difference between EBP and NP. This scheme is shown in
Fig.

Within EBP, since the meridional refinement is adopted, a similar scheme for
smooth transition is also used, as shown by the example grid in
Fig.

The meridional and zonal grid edge sizes of the sample grid are shown in
Fig.

Scales of the 1

In the sample grid, 93.8

Since the sample grid is an orthogonal curvilinear grid, it can be utilized by the majority of OGCMs. We evaluate the sample grid from the following aspects: (1) the static evaluation in terms of the maximum allowed time steps and comparison with existing dipolar grids, and (2) the application of the grid in POP.

With the assumption of an explicit (split) formulation, we show the global
map of the maximum allowed count of time steps per simulation hour for the
external gravity wave (i.e., the barotropic mode) in
Fig.

Similar to traditional dipolar grids (e.g., GX1), the critical area with
respect to barotropic time step size is in the Arctic, near Greenland, as the
result of (1) small zonal edge sizes and (2) the large external gravity wave
speed. The sample grid is comparable to GX1 in terms of the maximum allowed
time step count per simulation hour (87.7 vs. 58.5 for GX1, which has a lower
zonal resolution of 1.125

We further implemented the grid in POP. We generate the grid's depth mask
field by interpolation and discretization on 46 depth levels. The depth for
each vertical layer starts from 3 m for the surface layer to 250 m in the
deep ocean. The global maximum depth is 6000 m. Simulation with idealized
forcing is carried out to demonstrate that the sample grid can be easily
utilized by OGCMs. The configuration of the simulation is as follows: (1) the
potential temperature and salinity are initiated to a climatological profile

Analytical wind stress and surface heat forcing.

Model evaluation of sample grids with SCSC mapping.

In this section we propose the second grid generation method for OGCMs. It
targets the new trends of high-resolution and multi-scale ocean modeling
(items 8 and 9 in the list in Sect.

The outline of the grid generation method is as follows.

The manual selection of the polygonal boundary for each continental mass that is to be mapped to slits. The area enclosed by the polygon should be strictly in-land.

The construction of an MCSC mapping

According to the grid resolution requirements, a polar grid coordinate
system is generated on

The generation of land and depth masks.

The mapping scheme between

Conformal mapping between

Because of (1) the orthogonality of the polar coordinates on

For the numerical implementation, we use an adapted version of the MCSC
open-source software

Continental boundaries and slit information.

For the sample grid, we limit the choice of the polygonal boundaries for continents to those enclosed by manually picked points. The number of points per continental mass (i.e., the vertex count for the corresponding polygon) is kept small, so that it is feasible for a manual choosing process. It also ensures that the region enclosed by the polygon is strictly in-land, to guarantee that no oceanic region is mapped to slits. The scheme presented here is basic, and only used to demonstrate the grid generation methodology. More advanced schemes are also possible but beyond the scope of this paper, including (1) a spline-based smooth boundary generated from manually picked in-land points, or (2) automatically retrieved continental boundaries.

Each polygonal region, representing a continental mass, is mapped to either a
circular or radial slit on

We construct the sample grid with the four major continental masses as listed
in Table

Global grid with MCSC mapping.

We focus on two aspects for the basic evaluation of the sample grid: (1) the
effect of the continental area removal, and (2) the alignment of grid lines
to large-scale coastlines. As shown in Table

Figure

The actual land–sea distribution and grid cell edge sizes in the grid index
space are shown in Fig.

Through the static evaluation of the grid, we show that the sample grid satisfies the three major aforementioned features: (1) the removal of major continental masses from the grid index space, (2) the alignment of large-scale coastlines to grid lines, and (3) enhanced spatial resolution in coastal areas. The features are achieved at the same time, as a result of the conformal mapping and its harmonics behavior.

We evaluate the sample grid under the same protocol as in Sect.

Figure

Meridional and zonal grid step sizes in grid space.

Figure

In this article, advanced conformal mapping techniques, namely
Schwarz–Christoffel mappings, are used to generate orthogonal grids for
ocean general circulation models. These curvilinear orthogonal grids are
indexable in a regular Cartesian manner, and can be easily integrated with
existing OGCMs that already support orthogonal curvilinear grids, such as POP
or NEMO

Spatial discretization with irregular and non-orthogonal grids is adopted by
algorithms such as finite-element methods, which are widely used in
computational fluid dynamics and structure design. In recent years, these
non-structured grids have also been adopted in ocean modeling, such as FVCOM

Model evaluation of sample grids with MCSC mapping.

The sample grid in Sect.

In

The invariance of the solution of a Laplacian equation with first- or
second-type boundary conditions under conformal mappings could be further
utilized for the grid generation for OGCMs. Instead of the construction of
the conformal mapping, a numerical solution of a boundary value problem is
needed. The first- and second-type boundary conditions are equivalent to the
circular and radial slits as in the approach in Sect.

The proposed method in Sect.

In the supplements, we provide the grid input files for POP in binary format
for the two sample grids, with nominal 1 and 0.5

In this paper, we propose two new grid generation methods for global ocean
generation circulation models. Contrary to conventional dipolar or tripolar
grids based on analytical formulations, these new methods are based on
Schwarz–Christoffel (SC) conformal mappings with user-defined boundary
information. The first method improves the conventional dipolar grids. With
SCSC mappings, we construct an orthogonal North Pole patch with a smooth but
irregular southern boundary. By utilizing the disconnectedness of major ocean
basins in the mid-latitudes in the Northern Hemisphere, the scaling factors
across patch boundaries are kept low. In the sample grid, 93.8

The second grid generation method aims at more modern topics in ocean modeling: (1) the removal of major continental masses from the global grid, (2) better resolution at coastal regions with the alignment of large-scale coastlines to grid lines, and (3) the support for multi-scale ocean modeling. This is achieved by constructing an MCSC mapping, which maps the continental areas to slit regions. In the grid index space, these areas correspond to one-dimensional grid cells. The conformal mapping ensures the alignment of grid lines with the boundaries of these areas, hence achieving the approximate alignment with coastlines. Oceanic regions near these boundaries feature a higher density of grid points, which corresponds to better spatial resolution in coastal regions. Compared with conventional dipolar or tripolar grids, this method exploits more information on land–sea distribution in the form of user-prescribed continental boundaries.

Through static evaluation and simulation with the POP ocean model, we show that the sample grids can serve as swap-in replacements for existing grids for the majority of OGCMs that already support orthogonal curvilinear grids. The MCSC-based grid generation method could also be used in combination with other dynamic or static spatial refinement methods to achieve multi-scale ocean modeling. For the first aspect of the future work, we plan to further apply the proposed methods with refined continental boundary information for grid construction and long-term spin-up runs with realistic atmospheric forcings. As the second aspect, we plan to formulate the proposed methods into complete, open-source grid generation software for both global and regional ocean modeling.

We would like to thank the editors and the referees for their invaluable effort in improving this paper. This work is partially supported by the National Science Foundation of China under grant nos. 41205072 and 51190101, and the Joint Center for Global Change Studies (no. 105019). Edited by: D. Ham