A Curvilinear Ocean Model Using a Grid Regionally ...

VOLUME 31 JOURNAL OF PHYSICAL OCEANOGRAPHY OCTOBER 2001

A Curvilinear Ocean Model Using a Grid Regionally Compressed in the
South Indian Ocean

ROSS J. MURRAY

School of Earth Sciences, University of Melbourne, Melbourne, Victoria, Australia

C. J. C. REASON

School of Earth Sciences, University of Melbourne, Melbourne, Victoria, Australia, and EGS Oceanography Department,
University of Cape Town, Cape Town, South Africa

(Manuscript received 15 May 2000, in final form 8 January 2001)

ABSTRACT

It is shown that a global curvilinear grid with variable resolution is an efficient way of providing a high
density of grid points in a particular region. In equilibrium experiments using asynchronous time steps, this type
of grid has been found to allow a better representation of smaller-scale features in the high-resolution region
while maintaining contact with the rest of the World Ocean, provided that lateral mixing coefficients be scaled
with grid size so as to maintain marginal numerical stability. In this study, the region of interest is the southern
Indian Ocean and, in particular, that of the South Indian Ocean Current. In all experiments, decreased viscosities
and diffusivities generally led to increased currents and tracer gradients. In horizontal mixing simulations,
maximum current speeds in the frontal region were mainly determined by local (i.e., high-resolution region)
viscosities, while maximum temperature gradients were determined by local values of both lateral viscosity and
diffusivity. With eddy-induced transport experiments, maximum values were analyzed on isopycnal surfaces.
Isopycnal diffusivities were found to control tracer gradients on isopycnals but not isopycnal slopes, while
thickness diffusivities controlled isopycnal slopes but only to a small degree tracer gradients. Changes to mixing
coefficients in the coarse part of the grid had hardly any influence on the frontal properties examined, although
they did affect currents in the Indian Ocean to some extent via their control on size of the Antarctic Circumpolar
Current and the Pacific–Indian Throughflow.

1. Introduction designed to allow at least one boundary to follow a
smoothed coastline. In global models, the fitting of grid
In recent years, a number of programs for generating contours to coasts is not practical, but the provision of
orthogonal curvilinear coordinate grids and prognostic some, albeit limited and large-scale, control of grid res-
models for using them have been developed. One of the olution and alignment is.
obvious considerations for global ocean modeling has
been the desire to keep grid singularities away from the The ‘‘R21’’ (3.19Њ lat ϫ 5.63Њ long) atmospheric
ocean domain and avoid the numerical problems that spectral model grid has been used in ocean modeling
they cause. Grid generation programs that accomplish studies (e.g., Moore and Reason 1993) directed toward
this have been written by Madec and Imbard (1996), the goal of coupling with an R21 atmospheric model at
Murray (1996), and Smith et al. (1995). the Commonwealth Scientific and Industrial Research
Organisation Division of Atmospheric Research in Mel-
While keeping singularities away from ocean areas bourne, Australia. For much ocean modeling work, this
remains a desideratum, curvilinear grids also have the grid is too coarse, and in a number of subsequent stand-
advantage of allowing greater flexibility of grid orien- alone studies conducted both before and after this goal
tation, alignment, and density. Grids for the limited-area was achieved (Gordon and O’Farrell 1997), a double
models of Blumberg and Mellor (1987) and Haidvogel resolution R42 (1.59Њ ϫ 2.81Њ) grid was chosen (e.g.,
et al. (1991) are generated by a program written by Hirst and Godfrey 1993). The R42 model has the ad-
Wilkin (1987, unpublished manuscript) and specifically vantage of being able to resolve channels and represent
small-scale features, such as western boundary currents
Corresponding author address: Mr. R. J. Murray, School of Earth and oceanic frontal zones, better than the R21 model;
Sciences, University of Melbourne, Parkville, 3010, Victoria, Aus- however, it requires computations to be carried out at
tralia. four times as many grid points, which can be wasteful
E-mail: [email protected]

᭧ 2001 American Meteorological Society 2809

2810 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

for models in which enhanced resolution is only re- the Southern Hemisphere and 3.19Њ in the Northern
quired in a small portion of the World Ocean. Hemisphere. The changes in resolution are graded by
smooth transitions over the latitude band 23ЊS–0Њ and
The aim of this set of experiments is to show whether the longitude bands 0Њ–45ЊE and 135ЊE–180Њ. The com-
a curvilinear model employing a variable resolution grid pressed grid has an array size that (excluding the mar-
can be used to provide a fine resolution of oceanic fea- ginal rows) is only 0.48 of that of the R42 grid, which
tures in the region of interest, while adequately repre- has the same resolution in the southern Indian Ocean;
senting the interactions between this region and the rest nevertheless, it is characterized by much wasted reso-
of the global ocean. The model that we have used is lution and large aspect ratios in the zonal and meridional
described by Murray and Reason (2001a) who also dis- bands radiating from the focal region.
cuss the advantages and attractive resolution properties
of curvilinear grids in the North Pole region. To dem- Another way of achieving the desired variation of
onstrate the applicability of a curvilinear grid in a global resolution is by using a curvilinear grid. The asym-
ocean model, the region of interest in the present study metrical bipolar grid (Murray 1996) is one that is suit-
was chosen to be the southern Indian Ocean and adjacent able for this purpose; it is generated by reprojecting
Southern Ocean; and particular features that we wished spherical coordinates so that the poles be shifted toward
to assess the model’s ability to represent well are the one side of the globe, where the resulting grid concen-
currents and tracer variation along the Subtropical Front. tration is greater than on the opposite side; an equivalent
The ability of the curvilinear model to do this using a transformation is produced by applying the complex
grid designed to have approximately R42 resolution in projection of Schmidt (1977) about an axis through the
the focal region and R21 resolution elsewhere is inves- grid equator. The whole system can then be rotated to
tigated with reference to solutions of the regular lati- place the poles in any pair of chosen locations. Figure
tude–longitude models. The comparison is expected to 1b shows a grid with poles in Asia and Antarctica (50ЊN,
reveal whether the coarser resolution of the peripheral 75ЊE and 85ЊN, 75ЊE). The grid lines are actually all
part of the grid is likely to cause any remote effects that circles, although this is not easily recognized in a cy-
may degrade the solution in the focal region and also, lindrical equidistant projection. The degree of grid size
incidentally, whether there the curvilinear solution pos- variation depends upon the asymmetry of the poles. This
sesses any properties attributable to the varying orien- grid has poles 135Њ apart and a cell dimension ratio on
tation of the grid. the grid equator of 2.33:1. Transformed from a 2.5Њ ϫ
4Њ spherical grid, it has a cell size on the axis of con-
As will be shown presently, the answers to these ques- traction of 1.67Њ ϫ 2.67Њ (ϭ4.46 deg2), similar to that
tions depend upon the prescriptions chosen for the vis- of the R42 grid (4.48 deg2). The number of array points,
cous and diffusive coefficients. Simulations have been however, is only 36% of that of the R42 grid. The more
carried out with horizontal mixing and with isopycnal efficient utilization of grid points is brought about in
mixing and eddy induced transport, in both cases with two ways: first, because the need to perform calculations
constant and with spatially variable coefficients. The at a large number of closely spaced points in the Arctic
selection of the grid and the details of the model are region is avoided and, second, because there is a large
given in section 2, and the results of the simulations are number of potential array points in the land area sur-
compared in section 3. rounding the northern grid pole (and to a lesser extent
the southern pole), which does not need to be repre-
2. Methods sented by grid rows. The second point is particularly
relevant to vectorizing computers, which function most
a. Computational grid efficiently when computations are carried out over the
entire array, without skipping land points. The curvi-
Grid compression in zones or sectors of interest is linear grid also has the advantage over the compressed
one economy that was considered in the search for a latitude–longitude grid of being, by construction, free
suitable grid. One-dimensional enhancement of latitu- of large aspect ratio cells.
dinal resolution has for some time been practiced in
tropical models designed to resolve the equatorial wave- The bipolar grid is a geometric transformation of the
guide. For regional modeling, grid compression may be spherical coordinate grid. Nonspherical grids can also
applied in both meridional and zonal directions. Coˆte´ be transformed so as to have a regional concentration
et al. (1998) have used a doubly compressed global grid of grid cells; for instance, J. L. McGregor (personal
with resolutions in the focal region down to 0.0033Њ in communication) has used the Schmidt projection to
a semi-Lagrangian atmospheric model. Constraints on compress the fairly uniform conformal cubic grid of
the time step and exchange coefficients make this degree Rancˇic´ et al. (1996) for use in a curvilinear model based
of compression impractical in Eulerian models; how- on the CSIRO limited-area atmospheric gridpoint mod-
ever, we have tested a grid with a 2:1 ratio of com- el, with resolutions typically ranging between 75 and
pression (Fig. 1a). It has a longitudinal resolution of 800 km. Numerical problems that one might expect to
2.81Њ in the Indian and 5.63Њ in the Atlantic and Pacific develop near the eight corner points of this grid are
Ocean sectors and a latitudinal resolution of 1.59Њ in mitigated by the use of semi-Lagrangian advection and

OCTOBER 2001 MURRAY AND REASON 2811

FIG. 1. Compressed grids: (a) a latitude–longitude grid separately
compressed in the zonal and meridional directions, and (b) bipolar
and (c) multipolar curvilinear grids designed to give the same regional
compression. (d) A contour map of the larger grid cell dimension for
the (multipolar) curvilinear grid (contour interval 0.25Њ), the heavy
dashed line indicating where the grid spacing becomes larger in the
y (or approximately meridional) direction than the x direction as the
grid poles are approached. (e) The variable viscosity field prescribed
for the horizontal mixing experiments (contour interval 0.5 ϫ 109
cm2 sϪ1).

by the fact that the ‘‘focal’’ singularities found at these grid; more importantly, it produces a flatter contour field
points do not generate such severe convergence as do and a more uniform distribution of resolution in the fine
grid poles (Murray 1996). In Eulerian models, it is de- resolution area; this property is roughly based on the
sirable, nevertheless, to exclude both types of singular- principle of a parallel plate condenser. The resolution
ity from the computational domain, and in ocean models is a fairly constant ϳ1.7Њ ϫ 2.7Њ in the central Indian
there are several types of grid that can be designed to Ocean, and increases gradually by a ratio of 1:2.3 to
do this as well as giving some control over the distri- ϳ6.48Њ ϫ 4.06Њ in the north-central Pacific and Atlantic
bution of resolution. Oceans near Mexico (Fig. 1d).

The grid that was found best to suit the purposes of b. Model formulation
the present study refines the distorting facility of the
off-axis bipolar grid using the multipolar technique of Four horizontal grid configurations were used in these
Murray (1996) and is shown in Fig. 1c; it is generated experiments, three of them being latitude–longitude and
about two equal positive poles in Asia (45ЊN, 75Њ Ϯ one being curvilinear. The latitude–longitude grids were
30ЊE) and two equal negative poles in Antarctica (81ЊS, 1) the regular ‘‘coarse resolution’’ or R21 3.19Њ ϫ 5.63Њ
75Њ Ϯ 55ЊE). This also is a nominal 2.5Њ ϫ 4Њ grid (i.e., grid, 2) the regular ‘‘fine resolution’’ or R42 1.59Њ ϫ
would require 72 ϫ 90 cells to cover the globe) and 2.81Њ grid, and 3) the variable resolution ‘‘compressed’’
has a cell size at maximum contraction of 1.69Њ ϫ 2.62Њ grid. The appearance of the two regular grids is seen in
(ϭ4.43 deg2). The method of construction allowed the the southern Indian and North Pacific regions of the
positive, or north, poles to be chosen so as to place most compressed grid, respectively (Fig. 1a). The curvilinear
of Asia outside the grid array, resulting in a grid econ- grid is the one shown in Fig. 1c. Each horizontal grid
omy slightly better than for the bipolar grid and an array was used in conjunction with either a 12 or a 21 level
size only 34% of that of the 1.59Њ ϫ 2.81Њ spherical

2812 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

vertical grid. The vertical grids are the same as those has been recommended by Hirst and Cai (1994), who
used by Hirst and Cai (1994): the uppermost two layers found that they give better water mass properties when
of each have 25-m thickness, but below 160 m the 21 time-invariant boundary conditions are used. Experi-
level model has twice the resolution of the 12 level ments were run from an initial condition interpolated
model. The bottom topography for each horizontal and from the 3D Levitus annual average ocean climatology
vertical grid combination was interpolated after slight and integrated for ϳ220 surface years, after which there
smoothing of the Scripps 1Њ dataset and modified by was no perceptible change in intermediate water prop-
hand so as to resolve important channels and sills. An erties. Two sets of experiments were conducted, the first
effort was made to represent these features in the same employing horizontal mixing of tracers and the second
way with all grid configurations. employing isopycnal mixing of tracers with eddy-in-
duced transport.
The latitude–longitude grids were all used in the (lat-
itude–longitude) GFDL MOM1 model (Pacanowski et The horizontal mixing experiments employed 12 ver-
al. 1991), and the curvilinear grid, in the curvilinear tical levels, a fairly short restoration time of 4 days, and
model described by Murray and Reason (2001a). The a vertical diffusivity that was set to KH ϭ 1 cm2 sϪ1,
curvilinear model follows the coding of MOM1 and except between the uppermost pairs of levels, where it
differs from it only to the extent that spatial finite dif- was 20 and 1.5 cm2 sϪ1, as used by Moore and Reason
ferencing operators have been generalized to handle (1993). Although coarse vertical resolution can cause
transverse orthogonal curvilinear coordinates; it in- stability problems (Weaver and Sarachik 1990) and the
cludes the same basic numerical features as its spherical value of KH may be unrealistically high at intermediate
counterpart—an Arakawa B grid, constant depth levels, levels, these features were not considered relevant to
a rigid-lid condition, a no-slip lateral boundary condi- these experiments, which were designed to study de-
tion, centered advection, and leapfrog time stepping— pendencies related to horizontal resolution. Horizontal
and the options of using asynchronous time steps, im- viscosities and diffusivities must be chosen to meet sta-
plicit vertical diffusion, implicit Coriolis force, isopyc- bility requirements, which depend on resolution, and
nal mixing, and eddy-induced transport. hence varied between experiments; this question is dealt
with in the next subsection.
Lateral exchange coefficients and vertical diffusivity
were set constant in time but varied in space as described To obtain a better representation of interior water
in section 2c. Convective adjustment was simulated by masses, a second set of experiments was conducted,
using implicit vertical diffusion of tracers, with KH ϭ using isopycnal diffusion in conjunction with eddy-in-
106 cm2 sϪ1. Vertical viscosity was set to KM ϭ 20 cm2 duced transport. The latter was implemented as a skew
sϪ1 irrespective of stability. diffusion (Griffies 1998), and both within the framework
of Cox’s (1987) isopycnal numerics and slope limiting
In order to carry out a sufficient number of simula- scheme. These experiments used a 21 level grid, a var-
tions in reasonable time and to focus attention on the iable vertical diffusivity based on the profile of Kraus
grid-related aspects of the solutions, all simulations em- (1990), and a surface restoration time of 30 days. Hirst
ployed constant forcings and the accelerated integration and McDougall (1996) found that short restoring time-
procedure of Bryan (1984). A tracer time step of 2 days scales used in water mass studies to promote sufficient
was used at the surface, with acceleration factors in- deep water mass formation in horizontal mixing exper-
creasing in the lower layers from 1 to 8. A fairly short iments are not necessary or desirable when eddy-in-
momentum time step of 20 minutes was necessary with duced transport is used.
the large (up to 16 day) tracer time steps being used
because of the need to ensure internal gravity wave c. Lateral mixing coefficients
stability at the smallest grid spacings (Murray and Rea-
son 2001b); in some experiments, stability of the viscous Mixing coefficients are constrained by upper and low-
term may have been colimiting. In the latitude–longi- er bounds that are, among other things, increasing func-
tude model, Fourier filtering supplemented by coeffi- tions of the zonal and meridional grid dimensions, h1
cient tapering was applied north of 75ЊN, giving effec- and h2. At the upper bound, viscosities and diffusivities
tive minimum grid spacings of 1.5Њ and 0.7Њ for the and coefficients must be limited to satisfy the time-
coarse and fine resolution grids; in the curvilinear mod- dependent diffusive stability criteria, AM Յ ⌬2/(4␦tuv)
el, the minimum grid spacing was 0.9Њ, and no filtering and AH Յ ⌬2/(4␦tts), where ⌬2 ϭ 1/(1/h12 ϩ 1/h22), and
was applied. ␦tuv and ␦tts are the time steps for momentum and tracers.
In latitude–longitude models, these criteria (as well as
Each model was forced with annual average Heller- internal gravity wave and tracer advective stability cri-
mann and Rosenstein (1983) wind stresses interpolated teria) are limiting at the small zonal grid spacings ap-
to the model grid; for the curvilinear model, rotation proaching the poles, necessitating the use of Fourier
was also necessary. Tracers were restored to fields in- filtering (which increases the effective grid spacing) and
terpolated from the 10-m climatology of Levitus (1982) sometimes the local reduction (or ‘‘tapering’’) of mixing
for the winter months in each hemisphere, with values coefficients. Curvilinear model grids are normally de-
modified to correct inadequacies in the data in the
Greenland and Labrador Seas; the use of winter fields

OCTOBER 2001 MURRAY AND REASON 2813

signed to avoid small ocean grid cells and, thus, the in the tracer fields; they correspond to f H ϭ 0.32 cm
sϪ1 and are within the accepted range. In one of the
need for these expedients. For setting parameters in
R42 model experiments, the higher (i.e., coarse res-
models, these upper bounds, once satisfied, are of less
olution) values of viscosity and diffusivity were used.
relevance than the lower bounds for the largest cell size
In the case of the curvilinear model (and the com-
in the ocean domain (normally at the equator) since
pressed latitude–longitude model), the larger, R21 mod-
these normally impose values greater than the physical
el coefficient values must be used so as to give stability
eddy viscosity and diffusivity of the ocean and, hence,
at the largest grid spacings, similar to those of the R21
limit the degree of structure in the solution.
grid. This produces overly smooth fields in the fine-
The lower bound on viscosity is set by the Munk
resolution area. Since the aim of using this grid is to
condition, AM Ͼ ␤[͙3 max(h1, h2)/␲]3, which is usu-
ally more restrictive than the horizontal grid Reynolds resolve features more finely where the resolution allows
condition at resolutions coarser than about 1Њ (Bryan et
al. 1975). Following Moore and Reason (1993) and it, experiments were carried out with spatially variable
Hirst and Cai (1994), viscosities of 9 and 1.2 (ϫ109
cm2 sϪ1) were used for the R21 (5.63Њ) and R42 (2.81Њ) mixing coefficients as well as with the constant coef-
grids respectively; these conform to this formula and
ficients of the R21 model. Two variable coefficient pre-
are roughly in the ratio 8:1, reflecting the cubic size
scriptions were used. The first was based on the above
dependence.
stability criteria, namely,
Diffusivities used by modelers tend to follow a lin-
ear size dependence, AH ϭ f H max(h1, h2), which [ ΂ ΃ ]AM ϭ max ␤
reflects that of the grid Pe´clet condition for a maxi- ͙3 max(h1, h2) 3
␲
mum current speed of 2 f H. Because of the stabilizing , 1.2 ϫ 109 cm2 sϪ1 ,
effects of horizontal viscosity, values can be rather
(1)
lower than those corresponding to actual maximum
AH ϭ max[0.32 cm2 sϪ1 ϫ max(h1, h2), 1 ϫ 107 cm2 sϪ1].
modeled current speeds and are based on experience (2)

in suppressing noise (Bryan et al. 1975). These au- The constant minimum values are the parameters used
thors used AH ϭ 2.5 ϫ 107 cm2 sϪ1 with a 4.7Њ ϫ in the R42 model; these determined the coefficient
5.6Њ grid, as did Moore and Reason (1993) with a 3.2Њ values for grid spacings less than 2.8Њ and obtained
ϫ 5.6Њ grid; in both cases the choice corresponds to over most of the Indian Ocean. Elsewhere, the co-
f H ϭ 0.40 cm sϪ1. A horizontal or isopycnal diffu- efficient fields looked similar to that of the grid size
sivity of 1 ϫ 107 cm 2 sϪ1 has been used with grids variation (Fig. 1d). Spatial variation of the eddy vis-
having a variety of maximum grid dimensions: 2.8Њ cosity, when applied, generates extra metric terms,
(Hirst and Cai 1994), 3.0Њ (Duffy et al. 1997), and which are allowed for in the models (Murray and Rea-
4.0Њ (Danabasoglu and McWilliams 1995), corre- son 1999, 2001a).
sponding to f H being 0.32, 0.30, and 0.23 cm sϪ1,
respectively. In spite of these differences, it is as- In the horizontal mixing simulations, fields, espe-
cially that of the horizontal streamfunction, were noisy
sumed that the choice of diffusivity was governed near and downstream of Drake Passage and the New
Zealand Plateau. These are regions of complex topog-
largely by numerical considerations. The diffusivities raphy, and the noise was apparently due to grid Pe´clet
used in this study, 2 and 1 (ϫ10 7 cm 2 s Ϫ1 ) for the or grid Reynolds violations caused by large vertical ve-
R21 and R42 grids respectively, have been found locities. In order to quiet these motions without intro-
ducing any explicit current dependence, the geograph-
comfortably adequate for suppressing wavy contours ical variation was modified to

AM ϭ min{9 ϫ 109, max[12 ϫ 109(0.35 ϩ 0.65 cos␪), 1.2 ϫ 109]} cm2 sϪ1, (3)
AH ϭ min{1 ϫ 107, max[2.4 ϫ 107(0.6 ϩ 0.4 cos␪), 1 ϫ 107]} cm2 sϪ1, (4)

where ␪ is the great circle distance from 20ЊN, 105ЊW. able. Constant values were the same as those used for
The modified viscosity field is shown in (Fig. 1e).
These relations have the same general form as Eqs. AH in the horizontal mixing experiments. The quieting
(1) and (2) but have larger values in the southwestern effect of eddy-induced transport on vertical velocities
corners of the Atlantic and Pacific Oceans and more
clearly separate regions of constant low and high val- and the increased vertical resolution used in these ex-
ues.
periments tend to mitigate problems with noisy fields,
Isopycnal and thickness diffusivities in the curvilinear
model were also allowed to be either constant or vari- so viscosities and diffusivities in the variable coefficient

simulations were prescribed without enhancement, ac-
cording to Eqs. (1) and (2), f H ϭ 0.32 cm sϪ1 being
once again used, this time for AE and AI.

2814 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

FIG. 2. Velocities at 635-m depth shown as 600 day time-centered streamlines in the longitude range 80ЊW–125ЊW: (a) coarse resolution
model (expt 1) on its own grid and (b) interpolated to the curvilinear grid, (c) fine resolution model (expt 3) interpolated to the curvilinear
grid, (d) curvilinear model with constant AM and AH (expt 4), (e) expt 4 minus expt 1, (f ) curvilinear model with variable AM and AH (expt
7), (g) expt 7 minus expt 1, (h) expt 7 minus expt 3. The curvilinear grid on which (b)–(h) are plotted is half the resolution of the curvilinear
model grid. The bold lines represent the land boundaries of ocean ‘‘t’’ grid squares at the 635-m level.

TABLE 1. Quantities from horizontal diffusion model simulations for various grids (R21, R42, and curvilinear), horizontal viscosities (AM),
and horizontal diffusivities (AH): maximum 635-m level current speed (Uzm) and horizontal temperature gradient [(‫ץ‬T/‫ץ‬y)zm] at 55ЊE, and
transports of the Antarctic Circumpolar Current (ACC) and Pacific–Indian throughflow (PITF). For variable coefficients, the range is given,

with the high-resolution value before the dash and the coarse resolution value after it.

Expt Grid AM AH Uzm (‫ץ‬T /‫ץ‬y )zm ACC PITF
(109 cm2 sϪ1) (107 cm2 sϪ1) (cm sϪ1) (ЊC/deg) (Sv) (Sv)

1 R21 9 2 4.4 0.92 104 19
2 R42 9 2 3.6 0.84 101 12
3 1.2 1 8.5 1.66 180 24
4 curvilinear 9 2 3.6 0.83 99 9.4
5 1–2 3.8 1.08 107 8.8
6 1.2–9 2 6.4 1.12 130 19
7 1–2 7.3 1.51 145 17
8 1.2–9 1 7.2 1.50 148 18
9 1.2 1–2 7.4 1.55 154 17

OCTOBER 2001 MURRAY AND REASON 2815

FIG. 2. (Continued ) is within the range of depths in which the meridional
temperature gradient, at least in the model, was strong
3. Results but latitudinally confined. At shallow levels, the surface
relaxation reduces the sharpness of the front, and, below
a. Horizontal mixing simulations about 1500 m, the front disappears as the isotherms level
out and the interior starts to become dominated by deep
Simulations were carried out using horizontal diffu- water.
sion of tracers, with constant coefficients in the R21 and
R42 models and constant or variable coefficients in the The velocity fields at this level for the various ex-
curvilinear and compressed models. In order to compare periments are shown in Fig. 2 as time-centered stream-
simulations performed on different grids, the fields need lines. Coarse resolution model velocities (expt 1) are
to be interpolated to a single grid. The curvilinear grid shown on their own grid in Fig. 2a and as interpolated
has been chosen for displaying the horizontal fields be- to the curvilinear grid in Fig. 2b. For clarity in repre-
cause most of those illustrated have been modeled on senting velocities modeled on or interpolated to the cur-
this grid and because its resolution characteristics are vilinear grid, a grid of only half the model resolution
most relevant to this study; however, curvilinear fields in each direction was used for the streamline departure
have been reinterpolated to latitude–longitude coordi- points.
nates for obtaining meridional sections.
Two experiments were carried out with the fine res-
The coefficients used in these experiments and some olution model, one with coarse resolution coefficients
diagnostic quantities modeled using them are given in (expt 2) and one with the smaller coefficients allowable
Table 1. Note that all temperatures referred to in this at fine resolution (expt 3). Currents in both experiments
study are potential temperatures. Of particular interest differed from those in experiment 1 in certain places
in connection with the Indian Ocean is the representa- where the increased resolution had allowed channels to
tion of the Subtropical Front and the associated South be represented (e.g., the Mozambique Channel) that
Indian Ocean Current. This is quantified in terms of the were not resolvable on the coarse grid or to be repre-
maximum 635-m meridional temperature gradient and sented with more grid points (e.g., the Drake Passage).
maximum current speed found along the 55ЊE meridian The currents in experiment 2 (not shown) were else-
in the latitude zone 50Њ–35ЊS. The 55Њ meridian was where very similar to those of the coarse resolution
chosen because it is far enough east of the Agulhas model. By contrast, the currents experiment 3, shown
retroflection to show the effects of diffusion at the front as interpolated to the curvilinear grid Fig. 2c, were
but still somewhat to the west of Kerguelan Plateau stronger and narrower than in experiments 1 and 2, and
(69ЊE), where the southern part of the flow at this level showed a better retroflection of the Agulhas Current.
is perturbed. The 635-m level was chosen as one that Thus, the intensification of currents in experiment 3
relative to experiment 1 was not (or was very little) due
to the increased resolution per se, but rather to the re-
duced exchange coefficients that this allowed.

Four simulations were conducted with the curvilinear
model, using constant and variable horizontal viscosity
and constant and variable horizontal diffusivity. The
maximum grid spacing of the curvilinear grid is similar
to that of the R21 grid, and so with constant coefficients
(expt 4) the eddy viscosity and diffusivity needed to be
the same as used in the coarse resolution model. For
this simulation (Fig. 2d), the velocity field of the cur-
vilinear model looked very similar to that of the R21
model (Fig. 2b), even in areas where the resolution of
the curvilinear grid was greater.

Differences between the experiment 4 and experiment
1 model velocities are shown in Fig. 2e on the same
velocity scale. The velocity differences are mostly much
smaller than actual velocities, especially in the Pacific
and Atlantic Oceans, and there is no obvious systematic
change in the Antarctic Circumpolar Current (ACC), the
Kuroshio Current, or the Gulf Stream. Nevertheless,
there are some notable differences, for instance, to the
east of Drake Passage, around Madagascar, and south
of New Zealand. These can readily be understood as
being due to differences in the resolution and orientation
of topography and consequent changes in the direction

2816 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

and alignment of currents. The most extensive differ- ficients (Fig. 3d) also produced tighter tracer gradients
ence is the anomalous circulation southward around similar to those of the fine resolution model in those
eastern Australia, westward across the Southern Ocean, areas that fell in the fine part of the curvilinear grid.
northward around Madagascar, and eastward across the The sharpest gradients were found across the Agulhas
tropical Indian Ocean. This is due to the smaller channel retroflection and the Kuroshio Current.
width and, hence, decreased flow through the Indonesian
passages in the curvilinear model as compared to the Constant and variable coefficient experiments were
R21 model. Differences between the curvilinear and also carried out using the compressed latitude–longitude
R42 solutions with coarse resolution coefficients (not grid. Diagnostics for these experiments (not numbered
shown) were much smaller and the anomalous circu- or tabulated), showed the same sensitivities as in the
lation was less evident. corresponding curvilinear model experiments (expts 4
and 7), except in the case of some of the transports (the
Experiments 5–7 used a variable horizontal viscosity Pacific-Indian throughflow transports were 17 and 28
and/or diffusivity given by Eqs. (3) and (4). When both Sv: Sv ϵ 106 m3 sϪ1), which appeared to be rather
coefficients were made variable (expt 7), the curvilinear sensitive to the width of the channels and the viscosity
model velocities (shown in Fig. 2f), took on much of in them. The temperature field for the variable coeffi-
the character of those of experiment 3 (Fig. 2c) in the cient experiment was qualitatively very similar to that
Indian Ocean but remained more like those of experi- of experiment 7, but manifestations of noise were more
ment 1 (Fig. 2b) in the Pacific and Atlantic Oceans. The pronounced, possibly because of the large aspect ratios
separate effects of variable viscosity and variable dif- and, hence, wide variation of grid size in this model.
fusivity were learned from experiments 5 and 6, in
which only diffusivity or only viscosity was allowed to Experiments 5 and 6, in which only one of the co-
vary below the R21 value. The results of these exper- efficients, viscosity or diffusivity, was allowed to vary
iments (see Table 1) show that the intensification of the and, hence, assume low values in the Indian Ocean,
currents was almost entirely due to the change in the permit an estimation of the sensitivity of currents and
viscosity prescription, the change in the diffusivity pre- tracer gradients to changes in the mixing coefficient
scription having had only a minor effect. specification. The figures in Table 1 show that velocities
were mainly affected by changes in viscosity and were
The differences between the experiment 7 velocities fairly insensitive to changes in diffusivity. The response
and each of the interpolated spherical model velocities of temperature gradients was different, similar increases
(Figs. 2b and 2c) are shown in Figs. 2g and 2h. In the being brought about by variable, and hence locally re-
central and northeastern Pacific Ocean and the central duced, values of both diffusivity and viscosity.
Atlantic Ocean, the differences between the curvilinear
and coarse resolution velocities are small, but there are The effect of lowered viscosity on tracer gradients
differences in these regions compared to the fine res- along the section line can most easily be explained as
olution model, which has a more intense North Pacific being due to increased current speeds, which give dif-
gyre and Gulf Stream. In the Indian Ocean proper, the fusion less time to act. In the constant and variable
curvilinear solution differs considerably from the coarse viscosity experiments, core current speeds at 55ЊE (av-
resolution model, but rather little from the fine reso- erage for the two diffusivity prescriptions) were 3.7 and
lution model. The southern Atlantic, southwest Pacific, 6.9 cm sϪ1, respectively, which are roughly in inverse
and the western North Pacific Oceans are regions in proportion to the boundary current length scale
which the resolution of the curvilinear model is inter- (ϰ͙3 AM); these speeds correspond to travel times over
mediate between coarse and fine resolution, and the so- the 3500-km distance of 1400 and 600 days, respec-
lution differs from those of both latitude–longitude tively. Assuming an initially discontinuous temperature
models, although in opposite senses. The Drake Passage variation between the Agulhas and South Atlantic Cur-
occurs in one of these areas, so the ACC transport is rents and a constant eastward velocity u, the temperature
likewise intermediate between that of the R21 and R42 variation across the front will be of the form T(y) ϭ
models (see Table 1); this may be the reason for the ⌬T/2 erf [y/(͙2␦)]/(͙␲ /2), with a maximum gradient
large differences from both latitude–longitude models (‫ץ‬T/‫ץ‬y)max ϭ ⌬T/(2͙␲ ␦), where ⌬T is the total tem-
in the Southern Ocean. perature change across the front, y is the distance north
of the front, and ␦ ϭ ͙AHt ϭ ͙AHx/u is the half-width
Temperatures at the 635-m level are shown in Fig. 3. of the front due to diffusion, AH being either 2 or 1 ϫ
Again, the curvilinear model solution of experiment 4 107 cm2 sϪ1 in this part of the grid. Values of ␦ calculated
with constant (high) coefficients [Fig. 3c is similar to from this expression with constant coefficients, reduced
that of the R21 model (Fig. 3a)]. The temperatures in AM, and reduced AM and AH are 750, 500, and 300 km,
experiment 3 (Fig. 3b) differ from both in having sharp respectively; the maximum meridional temperature gra-
(or sharper) gradients in the western boundary currents dients at 55ЊE (1/134, 1/99, and 1/74 ЊC/km) show the
and in confluences of the ACC (near the Agulhas ret- expected inverse variation with ␦. Thus, when variable
roflection, to the east of New Zealand, and to a lesser coefficients are used, both AH and t, and hence also
degree at the Brazil–Falklands confluence). The cur- diffusive length, are (approximately) proportional to
vilinear model experiment with variable mixing coef- grid spacing.

OCTOBER 2001 MURRAY AND REASON 2817

FIG. 3. Temperatures, 635 m (contour interval 1ЊC) for (a) coarse

resolution model (expt 1), (b) fine resolution model (expt 3), (c)

curvilinear model with constant AM and AH (expt 4), and (d) curvi-
linear model with variable AM and AH (expt 7); also (e) curvilinear
model expt 9 minus expt 7 differences (contour interval 0.2ЊC) show-

ing effect of reduced remote diffusivities.

The above calculation does not take into consider- these experiments, one or the other coefficient is lower
ation preexisting temperature gradients or horizontal than normally considered desirable, and the simulations
convergence, which would tend to maintain the frontal did produce some noise; however, the simulations were
gradient, or quantitative errors due to the assumption of deemed to be adequate for these comparisons. Fields
uniform velocity; but the response does show the dom- from each experiment were compared with the results
inance of lateral mixing processes in controlling struc- of the variable viscosity and diffusivity experiment (expt
ture in the solution. It thus appears that the resolution 7), in which the coefficient values were the same in the
of currents and tracer gradients along fronts is able to finer part of the grid but different elsewhere. Most of
increase in step with grid resolution when mixing co- the tabulated quantities showed little sensitivity; the
efficient prescriptions similar to those suggested are put only quantity changing by more than about 2% was the
into practice. Regional grid compression coupled with ACC transport, which increased by 6% with a constant
resolution-dependent mixing could therefore be advan- low viscosity, probably owing mostly to altered fric-
tageous for improving the depiction of locally generated tional effects in Drake Passage, where intermediate vis-
small-scale features. cosities obtained in the variable coefficient experiments.
The temperature differences (expt 9 minus Expt. 7) in
The sensitivities so far discussed only relate to the Fig. 3e show the effects of reducing remote viscosities:
local coefficient changes, that is, those applied in the the main anomaly affecting the focal region was the
fine resolution region. There remains the possibility that band of negative values approaching 0.8Њ–1ЊC near
the solution may be sensitive to the resolution and, 40ЊS; this was caused by a northward displacement of
hence, coefficient values in other parts of the grid. The the currents, but was not associated with any change in
response to remote coefficient values was tested by run- maximum current speeds or temperature gradients at the
ning two experiments, one using a constant low diffu- Subtropical Front. With reduced diffusivities, temper-
sivity of 1 ϫ 107 cm2 sϪ1 and variable viscosity (expt ature differences from experiment 7 in the Indian Ocean
8) and the other using a variable diffusivity and a con- sector were slowly varying and less than 0.5ЊC, except
stant low viscosity of 1.2 ϫ 109 cm2 sϪ1 (expt 9). In

2818 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

TABLE 2. Quantities from eddy-induced transport simulations using the curvilinear model with various horizontal viscosities (AM), thickness

or eddy transport diffusivities (AE), and isopycnal diffusivities (AI): maximum 545-m level current speed (Uzm) and horizontal temperature
gradient [(‫ץ‬T/‫ץ‬y)zm] at 55ЊE; maximum ␴t ϭ 27.4 current speed (U␴m), isopycnal temperature gradient [(‫ץ‬T/‫ץ‬y)zm], and isopycnal slope (Ϫ(‫ץ‬z/
‫ץ‬y)␴m) at 55ЊE; and ACC transport. For variable coefficients, the range is given, with the high-resolution value before the dash and the coarse
resolution value after it. In the upper half, AE ϭ AI for all experiments; in the lower half, separate values are given for AE and AI (the latter

in parentheses), and the expt 10 figures are repeated for clarity.

Expt AM AE (AI) Uzm (‫ץ‬T /‫ץ‬y )zm U␴m (‫ץ‬T /‫ץ‬y )␴m Ϫ (‫ץ‬z /‫ץ‬y )␴m ACC
(109 cm2 sϪ1) (107 cm2 sϪ1) (cm sϪ1) (ЊC/deg) (cm sϪ1) (ЊC/deg) (m/deg) (Sv)

10 9 2 2.8 1.02 2.8 0.27 61 49
11 1–2 3.3 1.31 3.2 0.40 84 62
12 1.2–9 4.6 1.21 4.5 0.26 74 65
13 2 6.2 1.50 5.5 0.43 104 85
1–2
10 9 2.8 1.02 2.8 0.27 61 49
14 2 (2) 2.9 1.08 2.9 0.38 60 50
15 2 (1) 3.2 1.23 3.2 0.28 86 61
16 1 (2) 3.3 1.32 3.2 0.40 84 62
1 (1)

downstream of the Kerguelan Plateau, where the flow direction. In order to shed light on this, a set of simu-
was disturbed. In both comparisons, temperature anom- lations (expts 10–16) was carried out similar to those
alies north of about 35ЊS were less than 0.1ЊC and cur- of section 3a, but with the isopycnal and thickness dif-
rent anomalies less than 0.05 cm sϪ1. fusivities taking the place of horizontal diffusivity. As
b. Eddy-induced transport simulations the focus here was on the effects of different mixing
coefficient prescriptions rather on grid arrangement, all
Isopycnal mixing and eddy-induced transport, in ad- of the simulations were performed on the curvilinear
dition to possibly improving the quality of individual grid. The coefficient ranges used are shown in Table 2.
simulations, are expected to show sensitivities to chang-
es in mixing coefficients different from those found with In experiments 10–13, constant and variable viscos-
horizontal mixing. Much of the smoothing of tracer ities and diffusivities were used, as in section 3a, in
fields caused by horizontal diffusion is due to its dia- each case with AE ϭ AI. The general appearance of the
pycnal component; with pure isopycnal diffusion, the fields in this set of experiments is illustrated in Fig. 4a,
smoothing should be less. On the other hand, differences which shows temperatures for the simulation using var-
between experiments in the diffusion of tracers along iable viscosity and diffusivity, prescribed as in Eqs. (1)
isopycnals will generate a component in the horizontal and (2) (expt 13); this is for the 545-m level, which is
one of the two levels in the 21 level grid comprising
FIG. 4. Temperatures at 545-m depth: (a) variable AE and AI (1–2 the depth range of the 635-m level of the 12 level grid.
ϫ 107 cm2 sϪ1, expt 13) and (b) Levitus (contour interval 1ЊC). Compared to the equivalent experiment using horizontal
mixing (expt 7, Fig. 3d), the fields show some improve-
ments: the isotherms define the subtropical gyres as in
the Levitus data (Fig. 4b) better; the spurious warm
tongue in the southeast Indian Ocean, caused by rapid
downwelling off the West Australian coast in the hor-
izontal mixing experiment, is absent; and the fields near
the Kerguelan Plateau and in other places are a little
smoother.

The response of varying the local values of the co-
efficients can be seen from Table 2. Velocities and tem-
perature gradients at 545 m in the Indian Ocean in-
creased in response to reduced coefficients, as in the
horizontal mixing experiments, but with less change in
the horizontal temperature gradient in response to low-
ered viscosity. There was little response to changes in
remote values of the coefficients. In order to separate
the effects of isopycnal and thickness diffusivity, sim-
ulations were also carried out using either high or low
constant values of AE and AI (expts 14–16). Current
speeds were essentially independent of AI. Changes in
the isopycnal diffusivity affected the distribution of trac-
ers on isopycnals but not the density surfaces themselves
or the pressure gradients to which they gave rise. In

OCTOBER 2001 MURRAY AND REASON 2819

response to a halving of AE and the consequently re- FIG. 5. Isopycnal ␴t ϭ 27.4 (a) depths (contour interval 100 m)
duced flattening of the isopycnals, there was a 15% and (b) temperatures (contour interval 0.5ЊC) for expt 13; also (c)
increase in the maximum current speed at 55ЊE and a Levitus ␴t ϭ 27.4 temperatures.
25% increase in the ACC. Horizontal temperature gra-
dients were affected by changes in both diffusivities— (This equation strictly applies only to gradients at a
20% by a halving of AE and 7% by a halving of AI; single point, not to the tabulated maximum gradients,
however, these numbers are not very informative. As which occur at different latitudes.) Since neither (‫ץ‬T/
discussed in section 3a, there is a logical connection ‫ץ‬z)y nor (‫ץ‬z/‫ץ‬y)␴ changed appreciably with AI, the sen-
between horizontal diffusion and horizontal tracer gra- sitivity of (‫ץ‬T/‫ץ‬y)zm to AI was rather small (7%) com-
dients, but this connection does not hold when mixing pared to that of (‫ץ‬T/‫ץ‬y)␴m (40%).
is along isopycnals. In the latter case, it is better to
consider the slopes of isopycnal surfaces and the current On the isopycnal surface, the pattern of the isotherms
speeds and tracer gradients on them. was also different, maximum gradients on isopycnals
being more confined in space although less in magnitude
When this is done [see columns U␴m, (‫ץ‬T/‫ץ‬y)␴m, and than in the horizontal (cf. Figs. 4 and 5, noting the
Ϫ(‫ץ‬z/‫ץ‬y)␴m in Table 2], the separate effects of changing decreased contour interval). On the ␴t ϭ 27.4 isopycnal
AE and AI become strikingly simple: AI only affected surface, there were two regions of sharp gradients in
the sharpness of the tracer gradients on isopycnals but the 20Њ–90ЊE longitude sector at about 50Њ and 40ЊS,
not the isopycnal slope, while AE for the most part only corresponding to the polar and subtropical fronts, re-
affected the isopycnal slope and had rather a minor in- spectively. The separation of the fronts on the isopycnal
fluence on the isopycnal tracer gradient. These obser- surface can be understood from the section shown in
vations are as one would expect. Isopycnal diffusion Fig. 6. The fronts separate three water masses: the Ant-
causes mixing on isopycnals but, leaving aside nonlinear arctic Surface Water, the Antarctic Intermediate Water
effects, not the position of the isopycnals themselves. of circumpolar origin, and the warmer and more saline
The sensitivity to the halving of AI was about ͙2, as water carried by the Agulhas Current. The correspond-
the diffusive width formula given earlier would predict.
Eddy induced transport, by design, brings about a flat-
tening of isopycnal surfaces but should have no direct
effect on the mixing of tracers on isopycnals. One
would, however, expect an indirect effect due to the
increase in current speeds caused by reducing AE and
to the consequent decrease in diffusion time; the figures
suggest that this effect was present. The reduction in AE
caused the maximum isopycnal temperature gradient to
change by a factor of 1.04, which is approximately the
square root of 3.6/3.2 ϭ 1.1, the factor by which the
maximum current velocity changed. Note that the ap-
propriate current maxima are those that have been found
on the isopycnal surface, U␴m, which are not necessarily
the same as the maxima on the 545-m level, Uzm.

The depths of the ␴t ϭ 27.4 potential density surface
in experiment 13 are shown in Fig. 5a. This surface,
which is characteristic of Antarctic Intermediate Water,
outcrops in the Southern Ocean at about 60ЊS and reach-
es its greatest depths in the centers of the subtropical
ocean gyres. Temperatures on this surface are shown in
Fig. 5b; the salinity plot would be similar in pattern
because of the compensation of the variables on iso-
pycnals. The temperature change on the isopycnal over
the 10Њ of latitude between 35Њ and 45ЊS was only 2ЊC,
which was much less than on the 545-m horizontal sur-
face, where it was 10ЊC. In this latitude range there was
a considerable isopycnal slope, and most of the hori-
zontal change (about 3/4) was due to variation across
isopycnals, rather than along them, and was essentially
a vertical change, as can be understood from Fig. 6 or
the equation

(‫ץ‬T/‫ץ‬y)z ϭ (‫ץ‬T/‫ץ‬y)␴ Ϫ (‫ץ‬T/‫ץ‬z)y/(‫ץ‬z/‫ץ‬y)␴.

2820 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

FIG. 6. Temperature section along the grid meridian at ϳ54ЊE for expt 13. The bold line
indicates the ␴t ϭ 27.4 surface.

ing Levitus field (Fig. 5c) also shows the two fronts in to show a square root response to changes in current
this sector. speeds caused by changes in viscosity, but this is not
evident in the tabulated figures. Although maximum
In the model, the gradient across the Subtropical Front currents increased by 70% with a halving of ͙3 AM in
was very sharp south of Africa, but the isotherms fanned the Indian Ocean, maximum isopycnal temperature gra-
out because of isopycnal diffusion. North of the front dients (and hence diffusive widths) at 55ЊE were almost
the modelled isopycnal surface was almost isothermal. unchanged. The current and tracer fields shown in Fig.
Two features of the Levitus field were absent: the steady 7 help to explain this inconsistency. In the low viscosity
northward temperature gradient in the tropical and north experiment 13, the main part of the Agulhas Current
Indian Ocean and the warm saline tongue that distin- retroflected about 8Њ farther west than in the high vis-
guishes the Agulhas return flow from the cooler and cosity experiment 11 and also provided a 2 Sv leakage
fresher intermediate water filling the south Indian gyre to the Atlantic Ocean. Maximum gradients near South
after entering it from the southeast. These deficiencies Africa were indeed greater in experiment 13 than in
may be due to the absence of Red Sea and Persian Gulf experiment 11; however, because the Agulhas return
Waters, which do not have a source in the model (see flow was in proximity with water originating in the
Tomczak and Godfrey 1994). South Atlantic Ocean over a greater distance before

One would have expected isopycnal tracer gradients

FIG. 7. Isopycnal ␴t ϭ 27.4 temperatures (fine gray contours, contour interval 0.5ЊC) and currents (2400-day streamlines) for (a) variable
and hence low AM (ϭ1.2 cm2 sϪ1) in the Agulhas region (expt 11) and (b) constant AM (ϭ9 cm2 sϪ1) (expt 13), in both cases with variable

AE and AI (Mercator projection).

OCTOBER 2001 MURRAY AND REASON 2821

reaching the section longitude of 55ЊE, the effective TABLE 3. Approximate ratio of increase of the following quantities
diffusion time was about the same. Looking at this in
another way, one notes that the flow east of 40ЊE at for a halving of A1M/3, AH, AE, and AI (globally or, where variable, in
40ЊS was almost parallel in experiment 13, but was con- the fine resolution area) in horizontal mixing (HM) and eddy-induced
vergent in experiment 11 because of the smoothing of
the flow field by the higher viscosity. The divergence transport (EIT) experiments: maximum 545-m or 635-m current speed
of the isotherms at a front should be always greater than (Uzm) and horizontal temperature gradient [(‫ץ‬T/‫ץ‬y)zm] at 55Њ; maxi-
that of the streamlines, and the difference for an error mum ␴t ϭ 27.4 current speed (U␴m), isopycnal temperature gradient
function temperature variation can be shown to be given [(‫ץ‬T/‫ץ‬y)␴m], and isopycnal slope [(‫ץ‬z/‫ץ‬y)␴m] at 55Њ; and ACC transport.
by Ratios are rounded to the nearest 0.05, except for some below 1.1;

and those above 1.2 are shown in bold face.

Mix- Halving
ing of Uzm (‫ץ‬T/‫ץ‬y)zm U␴m (‫ץ‬T/‫ץ‬y)␴m (‫ץ‬z/‫ץ‬y)␴m ACC
HM
‫␪ץ‬T ‫␪ץ‬u AM A1/3 1.8 1.4 — — — 1.35
‫ץ‬y ‫ץ‬y u␦ 2 EIT M — 1.1

Ϫ ϭ , AH 1.1 1.3 — — 1.2 1.35
1.4 1.25
A1/3 1.8 1.15 1.7 1.0 0.98 1.02
M

where ␪T and ␪u are the orientations of the isotherms AE 1.15 1.2 1.1 1.04
and streamlines, respectively. In the region 40ЊS, 45Њ–
50ЊE, the relative divergence was greater in experiment AI 1.03 1.07 1.0 1.4
11 (1/1250 km) than in experiment 13 (1/2100 km),
which compensated for the lower current speeds. This modelers seem not to have taken this opportunity. In
explains the constancy of ␦ but not its quantitative value, practice, with both types of grid, stability violations due
which consists in choosing an appropriate value of u to rapid vertical movements over the rough topography
from a variable meridional profile. Farther south, the in the path of the ACC may be a limiting factor when
relative divergence in experiment 11 was actually less vertical resolution is coarse. Keeping the viscosity
than in experiment 13, so a further explanation is re- somewhat larger than the Munk condition dictates in
quired. The above equation is based on an isopycnal these areas can reduce these vertical velocities.
advective–diffusive balance and embodies the reason-
able assumption that diapycnal flow and, hence, dia- Variable resolution grids constructed in latitude–lon-
pycnal advection are negligible: diapycnal diffusion, gitude coordinates by separate compressions in the zonal
however, is not, and departures from the above predic- and meridional directions are capable of producing cred-
tions will occur where the meridional gradients of this ible simulations with both constant and variable coef-
term become comparable with those of the other terms. ficients. They are not so efficient or as nicely graded as
This has been diagnosed to occur in experiment 11 be- curvilinear grids, and are characterized by larger aspect
tween 45Њ and 50ЊS, where a strong positive gradient ratios. The problem with large aspect ratios is that the
of diapycnal diffusion helped to maintain the sharpness range of maximum and minimum grid sizes is increased,
of the front in spite of weak streamline convergence which makes it harder to satisfy stability criteria.
and, hence, isopycnal advection.
Comparisons between the curvilinear and regular lat-
4. Conclusions itude–longitude model solutions show that the two types
of model give similar results in areas where the mixing
A particular example has been used to show how a coefficients are similar and where the influence of to-
curvilinear grid can be used to effect a regional con- pography, which inevitably differs between models, is
centration of grid points within a global ocean model. not felt. The same qualitative conclusions could just as
Grid compression, however, is only useful if it results easily be demonstrated using latitude–longitude models
in a sharper delineation of features. Physical processes with displaced or differently resolved grids; the only
tend to smooth fields in large areas of the ocean but in additional generalizations with curvilinear grids being
regions of convergence will naturally tend to produce their changed orientation and their curvilinearity in the
sharp gradients. The chief limitation to resolving these meridional as well as the zonal coordinate direction.
gradients is the artificially high mixing coefficients nec- There is a possibility of some effect of grid orientation
essary to meet stability criteria. For coarse resolution near the equator, where the varying placement of grid
models, the Munk condition places a lower limit on the points relative to it could also cause some disorgani-
viscosity and a condition having the form of the grid zation of the rapid and highly sheared equatorial cur-
Pe´clet criterion places a lower limit on the diffusivity. rents in higher resolution simulations. This is not ex-
pected to be a problem in coarse resolution simulations,
With a variable resolution grid, it makes sense to which smooth out the currents anyway, and was not
prescribe mixing coefficients to have a spatial variation obvious in the current differences shown in Figs. 2e and
that just satisfies these lower stability limits at each grid 2g or in shallower fields that we looked at.
point. Variable coefficients can similarly be applied in
models using regular latitude–longitude grids when the The separate sensitivities of selected quantities to
greater cell dimension at the equator is in the zonal changes in AM and AH are summarized in Table 3 as
direction, and hence varies with latitude; however, ocean multiplicative factors for each of the listed variables for
a halving of each of the diffusivities and for a reduction

2822 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

of the viscosity by a factor of 1/8 (i.e., a halving of within limits to give acceptable transports for a partic-
ular grid. In the Subtropical Front region, changes to
A ,1/3 the boundary current length scale). remote viscosities caused some displacement of the tem-
perature and velocity patterns but hardly any change in
M the tabulated measures of frontal strength at 55ЊE. Ve-
locity and tracer fields north of 30ЊS in the Indian Ocean
Experiments with horizontal mixing showed that, showed almost no sensitivity to mixing coefficients out-
side the focal region. This insensitivity is fortunate be-
when both viscosity and diffusivity are reduced ac- cause it means that for studying processes connected
with the Subtropical Front and the interior of the Indian
cording to stability criteria with decreasing grid size, Ocean, the variable resolution curvilinear grid would
serve as well as a latitude–longitude grid of similar
frontal current speeds and horizontal temperature gra- resolution, with a consequent economy of computation.

dients increase in step with resolution. This, in itself, The conclusions of this study have been based on
experiments using a particular grid of a particular res-
indicates that these properties at fronts are being limited olution that happened to be suitable for modeling the
Indian Ocean; furthermore, the experiments employed
by resolution considerations rather than properly mod- constant forcings and accelerated tracer time steps be-
cause these methods allowed multiple experiments to
eled physical processes. Taken separately, the maximum be conducted in a reasonable time. If the above aspects
of the formulation were altered, the ocean climate could
current speeds and temperature gradients showed dif- well change somewhat; for instance, Danabasoglu et al.
(1996) have shown that an asynchronous seasonal so-
ferent sensitivities, the former being almost inversely lution will adjust on switching to synchronous integra-
tion because of phase and amplitude errors caused by
proportional to A1/3 and the latter being inversely pro- using differing momentum and tracer time steps. As the
M focus of the study is the ocean’s response to a variation
of resolution and mixing coefficients and not the details
portional to (A1M/3AH)1/2, in agreement with a simple the- of the solution in a particular region, there is no reason
ory of diffusive spreading. The frontal quantities have to suppose that the conclusions would not be applicable
to coarse resolution synchronous seasonal simulations
only been tabulated for the 635-m level, but the same or to grids having different variations of resolution and
emphasizing different regions of the World Ocean.
sensitivities have been found to apply, with some var- However, the conclusions do not extend to simulations
in which oceanic eddies are resolved because the role
iations, to levels between 200 and 1000 m at least. of applied diffusion in them is different. Latitude–lon-
gitude models become capable of resolving midlatitude
In the experiments employing isopycnal mixing and eddies when the resolution reaches about 1/2Њ; but a
curvilinear model whose resolution straddled this value
eddy-induced transport, it was considered appropriate could be eddy resolving in one part of the grid and not
in another—an interesting subject for investigation.
to find maxima on isopycnal surfaces. The sensitivities Also, while the conclusions may have some relevance
to a slowly varying ‘‘equilibrium’’ state, they are not
to diffusivity changes thus analyzed were completely in applicable to the propagation of the transients by which
the adjustment to equilibrium takes place; these tran-
accord with the way these parameterizations are sup- sients are either absent or poorly resolved and repre-
sented in the highly viscous and stationary type of so-
posed to work: isopycnal diffusivities controlled tracer lution considered in this study.

gradients on these in the same way as did horizontal Equatorially trapped waves are transient phenomena
of particular importance to climate variability. Models
diffusion on horizontal surfaces but had no effect on designed to resolve them require enhanced latitudinal
resolution near the equator; this is efficiently provided
isopycnal slopes; thickness diffusivities, on the other by latitude–longitude grids because of the way that their
alignment contains this resolution within the equatorial
hand, affected isopycnal slopes, but only exerted an in- zone. Curvilinear grids can also be designed to have an
east–west orientation at the equator, as they have been
direct influence on tracer spreading via the effect of by Madec and Imbard (1996) and Smith et al. (1995);
but this requirement severely limits the possibilities for
density gradients on currents speeds. Changes in vis-

cosity produced the same roughly proportionate changes

in currents as in the horizontal mixing experiments, but

their indirect effect on frontal width was found to be

modulated by concurrent changes to the flow and dia-

pycnal diffusion. Allowing for these effects, the simple

diffusive spreading model and the linear relation be-

tween structure and resolution that follows from it re-

main useful concepts for grid design.

The tabulated sensitivities mostly relate to frontal

properties in the southern Indian Ocean, which is the

region upon which the grid is focused. Because the larg-

er constant and the variable coefficient prescriptions dif-

fered mainly in this region, the ratios essentially gave

the sensitivity to local coefficient changes. Comparisons

between experiments using variable and smaller con-

stant coefficients, which differed only in the coarse res-

olution part of the grid, provided an indication of the

sensitivities to remote coefficient values. The ACC

transport, and hence current velocities in the southern

Indian Ocean, showed some sensitivity to viscosities

obtaining in Drake Passage, where they were a little

higher than in the Indian Ocean in some variable vis-

cosity experiments. However, this transport is also sen-

sitive to the model channel width and topography, which

never very adequately represent real channel structure

and may give the ‘‘wrong’’ transports even in a more

highly resolved model; they can reasonably be modified

OCTOBER 2001 MURRAY AND REASON 2823

creating a grid that concentrates resolution in a partic- Haidvogel, D. B., J. L. Wilkin, and R. Young, 1991: A semi-spectral
ular region. In general, a curvilinear grid will not align primitive equation ocean circulation model using vertical sigma
with the equatorial zone nor contain a band of high and orthogonal curvilinear horizontal coordinates. J. Comput.
resolution within it. For grids that have sufficient global Phys., 94, 151–185.
resolution not to need equatorial compression, this in
itself will not necessarily be a disadvantage; but a ques- Hellerman, S., and M. Rosenstein, 1983: Normal monthly wind stress
tion remains whether the oblique orientation of the grid, over the World Ocean with error estimates. J. Phys. Oceanogr.,
as well as possibly affecting the representation of equa- 13, 1093–1104.
torial currents, as previously suggested, could generate
spurious asymmetries in wave forms or propagation due Hirst, A. C., and J. S. Godfrey, 1993: The role of Indonesian Through-
to differing dispersion characteristics in different direc- flow in a global ocean GCM. J. Phys. Oceanogr., 23, 1057–
tions: this is a subject that may warrant further attention. 1086.

Acknowledgments. Funding from the Antarctic Sci- ——, and W. Cai, 1994: Sensitivity of a World Ocean GCM to chang-
ence Advisory Committee administered by the Austra- es in subsurface mixing parameterization. J. Phys. Oceanogr.,
lian Antarctic Division of the Department of the En- 24, 1256–1279.
vironment, Sport and Territories is gratefully acknowl-
edged. ——, and T. J. McDougall, 1996: Deep-water properties and surface
buoyancy flux as simulated by a z-coordinate model including
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