Ocean Modelling 72 (2013) 92–103
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Ocean Modelling
journal homepage: www.elsevier.com/locate/ocemod
Using a resolution function to regulate parameterizations of oceanic
mesoscale eddy effects
Robert Hallberg ⇑
NOAA Geophysical Fluid Dynamics Laboratory, 201 Forrestal Rd., Princeton, NJ 08540, USA
Atmospheric and Oceanic Sciences Program, Princeton University, Princeton, NJ 08540, USA
a r t i c l e
i n f o
Article history:
Received 12 April 2013
Received in revised form 16 August 2013
Accepted 20 August 2013
Available online 5 September 2013
Keywords:
Ocean model
Eddy parameterization
Resolution function
Eddy permitting
a b s t r a c t
Mesoscale eddies play a substantial role in the dynamics of the ocean, but the dominant length-scale of
these eddies varies greatly with latitude, stratification and ocean depth. Global numerical ocean models
with spatial resolutions ranging from 1° down to just a few kilometers include both regions where the
dominant eddy scales are well resolved and regions where the model’s resolution is too coarse for the
eddies to form, and hence eddy effects need to be parameterized. However, common parameterizations
of eddy effects via a Laplacian diffusion of the height of isopycnal surfaces (a Gent–McWilliams diffusivity) are much more effective at suppressing resolved eddies than in replicating their effects. A variant of
the Phillips model of baroclinic instability illustrates how eddy effects might be represented in ocean
models. The ratio of the first baroclinic deformation radius to the horizontal grid spacing indicates where
an ocean model could explicitly simulate eddy effects; a function of this ratio can be used to specify
where eddy effects are parameterized and where they are explicitly modeled. One viable approach is
to abruptly disable all the eddy parameterizations once the deformation radius is adequately resolved;
at the discontinuity where the parameterization is disabled, isopycnal heights are locally flattened on
the one side while eddies grow rapidly off of the enhanced slopes on the other side, such that the total
parameterized and eddy fluxes vary continuously at the discontinuity in the diffusivity. This approach
should work well with various specifications for the magnitude of the eddy diffusivities.
Published by Elsevier Ltd.
1. Introduction
Mesoscale eddies are ubiquitous in the ocean, and are of leading
order importance to the dynamics of major current systems, such
as the Antarctic Circumpolar Current (e.g. Hallberg and Gnanadesikan, 2006), the Kuroshio (e.g. Waterman et al., 2011), and the Gulf
Stream (e.g. Chassignet and Marshall, 2008). Credible models of the
ocean’s dynamics need to either explicitly resolve eddies or to
parameterize their effects.
The dominant spatial scales of baroclinic ocean mesoscale eddies can be broadly characterized by the first baroclinic deformation radius, which is the distance that a nonrotating first-mode
internal gravity wave would propagate in one inertial timescale
(e.g. Gill, 1982). With an appropriate regularization at the equator,1
the q
first
baroclinic
deformation
radius
is
given
by
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
LDef ¼ c2g = f 2 þ 2bcg , where cg is the first-mode internal gravity
⇑ Address: NOAA Geophysical Fluid Dynamics Laboratory, 201 Forrestal Rd.,
Princeton, NJ 08540-6649, USA. Tel.: +1 609 452 6508.
E-mail address: [email protected]
1
This is just a simple function that goes smoothly between the appropriate
equatorial and mid-latitude definitions of the deformation radius without the need
for any arbitrary transition latitude (see, e.g. Chelton et al., 1998).
1463-5003/$ - see front matter Published by Elsevier Ltd.
http://dx.doi.org/10.1016/j.ocemod.2013.08.007
@f
wave speed, f is the Coriolis parameter, and b ¼ @y
is its meridional
gradient.
In idealized models of baroclinic instability, the upper and lower bounds of unstable wavelengths are proportional to the deformation radius, while the most unstable wavenumber is the
inverse of the deformation radius (see, e.g., the textbook by Pedlosky (1987)). The observed dominant eddy length-scales in the
ocean vary more slowly with latitude than does the first baroclinic
deformation radius (Stammer, 1997), but this may reflect the
greater influence of higher baroclinic modes in the tropics and of
effectively barotropic eddies in higher latitudes.
Numerical ocean models need to represent the effects of mesoscale eddies, either by explicitly resolving them or via a suitable
parameterization, if they are to replicate the dynamical response
of the real ocean. As will be illustrated later, the ratio of a model’s
grid spacing to the deformation radius gives a good indication of
whether a model will be locally capable of explicitly resolving eddy
effects. However, as both the deformation radius and an ocean
model’s grid spacing vary in space, one should ask where, not
whether, a global ocean model can explicitly represent eddies.
Fig. 1 shows the ocean model horizontal resolution required for
the baroclinic deformation radius to be twice the grid spacing,
based on a nominally eddy permitting ocean model after one year
93
R. Hallberg / Ocean Modelling 72 (2013) 92–103
Fig. 1. The horizontal resolution needed to resolve the first baroclinic deformation radius with two grid points, based on a 1/8° model on a Mercator grid (Adcroft et al., 2010)
on Jan. 1 after one year of spinup from climatology. (In the deep ocean the seasonal cycle of the deformation radius is weak, but it can be strong on continental shelves.) This
model uses a bipolar Arctic cap north of 65°N. The solid line shows the contour where the deformation radius is resolved with two grid points at 1° and 1/8° resolutions.
of spin-up from climatology. At the coarse resolution that is typical
of the ocean components of CMIP5 coupled climate models (nominally 1° resolution), an ocean model only resolves the deformation
radius in deep water in a narrow band within a few degrees of the
equator; any important extratropical eddy effects will need to be
parameterized. At a much higher resolution, such as a 1/8° Mercator grid, the deformation radius is resolved in the deep ocean in the
tropics and mid-latitudes, but even in this case eddies are not resolved on the continental shelves or in weakly stratified polar latitudes. An unstructured and adaptive grid ocean model could help
to address this issue, but such models are not yet in widespread
use for global ocean climate modeling, and even then computational speed may dictate the use of models that do not resolve
mesoscale eddies everywhere.
In this paper, a series of numerical simulations of a variant of
the Phillips (1954) model of baroclinic instability are used to
examine the effects of resolution on a numerical model’s ability
to exhibit the net overturning circulation driven by mesoscale eddies. The effects of a commonly used parameterization of eddy effect, both on the models’ explicitly resolved eddies and on the net
overturning, are examined. Based on these results, a simple prescription is offered for the typical situation in global ocean models, where eddies are resolved in only part of the domain and in
that portion it is desired that the model be allowed to explicitly
simulate their effects, but in the remainder of the domain that
eddies be entirely parameterized. Specifically, the eddy diffusivities should be multiplied by a ‘‘resolution function’’, ranging from
0 to 1, of the ratio of the baroclinic deformation radius to the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e ¼ ðDx2 þ Dy2 Þ=2. The resolumodel’s effective grid spacing, D
tion function that works best for the cases presented here rapidly
makes a transition from 1 when this ratio is greater than a value
of about 2 (the exact value is not very important and can be chosen to be higher) to 0 for larger values. In the idealized case presented here, this prescription is found to give a reasonable
representation of the net eddy-driven overturning over a wide
range of resolutions.
2. The test configuration and model
Phillips (1954) analyzed the baroclinic instability that arises in
a simple two-layered quasigeostrophic model of a geostrophically
sheared flow in a reentrant channel. This problem has the advantage that many of the properties of the eddies, including necessary
conditions for the growth of instabilities, the growth rate, energetics and vertical structure of the exponentially growing linear
modes can be calculated analytically, as has been documented in
many textbooks on geophysical fluid dynamics (e.g. Pedlosky,
1987; Vallis, 2006).
This study examines instabilities of a stacked shallow water
variant of the Phillips problem, which is described by the momentum and continuity equations:
@un ^ r un un ¼ r M n þ 1 kun k2
þ f þk
2
@t
r T dn2 cD ku2 ku2 ;
ð1Þ
h i
@hn
þ r ðhn un Þ ¼ ð3 2nÞ c g3=2 x g3=2;Ref r K h rg3=2 :
@t
ð2Þ
Here un is the horizontal velocity in layer n, where n = 1 for the
top layer and n = 2 for the bottom layer. hn ¼ gn1=2 gnþ1=2 is the
thickness of layer n, which is bounded above and below by interfaces at heights gn1=2 and gnþ1=2 . These equations are solved in a
2000 m deep channel that is 1200 km long and reentrant in the
x-direction, and 1600 km wide in the y-direction with vertical
walls at the northern and southern boundaries. The Coriolis parameter, f, varies linearly in the y-direction between 6.49 105 s1
and 9.69 105 s1, following the common b-plane approximation. The horizontal stress tensor, T, is parameterized with a shear
and resolution dependent Smagorinsky biharmonic viscosity (Griffies and Hallberg, 2000). The Montgomery potentials,
M n ¼ p=q0 þ gz, in the two layers are given by a vertical integration
of the hydrostatic equation, so that
94
M 1 ¼ g g1=2
R. Hallberg / Ocean Modelling 72 (2013) 92–103
and M 2 ¼ M1 þ ðg Dq=q0 Þg3=2 ;
ð3Þ
where Dq ¼ 0:002q0 is the difference in density between the two
layers, g is the gravitational acceleration of 9.8 m s2, and q0 is
the mean density. These equations are solved numerically using
the Generalized Ocean Layered Dynamics (GOLD) model (Hallberg
and Adcroft, 2009), with 10 different horizontal resolutions ranging
from 2.5 km up to 80 km; because all of the relevant physical
parameters for this case are self-scaling (e.g., by using a Smagorinsky viscosity), the only model parameter that was changed between
the various resolutions was the time step.
These equations are essentially the standard nonlinear 2-layer
stacked shallow water model, but with three modifications. The
first is the inclusion of a Laplacian diffusion of the internal interface height, with coefficient K h . An eddy parameterization based
on the extraction of available potential energy via the diffusion
of isopycnal heights in layered models was the original inspiration
for the Gent and McWilliams (1990) parameterization in z-coordinate models, and this parameterization is the equivalent in this
two-layered system. The sensitivity of this system to the value of
K h is a large part of this study. The numerical model described here
does not require this term for numerical stability, and in the reference cases K h is set to 0. The second modification of a standard two
layer model is the inclusion of a large bottom drag, to prevent the
barotropization of the eddies and to keep these solutions in an
oceanographic regime. Mesoscale eddies in the ocean are observed
to have substantial geostrophic shears, even if they are often described as ‘‘equivalent barotropic’’ because the direction of the motion tends to be largely the same throughout the water column.
Arbic and Flierl (2004) demonstrate that a strong quadratic bottom
drag serves the purpose of giving idealized eddies this structure,
while noting that in the real ocean form drag exerted by rough
bathymetry could fill an analogous role.
The third modification is the inclusion of a transfer of mass between layers that acts to drive the zonal mean interior interface
height back toward a specified profile, shown in Fig. 2, with a rate
of c = 1/ (10 days). The specified density profile has a reversal in
the potential vorticity gradient of the lower layer between
y = 600 km and y = 1100 km, (Fig. 2)(b), and the flow is subject to
baroclinic instability. By damping the zonal mean interface height
via a zonally uniform mass flux, this forcing avoids damping baroclinic eddies, while energizing the flow and ensuring that the model equilibrates in a state of vigorous eddy activity. The damping
rate used here is strong enough that the results do not depend
strongly on its particular value. A similar forcing strategy was employed by Jackson et al. (2008) in their study of statistically equilibrated stratified shear instability.
As will be shown later, a key indicator of the model’s ability to
resolve baroclinic instability is the ratio of the first-mode baroclinic deformation radius to the diagonal grid spacing. In the two-layer
system studied here, the first-mode baroclinic gravity wave speed
is well approximated by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g Dqh1 h2
;
cg q0 ðh1 þ h2 Þ
Fig. 2. (Top) A side view looking westward of the interface height profile toward
which the zonal-mean interior interface heights are damped, along with the zonal
velocities (color) and free-surface height (exaggerated by a factor of 50) that are in
geostrophic balance with this density structure and no bottom flow. (Bottom)
Profiles of the deformation radius (blue) and layer potential vorticity (red) of the
reference profile.
domain and 19 km at the north (Fig. 2(b)), although in the baroclinically unstable latitudes the deformation radius is in the narrower
range of 22 to 39 km. The actual calculations presented here use
the local instantaneous deformation radius, as determined by iteratively solving for the eigenvalues of the modal decomposition equation. The relevant grid spacing for determining whether the
dynamics can be well represented is the coarsest direction, an
appropriate measure of which is proportional to the diagonal grid
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e ¼ ðDx2 þ Dy2 Þ=2. For the simple isotropic Cartesian
spacing, D
grid used here this is simply the grid spacing in each direction,
but with a locally anistropic grid it picks out the least-well resolved
direction. It will be shown later that there need to be roughly 2 grid
points per deformation radius to explicitly represent the eddy
transports, so the x-direction grid spacings that are required to adequately resolve the deformation radius in the baroclinically unstable zone range from 11 km to 20 km.
3. Results
ð4Þ
although in the general multilayered or continuously stratified case
cg is given by the second largest eigenvalue of the normal mode
decomposition equation (see chapter 6 of Gill, 1982). The expresqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sion for the deformation radius used here, LDef ¼ c2g = f 2 þ 2bcg ,
converts smoothly from an appropriate mid-latitude deformation
radius to the equatorial deformation radius, and can be applied
without modification in a global model; for the mid-latitude
b-plane used here, the b term in the denominator reduces the deformation radius by between 1 and 2%. The deformation radius of the
reference profile ranges between 47 km at the southern end of the
a. Simulations without parameterized eddy effects.
When there are no eddy parameterizations and at high resolution, the upper layer develops a vigorous baroclinic eddy field, as
illustrated by the upper layer velocities in Fig. 3. Small perturbations in the initial conditions grow to finite amplitude within about
150 days, and the solutions are in statistically steady states thereafter; the flow fields after 10 years, shown in Fig. 3, are qualitatively similar to other fields in the turbulent but statistically
steady period of the run. While there is a mature field of coherent
vortices up to resolutions of 10–20 km, even the coarser resolutions show some feeble fluctuations from the zonal mean. Recall
that the damping only applies to zonal mean properties, so even
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R. Hallberg / Ocean Modelling 72 (2013) 92–103
slowly growing fluctuations have time to become substantial. The
nature of the forcing also ensures that the zonal mean state is
strongly unstable throughout the run, and that the eddies are not
able to drive the solution into a state of marginal stability.
The upper layer relative vorticity field (Fig. 4) shows clear qualitative differences between all of the resolutions. Even the 5 km resolution run does not exhibit nearly as rich of a field of filaments and
coherent structures as does the 2.5 km run, and there is every reason
to expect that even finer resolution models would exhibit an even
richer small-scale structure. These runs are not converged in that
they do not resolve a full enstrophy cascade, and thus none of them
should be characterized as fully eddy resolving by all metrics.
However, for many ocean modeling purposes, it is not the eddy
properties themselves that are of primary interest, but rather the
effect that the eddies have on the large-scale circulation. One good
measure of the large-scale impact of the eddies is the long-term
mean overturning circulation that is driven by the eddies,
VðyÞ
¼
Z
t
v 1 h1 dx
;
ð5Þ
where the overbar denotes a time-average and v is the northward
velocity. As shown in Fig. 5, the highest resolutions exhibit very
similar overturning transports, but at resolutions coarser than
about 10–15 km, the eddy-driven overturning progressively weakens. There is no abrupt cut-off to the net eddy transport; in fact
even at the coarsest resolution (50 and 80 km), the weak meandering evident in Figs. 3 and 4 contributes much of the overturning that occurs. But if the objective is to simulate an eddy field
that reproduces the overturning of the fully resolved flows, this
is qualitatively achieved only out to resolutions of order 15–
25 km.
On timescales long enough to neglect storage of mass, the timemean northward transport balances the integrated diapycnal mass
flux, so that
Fig. 3. Instantaneous upper layer velocity (speed in color, with directions given by the arrows) at day 3650 at horizontal resolutions of (a) 2.5 km, (b) 5 km, (c) 10 km, (d)
20 km, (e) 33 km, and (f) 50 km. No parameterization of the eddy effects is used in these simulations. The lower layer velocities are much smaller in magnitude.
96
R. Hallberg / Ocean Modelling 72 (2013) 92–103
Fig. 4. Instantaneous upper layer relative vorticity normalized by the local Coriolis parameter at day 3650 at horizontal resolutions of (a) 2.5 km, (b) 5 km, (c) 10 km, (d)
20 km, (e) 33 km, and (f) 50 km. These are the same simulations and times as are depicted in Fig. 3.
VðyÞ ¼
Z
Z
t
v 1 h1 dx
y
Z
¼
Z
y
Z
t
wdxdy 0
Z
y
Z
0
@h1
dxdy
@t
t
t
wdxdy ;
ð6Þ
0
where w is an upward diapycnal velocity. For the 15 year averages
shown in Fig. 5, this balance holds quite well. Additionally, the
velocity and thickness can be decomposed into the zonal- and temporal-mean and anomalies from that mean, in which case the transport can be written as
VðyÞ ¼
Z
v 1 h1
t
dx ¼
Z
v 1 xt h1
xt
dx þ
Z
v 01 h01
t
dx:
ð7Þ
In all of the runs, the rectified anomalies contribute most of the
overturning; only in the lowest resolution runs is the transport by
the time-mean velocity (due to the viscous terms breaking geostrophy) noticeable.
b. Effects of a common eddy parameterization.
It is common in coarse resolution ocean models to use an isopycnal height diffusivity or its advective counterpart in depthspace to parameterize the restratifying effects of eddies (e.g. Gent
et al., 1995). This approach can emulate the adiabatic slumping
effects that eddies exert in isopycnal surfaces, and by design they
extract available potential energy from the mean flow. [Such a
parameterization is sometimes called a thickness diffusivity
(e.g. Eden et al., 2009), but this is a misnomer derived from
flat-bottom or reduced gravity models; with variable bottom
topography a literal interpretation of the isopycnal height diffusivity as a thickness diffusivity leads to a substantial near-bottom
source of available potential energy, and this interpretation
works contrary to the interpretation of such a parameterization
as defining a streamfunction at the interfaces (Ferrari et al.,
2010).] A parameterization of the eddy effects via an isopycnal
height diffusivity is also very effective at suppressing baroclinic
eddies, as is vividly illustrated with snapshots of the upper layer
relative vorticity with different values of the isopycnal height diffusion (Fig. 6). An isopycnal height diffusion directly extracts the
97
R. Hallberg / Ocean Modelling 72 (2013) 92–103
systems.) Fig. 7(b) illustrates the reason behind the behavior, by
decomposing the overturning transport into its explicitly resolved
portion and that due to the parameterization:
VðyÞ ¼
Z
v 1 xt h1
xt
dx þ
Z
v 01 h01
t
Z
@ g3=2 t
dx þ K h
dx:
@y
ð8Þ
The resolved eddy transports are strongly suppressed by even modest values of the diffusivity, while the parameterized transports increase nearly linearly with the diffusivity. The eddy length-scales
here are smaller than the scale of the front or the mean-free-path
of the eddies, so the Laplacian isopycnal-height diffusion is much
more effective at suppressing the resolved eddies than it is at mimicking their effects.
c. Introducing a resolution function.
No eddy parameterization perfectly captures all of the effects of
eddies, but unless the eddies are adequately resolved they cannot
be explicitly represented. As illustrated in Fig. 1, over a broad range
of resolutions, global ocean models are able to resolve the dominant eddy length scales over a portion of the globe, but do not resolve them in weakly stratified, high latitudes, or shallow waters.
Traditionally this has led to a compromise in choosing whether to
parameterize eddies. The results presented above suggest instead
that it might be preferable to choose where to parameterize eddies
and where to allow the model to represent them explicitly.
Although this paper is focused on the idea of introducing a resolution function to control where to apply eddy parameterizations,
the parameterization itself of the eddy effects clearly matters too.
Both the appropriate magnitude of an isopycnal height diffusivity
and its spatial structure depend on the physical system that is
being studied. Even with the simple system studied here, changing
parameters can greatly alter the appropriate diffusivity. This can be
illustrated by diagnosing an implied diffusivity from the eddy permitting cases using
Fig. 5. Time-mean zonally integrated overturning transport for the Phillips model
as a function of horizontal resolution, Dx, averaged over years 6–20 with no eddy
parameterization. (A) Shows the time-mean overturning transport as a function of
latitude for each of 10 different horizontal resolutions. (B) The heavy black line is
the peak value of the time-mean zonally integrated overturning, while the red lines
show the peak mean transport plus and minus the RMS variability; the blue dashed
lines show the maximum and minimum 30-day mean overturning transport over
the same period. (For interpretation of the references to colour in this figure legend,
the reader is referred to the web version of this article.)
available potential energy of baroclinic eddies, and even a modest
isopycnal height diffusivity substantially shortens the eddy lifetime and greatly reduces the structural richness of the eddy field.
Fig. 7 illustrates the effects that introducing an isopycnal-height
diffusion has upon the overturning transport in the 10 km resolution case, in which the eddies could be explicitly represented. With
small values of the diffusivity (here up to about 500 m2 s1), the
overturning is similar to its value without the parameterization,
but with modest values of the isopycnal-height diffusivity (here
1500 m2 s1 to 3000 m2 s1), the total overturning transport declines substantially before increasing almost linearly with the largest diffusivities. The model is able to reproduce the overturning of
a highly resolved model, either by omitting the parameterization
altogether or by using a sufficiently large diffusivity, here empirically determined to be about 8000 m2 s1. (The actual values of
the diffusivities at which these changes occur are a strong function
of the specific details of the configuration, and the choice of a forcing that keeps the flow in a strongly baroclinically unstable state;
the actual values should not be interpreted as relevant to the real
ocean, although it may be reasonable to expect that the qualitative
behavior of this model should be quite representative of other
K Implied
ðyÞ ¼
h
Z
v 01 h01
t
Z
t
@ g3=2
dx
dx:
@y
ð9Þ
Fig. 8 shows strong variations in the implied diffusivity from (9)
for three case – the standard case discussed previously, a case
where the intensity of the baroclinic instability is increased by
driving the zonal mean internal interface height toward its target
value 4 times more strongly than in the standard case, and a case
where the intensity of the baroclinic instability is greatly reduced
by reducing the amplitude of the target interface height changes
across the jet (shown in Fig. 2) from 600 m to 250 m. Increasing
the damping rate c in (2) by a factor of 4 increases the constant diffusivity that best matches the explicit eddy transport to about
11,000 m2 s1 (as determined from the equivalent of Fig. 7(a)) from
about 8000 m2 s1 in the standard case, while decreasing the baroclinicity of the system by reducing change in interface height
across the jet from 600 m to 250 m reduces this value to
1150 m2 s1. Fig. 8 also shows that, in each case, the implied diffusivity is broadly peaked around the baroclinically unstable latitude
range. As seen in Fig. 2, the meridional gradient of the lower layer
target potential vorticity is reversed (a necessary condition for
baroclinic instability) for y = 520 to 1050 km in the standard case,
but the elevated diffusivities in Fig. 8 extend about 100 km further
on either side. With the weakly unstable case the largest implied
diffusivity is substantially more localized in latitude than in the
standard case, primarily because in this case the target profile only
meets the necessary conditions for baroclinic instability for y = 650
to 950 km. As discussed later, a number of approaches have been
suggested for prescribing the spatial pattern and magnitude of diffusivities in eddy parameterizations, most of which are more consistent with the profiles in Fig. 8 than simply prescribing a spatially
constant diffusivity. However, to focus the discussion here on the
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R. Hallberg / Ocean Modelling 72 (2013) 92–103
Fig. 6. Instantaneous upper layer relative vorticity normalized by the local Coriolis parameter at day 3650 with 10 km horizontal resolution and isopycnal height diffusivities,
K h , of (a) 0, (b) 500 m2 s1, (c) 1000 m2 s1, (d) 2000 m2 s1, (e) 3000 m2 s1, and (f) 8000 m2 s1.
resolution function itself, the interface height diffusivities throughout this section are given by a constant diffusivity, as determined
from Fig. 7, times a function of the resolution relative to the deformation radius.
An appropriate measure of whether the baroclinic eddy dynamics are likely to be well resolved is the ratio, RH , of the horizontal
grid spacing to the first-mode deformation radius,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c2g = f 2 þ 2bcg
LDef
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi:
RH ¼
¼
e
ðDx2 þ Dy2 Þ=2
D
ð10Þ
When RH is small, the baroclinic eddy dynamics are not resolved by
the grid spacing, and an eddy parameterization is required. When
RH is large, an eddy parameterization is unnecessary and may be
counterproductive. These considerations suggest that the eddy diffusivity should be multiplied by a resolution function, FðRH Þ, which
decreases from 1 for small RH to 0 for large RH , making the transition
between the two when RH is of order 1.
One candidate resolution function that meets these criteria is
F 4 ð RH Þ ¼
1
1 þ 14 R4H
:
ð11Þ
This function goes smoothly between the correct limits, and has
been used in both NOAA/GFDL’s 1° ESM2G coupled climate model
(Dunne, 2012) and in 1/8° global ocean simulations (Adcroft et al.,
2010). The application of this resolution function to a hierarchy of
model
resolutions,
with
the
diffusivity
specified
by
K H ¼ 8000 m2 s1 F 4 ðRH Þ, is shown in Fig. 9. The dimensional constant, 8000 m2 s1, is empirically determined from the value in
Fig. 7 at which the parameterized eddies match the resolved overturning; subsequent studies will examine how the idea of using a
resolution function might fit with more sophisticated ideas for
determining this diffusivity. As shown in Fig. 9(a), this choice of a
resolution function is not particularly successful, in that there is a
strong resolution dependence of the overturning, with a minimum
at about 20 km resolution. Only the 10 km case is close to the
R. Hallberg / Ocean Modelling 72 (2013) 92–103
99
Fig. 8. The implied isopycanl height diffusivities as a function of latitude as
calculated by dividing the time-mean transient eddy volume flux by the slope of the
time-and zonal-mean internal interface height times the length of the domain from
simulations at 2.5 km resolution. The configuration used throughout this paper is
shown in black, while a blue line is from a case where the damping rate c in (2) is
increased fourfold, to 1/(2.5 days), and the heavy red lines is from case where the
baroclinicity of the system is reduced by decreasing the meridional drop in the
target interface height across the jet from 600 m to 250 m. The light dashed red line
is the same as the heavy red line but multiplied by a factor of 5 for easier
comparison with the other lines. Those latitudes where the magnitude of the slope
of the time- and zonal-mean interface is less than 2105 are excluded. (For
interpretation of the references to colour in this figure legend, the reader is referred
to the web version of this article.)
Fig. 7. Meridional peak value of the time-mean and zonally integrated overturning
transport, averaged over years 6–20 at 10 km horizontal resolution as function of
the isopycnal height diffusivity. (A) Shows the peak mean value (heavy black line)
along the mean plus and minus one standard deviation (red) along the minimum
and maximum 30-day average peak transports (dashed blue). (B) Shows the
meridional peak zonally integrated time-mean overturning total transport (black),
the peak overturning transport due to the parameterized eddies (red) and the peak
overturning transport due to the resolved eddies (blue). The red and blue lines do
not exactly add up to the black line because the peak transports can occur at
different latitudes, as shown later in Figs. 8 and 9. (For interpretation of the
references to colour in this figure legend, the reader is referred to the web version of
this article.)
2.5 km reference solution. The reason behind this failure can be
seen in the profiles of the mean diffusivity shown in Fig. 9(b). This
choice of resolution function varies so slowly that many of the simulations are using diffusivities that are large enough to suppress the
resolved variability but too weak to parameterize the true overturning. This is vividly illustrated by the 20 km resolution case, in which
the overturning is substantially weaker than the corresponding
solution shown in Fig. 5(a), which does not include any eddy
parameterization. As shown in Fig. 9(c), the diffusivity is strong enough to greatly suppress the explicit eddy overturning, but much
too weak to accomplish the overturning via the parameterized diffusive transport.
The proposed resolution function described in (11) can be made
more abrupt by increasing the power of RH in the denominator. Progressively increasing this power improves the solutions in the Phillips model test case presented here, which leads to the idea of
testing the limiting case of a step-function. Fig. 10 illustrates the
overturning transport with the step function resolution function
F Step ðRH Þ ¼
1 RH < RCrit
0 RH P RCrit
;
ð12Þ
with RCrit ¼ 2, and a total isopycnal-height diffusivity of
K H ¼ 8000 m2 s1 F Step ðRH Þ. The convergence of solutions across
resolutions in Fig. 10(a) is remarkable. The peak values agree closely
across resolution. The transports on the flanks of the jet are too
large in the cases that rely on a parameterization (e.g., 33 km resolution), but this reflects the use of a spatially constant diffusivity
that was chosen to match the peak transport instead of the tapered
profiles diagnosed in Fig. 8, and not a problem with the resolution
function (in these cases F Step ðRH Þ is 1 everywhere). The time-and zonal-mean diffusivities (shown in Fig. 10(b)) in intermediate cases
shows a smooth change in values due to the temporal and spatial
fluctuations of the front at which the deformation radius is large enough to be resolved. The total overturning fluxes due to combined
effects of the parameterization and the resolved eddies match and
blend smoothly (Fig. 10(c)).
Similar results to those shown in Fig. 10 hold over a wide range
of values of RCrit > 2 (for which values even resolvable eddies are
suppressed and parameterized), but degrades noticeably for smaller values of RCrit (when some unresolvable eddies are not parameterized). The test case presented in Fig. 10 is strongly
baroclinically unstable, and setting RCrit ¼ 1 gives an overturning
that is only modestly degraded compared with RCrit ¼ 2, probably
because there are broad range of unstable wavenumbers and the
longer waves are able to accomplish the eddy transport even when
the most unstable wavelength of the continuous solution is poorly
resolved. Setting RCrit ¼ 0:7 leads to a qualitatively dissimilar overturning, with a greatly reduced peak overturning for marginally resolved cases (similarly to Fig. 5) and an abrupt decrease in the
overturning where the resolution function (12) disables the
parameterized eddy fluxes.
The standard test case used here is strongly baroclinically
unstable, but it can easily be modified to be much more weakly
unstable by reducing the drop in the target interface height
across the jet from 600 m to 250 m. (Setting the drop in interface
height to just 150 m would have made the jet baroclinically stable.) Fig. 11 is the equivalent of Fig. 10 for this weakly unstable
case with the total isopycnal height diffusivity given by
K H ¼ 1150 m2 s1 F Step ðRH Þ with RCrit ¼ 2. Again, there is a strong
100
R. Hallberg / Ocean Modelling 72 (2013) 92–103
Fig. 9. Overturning transport averaged over years 6–20 when the isopycnal height
diffusivity is specified as K h ¼ 8000 m2 s1/ [1 + 0.25ðLDef Þ=ðDxÞ4 ]. (A) Time mean
zonally integrated overturning transport as a function of resolution; the 2.5 km
reference case (dashed) is the only one that does not use an isopycnal-height
diffusivity. (B) The time- and zonal-mean isopycnal height diffusivity. (C) The time
mean zonally integrated overturning at 20 km resolution due to the parameterized
diffusive transport (red), the explicitly resolved flow (blue) and the total (dashed
magenta). (For interpretation of the references to colour in this figure legend, the
reader is referred to the web version of this article.)
Fig. 10. Overturning transport averaged over years 6–20 when the isopycnal height
diffusivity is specified as K h ¼ 8000 m2 s1 where LDef < 2Dx and 0 elsewhere. (A)
Time mean zonally integrated overturning transport as a function of resolution. The
lines for the 25 km, 33 km and 40 km runs are nearly indistinguishable. (B) The
time- and zonal-mean diffusivity. (C) The time mean zonally integrated overturning
at 16 km resolution due to the parameterized diffusive transport (red), the
explicitly resolved flow (blue) and the total (dashed magenta). (For interpretation
of the references to colour in this figure legend, the reader is referred to the web
version of this article.)
similarity in the overturning transports across a wide range of
resolutions. The 16 km solution is somewhat of an outlier in
Fig. 11; in this case the parameterized eddy fluxes are only active
on the northern flank of the jet but the decision to use a constant
diffusivity appropriate to the center of the jet instead of a latitudinally tapered diffusivity, as diagnosed in Fig. 8, acts to strengthen the resolved baroclinic instability and associated transport in
the center of the jet itself. For this weakly unstable case, intermediate resolution solutions with values of RCrit below about 2 do
not reproduce the high-resolution transport. The difference in
the more restrictive appropriate value of RCrit between the
strongly and weakly unstable cases lies in the fact that the
long-wave cut-off of baroclinic instability and the wavelength
with the fastest growth rate both move to smaller scales for
R. Hallberg / Ocean Modelling 72 (2013) 92–103
101
this marginally unstable case also works for the strongly unstable
case, the value of RCrit ¼ 2 should be appropriate for realistic
ocean modeling.2
Using a step function change in the diffusivity might seem like a
terrible idea, because it introduces an artificial discontinuity to diffusivity in the solution. However, if the resolution is sufficiently
fine that eddies can grow, the total mass fluxes will vary continuously, because otherwise a growing interface height discontinuity
and strong associated baroclinicity would develop. In the runs
shown here, the continuous transition of the fluxes from the
parameterized to the explicit is accomplished by a flattening of
the instantaneous isopycnal slopes on the side with the strong
parameterization and some steepening (but not to an extent that
is highly atypical) on the side where the parameterization is suppressed. Another objection to an abruptly changing diffusivity
might be that gradients in a diffusivity are analogous to adding
an advective term to a spatial smoother, since
@
@x
j
@C
@x
¼
@ j @C
@2C
þj 2
@x
@x @x
ð13Þ
and an abrupt change in the diffusivity is therefore akin to adding
an infinite advective speed. However, this does not impose a limitation on the stability of the model, because when implemented in a
model these changes actually occur over a grid spacing, meaning
both that the effective cell Reynolds number is guaranteed to be
1, and that the CFL time-step limit on the apparent ‘‘advective’’ term
in (13) is equivalent to the time-step limit already imposed by the
diffusivity.
The simulations presented here strongly suggest that multiplication by an abruptly varying resolution function may be an ideal
way to automatically transition between using parameterizations
of eddy effects in one part of the domain, and allowing an ocean
model to represent eddy effects explicitly in other regions.
4. Discussion and summary
Fig. 11. Similar to Fig. 10, but for a weakly unstable case where the interface height
drops by only 250 m across the jet instead of 600 m. In this case, the isopycnal
height diffusivity is specified as K h ¼ 1150 m2 s1 where LDef < 2Dx and 0
elsewhere. (A) Time mean zonally integrated overturning transport, averaged over
years 6–20, as a function of resolution. The lines for the 25 km, 33 km and 40 km
runs are nearly indistinguishable. (B) The time- and zonal-mean diffusivity. (C) The
time mean zonally integrated overturning at 16 km resolution due to the
parameterized diffusive transport (red), the explicitly resolved flow (blue) and
the total (dashed magenta). (For interpretation of the references to colour in this
figure legend, the reader is referred to the web version of this article.)
weaker baroclinic shears relative to bL2Def . (See, for example, chapter 6.6 of Vallis (2006) for a thorough discussion of the linear
phase of baroclinic instability in the Phillips model.) Since much
of the ocean is only weakly unstable, and the critical value for
Most large-scale ocean models include both regions where spatial resolution is adequate for eddy effects to be captured explicitly
and regions where the dominant eddy scales are unresolved and
any significant effects of eddies need to be parameterized. The
common approach has traditionally been to make a choice between parameterizing eddies or not throughout the model’s domain. This paper instead proposes that eddies should be
explicitly represented in numerical ocean models in those portions
of the domain where the model’s resolution is sufficiently fine and
parameterized where it is not. This paper uses an idealized model
of baroclinic instability to demonstrate that this objective can be
accomplished by multiplying the parameterized eddy fluxes by a
function, ranging from 1 to 0, of the ratio of the baroclinic deformation radius to the model’s grid spacing.
Eddy parameterizations suppress mesoscale eddies. Real eddies
tend to be self-regulating because they modify their environment
to suppress the source of their own growth, and successful eddy
parameterizations do the same thing. For instance, one of the key
principles behind the successful Gent–McWilliams eddy parameterization is that it deliberately extracts available potential energy
without diabatic mixing (Gent et al., 1995). Eddy parameterizations also tend to be scale selective, operating much more rapidly
on smaller horizontal scales than larger scales. Because eddy
2
The appropriate value for RCrit will depend on the numerical methods and
closures being used and how well eddies with spatial scales close to the grid spacing
are described. The GOLD model configurations described here use an Arakawa C-grid
spatial discretization of the dynamic core with a biharmonic Smagorinsky lateral
viscosity. The tests described here would need to be repeated to determine the right
value of RCrit to use with other numerical discretizations and grid-scale closures.
102
R. Hallberg / Ocean Modelling 72 (2013) 92–103
spatial scales in the ocean are often comparable to or smaller than
the scales of the large-scale structures that drive the eddy instabilities, eddy parameterizations are typically at least as effective at
suppressing eddies as they are at reproducing their effects on the
large-scale structure. As a result of this eddy suppression, this
study obtained the best results when the resolution function was
chosen to change quite abruptly to fully disable the parameterizations where the eddy scales are adequately resolved.
Existing eddy parameterizations introduce only an imperfect
representation of some of the effects of eddies. Some improvements in large-scale measures of eddy effects, such as the Southern
Ocean overturning response to wind-stress changes, can be obtained by adopting parameterizations whose intensity and structure are free to vary greatly with the model’s simulated state
(Gent and Danabasoglu, 2011). However, other eddy effects are
not well captured by presently available parameterizations. For instance, in the case of the Southern Ocean, Hallberg and Gnanadesikan (2006) demonstrate that explicitly modeled eddies lead to a
coherent transport of water over significant distances, including
southward transports of watermasses that are lighter than any that
exist at a given location in the mean. [The atmospheric analog of
this effect is wintertime cold-air outbreaks (e.g. Held and Schneider, 1999).] This is something that no local diffusive parameterization will ever be able to capture. So even if ocean models had
greatly improved theories for predicting the dependence of the
parameterized eddy-transport strength and structure on the large
scale ocean state, there would still be a compelling justification
to rely upon numerical ocean models to explicitly represent eddy
effects where ever possible.
There are many different approaches for prescribing the intensity and structure of eddy effects parametrically based on the model’s local mean state (e.g. Visbeck et al., 1997; Held and Larichev,
1996; Treguier et al., 1997; Danabasoglu and Marshall, 2007,
etc.), or based on an auxiliary energy equation that allows the eddy
effects to be based on past states and to spread in space (e.g. Cessi,
2008; Eden and Greatbatch, 2008; Marshall and Adcroft, 2010).
Although the utility of the resolution function was demonstrated
here only for a very simple parameterization with a constant isopycnal-height diffusivity that was empirically determined for a
single specific configuration, the same idea will apply equally well
with any of these other more elaborate parameterizations. The
present study suggests a way to apply these earlier ideas only
where they are needed, and should be seen as complementary to
that past work.
The GOLD simulations presented here do not require any isopycnal height diffusivity for numerical stability. However, there
are substantial ancillary benefits of using a Gent–McWilliams diffusivity in depth-coordinate B-grid ocean models: it suppresses
the well-known checkerboard null mode in the density field, and
it suppresses numerical diapycnal mixing arising from advective
truncation errors arising from grid-scale tracer variations (Griffies
et al., 2000; Ilicak et al., 2012). When it is desirable to retain an
eddy parameterization as a numerical closure, it might be advisable to follow the suggestion of Roberts and Marshall (1998) to
use a more highly scale-selective biharmonic interface height diffusion (which is much less effective at suppressing well-resolved
eddies) with a strength based on numerical considerations, as a
supplement to a Laplacian-based parameterization of the physical
effects that is regulated by a resolution function.
The two-layered Phillips (1954) model of baroclinic instability
has been used for decades to understand the dynamics of baroclinic instability, but the simplicity of this model clearly imposes certain limitations on what it can be used to explore. With only two
layers, there is only a single baroclinic mode. By contrast, there
are countable baroclinic modes in a continuously stratified ocean,
and there is some evidence that these higher modes may be
important for shaping the properties of the eddy field and its rectified effects (e.g. Smith, 2007; Berloff et al., 2009). In such cases, it
may be advantageous to keep parameterizing the eddy effects until
a higher-mode baroclinic deformation radius is resolved, or equivalently until the first mode is very well resolved. In the examples
discussed here, the value of ratio of the horizontal resolution to
the deformation radius at which the transition occurs does not
matter too much, provided that the transition value of RH is greater
than about 2 and the transition is abrupt. The consideration of the
effects of the second or third baroclinic modes might argue for
choosing a transition value of order 4 or 6, since the higher mode
wave speeds scale approximately inversely with the mode number.
Other eddy processes with smaller spatial scales that are independent of the first baroclinic deformation radius, such as submesoscale restratification of the mixed layer (Fox-Kemper et al., 2008),
should be treated separately, perhaps using a different definition
of a resolution function based on their dominant spatial scales.
We have every expectation that the ideas presented here can be
straightforwardly generalized to apply to a broad range of situations where marginally resolved processes sometimes need to be
parameterized.
The ideas presented here have been tried in realistic global
ocean models at various resolutions ranging from 1° (Dunne,
2012) to 1/8° (Adcroft et al., 2010). When combined with an appropriate closure for the parameterized eddy intensity, the resolution
function offers the prospect of exploiting an ocean model’s full potential for explicitly representing oceanic processes while behaving
sensibly throughout its domain. By eliminating global decisions
about which processes to parameterize, this approach also offers
the prospect of greatly reducing the amount of parameter tuning
that has traditionally occurred in configuring global ocean-climate
models at various resolutions. Most global ocean models include
some regions where the mesoscale eddies are well resolved, but
other regions where they are not. Moreover, these regions where
the eddies are resolved will evolve in time with the stratification
of the ocean. (The first baroclinic deformation radius can be calculated quickly enough that updating it frequently as the model’s
state evolves is a trivial part of the cost of running a realistic ocean
model.) The application of a resolution function, as described here,
is a promising avenue for creating global-scale ocean models that
have a credible representation of eddy effects throughout their domain, via explicit resolution where possible and parameterization
where necessary.
Acknowledgments
I would like to thank Alistair Adcroft for numerous conversations on this topic and invaluable feedback, Sonya Legg for her
helpful comments on a preliminary draft, and Matthew Harrison
for his assistance in remaking one of the figures. I gratefully
acknowledge improvements prompted by three anonymous
reviewers. This work was carried out in part while the author
was a fellow at the Isaac Newton Institute’s 2012 programme on
‘‘Multiscale Numerics for the Atmosphere and Ocean’’.
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