Parameterization of subgrid stirring in eddy resolving
ocean models. Part 1: Theory and diagnostics
J. Le Sommera , F. d’Ovidiob,c , G. Madecb,d
a
LEGI, UJF/INPG/CNRS, Grenoble, France
LOCEAN, UPMC/IPSL/IRD/CNRS, Paris, France
c
ISC-PIF, Paris, France
d
NOCS, National Oceanographic Center, Southampton, UK
b
Abstract
Horizontal stirring by time-varying mesoscale flows contributes to forming submesoscale tracer filaments. In this paper, we propose a parameterization of the passive transport associated with filamentation by mesoscale
flows for use in O(10km) resolution ocean models. Theoretical motivations
are provided for modelling subgrid stirring by the resolved mesoscale flows
with an anisotropic generalization of Smagorinsky operator. For level coordinate models, an isoneutral formulaton of the proposed subgrid operator
is provided. The proposed subgrid operator is diagnosed with DRAKKAR
global 1/4◦ eddy-resolving model output. In the Southern Ocean, the parameterization is shown to provide diffusivities peaking at about 400 m2 s−1 .
If applied prognostically, the proposed subgrid operator could drive meridional heat transports of about .5P W at 45◦ S. This suggests that a significant
fraction of the transport by mesoscale flows could be associated with tracer
features of scale smaller than our model grid size (∼20km at 45◦ S). A large
contribution to this transport is associated with differential advection by
the time-mean flow at subgrid scale.
Key words: lateral mixing, anisotropic diffusion, eddy resolving ocean
models, large eddy simulation
1. Introduction
Mesoscale eddies play a key role in setting tracer distributions and shaping the stratification of the global ocean. Therefore, considerable effort has
been put into trying to include their effect in ocean climate models. Current
Preprint submitted to Ocean Modelling
March 31, 2011
parameterizations of mesoscale eddies combine variants of isoneutral diffusion, following Solomon (1971) and Redi (1982), and eddy-induced transport
as proposed by Gent and McWilliams (1990) and Gent et al. (1995). Parameterizations of mesoscale eddies are usually horizontally isotropic, with some
notable exceptions (e.g. Wajsowicz, 1993; Smith and McWilliams, 2003;
Smith and Gent, 2004). The refinement of horizontal grids and the advance
of high resolution remote sensing have recently highlighted the striking complexity of oceanic flows at submesoscale, revealing a full range of new physical processes (Capet et al., 2008b; Thomas et al., 2008; Klein et al., 2008).
As noted by Müller and Garrett (2002), new parameterizations are therefore crucially needed for eddy resolving models to account for those scales
of motion. On the one hand, such new parameterizations should aim at including the effect of new physical processes associated with those new scales
of motion (hereafter refereed to as submesoscale processes). In that respect
a recent example is the parameterization of the restratification due to mixed
layer eddies (see Fox-Kemper et al., 2008; Fox-Kemper and Ferrari, 2008;
Boccaletti et al., 2007). On the other hand, as the mesoscales become partly
resolved, the ocean modelling community is facing a conceptual transition
regarding the design of eddy closure schemes (Fox-Kemper and Menemenlis,
2008; Nadiga, 2008). Some valuable information being contained in the resolved mesoscales, ocean climate models now fall into the framework of large
eddy simulation models, or more precisely, the mesoscale ocean large-eddy
simulations regime (MOLES) as defined by Fox-Kemper and Menemenlis
(2008).
Observational and modelling evidence shows that geostrophic turbulence ubiquitously leads to the formation of submesoscale tracer filaments
in the global ocean. Tracer release experiments as performed during NATRE
(Ledwell et al., 1998) have shown the tendency of mesoscale eddies to form
elongated tracer features with width O(1km), usually referred to as submesoscale tracer filaments. These experiments have also shown how stirring by
mesoscale eddies is responsible for the density compensated nature of thermohaline filaments (Ferrari and Paparella, 2003; Smith and Ferrari, 2009).
The combined use of high resolution sea surface temperature, ocean color
satellite imagery and satellite sea level measurements have shown that advection by mesoscale eddies is responsible for the widespread nature of submesoscale tracer filaments in the global ocean (Abraham and Bowen, 2002;
d’Ovidio et al., 2004, 2009a; Waugh and Abraham, 2008) and for structuring
marine ecosystems at different levels of the trophic chain (Kai et al., 2009;
2
Cotté et al., 2011; d’Ovidio et al., 2011). Likewise, phytoplankton patchiness is now partly attributed to stirring by mesoscale eddies (Abraham,
1998; Martin, 2003; Lehahn et al., 2007). Submesoscale tracer filaments
are preferentially formed in strain-dominated regions of the ocean between
mesoscale eddies or along fronts where most of the stirring is concentrated
(Hua et al., 2001). This picture is consistent with what is expected from two
dimensional turbulence theory (Haller and Yuan, 2000; Klein et al., 2000).
As shown by McWilliams et al. (1994), at scales smaller than the largest
internal Rossby radius, geostrophic turbulence can be strongly anisotropic
(see in particular their Fig. 3). Accordingly, global satellite-based observations of oceanic currents have recently corroborated the picture of an intrinsically anisotropic oceanic mesoscale turbulent flow (Scott et al., 2008).
The effective diffusion due to mesoscale flows and submesoscale tracer filaments can therefore be suspected to be anisotropic too, as suggested by
some recent primitive equations numerical simulations (Kamenkovich et al.,
2009).
From a practical standpoint, an interesting property of ocean stirring is
that a component of the tracer submesoscale variability can be reconstructed
from two-dimensional time-dependent velocity fields coarse grained at the
mesoscale (see for instance Fig. 4 in Lehahn et al. (2007)). It is why relatively coarse grained altimetric data can be used to reconstruct some properties of submesoscale tracer filaments (Despres et al., 2009; d’Ovidio et al.,
2009a). This property is also what allows to estimate eddy-diffusivity or stirring efficiency from altimetric data (d’Ovidio et al., 2004; Marshall et al.,
2006; Waugh and Abraham, 2008). A flow such that small scale mixing is
governed through passive transport by the larger scales falls into the chaotic
advection regime (Aref, 1984). A geophysical example of this flow regime
is found in the lower stratosphere (Haynes, 1999; d’Ovidio et al., 2009b).
The relevance of the chaotic advection regime for oceanic flows is briefly
discussed by Marshall et al. (2006). Indications that stirring by mesoscale
flows fall into the chaotic advection regime are also provided by Waugh et al.
(2006) (see in particular their section 5). There is nonetheless a debate as to
whether oceanic flows fall into this regime. From a theoretical standpoint,
as first suggested by Bennett (1984), the ability of low-resolution velocity
fields to produce reliable estimates of small-scale stirring can be related to
the steepness of the horizontal kinetic energy spectra E(k). More precisely,
the steeper the kinetic energy wavenumber spectra1 , the more the spectrally
1
that is to say the larger the α with E(k) ∼ k −α
3
nonlocal dynamics controls tracer stirring, and the smaller the error in a
tracer field reconstruction based only the larger scales of the velocity field
(Bartello, 2000). In the surface ocean, wavenumber spectra have recently
been shown, on numerical and observational grounds2 , to be better aligned
with the predictions of the surface quasi-geostrophic (SQG) theory (Klein
et al., 2008; Le Traon et al., 2008) than the prediction of quasi-geostrophic
turbulence theory. This would imply that the kinetic energy spectra are
shallower in the surface layers with E(k) ∼ k −2 (Capet et al., 2008a). It is
therefore unclear to what extent chaotic advection regime holds on the surface layers. In the bulk of the ocean interior instead, in the absence of direct
observations of velocity wavenumber spectra, one should rely on model simulations. Recent submesoscale resolving simulations seem to show steeper
spectra at depth than in the surface layers (Klein et al., 2008) therefore
suggesting that the chaotic advection regime could better hold at depth.
Here, taking a pragmatic approach, we are interested in quantifying the
lateral stirring at submesoscales due to differential advection by mesoscale
flows. Obviously, other mechanisms than passive advection by mesoscale
flows, including intrinsic submesoscale processes, could also contribute to
tracer stirring at scales smaller than 10km since filaments of density are
dynamically active and generate submesoscale dynamics. Consistent with
a MOLES approach, in what follows we will assume that some information
regarding stirring by mesoscale eddies can be diagnosed from the resolved
flows of eddy resolving ocean models. The following sections propose a way
to use this information in order to constrain a closure for the missing tracer
advection in eddy resolving ocean models and of preserving the anisotropic
nature of mesoscale stirring.
The purpose of this paper is (i) to introduce the theoretical background
for parameterizing the effect of tracer stirring through passive transport at
subgrid scale in eddy resolving models with an anisotropic diffusion operator
and (ii) to estimate the order of magnitude of the tracer fluxes induced by
this operator in an eddy resolving model. In what follows, we will show that
the subgrid stirring by mesoscale flows can be represented by an horizontal
anisotropic diffusion operator of the form
h2
p r
K2d ≃
(1 + δ)
.
(1)
r q
2
2
Note that there are still some discrepancies between the spectra obtained with
altimeter-derived velocities and directly-measured velocities (Wang et al., 2010).
4
Here, h is the effective horizontal grid size and p, q refer to
√
√
p = r 2 + a2 + a and q = r 2 + a2 − a,
(2)
r = vx + uy is the rate of shear strain of the horizontal velocity field u and
a = ux − vy is the rate of normal strain. The non dimensional parameter δ
is a measure of the divergence of the flow, it is defined by
δ=√
d
r 2 + a2
(3)
with d = ux + vy referring to the divergence of the horizontal velocity field.
Our diagnostics of the above operator in an eddy resolving model show that
the stirring by mesoscale flows induces significant tracer fluxes at subgrid
scale. This suggests that the transport by mesoscale flows involves a wide
range of scales including the submesoscales.
In this first paper, we do not discuss the discretization of the proposed
operator as we believe that it is essential for a physical parameterization to
be derived in a continuous setting before being discretized according to the
specific numerics of a given ocean model. These aspects, together with the
effect of our parameterization on the large scale ocean circulation, will be
addressed in a second paper. The present paper is organized as follows. In
section 2, we show how to define an anisotropic diffusion operator in order
to mimic the effect of advection at the subgrid scale on tracer fields. The
proposed anisotropic diffusion operator and its rotation along the epineutral
plane in three dimensions are described in section 3. Various diagnostics
illustrating the properties of the proposed parameterization are presented
in section 4. A summary and a discussion are provided in section 5.
2. Parameterizing subgrid stirring with anisotropic diffusion
2.1. Principle of the parameterization
Consider a tracer field τ evolving in a two dimensional velocity field u
according to
∂t τ + u · ∇τ = 0.
(4)
Following classical textbook methods (see e.g. LeVeque, 2002), as a first
step toward the discretization of eq. (4), let us define the spatially filtered
5
quantities
τ̂ (x, t) =
û(x, t) =
Z
Z
τ (x + l)ϕ(l)dl
(5)
u(x + l)ϕ(l)dl
(6)
where ϕ(l) designates some low-pass spatial filter. In what follows, we
will assume that the spatially filtered quantities are the resolved variables
of some discrete formulation of eq. (4). Obviously, the spatially filtered
quantities do not generally follow a pure advection equation. More precisely,
∂t τ̂ + û · ∇τ̂ = M(τ, u) with M(τ, u) 6= 0.
(7)
Indeed, it is straightforward to show that M(τ, u) = û · ∇τ̂ − u\
· ∇τ . so
that this quantity should be parameterized. Hence, in what follows, our
purpose is not to include the effect of new physical mechanisms arising at
smaller scales but rather to model the tracer submesoscale cascade induced
by mesoscale turbulence and suppressed by the spatial filtering. In ocean
climate models, this missing term is classically parameterized with a combination of vertical diffusion, lateral diffusion and an eddy-induced transport
velocity (e.g. Griffies et al., 2000). The lateral diffusion is usually chosen
in the form of an horizontal (or isoneutral) isotropic Laplacian diffusion.
Note that all the above operators being linear in τ , M also exhibits a linear
dependence with respect to τ . It should also be stressed that, in practice,
since M depends on the spatial filter, a discrete version of this r.h.s. term
M will also depend on the advection scheme being used for discretizing
eq. (7).
In order to get a sense of the general form of M, consider the following
one dimensional sketch situation. Let us assume that u is linear with the
form u = r x with r > 0, and assume that ϕ is a gaussian kernel of half
width l0 . A straightforward Taylor expansion of eq. (5) tells us that
τ̂ ≃ τ +
l02 2
∂ τ.
2 x
(8)
Combining eq. (4) and eq. (8), noting that û = u in this particular case and
neglecting the high order terms, we get
l2
∂t τ̂ + û ∂x τ̂ ≃ 0 ∂x (r ∂x τ̂ )
6 2
(9)
which tells us that the missing term is a laplacian diffusion with constant
l2
coefficient κ = 20 r. Here, we note that the diffusion coefficient is proportional to the half width of the gaussian kernel. This suggests that the
missing term should depend quadratically on the size of the spatial grid use
for the discretization of eq. (4). This dependance is consistent with current
practice in ocean climate modelling (see Griffies et al., 2000).
Going back to a more general setting, we recall here some general concepts of chaotic stirring with the example of Fig. 1. The panels show a
mesoscale-resolving velocity field (red arrows on a 1/3◦ grid, black lines)
and a tracer anomaly (black dots) advected on a smaller scale grid, over
which the velocity field is linearly interpolated. Even though the velocity
field has no submesoscale components, tracer submesoscale filaments can
be generated by the stretching and folding induced by the mesoscale turbulence. This stretching and folding effect, when integrated over time (in this
case, three weeks) can create very complex tracer structures, not necessarily
aligned with the streamlines of the velocity field: see for instance the tracer
anomaly in (-16, -52.5), which has still the shape the eddy core from which
it has been ejected, even though it is in a jet-dominated region.
The proposed anisotropic mixing operator is designed to mimic the submesoscale tracer redistribution due to the mesoscale velocities fields. This
will be achieved by breaking down the tracer chaotic advection into the
stretching and folding events which compose it. This concept is shown
schematically in Fig. 2. Suppose for a moment that the velocity is nondivergent (we will relax this hypothesis later) and that we would like to
compute the submesoscale evolution of a tracer at a grid point under a
coarse grained velocity field. We aim to do this for a model time step, during which the mesoscale structure of the velocity field is invariant. The flow
being non divergent, the effect of advection at the subgrid scale is both to
stretch the tracer patch (along a locally defined stretching axis) and to compress it (along a locally defined compression axis). As previously discussed,
the spatially averaged quantities do not follow a pure advection equation
and additional terms are needed in order to mimic the effect of advection
at the subgrid scale. Nevertheless, as mentioned above, the information
required to specify the evolution of the tracer field due to subgrid advection
is (partially) contained in the coarse grained velocity field, which is obvious in Fig. 1. In what follows, we propose to model the grid box to grid
box tracer fluxes due to these missing dynamics. The missing dynamics
will be included in the form of anisotropic horizontal diffusion. The coming
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Figure 1: A mesoscale velocity field has the ability to generate submesoscale tracer
structures through the stirring process. In this example the velocity field on the depicted
grid (1/3◦ resolution) is interpolated on a smaller grid (1/12◦), where a tracer is advected.
Even if the velocity field does not contain submesoscale information, the tracer is stirred
into complex submesoscale filaments. Top: Initial condition. Bottom: Tracer distribution
and velocity field after 21 days.
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Figure 2: Schematic illustration of the proposed subgrid parameterization. A grid-size
tracer anomaly is stretched by the coarse grain velocity field along a direction when
advected at submesoscale resolution for short times (in this case, six days). An isotropic
diffusion operator would simply spread the tracer anomaly into a wider circle. Therefore
the subgrid stretching process cannot be represented by isotropic diffusion. Here we aim
to represent this subgrid stretching process as an anisotropic diffusion.
subsections will present how to define this operator.
2.2. General form of the anisotropic diffusion tensor in two dimensions
Our purpose is to include the effect of advection at the subgrid scale
in the form of anisotropic horizontal diffusion. Accordingly, we propose to
approximate the spatially filtered solutions of eq. (4) by discretizing the
following partial differential equation,
∂t τ̂ + û · ∇τ̂ = ∇ · (K2d · ∇τ̂ ),
(10)
where, as above, (û, τ̂ ) represent the spatially filtered quantities and K2d =
K2d (x, t, û, τ̂ ) is a two dimension matrix operator to be determined. That is
to say, M(τ, u) is parameterized in the form ∇ · (K2d ∇τ̂ ). In what follows,
we will also assume that K2d does not depend on τ̂ .
More precisely, we decide to model only the effect of stretching along
the principal stretching axis therefore neglecting the compression along the
principal compression axis3 . So, we choose a anisotropic diffusion tensor of
the form
ϑ
K2d = κ M2d
,
(11)
ϑ
where M2d
is associated with the unit two dimensional vector ϑ = (ϑ1 , ϑ2 )
according to
2
ϑ1 ϑ1 ϑ2
1 0
T
ϑ
.
(12)
R =
M2d = R
ϑ1 ϑ2
ϑ22
0 0
3
We will come back later on this assumption in section 5.
9
Above, R refers to the two dimensional rotation matrix operator which
transforms (1, 0) into ϑ = (ϑ1 , ϑ2 ). The relations eq. (11) and eq. (12)
define a two dimensional anisotropic diffusion tensor which diffuses a tracer
in direction ϑ at a diffusion rate κ. That is to say, following the dynamics
prescribed by eq. (10) and eq. (11) a Gaussian tracer patch of half width
σ(t0 ) would be spread over time in direction ϑ so that its half width in
direction ϑ goes as
σϑ (t0 + ∆t)2 = σ(t0 )2 + 2κ ∆t.
(13)
Now, our parameterization is fully defined provided we know how to specify
ϑ and κ at every grid point and time-step. In order to determine those
variables, we will examine in the following section the effect of differential
advection 4 on the separation of particle pairs. More precisely, we examine
this problem for initial separations of the order of the grid scale we target
and over short time intervals. This is because, eq. (11) being computed at
every time step, we only require ϑ and κ to be valid over short time intervals.
That is to say, our approach does not require a diffusive regime to be valid in
a statistical sense. Indeed, we only seek to define an instantaneous apparent
coefficient of horizontal diffusion (Okubo, 1966) in direction ϑ, following
1 d 2
σϑ ,
(14)
∆t→0 2 dt
which would be valid for short times. This point is at the very heart of
our approach. Indeed, there is no reason to expect subgrid advection to be
adequately modelled by diffusion with constant diffusivity over long times.
Nevertheless, there is observational, numerical and theoretical evidence supporting that, in the ocean, the square-separation of particle pairs initially
grows exponentially with time provided the particles are close enough to fall
into the so-called exponential regime of relative dispersion (Iudicone et al.,
2002; Ollitrault et al., 2005). In this regime, for very small ∆t compared
with the characteristic time of velocity inhomogeneity, that is to say for
κ = lim
ǫ = ∆t ||∇u|| ≪ 1,
(15)
the square-separation grows initially approximately linearly with time. One
can therefore expect to find κ and ϑ so that the square-separation in direction ϑ follows eq. (13). This is the purpose of the following subsection.
4
Herein, the term differential advection refers to the effect of spatial variations of the
velocity field on the dynamics of particles and tracers. This definition is therefore slightly
more general than the one used in atmospheric sciences.
10
2.3. Defining ϑ and κ for a two-dimensional velocity field
Consider two fluid particles in a two dimensionnal velocity field u, with
Jacobian matrix ∇u = [∂j ui ]i=1,2;j=1,2. Assume the particles are initially
close to each other and lay in the vicinity of (x0 , t0 ). Over a short time
period, the separation vector δx between the two points grows with t as
(Ottino, 1989)
δx(x0 , t0 + ∆t) ≃ δx(x0 , t0 ) + ∇u(x0 , t0 ) δx(x0 , t0 ) ∆t,
(16)
that is to say, with simplified notations,
δx ≃ S δx0
with S(x0 , t0 , ∆t) = I + ∆t ∇u.
(17)
The square separation therefore goes as
T
||δx||2 ≃ δxT
0 S S δx0 .
(18)
Note that, to first order in ǫ, we have
ST S ≃ I + 2 ∆t D
(19)
where D is the rate of strain tensor
1
D = (∇u + ∇uT ).
2
(20)
The above calculations are essentially equivalent to the one leading to the
definition of finite-time Lyapunov exponents (see e.g. Haller, 2001; Lapeyre,
2002). However, the validity of eq. (18) being restricted to short time
intervals, the expression now depends only on Eulerian quantities. The
tensor ST S being real and symmetric, it has two real eigenvalues. Let
Λ+ (x0 , t0 , ∆t) be the largest eigenvalue of ST S and ϑ+ (x0 , t0 , ∆t) the associated eigenvector5 .
Now the question is to determine the effect of differential advection on
a Gaussian tracer patch of half width σ(t0 ) over a short time interval. The
equation (18) essentially tells us that, in direction ϑ+ , the patch will be
deformed so that its half width is given by
σϑ+ (t0 + ∆t) = σ(t0 ) Λ+ (x0 , t0 , ∆t).
5
Note that eq. 19 implies that ϑ+ is also an eigenvector of D.
11
(21)
Note that the patch will also be deformed along the second principal direction of ST S. Nevertheless, here we would like to mimic only the stretching
effect of subgrid advection at the effective grid size h. This goal will be
achieved provided we can define ϑ and κ. We therefore choose ϑ = ϑ+ and
define κ in order to match eq. (13) and eq. (21). Doing so, our parameterization of subgrid advection is fully defined with
2
κ = h2 λ
and
λ=
Λ+ − 1
.
2∆t
(22)
Note that the above parameters can be used to approximate subgrid stirring
over short time intervals only. In practice, if one chooses to estimate these
parameters at every time-step, the Courant-Friedrichs-Lewy condition6 (see
e.g. LeVeque, 2002) which is enforced by the ocean model time-step will
insure that condition eq. (15) is satisfied. Importantly, it can be readily
shown that κ does not essentially depend on ∆t. Indeed, the eigenvalues
Λ± of ST S can be obtained to first order in ∆t by solving the characteristic
equation Det((1 − Λ± ) I + 2 ∆t D) = 0. Some algebraic manipulations leads
to
√
Λ± ≃ 1 + ∆t (d ± r 2 + a2 )
(23)
where d = ux + vy is the divergence of u, r = vx + uy is the rate of shear
strain and a = ux − vy is the rate of normal strain. So that, we finally have
√
(24)
λ ≃ d + r 2 + a2 .
Note that a priori λ can be negative for strongly convergent motions. It
can also be readily shown that λ/2 is the largest eigenvalue of the rate of
strain tensor D. Finally, our parameterization of subgrid advection takes
the form
√
ϑ
K2d = κ M2d
with κ ≃ h2 (d + r 2 + a2 ),
(25)
where ϑ is the eigenvector of ST S associated with Λ+ . In what follows, we
shall use the above formulation based on first order solutions. Obviously,
ϑ is also an eigenvector of the rate of strain tensor D. This point is in
agreement with Okubo (1966) who also noted that a tracer patch starts to
diffuse most rapidly in the direction of the major principal axis of D.
6
aka the CFL condition
12
3. Formulating the proposed subgrid operator
3.1. Two-dimensional formulation of the parameterization
At this point, it is worth formulating more explicitely the set of equations
that define our parameterization of subsmesoscale stirring. The proposed
anisotropic diffusion operator is obtained from equation (25) by explicitely
computing the eigenvector ϑ associated with λ. It is straightforward to show
from equations (19) and (23) that, to first order in ǫ parameter defined in
(15), the eigenvector ϑ = (ϑ1 , ϑ2 ) of ST S associated with λ verifies
−q ϑ1 + r ϑ2 = 0
(26)
r ϑ1 − p ϑ2 = 0
with p, q referring to
√
p = r 2 + a2 + a
and
q=
√
r 2 + a2 − a.
(27)
Here, as above, r = vx + uy is the rate of shear strain of the horizontal
velocity field u and a = ux − vy is the rate of normal strain. Assuming that
ϑ21 + ϑ22 = 1 and noting that r 2 = pq then leads to
ϑ21 =
p
,
p+q
ϑ22 =
q
p+q
and ϑ1 ϑ2 =
r
p+q
(28)
From equation (25), the two-dimensional formulation of the parameterization now takes the form
h2
p r
K2d ≃
(1 + δ)
,
(29)
r q
2
where, as above, h is the effective horizontal grid size. For conciseness, the
non dimensional divergence parameter δ has been introduced as follows
δ=√
d
+ a2
r2
(30)
with d = ux + vy referring to the divergence of the horizontal velocity
field. Note that this formulation has an obvious limit for weakly divergent
flows for which δ << 1. For instance, quasi-geostrophic flows will fall into
this regime. We also stress that this set of equations can alternatively be
considered as a diagnostic relation for estimating submesoscale stirring due
to mesoscale flows.
13
In practice, it is important to recognize that there is some arbitrariness
in definition of the the effective grid size h. One would obviously think of
using the actual horizontal grid spacing so that h2 = dx dy. But, as stated
above, the missing term M depends on the tracer advection scheme being
used for discretizing the tracer advection equation in the model. Therefore
the best choice for h should depend on the tracer advection scheme. In
particular, we think the effect of our parameterization of subgrid stirring
should be reduced if one uses some high order subgrid reconstruction as in
Prather (1986)’s advection scheme. For practical applications, we propose
to introduce an O(1), model-dependent constant C and define h2 = C dx dy.
3.2. Three dimensional formulation of the parameterization
In the previous subsection, we have proposed a parameterization for including the effect of subgrid stirring in a two-dimensional framework. We
now need to specify how this parameterization should be applied to a threedimensional primitive equation model. One possible approach for treating
the three-dimensional case could be to apply the methodology introduced
in the previous section for two-dimensional velocity fields to the full threedimensional velocity. Instead, one could simply use the two-dimensionnal
formulation and apply an epineutral rotation to the two-dimensional operator. Indeed, the current practice for developing a parameterization of
unresolved physical processes in ocean climate models is to distinguish diabatic and adiabatic processes (see e.g. Griffies, 2004). In the ocean interior,
the transport associated with mesoscale eddies is generally considered to
be an adiabatic process (Solomon, 1971) and should therefore be included
in models as fluxes along the locally referenced potential density surfaces,
or equivalently along the epineutral plane7 . As pointed out by Eden et al.
(2007), this does not imply that actual eddy fluxes in the ocean have no
component across locally referenced potential density surfaces. The socalled rotational component of eddy fluxes are indeed practically very significant. Nevertheless, McDougall et al. (2007) have clearly made the point
that those components should not be included in ocean models. In the
surface layers, the transport associated with mesoscale eddies is probably
a diabatic process because of the geometry of the mixed layer (Treguier
et al., 1997) and because of air-sea fluxes (Greatbatch et al., 2007). But
no clear consensus has emerged so far about how to specify this diabatic
7
The epineutral plane is defined locally; it is tangent to the locally referenced potential
density surface.
14
contribution in ocean climate models. Hence, according to current practice
in ocean modelling and from a practical standpoint, it is essential for a
parameterization of the stirring associated with mesoscale eddies not to induce fluxes across locally referenced potential density surfaces in the ocean
interior. This property is obviously achieved by an implementation of the
proposed parameterization in an isopycnal coordinate model to the extent
that its vertical coordinate variable matches the locally referenced potential
density. In σ or z-coordinate models, this property should be enforced by
rotating the fluxes along the locally referenced potential density surfaces.
We stress that we do not discuss in this paper how to connect the ocean
interior with the surface layers. This issue is the subject of a debate which
goes far beyond our present discussion (Gnanadesikan et al., 2007; Ferrari
et al., 2008).
The transformation which allows to rotate a two-dimensional diffusion
operator in the epineutral plane is originally due to Redi (1982) for isotropic
diffusion. This transformation, and its small slope version, is widely used
in the ocean modelling community. But, surprisingly, there is not not much
literature about rotating an anisotropic diffusion operator along the epineutral plane. Using the coordinate transform introduced by De Szoeke and
Bennett (1993) in the small isopycnal slope limit, Smith (1999) argues in
the appendix of his paper that a two-dimensional horizontal operator K2D
can be rotated into the three-dimensional operator
K2D
K2D · Σ
.
(31)
K3D =
K2D · Σ Σ · K2D · Σ
Here, Σ is the isopycnal slope vector Σ = (Sx , Sy ) = (−ρx /ρz , −ρy /ρz ) with
ρ referring to the locally referenced potential density. This transformation
is also used by Smith and Gent (2004). Going back to our derivation,
applying the transformation (31) to (25), a three dimensional version of our
ϑ
parameterization is obtained in the form K3d = κ M3d
with
 2

ϑ1
ϑ1 ϑ2 ϑ1 Sϑ
ϑ
ϑ22 ϑ2 Sϑ  ,
≃  ϑ1 ϑ2
M3d
(32)
2
ϑ1 Sϑ ϑ2 Sϑ Sϑ
where Sϑ = (ϑ1 Sx + ϑ2 Sy ) is the slope of isopycnal in direction ϑ = (ϑ1 , ϑ2 )
An alternate derivation of equation (32) following the geometrical approach
originally introduced by Redi is also given in appendix A. The three dimensional anisotropic diffusion tensor K3D is then fully defined from the
15
diffusivity κ, the direction of tracer stirring ϑ and locally referenced potential density gradients. It is important to stress that the decoupling of
horizontal spatial directions which is a convenient property of the small
slope limit of Redi’s operator no longer holds with the above formula.
In z−coordinate ocean models, we proposed to use the three-dimensional
epineutral formulation of the parameterization, which consists in a anisotropic
diffusion operator of the form


p
r
p Sx + r Sy
2
h
 , (33)
r
q
r Sx + q Sy
(1+δ) 
K3d ≃
2
2
2
p Sx + r Sy r Sx + q Sy p Sx + q Sy + 2r Sx Sy
where Sx , Sy are the neutral slopes in x and y directions respectively as
defined in eq. (44). The operator (33) has been obtained by replacing (28)
in the three dimensional form (32).
4. Diagnosing the subgrid operator in an eddy resolving model
simulation
Before actually implementing the parameterization in a fully prognostic model, worthwhile information can be provided by diagnosing an eddy
resolving ocean model simulation. In this section, we present diagnostics
based on a reference experiment of the global DRAKKAR-ORCA025 model.
The ocean sea-ice code is based on the NEMO framework (Madec, 2008).
The model configuration DRAKKAR-ORCA025 is described in Barnier
et al. (2006). This model configuration uses a global tripolar grid with
1442×1021 grid points and 46 vertical levels. The horizontal grid is isotropic
in the southern hemisphere and quasi-isotropic in the northern hemisphere.
The horizontal resolution is 1/4◦ at the equator and varies with the cosine
of the latitude. The vertical grid spacing is finer near the surface (6 m)
and increases with depth to 250m at the bottom. The reference experiment
considered herein has been performed by the DRAKKAR group. It was run
for almost 50 years (1958-2007) with an interannually varying forcing based
mostly on the ERA40 dataset as described in Brodeau et al. (2010). This
experiment8 has been presented and discussed in a series of papers focusing on both the global scales (see e.g. Lombard et al., 2009; Penduff et al.,
2010) and on regional scale processes, in particular in the Southern Ocean
8
which is known as DRAKKAR-ORCA025-G70 model run
16
(Treguier et al., 2007; Renner et al., 2009). The diagnostics presented in this
section are based on years 1990 to 1995 of the simulation. The model data
are available at the global scale, but in this paper we will only discuss the
subgrid diffusivities obtained in the Southern Ocean. This is because there
is a vivid debate about the intensity and the spatial distribution of lateral
stirring due to mesoscale flows in the Southern Ocean (Naveira-Garabato
et al., 2010; Marshall et al., 2006; Sallée et al., 2008; Shuckburgh et al.,
2009; Abernathey et al., 2010). Note that, due to the variation of the grid
size with latitude, the model resolution ranges from 24km at 30◦ S to 13.8
km at 60◦ S, and even gets to 7 km in the Weddell and Ross Seas. In the
Southern Ocean, the model configuration can therefore be considered to
be marginaly eddy resolving. As a first step, we only discuss the horizontal
subgrid operator K2d . Therefore, hereafter for conciseness, K refers to K2d .
The first property of the proposed subgrid operator we stress is its effect
on tracer variance. For a given tracer field τ , the associated tracer variance
is defined as φ = τ 2 /2. In practice, the subgrid operator we introduce is
almost always positive and can only decrease tracer variance. Indeed, as
demonstrated in appendice C, if one uses a formulation of the type eq. (10),
the local tracer variance production term has the form
Πφ = −(K ∇τ ) · ∇τ.
(34)
Now, if one uses a subgrid operator K of the form (25), it is straightforward
to show that
√
Πφ = −κ||∇τ ||2 sin2 θ with κ = h2 (d + r 2 + a2 ),
(35)
where θ is the instantaneous angle between ϑ and the ∇θ in the horizontal plane. The sign of Πφ is therefore only determined by the magnitude
of the horizontal velocity divergence. Diagnostic of the magnitude of the
horizontal velocity divergence |d| based on 5 days averages of DRAKKARORCA025 model run (not shown) indicate that the divergence is enhanced
close
to the boundaries9 but remains mostly smaller that the rate of strain
√
r 2 + a2 . More precisely the instantaneous value of κ are found to be
positive in about 99% of the grid boxes in the Southern Ocean sector10 of
9
that is to say, in the vicinity of the lateral boundaries and close to the ocean floor
where large deviations from the quasi-geostrophic regime are indeed expected.
10
In what follows, 30◦ S is considered to be the northern limit of the Southern Ocean
sector of the global ocean.
17
DRAKKAR-ORCA025 model configuration. We therefore conclude that
in practice, our subgrid operator is positive and therefore decreases tracer
variance.
We now turn to the actual values of our subgrid operator K as diagnosed
from DRAKKAR-ORCA025 model run. We stress again that the following
results are based on offline diagnostics of the model simulation based on
5 days averaged model outputs. The subgrid operator was not used in
the prognostic equations of the model during its integration. Fig. 3 shows
the values and spatial distribution of the norm11 of the subgrid operator
K at the surface of the Southern Ocean in DRAKKAR-ORCA025 model
run. Both the instantaneous diffusivity ||K|| and the mean diffusivity ||K||
exhibits locally large values of a few thousand m2 s−1 . Nonetheless, the peak
value of the histogram of surface diffusivity in the Southern Ocean (not
shown) is at about ∼400 m2 s−1 for both the instantaneous and the mean
fields. Both fields vary spatially over relatively short scales although the
mean diffusivity appears to be smoother than the instantaneous diffusivity.
In a broad sense, the spatial patterns we obtained are consistent with what
could be expected from previous estimates of surface diffusivities with large
values in regions of intense mean flow and eddy activity.
A most important property of the subgrid operator introduced herein is
its being essentially anisotropic. Indeed, the instantaneous operator K induces tracer transport in one single direction, namely the stretching eigendirection of the rate of strain tensor D. Still, when averaged over 6 years,
the subgrid operator is only weakly anisotropic as illustrated in Fig. 4. The
most frequent ratio between the two eigenvalues of K is about .5. In order
to better understand the physical meaning of our subgrid parameterization,
we can split the time-averaged anisotropic diffusion tensor K into
e
K = Km + K,
(36)
where Km is the anisotropic diffusion tensor diagnosed from the 6-years
time-averaged velocity field. The subgrid tensor due to the time-mean
flow Km accounts for the tendency of the time-mean flow to expand tracer
patches in certain specific directions due to a mechanism similar to sheardispersion. More precisely, shear-dispersion usually refers to the coupled
effect of differential advection and transverse mixing (Okubo, 1968; Young
11
In what follows, the norm of K refers to the value of its largest eigenvalue and
therefore represents its maximum rate of diffusivity.
18
Figure 3: Norm of the anisotropic diffusion tensor as computed with DRAKKAR 1/4◦
model outputs. top panel : diffusivity (m2 s−1 ) ||K|| diagnosed from a 5 days mean
centered on 4 June 1992. bottom panel: mean diffusivity (m2 s−1 ) ||K|| diagnosed from 5
days means and averaged over years 1990 to 1995 of the simulation. The time averaging
was performed on the two-dimensional operator before computing its norm. Note the
different colorscale in both plots.
19
Figure 4: Ratio of the diffusion rates λmin /λmax associated with the mean subgrid operator diagnosed from the 1990 to 1995 mean anisotropic diffusion tensor at the surface and
south of 30◦ S. λmin and λmax refer to the eigenvalues of K. As illustrated schematically,
the peak value of the histogram corresponds to a ratio of about 0.5. The time averaged
anisotropic diffusion tensor therefore appears to be only weakly anisotropic in most of
the Southern Ocean.
20
Figure 5: Schematic illustration of the shear-enhanced dispersion along the flanks of a
zonal jet. The jet is figured by the dashed isoline. The grey shaded zones are regions of
enhanced stirring. This illustration explains the spatial structure of the subgrid stirring
induced by the mean component of the flow Km .
et al., 1982). Here, we only discuss the effect of differential advection due to
spatial variations of the velocity field at the grid scale. A typical geometry
which leads to significant differential advection by the time-mean flow is
illustrated in Fig. 5, which shows how the mean shear is enhanced along
the flanks of √
mesoscale jets. The norm of our subgrid operator being proportional to r 2 + a2 , K is therefore usually found to be significant along
the flanks of the mean jets. Hereafter, we therefore refer to Km as to the
mean-flow shear-dispersion diffusivity. This part of K which describes the
enhanced stirring in certain specific directions is intrinsically anisotropic.
e that is to say the deviation to Km as
Now, it is interesting to look at K,
defined in eq. (36). It is straightforward to show that this eddy contribution
to K is actually an isotropic diffusion operator with a diffusion rate κ̃ of
the form
2 √
p
e = h ( r 2 + a2 − r2 + a2 ).
κ̃ = ||K||
(37)
2
Fig. 6 shows the magnitude and the spatial structure of κ̃ at the surface
of the Southern Ocean in DRAKKAR-ORCA025 model simulation. This
figure also illustrates the very close correspondence between the patterns of
κ̃ and of the square root of the eddy kinetic energy. The eddy contribution
e therefore appears to be mostly governed by the actual level of variance
||K||
in the mesoscale velocity field. Note that a similar correspondance is also
observed with some estimates of Taylor diffusivity in the Southern Ocean
21
Figure 6: Kinematic eddy-diffusivity and eddy velocity scale. top panel : diffusivity
e as diagnosed with DRAKKAR 1/4◦ global model over years 1990-1995.
(m2 s−1 ) ||K||
bottom panel : square root of the eddy kinetic energy (m s−1 ) over the same time period.
Note the close correspondance of the patterns of the two fields.
22
Figure 7: Vertical profile of the diffusivity (m2 s−1 ) diagnosed with DRAKKAR 1/4◦
global model over years 1990-1995 and averaged at constant pressure south of 30◦ S.
(Sallée et al., 2008)12 . In what follows, we therefore refer to this contribution
e as to the kinematic eddy-diffusivity. We stress that this part of K, which
K
accounts for the correction with respect to Km due to the fluctuations of
the velocity field, is instrinsically isotropic.
There is an ongoing debate about the vertical structure of lateral diffusivity in the Southern Ocean (Naveira-Garabato et al., 2010). Of particular
interest is the question of the possible enhancement of lateral diffusivity
near the steering level of linear perturbations at middepth within the ACC
(Smith and Marshall, 2009; Abernathey et al., 2010; Ferrari and Nikurashin,
2010). Fig. 7 shows the vertical profiles of the various contributions to our
subgrid operator averaged over the Southern Ocean sector of DRAKKAR
1/4◦ global model. This figure illustrates that, on average over the Southern Ocean, all the contributions to the instantaneous diffusivity are surface
intensified and do not exhibit any intensification at mid-depth. Such a
vertical structure of lateral diffusivity in the Southern Ocean is consistent
with several other studies. Surface intensification of lateral diffusivity is
found in the high resolution model study of Griesel et al. (2009). Some
12
However, Griesel et al. (2010) do not find a similar correspondence, despite using an
approach similar to Sallée et al. (2008).
23
indications suggesting a surface intensification of lateral diffusivity are also
provided by Zika et al. (2009). Interestingly, the surface intensification of
lateral tracer diffusivity found in our study is reminiscent of the vertical
profile of Gent-McWilliams diffusivity inferred by several authors with inverse methods (Ferreira et al., 2005; Olbers and Visbeck, 2005) and used
in some theoretical studies (Marshall and Radko, 2003)13 . We stress that
the profiles presented in Fig. 7 are obtained with a spatial average over the
whole Southern Ocean and might not be representative of individual profiles
at specific locations14 .
From a physical perspective, for a given tracer τ , the only relevant quantity is actually the divergence of the tracer flux associated with the subgrid
operator ∇ · (K ∇τ ) and not the diffusivity itself. One might also be more
interested in the contribution of the subgrid tracer fluxes to the time-mean
tracer budget. We therefore introduce herein the decomposition of the timemean tracer flux into contributions associated with the diffusivities introduced previously, namely
e ∇τ + K′ ∇τ ′ ,
K ∇τ = Km ∇τ + K
(38)
∇ · (K′ ∇τ ′ ) ≃ ∇ · (Kτe ∇τ ).
(39)
where the primes designate the deviations to the time-mean quantities. For
instance, K′ is the deviation to K according to K = K + K′15 . Eq. (38)
illustrates that besides the contributions of the mean-flow shear-dispersion
diffusivity and the kinematic eddy-diffusivity, there is an additional term in
the time-mean tracer bugdet which involves the correlations of the subgrid
operator with the tracer gradients. One could tentatively define a diffusivity
operator Kτe to account for this additional term so that
We stress that such a diffusivity operator Kτe is a priori dependent on the
tracer field being considered. In what follows, we therefore refer to the last
r.h.s term of eq. (38) as to the effect of a tracer-correlated eddy-diffusivity
operator. Of much interest for ocean modellers is the potential impact of a
subgrid transport parameterization on the transport of heat in a model simulation. Getting this information obviously requires implementation of the
13
One should nonetheless be aware that lateral tracer diffusivity and Gent-McWilliams
diffusivity are only equal under certain very specific circumstances.
14
The surface intensification might also not hold for the tracer-correlated eddydiffusivity introduced below.
15
e + K′ .
Note that we now have K = Km + K
24
Figure 8: Depth-integrated northward heat transport (PW) associated with the proposed
◦
subgrid
R y ′ R parameterization as diagnosed with DRAKKAR 1/4 model. The heat transport
dy dz ρ0 Cp Fθ · ny (with ny pointing northward) has been plotted for each flux
contribution Fθ indicated in the legend.
subgrid parameterization prognostically in an ocean model. Nevertheless,
we could expect to get some indications of the spatial structure and order
of magnitude of the subgrid operator from diagnostics of model simulations
which have been run without the subgrid parameterization. Fig. 8 presents
estimates of the depth-integrated northward heat transport associated with
the various flux contribution in eq. (38) for potential temperature θ as diagnosed in the Southern Ocean sector of DRAKKAR 1/4◦ global model. First,
Fig. 8 shows that the time-mean heat transport associated with our subgrid
operator is comparable in magnitude with the actual explicit contribution
of eddies fluxes in the model, namely ∇ · (u′ θ′ ), with zonally-integrated
values of about .5PW. Notably, at the latitude of Drake passage (55◦ S65◦ S), the value of about .1PW is consistent with the estimates of subgrid
diffusive heat transport with this particular model configuration (Treguier
et al., 2007). The divergence of the depth-integrated heat flux corresponding to our operator exhibits local maxima of a few thousand W.m−2 (not
shown). The above values should be compared with existing estimates of
eddy heat transport and eddy heat flux divergence, as, for instance, the one
provided by Stammer (1998), Jayne and Marotzke (2002) or Meijers et al.
25
(2007). Our result are therefore suggestive that a significant part of the heat
flux carried by mesoscale flows is actually occurring at smaller scale than
the resolution required for explicitly resolving the mesoscale flows in ocean
models. Second, for what concerns heat transport, a large contribution to
the proposed subgrid tracer flux is due to the mean-flow shear-dispersion
diffusivity. Third, the contribution of the tracer-correlated eddy-diffusivity
to heat transport appears to be notably smaller than the two other contributions. It is not clear to what extent this last result would hold for
other tracers as well. This question obviously requires further work. We
also acknowledge that the above picture might be different if the subgrid
parameterization had been used in the prognostic equations of the model
during its integration. Still we think that the above results are suggestive
that the effect of our parameterization of subgrid stirring might be significant on the large scale heat budgets.
5. Conclusion and discussion
In this paper, we have proposed a parameterization of subgrid tracer
stirring for use in eddy-resolving ocean climate models. This parameterization is meant to make use of the information about tracer stirring at
the submesoscale which is held by the time-varying mesoscale velocity field
computed by the ocean model. We have shown that for small enough time
intervals, the stretching effect of stirring by the mesoscale flow is robustly
equivalent to the effect of an anisotropic diffusion operator whose parameters can be readily estimated from the resolved velocity field. This leads
to the two-dimensional version of the proposed parameterization of subgrid
stirring with a time-varying anisotropic diffusion operator. In three dimensions, the two dimensional anisotropic diffusion tensor should be rotated in
the epineutral plane. We have provided the formulae for performing this
rotation in the epineutral plane. The elements of the proposed anisotropic
diffuson operator have been diagnosed from the outputs of a 1/4◦ resolution
global simulation performed by the DRAKKAR group (Barnier et al., 2006).
These diagnostics have shown that, for a 1/4◦ resolution ocean model, the
proposed parameterization produces strongly inhomogeneous diffusivities
with magnitude up to a few thousands m2 s−1 . The most common values
have been shown to be ∼ 400m2 s−1 which is close to the value which is
usually used for this model configuration. Our offline diagnostics have revealed that the anisotropic diffuson operator remains strongly anisotropic
even when averaged over long times. In practice, our subgrid parameter26
ization is positive definite. The time-averaged effect of the proposed subgrid operator on tracer budgets has been shown to combine the effect of
the anisotropic mean-flow shear-dispersion diffusivity, of the isotropic kinematic eddy-diffusivity and of a tracer-correlated eddy-diffusivity. The first
two contributions are found to be mostly surface intensified in the Southern
Ocean. The kinematic eddy-diffusivity is due to the nonlinearity of subgrid
stirring with respect to the velocity field and therefore appears because of
the time-averaging. The kinematic eddy-diffusivity is then governed by the
level of variance of the velocity field and appears to be mostly proportional
to the square root of eddy kinetic energy. Further diagnostics suggests
that the combination of the above diffusivities might impact significantly
on heat budgets of eddy resolving ocean models. The subgrid parameterization should therefore be tested in prognostic mode. This is the purpose
of a companion paper.
Some remarks are necessary concerning the relation of our parameterization with some subgrid operators proposed in the literature. First, our
parameterization is obviously closely related to Smagorinsky (1963, 1993)’s
seminal papers. Indeed, Smagorinsky introduced the idea that lateral diffusivity16 should be parameterized with an isotropic operator following the
scaling
√
κsm ∝ h2 r 2 + a2
(40)
with the notations of section 3. Our two dimensional parameterization
(eq. (29) and following equations) can therefore be seen as a generalization
of Smagorinsky’s idea with two new features. On the one hand, our parameterization also depends on the divergence of the velocity field. On the other
hand, the subgrid operator we propose is intrinsically anisotropic which is
not the case of eq. (40). There is also a close link between our approach and
the strain diffusivity operator originally introduced by Leonard and Winckelmans (1999) and further developed by Eyink (2001), Dubos and Babiano
(2002), Bouchet (2003) and Chen et al. (2003). Within the framework of
two-dimensional turbulence theory, the above authors introduced the idea
of parameterizing M(τ, u) (see eq. (7)) in the form
M(τ, u) = ∇ · (KSD · ∇τ )
with
KSD ∝ h2 D.
(41)
In particular, Dubos and Babiano (2002) (see also Dubos, 2001) derived
this model for M(τ, u) with an asymptotic expansion in the filter size of the
16
The most common implementation of Smagorinsky’s idea is for viscosity but the
scaling can be applied to tracer diffusivity as well.
27
spatially filtered advection equation (7). Herein, our formulation is derived
with an implicit asymptotic expansion for small time intervals. Still, our
parameterization can be considered as a modified strain diffusivity. Indeed,
as noted previously, eq. (20) and eq. (19) obviously imply that ϑ is an
eigenvector of D. One can straightforwardly show that the anisotropic
ϑ
diffuson tensor K2d = κ M2d
we propose is essentially a projection of KSD
which retains only the effect of KSD along the eigen-direction17 associated
with the eigenvalue λ. Interestingly, numerical tests performed by Dubos
(2001) have revealed that strain diffusivity was numerically unstable because
of the tracer variance production that arises along the compression axis of
D. This ill-conditionned nature of the strain diffusivity operator is certainly
a reason why it has not been tested in ocean models yet. As shown in
section√4, our two-dimensional operator decreases tracer variance provided
−d < r 2 + a2 which is typically verified in oceanic mesoscale flows. The
constraint on variance production a posteriori justifies our neglecting the
compression along the compression axis of D in section 2.2. Therefore
our two dimensional parameterization is essentially a regularization of the
strain diffusivity operator.
A consequence of the relation of our parameterization with both Smagorinsky diffusivity and the strain diffusivity operator is that although our
heuristic derivation illustrated in Fig. 1 is a priori valid for tracer advection only, we can think of applying a similar procedure to the momentum
equation too. Indeed, both Smagorinsky’s scaling and Dubos and Babiano
(2002)’s asymptotic expansion should in principle hold for the momentum
equations as well. A promising approach for applying our subgrid closure
to the momentum equations is to consider mixing of potential vorticity instead of momentum, as first proposed by Welander (1973). Indeed, one
could think of applying our parameterization to both, passive tracers, as
considered in this paper, and potential vorticity q. If applied with care regarding momentum budgets, such a procedure provides a parameterization
of momentum transfers, and even allows to represent up-gradient momentum fluxes (Wardle and Marshall, 2000). One could therefore expect that
applying our subgrid closure to potential vorticity could affect the appearance and dynamics of zonal jets (Eden, 2010). The relevance of applying an
operator of the form (25) as a closure for potential vorticity is supported by
the results of Chen et al. (2003) who suggested that the stretching of vorticity gradients along the stretching axis of the rate of strain tensor D is the
17
For weakly divergent flows this is essentially the stretching axis
28
main mechanism responsible for the enstrophy cascade in physical space.
The alignment of subgrid eddy fluxes of potential vorticity with the nonlinear combination of resolved gradients D · ∇q observed in direct numerical
simulations of geostrophic turbulence by Nadiga (2008) is also encouraging.
Nevertheless, extending our parameterization to the momentum equations
is not straightforward. Indeed, potential vorticity is not a prognostic variable of ocean climate models. Therefore, care has to be taken in order not to
introduce spurious forces in the momentum budgets when applying an eddy
potential vorticity closure. This problem has already been noted and several
different approaches have been proposed for enforcing this constraint (Wardle and Marshall, 2000; Eden, 2010). Still, no clear consensus has emerged
so far as to which of these methods is more appropriate. Another potential
problem is related to the relevance of the quasi-geostrophic approximation
at the subgrid scales being considered (<10km). Indeed, turning a potential
vorticity flux into a momentum flux requires to use a balance condition for
applying the invertibility principle for potential vorticity. Obviously, it is
unclear whether the geostrophic balance is applicable for inverting potential
vorticity at the subgrid scale (<10km) in eddy resolving models. For all the
above reasons, further work is required for applying our subgrid closure to
momentum equations.
As a concluding remark, we emphasize that the two-dimensional anisotropic diffusion operator defined in eq. (29) can also be considered as a diagnostic relation for estimating how mesoscale flows are stirring tracer fields
at submesoscales from model data or from observations. More precisely, our
diagnostics allow to quantify a fraction of the stirring by mesoscale flows,
namely the fraction of mesoscale stirring occurring in flow features smaller
than a given scale h. In that sense, the diagnostics performed in section 4
can be put into perspective with published estimates of lateral eddy stirring in the Southern Ocean. Obviously, the comparison should be taken
cautiously because (i) our approach does not quantify the stirring at scale
larger that h and (ii) intrinsic submesoscale dynamics might also affect
tracer transport at submesoscales. Various methods have been used in the
literature for estimating lateral diffusivity in the Southern Ocean from satellite altimetry, float trajectories or eddy resolving numerical model output.
The methods are based on very different theoretical backgrounds which fall
into three distinct categories. A first set of studies rely on scalings based
on mixing length arguments to estimate lateral diffusivity from eddy statistics (Keffer and Holloway, 1988; Stammer, 1998; Naveira-Garabato et al.,
29
2010). A second thread of methods rely on Taylor’s theory of homogeneous
isotropic turbulence for estimating lateral diffusivity from Lagrangian velocity autocorrelation (Sallée et al., 2008; Griesel et al., 2010). A third
approach is based on Nakamura’s approach which diagnoses the effective
diffusivity from tracer contour elongation in flows dominated by advection
(Marshall et al., 2006; Shuckburgh et al., 2009; Abernathey et al., 2010).
There is still no consensus on the amplitude and on the patterns of lateral diffusivity in the Southern Ocean. All the above studies indicate that
stirring is spatially inhomogeneous. Most of them assume18 to a certain
degree that the lateral stirring can be modelled with an isotropic diffusivity operator. The diagnostics presented in section 4 quantify the lateral
stirring occurring at scales smaller than a given scale. The process we diagnose is therefore a fraction of the stirring by mesoscale flows. This fraction
depends on a given cut-off scale h. It is therefore not obvious how our estimates should be quantitatively compared with other published estimates of
lateral stirring. Nonetheless, our study brings several new pieces of information in the debate. First, our diagnostics suggest that a significant fraction
of the transport by mesoscale flows in the Southern Ocean could be associated with tracer features of scale smaller than our model grid size (namely
about 18km at 50◦ S). Second, our results show that shear-dispersion by the
time-mean mesoscale flow, that is to say the mean jets, is a important contribution to the stirring by mesoscale flows in the Southern Ocean. Third,
our results unequivocally show that the subgrid transport operator in eddy
resolving ocean models should be anisotropic because the stirring at submesoscale by mesoscale flows is intrinsically anisotropic even when averaged
over long times. We therefore postulate that some of the discrepancies between the lateral diffusivity estimates in the Southern Ocean might be due to
the extent to which a given lateral diffusivity estimation method projects
this anisotropic information onto an isotropic operator. Clearly, further
work is required in order to fully understand the relation of our diffusivity
estimates with Nakamura diffusivity and Taylor diffusivity. In that perspective, we stress that our estimate is based on strictly the same information as
Nakamura diffusivity estimates, namely the time-varying mesoscale velocity
field. It should therefore be possible to reconcile both approaches. We also
stress the encouraging similarity between the kinematic eddy-diffusivity we
introduced and Taylor diffusivity estimates (see e.g. Sallée et al., 2008). As
18
This is not the case of the studies based on Nakamura’s approach which intrinsically
quantifies the cross-stream diffusivity.
30
a conclusion, we have noted above a significant fraction of the stirring by
mesoscale flows is actually occurring at scales smaller than the mesoscales.
Our work provides a method for estimating these submesoscale tracer fluxes
from altimetric data with the operator (29). From an observational standpoint, the additional contribution K∇τ to eddy tracer flux estimates should
be added to the more classical contribution u′ τ ′ . The study of this contribution from satellite data is the purpose of an ongoing study to be published
elsewhere.
Acknowledgement
The authors made extensive use of some tools provided by the open
source software community. In particular, symbolic computations have been
performed with Sage (Stein et al., 2009). The diagnostics and plots have
been produced with Python (Oliphant, 2007) together with the packages
Numpy and Matplotlib. JLS hereby thanks J.M. Brankart for his helpful
advice on algebraic manipulations, J. Zika for his various comments and E.
Shuckburgh for a discussion on the details of effective diffusivity diagnostics.
FdO wishes to thanks A. Babiano for fruitful discussions. The authors
would also like to thank the three anonymous reviewers and S. Griffies whose
comments and suggestions during the review process have notably improved
earlier versions of the manuscript. JLS, FdO and GM are supported by the
CNRS. This work was funded by the ANR through contract ANR-08-JCJC0777-01.
A. Epineutral anisotropic tracer diffusion in z-coordinate ocean
models
In this section, we describe how to perform the rotation of an anisotropic
diffusion operator along the epineutral plane in level-coordinate models following the approach originally introduced by Redi (1982). Following Redi
(1982), given a two-dimensional anisotropic diffusion tensor of the form
ϑ
K2d = κ M2d
, one can apply this operator in the epineutral plane by using
ϑ
a three-dimensional anisotropic diffusion operator of the form K3d = κ M3d
ϑ
where M3d is defined as
!
ϑ 0
ϑ
M3d
= RS M2d 0 RS T .
(42)
0 0 0
31
Figure 9: The rotation operators Rα , Rβ and Rγ and the relative orientation of the
coordinate systems.
Here RS is the matrix operator which rotates the coordinate frame from the
epineutral coordinate system (xn , yn , zn ) to the geodesic coordinate system
(xg , yg , zg ). More precisely, RS combines three successive elementary rotations as illustrated in Fig. 9 : a rotation by −γ about axis zn to transform
a vector in the epineutral coordinate system into the intermediate coordinate system (x̃n , ỹn , zn ) where x̃n is pointing down the epineutral plane
when viewed from the geodesic system; a counterclockwise rotation by −β
about axis ỹn to get the vector in the coordinate system (xα , ỹn , zg ); and a
rotation (positive if counterclockwise) about zg by angle −α which finally
rotates the vector into the geodesic coordinate system. That is to say, RS
is given by the matrix product




cos α − sin α 0
cos β 0 − sin β
cos γ − sin γ 0
1
0   sin γ cos γ 0 .(43)
RS =  sin α cos α 0  0
0
0
1
sin β 0 cos β
0
0
1
The above notations have been chosen in order to follow closely Redi (1982)’s
original paper. In particular, we stress that α and β have the same definition herein and that the operator R introduced by Redi (1982) is equal to
the product Rαβ = Rα Rβ with our notations. In our case, the additional
ϑ
elementary rotation Rγ is necessary because M2d
is not isotropic contrary
to the operator considered in Redi (1982). Obviously19 , in the limit of small
19
This is true by continuity because we have RS = I for β = 0
32
β, we should have α ≃ −γ. Following Redi (1982), we also introduce
Sx = cos α tan β = −ρx /ρz
and
Sy = sin α tan β = −ρy /ρz ,
(44)
where ρ designates the locally referenced potential density. Importantly,
ϑ
note that the general form of M2d
is preserved by a rotation in the (xn , yn )
plane. We can therefore define ϑ̃ = (ϑ̃1 , ϑ̃2 ) so that
Rγ
ϑ 0
M2d
0
0 0 0
!
ϑ̃ 0
M2d
0
0 0 0
Rγ T =
!
.
(45)
ϑ
This notably simplifies the calculation of M3d
from eq. (42) since we now
ϑ
only need to combine M2d with Rαβ as a first step and then replace ϑ̃ by its
value as a function of ϑ. Moreover, to first order in β, one can show that,
ϑ̃1 ≃ (ϑ1 Sx + ϑ2 Sy )/S
where
S=
and
q
ϑ̃2 ≃ (ϑ2 Sx − ϑ1 Sy )/S
Sx2 + Sy2 .
(46)
(47)
Some cumbersome but straightforward algebra then leads to equation(32)20 .
B. Epineutral diffusion with κx 6= κy
Although implicitly implied by Smith (1999)’s apendix, the epineutral
diffusion operator with different rates in the zonal and meridional directions
is, to our knowledge, not described practically in the literature. Redi (1982)
has explained how to rotate an isopycnal diffusion operator of the form
1 0
.
(48)
K2d = κ
0 1
In this paper, we have treated the case where
2
ϑx
ϑx ϑy
ϑ
K2d = κ M2d = κ
ϑx ϑy
ϑ2y
20
(49)
For deriving equation eq. (32), we assumed that cos β ≃ 1 which is essentially similar
to a first order Taylor expansion for small β. This is equivalent to the small slope limit
of Redi (1982).
33
with ||ϑ|| = ϑ2x + ϑ2y = 1. The purpose
any symmetric operator of the form
a
K2d = κ
c
of this appendice is to stress that
c
b
(50)
can be decomposed into a sum of operators of the form eq. (49). In particular,
κx 0
K2d =
(51)
0 κy
1,0
0,1
with κx 6= κy , can be decomposed into K2d = κx M2d
+ κy M2d
. Therefore,
we get


κx
0
κx Sx

κy
κy Sy
K3d ≃  0
(52)
2
2
κx Sx κy Sy κx Sx + κy Sy
with
Sx = −ρx /ρz
Sy = −ρy /ρz .
Note that this result could have been expected from symmetry considerations but to our knowledge no detailed proof was available in the literature
so far.
C. Tracer variance equation
We introduce in this apendice the tracer variance production Πφ used
in section 4. Consider a tracer τ , the evolution of which follows
∂t τ + u · ∇τ = ∇ · (K ∇τ ),
(53)
where u refers to some velocity field (either two-dimensional or three dimensional), and K is a subgrid tensor (with the corresponding dimensionality).
We seek the evolution equation for the tracer variance defined as φ = τ 2 /2.
Following Dubos (2001) (especially eq. 4.13 and 4.14), let us introduce the
linear operator L such that for a given scalar field a,
La = ∂t a + u · ∇a − ∇ · (K ∇a).
34
(54)
Now consider two tracer fields a and b which follow the advection diffusion
equation (53). Obviously we have
La = 0 and Lb = 0.
(55)
The operator L being linear, it is straighforward to show that
L a b = aLb + bLa − (K ∇a) · ∇b − (K ∇b) · ∇a.
(56)
The evolution equation for tracer variance φ is then simply given by the
above relation with a = b = τ , i.e.
Lφ = −(K∇τ ) · ∇τ,
(57)
that is to say,
∂t φ + u · ∇φ = ∇ · (K ∇φ) + Πφ
with Πφ = −(K∇τ ) · ∇τ.
(58)
Hereabove, Πφ can be interpreted as a tracer variance production term.
Indeed, the form of the above equation has the interesting property that,
if the flow is assumed to be non-divergent, the evolution of the total tracer
variance in a given control volume is only due to the combination of variance
fluxes at the boundary (both advective and diffusive) and internal variance
production associated with Πφ .
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