that is the three-dimensional extension of the successful zonal-residual-mean theory of atmospheric
science. The landmark paper of Gent and
McWilliams (1990) marked the beginning of the
oceanic work on this topic and since that time
oceanographers have accepted that scalar properties should be advected by a velocity other than
the Eulerian-mean velocity.
Mesoscale eddies act to mix fluid parcels in a
way that is highly constrained by the stratified
nature of the fluid. The residual-mean theory provides the link between the different views that are
apparent from averaging these turbulent flow
fields in height coordinates and in density coordinates. It reduces the parameterization problem
from three dimensions to two dimensions and it
shows how the eddy fluxes are skew-symmetric in
height coordinates so that the total advection
velocity can be adiabatic. While mesoscale eddies
cause a very significant southward flux of heat in
the Southern Ocean, should this flux be regarded
as a mixing or a stirring of heat? To the extent
that there are balancing gradients of potential temperature and salinity along isopycnals, this component of the heat flux will eventually lead to mixing
of heat at the very smallest scales of motion and
hence we might call this aspect of the mesoscale
heat flux ‘mixing’. The rest of the heat flux that is
observed in Cartesian coordinates does not appear
as a heat flux in density coordinates, but rather is
associated with an extra horizontal velocity that
advects all fluid properties along isopycnals. This
part of the mesoscale eddy heat flux might more
properly be labelled ‘stirring’.
To understand how an eddy flux in Cartesian
coordinates might induce an extra fluid velocity,
we appeal to residual-mean flow theory, which was
originally developed in the atmospheric literature
where the averaging operator was a zonal average.
In developing the three-dimensional residual-mean
theory it is convenient to first deal with the density
conservation equation and to ignore any non-linearity in the equation of state. The Eulerian-mean
density conservation equation reads (the overbar
operator describes a low-pass temporal average)
–
t ;
H и(V
–
– );(w –
– ) z
:Q
– 9
H и(VЈЈ
—– )9(wЈЈ
—– ) z
(5.2.1)
where Q is the instantaneous diapycnal source
term and is a form of density that excludes the
effects of the compressible nature of seawater. Our
quest is to gain an understanding of the threedimensional turbulent density flux and its divergence that appears in this equation. Less than a
decade ago it was commonplace in ocean modelling to assume that the horizontal density flux,
VЈЈ
—– , could be parameterized as a down-gradient
Fickian flux, while the vertical turbulent density
flux was ignored except for the part attributed to
small-scale diapycnal mixing. This exactly horizontal mixing led to fictitious diapycnal fluxes of
density (the Veronis effect) and caused serious
problems such as spurious vertical motion in
western boundary currents. The temporal-residualmean theory demonstrated the inadequacy of this
prior approach and provided a route forward that
avoided such fictitious diapycnal density fluxes.
In order to develop a residual-mean conservation equation for, say, density, we need to realize
that the Eulerian-mean density is not the most
appropriate mean density to appear in the mean
density conservation equation that is carried by an
ocean model. The Eulerian-mean density, – (x, y, z, t),
describes a density surface whose average height is
not that of the original Eulerian averaging, namely
z. The appropriate mean density for our purposes
is the one whose surface is, on average, at the
height of the averaging. This density can be
expressed in terms of – by
ϳ : – 9(
– / –
z ) z ;O(
3 ),
where
– ϵ ᎏ
1
2
ᎏ Ј
2
— , is half the density variance measured at a fixed point in space and the terminology
O(␣
3
) indicates terms that are of cubic or higher
order in perturbation amplitude.
McDougall and McIntosh (2000) rewrote the
mean density conservation equation (5.2.1), in
terms of the modified mean density,
ϳ , as
ϳ
t ;
H и(V
–
ϳ );(w –
ϳ ) z
:Q
– # 9
иF
M
;O(
3
)
(5.2.2)
and showed that the modified density flux, F
M can
be expressed as
F
M
: ϳ U
;
;M;O(
3
)
:9A
ϳ ;N;O(
3
)
(5.2.3)
where M and N are both non-divergent density
fluxes and the antisymmetric matrix A (see
McDougall, 1998) is defined in terms of the two
components of the quasi-Stokes streamfunction,
5.2 Mixing and Stirring in the Ocean Interior
341
Toole and McDougall
science. The landmark paper of Gent and
McWilliams (1990) marked the beginning of the
oceanic work on this topic and since that time
oceanographers have accepted that scalar properties should be advected by a velocity other than
the Eulerian-mean velocity.
Mesoscale eddies act to mix fluid parcels in a
way that is highly constrained by the stratified
nature of the fluid. The residual-mean theory provides the link between the different views that are
apparent from averaging these turbulent flow
fields in height coordinates and in density coordinates. It reduces the parameterization problem
from three dimensions to two dimensions and it
shows how the eddy fluxes are skew-symmetric in
height coordinates so that the total advection
velocity can be adiabatic. While mesoscale eddies
cause a very significant southward flux of heat in
the Southern Ocean, should this flux be regarded
as a mixing or a stirring of heat? To the extent
that there are balancing gradients of potential temperature and salinity along isopycnals, this component of the heat flux will eventually lead to mixing
of heat at the very smallest scales of motion and
hence we might call this aspect of the mesoscale
heat flux ‘mixing’. The rest of the heat flux that is
observed in Cartesian coordinates does not appear
as a heat flux in density coordinates, but rather is
associated with an extra horizontal velocity that
advects all fluid properties along isopycnals. This
part of the mesoscale eddy heat flux might more
properly be labelled ‘stirring’.
To understand how an eddy flux in Cartesian
coordinates might induce an extra fluid velocity,
we appeal to residual-mean flow theory, which was
originally developed in the atmospheric literature
where the averaging operator was a zonal average.
In developing the three-dimensional residual-mean
theory it is convenient to first deal with the density
conservation equation and to ignore any non-linearity in the equation of state. The Eulerian-mean
density conservation equation reads (the overbar
operator describes a low-pass temporal average)
–
t ;
H и(V
–
– );(w –
– ) z
:Q
– 9
H и(VЈЈ
—– )9(wЈЈ
—– ) z
(5.2.1)
where Q is the instantaneous diapycnal source
term and is a form of density that excludes the
effects of the compressible nature of seawater. Our
quest is to gain an understanding of the threedimensional turbulent density flux and its divergence that appears in this equation. Less than a
decade ago it was commonplace in ocean modelling to assume that the horizontal density flux,
VЈЈ
—– , could be parameterized as a down-gradient
Fickian flux, while the vertical turbulent density
flux was ignored except for the part attributed to
small-scale diapycnal mixing. This exactly horizontal mixing led to fictitious diapycnal fluxes of
density (the Veronis effect) and caused serious
problems such as spurious vertical motion in
western boundary currents. The temporal-residualmean theory demonstrated the inadequacy of this
prior approach and provided a route forward that
avoided such fictitious diapycnal density fluxes.
In order to develop a residual-mean conservation equation for, say, density, we need to realize
that the Eulerian-mean density is not the most
appropriate mean density to appear in the mean
density conservation equation that is carried by an
ocean model. The Eulerian-mean density, – (x, y, z, t),
describes a density surface whose average height is
not that of the original Eulerian averaging, namely
z. The appropriate mean density for our purposes
is the one whose surface is, on average, at the
height of the averaging. This density can be
expressed in terms of – by
ϳ : – 9(
– / –
z ) z ;O(
3 ),
where
– ϵ ᎏ
1
2
ᎏ Ј
2
— , is half the density variance measured at a fixed point in space and the terminology
O(␣
3
) indicates terms that are of cubic or higher
order in perturbation amplitude.
McDougall and McIntosh (2000) rewrote the
mean density conservation equation (5.2.1), in
terms of the modified mean density,
ϳ , as
ϳ
t ;
H и(V
–
ϳ );(w –
ϳ ) z
:Q
– # 9
иF
M
;O(
3
)
(5.2.2)
and showed that the modified density flux, F
M can
be expressed as
F
M
: ϳ U
;
;M;O(
3
)
:9A
ϳ ;N;O(
3
)
(5.2.3)
where M and N are both non-divergent density
fluxes and the antisymmetric matrix A (see
McDougall, 1998) is defined in terms of the two
components of the quasi-Stokes streamfunction,
5.2 Mixing and Stirring in the Ocean Interior
341
Toole and McDougall