14 Sv of zonally averaged diapycnal transport in
the Southern Ocean in both the Eulerian-mean and
the quasi-Stokes circulations. The Gent and
McWilliams (1990) scheme is often described as
being a parameterization of ‘bolus transport’, but
this is not the case since, by definition, the bolus
transport occurs along isopycnals and has no
diapycnal component. There is another compelling
reason why the eddy-induced velocity of Gent et al.
(1995) is not the down-gradient thickness flux or
bolus velocity: namely that the quasi-Stokes velocity is three-dimensionally non-divergent whereas
the bolus velocity is three-dimensionally divergent
(McDougall and McIntosh, 2000). Rather, the
Gent et al. (1995) scheme is the community’s first
attempt at parameterizing the quasi-Stokes streamfunction of the TRM theory.
Under the assumption that the horizontal flux
of density is directed down the horizontal density
gradient, the quasi-Stokes streamfunction, equation (5.2.4), of the TRM theory is virtually the
same as is used in the Gent et al. (1995) scheme.
While this down-gradient assumption on density
is a common assumption, several authors have
recently made the point that there may be better
theoretical support for the flux of potential vorticity along neutral density surfaces being directed
down the epineutral gradient of potential vorticity.
This naturally leads to a parameterization for
the bolus velocity and then one has to address the
questions of applying boundary conditions at the
top and bottom of the ocean and of avoiding a
singularity at the equator. The theory of Killworth
(1997) and other parameterization schemes based
on the down-epineutral-gradient of potential vorticity are, by construction, applicable to the parameterization of eddies in eddyless, density-coordinate
models. In such density-coordinate models, the
additional velocity that is required is the bolus
velocity and this velocity is three-dimensionally
divergent and adiabatic. It is not obvious a priori
that such a parameterization scheme should be
applied to a z-coordinate model where the use of
the continuity equation ensures that whatever
choice is made for its horizontal components, the
extra velocity is three-dimensionally non-divergent
and, most importantly, diabatic at leading order.
The down-gradient potential vorticity parameterization can also lead to a fictitious torque that
spontaneously generates angular momentum
(Cummins, 2000). In any event, this chapter does
not address in detail the various options that have
been proposed for parameterizing the quasi-Stokes
streamfunction, but instead we have concentrated
on describing the theoretical framework in which
such a parameterization will be used.
Treguier (1998) has analysed a high-resolution
primitive equation model for the zonally averaged
bolus velocity and has found a small diffusivity for
use in the Gent et al. (1995) scheme (the quasiStokes diffusivity). If it can be shown that it is
appropriate to use such a small diffusivity for the
quasi-Stokes streamfunction, then this should have
some benefits so long as the models remain stable
with these smaller diffusivities. The two benefits
that come to mind are avoiding the slowing of the
horizontal circulation of the subtropical gyres that
occurs with the larger values of the quasi-Stokes
diffusivity, and reducing the intrusion of Antarctic
Bottom Water into the North Atlantic, which is
too strong with present values of the quasi-Stokes
diffusivity (Hirst and McDougall, 1998).
At the large horizontal scales appropriate to
global circulation problems, the TRM flow can be
parameterized independently of the interaction of
the flow with bottom topography as described by,
for example, Holloway’s (1997) Neptune Effect.
This is because at these scales the Neptune Effect
forces a barotropic flow whereas the quasi-Stokes
circulation of the TRM theory is baroclinic, having
no depth-averaged component at each location.
There is also the interesting issue of whether the
present implementation of the TRM circulation in
ocean models is displaying improvements for the
correct reasons. Several authors have described
substantial improvements including more realistic
deep water properties, less unwanted deep convection and less drift in coupled atmosphere–ocean
models (see Hirst et al., 1996). An important common element of these improvements is that the Bottom Water of the world’s oceans has been able to
sink from the surface to the ocean bottom with
very little dilution. In fact, in the work of Hirst and
McDougall (1996) it was found that there was
insufficient diapycnal mixing occurring in the overflow regions. Previously such a result had only
been possible using a density-coordinate model. In
this way, a coarse-resolution height-coordinate
model has been shown to be sufficiently ‘adiabatic’
for the purposes of climate modelling.
In practice, the sinking of Deep and Bottom
Water frequently occurs in continental slope
SECTION 5 FORMATION AND TRANSPORT OF WATER MASSES
344
the Southern Ocean in both the Eulerian-mean and
the quasi-Stokes circulations. The Gent and
McWilliams (1990) scheme is often described as
being a parameterization of ‘bolus transport’, but
this is not the case since, by definition, the bolus
transport occurs along isopycnals and has no
diapycnal component. There is another compelling
reason why the eddy-induced velocity of Gent et al.
(1995) is not the down-gradient thickness flux or
bolus velocity: namely that the quasi-Stokes velocity is three-dimensionally non-divergent whereas
the bolus velocity is three-dimensionally divergent
(McDougall and McIntosh, 2000). Rather, the
Gent et al. (1995) scheme is the community’s first
attempt at parameterizing the quasi-Stokes streamfunction of the TRM theory.
Under the assumption that the horizontal flux
of density is directed down the horizontal density
gradient, the quasi-Stokes streamfunction, equation (5.2.4), of the TRM theory is virtually the
same as is used in the Gent et al. (1995) scheme.
While this down-gradient assumption on density
is a common assumption, several authors have
recently made the point that there may be better
theoretical support for the flux of potential vorticity along neutral density surfaces being directed
down the epineutral gradient of potential vorticity.
This naturally leads to a parameterization for
the bolus velocity and then one has to address the
questions of applying boundary conditions at the
top and bottom of the ocean and of avoiding a
singularity at the equator. The theory of Killworth
(1997) and other parameterization schemes based
on the down-epineutral-gradient of potential vorticity are, by construction, applicable to the parameterization of eddies in eddyless, density-coordinate
models. In such density-coordinate models, the
additional velocity that is required is the bolus
velocity and this velocity is three-dimensionally
divergent and adiabatic. It is not obvious a priori
that such a parameterization scheme should be
applied to a z-coordinate model where the use of
the continuity equation ensures that whatever
choice is made for its horizontal components, the
extra velocity is three-dimensionally non-divergent
and, most importantly, diabatic at leading order.
The down-gradient potential vorticity parameterization can also lead to a fictitious torque that
spontaneously generates angular momentum
(Cummins, 2000). In any event, this chapter does
not address in detail the various options that have
been proposed for parameterizing the quasi-Stokes
streamfunction, but instead we have concentrated
on describing the theoretical framework in which
such a parameterization will be used.
Treguier (1998) has analysed a high-resolution
primitive equation model for the zonally averaged
bolus velocity and has found a small diffusivity for
use in the Gent et al. (1995) scheme (the quasiStokes diffusivity). If it can be shown that it is
appropriate to use such a small diffusivity for the
quasi-Stokes streamfunction, then this should have
some benefits so long as the models remain stable
with these smaller diffusivities. The two benefits
that come to mind are avoiding the slowing of the
horizontal circulation of the subtropical gyres that
occurs with the larger values of the quasi-Stokes
diffusivity, and reducing the intrusion of Antarctic
Bottom Water into the North Atlantic, which is
too strong with present values of the quasi-Stokes
diffusivity (Hirst and McDougall, 1998).
At the large horizontal scales appropriate to
global circulation problems, the TRM flow can be
parameterized independently of the interaction of
the flow with bottom topography as described by,
for example, Holloway’s (1997) Neptune Effect.
This is because at these scales the Neptune Effect
forces a barotropic flow whereas the quasi-Stokes
circulation of the TRM theory is baroclinic, having
no depth-averaged component at each location.
There is also the interesting issue of whether the
present implementation of the TRM circulation in
ocean models is displaying improvements for the
correct reasons. Several authors have described
substantial improvements including more realistic
deep water properties, less unwanted deep convection and less drift in coupled atmosphere–ocean
models (see Hirst et al., 1996). An important common element of these improvements is that the Bottom Water of the world’s oceans has been able to
sink from the surface to the ocean bottom with
very little dilution. In fact, in the work of Hirst and
McDougall (1996) it was found that there was
insufficient diapycnal mixing occurring in the overflow regions. Previously such a result had only
been possible using a density-coordinate model. In
this way, a coarse-resolution height-coordinate
model has been shown to be sufficiently ‘adiabatic’
for the purposes of climate modelling.
In practice, the sinking of Deep and Bottom
Water frequently occurs in continental slope
SECTION 5 FORMATION AND TRANSPORT OF WATER MASSES
344