canyons and across sills, bathymetric features not
well represented in coarse-resolution models. The
TRM advection scheme achieves this ‘adiabatic’
sinking motion because of two almost equal
effects, as demonstrated by Hirst and McDougall
(1996). First, the unwanted horizontal diffusion is
eliminated, and second, an extra advection (or
skew diffusion) is added that assists in the transport of water from the surface to the deep. The
elimination of horizontal diffusion is thought to be
physically required, but the extra advection at the
bottom of the ocean seems to be more an artefact
of the bottom boundary condition on the quasiStokes streamfunction than a representation of the
actual boundary-current mechanisms that achieve
the transport of bottom water. In this way it may
be that half of the benefits that we are seeing to
date have been obtained by stealth: obtaining the
right results for the wrong reasons. It is not at all
clear that such cancellation of error will hold for
circulations in alternate climate states as, for
example, might result from global warming.
We concentrated here on the density equation
with the implicit assumption that the Eulerian-mean
velocity is available as model output of the momentum equations. This is usually a good assumption
because of the dominance of the geostrophic balance at the large scales relevant to the global ocean
circulation. There is, however, an alternative way
of implementing a TRM parameterization: one can
apply the TRM averaging approach to the horizontal momentum equations, thereby obtaining a
forcing term on the right of these equations that
looks somewhat similar to the Eliassen–Palm flux
of the zonal-averaging literature. The key result of
that literature is that while the Eliassen–Palm flux
is of second order in perturbation quantities, its
divergence is one order higher and so can often be
ignored, or at least assumed to be equal to the
northward flux of potential vorticity. No such
result has been proven for the three-dimensional
problem under temporal or ensemble averaging.
Indeed, it seems very likely that the divergence of
this flux is still of second order in perturbation
quantities. Nevertheless, if one boldly assumes this
result by a crude analogy with the zonal averaging
literature, the parameterization would effectively
be a form drag in the horizontal momentum equations and the tracer equations would not need any
extra skew flux term. The results of this approach
are expected to be very similar to the more
conventional method of implementing the TRM
scheme in the tracer equations.
In summary, the TRM approach tells us how
we should interpret the variables that are carried
in an eddyless ocean model, it reduces the eddyparameterization task from a three-dimensional to
a two-dimensional task, and it provides physical
insight that dictates that the quasi-Stokes streamfunction should be smoothly tapered to zero as the
sea surface and the ocean floor are approached.
The parameterization task is now quite specific:
we must learn how to parameterize the quasi-Stokes
streamfunction, equation (5.2.4), and we now
know that this is not the same as parameterizing
the bolus velocity. If we are successful in parameterizing equation (5.2.4) we will be able to run an
eddyless height-coordinate model knowing that it
is equivalent to first running a density-coordinate
model at high resolution and then averaging over
the mesoscale eddies.
5.2.4 Lateral dispersion between the
mesoscale and the microscale
Eddy motions also strain passive tracers on isopycnals, cascading variance towards smaller spatial
scales where it is eventually dissipated by molecular diffusion. Recent studies are beginning to illuminate the connections between the mesoscale and
microscale. Ledwell et al. (1998) inferred lateral
diffusivities as a function of spatial scale from the
lateral spread of the WOCE North Atlantic Tracer
Release Experiment (NATRE) tracer patch with
time. Initially, when the bands of SF 6 had horizontal scales of 100–1000 m, the tracer appeared to
spread with a lateral diffusivity of ϳ0.07 m
2 s
91
.
This rate is consistent with shear dispersion due to
near-inertial internal waves (Young et al., 1982).
After about 6 months, the tracer had been drawn
into much longer streaks with horizontal widths of
1–30 km. The width-to-length ratio of the streaks
suggested lateral diffusion on these scales was
ϳ2 m
2 s
91
, far stronger than can be accounted for
by internal wave shear dispersion (Sundermeyer
and Price, 1998). Polzin et al. (2000b) proposed
that vortical mode motions (fine-scale structures
with non-zero Ertel potential vorticity) might be
responsible for the dispersion on these scales. For
the period when the tracer patch was 30–300 km
in width and thus had comparable size to individual mesoscale eddies, Ledwell et al. felt an eddy
5.2 Mixing and Stirring in the Ocean Interior
345
Toole and McDougall
well represented in coarse-resolution models. The
TRM advection scheme achieves this ‘adiabatic’
sinking motion because of two almost equal
effects, as demonstrated by Hirst and McDougall
(1996). First, the unwanted horizontal diffusion is
eliminated, and second, an extra advection (or
skew diffusion) is added that assists in the transport of water from the surface to the deep. The
elimination of horizontal diffusion is thought to be
physically required, but the extra advection at the
bottom of the ocean seems to be more an artefact
of the bottom boundary condition on the quasiStokes streamfunction than a representation of the
actual boundary-current mechanisms that achieve
the transport of bottom water. In this way it may
be that half of the benefits that we are seeing to
date have been obtained by stealth: obtaining the
right results for the wrong reasons. It is not at all
clear that such cancellation of error will hold for
circulations in alternate climate states as, for
example, might result from global warming.
We concentrated here on the density equation
with the implicit assumption that the Eulerian-mean
velocity is available as model output of the momentum equations. This is usually a good assumption
because of the dominance of the geostrophic balance at the large scales relevant to the global ocean
circulation. There is, however, an alternative way
of implementing a TRM parameterization: one can
apply the TRM averaging approach to the horizontal momentum equations, thereby obtaining a
forcing term on the right of these equations that
looks somewhat similar to the Eliassen–Palm flux
of the zonal-averaging literature. The key result of
that literature is that while the Eliassen–Palm flux
is of second order in perturbation quantities, its
divergence is one order higher and so can often be
ignored, or at least assumed to be equal to the
northward flux of potential vorticity. No such
result has been proven for the three-dimensional
problem under temporal or ensemble averaging.
Indeed, it seems very likely that the divergence of
this flux is still of second order in perturbation
quantities. Nevertheless, if one boldly assumes this
result by a crude analogy with the zonal averaging
literature, the parameterization would effectively
be a form drag in the horizontal momentum equations and the tracer equations would not need any
extra skew flux term. The results of this approach
are expected to be very similar to the more
conventional method of implementing the TRM
scheme in the tracer equations.
In summary, the TRM approach tells us how
we should interpret the variables that are carried
in an eddyless ocean model, it reduces the eddyparameterization task from a three-dimensional to
a two-dimensional task, and it provides physical
insight that dictates that the quasi-Stokes streamfunction should be smoothly tapered to zero as the
sea surface and the ocean floor are approached.
The parameterization task is now quite specific:
we must learn how to parameterize the quasi-Stokes
streamfunction, equation (5.2.4), and we now
know that this is not the same as parameterizing
the bolus velocity. If we are successful in parameterizing equation (5.2.4) we will be able to run an
eddyless height-coordinate model knowing that it
is equivalent to first running a density-coordinate
model at high resolution and then averaging over
the mesoscale eddies.
5.2.4 Lateral dispersion between the
mesoscale and the microscale
Eddy motions also strain passive tracers on isopycnals, cascading variance towards smaller spatial
scales where it is eventually dissipated by molecular diffusion. Recent studies are beginning to illuminate the connections between the mesoscale and
microscale. Ledwell et al. (1998) inferred lateral
diffusivities as a function of spatial scale from the
lateral spread of the WOCE North Atlantic Tracer
Release Experiment (NATRE) tracer patch with
time. Initially, when the bands of SF 6 had horizontal scales of 100–1000 m, the tracer appeared to
spread with a lateral diffusivity of ϳ0.07 m
2 s
91
.
This rate is consistent with shear dispersion due to
near-inertial internal waves (Young et al., 1982).
After about 6 months, the tracer had been drawn
into much longer streaks with horizontal widths of
1–30 km. The width-to-length ratio of the streaks
suggested lateral diffusion on these scales was
ϳ2 m
2 s
91
, far stronger than can be accounted for
by internal wave shear dispersion (Sundermeyer
and Price, 1998). Polzin et al. (2000b) proposed
that vortical mode motions (fine-scale structures
with non-zero Ertel potential vorticity) might be
responsible for the dispersion on these scales. For
the period when the tracer patch was 30–300 km
in width and thus had comparable size to individual mesoscale eddies, Ledwell et al. felt an eddy
5.2 Mixing and Stirring in the Ocean Interior
345
Toole and McDougall