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Thus, for strongly supercritical baroclinic flows, the
annulus results obtained so far would seem to support a
parameterization approach based on the potential vortic-
ity diffusion hypothesis proposed by
Treguier et al.
[1997]
(hereafter referred to as THL97) and
Killworth
[1997],
with a closure for eddy diffusivity that is consistent with
Visbeck et al.
[1997]. Closer to marginal instability, how-
ever, a different closure would seem to be preferred that
results in an increasing efficiency parameter
α
though not
entirely following the simple, weakly nonlinear recipe of
Pfeffer and Barcilon
[1978]. Therefore, there would still
seem, to be a number of unresolved issues underlying this
somewhat unexpected behavior close to marginal instabil-
ity. In addition, the approach of
Visbeck et al.
[1997] is
overtly based on an application of linear instability theory
in a regime that is far from where linear theory should be
valid. The theoretical basis for this clearly deserves more
attention in future work.
10
-2
2.5
×
2.0
1. 5
1. 0
0.5
0.0
2
4
6
8
Boundary layer ratio,
P
Figure 1.24.
Dependence of eddy diffusivity for quasi-
geostrophic potential vorticity on boundary layer
ratio
1.4.5. Implementing Eddy Parameterizations
in an Annulus Model
(pro-
portional to
) derived using data from correlating
(u
q
)
against
∂q/∂r
in the rotating annulus model simulations of
Pérez et al.
[2010].
P
The diagnostic approach discussed above using DNS is
useful for investigating some of the underlying assump-
tions behind various approaches proposed for eddy
transport parameterizations, especially those relating to
the family of parameterizations following
Gent and
McWilliams
[1990]. But in some respects the ultimate
test of any given approach to this problem is actually
to implement the parameterization in a numerical model.
Although this has been common practice in generations
of ocean circulation models for many years [e.g.,
Danaba-
soglu et al.
, 1994], this is a relatively novel approach in
the context of rotating annulus experiments and model
simulations. Parameterizing turbulent transfers in rotat-
ing flows is, of course, of major importance for many
engineering problems, e.g., in turbomachinery, where it
has been customary for many years to employ large
eddy simulation (LES) methods coupled with turbulence
models based on rotational modifications to the classical
Reynolds Averaged Navier-Stokes (RANS) model [e.g.,
see
Cazalbou et al.
, 2005, and references therein] to rep-
resent the effects of shear instabilities in the presence
of background rotation. But the transformed Eulerian
mean approach underlying the Gent-McWilliams fam-
ily of parameterizations does not yet seem to have been
adopted within the engineering community to represent
unresolved eddy transports in stably stratified turbulence
in rotating cavities.
Recently, however,
Pérez
[2006] has taken the first
preliminary steps toward investigating this approach by
implementing two forms of Gent-McWilliams parame-
terization in his 2D (axisymmetric), Boussinesq Navier-
Stokes model of thermally driven flow in a fluid annulus
to equation (1.30) with
L
represented by the width of the
baroclinic zone (i.e., the annulus gap width) and
U
eddy
given by a thermal wind scale
LM
2
N
U
eddy
,
(1.58)
where
N
is the Brunt-Väisäla frequency,
M
2
=
g
ρ
∂ρ
∂r
,
(1.59)
2
L/
√
Ri
such that
U
eddy
L/τ
Eady
,
τ
Eady
is the lin-
ear growth time scale for the Eady model of baroclinic
instability, and Ri is the characteristic shear Richardson
number Ri =
N
2
/(∂u/∂z)
2
for the flow. In the event, no
single closure seemed to apply across the whole param-
eter range. Such a result is not unduly surprising, since
existing closures generally make assumptions based on
either linear instability theory (which one might expect
to hold close to marginal instability) or weakly nonlin-
ear theory [e.g.,
Pfeffer and Barcilon
, 1978]. Their results
led to the conclusion that the observed variation of
∼
K
q
with rotation was broadly consistent with weakly nonlin-
ear theory close to conditions of marginal instability, with
efficiency parameter
α
increasing roughly linearly with
c
as suggested by
Pfeffer and Barcilon
[1978]. Under
more strongly supercritical conditions,
α
appeared to con-
verge to a roughly constant value that was consistent with
the value obtained for example, by
Visbeck et al.
[1997],
even to the extent of close quantitative agreement (0.013,
cf. Visbeck et al.'s value of 0.015).
−
