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rotating at angular velocity
. As with his earlier diag-
nostic work, Perez's model was based on the conser-
vative, finite-difference model described in detail by
Hignett et al.
[1985], which solves the full Boussinesq
Navier-Stokes equations together with continuity and
temperature advection-diffusion equations in cylindrical
annular geometry using an exponentially stretched mesh
in
r
and
z
to ensure adequate resolution of boundary
layers. Nonslip, impermeable boundary conditions were
applied at each boundary of the cavity, with fixed isother-
mal conditions at the inner and outer sidewalls and ther-
mally insulating conditions on the horizontal boundaries.
Fairly coarse resolution was adopted (32
spurious numerical instabilities within the 2D parameter-
ized model in the most strongly super critical simulations,
even though the total heat transport of the 3D flow was
reproduced quite accurately in most cases.
Although the alternative THL97 PV-based parameter-
ization was not able to match the total Nusselt number
of the full 3D simulations as accurately as the GM90
scheme, it did result in much more realistic zonally aver-
aged temperature fields in the 2D parameterized model.
Figure 1.25 shows examples of (a) the eddy-induced TEM
stream function
χ
q
, (b) the resulting parameterized TEM
radial velocity
u
∗
, and (c) the eddy-induced TEM radial
velocity diagnosed from the corresponding fully 3D, eddy-
resolving model simulation at
(
,
32 points in
(r
,
z)
) in all cases. Each parameterized simulation was ini-
tialized from an isothermal state at rest in the rotating
frame and then first run to equilibrium while holding
the boundary conditions fixed. The eddy parameteriza-
tion was then activated by adding parameterized vertical
and radial velocity components
u
∗
, representing the trans-
formed Eulerian mean (TEM) velocity field [e.g.,
Andrews
et al.
, 1987] induced by the presence of baroclinic eddies,
to the 2D axisymmetric velocity field that was used to
advect momentum and temperature, and the model was
then integrated to its modified equilibrium.
The
u
∗
parameterization was derived from the instanta-
neously computed velocity and temperature fields within
the model either (a) using the original Gent-McWilliams
method based on equations (1.25) and (1.27) or (b) derived
from the zonal mean quasi-geostrophic potential vortic-
ity field following THL97 based on equations (1.32) and
(1.34). Because the zonally averaged isotherms become
very steep as the sidewall boundary layers are approached,
it was necessary to place limits on the isotherm gradient
utilized in the eddy parameterization.
Pérez
[2006] used
the method of
slopetapering
as advocated by
Danabasoglu
and McWilliams
[1995] for use in ocean circulation mod-
els. This method was also used to control the PV gradients
used in the THL97 parameterization, especially close to
the boundaries of the domain. The closure used for
×
10
6
)
.
χ
q
takes the form of a simple overturning aligned along
the principal direction of the isotherms and in the sense
required to advect them toward the horizontal. The corre-
sponding parameterized
u
∗
resembles the diagnosed TEM
radial velocity quite closely except close to the bound-
aries of the domain, where isotherm and PV gradients in
the main fields become very large and quasi-geostrophic
theory is no longer valid.
Examples of some simulated temperature fields are
shown in Figure 1.26, which shows (a) the zonally aver-
aged equilibrated temperature field in
(r
,
z)
from a fully
3D eddy-resolving simulation of moderately supercritical
flow at
(
,
T
)
=
(
1.22,1.6
×
10
6
)
, (b) the equilibrated
temperature field from a 2D axisymmetric simulation
under the same conditions as in (a), and (c) the cor-
responding equilibrated temperature field from a 2D
simulation using the THL97 eddy parameterization imple-
mented by
Pérez
[2006]. Under these conditions, the
axisymmetric isotherms (b) are much more steeply sloped
than obtained in the eddy-resolving 3D model (a), where
fully developed baroclinic instability acts to release a lot
of stored available potential energy. This is well reflected
in the parameterized simulation, where the additional
eddy-induced component of the meridional circulation
has strengthened the advection of temperature sufficiently
to relax the isotherm slope toward the horizontal in a
way that emulates quite accurately the effects of baroclinic
eddies on the zonal mean flow in the 3D eddy-resolving
simulation, even to the point of retaining the static stabil-
ity structure. The total Nusselt number in the parameter-
ized simulation was 9.5 compared with a time-mean value
of 10.1 in the 3D eddy-resolving simulation, indicating a
tendency for parameterized simulations to underestimate
eddy heat transfer by around 20%.
This tendency becomes more pronounced in more
strongly supercritical conditions, with a parameterized
Nusselt number of 7.5 compared with a 3D Nusselt num-
ber of 9.1 at the most extreme conditions investigated by
Pérez
[2006] at
(
,
T
)
=
(
0.599,3.26
×
was
either based on the formulation by
Visbeck et al.
[1997] or
(for potential vorticity) a constant diffusivity equivalent to
the value diagnosed from the fully 3D model simulations.
In practice, the original GM90 method was found to
be capable of matching the total Nusselt number of the
fully 3D simulated flows across the full range of param-
eters. However, apart from at the lowest rotation speeds
(close to marginal instability), the resulting temperature
fields did not match closely the zonally averaged fields
obtained in the full 3D eddy-resolving model. This was
almost certainly a reflection of the relative
ly w
eak co
rre
la-
tion found by
Pérez et al.
[2010] between
(u
T
)
and
∂T/∂r
in the full 3D low, so that the parameterized eddy-induced
circulation did not accurately reflect the real TEM circula-
tion in the 3D low. This actually led to the development of
K
10
8
)
.Asremarked
earlier, however, at these more extreme parameters the
T
)
=
(
0.017,1.17
×
