Etudes et évaluation de processus océaniques par des hiérarchies ...

4.5. MEAN CIRCULATION AND STRUCTURES OF TILTED OCEAN DEEP CONVECTION107
806 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 38
tel-00545911, version 1 - 13 Dec 2010
confirmed by our results presented in section 4), as are
the values of real oceanic deep convection. Below we
will thus only consider the case of three-dimensional
scaling.
The scaling for reduced gravity g 3D B 0 /u 3D (B 2 0/
H) 1/3 leads to a scaling for temperature anomalies,
given by T 3D (B 2 0/H) 1/3 /(g).
The values for the parameters considered in this
work are a subset of those observed in actual convection
processes in the World Ocean (see section 3a).
3. The model
In section 3a, we start by describing how the abovementioned
scientific results and questions guided our
choice of the physical problem adapted to explore the
dynamics of tilted convection. In section 3b we then
give a detailed explanation of the mathematical model
best suited to investigate this physical problem. The
numerical implementation employed to solve the mathematical
model is described in section 3c.
a. The physical experiment
The aim of this paper is the investigation of the openocean
deep convection process, which is the violent vertical
exchange of water masses when rotation and gravity
are not aligned. In the case of an alignment of rotation
and gravity, this process is local at the plume
scale, with either no or only a negligible effect on larger
scales (see the previous section). This important finding
allows us to consider a biperiodic horizontal slice of the
ocean, measuring only a few times the plume size in the
horizontal directions. The typical plume size with such
forcing is several hundreds of meters and descends at a
speed of a few tenths of a centimeter per second (see,
e.g., Wirth and Barnier 2006). We found in numerical
experiments that a domain spanning 8 km in each horizontal
direction allows for a sufficient space for a few
plumes, spanning a few hundreds of meters in the horizontal
direction, to descend independently. The atmospheric
forcing varies on horizontal scales of 100–1000
km so that it is adapted to use a homogeneous cooling
on the surface. An integration representing a few days
of dynamics is necessary to account for the descent of
several plumes and to obtain a statistically stationary
dynamics. The domain size of 8 km in both horizontal
directions is, however, far too small to obtain statistically
significant results for even the lowest-order quantities.
We thus continued the experiment after a statistically
stationary state was obtained for a couple of days
and used the (supposed) ergodicity of the dynamics by
averaging over space and time to obtain converged statistical
estimates.
Furthermore, the model domain spans exactly what
we intend to parameterize, that is, the convective dynamics
at a single grid point in a high-resolution
OGCM [O(10 km)]. Such a domain size is clearly too
small to investigate the restratification process that follows
the convection regime, which can be studied by
significantly coarser hydrostatic ocean models (see
ChBa).
To obtain information about the scaling behavior, we
performed calculations for three different values of
cooling (1000, 500, and 250 W m 2 ) for the case of
nontilted (North Pole) and tilted (Gulf of Lions, 45°)
convection.
b. The mathematical model
To study the convection process a nonhydrostatic
ocean model is essential. The mathematical model of
the herein-presented ocean convection experiments are
the Boussinesq equations [Eq. (1)] of an incompressible
flow [Eq. (2)] in a rotating frame, supplemented by
boundary conditions. The flow field is given by u and
the scalar (temperature) field by T. The buoyant scalar
is transported by the flow [Eq. (3)] and diffuses at the
rate :
t u u · u 2 u P gTe ⊥ 2 u,
· u 0,
t T u · T 2 T G.
The source term G ensures the prescribed heat fluxes
through the upper boundary, the equation of state is
linear with the expansion coefficient , and g 9.81
m s 2 denotes gravity. The ocean surface is modeled by
a free-slip rigid-lid boundary condition, while the ocean
floor is modeled by a no-slip boundary condition. By
using a no-slip boundary condition to model the ocean
floor we introduce an additional numerical difficulty
because we now have to resolve the thin Ekman layer
on this boundary. The role of this Ekman layer is, however,
of paramount importance to the convective dynamics
because it helps to relax the Taylor–Proudman–
Poincaré constraint, which inhibits all vertical motion;
we refer the reader to Bush et al. (1992) for a detailed
discussion of this point and to Wirth (2004) for a numerical
investigation thereof.
Because we are considering only a small slice of the
ocean (spanning 8 km in the horizontal directions), we
can safely neglect the sphericity of the earth by using
Cartesian geometry and we can also neglect the variation
of the direction of the rotation vector with the
latitudinal direction (). This geometry is called the f–F
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