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

108 CHAPITRE 4. ETUDES DE PROCESSUS OCÉANOGRAPHIQUES
APRIL 2008 W I R T H A N D B A R N I E R 807
TABLE 1. Physical parameters common in all experiments.
L x L y L z |f| c p
8 km 8 km 3.5 km 1.45 10 4 s 1 2 10 4 K 1 1000 kg m 1 3900 J K 1 kg 1
tel-00545911, version 1 - 13 Dec 2010
plane, where f and F are twice the normal and horizontal
components of , respectively; that is,
f 0 F
f 2 0
cos
sin .
The f–F plane allows for the implementation of periodic
boundary conditions in both horizontal directions.
The source term,
G 0
G 1
D expzD2 ,
represents the surface cooling that decreases exponentially
with depth on a characteristic scale D 90 m,
mimicking the violent mixing in the upper ocean at
convection sites due to high waves and mixed layer
turbulence.
The white-in-time-and-space noise (x, t), with smallamplitude
, is added. The amplitude is chosen such
that with a time step of 30 s the variance of the noise
term equals 10 2 ; that is, 0.1 (30) 1/2 . In the
spatial dimensions the noise is chosen independently at
every grid point using a pseudorandom number generator.
1
When the buoyancy density
gT
is used, rather than the temperature T, the constants
and g disappear from the governing equations. The surface
forcing is written as a surface buoyancy flux B 0
g/(c p )H 0 , where c p 3900 J (K kg) 1 is the heat
capacity, 1000 kg m 2 is the density of water, and
H 0 denotes the surface heat flux, measured in watts per
squared meter. We thus have B 0 H 0 (5.031 10 10 m 4
W 1 s 3 ) and G 0 H 0 /(c p ). The values are listed in
Tables 1 and 2.
1 More precisely, (x, t) is the time derivative of an ensemble of
Wiener processes W i, j,k (t) indexed by the grid points (i, j, k) and
[W i, j,k (t)] 2 t. Different pseudorandom number generators of
different complexities were used, ranging from a quasi-periodic
function to the “Mersenne twister” (Matsumoto and Nishimura
1998); the herein-represented results are found not to depend on
the actual generator employed.
4
5
6
The flux Rayleigh number is kept constant in all of
the experiments [Ra f (BH 4 )/( 2 ) 6.04 10 8 ], the
Prandtl number is one, and the natural Rossby numbers
Ro* are 0.0580, 0.0821, and 0.116 in experiments Ex1,
Ex3, and Ex4, respectively (where x 1 or 3).
We thus obtain the following numerical values for
the scaling variables: u 3D (B 0 H) 1/3 1.21 10 1
m s 1 , g 3D B 0 /u 3D 4.17 10 5 m 2 s 1 , and T 3D
g 3D /(g) 2.12 10 2 K.
c. The numerical implementation
The mathematical model is solved numerically using
a pseudospectral scheme entirely based on Fourier expansion.
The boundary conditions are implemented using
a method inspired by the immersed boundary technique.
For a detailed discussion of the model and the
new boundary technique we refer the reader to Wirth
(2004). Our model will forthwith be called the Harmonic
Ocean Model (HAROMOD).
The coherent structures dominating the convective
dynamics are known to have comparable horizontal
and vertical scales, a fact that has to be reflected in the
aspect ratio of the numerical grid. Because it is also
important to numerically resolve the nonlinear dynamics
of the plumes, a grid size of a few tenths of meters
in all three dimensions is required. The need for resolving
the bottom Ekman layer motivated the choice of a
finer resolution in the vertical direction, and we thus
have x y 2z 31.25 m, which corresponds to
256 256 224 grid points. The friction coefficients
are equal in the horizontal and vertical directions; they
are chosen such that the flux Rayleigh number is equal
in all experiments. The integration represents the dynamics
of ocean convection during 168 h after the onset
of cooling. Snapshots are stored every 3 h and data
from the last 90 h are used to obtain time-averaged
TABLE 2. Physical parameters varied in the experiments.
Expt Surface heat flux H 0 Latitude
E01 1000 W m 2 90°
E03 500 W m 2 90°
E04 250 W m 2 90°
E31 1000 W m 2 45°
E33 500 W m 2 45°
E34 250 W m 2 45°