Model
We used a three dimensional nonhydrostatic
numerical model, which has 128*128 grids
horizontally and 22 grids vertically. Grid
intervals are 50 m in horizontal and vertical
directions, unless otherwise noted. The width
of the model region is therefore 6.4 km*
6.4 km, and the ocean depth,
, is 1000 m. Cyclic boundary condition is
applied to the horizontal direction, and
slip boundary condition is applied to the
top and bottom boundaries. The model equations
are incompressible Navier-Stokes equations
of the rotating fluid with the Boussinesq
and rigid-lid approximations. The diffusivity
and viscosity terms of the equation are expressed
in a harmonic form with constant diffusion
and viscosity coefficients.
The equations are expressed by the finite
difference, and solved by a simplified marker
and cell method, with an Adams-Bashforth
scheme for the temporal integration. The
density equation is approximated so that
the density is inversely proportional to
temperature with the expansion coefficient
of 2.0*10-4K-1. The eddy diffusivity (
) and viscosity (
) are taken to be the same, with the horizontal
and vertical diffusivity (and hence viscosity)
being assumed to be identical. The external
forcing is uniform body cooling applied to
the upper 200 m layer, which is assumed to
be the model mixed layer. The present setting
is similar to that of KM and Jones and Marshall
(1993), though the most of their experiments
are conducted for the 2000 m depth ocean.
With the ideal setting, the controlling parameters
are the following three parameters: Coriolis
parameter (
), eddy diffusion coefficients (), and buoyancy flux (
). Each numerical experiments are integrated
at least for 144 hours, but some experiments
are integrated for a longer time for a visualization
purpose.
According to the previous studies (e.g.,
KM, Maxworthy and Narimousa 1994), we examine
the 2-D and 3-D regime dependencies on two
nondimensional parameters, which are flux
Rayleigh number 
and natural Rossby number
. These nondimensional parameters are uniquely
determined by external parameters. An appropriate
parameter range of the natural Rossby number
may be 0.01 < 
< 1 for the open-ocean deep convection as suggested by KM. In contrast, the order
of for the typical atmospheric convection is
, and hence the rotation can be neglected
for the atmospheric convective elements,
but not for the oceanic ones.
The horizontal length scale of plumes in
an experiment are determined as a horizontal
distance, at which spatial lag correlation
function at the middle depth takes the value
of 0.5 according to KM. The experiments whose
scales are too small (smaller than 50 m)
or too large (larger than 640 m) are not
included in the analysis, and the number
of experiments accepted for the analysis
is 157. The given parameters of all the experiments
are summarized in Table 1.