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.