Results 1, Separation Curve

First, we examine the separation between the 2-D and 3-D regimes in the nondimensional parameter space. The explained variance in the parameter space is shown in Fig. 2, where the contour of 60 % is the separation curve between the 2-D and 3-D regimes according to KM. The separation curve and KM's separation line are both consistent with the KM's 19 experiments shown by red circles, but more detailed structure of the curve is captured in the present study by a one-order larger number of experiments denoted by the blue marks. In particular, KM's separation line is not a good approximation for the separation curve for > 0.1, where gradients of the separation curve and separation line are opposite. The appropriate range of the natural Rossby number to open-ocean deep convection is 0.01 < < 1 (KM). Thus, the difference between the separation curve and line for > 0.1 is meaningful for the understanding of the deep convection. Consequently, the separation line proposed by KM is a linear approximation of the more complex separation curve in a relatively narrow range of the natural Rossby number.

For the mechanism which determines the 2-D or 3-D regimes with given external parameters, KM suggested that an instability of the first vertical mode is important, and they estimated from a scale analysis that the regime transition should be accompanied by a 2-D and 3-D separation line whose gradient is -1/2 (). Consistent with their scaling analysis, the separation line proposed by KM has the slope of -1/2. However, the difference between the present separation curve and KM's separation line suggests that the applicability of the instability analysis is significantly limited, probably due to complex structures in the horizontal and temporal directions even for the 2-D regime, which will be shown latter.

Figure 2. Explained variance (contour) of the vertical velocity by the first EOF mode in the common-logarithm parameter space of the natural Rossby number (ordinate) and the flux Rayleigh number (abscissa). The blue marks (crosses, circle, triangle, and rectangle) indicate the nondimensional parameters employed by the experiments in the present paper, while the red circles indicate the parameters used by KM. The 60% contour of the explained variance (shown by blue curve in the final panel) is the separation curve between the 2-D and 3-D regimes, while the red dashed line is the separation line proposed by KM. The experiments shown by blue circle, triangle and rectangle will be closely examined for their structure later, and the plume structures of the respective experiments will be referred to as the roll structure, the mushroom structure and the (conventional) heton structure, respectively.