Introduction
Open-ocean deep convection is a phenomenon in which vertical water movements penetrates well below the surface mixed layer caused by surface buoyancy loss in open ocean. The "open-ocean" means that the convective phenomena are not significantly influenced by lateral boundaries. Open-ocean deep convection supplies dense waters, which play a significant role in the earth's heat transport and hence heat balance.
Numerical simulation of open-ocean deep convection has been studied by a number of papers (see Marshall and Schott (1999) for a review) after the pioneering work by Johns and Marshall (1993). The oceanographic deep convection have different physical parameter ranges from those in the atmospheric convection as noted by Klinger and Marshall (1995) (hereafter KM); the atmospheric convection hits the ceiling (tropopause) before it feels the effect of rotation, but the open-ocean deep convection feels the earth's rotation before convective elements reach the ocean bottom. Therefore, the oceanic deep convection forms a relatively new class of convective problems in geophysical fluid dynamics.
A convection phenomenon usually occurs in a wider area than the ocean depth, and the spatial domain where convection occurs is called convection patch whose diameter is a few tens of kilometers. The water movements in a convection patch is characterized by a number of plumes, whose diameter is typically several hundreds meters. The plumes are convective elements that moves vertically, and play substantial roles in vertical water mixing in the convection patch.
Previous numerical and laboratory studies
have shown that plumes exhibit qualitatively
different two regimes, which are referred
to as a two dimensional (2-D) and three dimensional
(3-D) regimes. In a 2-D regime, velocity
fields are dominated by the first vertical
mode. Thus, the terminology, 2-D regime,
comes from the fact that one can approximately
know a whole (three dimensional) flow field
by obtaining one horizontal cross section
(one 2-D plane, i.e., at a quarter depth)
and the fixed vertical structure. On the
other hand, in a 3-D regime, the vertical
structures of the current fields vary spatially
and temporally, and hence one cannot extract
a dominant single vertical structure. KM
proposed that the 2-D and 3-D regimes can
be uniquely separated by a line in a two
dimensional logarithmic parameter space of
flux Rayleigh number and natural Rossby number.
However, because the number of the KM's experiments
(19) is not enough to determine whether or
not the separation line is a part of a more
complex curve. In order to address this question,
we have conducted a larger number of numerical
experiments (157).
The mechanism separating the 2-D and 3-D
regimes is not clear. We examine this problem
from a point of view of the entropy increase
rate of the system. It is known that behaviors
of some complex systems are explained by
the maximization hypothesis of entropy
increase rate. The hypothesis states that
the behavior of the system is adjusted so
as to maximize the entropy increase rate.
The hypothesis has been successfully applied
to a number of phenomena, which have various
time and spatial scales, i.e., the earth's
climate (e.g., Paltridge 1975, Ozawa and
Ohmura 1997) and frost heaving (Ozawa 1997).
In particular, for a system that has possible
different regimes corresponding different
entropy increase rates, a regime which has
a larger entropy increase rate than others
should be selected in reality, as far as
the maximization of the entropy increase
rate is applicable (Sawada 1981). A conceptual
example of the hypothesis for multiple regimes
is illustrated in Fig. 1. Thus, if the 2-D and 3-D separation is
consistent with the hypothesis, the entropy
increase rate should exhibit gradients shown
in either left or right panel in Fig. 1. We will examine whether this is the case
or not. Such approach examining the entropy
is a macroscopic approach, which would play
a complementary role to microscopic approaches
in understanding the regime transition. An
example of the microscopic approach is employed
by KM, who suggested that the instability
of the first vertical mode in the 2-D regime
causes the transition from the 2-D regime
into the 3-D regime.
In addition to the problems associated with
the 2-D and 3-D regime transition, we will
examine the structures of the plumes in the
2-D regime. Sometimes, the 2-D regime is
assumed to be characterized by quasi-stable
vortex pairs rotating different directions
in the top and bottom of the fluid. Such
a stable vortex pair is called as a heton.
For example, KM called the 2-D regime the
heton regime. However, it is not still clear
whether or not the major structures of all 2-D regimes are hetons. In the present paper,
therefore, we investigate the flow structures
in the 2-D regime in more detail than in
the previous studies by employing visual
inspections of vorticity and velocity structures
including animations. Apparently, the present
journal, Nagare Multimedia, has great advantages for the visual inspection
of time varying complex structures over conventional
paper journals.
Figure 1. Schematic explains how a regime, which has
a larger entropy increase rate than the other
potentially possible regime, is selected.
The solid and dashed curve indicates the
entropy increase rate in a two potentially
possible regimes as a function of a parameter
value. A regime having a larger entropy increase
rate (red solid or red dashed curves) should
be selected to occur in reality, as far as
the maximization hypothesis of entropy increase
rate can be applicable to the system. The
left panel is the case where the entropy
increase rate has its minimum at the transition
point between the two regimes. However, the
occurrence of the minimal value is not necessary
as illustrated in the right panel, where
both the entropy increase rates of the two
regimes are given by the decreasing functions
of the parameter.