26
Homogeneous Models of the Ocean Circulation
this dissipation reached its culmination, perhaps, in the 1960s. The very interesting pioneering paper of Carrier and Robinson (1962), in which the explicit role of dissipation is essentially ignored in favor of an attempt to
construct a purely inertial model of the circulation, is a good example of such
confidence. The hope in the inconsequential role for friction is partly related to
our inability to specify with confidence either an appropriate mathematical
representation of the role of small-scale friction or its magnitude. It is an
understandable human inclination to hope that what we cannot do is unimportant.
Since most parameterizations of small-scale turbulent dissipation lead to
mathematical representations, such as the diffusion of large-scale momentum,
that raise the mathematical order of the governing equations, additional
boundary conditions must also be specified. Various boundary conditions
could be applied in addition to the kinematic condition of no normal flow, such
as a condition of no slip on the tangential velocity, a condition on the
boundary stress, and others which we discuss below. Several are plausible;
none is rigorously required from first principles for the approximate equations
governing only the large-scale flow. It is unclear how to chose between them.
Along with the hope that friction is physically inconsequential if only small
enough, there is also frequently expressed the hope that the overall circulation
is largely independent of the boundary condition chosen to accompany the
raised order of the equations since there is rarely any determining argument
that forces us to choose one or the other of these proposed boundary conditions. Often the conditions are chosen for convenience of calculation or from a
subjective preference for the results eventuating from one or another of the
choices.
Some reflection indicates that these hopes are all likely to be disappointed.
The fact that the circulation of the ocean actually consists of an endless recirculation means that no matter how small the dissipation is, the fluid has
substantial time to experience the action of dissipative forces. Stated differently, if the forcing by the wind continuously puts vorticity into the fluid,
dissipation must continuously take it out. Since each fluid element is directly
acted on by the wind during at least part of the time that it traverses the gyre,
each fluid element receives an input of vorticity during its repetitive journey
which it must somehow contrive to lose in the remaining part of its circuit
around the gyre if a steady state is to be established. Each fluid element must
therefore pass through regions where dissipation is important so that the input
of vorticity and energy by the wind can be balanced by a dissipative drain of
the same quantity. Since the input is order 1 (in the sense that the flow is driven
by this input) the dissipation, somewhere must be order 1 for each fluid element
no matter how small the dissipation is estimated to be for the large-scale
motion.
The question that then arises is not whether this dissipation occurs. It
must. The dynamical issue of importance is whether in contriving a way to rid
itself of the vorticity and energy continuously added by the wind, the resulting
Homogeneous Models of the Ocean Circulation
this dissipation reached its culmination, perhaps, in the 1960s. The very interesting pioneering paper of Carrier and Robinson (1962), in which the explicit role of dissipation is essentially ignored in favor of an attempt to
construct a purely inertial model of the circulation, is a good example of such
confidence. The hope in the inconsequential role for friction is partly related to
our inability to specify with confidence either an appropriate mathematical
representation of the role of small-scale friction or its magnitude. It is an
understandable human inclination to hope that what we cannot do is unimportant.
Since most parameterizations of small-scale turbulent dissipation lead to
mathematical representations, such as the diffusion of large-scale momentum,
that raise the mathematical order of the governing equations, additional
boundary conditions must also be specified. Various boundary conditions
could be applied in addition to the kinematic condition of no normal flow, such
as a condition of no slip on the tangential velocity, a condition on the
boundary stress, and others which we discuss below. Several are plausible;
none is rigorously required from first principles for the approximate equations
governing only the large-scale flow. It is unclear how to chose between them.
Along with the hope that friction is physically inconsequential if only small
enough, there is also frequently expressed the hope that the overall circulation
is largely independent of the boundary condition chosen to accompany the
raised order of the equations since there is rarely any determining argument
that forces us to choose one or the other of these proposed boundary conditions. Often the conditions are chosen for convenience of calculation or from a
subjective preference for the results eventuating from one or another of the
choices.
Some reflection indicates that these hopes are all likely to be disappointed.
The fact that the circulation of the ocean actually consists of an endless recirculation means that no matter how small the dissipation is, the fluid has
substantial time to experience the action of dissipative forces. Stated differently, if the forcing by the wind continuously puts vorticity into the fluid,
dissipation must continuously take it out. Since each fluid element is directly
acted on by the wind during at least part of the time that it traverses the gyre,
each fluid element receives an input of vorticity during its repetitive journey
which it must somehow contrive to lose in the remaining part of its circuit
around the gyre if a steady state is to be established. Each fluid element must
therefore pass through regions where dissipation is important so that the input
of vorticity and energy by the wind can be balanced by a dissipative drain of
the same quantity. Since the input is order 1 (in the sense that the flow is driven
by this input) the dissipation, somewhere must be order 1 for each fluid element
no matter how small the dissipation is estimated to be for the large-scale
motion.
The question that then arises is not whether this dissipation occurs. It
must. The dynamical issue of importance is whether in contriving a way to rid
itself of the vorticity and energy continuously added by the wind, the resulting