34
Homogeneous Models of the Ocean Circulation
(2.4.3)
on the boundary.
The presence of viscous diffusion of vorticity implies the need for an additional dynamic boundary condition on the walls of the basin. If we interpret
(2.3.2) as applying to all scales of motion with An as the actual coefficient of
viscosity, the standard condition of no slip on the boundary would be required.
This would be:
u·t=O
(2.4.4)
where i is a unit vector tangent to the boundary. In terms of tjJ this becomes:
fz. \ltjl = 0.
(2.4.5)
It is not clear, however, that (2.4.5) is the correct boundary condition when
our equations explicitly resolve only the large-scale motion. The no-slip condition is a reflection of the interaction of the fluid with a solid boundary on the
microscopic level, and there is no reason why motion on scales of the order of
hundreds or thousands of kilometers should mimic that interaction. We might
even argue instead that the small-scale processes that occur right against the
wall buffer the large-scale flow and allow it to slip smoothly along the wall. A
condition of no tangential stress for the large-scale flow might be equally
logical then. For a north-south wall, for example, this would require:
OV+OU=O
ax ay
at the wall. Since u is already zero there by (2.4.1 ), it follows that 8v / 8x itself
must vanish there. We can write this boundary condition, which we call the slip
condition, in a form that is independent of the coordinate system by recasting
the condition in the form:
(2.4.6)
on each boundary.
On the other hand, if the lateral diffusion term in (2.3.2) is considered to be
a parameterization of small-scale mixing of vorticity by an unresolved eddy
field, we could argue that we are actually specifying a relation between the eddy
flux of total vorticity and the gradient of the large-scale vorticity field, i.e.:
-it'('= An\l(( + f3y)
(2.4.7)
whose divergence yields the diffusion term in (2.3.2). At a solid boundary the
normal component of the eddy flux of vorticity must vanish since the eddy
normal velocity must vanish there. This would imply that:
fz. \l(( + f3y) = 0.
(2.4.8)
On western boundaries this would in particular require that:
Homogeneous Models of the Ocean Circulation
(2.4.3)
on the boundary.
The presence of viscous diffusion of vorticity implies the need for an additional dynamic boundary condition on the walls of the basin. If we interpret
(2.3.2) as applying to all scales of motion with An as the actual coefficient of
viscosity, the standard condition of no slip on the boundary would be required.
This would be:
u·t=O
(2.4.4)
where i is a unit vector tangent to the boundary. In terms of tjJ this becomes:
fz. \ltjl = 0.
(2.4.5)
It is not clear, however, that (2.4.5) is the correct boundary condition when
our equations explicitly resolve only the large-scale motion. The no-slip condition is a reflection of the interaction of the fluid with a solid boundary on the
microscopic level, and there is no reason why motion on scales of the order of
hundreds or thousands of kilometers should mimic that interaction. We might
even argue instead that the small-scale processes that occur right against the
wall buffer the large-scale flow and allow it to slip smoothly along the wall. A
condition of no tangential stress for the large-scale flow might be equally
logical then. For a north-south wall, for example, this would require:
OV+OU=O
ax ay
at the wall. Since u is already zero there by (2.4.1 ), it follows that 8v / 8x itself
must vanish there. We can write this boundary condition, which we call the slip
condition, in a form that is independent of the coordinate system by recasting
the condition in the form:
(2.4.6)
on each boundary.
On the other hand, if the lateral diffusion term in (2.3.2) is considered to be
a parameterization of small-scale mixing of vorticity by an unresolved eddy
field, we could argue that we are actually specifying a relation between the eddy
flux of total vorticity and the gradient of the large-scale vorticity field, i.e.:
-it'('= An\l(( + f3y)
(2.4.7)
whose divergence yields the diffusion term in (2.3.2). At a solid boundary the
normal component of the eddy flux of vorticity must vanish since the eddy
normal velocity must vanish there. This would imply that:
fz. \l(( + f3y) = 0.
(2.4.8)
On western boundaries this would in particular require that: