Scaling for Sverdrup Theory
5
the dynamics of the oceanic interior. The theory consists of two parts. The first
is the derivation of a simplified vorticity balance for the interior region, and the
second is a vertical average of this balance which, under appropriate
conditions, leads to a remarkable relationship between the vertically averaged
meridional flow and the local value of the wind stress on the ocean surface.
Our starting point is the momentum equation written in a frame of
reference rotating with the earth at an angular velocity Q (7.3 x 10- 5 s- 1 ), i.e.:
Di1 2i\ -
\lp -
p
p
(1.2.1)
where i1 is the velocity seen in the rotating frame, p is the pressure, p is the
density, g is the acceleration due to gravity, and <;.)
generally, to diffuse the momentum of the large scale flow. The momentum
equation in the form ( 1.2.1) is far too general to give much insight into the
dynamics of the ocean circulation. To proceed further we require approximations specific to the oceanic interior, and it is in the act of making these
approximations in a systematic way that the dynamical character of Sverdrup
theory really becomes apparent.
Let us suppose that U is a characteristic value for the midocean horizontal
velocity (i.e., the velocity tangent to the earth's surface) outside western
boundary currents, and that L is a characteristic horizontal scale of that
circulation. Assuming that the large-scale flow pattern is essentially stationary
over the circulation time, L/ U, of the interior flow, the ratio of the relative
acceleration to the Corio lis acceleration in (1.2.1 ), i.e., the ratio of the first to
the second term in the momentum equation is:
UU/L = U = Ro
2QsinOU fL(1.2.2)
where e is the latitude and is the complement of the angle between the rotation
vector and the local normal to the earth's surface. The parameter f is the
Coriolis parameter:
f= 2Qsin0.
(1.2.3)
The parameter Ro is the Rossby number. For large-scale flows R0 is very small.
For example, if U, L, and fare 1 cm/s, 1000 km and 10- 4 s- 1 , respectively
(typical values at midlatitudes), then Ro is 10- 4 .
If we consider the horizontal equation of motion, i.e. the equations for the
motion tangent to the earth, the frictional force
horizontal surfaces in the flow, while the second part is the horizontal
divergence of the horizontal stress acting on vertical faces of the fluid element
so that:
5
the dynamics of the oceanic interior. The theory consists of two parts. The first
is the derivation of a simplified vorticity balance for the interior region, and the
second is a vertical average of this balance which, under appropriate
conditions, leads to a remarkable relationship between the vertically averaged
meridional flow and the local value of the wind stress on the ocean surface.
Our starting point is the momentum equation written in a frame of
reference rotating with the earth at an angular velocity Q (7.3 x 10- 5 s- 1 ), i.e.:
Di1 2i\ -
\lp -
p
p
(1.2.1)
where i1 is the velocity seen in the rotating frame, p is the pressure, p is the
density, g is the acceleration due to gravity, and <;.)
generally, to diffuse the momentum of the large scale flow. The momentum
equation in the form ( 1.2.1) is far too general to give much insight into the
dynamics of the ocean circulation. To proceed further we require approximations specific to the oceanic interior, and it is in the act of making these
approximations in a systematic way that the dynamical character of Sverdrup
theory really becomes apparent.
Let us suppose that U is a characteristic value for the midocean horizontal
velocity (i.e., the velocity tangent to the earth's surface) outside western
boundary currents, and that L is a characteristic horizontal scale of that
circulation. Assuming that the large-scale flow pattern is essentially stationary
over the circulation time, L/ U, of the interior flow, the ratio of the relative
acceleration to the Corio lis acceleration in (1.2.1 ), i.e., the ratio of the first to
the second term in the momentum equation is:
UU/L = U = Ro
2QsinOU fL(1.2.2)
where e is the latitude and is the complement of the angle between the rotation
vector and the local normal to the earth's surface. The parameter f is the
Coriolis parameter:
f= 2Qsin0.
(1.2.3)
The parameter Ro is the Rossby number. For large-scale flows R0 is very small.
For example, if U, L, and fare 1 cm/s, 1000 km and 10- 4 s- 1 , respectively
(typical values at midlatitudes), then Ro is 10- 4 .
If we consider the horizontal equation of motion, i.e. the equations for the
motion tangent to the earth, the frictional force
horizontal surfaces in the flow, while the second part is the horizontal
divergence of the horizontal stress acting on vertical faces of the fluid element
so that: