8
Sverdrup Theory
- OW
V'. u + oz = 0.
(1.2.11)
Here we have neglected a term of order wjR, where R is the earth's radius, in
comparison with owfoz, which yields an error ofO(DjR) which is very small. It
is important to recall that it in ( 1.2.11) is the horizontal velocity, and that V' is
the two-dimensional version of the gradient operator. Using (1.2.11) to
eliminate the horizontal divergence we obtain:
ow 0 A
pf3v = p f - + -k · \7 X r
oz oz
where f3 is the northward spatial derivative off That is:
13 = !._ of = 2n cos e
RoO
R
(1.2.12)
(1.2.13)
while below the mixed layer in the geostrophic region (1.2.12) becomes simply:
OW
{3v =f oz.
(1.2.14)
In this approximation the vorticity equation is reduced to the particularly
simple form (1.2.14). Its physical interpretation is simply that when a fluid
column is stretched in the vertical direction, i.e., when ow/ oz > 0, vorticity is
produced at a rate equal to this stretching times the planetary vorticity,/, itself.
In the midoceanic interior approximation the only vorticity which the fluid has
is the planetary vorticity, f The vorticity must increase if the stretching is
positive. However, since the relative vorticity is negligible, the fluid elements, to
increase their vorticity, must move northward where the planetary vorticity is
greater in order that the planetary vorticity which they possess becomes
greater. Thus v must be positive. This balance of vorticity production by the
stretching of planetary vorticity filaments f and the accession of greater
vorticity by meridional motion in the planetary vorticity gradient {3, is called
the Sverdrup relation. It is important to note that it is a local differential
balance in the sense that it holds on each fluid element beneath the mixed layer.
In the mixed layer the effect of stretching is augmented by the direct
differential torque due to the fluid turbulent stresses which also act to alter the
planetary vorticity of the fluid, and this enters as an additional source on the
right side of (1.2.12).
We discuss more completely in Section 1.4 further conditions for the
validity of (1.2.14) beyond those that we have so far described. The reader may
well ask why any further conditions beyond the smallness of Ro and En,
already described, are necessary. The need for further discussion is, in fact, a
measure of the real subtlety of the balance, which at first glance appears so
simple and even obvious. As we see below, it is actually neither.
Sverdrup Theory
- OW
V'. u + oz = 0.
(1.2.11)
Here we have neglected a term of order wjR, where R is the earth's radius, in
comparison with owfoz, which yields an error ofO(DjR) which is very small. It
is important to recall that it in ( 1.2.11) is the horizontal velocity, and that V' is
the two-dimensional version of the gradient operator. Using (1.2.11) to
eliminate the horizontal divergence we obtain:
ow 0 A
pf3v = p f - + -k · \7 X r
oz oz
where f3 is the northward spatial derivative off That is:
13 = !._ of = 2n cos e
RoO
R
(1.2.12)
(1.2.13)
while below the mixed layer in the geostrophic region (1.2.12) becomes simply:
OW
{3v =f oz.
(1.2.14)
In this approximation the vorticity equation is reduced to the particularly
simple form (1.2.14). Its physical interpretation is simply that when a fluid
column is stretched in the vertical direction, i.e., when ow/ oz > 0, vorticity is
produced at a rate equal to this stretching times the planetary vorticity,/, itself.
In the midoceanic interior approximation the only vorticity which the fluid has
is the planetary vorticity, f The vorticity must increase if the stretching is
positive. However, since the relative vorticity is negligible, the fluid elements, to
increase their vorticity, must move northward where the planetary vorticity is
greater in order that the planetary vorticity which they possess becomes
greater. Thus v must be positive. This balance of vorticity production by the
stretching of planetary vorticity filaments f and the accession of greater
vorticity by meridional motion in the planetary vorticity gradient {3, is called
the Sverdrup relation. It is important to note that it is a local differential
balance in the sense that it holds on each fluid element beneath the mixed layer.
In the mixed layer the effect of stretching is augmented by the direct
differential torque due to the fluid turbulent stresses which also act to alter the
planetary vorticity of the fluid, and this enters as an additional source on the
right side of (1.2.12).
We discuss more completely in Section 1.4 further conditions for the
validity of (1.2.14) beyond those that we have so far described. The reader may
well ask why any further conditions beyond the smallness of Ro and En,
already described, are necessary. The need for further discussion is, in fact, a
measure of the real subtlety of the balance, which at first glance appears so
simple and even obvious. As we see below, it is actually neither.