Midocean Approximations
183
4.3 Midocean Approximations
We are interested in situations in which the overall variation of the depth of a
density interface in midocean varies by order one over the extent of the gyre.
However, the midocean slope of the interfaces remains locally small as long as
Ro = U I fL remains small. Estimates for Ro were given in Chapter 3 and we
saw that in the midocean Ro is of the order of 0.001 for the velocities of the
order 10 cm/s and length scales of the order 1000 km. Of course, as the equator
is approached and the Coriolis parameter goes to zero, the local Rossby
number becomes large. Thus, the considerations of the present chapter will
exclude the equatorial zone. We return to the question of equatorial dynamics
in Chapter 6.
The ratio of the relative vorticity to the variable part of the planetary
vorticity (which is what counts in estimating the importance of the relative
vorticity in the overall vorticity balance) is of the order, UIPL 2 • For the same
scales used to estimate R 0 this parameter is of the order IQ- 2 • Hence to an
excellent approximation for the midocean the relative vorticity can be ignored
with respect to the variable part of the planetary vorticity in the expression for
the potential vorticity, i.e., to the order U I PL 2 :
(4.3.1)
In distinction to quasi-geostrophic theory we do not assume that h,. varies
only slightly from a mean value H,., nor do we assume that f can be linearized
about a central value. We are examining motions on truly planetary scales for
which those two approximations are quantitatively inaccurate. More importantly qualitatively, we examine the case in which the layer thickness vanishes
at the sea surface when the layer beneath it outcrops, i.e., when the interface
between the two layers reaches the sea surface.
The ratio of the kinetic energy to the pressure in the definition of the
Bernoulli function is:
p,.lu,.lz =a( puz ) = Ro ~ I
Pn
pUfL
(4.3.2)
so that to a first approximation, B,. = p,..
For small Rossby numbers the first approximation to the momentum
equation is the geostrophic balance:
! - - kA V'p,.
u,.- x-Po
( 4.3.3)
where we replace the variable density, p,., in the horizontal momentum
equation with the mean density p 0 , with an error of 0(10- 3 ).
183
4.3 Midocean Approximations
We are interested in situations in which the overall variation of the depth of a
density interface in midocean varies by order one over the extent of the gyre.
However, the midocean slope of the interfaces remains locally small as long as
Ro = U I fL remains small. Estimates for Ro were given in Chapter 3 and we
saw that in the midocean Ro is of the order of 0.001 for the velocities of the
order 10 cm/s and length scales of the order 1000 km. Of course, as the equator
is approached and the Coriolis parameter goes to zero, the local Rossby
number becomes large. Thus, the considerations of the present chapter will
exclude the equatorial zone. We return to the question of equatorial dynamics
in Chapter 6.
The ratio of the relative vorticity to the variable part of the planetary
vorticity (which is what counts in estimating the importance of the relative
vorticity in the overall vorticity balance) is of the order, UIPL 2 • For the same
scales used to estimate R 0 this parameter is of the order IQ- 2 • Hence to an
excellent approximation for the midocean the relative vorticity can be ignored
with respect to the variable part of the planetary vorticity in the expression for
the potential vorticity, i.e., to the order U I PL 2 :
(4.3.1)
In distinction to quasi-geostrophic theory we do not assume that h,. varies
only slightly from a mean value H,., nor do we assume that f can be linearized
about a central value. We are examining motions on truly planetary scales for
which those two approximations are quantitatively inaccurate. More importantly qualitatively, we examine the case in which the layer thickness vanishes
at the sea surface when the layer beneath it outcrops, i.e., when the interface
between the two layers reaches the sea surface.
The ratio of the kinetic energy to the pressure in the definition of the
Bernoulli function is:
p,.lu,.lz =a( puz ) = Ro ~ I
Pn
pUfL
(4.3.2)
so that to a first approximation, B,. = p,..
For small Rossby numbers the first approximation to the momentum
equation is the geostrophic balance:
! - - kA V'p,.
u,.- x-Po
( 4.3.3)
where we replace the variable density, p,., in the horizontal momentum
equation with the mean density p 0 , with an error of 0(10- 3 ).