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uu + vu ! fv = !gh ! gh
x y x s x
(10.2.9 a, b)
uv + vv + fu = !gh ! gh
x y y s y

where we have defined

p
h = s (10.2.10)
s
!g

If we eliminate the pressure gradient terms in (10.2.9 a,b) by cross differentiating, we
obtain,
x (
!
(
ui!(" + f ) + f +") u + v ) = 0 (10.2.11)
y
while the equation for mass conservation is ,

(
( uH) + vH) = 0 (10.2.13)
x y
which when combined with (10.2.11) yields the conservation of potential vorticity in the

form we discussed in section 8.3, namely, for steady flow,

! " + f
ui! = 0 (10.2.14)
H
From (10.2.13) we can define a stream function for the horizontal transport,

uH = !" , vH = " (10.2.15)
y
x
or in vector form,
!
ˆ
uH = k ! "# (10.2.16)

f +!
Since the potential vorticity q = is constant along streamlines,
H

Chapter 10 9