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! + f
q = = q " ( ) (10.2.17)
H

Applying the identity (7.7.1) to (10.2.9) shows that we can write the momentum
equations for steady flow as,

! 2
! ! % u (
'
! " u = #$ g{h + h } + * (10.2.18)
a & s 2 )

!
(
ˆ
where ! = k " + f ) while the dot product of that equation with the velocity yields the
a
Bernoulli theorem for the shallow water model,

! 2
! " u %
ui! g(h + h ) + ' = 0,
$
$ s 2 ' &
#
(10.2.19 a, b)
! 2
u
( B ) g(h + h ) + = constant on streamlines
s 2

so that the momentum equation (10.2.18) is actually,

! !
! " u = #$B (10.2.20)
a

But since the velocity is given by the streamfunction as in (10.2.16),

)
a (
! ! ! ( & + f )
ˆ
! " u = ! " k " #$ / H = % #$ (10.2.21)
a
H
which means that the momentum equation is just,

! + f
"# = "B (10.2.22)
H

but since B is a function only of ψ , !B = dB !" which with (10.2.22) yields a rather
d"

remarkable connection between the potential vorticity and the Bernoulli function,

Chapter 10 10