420
changes in the slowly varying baroclinic part. We anticipate therefore
a diagnostic balance for the barotropic QPV. In addition, the short baroclinic Rossby waves are filtered out and only the long nondispersive waves
with phase speed {3R~nt are retained.
We transform to barotropic and baroclinic streamfunctions <1? and 8,
defined by
(19)
The corresponding barotropic and baroclinic QPV balances will be given in
dimensionless form. We define a timescale 1/1/01 and a lengthscale L = Y/7r
and scale the dependent and independent variables according to
t' = tl/ol
( ' ') _ (x, y)
x,Y -
L
(<1?',8') = 16~2 (<1?,8)
The dimensionless width of the channel is then 7r and the length is 27rf
where f = X/2Y is a ratio of the channel dimensions. We also define
dimensionless quantities by
B' = BIH
r' = rIUJLH)
{3' = L{3/l/ol
E' = E/i/ol
M' = M/l/ol
and omit the primes in the following for brevity. The barotropic and baroclinic QPV balances then take the form
a<1?
(3 ax - .J(<1? - h8, B) -
-! (a8 + .J(<1?, 8)) + (3~(<1? + 8)
(j
at
ax
curlr - _E_V2(<1? - h8X20)
l+h
l+h
-h-curlr - (1 + h)MV 2 The barotropic balance (20) is diagnostic and linear. These noteworthy
properties are a consequence of the neglection of the relative vorticity.