164
3. Beyond the One-Way WaveEquation
and (3.122) may be written as
(3.123)
where J is the Jacobian operator
Bp 8q
8p 8q
J(p, q) = - - - - - .
8x 8y
8y 8x
In atmospheric science, (3.123) is known as the barotropic vorticity equation.
Pjertoft (1953) demonstrated that if the initial conditions are smooth, solutions
to (3.123) must remain smooth in the sense that there can be no net transfer of
energy from the larger spatial scales into the smaller scales. Fjertoft's conclusions
follow from the properties of the domain integral of the Jacobian operator. Let p
denote the domain integral of p, and suppose, for simpiicity, that the domain is
periodic in x and y . Then by the assumed periodicity of the spatial domain?
J(p,q) = - 8 (8 p -
q ) - - 8 ( 8
8x
8y
8y
q
p - ) =0.
8x
(3.124)
(3.125)
As a consequence,
and
pJ(p , q) = J(p2/2 , q) = 0,
qJ(p, q) = J(p , q2/2) = O.
The preceding relations may be used to demonstrate that the domain-integrated kinetic energy and the domain-integrated enstrophy (one-half the vorticity squared)
are both conserved. First consider the enstrophy, /2 = (V 21/1)2 /2. Multiplying
(3.123) by V21/1 and integrating over the spatial domain yields
which using (3.125) reduces to
.
The conservation of the domain-integrated kinetic energy, U . u/2 = V1/1 . V1/1/2,
may be demonstrated by first noting that the vector identity
V . (cea) = Va · a + a(V . a)
9Equivalent conservation properties hold in a reetangular domain in which the normal veloc ity is
zero at all points along the boundary.