3.6 Nonlinear Instability
165
implies that
8'111/1) _
1/1 81
81
('111/1 . '111/1).
2
8'11
21/1 = v. (
1/1 81
Then multiplying (3.123) by 1/1, using the preceding relation and integrating over
the periodic spatial domain one obtains
81
2
(:'1/1 . '111/1) = O.
Now suppose that the stream function is expanded in Fourier series along the x
and y coordinates
1/1 = L I>k,fei(kx;Hy) = L 1/Ik,f,
k e
k.e
and define the total wave number K such that K 2 = k 2 + f2. By the periodicity of
the domain and the orthogonality of the Fourier modes,
i.e
n · n = ' 111/1 . ' 111/1 = v · (1/1'111/1) -1/1'11 21/1 = - 1/1' 11 2 1/1 = LK 2 1/1l e
K 4 1/1l e .
k,f
and
= ('11 21/1)2 = L
The two preceding relations may be used to evaluate an average wave number,
- )1/2
K avg, given by the square root of the ratio of the domain-integrated enstrophy to
the domain-integrated kinetic energy,
n·n
Kavg =
(
Since the domain-integrated enstrophy and the domain-integrated kinetic energy
are both conserved, K avg does not change with time. Any energy transfers that
take place from larger to smaller scales must be accompanied by a second energy
transfer from smaller to larger scales to conserve K-there can be no systematic
energy cascade into the short-wavelength components of the solution .
Suppose that the barotropic vorticity equation (3.123) is approximated using
centered second-order differences in space and time such that
where the numerical approximation to the horizontal Laplacian operator is