Suppose that the vortex sheet is identical with a line of latitude, namely θ(α,t) = θ0 and φ(α,t) = α, then the right hand sides of both equations become
Hence, the vortex sheet θ = θ0 and φ = α + V0t is a steady solution rotating with the constant speed V0. We study the linear stability of this solution.
Assume that the solution is disturbed slightly, that is to say
Then, we expand the equations in terms of
So we get the equations :
Using the perturbation expansion with respect to small disturbances,
Integration gives the followings :
Let the small disturbances be Fourier series such as below :
Perturbation expansion of the equation of phi should be presented here
Therefore the eigenvalues are as follows :
Thus, if the mode n satisfies
then Fourier coefficients θn(t) and φn(t) become neutrally stable. On the other hand, for sufficiently large n, since the positive eigenvalue approaches asymptotically to
, a disturbance of high wave number grows like Kelvin-Helmholtz instability for planar flow.
We apply this stability condition to two special cases.
First, when the strengths of both pole vortices are identical, namely κ2 = 0, the stability condition is reduced to
It indicates that when there is no pole vortex on the spheroid, i.e., κ1 = 1 , the number of stable spectra depends on the value of 'a'. The number of stable spectra decreases as 'a' increases and it increases as 'a' decreases to -1. For fixed κ1, especially when the latitude is pi over 2, the # of stable spectra becomes indefinitely large as a decreases to -1.
Next, when the total vorticity on the spheroid is zero, namely κ1 = 0, the stability condition becomes
This means if the strength of the north pole vortex is greater than that of the south pole vortex, i.e., ΓN > ΓS , the vortex sheet in the northern hemisphere region has some neutrally stable spectra, while the vortex sheet in the southern hemisphere region has no stable spectra. Therefore, the vortex sheets in the northern hemisphere region evolve more stably than those in the southern hemisphere region at the initial moment of their evolution.
In what follows, we verify numerically the stability of the vortex sheet on the line of latitude,
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