In the quest to understand what happens to the stability as the strength of the vorticity increases, it’s important to predict precisely where the first instability will arise (in terms of the Floquet parameter \mu.

Intersecting lines of opposite signature could give rise to a potential instability.  The two black dots follow the first to collisions; both of which yield instabilities for non-trivial solutions.

If we seek Floquet parameters that result in the collision of eigenvalues of opposite signature for the trivial solutions, we can adapt our stability calculations to capture the instability.  It’s well known that for irrotational fluids on infinite depth, a Floquet parameter of \mu = .25 results in instability (one of the few cases where this can be found analytically).

For linear shear, the eigenvalues for the trivial solution take the form

\displaystyle \lambda^\pm = i \left(c k -\frac{1}{2}\left(\text{sgn}(k)\omega \pm \sqrt{\omega^2 + \vert k \vert g}\right)\right)

where c is the wave-speed from which we are bifurcating.  We can easily calculate the Krein signature.  As we increase the vorticity, the location of the collisions in the Floquet parameter changes (see the figure below).

In the figure above, I’ve plotted the eigenvalues -i \lambda_k^{\pm} as a function of \mu  in the form

\displaystyle \lambda_k^\pm = \left(c (k+\mu) -\frac{1}{2}\left(\text{sgn}(k+\mu))\omega \pm \sqrt{\omega^2 + \vert k+\mu \vert g}\right)\right)

This graph is a good reminder of a few different things.

1. The curves “split” from the irrotational case. It is probably worth discussing how the branch of eigenvalues does seem to separate due to the \vert k \vert terms that appear in the expression.  I haven’t thought about this too much, but (\mu,k)\to(-\mu,-k) implies \lambda^+ \to -\lambda^- (I think… double check this statement and its implications).

2. The first collisions occur at increasing large values of \mu.  This means that in order to see instabilities that arise for a specific \mu such that \lambda_n^+=\lambda_m^-, I will need to make sure that my truncation is at least as large as N > n - m + 1 + \lfloor\mu\rfloor.  This shouldn’t really be a problem in general.  However, there may be some issues with the map from \eta \to q_x that essentially cause the overall solution to be under-resolved.  I spent a lot of time controlling error for large k in terms of \hat\eta_k.  However, I never really looked at the derivatives and the map back to q_x.

3. Integer values of \mu.  There seems to be a potential numerical issue near \mu = 2.  For the irrotational case, the Krien signature doesn’t provide useful information when \mu takes on an integer value and \lambda =0.  As the vorticity increases, we see that what was originally a crossings at an integer multiple of \mu now gives rise to instabilities (yes, I checked in the nonlinear problem).  Is there a connection now between sub- and super-harmonic instabilities and these?  This is worth looking into.

4. The case where \mu = 0More concerning is what happens near \mu = 0.  As you can see, those lines seem to asymptotically approach each other.  There is still a unique “intersection” at \mu = 0, but they are definitely approaching each other. Perhaps this is related to the issues that I observe in my numerics in that “junk” seems to be associated with integer values of the Floquet parameter \mu.  I’ll add images of this soon.

Definitely some things to think about here.

3 thoughts on “Constant Vorticity (Update)

  1. Comments from chatting with Richard: Try increasing the number of resolutions by 10-15 modes and see what happens. That is, do I see 4 levels? Does it converge? Think back to the inc_modes_4_modes_amp_0_040_omega_3_h_infinity_loop.PNG

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