When the constant vorticity is strong enough to induce a region of lower pressure under the crest, is the wave stable?

Streamlines (top) and pressure contours (bottom) for strong constant vorticity.  Is this wave stable?

This is something that has been bothering me for a while.  I have spectral stability code that seems to work, until I begin considering cases where a pressure bubble develops.

The Good

In testing my numerical code, I’m able to recover precisely the NLS cut-off for the presence of the Benjamin-Feir instability in infinite depth as outlined in the work of Thomas, Manna, and Kharif as shown in the images below.

In infinte depth, the BF instability is not present for \omega>2/\sqrt{3}.
The threshold for finite depth – in agreement with NLS approximations


Even more important is the fact that we do see bubbles of instability as predicted by the collision of eigenvalues of opposite Krein signature.  All of these items point towards consistent numerics.

The Bad

For large values of vorticity, the numerics just don’t make sense!  The spectra is calculated by linearizing about a traveling wave solution.  For large values of the vorticity, say \omega = 3, the solutions that have pressure sinks are surprisingly tame.  So what’s going wrong?

About the solutions

In short, a solution may only contain 16 modes that register with amplitudes larger than 10^{-14}.  However, the normal derivative of the pressure at the surface behaves in a way the “seems under-resolved” in some cases.  Though, for strong vorticity, things are well resolved as seen in the figure below.

A sample solution when \omega = 5, and h = \infty.  This solution contains a pressure bubble and less than 10 modes in both the free surface \eta(x) and \partial_n p(x,\eta) = 0.

The type of behavior that I see in the stability spectra is precisely what I see when solutions or the spectra are under-resolved.    This could mean that while I’m seeing exponential decay in the tails of the Fourier coefficients of the traveling wave solution, perhaps that’s not really enough; maybe there is more info in the tails than I think.

However the way in which the spectra breaks down is itself interesting.  I’ll post pictures once I’ve finished generating a few, but there are interesting gaps that form.

Things I’ve tried

  • Care when dealing with k = 0.  This has been a somewhat painful process in  making sure that the “0” mode is handled correctly.  I’ve definitely learned a few things along the way on this one.  
  • Switching to the canonical Hamiltonian Coordinates.
  • Use of adaptive mesh techniques to increase accuracy of rendered solutions.
  • Examined different eigenvalue solvers for the generalized problem.  This includes checking symmetry structures and choosing the appropriate algorithms.
  • Triple checked the problem formulation.

Further Items to Consider

  • What is happening with the Hamiltonian structure?  Does it make sense when \partial_np(x,\eta) = 0?
  • Are the appropriate symmetries still present?
  • What if we were to reformulation this problem via a DNO expansion


2 thoughts on “Stability and Constant Vorticity

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