Looking at the real part of the growth rate \left(\mathcal{R}\left\lbrace\lambda\right\rbrace\right) vs. the Floquet parameter \left(\mu\right) for increasing matrix truncations sizes \left(N\right)

The above picture shows the real part of the growth rate \lambda vs. the Floquet parameter \mu for increasing truncation of the Floquet/Fourier/Hill matrix.

After a lengthy discussion with Richard Kollar, we have agreed that there seem to be two separate issues here.  The first is that one must be careful when restricting the Floquet Multiplier \mu \in \left[0,\frac{1}{2}\right] in conjunction with the size of the truncated matrix for the Fourier/Floquet/Hill method.  The second is that there is still some other type of instability that may be present (the “moving” instability).

Convergence, Truncation, and Krein Signatures

This seems obvious.  In the above scenario (for simplicity, the vorticity is \gamma = 3 and \vert \eta\vert_\infty = 0.04), examining Krein signatures for the trivial solutions indicates that we may detect instabilities at the following parameter values where (k_1,k_2) correspond to the k values for \lambda^+ and \lambda^-

Collision (1 + \mu, -1+\mu)Collision (1 + \mu, -2+\mu)Collision (1 + \mu, -3+\mu)
\mu = 8.681245184895484\mu = 22.816653826071420\mu = 42.406225916615078
\lambda \approx 30i\lambda \approx 75i\lambda \approx 138i

While the above table is only valid for non-zero amplitude solutions, we consider a relatively small amplitude solutions that is fully resolved by 7 Fourier modes up to machine precision.  That is,

\[ \eta(x)  = \displaystyle\sum_{n = -7}^{n=7} \hat{\eta}_n\,e^{inx} + \mathcal{O}\left(10^{-14}\right)\]

So, if we even expect to see an instability arising from the collision from (1,-1) for the Floquet parameters restricted to the interval \mu \in \left[-\frac{1}{2},\frac{1}{2}\right) then we must make sure to include 11 modes in the truncation of our Fourier/Floquet/Hill method (making it equivalent to 12-wave mixing – did I get that count right Olga?).  Thus, when considering instabilities for extremely large values of vorticity, the number of modes required for convergence will increase dramatically.  See the image in Constant Vorticity (Update) for additional details.

Other Questions/Comments

Of course, there are other lessons that have been learned along the way, as well as other questions.

Does the method you use to calculate the linearization matter?

This question relates to use of the FFT vs. directly calculating the integrals.  I’ve tried  both methods.  Despite the fact that the integral computations run significantly slower, the two methods provide the same results! Poctures comparing are shown below.  However, it is worth noting the following…  The banded structure of the matrices could be playing a role in this behavior (spoiler: it’s not).

What if you increase the range of Floquet Parameter?

N = 4 truncation using full integration
N = 4 truncation using FFT
N = 8 truncation

Why do you only see this phenomena for large positive vorticity?

Perhaps the easiest way to explain this is that it is due to the fact that you need a significant number of wave mixing in order to see the instability.  But why is the instability significantly stronger despite the solutions being more “linear”? And how does the instability grow as a function of either the vorticity, or the normal derivative of the pressure along the interface (if there even is some sort of relationship between the two).

Are there differences in the eigenfunctions?

Notices that there is a “traveling instability” that leads the introduction of each new instability.  This instability doesn’t appear in the appropriate location and moves as the truncation size is increased.  This movement happens in a consistent manner.

What about that damn pressure bubble?

Yeh, what about it?  I’ll get there (hint: right now, it’s a complete mess!)

Related Posts

Stability and Constant Vorticity (July 14, 2018)
Constant Vorticity – Update (July 18, 2018)

2 thoughts on “Constant Vorticity – The Saga Continues…

    1. I’m still trying to precisely figure out what I mean by that. It’s not the “typical” definition, so perhaps I should find better terminology to describe what I mean. Will think about that after my talk 🙂

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