Relating the conservation laws to the Bernoulli Equation

One of the interesting things that I realized when reading Olver, 1983 is his suggestion to investigate conservation laws as they relate to the quantities \phi_n^m where

\phi_n^m = \partial_x^n\partial_t^m\phi

He also suggested looking at these quantities for the stream function \psi as well.  Given that if \phi is harmonic, so is \phi_t, I replaced \phi with \phi_t in the nonlocal equation.  This results is a more complicated expression given by (modulo a sign error or two)

\displaystyle \frac{d}{dt}\int\phi\frac{d}{dn}\varphi_z\:dx = \int \phi_t \frac{d}{dx}\varphi_x - \eta_t \phi_x \frac{d}{dx}\varphi_z + \varphi_{zz}\eta_t^2\:dx

Double check that there isn’t a typo in the above equation.

In this case, you are able to get the other conservation laws (specifically, ones that I haven’t been able to generate so far such as the impulse AKA T_1). This also gives a very straightforward way to incorporate the Bernoulli equation through the \phi_t term on the right-hand side of the equation.

Of course, at this point, this isn’t taking into account any of the important simplification notes are used in Olver’s paper.  However, this is yet another way to consider attacking this problem.

Connecting Olver [1983] with AFM [2006]

Thinking more about these conservation laws and the connection with the nonlocal formulation, consider a harmonic function \varphi in conjunction with the velocity potential \phi and the stream-function \psi.

It is straightforward to show that (x + i\eta)^n are indeed conserved densities in the sense of Olver, 1983.  But do these make sense?  Well this was an embarrassing mistake, but the formulation is still helpful!  Details follow below.

Continue reading “Connecting Olver [1983] with AFM [2006]”

Constant Vorticity (Update)

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.

Continue reading “Constant Vorticity (Update)”