Congrats to Sal!

Last week, Salvatore Calatola-Young presented his Senior Synthesis project: “A Method for Deriving Conservation Laws for Water Waves”. In this talk, he presented the culmination of 3 years of research with me during his time as an undergraduate.

It wasn’t easy work, but he nailed it.

In summary, Sal showed how Olver’s conserved densities for the water-wave problem can be derived directly from a weak formulation without using the Lie Symmetries. Of course, Sal didn’t stop there! He extended the work for waves traveling over linear shear (constant vorticity), as well as multilayer flows. There’s a lot left to be explored here – and we’re working to bring it all together.

You can find the first in a series of papers available on the ArXiV (https://arxiv.org/abs/2105.07580).

Finding Conservation Laws

In joint work with Sal, we have finally written down the systematic process for deriving all of Olver’s conservation laws for the water-wave problem from our nonlocal-nonlocal formulation. This post outlines the current state of deriving the conservation laws from this Nonlocal-Nonlocal formulation discussed in the A Weak Formulation of Water Waves in Surface Variables. Click Continue Reading… to learn a little bit more about the process.

Snapshot of fluid domain
Continue reading “Finding Conservation Laws”

Thinking more about multiple scales

[mathjax] Perhaps this is widely known, but I got a bit stuck-in-the-weeds thinking about applying multiple scales to problems with nonlocal operators. I have honestly been a bit unsatisfied with the literature that I have found on this scenario – that said, if you have any suggestions or references, please feel free to leave them in the comments! Continue reading to see what I’ve been thinking about.

Continue reading “Thinking more about multiple scales”

A Weak Formulation of Water Waves in Surface Variables

In 2006, Ablowitz, Fokas, and Musslamini published a paper where they reformulated the water-wave problem for irrotational fluids as a system of equations in surface variables given by

[mathjax]\begin{align*}
\displaystyle 0&= q_t + \frac{1}{2}q_x^2 + g\eta -\frac{1}{2}\frac{(\eta_t+q_x\eta_x)^2}{1 + \eta_x^2}\\
\displaystyle 0&= \int_{-\infty}^{\infty} e^{-ikx}\left(\eta_t\cosh(k(\eta+h)) + iq_x\sinh(k(\eta+h))\right)\:dx, \qquad \forall k \in \mathbb{R}.\end{align*}

In this formulation, \(q(x,t)\) is the trace of the velocity potential along the free surface \(z = \eta(x,t)\).

In A Weak Formulation of Water Waves in Surface Variables (short note posted on arXiv.org), I take a look at extending their formulation to a coupled system of integro-differential equations given by

\begin{align*}
0&=\int_{-\infty}^{\infty} e^{-ikx}\left(\left(q_x\eta_t – \eta_x(q_t+g\eta)\right)\cosh(k(\eta+h)) -i\left(q_t + g\eta\right)\sinh(k(\eta+h))\right) dx\\
0&=\int_{-\infty}^{\infty} e^{-ikx}\left(\eta_t\cosh(k(\eta+h)) + iq_x\sinh(k(\eta+h))\right)\:dx, \qquad \forall k \in \mathbb{R}.\end{align*}

You might be asking, why?

Continue reading “A Weak Formulation of Water Waves in Surface Variables”

En Route to Conservation Laws

Deriving conservation laws from the nonlocal formulation of Ablowitz, Fokas, and Musslimani has been a little bit of a pet project of mine for exactly 2 years now. Finally, things have come together and I’ve started documenting things. What I was hoping would be a short paper has now turned into a series of papers on various different topics. I just posted a short note ot the ArXiV that lays out the first step – recasting the AFM formulation into a coupled system of nonlocal equations that encorporate both the kinematic and dynamic boundary conditions. I’m hoping to have the remaining papers posted in the near future. Once the papers are approved on the ArXiV, I’ll post them here along with some exposition. Excited to finally have a lot of this documented.