PhD Thesis

Louis Emerald completed his PhD thesis under my supervision in October, 2021. Most of his research work was dedicated to the study and the rigorous justification of fully dispersive models for the propagation of surface gravity waves.

Fully dispersive models are, by definition, models for the propagation of surface gravity waves whose linearization about the rest state coincides (or concurs) with linearized (Airy) water waves system, so that their frequency dispersion agrees exactly with the picture described by Wikipedia. This is typically not the case for standard models in oceanography (Korteweg-de Vries, Boussinesq, Green-Naghdi…) where the dispersion agrees only approximately.¹

Fully dispersive models are typically constructed from standard models by artificially tweaking the dispersive component so as to force the desired full dispersion property. One celebrated example is the so-called Whitham equations, which reads as follows: $$\partial_t\zeta+\sqrt{gd}\ \partial_x\big((\tfrac{\tanh(d|D|)}{d|D|})^{\frac12}\zeta+\tfrac3{4d}\zeta^2\big)=0,$$ and has been introduced by G. W. Whitham² as a fully dispersive counterpart of the Korteweg-de Vries equation: $$\partial_t\zeta+\sqrt{gd}\ \partial_x\big(\zeta+\tfrac{d^2}6 \partial_x^2\zeta+\tfrac3{4d}\zeta^2\big)=0.$$ Above, $g$ is the gravity acceleration, $d$ the depth of the fluid layer at rest, $\zeta(t,x)$ the surface deformation, and we use the following notation for Fourier multiplier operators: $$\widehat{f(D)u}(t,\xi)=f(\xi)\widehat{u}(t,\xi)$$ where $\widehat{u}(t,\xi)$ is the Fourier transform of the function $u(t,x)$ with respect to the variable $x$.

It turns out that the Whitham equation, contrarily to the Korteweg-de Vries equation, is able to reproduce qualitatively important features of the water waves system such as wavebreaking and non-smooth maximal-amplitude travelling waves. This recent discovery³⁴ triggered a renewed activity and fully dispersive counterparts of the Boussinesq and Green-Naghdi systems were quickly derived, studied and (numerically) compared.

Louis’ focus consisted in deciding whether these fully dispersive models could be rigorously justified as asymptotic models for the water waves system with improved precision with respect to their standard counterparts. This was made possible because, contrarily to previous works, Louis derived models directly starting from the water waves system rather than using an other model as intermediary.

The upshot of his analysis is that the Whitham equation (respectively Boussinesq-Whitham systems) improve the precision of the Korteweg-de Vries equation (respectively Boussinesq systems) when the dimensionless nonlinearity parameter is asymptotically smaller than the dimensionless shallowness parameter. Yet a deeper conclusion is that the Whitham equation (respectively Boussinesq-Whitham systems) are in fact really corresponding to the Hopf equation (respectively the Saint-Venant system), that is hydrostatic (non-dispersive) systems, improving their precision as soon as the dimensionless nonlinearity parameter is asymptotically small. This is also the way we can interpret the Whitham-Green-Naghdi model introduced in a preceding work⁵ and described here.