The Kakinuma model
Higher order models for interfacial gravity waves
The IsobeKakinuma model presented in the previous post has a natural extension to the framework of interfacial gravity waves, that is two layer of homogeneous immiscible fluids separated by a free interface. This hierarchy of partial differential equations was proposed and numerically studied by T. Kakinuma^{1} and, together with T. Iguchi^{2} we aimed at extending to this bilayer framework the recent results obtained on the IsobeKakinuma model.
In particular, we proved that
 For any sufficiently regular initial data satisfying suitable compatibility conditions and a hyperbolicity criterion, there exists a unique solution to the IsobeKakinuma model on a relevant timescale
 This solution is at a distance $O(\mu_1^{2N+1}+\mu_2^{2N+1})$ to the corresponding solution to the interfacial gravity waves equations, where $\mu_1$ and $\mu_2$ are dimensionless shallowness parameters with respect to the upper and lower fluid layers.
I will not write down the equations, which are similar in structure with the IsobeKakinuma model displayed in the previous post. In particular, the hypersurface $t = 0$ in the spacetime is characteristic in the sense that, since there are several equations prescribing the evolution of the quantity describing the free interface, some compatibility conditions must be enforced initially (and the propagate for positive times). Taking into account these compatibility conditions, the equations can be reformulated as two evolution equations demonstrating a canonical Hamiltonian structure of the equations which is analogous to the one of the interfacial gravity waves equations exhibited by by T. B. Benjamin and T. J. Bridges^{3}.
Our study bears some similitude with the ones on the IsobeKakinuma model, and in particular the latter one of T. Iguchi^{4}, in fact some pieces of arguments can be borrowed from the latter work. There is however additional difficulties in the bilayer framework in particular in order to deal with the additional constraint due to the rigidlid framework^{5}. I would like to concentrate below on a striking difference between the surface gravity waves framework and the interfacial gravity waves framework, namely KelvinHelmholtz instabilities.
I have already discussed KelvinHelmholtz instabilities in this post but let me recap. In our situation of a sharp interface between two homogeneous fluids and in the absence of regularizing effects such as interfacial tension, the interfacial gravity waves equations suffers from severe instabilities at high spatial frequencies. Shortly put, the initialvalue problem is mathematically illposed in a very strong sense: as proved by V. Kamotski and G. Lebeau^{6}, any solution with a minimal level of regularity needs to be in fact infinitely smooth. This means that the modelling assumption of a sharp interface between two immiscible fluids must be questioned. In practice, any amount of shear velocity generates some mixing. Yet one observes that when shear velocities are weak, the mixing layer (the pycnlocline) remains thin and stable, and one can expect that an assumption of sharp interface can be reasonable to describe the behavior of the flow at large spatial scale, where KelvinHelmholtz instabilities are weak or inexistent.
On the other hand, shallow water models are precisely dedicated to describe approximately the large scale behavior of the flow, hence it can be expected that they do not suffer from highfrequency instabilities. Such is the case for the bilayer SaintVenant system (displayed in this post), for which wellposedness in finiteregularity (Sobolev) spaces does hold under hyperbolicity conditions^{7} which correspond to the absence of KelvinHelmholtz instabilities at low spatial frequencies. Yet this nice property is by no means automatic, and it has been discussed in this post that the bilayer extension of a natural and widelyused model including nonhydrostatic dispersive effects unfortunately turns out to overestimate KelvinHelmholtz instabilities. The outcome of our analysis on the Kakinuma model is that it inherently incorporates a stabilizing effect similar to the bilayer SaintVenant system. The strength of this stabilizing effect diminishes as the size of the model increases, as the hyperbolicity domain for the wellposedness of the initialvalue problem shrinks. This is consistent with the expectation that the accuracy with respect to the full model for interfacial waves improves, so that KelvinHelmholtz instabilities need to enter into play below a certain precision threshold.

T. Kakinuma, [title in Japanese], Proceedings of Coastal Engineering, Japan Society of Civil Engineers, 47 (2000)
T. Kakinuma, A set of fully nonlinear equations for surface and internal gravity waves, Coastal Engineering V: Computer Modelling of Seas and Coastal Regions (2001)
T. Kakinuma, A nonlinear numerical model for surface and internal waves shoaling on a permeable beach, Coastal engineering VI: Computer Modelling and Experimental Measurements of Seas and Coastal Regions(2003) ↩︎ 
V. Duchêne and T. Iguchi, A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part I : Structures and wellposedness, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire.
V. Duchêne and T. Iguchi, A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part II : Justification as a shallow water approximation, arXiv preprint 2212.07117 ↩︎ 
T. B. Benjamin and T. J. Bridges, Reappraisal of the Kelvin–Helmholtz problem. Part 1. Hamiltonian structure, J. Fluid Mech., 333 (1997) ↩︎

T. Iguchi, A mathematical justification of the Isobe–Kakinuma model for water waves with and without bottom topography, J. Math. Fluid Mech., 20 (2018) ↩︎

One could of course use the freesurface framework, but the situation would be in fact more complex due to the simultaneous presence of both surface (barotropic) and interfacial (baroclinic) modes of wave propagation. ↩︎

V. Kamotski and G. Lebeau, On 2D Rayleigh–Taylor instabilities, Asymptotic Analysis, 42 (2005) ↩︎

Mathematically speaking, these conditions guarantee that the equations are hyperbolic symmetrizable. Physically speaking, these amount to assume that the shear velocity (that is, roughly speaking, the jump of horizontal velocities at the interface) is not too strong. ↩︎