The Kakinuma model

The Isobe-Kakinuma 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. Kakinuma¹ and, together with T. Iguchi² we aimed at extending to this bilayer framework the recent results obtained on the Isobe-Kakinuma model.

In particular, we proved that

1. For any sufficiently regular initial data satisfying suitable compatibility conditions and a hyperbolicity criterion, there exists a unique solution to the Isobe-Kakinuma model on a relevant timescale
2. 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 Isobe-Kakinuma model displayed in the previous post. In particular, the hypersurface $t = 0$ in the space-time 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. Bridges³.

Our study bears some similitude with the ones on the Isobe-Kakinuma model, and in particular the latter one of T. Iguchi⁴, 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 rigid-lid framework⁵. I would like to concentrate below on a striking difference between the surface gravity waves framework and the interfacial gravity waves framework, namely Kelvin-Helmholtz instabilities.

I have already discussed Kelvin-Helmholtz 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 initial-value problem is mathematically ill-posed in a very strong sense: as proved by V. Kamotski and G. Lebeau⁶, 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 Kelvin-Helmholtz 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 high-frequency instabilities. Such is the case for the bilayer Saint-Venant system (displayed in this post), for which well-posedness in finite-regularity (Sobolev) spaces does hold under hyperbolicity conditions⁷ which correspond to the absence of Kelvin-Helmholtz 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 widely-used model including non-hydrostatic dispersive effects unfortunately turns out to overestimate Kelvin-Helmholtz instabilities. The outcome of our analysis on the Kakinuma model is that it inherently incorporates a stabilizing effect similar to the bilayer Saint-Venant system. The strength of this stabilizing effect diminishes as the size of the model increases, as the hyperbolicity domain for the well-posedness of the initial-value problem shrinks. This is consistent with the expectation that the accuracy with respect to the full model for interfacial waves improves, so that Kelvin-Helmholtz instabilities need to enter into play below a certain precision threshold.