PhD Thesis

Mahieddine Adim completed his PhD thesis under my supervision in July, 2024. The goal of his thesis was to build bridges between geophysical models associated with (continuously) density-stratified flows and layered models.

The latter models are often used as simplified versions of the former. Indeed, the great benefit of layered models is that quantities at stake do not depend on the vertical space variable (dimension reduction), with the reasonable price to pay of an augmented number of equations (depending on the number of layers), and of course a loss in accuracy. Additionally, in situations where the flow profile can be approximately described as piecewise constant in the vertical variable (say a layer of fresh water on top of a layer of salted water), layered models help to focus on quantities of main interest and in particular in the behavior of the thin “interface” (the pycnocline). Yet in the study of such layered models (see this post and this one for some references) the description of the flow as a superposition of immiscible fluid layers is generally introduced as an ad hoc assumption and its validity is rarely discussed.

Mahieddine Adim has been able to quantify the “error” between layered models and corresponding equations for continuously stratified flows in two directions:

1. From multilayer to continuously stratified: continuously stratified flows can be approximated through multilayer models and the difference vanishes as the number of layers in the multilayer model grows.¹
2. From continuously stratified to bilayer: Multilayer models provide accurate description of stratified flows when the density and velocity profiles are only approximately piecewise constant.²

These two lines of results have been made possible using several key ingredients. Firstly, the assumption of hydrostatic equilibrium (which is only an approximation in a dynamical context) has been made. This is somewhat mandatory as Kelvin-Helmholtz instabilities prevent the propagation of sharp interfaces without the hydrostatic assumption (see this post). This is also motivated by the fact that the hydrostatic assumption is ubiquitely used in the geophysical context and in particular in primitive equations (see this post). Secondly, the comparison between the layered models and corresponding continuously stratified equations has been eased by the use of a reformulation of the latter using isopycnal coordinates, as in the previous work of Roberta Bianchini and myself³ (see this post). These isopycnal coordinates can be interpreted as considering the fluid domain as a foliation of an infinite number of (infinitesimally thin) layers with given constant densities (hence the framework of density-stratified flows) and offers a framework which unifies continuously stratified equations and (finitely) layered models. Finally, mostly motivated by the fact that the stability of the hydrostatic Euler equations is poorly understood (but also inspired by parameterizations used in the geophysical context), and again as in [³], the thickness diffusivity effect of the Gent and McWilliams parameterization plays a crucial role.

Hence the equations considered in the work of Mahieddine are the following ones. $$ \begin{aligned} \partial_t h + \partial_x ((1+ h) ( \underline{u} + u)) & = \kappa \partial_x^2 h, \\ \partial_t u + \left( \underline{u}+ u +u^\star\right)\partial_x u +\frac1{\underline{\rho}} \partial_x \Psi &=0, \end{aligned} $$ where $u^\star=- \kappa \frac{\partial_x h}{1+ h}$ is often referred to as the bolus velocity and the Montgomery potential $\Psi$ is given by $$ \Psi(t,x,r)= \underline{\rho} (r) \int_{-1}^r h (t,x, r’) {\rm d} r’ + \int_r^0 \underline{\rho} (r’) h (t,x, r’) {\rm d} r’. $$ The variable $h$ represents the deviation of the infinitesimal thickness of isopycnals from the reference value $1$, and $ u $ is the deviation of the horizontal velocity of the fluid particles from the reference value $\underline{u}$, and $\underline{\rho}$ their density. The unknowns $h$ and $u$ depend on the time $t$, the horizontal space $x$, and the variable $r\in(-1,0)$ referring to the isopycnal line at stake, while $\underline{u}$ and $\underline{\rho}$ are given and depend only on $r$. Finally $\kappa>0$ is the thickness diffusivity coefficient.

The corresponding layered equations are $$ \begin{aligned} \partial_t H_i + \partial_x (H_i U_i) & = \kappa \partial_x^2 H_i, \\ \partial_t U_i + \left( U_i +U^\star_i\right)\partial_x U_i +\frac1{\rho_i} \partial_x \Psi_i &=0, \end{aligned} \quad (i=1,\dots,N) $$ where $U^\star_i=- \kappa \frac{\partial_x H_i}{H_i}$ and $$ \Psi_i= \frac{1}{N}\sum\limits_{j=1}^i\rho_j H_j + \frac{1}{N}\rho_i\sum\limits_{j=i+1}^N H_j. $$ The variable $H_i$ represents the thickness of the $i^{\rm th}$ layer (counting from top to bottom) and $U_i$ its horizontal velocity, and $\rho_i$ the density of fluid particles. The unknowns $H_i$ and $U_i$ depend only on the time $t$ and the horizontal space $x$, while densities $\rho_i$ are constants.

While these two sets of equations have an obviously similar structure, it takes some effort and care (and some physically relevant assumptions) to suitably quantify the difference between the solutions they produce in the limits as the number of layers $N$ goes to infinity (direction 1.) and in the limit of sharp stratification, that is when $(\underline{\rho},\underline{u},h,u)$ approach piecewise constant profiles with respect to the variable $r$ (direction 2.).