So good afternoon everyone and welcome to this week's session of the C

So good afternoon everyone and welcome to this week's session of the C

and I seminar series. We have today Dr. Roberta Biagini from the Italian National Research Council.

and she will be speaking today about some problems that we have with the

and she will be speaking today about some problems that we have with the

research of the Italian National Research Council.

So, I would like to start by saying that we have a very good

research team here at the University of Italy.

She will be speaking today about some problems in the dynamics of stratified fluid.

Please, Roberta, you have the floor. Thanks for the presentation, thanks for this

kind invitation, for organizing this. So, yes, I will talk about some problems

for stratified fluids. So, essentially two recent results.

I will discuss about, I will present the results and also

mention my collaborators when I will talk about

the topics. But first, let me introduce what we are talking about,

stratified fluids. So, essentially we start from this

equation, which have the incompressible noyostokes equation,

okay, in this case with thermal diffusion in 2D.

So, here what you see is the density, of course, the equation for

the density and the equation for the momentum. So, rho is the usual density.

Here u is in 2D, so it's the velocity field with two components.

p is the pressure, which is related to the divergence free condition here.

And then there is this g, which is gravity.

So, about the system, in particular, what we are interested in is to study

some situation where you have essentially data study equilibrium.

So, meaning that you start from some steady solution, which are this one in the bottom of

the slide. So, you have this density rho bar y, which is only depending on the vertical component

or vertical coordinate. And, okay, u bar. And then you have this, this is what is called the

study balance. So, where you have this balance pressure. So, in this case, what we will assume

all the time is that this rho bar, this stratification, this background stratification

is stable. And with stable stratification, we mean that the derivative with respect to y

is negative. So, we will see in a moment why we assume this. But something that you can also,

I mean, imagine from experience in the sense that it's a kind of a usual situation in the sense that,

for instance, when you are in the sea, you can feel that there is a pressure, which is a kind of

diminishing when you go up. So, this is essentially a situation which is due to gravity, where the

lower density fluid is above, and then the density is decreasing with height. So, we assume to be in

a setting where the stable stratification is stable, and then this derivative is negative.

So, the first problem that we will consider in this context of stratified fluids in the stable

regime is the linearization around the, okay, the static balance, the equilibrium. So, meaning that

we will consider this steady solution here, this robot y, the velocity field is zero, and the

pressure is given by this hydrostatic balance. And then, after we have this steady solution, we

add just a small perturbation, this tilde here. And we plug the, this kind of linearization,

this perturbation inside the incompressible Navier-Stokes equation. So, essentially,

this is what we do in order to obtain this system that you see here, this b.

Just by adding other hypotheses, which is the Boussinesq's hypothesis, in the sense that

when you take this perturbation here, this expansion here, you plug this in the Navier-Stokes

equation, you have to think about the fact that the density fluctuation actually can be

neglected in all the terms which are the inertial terms of the momentum. And you have to consider

this fluctuation only in the terms which are important for gravity. So, this is actually the

Boussinesq's approximation, the fact that density variation are all important in the gravity terms,