 Shall I start? So good morning. So this is work done by my postdoc, Jin Han-Shi, who's unfortunately not able to be here. And so he asked me to give this talk. In fact, the slides are his, so any errors, I'm not going to be directly mine, at least. And this is a collaboration with Keith Julian in Boulder and his postdoc, Benjamin Michele. So I'm going to talk about a phenomenon called salt fingering, or salt finger convection, which is a very important process in the oceans. So let me explain the basic physics behind this. So I am imagining that I have a situation where I have a linear salinity profile. So the salinity increases from bottom to top, for example, due to evaporation in the equatorial regions. And I also have a temperature that increases linearly from bottom to top. So it's cold here, warm at the top. But in such a way that the overall density, which, of course, has contributions from both the temperature and the salinity, is still decreasing upward. So this is a so-called statically stable situation. It doesn't overturn. But nonetheless, you have some potential energy in the salinity field that can drive an instability in the presence of diffusion. And so I want to talk about this diffusive instability, which is the salt finger instability that arises in this case. So the important parameter here is going to be something called a density ratio. If you imagine that you write down an equation for how the density varies with temperature and salinity and linearize that relation, these are the two contributions to the density changes. And the ratio tells me whether I have enough energy in the salinity field to actually drive the instability. And then the other important parameter is this parameter tau, which is the ratio of the diffusion coefficient of salt relative to the diffusion coefficient of temperature. And in the ocean, that's a small quantity. It's something like 1 over 100. And in the talk, I will make use of the fact that this is a small parameter to simplify the primitive equations. So the basic physics is if I take a fluid parcel here and I move it downwards, so I'm taking something that is hot and salty, I move it to a region which is cold and less salty, then I want to find out what happens in its new location. Well, in its new location, temperature equilibrates very rapidly because it diffuses much more rapidly than salt. And that means the parcel here is going to maintain its salt content from here. But now it's in a region where the salt concentration is less, and so it maintains its negative points and keeps sinking. That's the notion of a salt figure. So this is an important process. So these are some oceanic measurements that we are interested in. So what you see here is the depth and temperature on the horizontal axis. And what you see is that the instability generates this kind of staircase structure in the cylinder profile. And this, of course, is important for mixing of the upper regions of the ocean. And these are very stable. These can persist for months or even years, this staircase structure. So I'm not going to talk about the staircases, but I'll just talk about what leads to the initial mixing via the salt figure instability. And this is what the vertical unit was the depth. So I'm just going down from 200 meters down to 500 meters. So this is what the instability looks like. This is at 2D calculations, but 3D calculations are also available. So you set up these density and salinity profiles in the way I indicated on the first slide. And then you run an initial value problem, and you see the formation of these structures, depending, of course, on the parameter RO. That was that density ratio I mentioned. And you see different structures. So here, for larger values of RO, you see nice fingers. They are the types of things that I'm talking about. And then you have the cold, fresh fingers rising in between. So that's the process. It's driven by diffusion. It's not a dynamical instability. So there's been a lot of work on this, and I don't have time in this brief talk to describe all the various approaches that have been adopted to study the consequences of this kind of instability. But I'm going to be interested here in trying to get a simplified description of this instability that will enable me to determine what is the saturated state, or statistically stationary state of this instability. So let's go to the next slide. So what are the basic equations? So I'm going to just do this in 2D, but we can do everything I'm describing in 3D. So here are the equations for the fluctuation in temperature and fluctuation in saliency relative to these linear profiles that I introduced at the beginning. Here are the dimensional equations in 2D. And 2D is somewhat simpler. What I've done here in the first equation is I've eliminated the pressure by taking the curl of the Navier-Stokes equation that allows me to introduce a stream function, which is convenient for this purpose. Here are the buoyancy terms that drive the flow. Here is viscous dissipation. And then I have equations for the advection and diffusion of that temperature and salinity fluctuations. Here are the background gradients. And these, of course, diffuse at different rates, as I indicated. Heat diffuses more rapidly than salt. And that's going to be important. So we always want to non-dimensionalize these kinds of equations. And we need to have a characteristic length scale for the fingers. And it turns out the characteristic length scale for the finger is given by this combination of parameters. So nu is the viscosity, thermal diffusivity, acceleration to gravity, thermal expansion coefficient. And here is this linear temperature gradient. This is going to give you the right order of magnitude for the finger size, which you can get out of linear theory. And so we're going to use this scale d to non-dimensionalize the equations. And then we have other scales for the temperature, salinity, the magnitude of the stream function, and time. And when I'm done, I'm going to introduce the following dimensionless quantities that will determine what happens. There is this parameter tau, which is the ratio of the diffusivities. That's the small parameter. Then I have another quantity called the Schmidt number. That's the ratio of viscosity to thermal diffusivity. This is large in the ocean, but in astrophysical applications, this could be of order 1. And then we have this buoyancy ratio that I've already introduced. OK. It should not be very high, the Schmidt number of the salinity. In the ocean, the Schmidt number is high. It's hundreds, probably 500, 600. But in astrophysical applications, where viscosity is much lower because it's photon, viscosity, you will have lower values of this quantity. And then it's also useful to think in terms of Rayleigh numbers. So this is the thermal Rayleigh number and the salinity Rayleigh number. And the way I've defined this quantity d, the thermal Rayleigh number is just h over d. So it gives you the kind of aspect ratio of the system. So if I non-dimensionalize the equations, as I indicated, here are the equations. Here are these dimensionless parameters, tau here. And then the Schmidt number appears over here. And then the driving is through this density ratio, r o. So here are the equations. These are effectively the primitive equations for the problem. And of course, we can simulate them. But the simulation is a little bit difficult when tau is very small, like 1 over 100. And so it's good to get rid of this tau dependence. And so I'm going to do that in the next few slides. So first, we're going to do linear theory. So linear theory, I just perturb about these linear profiles that we saw. And then I get a cubic dispersion relation. This is all very standard. I get a growing soul-finger instability. If this last term in the dispersion relation is negative, and that's the case when this parameter r o is greater than 1, so that's your statically stable. But it's less than 1 over tau. That's the diffusive instability threshold. So we're going to be interested in the situation where tau is very small. So if this is my parameter plane, so this is tau, this is 1 over r o, I'm going to be interested in small tau and large driving. So large r o means 1 over r o is small. So I'm interested in this regime. And that is of interest actually in a number of applications. In particular to the ocean. So the basic idea is to take the limit tau goes to 0. r o is 1 over tau, so it goes to infinity in such a way that the product here r o times tau remains of order 1. That's the idea. So we do that here. So we rescale the temperature. The temperature fluctuation is going to be small because thermal diffusivity is very strong in this limit. We rescale s with this parameter. Then we take the limit r goes to 0, tau goes to 0, excuse me. And then we get a very simple equation that relates the stream function to the temperature. So we can eliminate the temperature and we get this set of equations. So we go from three PDEs down to two PDEs. The errors are small. They order tau. And so this is like the equation for Benard convection. It's driven by here by salt gradients, not temperature gradients. The interesting thing here is a new term that gives you damping at large scales due to the thermal effect in addition to viscous damping at small scales. And so the balance between those two effects is what gives you the salt finger scale, in fact. And then you have an equation for the advection diffusion of salt. So this is for Schmidt number of order 1. If I take this oceanic limit I've mentioned and that is the Schmidt number much bigger than 1. Typically it's 500 or something like that. Then I get rid of this term, the inertial term. And I end up with an even simpler problem, which is the one we actually studied in detail. I just have a simple equation for advection diffusion of salt and it's coupled to the stream function. So I have a dynamical equation for the salt. I have a so-called diagnostic equation for the velocity field, OK? There is no restriction. Sorry? Well, the energy is conserved in the absence of diffusion, but everything is driven by diffusion. So energy conservation is not. Potential energy is released by the instability. That's what drives the instability. So if you maintain the gradients that I've introduced at the beginning, if you maintain the gradients, then there will be a statistically stationary state because you're not using up the gradients as the instability proceeds. Am I answering your question? So this is the problem I want to talk about. The other ones I don't have time to discuss. So let's just show that there are some exact nonlinear solutions of this. They take the form of, we call them elevator modes. So these are these descending salt fingers like this and the rising fresh fingers. So there's a wave. What is x? x is the horizontal direction. Horizontal. And z is vertical. z is vertical. So when m is 0, it means there is no vertical dependence. So these are just structures that are going up and down like this. And you can calculate the growth rate very easily. And it shows that the optimal growth rate is at m equals 0. So these elevator modes that have no structure in z grow the fastest. And they have this particular wave number. It's the horizontal wave number, as you can see here, for different values of rA. And you can calculate the optimum mode, the wave number, and also the growth rate. And so this just shows you what the optimal growth rate looks like. It has a function of the super-curriculity, which is this Rayleigh ratio minus 1. Critical value of that is 1. We can also get stability of these states. We're using Floquet theory. And this just compares what happens in this reduced model, that last set of equations that I talked about. This is that modified Rayleigh-Benard problem with this large-scale dissipation term. And then the original primitive systems. And you can see that we have retained in a highly reduced system, essentially the full properties of the original primitive equations. So let's talk about the properties of the reduced system now and the statistically steady states. So this is what you get if you take this reduced system, has this prognostic and diagnostic structure, and you integrate the equation starting from random initial conditions, small amplitude initial conditions. Well, first you get this very abrupt growth of this instability. You generate lots of fingers. And then the interactions between the fingers lead to this statistically stationary state. So this is a picture of what the solution looks like here at the peak energy. So this is depth or height. This is the horizontal scale. And then over here you see the statistically stationary state where you have little fingers that are descending of the salt fingers and rising structures in between. So that's the state. So here is the energy in the salinity field as a function of the super-criticality. So this curly r is rA minus 1. It just tells you how far above threshold for the instability you are. And you see different regimes here indicated by these lines, which are power law fits to the data that you get from the simulations. So I want to just say a few words about how we determine these power law scalings in this regime. And of course, the reason for doing this is because we want to understand how effective is this process. For example, for sub-grade scale modeling, for example, ultimately in GCMs where these kinds of small scale processes are ignored at the present time. So this is, again, a picture of what the structures look like in this statistically stationary state. You see, actually, they are patchy. You can get these regions where you have a lot of activity. Lots of descending fingers are generated here in other places you don't. And this is for as large a value of the driving or the salinity gradient that drives the instability. So that's the. And so what do we find? Well, we do a spectral analysis of these pictures that I've just shown you. And we try to look at the spectral peaks. And that defines, of course, a wave number that characterizes the scale of the structure. And that wave number called k-finger here are these data points from the simulations. And you see they track very closely the wave number that you get from this optimal theory, just looking at these elevator modes that have no vertical structures. So we use that as an ingredient in the analysis. And the analysis is based on looking at our reduced equations and trying to see which terms dominate in which regime. And that, of course, depends on the strength of the forcing as determined by that curly r parameter. So here at low forcing, what you see in the diagnostic equation, you see that the terms that balance very closely are the terms that involve the salinity and the stream function here. This term is damped very strongly because at these slow forcings, viscosity is unimportant. And you see here the prognostic, sorry, this is the diagnostic equation at the larger value of rA. And you see the balance is a little bit different. You have a balance between this term. And now this term becomes also significant. And in fact, it is this term that becomes small. So we use these balances between the terms to look at which terms dominate in which regime. And we do the same thing for the prognostic equation. Here is the prognostic equation. That's the time evolution equation. There's a nonlinear term and some linear terms. And again, you can see here, for example, that the dominant term for this value of rA is the nonlinear term. And then this time dependence, that's because you have an instability that grows, is growing. So you have a, sorry, how should I say this? In the statistically stationary regime, the time scale is slow. So you have a balance between these two terms. And then at larger values of forcing, you have a balance between these terms. And this term, the linear term in this equation becomes unimportant. So those are the ingredients that we use. So for example, in regime one, well, we know what is the characteristic length scale in the horizontal and also in the vertical direction. It goes like the one over four power of this super-curiality. And so we're going to use this fact to try to get the scaling laws for the salinity perturbation and for the stream function psi with this parameter r. And so the first regime is when r is small, you're relatively close to threshold. And then the balances give you these relations. This one is from the diagnostic equation. This one is from the prognostic equation. And we use these scaling laws in these equations and they gave us relations between these powers. And those relations can be solved and they gave us predictions A three over four and B equals one. That is exactly what was observed in the simulation. So that approach works. Now we go to larger forcing and when you look at larger forcing, we look at spectra. This is wave number horizontally. Energy in the salinity field. I show two things here in solid line, which is the interesting. Now I integrate the spectrum with respect to m. That's the vertical wave numbers. I get a 1D projection if you like of the spectrum and I pick out a peak at small scales. That's the soul finger scale. And then I also have a lot of power at large scales. So I have to look at the interaction between small scales and large scales and look at the dominant balances that arise from that. And that is summarized over here. I decompose the salinity and the velocity field of stream function into large scale components and also small scale components. And I look at dominant balances using the same procedure I just outlined. And this figure just shows where most of the activity is that's the peak of this one dimensional spectrum. And then at the end, I'm skipping a few details here. You end up with scaling laws like this. And if you look at what these things look like in when r is over the one, you get scaling that looks like this. And when r goes to infinity, you get a scaling that looks like this. And if I just go back, these scaling laws agree very well with these are the straight lines in fact that I've put on this figure. And you see that they match the characteristics of this flow rather well. So I just have one more thing to show you. We've also looked at PDFs and the various quantities in the flow. Of particular interesting is the salt gradient, the distribution of salt gradients. And you see that in contrast to these other quantities, this is the salt distribution, this is the vertical velocity distribution, which are nice in Gaussian. These salt gradients are highly anisotropic. You have this stretch Gaussian form. And we believe that this shape is a signature of the basic process that leads to saturation of the salt finger and has to do with the fact that I have descending salt fingers from above and I have rising freshwater fingers. They collide and that is the process that limits the length scale of the fingers in this fully developed regime. And that translates into this asymmetry between positive gradients and negative gradients. Because when the fingers collide, you create locally very strong cylinder gradients. So my conclusions are here. So I derived a hierarchy of models, reduced models, simplified that are valid in this limit of small diffusivity ratio. I talked about one of these models, the simplest one of those, which applies in the oceans where the Schmidt number is large. So I formally took the limit of infinite Schmidt number. And we saw structures that resemble simulations of the primitive equations. And we believe they also represent what happens under realistic conditions in the oceans. So we saw that the reduced system retains many of the properties of the primitive system, but of course it depends on fewer parameters. So it's easier to characterize the saturated state. And then I computed how the saturated state depends on the driving through this Rayleigh ratio. And I also showed you some PDFs of the various quantities that characterize this flow. And there is a paper on this that appeared earlier this year. So thank you very much. Thank you. I'm sorry, I think it used to be a part of our question. Alino? Yeah, please. At the beginning you show us three or four pictures with this solidity. They were BMS or whatever, simulation. Yeah, simulation. The structure over there are much bigger than the structure you obtain with your used model. No, because the DNS necessarily is in a smaller domain. So the fingers are relatively large compared to the domain size. One of the things that we can do here very easily is we can look at very large domains and therefore get a much better understanding of the statistically stationary state. In fact, if the domain is too small, you get other types of saturation. And particularly you get this kind of irregular bursting which doesn't occur if the domain is larger. So you can be misled as to what is the saturation mechanism. So the DNS moves at zoom of your... It's a zoom, yes. Yes, it's a zoom, yeah. Okay. That way you get a slow equivalent. So don't you have a slow as the networks and kind of... Realizations solutions? Well, yeah, okay, that's an interesting question. So one way of course is to rather take this limit which what I've taken formally is the fast limit. So I'm just looking at processes on a certain scale but you're right that there are longer time scales in the process because of the large value of the Schmidt number. So we haven't looked at that. Not in experiments but in simulations if you make the domain too small, you do get relaxation oscillations. So what do you have? Well, so in our experiments, if you use the same parameter values but a small domain, we have bursts which I didn't show you, I'm sorry. If you make this for the same parameter values if we increase the domain, the bursts disappear. So the bursts are clearly due to the domain size and so in that sense, I don't think they would be relevant to the oceanographic application.