 So, what I want to do is tell you about some recent progress in understanding a type of turbulence that we call geostrophic turbulence, and the equations that we use to describe it. And then I want to show you that the state of geostrophic turbulence is itself unstable to the formation of large-scale barotropic vortices. So this is work done with my colleagues mostly in Boulder in Colorado. Keith Julian is my co-PI on this project, and this is the motivation for our work. So we're interested in studying large-scale structures in the earth. So this is the standard kind of problem that is described by the equations of quasi-geostrophy that we heard about in the previous talk. So here is the eastern seaboard of the United States. Here is the Gulf Stream, and you can see the Gulf Stream forms these large meanders. The important thing about these structures is that they are of a sufficiently large scale so that if you define the Rospin number, that's going to be the important parameter in this talk, to the ratio of the characteristic velocity divided by the local rotation rate times the horizontal length scale, like the diameter of one of these meanders, that Rospin number is actually small compared to one. So what that means is that these structures, because they have a large scale, are strongly affected by rotation. Stratification can also be important. That's measured by something called the Froude number is the ratio of the same characteristic velocity divided by the Brunt-Bissala frequency, that's the oscillation frequency in a stratified atmosphere, and again the length scale, and this could be small or comparable to one depending if you're talking about the atmosphere, for example, or the oceans. Now the important thing for quasi-geostrophy is that you make this assumption, namely that the height of the layer that you're interested in divided by L is small compared to one. And of course, that's good for the oceans, for example, typical depth of the ocean is say four kilometers, these meanders may be 100 kilometers across, so H over L indeed is small. And that's the basic input that goes into quasi-geostrophic approximation. Now the consequence of this quasi-geostrophic approximation is that the motion is primarily horizontal, and you have hydrostatic balance in the vertical, and that's very restrictive because you cannot, using that kind of theory, describe convective phenomena. And convective phenomena are important in a variety of contexts in the atmosphere, for example, formation of cumulus clouds or the downwelling of the thermohaline circulation in the Labrador sea, things of that type. So what we want to do is we want to study structures that have significant vertical motions, and that means we have to relax the conditions under which quasi-geostrophic approximation applies. So I'm going to be studying things like this. For example, I have a descending plume, this could be in the thermohaline circulation in the Labrador sea, and it's descending, it's quite narrow in the horizontal direction, and it's extended in the vertical direction. So it's a columnar structure. So the aspect ratio now is quite different from the previous picture, I have H over L comparable to one or greater than one. So these are going to be thin structures and tall in the vertical. But we're going to do the same kind of analysis that you use for quasi-geostrophic, so we're still going to assume that the rosby number is small on these scales, and that's reasonable actually for this particular example, where the rosby number is about 0.2. And again, we're going to have some kind of fruit number describing stratification. So these kinds of structures have been studied by many people, and I just want to show a couple of pictures from a very nice experiment by Sakai, published already 20 years ago. So I have, this is a side view of a rotating tank, it's rotating about the vertical direction, it's visualized through a suspension of liquid crystal. The liquid crystal has the unfortunate property that it turns red when the liquid is locally cold, sorry about that, and it turns blue when it's warm, okay? So these red structures are not, you know, hot descending plumes or something like that, they are actually rising, excuse me, they're descending cold plumes. So by this is a column, it's a vortex, and here is another vertical structure that's rising because it's blue, it's warmer, right? So I have these structures that penetrate from top to bottom and bottom to top. And this is what it looks like from above, you have a sea of these vortices that just move around in an irregular fashion, and these structures are going to be calling tailor columns because they extend all the way from the bottom to the top of the layer. What is it that I really want to do? Well, this is the basic parameter space for this kind of rotating convection problem of buoyancy-driven flow. On the horizontal axis I have a measure of the rotation rate, here I'm using the tailor number, and on the vertical axis I have the forcing of the flow measured by the Rayleigh number, the temperature difference that is applied across the system. Down here in this region the Rayleigh number is not enough, so I have conduction, and then when I cross this black line convection sets in, and of course convection is delayed over here because of the presence of rotation. The rotation is stabilizing, so I have to heat more in order to get convection, and I'm going to be interested in the region that's strongly affected by rotation, and that means that the so-called convective Rosby number is going to be substantially less than one. And so that means if I calculate the Rosby number by calculating the typical velocity that's driven by buoyancy and use that to define the Rosby number, then Rosby number one is along this dashed line, Rosby number point naught one is along this line, I want to go into in this region, and of course I have to heat more and more strongly in order to get convection in the first place, right? So I'm going to be interested in the rapid rotation strong heating limit, right? That's this region here, Sakai's experiment barely get into this region, they're a nice experiment by Peter Vorobiev and Bobecki, they also just barely get into the region, I want to go in this direction, but I'm not going to be able to do that numerically because as we've heard under geophysical and astrophysical conditions, the parameters are very extreme, equipment numbers 10 to the minus 15 as we heard in the previous talk, so I want to take the primitive equations and I want to simplify them to be able to treat these inaccessible regions of the parameter space, okay? That's the basic idea of this talk, so what are the basic equations? Well, here are the primitive equations, this is just the Navier-Stokes equation, this is the Coriolis term, it's been written in terms of the Rosby number that we already saw and I did that by non-dimensionalizing the velocity in terms of some kind of characteristic velocity scale which is so far arbitrary, I'm just calling it U, and I have a characteristic length scale L as we saw in the previous pictures and I use these parameters to non-dimensionalize everything and that introduces this quantity 1 over R, remember I'm interested in low Rosby number, so this is going to be a large term, I'm also interested in situations where viscosity is not so important, so this is one over the Reynolds number, so this is going to be a small term, likewise thermal diffusion is going to be a small term, so I have a numerical problem because I need to solve these equations and I have these very small coefficients and I have these very large coefficients and how do I do that, right? So this is a problem because I need tremendous spatial resolution, this is a problem because I need to track fast inertial waves and I'm not interested in what the inertial waves are doing, I want to know what happens over longer time scales, so I want to get rid of these awkward parameters in order to be able to treat this problem and the way we do that is through an asymptotic expansion that simplifies the equations of motion and this is the basic idea, so I'm going to do an expansion in the Rosby number, remember rapid rotations mean small Rosby number, so to remind me that the Rosby number is small I'm going to call it epsilon, right? epsilon is always small and in order to make progress I need to link the horizontal length scale to the vertical length scale through and some kind of assumption that depends on epsilon and the one that actually works is H over L is 1 over epsilon, right? 1 over epsilon means that the horizontal scale is epsilon times the vertical scale, so that's exactly these columnar structures that I showed you on the previous slides, okay? So once we decide this is a good thing to do, then we're going to do a multi-scale expansion, right? I'm not going to give details of how we do that but the basic idea is that I'm going to rescale the horizontal scales x and y through this parameter 1 over epsilon and of course because it's going to be a three-dimensional flow I also need a similar term for the vertical direction but I'm driving the system over these very long scales in the vertical, right? Because my scales are small and I'm imposing boundary conditions at top and bottom so I also need a slow vertical scale, large vertical scale that's called capital Z here, right? So that's my spatial scaling and then corresponding to that there is some temporal scaling and then I have to decide how strong are the fields that I want to study and I've already said I want to look at non-hydrostatic motion and that means I want substantial vertical velocities called W here so I want to assume all of these are comparable in size so I'm getting away from this hydrostatic assumption that's part of the QG theory I'm going to divide the temperature into a horizontal average part and fluctuations, theta and of course I have to force the system strongly so my forcing measured by this parameter gamma has to be big one over epsilon that's just to get convection going the pressure of course is going to be large also and that if I put this into the equations then at leading order I get something I like and that is I see that I have a balance between the Coriolis force and the pressure I have the flow is horizontally non-divergent that means I can introduce a stream function and the stream function is nothing but the pressure this is the standard thing that you see in a weather map it simply says that rather than the flow being for example towards regions of low pressure it goes around the region of low pressure in this cyclonic fashion okay so we know all about this but now I want to get the evolution equation for psi and I'll go to next order in my small parameter epsilon and at that order I get couple of equations for the vertical velocity omega and the vertical velocity w okay so in this equation I have an equation for omega it's horizontally affected by this pressure field if you like psi there is some contribution from stretching in the vertical direction some dissipation right I have an equation for the vertical velocity w it's horizontally affected this is nothing but the vertical pressure gradient remember psi is p I have some buoyancy forcing and I have some dissipation right and then that's still not enough because I don't have an equation for theta so I have to go to yet further order in epsilon and that's why I'm not giving you details and then you find that theta is again horizontally affected there is a contribution to the thermal fluctuations from the vertical advection of the vertical gradient of the mean temperature some thermal diffusion and then there's an equation for the evolution of this horizontally average temperature which occurs on a slower time scale here is the vertical heat flux and some dissipation okay so those are the equations that I get the nice thing about this is they are actually a closed set of equations right if you make other assumptions you would not necessarily get closed equations so I'm going to rewrite them in a way maybe that makes them a little more familiar and that is I'm going to make a choice of the horizontal scale L I'm going to pick the scale that's predicted to be the one that first sets in when I cross the threshold for convective instability and that tells me that this epsilon that I picked is actually equine number to the one-third okay so here's the definition of the equine number equine number is small when rotation is large okay and then I'm also going to pick a characteristic velocity scale the viscous scale and then the equations look like the equations for Rayleigh-Bernard convection more or less here are the equations for the vertical vorticity omega equation for the vertical velocity here is the buoyancy term notice you have this combination of large R a small e I'm assuming this is of order one to drive turbulence and then I have an equation for the temperature fluctuation and for the evolution of the mean temperature and there is a new parameter that appears it's called sigma here that's the frontal number it's the ratio of the viscosity to thermal diffusivity I'm going to vary that parameter but for water it's 7 for air it's roughly 1 so that's typical values of sigma the point about this is that there are no small or large parameters anymore right so these equations are easier to time step that was the whole purpose of doing this calculation and the same expansion also gives you boundary conditions turns out the boundary conditions are nice they are free-slipped boundary conditions at top and bottom that are consistent with this asymptotic limit and the other thing I want to say about these equations is that they are quite interesting because they actually have two reflection symmetries and so there's actually no asymmetry between cyclonic and anti-cyclonic motions in this rapid rotation limit and so the first prediction I would make is that in that limit you should see equal numbers of cyclonic and anti-cyclonic vortices that's to be confirmed experimentally perhaps although there's some evidence that that is in fact the case so what do these equations show us? Well here is a regime diagram so I'm showing here the frontal number 7 remember is water this is the forcing parameter that combination of large r a small e small ekman number and so I'm increasing that along this axis and what you see here is that when I increase that parameter I get a cellular convection first that small combinations of this of these parameters then I get these Taylor columns that I showed you in the Sakai experiment eventually when I force it too strongly the Taylor columns start breaking up into plumes that's what p stands for okay if I do it at low frontal numbers then the plumes come in almost immediately the Taylor columns are presumably unstable and then there is a transition gradual transition into a state which we call geostrophic turbulence so it's a turbulent three-dimensional state but on all scales of the motion I have geostrophic balance that's the balance between the Coriolis terms and the pressure gradient on all scales so it's not the usual kind of turbulence sorry oops I'm going in the wrong direction here I just want to show you some pictures of what these states look like so on the left I have frontal number seven I'm increasing the Rayleigh number going in this direction these are the Taylor columns here are the plumes they detach from the top boundary for example the cold plumes that start descending but they no longer make make it all the way through okay so I'm calling them plumes likewise I have hot plumes that detach from the bottom they start rising don't make it all the way so that's the plume regime here is the corresponding thing for a particular choice of this parameter RA bearing the frontal number and when the frontal number is large well comparable to water I have these Taylor columns then I go into the plume regime and this state up here at low frontal number is the geostrophic turbulence regime so this is just a snapshot of what that would look like this is a periodic box computation in three dimensions it's a little bit fuzzy and that's deliberate because this is for sigma is 0.3 so thermal diffusion is stronger than viscosity and so the thermal field is a little bit fuzzier than the velocity field would be so there are some nice things I can tell you about the properties of this equation so if you take the equation for the temperature fluctuation multiplied by theta integrate over the domain you get a nice integral relation if you look at the saturated steady state the stationary state of the turbulence you can integrate the equation for the mean temperature and then you get the equation shown here there's just a quadratic equation you can solve it for the gradient of the mean temperature and you get a nice equation and when this factor vanishes that represents the boundary between the thermal boundary layer near the top and bottom of the of the cell and the bulk so we use that to characterize what we mean by the thermal boundary layers which are still in the system they get thinner as the Rayleigh number increases and the bulk region which is outside of these thermal boundary layers and the transition is basically when you know the bulk and the boundary layers carry half the heat flux measured here by the massive number so what about these boundary layers the boundary layers are important because I increase the Rayleigh number they first lose stability when I'm still not very close to isothermal in the core so this is high this is temperature the boundary layers become unstable here become unstable here and so that when I have a state that looks something like this at a larger value of the Rayleigh number these boundary layers are unstable but the core is actually stable right it's very opposite to standard non-rotating Rayleigh-Bernard convection and this just shows the eigen modes of the boundary layer instability just to convince you that that really happens and so we'd like to know something about the boundary layers because it's the boundary layers that are going to actually be very important in the rest of this talk so we assume that when we have large values of this parameter so being some kind of asymptotic turbulence regime all the quantities in the problem scale with some powers of this parameter capital R right so that's the horizontal scales this is the magnitude of the stream function and so on okay and then we look at what happens when R goes to infinity of course we can't do that numerically but we look at large values of R and we want to see if there are any terms that drop out of our reduced equations and we find that in the boundary layers they all remain roughly over the same order of magnitude and therefore we must have some relations between these exponents right and when we put all this together for example this is the Nusselt number here goes like R to the eta minus delta and if I take this expression for for for delta and for eta here I see that the Nusselt number has to go like R to this power four S minus one okay and so we'd like to check whether this in fact is reasonable and perhaps use numerics to determine S or use theory to determine S and so here is the numerics so this is a compensated plot of the Nusselt number and we see that when the exponent is R to the minus three halves these curves level off as I increase R and therefore that the appropriate power is Nusselt goes like R to the three halves okay and this shows the different numbers for the different leveling off for different frontal numbers for the larger frontal numbers it hasn't leveled off and that's because I haven't reached this geostrophic turbulence regime I would have to increase R even more presumably to reach that state so we get this prediction by matching the numerics to the previous theory and we also have an independent way of getting this result we assume that the Nusselt number goes like some power of of of Rayleigh times equine number to another power this has been these kind of relations been studied extensively when there's no rotations of beta is zero in the case of rapid rotation the situation is very different from the non-rotating case and that's because the temperature gradient at mid-height does not saturate as I increase Ra in fact what happens is that's because of rotation and so the bulk of the of the layer controls the heat flux in really non-rotating Rayleigh-Benard convection the flux is controlled by the boundary layers the ability of the boundary layers to transport heat so this is very different and then if we do use a Kolmogorov type argument saying that when Ra is large the transport properties must be independent of the microscopic diffusion coefficients it turns out we get a prediction exactly like this which is what we saw in the numerical computation so that's consistent that's nice and this is just to convince you that what I've just said is correct so this shows the mean temperature mid-level temperature gradient as a function of this parameter R and you see for example when I'm in the geostrophic regime that gradient just saturates right doesn't come closer and closer to isothermal and this just shows that the the bulk does in fact contribute most of the heat flux the boundary layers are insignificant and this is just to convince you that the scaling for the temperature fluctuation for the vertical velocity that we get from that theory are really obeyed in the numerical simulations here I'm just showing what happens as I vary the rotation rate so this is the Nassau number versus Rayleigh number this is the non-rotating curve that's been studied as our increased rotation this thing peels off with different rates as the rotation rate increases I don't have time to discuss that figure in more detail so the last thing I want to say is telling about is the instability of this geostrophic turbulence we discovered this serendipitously we integrated the equations and we found that things weren't quite saturating there was a drift and when we looked at the data it's always a good idea to look at your data we discovered the formation of these large structures that emerge out of the turbulent state sorry so this is what it looks like so I started some random initial conditions I settled towards the saturated state here is the drift I was describing so and we saw that this drift had to do with as a function of time the evolution of these large scale structures and this shows the kinetic energy in the three-dimensional fluctuations I'm going to call those the baroclinic mode or baroclinic state and this shows the energy in a barotropic state that's a vertically you know integrated state that doesn't have a vertical structure and you can see that the energy of that state increases continuously as these vorices grow so this is what they look like like the turbulent structures that live on top of this turbulent state and here you can see them penetrating all the way across so these are the kind of things I'm talking about here again is another picture that shows this in a in a in a different way right so so this is an example I just want to compare this with some of the other talks we've heard today we're not talking about small vorices combining into larger vorices larger vorices combining into yet larger vorices this is not 2d right this is not 2d hydrodynamics this is fully 3d and the energy goes directly from small scales into the largest available scale in the system and the way that happens is that you align the phases of the small scale modes and then the small scale modes interact and they can put energy directly into the large scale mode right that's example of spectral condensation that somebody mentioned also today so here are the spectra that you get so these are the baroclinic states three-dimensional fluctuations they have a more or less called mogoro spectrum this is the the barotropic contribution so this is the n-strophic cascade that we've been talking about there's a turnoff over here but it's a function of time that turnoff gets contaminated by these large scale vorices and this k to the minus three spectrum is a signature of the large scale vorices so that's what the spectral picture looks like this is what it looks like in time I start with some different modes so this is the baroclinic state you see the energy in the baroclinic state hardly varies this solid line is the largest available barotropic mode that fills the box right that's the vortex mode and you can see that initially it does nothing but then it just takes off and and dominates everything as time increases and that's the spectral condensation that we've been talking about okay so how do we quantify this I just have a couple more slides is that okay so we're going to take the vorticity and the stream function and we're going to divide into a barotropic component that's vertically averaged so that's two-dimensional and then I have a three-dimensional fluctuation that's the baroclinic component okay so this is standard barotropic baroclinic decomposition and then I can get equations for the barotropic part that's the 2d part except that there is forcing from the fluctuations the 3d fluctuations if I didn't have the 3d fluctuations this would just be 2d Euler or 2d hydronomics and I would have the usual you know inverse energy cascade with vortices combining into bigger vortices dot dot dot right this is different because of this term and this term you know it's computed from the equation for the fluctuations the baroclinic contribution right so that's one way of thinking thinking about it or you can look at the same thing in terms of what happens in Fourier space and maybe that's more familiar to people here and then I can get an equation for the energy in the barotropic component so rate of change of the energy well there's a contribution from barotropic barotropic interaction there is a contribution from baroclinic barotropic contribution that's these fluctuations and then there is some dissipation right and so these transfer rates can be computed from our simulations and I think the next picture shows what that looks like so here here is the fk that's the baroclinic contribution to the barotropic mode and you can see the function of time from region 1 to region 2 to region 3 the peak of that contribution I'm summing over all the baroclinic modes horizontally that peak moves towards k equals 0 and that's because you're putting all the energy from the small scales into the large scales right there's no intervening cascade of vortices there's no you know there's no inverse energy cascade as such so that's this so let me just conclude now by mentioning that since we did this work the people have solved primitive equations for actually very low equine number like 10 to the minus 7 that was done by Stefan Stelmach in Münster and he found qualitatively a similar sequence of transitions this is a side view this is a top view here he has cellular convection these are the Taylor columns that I showed you this is what it looks like from above like in the experiment of Sakai then he has the plume regime where the Taylor columns break down and then finally he gets something like the geostrophic turbulence and if I translate his parameter values into my combination of parameters this ekman to the four thirds times really you see that spanning the same more or less the same parameter regime that I've been talking about all along so I think that's interesting because you know the theory was designed to take these extreme limits to get to geophysically or astrophysically relevant regimes this is not 10 to the minus 15 but it still seems to seems to more or less work so I call this the unreasonable effectiveness of asymptotics it's a good thing that happens right so here are my conclusions I showed you that these reduced equations are useful because they can be integrated and they describe what happens in these extreme parameter regimes I showed you for example that the natural number scales with the Rayleigh number in this fashion including the pronto number dependence and the rotation rate dependence and I showed you that the scaling had to do with the fact that the bulk of the convection outside of the thermal boundary layers was the thing that was limiting the efficiency of heat flux transport through the layer I showed you that geostrophic turbulence was unstable to these large-scale barotropic modes that evolved directly from the turbulent state and I showed you the the barotropic baroclinic spectra and including the signature of these vertices at large scales and finally I think I have a couple of references to to the more recent work by Favi et al and Goreville et al where the primitive equations have been solved and these large-scale vortex structures have also been detected and so there are some references in case people are interested in more details thank you very much