 Okay, so this is, sorry about the title, it's very long. So anyway, so this is an extension of some work that I did with my colleague who was my PhD supervisor, Jorgen Fredrickson. And so a few years ago we developed an extension of the direct interaction approximation to inhomogeneous flows that have some application for problems in the atmosphere. And so just recently we did some work mycovianizing that approach. And so that's really what I'm going to talk about. Okay, so Creighton's DIA has largely been abandoned over recent years due to the problem of incorrectly predicting the inertial range exponent of the power law decay, and that's for both two and three-dimensional turbulence. So there's lower order statistical closures that don't accurately describe the higher order correlations associated with the formation of coherent structures in 2D turbulence. But for turbulent geophysical flows where rosby waves are present, these linear wave effects tend to inhibit the formation of those coherent structures. And so we think there's some utility to applying 2.2 time closures such as the DIA for geophysical transport calculations. Application of the DIA to inhomogeneous turbulence, it's a difficult problem. Part due to memory requirements, if we're looking at synoptic formation of say blocks and other large scale features in the synoptic atmosphere, it requires about five days of memory to capture the growth of those instabilities. So any 2.2 time closure, inhomogeneous closure, you have to carry around that kind of memory. So Markovian closures where only equal time correlation data is predicted and time history effects are approximated by a triad interaction times and knowing only the current value of the triad relaxation time and the other state variables. These have proved very popular and of great utility and so we're motivated to try and extend this approach to those problems. So just to sort of put a bit of background. So as I said, the non Markovian closures and there's the three more general 2 time isotropic approaches, which is the Creighton's Direct Interaction Approximation, Jack Herring's Self Consistent Field Theory and David MacCone's Local Energy Transfer Theory and I've included our inhomogeneous form there. They're computationally costly. So they go like order of N cubed where N is the number of time steps. They're Galilean invariant in that they're invariant to translation, rotation and uniform motion but they tend to violate Galilean invariance so they're realisation dependent and that's this idea that the spurious convection effects on the small scale eddies by the large scale eddies lead to the incorrect modelling of the inertial range kinetic energy spectra. So another way to think of is that these 2 time correlation functions spuriously carry the interactions between the large and small scales into the equal time correlation functions. So the Markovian closures assume that the rate at which the memory integral of the gaze is much faster than the time scale in which the co-variance has evolved. So the only equal time correlation functions, they satisfy random Galilean invariance. So some work in 1993 by John Bowman, John Cromies and Otto Viani, they postulated that the E.D. Q&M is not, which is one of these Markovian closures, is not realisable in the presence of waves and so they were looking at the drift wave problem Husser-Gallon-Mimmer equation. So we haven't actually tested that whether it's true or not, I think that they probably are realisable if you include the empirical tuning parameter. So Bowman postulated that a realisable Markovian closure that's realisable in the presence of waves but in general the problem is one of finding a principle to determine the decay rate of these response functions. So we're going to apply the fluctuation dissipation ansets of Bowman to our inhomogeneous closure and derive a Markovianised variant. Okay, so just to set more of the background I'm going to spend a bit of time describing what this quasi-diagonal direct interaction approximation is that forms our inhomogeneous version of the DIA. So it's an outstanding problem of how to generalise isotropic self consistent field theories of Creighton and others to these general inhomogeneous flows including realistic topography. The other thing is how to incorporate large scale rosby waves say on a beta plane and how to deal with long integrations of time history integrals. So as I've indicated atmospheric regime transitions typically require time history information over many days. Formerly we also have to deal with this problem of vertex renormalisation so it's the second order propagator renormalised closure if approximated some of the higher order moments to some approximation but formerly the vertex terms a lack of vertex renormalisation leads to this underrepresentation of the small scale kinetic energy. So this is a general problem of quadratic non-linearity and strong coupling. So for infinite resolution and moderate Reynolds numbers the DIA underpredicts the inertial range kinetic energy as do the other variants such as the let theory and self consistent field theory. So I show that these three closures are actually based on differing interpretations or applications of the fluctuation dissipation theorem. For finite resolution and moderate Reynolds number all of these homogeneous two point non-Marcovian closures underestimate the small scale kinetic energy and dramatically underestimate the skewners. So this is due to the fact that as I said we've renormalised the propagators the vertices are bare and the decay times of the two time cumulant response functions are determined incorrectly by the energy containing range rather than by the local excitation levels. So a vertex renormalisation would include the damping effect or include these indirect interactions thereby imposing interaction of eddies with those in a nearby local wave number space and while at the same time restricting any spurious non-local interactions. So just to begin with we're going to start with the barotropic vorticity equation. So all the results I'll present will be two-dimensional turbulence in spectral space with circular truncation. So we start with the barotropic vorticity equation. This is the Jacobian so the vorticity is related to the stream function by the Laplacian. In the absence of forcing in dissipation the two quadratic invariance once we discretise would be energy and enstrophy. So just to emphasise that this is an equation without a mean field so that these are all transients. I'm going to use the notation where the diagonal cumulant or covariance if you like but here the cumulant is in this form and the off-diagonal terms are related by these wave numbers P and Q. So I'm also going to use a notation where I'll refer to the right-hand side of this tendency equation where it's homogeneous by this H term. And so you can see straight away that we have this problem of dealing with this 3.2 time cumulant. So this is just the mapping to spectral space and as I said we're going to consider KXKY space. Zeta is the spectral component of the vorticity and then we have this dissipation operator and here so we're going to be on the doubly periodic plane and so it's K squared. So the other thing is we have these interaction coefficients that couple the triad interaction coefficients which have this form and I'm going to use this notation for the conjugate. Okay so I'm going to use the direct interaction approximation from Creighton and so here I'm not going to go through it but I'm just going to describe it. So we replace the 3.2 time cumulant by the DIA which has a nonlinear noise term and a nonlinear damping term. Let's have an equation for the response function. So the response function describes the effect of an infinitesimal perturbation at wave number L and at some time T prime on the field at K and time T. So in the DIA it takes this following form. Here we again have this nonlinear damping term and the response function satisfies certain properties. The identity, it's causal, it's transitive and it's statistically independent and so in the notation that we're using here the nonlinear noise and the nonlinear damping and the nonlinear noise take the following forms acting on these diagonal terms. We also have a single-time cumulant equation. Here the right-hand side turns out to be real and similarly there's a single-time equation for the response function. So as I said the main homogeneous closures or isotropic closures are related by application or choice of the fluctuation dissipation theorem. So the FDT connects the linear response relaxation from a non-equilibrium state to its classical fluctuation properties in equilibrium. So really there's an infinity here but we choose to apply this at some finite time, at some distance away from equilibrium. So the principle here is the idea is to find a principle to determine the decay rates. So the self-consistent field theory relates to the DIA simply by replacing the two-time cumulant equation with this fluctuation dissipation ansatz and the local energy transfer theorem replaces the two-time response function of the DIA with this choice of the FDT. So for the EDQNM, so in the absence of fourth-order cumulant, so the EDQNM says that the DDT of the third moment is zero. So we replace fourth-order cumulant and it removes the damping mechanism necessary to bound the third-order cumulant. So this replaces linear viscosity with an eddy viscosity, so linear plus turbulent viscosity. So according to Leith, the EDQNM is obtained by making the best Markovian fit to the DIA consistent with the underlying longevity of representation. So in the absence of waves, the EDQNM has an exact stochastic model, so it's realizable. The term EDQNM refers to an entire family of closures that depend on the choice of an eddy damping parameter and it can be freely adjusted to match the phenomenology of the inertial range and you know this is proved to be a powerful tool. So the FDT ansatz for the EDQNM is in this following form. So the two-time cumulant equation the EDQNM is we have a modified form of the FDT and assuming that the exponential decay rate for the response function takes the following form here, so this would be on the sphere, that's why we've got k plus one rather than k squared. So here the dissipation operator and the eddy damping is given by this empirical form here which has a tuning parameter at the front and so it's been found that for two-dimensional turbulence a choice of 0.6 gives some pretty good comparison with direct numerical simulation. So Bowman or Schoed or proposed that in the presence of waves or complex mode coupling phenomena this above FDT ansatz can lead or has the possibility of leading to the EDQNM being non-realizable. So instead they propose a modified FDT which takes the following form here and here ck to the half is the principal square root of the real part of the diagonal cubulin and so we're going to apply this ansatz to derive a realizable Markovian two-point closure for two-dimensional turbulence in the presence of topography and Rosby waves. So the equation that we're going to use is an extension of the barotropic equation. So here we've included a large-scale east-west flow we have some scale topography we have the beta effects we have differential rotation and we include an extra term which corresponds to the solid body zonal rotation vorticity so it's a small term but it allows us to have certain symmetry properties with the interaction coefficients and I'll show you the result of that. We also have a form drag equation for the large scales and so here s is the doubly periodic domain we have a momentum flux and we can also include some relaxation towards some observed flow so again the invariance in the absence of forcing a dissipation now take the following form for the large and small scale kinetic energy and now in terms of the potential entropy and there's actually a direct one-to-one mapping to the sphere. Okay so if we write this equation in Fourier space so we have the equation for the small scales we have the interaction coefficients a self-interaction term a term that couples to the topography and then we have you know this extension to allow for the inclusion of large-scale Rosby waves and an east-west flow and so I won't go through this but this is the form that it takes and here f0 is the bare forcing and we have a complex dissipation which is related to the bare viscosity and an intrinsic Rosby wave frequency by that expression so just a couple of points on the dispersion relation so as I said here this is the bare forcing we have a complex viscosity we've also defined a z to zero mode okay so in here as you saw earlier there was a h0 term which is taken to be zero but could more generally be related to a large-scale topography so we note that the east-west flow is real and we've defined this term here to be imaginary so this ensures that all the interaction coefficients are real and then it's possible to extend the range of the interaction coefficients over P and Q to include this new vector zero and so the point of that is that we can now write interaction coefficients that involve the main field and the topography in this form and then with this extension to include this large-scale east-west flow and the beta effect we still satisfy all the properties of the cyclic properties of the of the interaction coefficients but we can collapse all of this back into the same form that we had for the f-plane right so we have to do the same thing for the large-scale flow you can take the large-scale form drag equation and pretty much do the same thing collapse it into this standard form where all of the complexity is moved to the choice of the interaction coefficients or the definition of the interaction coefficients so how do we get to the closure ok so we write an equation for the main field and the transients again in this following form and you immediately see that we have a covariance matrix or a two-point cumulant and so the and here we make the assumption that the variation in the topography is small so we define h to be the scaled spatial variation of the height of the topography relative to the total depth so the equation for the diagonal cumulant now has an inhomogeneous component that's added to the homogeneous term and this involves the covariances or 2.2 time cumulants interacting with the topography and interacting with the main field so the problem is how to get a closure for those terms so we do take the standard approach we do a perturbation expansion here we go to second order then we truncate and we renormalize so in keeping with the spirit of the DIA but we make a couple of assumptions so we assume that the perturbations are initially multivariate Gaussian and that the zero thought they're diagonally dominant so that allows us to write to first order the 2.2 time cumulant in the following form and then we renormalize and so as I said sufficient conditions for diagonal dominance the topography and the main field is sufficiently small however canonical equilibrium the off diagonal elements of the equal time covariance vanish and that's regardless of the size of the main field in topography so the expression that we get for the 2.2 cumulant equations is in the following form and here I've included a cumulant update so I'll talk about that a little bit later but it's a generalization of the approach of Harvey Rose to truncating these long time history integrals and approximating the off diagonal information in terms of diagonal elements and similarly the response function now includes this additional term that's now coupling to the main field as I said we're using the DIA for the 3 point term and we're using this cumulant update in the form that Harvey Rose proposed and so what we have here is that the QDIA equations include off diagonal and non-gasian initial conditions through these initial terms here and through the cumulant update terms for the 2 point cumulant we allow initial information about off diagonal elements in the 2 point cumulant right so this would be in a more formal setting and generalization of the operator formalism of Decker and Harker and Rose so equation for the main field now takes the following form we have a minus P minus Q term a term coupling to the topography and now we have a term for the effect of the non-linear damping on the main field and essentially a main field topographic interaction term and a main forcing the tendency equation for the 2 time cumulant now takes the following form so we have this non-linear noise term and an inhomogeneous analog we have a non-linear damping term and an inhomogeneous analog we have this update setting which allows us to periodically collapse these long time history integrals so this is just the form of the P's and the Pi's the inhomogeneous contributions now to the tendency so the response function also has a contribution from the interactions between the eddies and the main field and topography and again as I said the right hand side of the diagonal single time cumulant is again real so I'm just going to show a few results of what this turns out to do and so the diagnostics I'm going to use essentially the transient energy enstrophy, the dissipation a large scale Reynolds number and we're also going to look at skewness okay so okay before I get to that so if we just look at the homogenous form of going back to the DIA if we look at decaying isotropic turbulence so what you find is so here's the initial spectrum so this is the direct interaction approximation so here we have the DNS and the dotted line here is a regularised form of the direct interaction approximation so regularisation is just a simple procedure to approximate these higher order vertex terms so all we do is we choose to localise zero the interaction coefficients kpq if p is less than k on alpha one or q is less than k alpha one in these interaction coefficients for the two time cumulant terms and response functions so this is just comparically imposing this localisation of interaction across wave numbers so what we found was that there was almost a universal choice of this parameter and it didn't matter whether we were doing the inhomogeneous problem or the homogenous problem or in fact the strength of the topography so it turns out to be a pretty efficient approach well maybe it's a pragmatic approach to deal with this difficult problem of what to do with vertex renormalisation so as I said regularisation inserts the transfers between the large the small scales proceed by a cascade that's local in wave number and it also localises the eddy meanfield and eddy topographic transfers so one of the experiments was here here we have an again this is a comparison to ensemble average direct numerical simulation so a couple of thousand realisations starting from an initial state where we specified a single realisation of a random topography with the following functional form and we start with a mean field of the following form so the meanfield is dominating the large scales and this is the initial transient spectrum and so here we've evolved this thing out for quite of time so the initial transient kinetic energy spectrum has come up it sucked energy out of the large scales and you can see there's a very good correspondence between the DNS and this two point closure so if you look at the skewness though and these jumps these restarts in the skewness where we're actually under representing the skewness but the point here is that small scale large amplitude topography is actually acting to localise transfers but if we go to a spectrum that's more like the atmospheric state where the small scale mean field is actually orders of magnitude weaker than the small scale transient field and then we evolve this thing forward what you find is we have a dramatic underestimation not only in the transient but also in the eddy meanfield contributions from eddy meanfield eddy topographic transfers and so we have this huge underestimation of the skewness so when we apply this empirical vertex renormalisation we can actually correct for that and we get a very good agreement at all scales and to some extent you have to do a lot of realisations of DNS to resolve some of these small scale features so it's kind of a curious thing because the the DIA the physical argument is it's this spurious that the large scale eddies are interacting directly with the small scale eddies and destroying them but none of that applies for these eddy meanfield eddy topographic interaction terms and so just sort of in a simple diagrammatic way we can show that it's almost a topological result of having again that the terms involving the meanfield topography essentially form a propagator and you have these single loop expressions for the inhomogeneous terms. So to Markovianise this system we applied the Bowman et al. ansatz for the FDT and this big white gap here is that when you apply this and rewrite the equation for the single time diagonal cumulant you can derive a unique triad interaction terms for all of these terms that involve meanfields and the inhomogeneous terms and so rather than do that I'll just write this in the simplified form here and so we have a new equation for the single time diagonal cumulant and we have to replace the response function by this Markovianised variant okay and so Bowman suggested that this is a natural consequence of supposing that self-energies are Markovian but it does seem to be something stronger so rather than showing you many pages of mathematics to derive these unique triad interaction times a simple way to do this and to do it numerically is to do essentially what you would do with going from the DIA to the SCFT or the LET so we're going to replace the QDIA response function with this Markovianised variant the two-time cumulant by the FDT right and then that will all flow through and we get a Markovianised variant but I will say one thing so this is really highlighting the mean field equations so if we rewrite the mean field equations again using this FDT ansatz of Bowman they have these two terms that pop out and in fact there's another one and from here you can see that you're going to get unique triad relaxation times so the QDIA itself is actually realisable it has an underlying or generalised Longivon equation it has an exact plastic model representation which takes this form and I'll just jump to some results so here we have an initial topography which is a conical mountain centered at 30 north it's 2500 metres in height we start with an initial spectrum that's where the electricity is anti-correlated with this structure so a kind of economical equilibrium type initial state and then we start the thing off so we have a 7.5 metre a second eastward large scale flow and so it impinges on this conical mountain and after about 10 days we spin up these stationary rosby waves and so the correlation between the QDIA and 1800 realisations is very high and to look at the kinetic energy spectrum so this is our initial mean field this is our initial transient field and so we're sucking energy out of the transient field and generating this large scale stationary rosby wave at this mode here so there's a very close agreement there so if we look at the Markovian ice QDIA so in the QDIA here this is a full 10 day simulation we haven't used the restart procedure we've contained all of the information in the time history integrals and the Markovian ice QDIA again and so they're largely indistinguishable and so we look at the kinetic energy spectrum there's good agreement there's some slight agreement where the transients are resonant with this rosby wave but if we look at the polyinstrophe then you can actually see that you know there's very accurate agreement so these are our initial results at fairly large scales but if we compare them to some earlier work on looking at comparisons of the DIA renormalised Markovian enclosure and the self consistent field theory at for three waves for similar low resolution studies these are pretty reasonable results so I think rather than so rather than go on and talk about sub grid terms I think I might just end that there thank you very much questions