 Thank you for choosing this talk about the other one and today I will talk about transition from direct and is cascade that is from large scale to small scale to inverse that is from small scale to large scale in homogeneous and isotropic three dimensional turbulent flows. Okay so I will give a brief introduction to the energy transfer from my helicity point of view and then I will talk about a transition with helical decimation where I remove some of the helical modes from the system then I will talk about a transition where we have all the modes intact but we will see a transition in three dimensional turbulence. Okay so turbulence is everywhere and it is they are believed to be the solutions of the Navier-Stokes equations with suitable boundary conditions and forcing mechanism. The question we ask here is that can we split these Navier-Stokes equations into its elementary constituents and try to understand try to develop a deeper understanding of the mechanism involved in one of the central problem of energy transfer in turbulent flows and so we will be talking about this we will mainly talking about the triadic interactions in Navier-Stokes equations. So when you talk about direct transition it is mainly associated with three dimensional flows and when you talk about the inverse transfer it is mainly associated with the two dimensional flows. The fundamental difference between two dimensional and three dimensional is that in the three dimensions we have in two dimensions you have two conjured quantities that is energy and entropy both of them are sign definite and that restricts energy to go from small scale to large scale and entropy goes from large scale to small scale. However in three dimensions two conjured quantities in ideal case of course no dissipation no forcing two conjured quantities are the energy and the other one is the helicity and helicity can be helicity is not sign definite it can be either positive or negative and therefore it does not put any such restriction on the energy flow from small to large to small it is only our empirical observation that energy flows from large scale to small scales in three dimensional Navier-Stokes. However there are situations where if you have strong rotation or I will come to that in a minute or external field like magnetic field there we have observed energy going from small scale to large scale. So here are some previous results where you reduce the aspect ratio of your simulation domain across one direction you will see the more and more energy accumulates at the large scale and there are also experimental observation in the thick fluid layers where you have seen energy going from small scale to large scales. So the question is many of these are not our summation but yes it is periodicity yes. So the question is there any transition between 2Td to 3d what when you look at many flows in geophysics or astrophysics astrophysical flows if you look at aspect ratio the scenic flows atmosphere is very much like two dimensional but there is not much difference. So it is basically depends upon which scale you are looking at then when you have strong rotation in the system then you see large two dimensional structures and also if you have strong shear flow there also people have observed split energy cascade towards the large scale. So now we want to understand what is the role of helicity here because helicity is something which doesn't exist in 2d but exist in 3d. So it is basically what I want to show here is that there are some interactions which are naturally at the part of the three dimensional Navier-Stokes equation and which can transfer energy from small scale to large scale backwards and that can be triggered by dynamics of helicity. So the question is as I said since helicity can be either positive or negative it does not put any restriction on the energy cascade. So the question is can we make helicity positive and definite and see inverse transform? Answer to that is coming before that what I want to what I do here is basically I solve the Navier-Stokes equation Fourier space we do direct numerical simulations using a pseudo spectral code and this is the Navier-Stokes equation Fourier space and the incompressibility condition gives you leaves two degrees of freedom in the Fourier space for each Fourier component and we choose these two degrees of freedom of the velocity component, velocity Fourier mode to be the plane polarized helical waves that are the solution of the core operator. If you have a velocity Fourier component, if you have a velocity Fourier component you split them into two parts one with positive velocity, one with the negative velocity and h plus h minus are the Eigen factors of the core operator here already written in Fourier space and you write then you have the equations for the positive helical modes and negative helical modes separately then you can split your energy into two parts positive velocity and negative velocity and like that then since you have all non-linear process all energy cascade is happening through this triad as I showed you before since now each Fourier mode has two degrees of freedom this triad can go to eight and since the positive negative is the symmetry there so we have four classes of triads this single triad now becomes four classes of fundamental interactions. Now some of them are some of these triads if you do linear stability analysis then we will see that this class of triad where this I call homo kyrals because they are made with this Fourier modes it is same sign of felicity these are called homo kyrals and they are capable of transferring energy from large wave number that is small scale to small wave number that is large scale so these class of triads are capable of transferring energy from small scale to large scale this is also partly and mainly two small scales but partly two large scales and these two three and four they only transfer to the small scales which are very much dominant in your standard Navier-Stokes equations but however there is a part which can transfer energy to large scales and in 2012 in this paper we reported that if you have if you remove they are supposed to be removed ok so if you remove these triads and only keep this one then you will see if you still the system is three dimensional and isotropic but you put energy at this scale and energy goes to large scales this is clear inverse cascade only if you keep these triads that is only positive helical modes in the system. Now we are doing some unconventional numerics here and some sophisticated numerics where you can basically remove the triads of your choice then you can understand the dynamics of each individual triad you can remove all the inverse cascade triads and only keep this or you can do the other one. So these are the games we play here so this is not only the mathematical or numerical things which are we are doing which has no reality but in fact it is observed that if you have a strong rotating flow you and you put energy here at this scale and if you have a if you have a this is the rotation rate if you have strong rotation then you see energy goes to the large scale and for when the energy goes to large scale you will see the contribution for the energy flux is mainly coming from the flux carried by the homo-carrot triads. So these are very much comparable the energy contribution coming from the energy flux coming to the large scale is only due to the first set of triads. This is a basically a numerical verification of the theory. So what we do basically if you have all the triads you see energy going from large scale to small scale and if you have only homo-carrots like helicity modes with one sign of helicity then energy goes from small scale to large scale. So where is the transition happening? So what we do for that is basically we kept all the positive modes then we start adding okay so we kept all the modes we started removing modes of one sign here we remove the negative helical modes from the system and we introduced a parameter alpha which varies from 0 to 1. If alpha equal to 0 that means I have not touched the system if alpha is equal to 1 I have removed a helicity mode of one sign if alpha is 0.3 that means I have removed 30% of the negative helical modes from the system randomly from all scales okay. So that means that gives a probability to the first set of triads one and all others with reduced probability. So and we saw that only when alpha is very close to 1 that is I remove all the modes of one sign of helicity energy still goes to small scales it only goes to the large scales when you remove all the modes otherwise it does not. So the critical value of alpha is basically 1. So we see that the forward cascade mechanism is very robust okay and we reported this work few years ago and so remember what I did is basically I removed negative helical modes from the system randomly at all scales okay. So that is the clue here so what we did again is that we kept all the positive modes where we see inverse cascade but then we introduced the negative helical modes at a particular scale every removed everywhere but introduced only at one scale to see what happens earlier I removed from all scales randomly that means it was also present at all scales even if it is 0.999 that is 0.001% it is lot of triads for simulations with 512q okay. So that means there is a pretty good chance that negative helical modes exist at all scales. So to check what happens if you remove if you keep the negative helical mode at only one scale. So here is the scale where I put energy and here are two cases where I put negative helical modes at the scale larger than the forward scales and here I put negative helical modes at a scale smaller than the energy injection scale. So we will see that eventually wherever I put the energy the energy accumulates only in the mode where the negative helical mode is there. Therefore formation of large scale structures or large scale coherent structures or small scale coherent structures these are these are the process you form large condensed sets. Energy accumulates only at the scale where you put the energy. So you can control where you want to form a condensed set. You can create a structure where you want at whichever scale you want. So these are the things we learned. So we learned that the negative that helicity modes of both kinds are essential for the forward energy transfer and if you remove if you keep a mixed helical modes at one scale a condensed set is formed there and this is what we learned with helicity decimation where you remove some of the helicity modes. But what I am doing there is basically I am removing the degrees of freedom from the system that I am restricting my dynamics of the Navier stokes. Of course I am able to mimic I mean when I showed you the slide where I saw that the contributions from the homo karel triads matches with the inverse flux in the strong rotation case that puts my case that it is agreement. But here in this case is I am removing some of the degrees of freedom. So what we wanted to do that without removing the degrees of freedom we wanted to change the non-linear term in a way that I am able to see the transition which usually we see in cases like thick fluid layer or rotation or anything. I want to see how the non-linear term changes basically I change the weightage of this triad, different classes of triad if I can change the energy transfer direction. So for that what I did what I did I have this non-linear term then I kept the homo karels with a this with a probability 1 and this all other triads with a probability with a probability lambda that means the weight before all these triads has changed. Now I change this parameter lambda from 1 to 0 when lambda is 1 I have all the triads like pure Navier stokes and I have photodynamic transfer when lambda is 0 I have only this triad which will give me inverse transfer. In this case the transition did not happen at lambda extreme value of lambda in fact it happened at a discontinuous value where lambda is close to 0.3 and we have all the we do not break any dimensionality we do not break any ideal invariant there is no broken symmetry nothing but we still could observe a transition from forward to inverse and this was reported recently that is all and what I want to say is that in three dimensions there are inverse there are interactions which are responsible for inverse energy transfer and those triadic interaction can be enhanced naturally depending upon the boundary conditions or forcing mechanism you can still get transfer of energy from small scale to large scale and the flow can still assume three dimensional configurations where you have inverse energy transfer and there is always a competition among the interaction that transfers energy in both directions. Thank you very much for your attention. There is one kind of philosophical look at that which is very often is missed when you start removing the molds let's say you have turbulence with a particular correlation like if you start removing the molds in the fuller sense with the scales are larger than the natural correlation like you know what happens you are inducing artificial correlation like very good in this large scale model and so how much is what you are doing here is that to this basically more or less numerical effect by effectively increasing the correlation like of your simulations rather than moving into physics of your problem. Yes of course when you do different kind of filters where you remove you project your system into some kind of manifold where you exactly it's not very different from that but we it's not all kind of projections into different manipulatory changes the correlation length and sometimes it does and we have reported how it changes in one of the cases and it's basically changes the intermittency of the system. So in the case where you remove the negative physical modes we have seen the changes in intermittency intermittency changing drastically and yeah we also looked at intermittency but here in this talk I just but it is all there in our paper we talked about intermittency. So the large when energy goes from small scale to large scale we basically see the large scale structures are formed basically instead of energy being distributed at the small scale you have more energy at the large scale so you see more and more large scale structures from like cyclones is formed with a collected energy from the small scale to basically large scale structures. So you will see more prominent structures. Yes for the noise. Yes. Yeah so the there was a talk furthering on the first day so like helicity everywhere or helicity nowhere. The helicity is conjured in total. So you can still have a region you can still have a flow where the two regions are separated like the net helicity is still zero but you have regions where you have a positive helicity or negative helicity like in fact if you go to very large Reynolds number the positive helicity and negative helicity that tend to go to infinity and in finite Reynolds number case also you can have different regions in the flow with different sign of helicity. So we will see helicity effects there. This is very prominent in magnetodynamic flows.