 based on my long-term collaboration with many scientists from USA and now some of them in France and China and also in Korea. So I talk about turbulence spreading first then avalanches and for each topic I start with a simple physics model and some simulation readers and relevance to experiment but due to time limits probably are focused on simple physics side. The turbulence spreading is studied using very simple non-linear theory and also global gyrokinetic simulation and what I mean by turbulence spreading is fluctuation amplitude in the linearly stable zone can be significant due to turbulence spreading so it can affect confinement scaling with respect to machine size in magnetic fusion business is very important to know how future large machines will coast. So spreading of edge turbulence into the interior the core can exceed local turbulence in connection region and this is called short for problem in no man's land and I'll skip some topics. So we are dealing with magnetically confined plasma so you heard from previous speaker that's rather simple physics machine but they are larger machine which is more core oriented for future magnetic fusion. So we want to know how to determine the fluctuation amplitude in such plasma and conceptually it's still underlying the physics is as follows probably due to Boris Kadomshev. So if you plot the instantaneous growth rate gamma as a function of fluctuation amplitude if you do linear theory of course fluctuation amplitude the infinite decimally smaller than linear growth rate is well-defined and typically for specific mode K the fluctuation will grow exponentially in time first then it will slow down due to interaction non-linear interaction with the other mode and finally saturate so if you just measure growth rate it will go down to zero and this one will determine the fluctuation amplitude at non-linear saturation. So this can be schematically written as the effective growth rate as a linear growth rate minus the non-linear damping term which goes like k perpendicular square times turbulent diffusivity due to non-linear interaction. So now there are many transport models for Tokamaks but if you look at it in detail actually this is still the conceptual foundation of most transport models and transport models have been refined with more you know powerful simulation and etc so actually those are successful in explaining whether well-known and well understood experiment but still there are significant number of examples it cannot really we cannot really understand based on this paradigm that include anomalous transport in the region where there's no linear instability and also turbulent spreading into less unstable zone so when we start talking about turbulent spreading surprisingly many people had a strong objections and that was rather surprising they called it rather what did I call I'm still sleepy it's consulting behavior but if you think about it in case space this is well known nobody will object that due to non-linear transfer of energy in case space some linearly damped mode will be excited due to overspill of spectrum so this kind of calculation in case space can be also performed in magnetized plasma at least in the weak turbulence regime and these are examples and mostly we are talking about the leading magnetic fusion experiment concept is Tokamak which is toroidal so unlike the previous example LAPD which is cylindrical with almost straight magnetic field in Tokamaks magnetic fields are helical and their pitch is a function of radius so you can have helical magnetic field and sometimes at some location it can close on itself and that's called more the rational surface so we have a rather complicated situation when we do the fully analysis of fluctuation and look at each colloidal harmonics or the numbers and etc so if you think about this fluctuation sitting on each rational surface where landau damping is minimal you can easily accept the notion of the non-linear coupling of different mode can lead to radial diffusion of fluctuation itself due to non-linear interaction of fluctuation so in the simplest non-linear closure of the E cross B non-linearity which is the dominant non-linearity in plasma dynamics usually we have only this kind of k per square times diffusivity non-linear damping term but if you do more elaborate closure you can easily find there is a radial diffusion of fluctuation itself so around 2004 we have developed a very simple non-linear model so I capital I here is the turbulent intensity of the patch or collection of turbulence not just each k mode but collection of those mode so you can consider it as an envelope of turbulence so if you keep only this term this is linear growth and this is non-linear damping goes like ice pier then radial diffusion of fluctuation itself will appear as a non-linear what happened non-linear diffusion of the fluctuation itself so if you look at this equation this gradient of fluctuation will play a crucial role in future a spatial temporal evolution of fluctuation envelope so if we do the radial integration for some small radial extent we can rewrite that equation schematically like that and we can define this term jump of the gradient of the intensity from the right to the left as delta prime so you heard about delta prime who are doing plasma theory this is the key parameter for carrying mode instability carrying mode is resistive kink instability previous speaker talk about ideal kink but when even kink mode is stable according to ideal mhd inclusion of resistivity can make this system unstable to tearing mode so this is one of the classical paper in plasma theory so delta prime basically physics of delta prime is the characterization characterization of the net flux of turbulence into this region or the free energy stored in current gradient equilibrium current gradient injecting energy to the fluctuation so sine of delta prime determines the condition of the growth and from dimensional analysis the growth rate should go like delta prime over the radial extent and also propagation velocity inverse should go like radial velocity over radial extent so if you use only two this two relation we can get the extent of the turbulence spreading also we noted that first Kelvin Rosenberg's tearing mode predicts gamma going like resistivity three fifths power delta prime four fifths power and also delta x going like fractional power if you multiply it you get the same relation so so this is nothing but elaborate dimensional analysis in the case of this rather elaborate boundary layer theory so why this one okay so why do we care about turbulence spreading because if we totally ignore turbulence spreading probably we will get something called gyro bone scaling where turbulent diffusion coefficient reduce as we build larger and larger motion a is the minor radius but due to turbulence spreading or even before theorists found not found but insist we if we do some theory usually we get gyro bone scaling but experiment up to now has seen boom like scaling which is more pessimistic even if we build larger motion turbulent diffusivity does not decrease much so I cannot discuss this in detail but in summary if we did not have the turbulence spreading actually from theory regardless of what experiment will truth is if we do theory somehow we get gyro bone scaling but due to turbulence spreading when system size is rather small we get boom like scaling because extent of the turbulence spreading scales with gyro radius not the system size so we can do numerical experiment of turbulence spreading by making the gradient of ion temperature rather simplified so that we can understand physics easily and so varying this the width of this linearly stable zone or in other words making dissipation stronger we can see the scaling so from previous slide I have shown you that only from the dimensional analysis we can we can find propagation time of turbulent patch into the linearly stable zone goes like radial extent of turbulence spreading divided by radial group velocity and also this one should be balanced when the turbulence get damped by linear damping rate here or this patient so damping rate gets higher and higher as you go deeper into the linearly stable zone so I tailor expanded linear damping rate so we match these two a timescale when turbulence spreading stops so from this simple relation I derived we get turbulence spreading going like 18 gyro radius for this specific set of parameters while first principle gyro kinetic simulation has shown 25 ion gyro radii so it is so given the simply 30 of our approach this is pretty good so we kept on doing this theory so this relation we okay 15 minutes so this simple relation is only gotten from balancing the radial flux of turbulence from the very turbulence zone into the propagation into the dissipation so we heard from previous speakers the second speaker that the core model of this patient's care plays very important role even in complicated situation like viscosity is not uniform in space so core model of this patient's care is nothing but the we get this from the balance between energy input from nonlinear interaction in inertial range and energy drain from viscous dissipation so this is not really identical but was turbulent intensity and the energy injection rate fractional power and also downstairs is this patient's fractional power so there are some similarity to more well-known and generic situation so so turbulence spreading is not something you know only a few people who who don't like local simulation would do this is quite generic so turbulence spreading after our work actually it was initiated by French group led by Céviève Gabet it was somehow not many people followed but after our work many field people did both simulation and theory and some of them even try to measure the turbulence spreading but fortunately one expertise here I cannot go through it or George Mackey and another one who had a good chance of measuring turbulence spreading was Edna Snakowski he worked on TFTR in Princeton but rather than doing research he went to US Department of Energy and he managed our research but I hear he could right okay so that now even less chance of you know measuring turbulence spreading so second topic is avalanches I talk about major scale structures first in droid geometry those are streamers and convective cells and of course zonal flows I did not really mention and also up to this this is just the structure of fluctuation but fusion business we have to talk about the transport because our goal is to confine high energy ions so that they can do fusion reaction so we need to confine high temperature plasma and we'll talk about self-organized criticality and things like that so since early days of nonlinear simulations using fluid models toroidal simulation has seen something called streamers which is really elongated highly anisotropic in poloidal cross-section but it still aligns with helical magnetic field so due to some reason this is quite natural nonlinear structure and also linearly this is the global eigen mode which is due to the coupling of different poloidal harmonics and more recently from gyrokinetic simulations many simulation has identified two different characteristic special scale the microscopic scale is about several iron gyro radii which is well known thanks to the experts like Ray Funk and George Mackey and so on but simulation began to see more larger scale so we call it mesoscale structure because this is somewhat smaller than the system size which is macroscopic and also when TFTR shut down many of us Princeton people went to Japan and worked on JT-60U plasma experiment so this is USA teams effort on Japanese machine and in retrospect it has seen really large scale means a scale fluctuation before the formation of the Internet transport barrier and more recently Pascal Anakin French scientist working on Aztecs upgrade German machine has seen these two scale both micro scale and mesoscale turbulence structures so both cases one case is American physicist working on Japanese machine another French physicist working on German machine so I think there must be some truth in that because they will see this kind of structure so I didn't have time to talk about zonal flow but zonal flow is driven by turbulence itself and is highly anisotropic as well this is only really localized this is toroidally symmetric and also equals to be poloidal flow poloidally symmetric so highly anisotropic but some plasma wants to have this kind of radially elongated streamers so it's either horizontal or vertical that's natural state for me my natural state right now is horizontal one because it's like three in the morning in South Korea but somehow I sustain myself in vertical position but I don't know how long I can do it so anyway but those are not the only natural mesoscale structure actually in Japan there's a really even more complex geometry fusion machine called large helical device and when I first saw the report by in our case on about this very large fluctuation structure with toroidal and poloidal mode number around one I like Japanese cuisine so much that gave me an impression about very large futomaki or something like that so so I call it was largest futomaki and also you will hear from our chairman professor Sidorat probably during this conference from LAPD he also has seen some kind of convective cell from kinetic elephant turbulence so so this is from UCLA motion LAPD so you may think this LA means Los Angeles but you know this is large aspect ratio so this is large large and this structure is also large so there's no shortage of large structures in large motion and it has been also observed by wishing one so many simulation has seen also relaxation at all special scale and also time scales almost the same similarity and also many fluid simulation has seen it but just seeing it is not enough because sometimes you can see it but if it's not dominant that has nothing to do with fusion so we can publish a few papers but you know in our fusion business we really want to contribute to the energy crisis so more recently this is simulation from University of Kyoto group really powerful global flux driven gyrokinetic simulation and they look at the heat flux avalanche event statistics looking at PDF of that and rather surprisingly rather small heat transport event goes not only going outward sometimes going ignored and small-scale heat flux event almost cancer with each other and only the largest scale once will determine the next sign or direction of transport which is of course outward because it's coming from hot interior to cold exterior so if this is really true of course there are many ways of doing statistical analysis so this shouldn't be taken as a Bible but but quasi linear calculation usually think at each K there is a rather smooth you know outward flux which add up to total heat flux so this is rather drastically different result from that so so in relation this many theorists and traditionally found linear instability sort of some plasma governing equation and you look at linear dispersion relation and you look at imaginary part of omega and that can be unstable and there are really variety many many different instabilities and this is getting late but actually I heard from professor Ronald Sargadev that in probably mid 60s actually many giants in plasma theory like Marsha Rose and Bruce and Sargadev and Bruno Kopi and so on gathered in this place and they work and in those days they found new instability almost every week and there was really productive time but they really made our life of it you know difficult because all those linear instabilities are you know found in 60s 70s so we have to do no linear theory and simulation but apart from this we got some hint from self-organized criticality in different disciplines such as sand pile model by far and we applied this and construct heat flux expression only from the conservation of symmetry and conservation law not from the linear dispersion relation so so they kind of construction based on conservation or symmetry usually give us some self similarity or power law you know things like one of the f spectrum transport the event at every scale and so on and actually this is either one or another gyrokinetic simulation expert has produced this plot and this in some frequency range it also shows one of our f spectra so even from the you know most first principle gyrokinetic simulation kinetic equation can exhibit something like this not only really over simplified sand pile model of self-organized criticality and also the underlying symmetry is the joint reflection symmetry in the case of sand pile model which means the bump goes down but this kind of whole negative fluctuation should go up the gradient and even more complicated situation gyrokinetic simulation tends to see something like this so so our lenses sometimes exhibit joint reflection symmetry and there are more examples and also this kind of heat transport the event has been observed from experiment from electron cyclotron emission in D3D plasmus so do I still have time okay so this is most recent twist so if you look at or we draw or we we assume in a tokamak pressure profile is monotonically going down rather smoothly and assuming that gradient of pressure profile we calculate linear instability and we calculate heat flux from gyrokinetic simulation etc but more recently some theorists have argued actually tokamak plasmus should exhibit something like a cross speed staircase which is the pressure profile becomes rather corrugated and there will be rather sharp pressure gradient region something like this and which will lead to equal speed shear flow due to the you know diamagnetic flow due to radiate force balance and in between there we rather smooth pressure profile and heat flux avalanche can only last between this equal speed shear layer because equal speed flow can really stabilize the turbulence and transport but I don't want to underestimate the you know artistic ability of French people but actually this is from the paper not not not just as you know his seminar GMD Pradelier had this plot in his paper so so I had want to call it as BDS which is in English scientific cartoon and in Japanese this is some manga and what I mean by BDS is you know who can who can really be as you know so so actually not only he draw that our scientific cartoon he actually did very very powerful and very illuminating flux driven gyrokinetic simulation as seen equal speed staircase and and Koska and Diamond wanted to understand it so this is basically on extension of the sand pile model I told you before so this kind of heat flux expression has joined to reflection symmetry and the governing equation we get from this is nothing but the burgers equation and if you include finite response time over the heat from instantaneous heat flux to the average heat flux you get nonlinear telegraph equation and you can find something like equal speed staircase so this is finite response time is analogy to the traffic jam the reasonable driver can respond to the you know other traffic flow so that will lead to the traffic jam but the second or third Patrick diamond to my knowledge he he doesn't know how to drive so I don't know whether you want to believe he's argument about driving drivers you know reaction time or not so anyway something similar has been also observed from Ido Morazan simulation that this is rather complicated but instantaneous heat flux can occur first and it will take finite time to respond to the average heat flux level so this will lead to some modification of earlier assessment we did using different gyrokinetic simulations so this kind of finite time delay may have some truth and also people began to observe equal speed staircase in this was in tall Supra I forgot whether it was aspects of great only plasma so in conclusion the topics I have discussed research have been led by mostly theorists and simulation expert and we think it really contributed to our enhanced understanding of very complex plasma turbulent system and more dedicated experiment addressing these issues to be desirable okay thank you again for staying