Coarse-graining as a technique to reveal sub-grid scale fluxes and test the diffusive assumption.

PHD Researcher yet there I P and I'm associated to add to this transregional project which concerns itself with an energy transfers and yet but in the ocean and as a modeler I'm interested in very specific energy transfer

namely that 1 between the resolved to its scales and the unresolved separate scales in model so when we model and the fluid equations numerically we have to do some sort of a filtering

process before we run these equations and as not affect all of and the filtering process we have to account for things we overseas you can easily imagine if I average of a forest I don't get to see the friction the force and useful main flow and have to parametrize this and

so I'm going to give you a little rundown so what happens if I felt the that these equations and what are the consequences of this filtering operations and then how do we go into the parametrization of these processes and then lastly I'm going to show you an experiment I did and I ask you a bunch of questions because I have more questions than answers in this talk say on regarding the for the method as we already heard yesterday and we can decompose each flow variable into a meaningful variable and fluctuations from these variable and the mean flow is denoted with the fat and just to avoid confusion I also have is the box I can also introduce hats which is then the density weighted average of this mean flow variable which is really nice full-flow atmospheric Sciences and so if I now applied this

averaging methods to my time evolution of any variable what I get is the convolution all of the mean variable DOS this factor and this is

the 1 which should we usually have to parametrize because as you see here we have a function of the mean variables and we have an average of defining variables which we cannot resolve so this 1 right here is an unknown usually treat this 1 as 1 single thing and for some of are going to the momentum equation and we see here this is from the stress tensor and the Coriolis force term and the 3rd dear potential yes you school and agrees with you know how you this 1 and the yeah I by just applied the filtering to yes sorry and this results from the continuity equation so by going average the continuity equation and I can basically and substitute this 1 for this

term that that's how it turned so yeah thanks for the question

and so the on and going to leave out most

of the terms of most of the equations so you think and equations are from beyond recognition that's because I'm leading out everything that's not concerning my particular studies now and so if I averaged the momentum equation when I get this is this term and which I will call the 2 ferment influx tensor and and this is the part of the equations which we need to parametrize and now to learn how we can go and parametrized a spot we have to look at their energy equations so if I averaged the energy equations I get and the kinetic energy budget for my course grades I also get a canon at turbulence kinetic energy equation which is basically the equation which tracks the energy which is in the considered to be inside and to motion below Magritte's cases something I don't see but can still keep track all that energy and then they get in the internal energy budgets of full for might grid-scale basically and now you can see here that I have these these terms which appear in the canadian energy equation the 2 kinetic energy equation and again was in the internal energy equation and now the whole approach of parameterizing this guy it depends on how it decides to to approach these equations the canon to the kinetic energy equation and the internal energy equation if I decide to parametrize this 1 this is some sort of a flux which goes from results they on into the result scale but it may go both ways if I decide that I want to save me some numerical computation time and they include the 2 LAN kinetic energy equation into the internal energy equation I just some these 2 up this 1 physically and books out and this 1 turns into a factor which goes directly into the and internal energy and if I do this and I'm going from kinetic energy Darroch to you dirty to temperature and that's the good old story I throw stones the stone is the wall the wall gets hot the will never gets school and the stone never flies back at me so this is the 2nd law of term of enemies that may not occur no yes yeah I just left them all in the box basically because this is the 1 I want to talk about so I just mind of all of them and just left very transfers in that the transfer and specific and so now I have an idea of how to approach it you I have to have I have to style but parametrization for the tubulin momentum of flux tensor in such a way that this tensor product is greater or equal 0 and now we come in and say OK how are we going to fashion this so this would be the when she had this 1 and this is my Tao Sagan and basically designed Macao in such a way med I gets and the credit but product of this term so this is the 1st part I put into my Tao vying had scored a great product this is sure to be positive of multiply this to Nobel the left with MIT is that's always positive and I have a little perimetre which I can just to be always positive I'm saying mice now I still have sort of a uniform and parameter over my whole simulation so I can go and make it even better I take those keep approach we basically said or the ideas basically each package of to the motion has sort of a mean free path and after you traveling along this path it will disperse and turn into internal energy so he just went in and similar approach and introduces all of filtering scales as basically how calls are we making our grid and and this is my Rinsky permitted this yes so now we have our children for mental flux tensor depending on the ions from Rinsky para meter and on the when she attends so basically those really straightforward and I get even more local variability so I don't have 1 uniform friction will migrate but have adaptable friction basically now this can be taken even 1 step further and I can go into the dynamic anarchist Marinsky so I guess since Nostrum engage there's been a lot of research on the facts the general ideas that and if I look at the energy within each scale or each wavelength and the and the atmosphere I see that the the kinetic energy stored within always decreases so and that with the the constant slope so if I transition from this gave example than here a certain amount of kinetic energy must have diffused until there so I can shape my and my parameters such that basically follows this the Dominicans Marinsky basically introduces the idea they can introduce a 2nd filter which is larger than the Grateful Dead and basically diagnostically gets my my permit this C. S. and from that and then I can go and say well if I take my equations which I have for this great and I average once more I gets the the momentum flux which I had for the original equations with the with another average over them and plus another new driven momentum fast exchange but this time with variables I have all resolve basically because I'm just averaging over the grade and of course I get on the tool and pimenton flux Tenzer if I put directly go the consequent how can this be just go and say OK well and the friction sort of induced by these 2 terms must be equal to the friction induced by this term so set them equal and now the next step out would be to assume that the 2 these 2 C S's or equal than I tens equation which I do need to solve somehow the straightforward approach would be to take some sort of tense on that would be the mathematicians approach to solve this equation and the physicists approach would be to look at the energy equations so I just take the tens of product with their own with win stress and from there by basically gets yeah that the the terms just

from the kinetic energy equation and I can display sold for the CS square and this is

basically when my reception research came in I this we went out and wanted to look at this guy here so they take this approach by multiplying with all by taking the tensor product with the delta the and I'm not sure that this guy here is always negative will will always positive prosaic and so I might get problems my assumption because I want to ensure diffusivity right if they go down and this 1 has the wrong sign I all of a sudden get energy which was up from from lower goes upwards so this is where I come

in so 1st of all I want to apply this Marinsky will we want applied this Marinsky to the can model and the Yakin model uses a grid space model this Marinsky has been developed for all the spherical harmonics model which also apply this the whole thing in the frequency realm and then was the married of our and new proposed way of defining the CS cred instead of using and tens of long using and the modification bed with the dead the the and then what did the dynamics of this 1 look like so what so the picture get do we get it when we when visualize this 1 and of course we would like it to have a lot to and she a production to be sort of in a disorganized state because remember earlier we had the idea to use so the diffusion coefficient which gives us sort of the the idea of a randomly distributed it's set of pluses and minuses in our equations so went used I can I P which we have feared that the Institute just maintain bond gas as axonal bridge Scuds structured and unstructured of course it's monitor static and it's run dry sets the fair from Excel approximation and I went in

and computer myself which took me quite some time and methods to basically just detained rings around center sells an average over these these start area to get my my calls grand variables basically my filtered variables and then I went in and computer this 1 rather straightforward only I did this using a model wrong

with that a Canada's mean grid spacing and then reduced the resolution by filtering to 200 cm grid spacing and with about 70 who brought tickets that goes up to some 35 Canada's roughly and on a visiting Edinburgh Clinic wife cycle life experiment found

the results which I get looks something like this showing little moving

and so as the burden ink wave

progresses physically you can see in in the contour lines you can see the vorticity and the red and blue colors which

you see on that and diffusion of horizontal momentum this this guy right here computed for this coarse-grained process and what we see is that we have negative as well as positive values everything is all right with the negative values the positive ones are concerning because that is quite the opposite of what we wanted to be the 2nd thing is that we have sort of large-scale structures we also so the contradicts our initial idea of having sort of a diffuse of process which is random organized instead we get is large space structures sort of like small-scale structures which we can think of being averaged over going to 0 this 1 little thing we can take a spike and which is dead and that it pays to run over all these points it is it has its center of weight and the positive so we didn't meet next positive effect which means we obey around the the whole idea of the bound cascade so we have a net dissipation of energy in this whole process but still we had some some energy which is taken down from all upgrades gives to the grid scales and have energy bill goes down again and this basically leads to to to the

questions with that I have so too is approach we've taken for using our ears and separates K privatization the dynamic as Marinsky is a flawed in some sense should be rather use depends on on should we go in and use determine kinetic energy equation to allow for these exchanges with we see or if we use that an amicus Marinsky and we have these negative and positive values in it I can we then extend the comedian even verify the whole idea of being able to project our and the fusion power meter from what resolvent what we don't resolve which is the whole basic idea of that an anarchist Lewinsky I hope you can help me in some sense thanks for your attention so far the core of the data in

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