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Discretization of generalized Coriolis and friction terms on the deformed hexagonal C-grid

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tokenization of the generalized coherence term that on the deformed Texaco no secret so most important
the interesting OK we are in numerical modeling of atmospheric flows and
atmospheric of those have the property that a appeal and assumed and for in former times in America modeling for weather prediction and climate science it has been done on the not geographically in a geographical coordinate system in which all of you know that do meridiennes converge at the pool and so there was an attempt to get rid of the so-called poll a problem because if you approach a pool and the euro could spaces become smaller and and you times the cause of the good or become smaller and therefore you lose efficiency and therefore in recent the recent decades decades several approaches a well tried to get rid of after a week in LA let long coordinate system and instead to use alternative switch up I'd other cube sphere then you stay was the x y coordinate system all you go to attitude orders so-called geodesic rates and this is followed by for instance the high commodity DWT and MPI what's been instituted book and data defined the other groups like that that they say OK DeMoss points are inside of 2 triangles in the center of a triangle and I have been the divergent of the velocity U defined over triangle and the vorticity I have defined around us and a hexagonal and to that is kind of critter staggering sorry popular and meteorology a because they allow for but of a Ph propagation properties so there's also another attempt which is followed by the time and by the United States model Ampacet's as or model prediction across scales and it said that it has the property that debt it also can somehow all collect mortgage points over the United States for instance and to release the grid space a lot of regions and this is the same approach as I use in my model there's so-called I convex model where I just do the same as like a mother DWT but I exchange dear the grid by using that what I called it doing good as a primer cord and vise versa so I have did I virgins on the hexagonal mesh and the vorticity at 1st glance at our triangular mesh so I found a problem with such kind of
that is bad you have 3
so you have 3 horizontal velocity components we have 1 direct and in this direction and this direction and 1 in this direction and this is not like you have an you
should coordinate system was x and y coordinates so you you can reach a point just by going some of some steps to this point and here from on the X. axis of just 2 is the the steps in the x direction so if you have to stream area at coordinate system as we have on this field on this dude as it prints out we can either say OK via reach a point P you by going some of those steps parallel to the XY direct X 2 directional or some steps plummeted to the X 1 direction but this is not what you want we have 3 L a core of basic to coordinate directions and assert direction is actually a linear dependent often on that 2 was so you can reach so did or you can imitate stakes 3 basis spectra by going X 1 plus X 2 0 and might apply whereby by minus 1 and this is also a valid for the velocity components U 1 U 2 when he was 3 so therefore if you use such kind of coordinate systems on a geodesic red you get an additional constraint which I call the linear dependency constrained namely that Europe velocity components on a regular mesh are not independent but they depend on each other and this makes quite a lot of headaches because if you have fewer Navier-Stokes equations or whatever iterations you want to simulate on this year they already completely determined by physical properties but no you and mathematical constraint namely the linear dependency constraint and plot what does it do to your equations and how important is this linear dependency constraint and this is the main message of my talk but before we go to the Sierra dealer stay was the regular of equilateral mesh maybe have dis- basis vectors in 1 direction 2 directions we direction so and we can relate to studio x y coordinate system be have display most linear dependency of the basis vectors and we also have to linear dependency of 2 vector on a components so of but how important is this linear dependency of the relation to the that we will inspect but 1st we have to see a whole weekend linear the dependency relation onto our great because we have staggered great they're not all all components are co-located and that means we have here the hexagonal and today you 1 direction or the black arrows they was created here and you to direction look at T and its redirection is located here so therefore this linear dependency relation which I a whole I call it it is defined in the center of X and so I have to average 1st these components to the center of a hexavalent and then I ask myself OK these are vector of components of the 1 you he was to you to use 3 vectors and how can I prove that day really at the I averaged just simply from the edges to the center of 2 0 day used this kind of relations and if I just was simply average just say average you 1 over the U 1 direction are you won't get this linear dependency relation and Johnson found that and if you changed is averaging so that take into account also parts of this vector and this vector for the U 1 ever for the 1 direction averaged then you will need to this linear dependency constraint and to add this can be proven and by considering it's a composition of 2 of of the vector so expressing the vector with the same function and a velocity potential and you do not have any boundary condition because you are on this uracil that you have a periodic domain and then you can say OK I have a stream function you know and the velocity component components here and I think it can express this was this kind of and what's the composition and then I have to prove for each type of 2 rooms and that day at 2 0 and order to the averaging and so 1st you can see I have to put the stream function variables to the center of a hexagon the which is Connes on my intuition to put the stream function at the corner of a of a hexavalent so I have to put it to you because otherwise I cannot show that this that term here cancers all this term so you can see here there's a spree over each year and of 1 of Ritchie and i've enters under the averaging and dispose concert all and so it is very easy to show for the for the stream function contributions that I have just linear dependency constraint and it is more difficult to prove it for them for the tire supportive for the velocity potential but here I have visualized this iteration in terms of the coefficients of Kaio which appear here under the averaging and you see that it at every point these contributions but of at 2 0 that means that this relation is indeed fit so this is not yet and this is only the mathematical constraints there's no physics and and no I I want to see what happens if I do
if I if I do not food through this linear dependency constraint how does my
that's my the velocity you might look like so this is all linear dependency constraint and what happens if it is not fully fit I can express this linear dependency constrained was the head of the what his cities so of what is it is a rotation just around a triangle here and you can see there are a bunch of triangles and lower tip triangles and if I CompuAdd the vorticity around a blue trying to for instance I have yielded a plus and if I had a computed around right triangle I have you the minus and to using this relation I can call instead for a heck circle the center which is I found dead the sum offered up a tube triangles the vorticity and of all of the above 2 triangles is equal to the vorticity of law triangle so and he there's a typical pattern that this is not the case if for instance the orbit of triangles there are always positive values as as you find for instance Europe then you have a right positive positive positive and also around is trying its they all positive fare as the launch of triangles they are almost 0 so that means that but the linear dependency relation here that is equal to the generation of a checkerboard pattern so that means if I can't prove for my iterations that I have to linear dependency constraint fulfilled then I will see that I have no checkerboard pattern in the velocity it in the vorticity field therefore it is very
important all my initial iterations that I obey the linear Connes dependency constraint for every sup a term which I have and creation here and the which are the gradient terms that what is achieved flux term and and diffusion tram so and as I have already proven for the gradient that under linear dependency relation is fulfilled soul which of these posts rows and engines burn has proven that the linearized CoreOS term gifts this linear combiner dependency constrained what what has not been proven is if I linearize this what is city flexed around a constant so no current you but then I do the vorticity you may vary and the velocity feud is here hold constant and this is this part of the equation and nobody has yet looked at whether this this kind of linearization cannon of with this linear dependency relation and this is where we want to have a look on that but 1st we have to prove that also do diffusion to come in It's linearized version which is not the nonlinear numbers to extend the what a linearized version that I can write it down and is dependent on the divergent and the vorticity and this term is very very is similar to the the
composition because if I apply in nebula if apply apply a allow us in here the blossom applies to all this guy's here and it gives me a vorticity here enters into divergent CIA and so so if
I can expressed this yeah devote Citysearch appear here as a lab bastion of the stream function than I am on the right side and indeed I can show that the vorticity here is equal to overlap plus and apply to the side averaged over divide but then D.'s what is a C are no longer on triangles stay out on a run by here and all important thing for the time which I mentioned for this what is at linearize what is it flux through is that I get from this relation that I note that the vorticity 1 is the Laplacian often so of 1 an average of 2 of those stream function I found a relation that I can say OK did to the average of 2 in this redirection of 2 1 of what is city is the same as detailed average over the 1 direction of does 3 velocity and because of that because of this permutation here which becomes possibly because of this relation I can prove that this Berlin arise what is it flux Terman this form exactly this form was this usage of the what is the trees which is just plucked in here was all prove I can prove that this is the only possibility for the usage of the vorticity in this linearized what a city flux term here and no I kind look
what honor schemes are doing for this linearized what is it flex and I know it is kind of awarding at-risk scheme which are a known in the literature images Anne-Sophie conserving scheme of energy conserving scheme and also myself invented this scheme and if I know check for the linear dependency constraints that all too with some kind of arrows and only this new approach which I showed on the previous slide gives us 0 here and
no I can this but this is but this is
only the linearized version of this what is it flux stem and I have to marry its with
the was the listed gradient of the kinetic energy and also this the Coriolis comes so I have to put this as a food nonlinear to and there is a way to specify and this so that and so that the linear dependence constraint is for food to for the what is it flux term and also for the gradient of the kinetic energy and I have to rewrite it now in for informal off the advection so because I know ordered the gradient of the kinetic energy plus the what is a T times the velocity which is just a lump transform must be the same as the advection 2 but I found it is not the same there's something which is not consistent and occur the question is no is it important that to do so I do not achieve was my discretization from which I know it is conserving and at obeys the linear consist dependency constraints it does not arrive as the in the directive form I have an arrow does matter or does it not matter of
he is about you see it different a version or different implementations of this what is it flux
1st list scheme which is in them
in this role he helped at-risk energy-conserving scheme and the 2nd scheme I shows this scheme which I invented some years ago and this is the new and and the new scheme and this is the so-called per
clinic of a pH test by I have about clinically unstable flow and I let the model run and this was the vertical velocity and see clearly some really ugly vertical velocity fields which damage my assimilation completely and I know claim that this is because the linear dependency is not fulfilled as so I get a checkerboard pattern and of a lot in the in the vorticity and this also in is inherited to the vertical velocity so form for the other schemes have achivement invented some years ago
discoverable smaller here was is free of all 4 here was a 1 over 4 and therefore
of this instability is not showing up at least not at that time after 7 days and T I have 2 new scheme and if I let them on the run and sometimes longer up and and go no toward a southern hemisphere that I have not introduced a perturbation on the barrel clinically unstable flow that means I do not want to generate of wave at all I see that was my all scheme I still have this kind of instabilities and I know that I still have this arrow and a linear dependency but for the new scheme I do not have to in the linear dependency so I do not have disk small scales of arrows you but I still have is achieved which tries to are ordered the remaining arrows in the which are introduced because of this bending coordinates lines day are large enough to trigger approximately race but after 15 days this as there are negligible so we are good was this scheme that doesn't make any kind of the small-scale structures so an old
I have and the I investigates to and the fraction tens or and the problem here is that we have to specify this friction 10 so he also to be a trace free and symmetric that is a point which we had already in a talk of what the file but this depends no also on the on the coordinate direction so I if you add up the terms you and you see that you get a 0 trace which is all right and you can also see that that this is symmetric if you compare these elbows terms and this this these are the physical properties you want to have and you can also show that the entropy production is positive for that but now if you assume that you have a constant diffusion coefficient you can put it before this term Floyd for and of this term and then you would you would think that you would just end up with a low plaster and for the for the U 1 component or for the velocity component but it this that this is not the case affects does that we use this described tree very odd coordinate system introduces a new term here Our which was not they are if you would do it on the quadrilateral a great so and this is the same kind of error is the loss of as
was found here if you do something and so-called vector invariant form it does not it is not to say in the in the in that advective formant as in the same problem we
have here was the was the tens or form which is physically right but it does not do the same if I rewrite it in a very which I know so that is really this is that so de Matemática constraints of some whole why only it's what we know from physics and I don't know whether this is a problem or not at least it isn't something to document and now I would
come to the point of after deformed measures and so the question is always or at least in this 1st scheme the question is how do I reconstruct this horizontal velocity here in the row and the in the horizontal velocity iteration and the trust schemes say OK just derived vorticity equation and then you see here the divergence operator and you have you have a flux divergency and if that if you do not know how to reconstruct the velocity here just a reconstructed in that way that you have for this triangular velocity that you have a closed boundary which I have here in green which which is which is it which represents the the guy was the the ory around a green curve here but the point is the 7 ring LANs them Oracle and clamp scheme goes toward goes toward a vorticity equation and takes the vorticity equation in order to your Iife the whole I and reconstruct its velocity components but on the other side I could also say OK I go to the divergent situations and do the same thing and for a constant Coriolis fossil of forget about this feature here I get yeah something which has to depend on the rotation of we and this is shown here are shown here so I for for the linearized cases I can derive a divergent situation and this term here surely has to represent the vorticity namely an average WiTricity all these kind of see and then some this would result in the fact that I have to reconstruct my velocity components out ofthis blue vector components there as if I would go over there of what is city equation I would to reconstruct my velocity components order of direct contributions and you see there down lot of velocity components which are the same namely this this this this and these Fourier pending on the scheme all depending on the decision but I want to obey what is it a creation of our divergent equation I I don't know gets this edge involved on this edge involved and so I'm not sure about dichroic answer here and then I just wanted to
say something about the collaboration between s and I appeal so the problem must hold true Rica hold to computer these kind of a deformations on a deformed red and so Alexander helped me out to defined a good vectored Inc reconstruction which also these major discovered HIV which are needed here because I kind to compute these kind of generative physicals all stocks of but with this a suggestion that works no fine so this is so was my talk and a summary is the 10 of you have a mathematical constraint but this mathematical constraint is not always or at some hole why only it's physical things which we know so for instance we have a problem with the stress tensor and we also have this problem with a correction term which are not exactly consistent and aggression is is is important or not and what does it tell us and then if we are on the on the illegal a mesh they have deformed measures we cannot tell anything really you exactly but apart from the main message is that nobody has yet the I found this kind of generally is a generalized to call books correctly so this is the 1st time that I showed this in a talk at the
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Metadaten

Formale Metadaten

Titel Discretization of generalized Coriolis and friction terms on the deformed hexagonal C-grid
Serientitel The Leibniz "Mathematical Modeling and Simulation" (MMS) Days 2018
Autor Gassmann, Almut
Mitwirkende Leibniz-Institut für Oberflächenmodifizierung e.V. (IOP)
Leibniz-Institut für Troposphärenforschung (TROPOS)
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/35348
Herausgeber Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS), Technische Informationsbibliothek (TIB)
Erscheinungsjahr 2018
Sprache Englisch
Produktionsort Leipzig

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik
Abstract The talk discusses the generalized Coriolis and friction terms from two perspectives: (i) within the linearized discretized momentum equations on an equilateral grid, and (ii) as nonlinear terms on a distorted mesh. The discrete linearized momentum equations are formulated using a trivariate coordinate system. The tendencies of the different forcing terms for each wind component must be linear dependent. This constraint determines unique discretizations for each term. The linearized vorticity flux term around a zonal mean current requires only the four rhombus PVs next to an edge. The vector Laplacian must be formulated with the vorticity on vertices defined as the average of three rhombus vorticities. A modified generalized Coriolis term is defined on the deformed mesh. The baroclinic wave test on the sphere does not reveal any sign of a non-linear Hollingsworth instability, even though it is demonstrated that the Lamb form and the advective form of momentum advection are not equivalent. Physical constraints determine the shape of the stress tensor. These are invariance to the addition of solid body rotation and a resulting positive definite dissipation rate. An appropriate stress tensor formulation does not deliver a Laplacian momentum diffusion in the linear case. On the deformed mesh, parts of this stress tensor are obtained by a least squares reconstruction of wind gradients. This approach avoids spurious deformations diagnosed for constant flow in the vicinity of pentagon cells. It is impossible for both terms to meet all physical requirements and the additional numerical linear dependency constraint. The topic is of general interest, because the linear dependency of overspecified dynamics may play a role also in other modeling fields.

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