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

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Discretization of generalized Coriolis and friction terms on the deformed hexagonal C-grid
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20
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CC Attribution 3.0 Germany:
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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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Standard ModelHexagonTerm (mathematics)Helmholtz decompositionLogical constantElectric currentEquationVelocityLinear mapTerm (mathematics)Numerical analysisVortexDivergenceComputer animation
Vector potentialFunction (mathematics)VelocityStandard ModelHelmholtz decompositionTime domainHexagonTerm (mathematics)VortexDivergenceAverageFunctional (mathematics)FluxDirection (geometry)TriangleLinearizationTheory of relativityDivisorMultiplication signPermutationLaplace-OperatorRhombusTerm (mathematics)VelocityComputer animation
Standard ModelLinear mapVelocityEquationHexagonTerm (mathematics)Helmholtz decompositionFluxModulformVortexTerm (mathematics)Computer animation
8 (number)Term (mathematics)HexagonLinear mapLinear independenceFood energyFluxVortexStandard errorArrow of timeSlide ruleComputer animation
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Invariant (mathematics)AdditionSymmetric matrixRotationSolid geometrySign (mathematics)CoefficientLogical constantStandard ModelEquationVelocityHexagonEntropyStandard errorDirection (geometry)Mass diffusivityTerm (mathematics)QuadrilateralFraction (mathematics)Point (geometry)10 (number)VelocityProduct (business)FrictionLogical constantPhysical systemCategory of beingConnectivity (graph theory)Laplace-OperatorTensorBounded variationComputer animation
Food energyLinear mapStandard ModelVelocityEquationHexagonTerm (mathematics)FluxPerspective (visual)ConsistencyConservation lawInvariant (mathematics)RotationSolid geometrySymmetric matrixAdditionSign (mathematics)CoefficientLogical constantInvariant (mathematics)Vector spaceModulformTensor10 (number)PhysicalismComputer animation
Helmholtz decompositionTriangleTheoremVotingDivergenceFluxEquationHexagonStandard ModelTangentTerm (mathematics)WeightStokes' theoremMathematicsRegular graphLogical constantVector spaceDivergenceConnectivity (graph theory)FluxVelocityCurveVortexGreen's functionPoint (geometry)Decision theoryGauß-IntegralsatzTerm (mathematics)EquationOrder (biology)Operator (mathematics)Ring (mathematics)IterationEuclidean vectorRotationBoundary value problemDirection (geometry)Computer animation
HexagonScherbeanspruchungTensorEntropySquare numberDerivation (linguistics)Stokes' theoremStandard ModelMultivariate AnalyseRegular graphMultiplication signStress (mechanics)Derivation (linguistics)Vector spaceGoodness of fitTerm (mathematics)PhysicalismGenerating set of a groupTensorMeasurementComputer animation
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Transcript: English(auto-generated)
of the generalized Coriolis term on the deformed hexagonal secret. So most important, okay, we are in numerical modeling of atmospheric flows, and atmospheric flows have the property that they appear on the sphere, and
in former times numerical modeling for weather prediction and climate science has been done on the geographical coordinate system, in which all of you know that the meridians converge at the pole, and so there was an attempt to get rid of the so-called pole problem, because if you
approach the pole and your grid spaces become smaller, and then your time stack also becomes smaller, and therefore you lose efficiency. And therefore in recent, in the recent decades, two decades, several approaches were tried to get rid of the regular
lat-long coordinate system, and instead use alternatives, which are either the cubed sphere, then you stay with the XY coordinate system, or you go to the, to the so-called geodesic grids, and this is followed by, for instance, the
icon model at DWD and MPIM, Max Planck Institute in Hamburg, and they define the grids like that, that they say, okay the mass points are inside of the triangles, in the center of a triangle, and I have then the divergence of the
velocity defined over a triangle, and the vorticity I have defined around as a hexagon. And this kind of stack rings are very popular in meteorology, because they allow for better wave propagation properties. So there's also another attempt which is followed by the, by the United States
model, MPAS, it's a model prediction across scales, and it has the property that it also can somehow collect market points over the United States, for instance, and relays the grid space over other regions. And this
is the same approach as I use in my model, the so-called icon hex model, where I just do the same as the icon model at DWD, but I exchange the grid by using that what I call the dual grid as a primal grid and vice
versa. So I have the divergence on the hexagonal mesh and the vorticity at first glance at a triangular mesh. So the problem with such kind of grid is that you have three, you have three
horizontal velocity components, so you have one direct in this direction, one in this direction, and one in this direction. And this is not like you have in a usual coordinate system with x and y coordinates, so here you can reach a point just by going some steps to this point, and here from on
the x-axis just two, three steps in the x direction. So if you have this three variate coordinate system, as we have on the sphere, on this geodesic grids, we can either say, okay, we reach a point P by going some steps parallel to the xy direct, x2 direction, or some
steps parallel to the x1 direction, but this is not what we want. We have our three basic coordinate directions and the third direction is actually linear dependent on the other two, so you
can reach, or you can imitate the x3 basis vector by going x1 plus x2 and multiply by minus one. And this is also valid for the velocity components u1, u2, and u3. So therefore, if you
use such kind of coordinate systems on a geodesic grid, you get an additional constraint, which I call the linear dependency constraint, namely that your 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
your Navier-Stokes equations, or whatever equations you want to simulate on the sphere, they are already completely determined by physical properties. But now you add a mathematical constraint, namely the linear dependency constraint, and 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 sphere, we stay with the regular equilateral mesh, where we have the basis vectors in one direction, two directions, and three directions, and we
can relate this to the xy coordinate system. We have this famous linear dependency of the base vectors, and we also have the linear dependency of the vector components. So, but how important is this linear dependency relation that
we will inspect? But first we have to see how we can translate this linear dependency relation onto our grid, because we have a staggered grid, where not all our components are co-located. That means you have here the hexagon,
and the u1 direction are the black arrows, they are located here, and the u2 direction are located here, and the three direction is located here. So therefore, this linear dependency relation, which I how I call it, it is defined in the center of a hexagon. So I have to average first these
components to the center of a hexagon. And then I ask myself, okay, these are vector components of u1, u2, u3 vectors, and how can I prove that they really, if I average them just simply from the edges to the center, do
they yield this kind of relations? And if I just do a simple average, just say average u1 over the u1 direction, you won't get this linear dependency relation. And John Silgan found that if you change this averaging, so take into account also
parts of this vector and this vector for the u1, for the one direction average, then you will meet this linear dependency constraint. And this can be proven by considering Helmholtz decomposition of the vector, so expressing the vector with the same function and the velocity potential, and
you do not have any boundary condition because you are on the sphere, so you have a periodic domain. And then you can say, okay, I have a stream function here, and a velocity component here, and I can express this with this kind of Helmholtz decomposition.
And then I have to prove for each type of terms that they add to 0 under the tilde averaging. And first you can see, I have to put the stream function variables to the center of a hexagon, which is counter my intuition to put
the stream function at the corner of a hexagon. So I have to put it here because otherwise I cannot show that this term here cancels out with this term. So you can see here there's a three average here and a one average here, and if I add this under the averaging, these both cancel out. And so
it is very easy to show for the for the stream function contributions that I have this linear dependency constraint, and it is more difficult to prove it for the for the chi's, for the for the velocity potential. But here
I have visualized this equation in terms of the coefficients of chi which appear here under the averaging, and you see that at every point these contributions add to 0. That means that this relation is indeed fulfilled. So
this is not yet, this is only the mathematical constraints, there is no physics in. And now I want to see what happens if I do not fulfill this linear dependency constraint. How does my does my velocity field might
look like? So this is our linear dependency constraint, and what happens if it is not fulfilled? I can express this linear dependency constraint with the help of the vorticities. So a vorticity is a rotation just around a
triangle here, and you can see there are upper tip triangles and lower tip triangles. And if I compute the vorticity around the blue triangle, for instance, I have here the plus sign, and if I computed around a red triangle, I have here the minus sign. And using this relation I can found that for a
hexagonal center, which is here, I found that the sum of the upper tip triangles, the vorticity of the upper tip triangles, is equal to the vorticity of the lower tip triangle. So and here there is a typical pattern where this is not the case. If, for
instance, the upper tip triangles are always positive values, as you find, for instance, here, then you have a red positive, positive, positive, and also around these triangles they are all positive, whereas the lower tip triangles they are almost zero. So that means
that the linear dependency relation here is equal to the generation of a checkerboard pattern. So that means if I can prove for my equations that I have the linear dependency constraint fulfilled, then I will see that I have no checkerboard pattern in the
velocity, in the vorticity field. Therefore it is very important for my shallow water equations that I obey the linear dependency constraint for every sub-term which I have in the equation here, and which are the gradient terms, the vorticity flux term,
and the diffusion term. So and I have already proven for the gradients that the linear dependency relation is fulfilled, so which are these both rows, and then John Subban has proven that the linearized Coriolis term gives
this linear dependency constraint, but what has not been proven is if I linearize this vorticity flux term here around a constant sonar current U, then the vorticity may vary and the velocity field is here hold constant,
and this is this part of the equation, and nobody has yet looked whether this this kind of linearization can fulfill this linear dependency relation. And this is where we have want to have a look on it, but first we have to prove that also the diffusion term in its
linearized version, which is not the nonlinear Navier-Stokes sensor, but a linearized version, I can write it down as dependent on the divergence and the vorticity, and this term is very very similar to the Helmholtz
decomposition, because if I apply a Laplacian here, the Laplacian applies to all these guys here, and it gives me a vorticity here and divergence here. And so if I can express this here, the vorticities which
appear here as a Laplacian of the stream function, then I am on the right side, and indeed I can show that the vorticity here is equal to a Laplacian applied to the psi averaged over the one. But then these vorticities here are no longer on triangles, they are
on rhombi here. And the important thing for the term which I mentioned for this linearized vorticity flux term is that I get from this relation that I know that the vorticity one is the Laplacian of the one average of the stream function, I found a
relation that I can say, okay, the tilde average in the three direction of the one vorticity is the same as the tilde average over the one direction of the three velocity. And because of that, because of this
permutation here, which becomes possible because of this relation, I can prove that this linearized vorticity flux term in this form, exactly this form with this usage of the vorticities which is just plugged in here without proof, I can prove that this is the
only possibility for the usage of the vorticity in this linearized vorticity flux term here. And now I can look at what other schemes are doing for this linearized vorticity flux term, and I know this kind of Trisk scheme
which are known in the literature, which is the enstrophy conserving scheme or the energy conserving scheme, and also I myself invented the scheme, and if I now check for the linear dependency constraints, they all do some kind of errors. And only this new approach, which
I showed on the previous slide, gives a zero here. And now I can, this is, but this is only the linearized version of this vorticity flux term, and I have to marry it with the, with the, with the gradient of the kinetic energy, and also with the Coriolis term, so I have to
put this as a full nonlinear term, and there is a way to specify this so that the linear dependency constraint is fulfilled for the vorticity flux term and also for the gradient of the kinetic energy, and I have to rewrite it now in form of
the advection term. So because I know that the gradient of the kinetic energy plus the vorticity times the velocity, which is just the lump transform, must be the same as the advection term, but I found it is not the same. There is something which is not consistent, and
the question is now, is it important that this, I do not achieve with my discretization, from which I know it is energy conserving and it obeys the linear dependency constraint, it does not arrive as the, in the advective form. I have an error. Does it matter or does it not matter? Here you see
different version or different implementations of this vorticity flux term here. First with the scheme which is in this row here, the Trisk
energy conserving scheme, and the second scheme I show is this scheme which I invented some years ago, and this is the new scheme. And this is the so-called baroclinic wave test, where I have a baroclinically unstable flow, and I let the model run, and this shows the vertical velocity, and I see clearly
some really ugly vertical velocity fields which damage my simulation completely. And I now claim that this is because the linear dependency is not fulfilled. So I get a checkerboard pattern in the vorticity, and this also is inherited to the
vertical velocity field. So for the other scheme, which I invented some years ago, this error was smaller. Here was a 3 over 4, here was a 1 over 4, and therefore this instability is not showing up, at least not at the time after seven days. And here I have the
new scheme. And if I let the model run sometimes longer, and go now to the southern hemisphere where I have not introduced a perturbation on the baroclinically unstable flow, that means I do not want to generate a wave at all. I see that with my old scheme, I still have this kind of
instabilities, and I know that I still have this error in the linear dependency. But for the new scheme, I do not have the error in the linear dependency, so I do not have this small-scale errors here, but I still have a baroclinicity which tries to,
or the remaining errors in the, which are introduced because of this bending coordinate lines, they are large enough to trigger baroclinically waves, but after 15 days, this is negligible. So
we are good with this scheme, it doesn't make any kind of the small scale structures. So and now I have investigated the friction tensor, and the
problem here is that we have to specify the friction tensor here also to be trace-free and symmetric. That is the point which we had already in the talk of Wolfgang Threier, but this depends now also on the coordinate direction. So if you add up the terms here, you see that you get
a zero trace, which is alright, and you can also see that it is symmetric if you compare these both terms, and this is, 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, or in front of this term, and then you would, you would think that you would just end up with a Laplacian for the U1 component, or for the velocity component. But it is, this is not the case. The fact that
we use this tri- three variate coordinate system introduces a new term here, which was not there if you would do it on a quadrilateral grid. So, and this is the same kind of error it was, as was found here. If you do something
in the so-called vector invariant form, it does not, it is not the same in the, in the advective form, and this, and the same problem we have here with the, with the tensor form, which is physically right, but it does not do the same if I rewrite it in a way which
I know. So that is really, this says that the mathematical constraints somehow violates what we know from physics, and I don't know whether this is a problem or not, at least it is a something to document. And now I will come to the point of the deformed
meshes, and the question is always, or at least in this thrisk scheme, the question is how do I reconstruct this horizontal velocity here, in the, in the, in the horizontal velocity equation, and the tris scheme say, okay, just derive
the vorticity equation, and then you see here the divergence operator, and you have, you have a flux divergence here, and if you do not know how to reconstruct the velocity here, just reconstruct it in that way, that you have for this triangular velocity, that you have a closed boundary, which I have
here in green, which, which, which is a, which represents the Gauss theorem around the green curve here. But the point is, the Subban, Ringler, and
Skammerer and Klamp scheme goes to the, goes to the vorticity equation, and takes the vorticity equation in order to derive how I reconstruct this velocity components. But on the other side, I could also say, okay, I go to the divergence equations, and do the same thing, and for a constant Coriolis
force, so forget about this theta here, I get here something which has to depend on the rotation of v, and this is, I shown here, I shown here, so I, for, for the linearized case, I can derive a divergence equation, and this term here
surely has to represent a vorticity, namely an averaged vorticity over these kind of thetas, and then this would result in the fact that I have to reconstruct my velocity components out of this blue vector components, whereas
if I would go over the vorticity equation, I would reconstruct my velocity components out of the red contributions, and you see there are a lot of velocity components which are the same, namely this, this, this, this, and these four
here, but depending on the scheme, or depending on the decision whether I want to obey a vorticity equation, or a divergence equation, I either get this edge involved, or this edge involved, and so I'm not sure about the correct
answer here, and then I just wanted to say something about collaboration between MMS and IRP, so the problem was how to, how to compute these kind of deformations on a deformed grid, and so Alexander helped me out to define a
good vector reconstruction, which also gives me the derivative, which are needed here, because I cannot compute these kind of derivatives with the Gauss-Ostokes theorem, but with this suggestion, it works now fine. So this is, was my talk, and the
summary is, we can, we have a mathematical constraint is not always, or somehow violates physical things which we know, so for instance we have the
problem with the stress tensor, and we also have this problem with the advection term, which are not exactly consistent, and the question is is this important or not, and what does it tell us, and then if we are on the, on the irregular mesh, where we have deformed meshes, we cannot tell
anything really exactly, but the main message is that nobody has yet found this kind of generalize, generalize the Coriolis term works correctly, so this is the first time that I showed this in a talk.