talk gassmann

Principles for the discrete formulation of an NH global model by using Nambu brackets: 

Principles for the discrete formulation of an NH global model by using Nambu brackets Almut Gassmann Max Planck Institute for Meteorology

Slide2: 

Nambu bracket formulation of the NH dynamics We learned from the talk of H. Herzog: The non-hydrostatic dynamics takes the following form in Nambu bracket formulation: Why is this formulation so essential for the numerics?

My personal background: 

My personal background The experience with the Lokal-Modell (LM) learned me that consistent numerics is the most essential! Examples: 1)The „Day after tomorrow“ effect in the ClimateLM (CLM). 2)Wrong flow past a hilly mountain if the wave solver is not properly set up.

Historical background – Arakawa Jacobian: 

Historical background – Arakawa Jacobian The question of numerical consistency arose already in the early stages of NWP. One of the first great achievements was the Arakawa Jacobian for the barotropic vorticity dynamics. Seen with today‘s eyes the Arakawa Jacobian is the first application of discretised Nambu brackets in meteorological modeling. Global conservation of... circulation kinetic energy enstrophy

Historical background – Arakawa Jacobian: 

Historical background – Arakawa Jacobian The general barotropic dynamics may be written with the aid of the functional derivatives of the underlying conserved quantities and one yields wherein the Nambu bracket is defined. Setting F to z, , one obains directly the vorticity equation, since in this case The Nambu bracket is antisymmetric. The elements permutate circularly without change.

Historical background – Arakawa Jacobian: 

Historical background – Arakawa Jacobian The antisymmetric property is essential to prove the integral constraints That‘s why this property should be kept when discretising the Jacobian. Salmon (2005) proposed to keep it by explicitly demanding it by weighting each permutation with 1/3 For discretisation in a numerical model, the integral becomes a sum over all gridpoints. We consider a rectangular grid with indices i,j that represent an area element Wi,j: i,j i,j-1 i,j+1 i+1,j i-1,j J(a,b)(i,j)= ((a(i+1,j)-a(i-1,j))/2dx *(b(i,j+1)-b(i,j-1))/2dy) -((b(i+1,j)-b(i-1,j))/2dx *(a(i,j+1)-a(i,j-1))/2dy) W(i,j)=dx*dy

Historical background – Arakawa Jacobian: 

Historical background – Arakawa Jacobian For the integration of the vorticity equation at a grid point i,j: set F to zi,j sum over all elements the delta function picks out a factor 1 at the point i,j and zero elsewhere reorder the terms The result is the Arakawa Jacobian! Remark: Even if the area elements are not of the same size, this method will work. Then, weighting factors according to the relative size of the area elements will appear.

What we learned so far: 

What we learned so far The Arakawa Jacobian is able to avoid nonlinear instability by obeying the antisymmetric structure of the true Jacobian conserving quadratic conservation quantities that belong to the barotropic dynamics (kinetic energy, enstrophy) avoiding wrong scale interactions so that an erroneous energy cascade to small scales does not take place The Arakawa Jacobian imitates the barotropic vorticity dynamics to a very high degree. The reason for that success is the close accordance with the Nambu bracket formulation. Construction rule: Approximate the integral as a sum!

Slide9: 

Application of the construction rule to the NH equations We start with repeating the Nambu-brackets for the NH dynamics without forcing terms A lot of contributions of the cyclic permutations vanish, because the functional derivative vanishes the functional derivative is unity and vanishes when taking the gradient or the divergence That‘s why Salmon‘s (2005) rule is not necessary here. The mass and Q brackets are automatically 3fold antisymmetric.

Slide10: 

Application of the construction rule to the NH equations As a first step, we have to specify the computational grid. Without loss of generality we choose a C-grid in the horizontal and an L-grid in the vertical direction on a cubic grid. This cube has the reference volume for integration: V=dx*dy*dz The absolute vorticity and velocity components, and the mass points are not collocated! That drives us into subsequent truble because a lot of averaging is required. The constituting global quantities must be defined so that functional derivatives are defined at appropriate points:

Slide11: 

Application of the construction rule to the NH equationscc Derivation of the helicity bracket: The point is here, that the scalar triple product must vanish if two arguments are equal. Comparing the result with Sadourny‘s (1975) enstrophy conserving scheme, a similarity exists if double averaging is dropped. (But care: The Nambu brackets of the SW system are completely different!) Setting F to v and assuming incompressibility:

Slide12: 

Application of the construction rule to the NH equations For the derivation of the discrete M and Q brackets, it is sufficient to investigate a 1-dimensional problem as a prototype. The choice of the grid staggering leads to the definition of the energy functional and its derivatives on the grid. Note: The kinetic energy term is obtained by first squaring, and averaging subsequently. Defining the H functional differently by firstly averaging the velocity and then squaring it, leads also to a consistent formulation, but with more averaging in the continuity equation.

Slide13: 

Application of the construction rule to the NH equations With a similar approach we obtain the discretisation of the Q bracket with the following functional derivatives: Together the M and Q brackets give the following second order spatial differences: Note: The pressure gradient term appears necessarily as the gradient of the Exner pressure. In the thermodynamic equation, the mass flux is multiplied with and thus on the left hand side is not the same as on the right hand side.

Slide14: 

Temporal discretisation So far, all Nambu brackets are discretised. Spatial differencing is no longer questionable. But what for temporal discretisation? Nambu discretisation does not help in that context. But, if we can rely on spatial correctness, then we can search for temporal discretisations independently. For nonlinear quantities we find two possibilities to obtain conservation 1.) 2.) In general, the following analytic rule is not maintained by the discretisation: which is needed when in our case for , where e is the internal energy

Slide15: 

Temporal discretisation We consider the dynamics given only through mass bracket {F,M,H} (without F) and apply rule 1.) We obtain four possibilities for the temporal discretisation: Inserting the equations for v and r, and assuming that discrete spatial differentiation rules work as in the analytic case, the following temporal weights are found for the kinetic energy and the mass flux terms: Example: (next slide)

Slide16: 

v explicit v implicit v explicit, r as in impl. case v implicit, r as in explicit case Temporal discretisation

Slide17: 

Temporal discretisation Mass bracket time discretisation is solved Implication for Q-bracket: implicit (because of wave propagation) Helicity bracket: open problem But in all, it seems to become an implicit nonlinear problem. Questions: How to solve it? How to simplify the problem (e.g. take advantage of the geostrophic and hydrostatic balances via regularisation (Reich), this requires p and Q as prognostic variables) An in detail discussion with mathematicians is needed! WORKSHOP