Combining the VP and EVP models to solve the sea ice momentum equation Jean-François Lemieux Dana Knoll, David Holland, Elizabeth Hunke AOMIP workshop 20 October 2010

Pros and cons of the VP and EVP models VP solution! Implicit approach (no stability issue) Issues with parallelization Slow numerical convergence VP EVP Naturally suited for parallelization Easy to implement Undamped elastic waves

The goal: combine the VP and EVP models VP+EVP

The momentum equation where We want to solve this implicitly at time t:

The nonlinear system of equations

We want to solve Picard JFNK do k=1, kmax Solve if stop enddo

Comparison of the JFNK and Picard solvers 40 km. Picard JFNK Lemieux et al. 2010

The preconditioned FGMRES method where The solution is approximated in the subspace: where

The preconditioning operator Should be efficient Given a Krylov vector y, we get the vector z as

Jacobi as a preconditioner Recall we want to solve: We can use A for the preconditioning step: do p=1, 10 enddo

EVP as a preconditioner Recall that with Jacobi we have: With the EVP, we time-step the following: do p=1, 10 enddo

Computational efficiency

Is our treatment of the inertial term inconsistent? With Jacobi: With EVP:

EVP as a preconditioner (2nd try) Recall that with Jacobi we have: With the EVP, we now time-step the following: do p=1, Np enddo

Conclusions Possible to recast EVP as a preconditioner EVP as a preconditioner is however less efficient than a Jacobi iteration We think there is an inconsistency in our mathematical derivation Please call 212-998-3331 if you know what’s wrong!!! Should I remove the 1st conclusion.

Thank you! With the tanh. Quadratic in the vicinity of the solution.

With the tanh. Quadratic in the vicinity of the solution.

Viscous-plastic formulation q -P/2 . ridge lead Advantage of yield curve: need two stresses to represent the failure. Eps are the strain rates – function of the velocity. Very little tensile strength. Can resist large stresses before yielding in compression. Hence, for a slightly divergent field, the stress is close to zero while it can be very high for a slightly convergent field. Stresses should be on the yield curve. Hibler, 1979

The model Dynamic/thermodynamic VP rheology, ellipse (Hibler, 1979) Domain: Arctic, North Atlantic and CAA Resolutions: 10, 20, 40 or 80 km (C-grid) Coupled to a slab ocean model Forcing: - geostrophic winds NCEP/NCAR (6h) - climatological currents Advection of momentum is neglected. Thermo model also but we are here interested in the dynamic.

Computational gain of JFNK over the standard solver Quality of approx. solution Cpu gains are multiplicative

Typical shear deformation field (10 km)

Failures of the two solvers

In matrix form… Newton method Standard method (iteration k) F(u)=A(u)u-b=0 Standard method (iteration k) Newton method (iteration k) A(uk-1)uk = b J(uk-1)duk = -F(uk-1) uk = uk-1 - A-1(uk-1)F(uk-1) uk = uk-1 - J-1(uk-1)F(uk-1) uk = uk-1 - (A(uk-1)+G(uk-1))-1F(uk-1)

Failure of the line search method