Using the Jacobian- free Newton- Krylov method to solve the sea- ice momentum equa<on

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1 Using the Jacobian- free Newton- Krylov method to solve the sea- ice momentum equa<on Jan Sedláček1 and Jean- François Lemieux2 1ETH Zürich; 2Environment Canada

3 The sea- ice momentum equa<on AssumpFon: Long enough Fme step i hf k u i + a( a w )+r i hgrh d =0

4 The sea- ice momentum equa<on AssumpFon: Long enough Fme step Coriolis force rheology i hf k u i + a( a w )+r i hgrh d =0 wind stress water drag sea surface Flt

5 The sea- ice momentum equa<on Note: wind speed generally much larger than sea- ice speed non- linear terms rheology i hf k u i + a( a w )+r i hgrh d =0 water drag

6 Introduc<on to rheology 1- D Example: stress < crifcal value elasfc material deformafons reversible stress = crifcal value plasfc material deformafons not reversible stresses cannot be larger than crifcal value

7 Introduc<on to rheology 1- D Example: stress < crifcal value elasfc material deformafons reversible stress = crifcal value plasfc material deformafons not reversible stresses cannot be larger than crifcal value BUT: storage and numerically expensive sea ice modeled as viscous- plasfc material (very viscous / creep flow)

8 Yield curve cavitafng fluid q ice cream cone q -p -p granular Hibler model; EVP q q -p -p

9 Cons<tu<ve law (i.e., rela<on between stress and strain rate) and normal flow rule σ ij =2η ɛ ij +[ζ η] ɛ kk δ ij Pδ ij /2

10 Cons<tu<ve law (i.e., rela<on between stress and strain rate) and normal flow rule ( ) ice concentrafon ice strength parameter P = P hexp[ C(1 A)] ice thickness σ ij =2η ɛ ij +[ζ η] ɛ kk δ ij Pδ ij /2 ice concentrafon parameter

11 Cons<tu<ve law (i.e., rela<on between stress and strain rate) and normal flow rule strain rates ɛ 11 = u x,, ɛ 22 = v y, ɛ 12 = 1 ( u 2 y + v ) x ɛ kk = ɛ 11 + ɛ 22 σ ij =2η ɛ ij +[ζ η] ɛ kk δ ij Pδ ij /2 ( ) ice concentrafon ice strength parameter P = P hexp[ C(1 A)] ice thickness ice concentrafon parameter Kronecker Delta

12 Cons<tu<ve law (i.e., rela<on between stress and strain rate) and normal flow rule strain rates ɛ 11 = u x,, ɛ 22 = v y, ɛ 12 = 1 ( u 2 y + v ) x ( ɛ kk = ɛ 11 + ɛ 22 ( ( ) σ ij =2η ɛ ij +[ζ η] ɛ kk δ ij Pδ ij /2 ( ) ice concentrafon ice strength parameter P = P hexp[ C(1 A)] ice thickness ice concentrafon parameter bulk viscosity ζ = P 2 shear viscosity η = ζe 2 Kronecker Delta =[( ɛ ɛ2 22 )(1 + e 2 )+4e 2 ɛ ɛ 11 ɛ 22 (1 e 2 )] 1 2

13 Cons<tu<ve law (i.e., rela<on between stress and strain rate) and normal flow rule strain rates ɛ 11 = u x,, ɛ 22 = v y, ɛ 12 = 1 ( u 2 y + v ) x ( ɛ kk = ɛ 11 + ɛ 22 ( ( ) σ ij =2η ɛ ij +[ζ η] ɛ kk δ ij Pδ ij /2 ( ) ice concentrafon ice strength parameter P = P hexp[ C(1 A)] ice thickness bulk viscosity shear viscosity ζ = P 2 η = ζe 2 =[( ɛ ɛ2 22 )(1 + e 2 )+4e 2 ɛ ɛ ( 11 ɛ 22 (1 e 2 ) )] 1 2 ice concentrafon parameter Kronecker Delta using: ζ = min ( ) P 2, ζ max η = min ( ) P 2e 2, η max

14 and now graphically failure in shear pure compression viscous flow outside yield curve: not physical

15 Brief history 1979: viscous- plasfc sea ice model outer loop successive over- relaxafon, linear relaxafon 1992: cavitafng sea- ice model only pressure 1997: elasfc- viscous- plasfc sea- ice model arfficial elasfc term explicit conjuate gradient

16 Difference SOR vs. GMRES only for the linear solver

17 Outer loop convergence KineFc energy

18 Outer loop convergence KineFc energy OL iterafons 2 OL iterafons 10 OL iterafons 40 OL iterafons

19 Removing discon<nuity ( ) ζ = min ( ) P 2, ζ max ( P ζ = ζ max tanh 2 ζ max ) + ζ min

20 Full Jacobian- free Newton- Krylov

21 Full Jacobian- free Newton- Krylov

22 The final result simulafon with 10 km resolufon sea- ice velocity shear deformafon

23 References Lemieux, J.- F., B. Tremblay, J. Sedláček, P. Tupper, S. Thomas, D. Huard, and J.- P. Auclair (2010), Improving the numerical convergence of viscous- plasfc sea ice models with the Jacobian- free Newton Krylov method, J. Comput. Phys., 229(8), , doi: /j.jcp Lemieux, J.- F., and B. Tremblay (2009), Numerical convergence of viscous- plasfc sea ice models, J. Geophys. Res., 114(C5), doi: /2008jc Lemieux, J.- F., B. Tremblay, S. Thomas, J. Sedláček, and L. A. Mysak (2008), Using the precondifoned Generalized Minimum RESidual (GMRES) method to solve the sea- ice momentum equafon, J. Geophys. Res., 113(C10), doi: /2007jc Lemieux, J.- F., D. A. Knoll, B. Tremblay, D. M. Holland, and M. Losch (2012), A comparison of the Jacobian- free Newton Krylov method and the EVP model for solving the sea ice momentum equafon with a viscous- plasfc formulafon: A serial algorithm study, J. Comput. Phys., 231(17), , doi: /j.jcp

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