Geophysical Ice Flows: Analytical and Numerical Approaches

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1 Geophysical Ice Flows: Analytical and Numerical Approaches Will Mitchell University of Alaska - Fairbanks July 23, 2012 Supported by NASA grant NNX09AJ38G

2 Ice: an awesome problem...velocity, pressure, temperature, free surface all evolve

3 Outline I. Introduction to viscous fluids II. Exact solutions III. Finite element solutions

4 Stress: force per unit area A tornado sucks up a penny. At any time: The fluid into which n points exerts a force on the penny Force / area = stress The stress vector is a linear function of n In a Cartesian system: stress = σ n σ is the Cauchy stress tensor Quiz: Suppose there is no p 0 such that σ n = pn. Physical interpretation?

5 Decomposition of stress In a fluid at rest, σ n = pn, so σ = pi In general, choose p = Trace(σ)/d, so σ = pi + τ where τ has zero trace.... this defines pressure p and deviatoric stress τ

6 Strain rate let u be a velocity field the gradient of a vector is the tensor ( u) ij = u j x i define Du = 1 2 ( u + ut ) 2 u 1 u 2 + u 1 in 2D: Du = 1 2 x 1 x 1 x 2 u 1 + u 2 2 u 2 x 2 x 1 x 2 Du is the strain rate tensor. True or False: Since Du is a derivative of velocity, it measures acceleration.

7 Constitutive Laws: Newtonian How does a fluid respond to a given stress? For Newtonian fluids (e.g. water) a linear law: τ = 2µDu The proportionality constant µ is the viscosity.

8 Constitutive Laws: Glen s For glacier ice, a nonlinear law. define assume τ = the law is either of Tr(τ T τ) and Du = 2 Tr(DuT Du) Du = A τ n τ = (A τ n 1 ) 1 Du τ = A 1/n Du (1 n)/n Du A is the ice softness, n 3 is Glen s exponent.

9 Stokes Equation What forces act on a blob occupying a region Ω within a fluid? body force, gravity: Ω ρg force exerted by surrounding fluid: Ω σ n = Ω σ Force = rate of change of momentum ρg + σ = Ω t = In glaciers, Fr= D Dt ρu : p < so Ω ρg + σ = 0. Ω ρu D Dt ρu.

10 Incompressible Stokes System (two versions) { ρg + σ = 0 u = 0 σ = τ + p = 2µ ɛ p = µ ( u) + µ ( u T ) p ( u) = ( u T ) = u j = ( u) = 0 x j x i x i u i = u x j x j { µ u + p = ρg u = 0

11 The Biharmonic Equation for 2D, incompressible flow: u = (u, 0, w) and u = 0 there is a streamfunction ψ such that ψ z = u, ψ x = w. take the curl of the Stokes eqn [ ] µ u + p = ρg to get the biharmonic equation ψ xxxx + 2ψ xxzz + ψ zzzz = 0 or ψ = 0. Quiz: give an example of a function solving the biharmonic eqn.

12 Ice: an awesome problem...velocity, pressure, temperature, free surface all evolve

13 Slab-on-a-slope: a tractable problem g = (g 1, g 2 ) = ρg ( sin(α), cos(α) )...no evolution

14 Stokes bvp find a velocity u = (u, w) and pressure p such that p + µ u = g on Ω u = 0 on Ω u(0, z) u(l, z) = 0 for all z u x (0, z) u x (L, z) = 0 for all z u = f on {z = 0} w = 0 on {z = 0} w x + u z = 0 on {z = H} 2w zx (u xx + u zz ) = g 1 /µ on {z = H} where f (x) = a 0 + n=1 a n sin(λ n x) + b n cos(λ n x).

15 biharmonic bvp find a streamfunction ψ such that ψ = 0 on Ω ψ z (0, z) ψ z (L, z) = 0 for all z ψ xz (0, z) ψ xz (L, z) = 0 for all z ψ x (0, z) ψ x (L, z) = 0 for all z ψ xx (0, z) ψ xx (L, z) = 0 for all z ψ(x, 0) = 0 for all x ψ z (x, 0) = f for all x ψ zz (x, H) ψ xx (x, H) = 0 for all x 3ψ xxz (x, H) + ψ zzz (x, H) = g 1 /µ for all x. (1)

16 biharmonic bvp, subproblem: f = 0 ψ(x, z) = g 1H 2µ z2 g 1 6µ z3 g 1 H u(x, z) = µ z g 1 2µ z2 w(x, z) = 0...this is Newtonian laminar flow, a well known solution.

17 biharmonic bvp, subproblem: f 0 strategy: separate variables: ψ(x, z) = X (x)z(z) periodicity: take X (x) = sin(λx) + cos(λx) for λ = 2πn L the biharmonic eqn reduces to an ODE: for λ > 0 this gives 0 = 2 (XZ) = X [λ 4 Z 2λ 2 Z + Z (iv)] Z(z) = a sinh(λz) + b cosh(λz) + cz sinh(λz) + dz cosh(λz) homogeneous bcs determine b, c, d in terms of a weighted sum gets the nonzero condition

18 Exact Solutions Horizontal Component of Velocity: u(x, z) = a 0 + g 1H µ z g 1 2µ z2 + λ n H 2 (a n sin(λ n x) + b n cos(λ n x)) λ 2 nh 2 + cosh 2 Z n(z) (λ n H) n=1 where Z n(z) = 1 ( ) H cosh(λ nh) sinh(λ n (z H)) + λ n z cosh(λ n (z H)) + cosh(λ nh) λ n H sinh(λ n H) ( λ n H 2 cosh(λ n (z H)) ) + λ n z sinh(λ n (z H)) + λ n cosh(λ n z).

19 Exact Solutions Vertical Component of Velocity: w(x, z) = where n=1 λ 2 nh 2 λ 2 nh 2 + cosh 2 (λ n H) (b n sin(λ n x) a n cos(λ n x))z n (z) Z n (z) = sinh(λ n z) 1 H cosh(λ nh)z sinh(λ n (z H)) ( cosh(λn H) + λ n H 2 sinh(λ ) nh) z cosh(λ n (z H)) H

20 Exact Solutions Pressure: p(x, z) = g 2 z g 2 H λ 3 + 2µ nh(a n cos(λ n x) b n sin(λ n x)) λ 2 nh 2 + cosh 2 (λ n H) n=1 [ sinh(λ n z) cosh (λ nh) cosh (λ n (z H)) λ n H ].... this is new.

22 The finite element method numerical approximation of p and u requires a mesh of the domain: leads to a system of linear equations Ax = b

23 Variational Formulation: incompressibility Incompressibility: u = 0. We seek a u H 1 (Ω) such that for all q L 2 (Ω), we have q u = 0. u is a trial function; q is a test function. Ω

24 Variational Formulation: Stokes Put σ = τ pi in the Stokes equation: 0 = τ p + ρg Dot with v H 1 and integrate over Ω. Integration by parts gives τ : v p v n σ v = ρ g v. Ω Ω Ω Ω More manipulation gives 1 ( ) ( ) 2 µ u T + u : v + v T Ω Ω p v = ρ g v. Ω

25 Variational Formulation Find (u, p) H 1 E L2 such that for all (v, q) H 1 E 0 L 2 we have 1 ( ) ( ) 2 µ u T + u : v + v T p v+ q u = ρ g v. Ω Ω Ω Ω Still a continuous problem: (u, p) satisfy many conditions. Make a discrete problem using finite-dimensional spaces.

26 Pressure Approximation space Continuous functions that are linear on each triangle: a 16-dimensional space with a convenient basis.

27 Velocity Approximation space Continuous functions that are quadratic on each triangle: a 49-dimensional space (per component) with a convenient basis.

29 Implementation I...based on [Jar08] but with an important difference 1 from d o l f i n i mport 2 #Set domain p a r a m e t e r s and p h y s i c a l c o n s t a n t s 3 Le, He = 4e3, 5 e2 #l e n g t h, h e i g h t (m) 4 a l p h a = 1 p i /180 #s l o p e a n g l e ( r a d i a n s ) 5 rho, g = 917, #d e n s i t y ( kg m 3), g r a v i t y (m sec 2) 6 mu = 1 e14 #v i s c o s i t y ( Pa s e c ) 7 G = Constant ( ( s i n ( a l p h a ) g rho, cos ( a l p h a ) g rho ) ) 8 #D e f i n e a mesh and some f u n c t i o n s p a c e s 9 mesh = R e c t a n g l e ( 0, 0, Le, He, 3, 3 ) 10 V = V e c t o r F u n c t i o n S p a c e ( mesh, CG, 2) #pw q u a d r a t i c 11 Q = FunctionSpace ( mesh, CG, 1) #pw l i n e a r 12 W = V Q #p r o d u c t space 13 D e f i n e the D i r i c h l e t c o n d i t i o n at the base 14 d e f LowerBoundary ( x, on boundary ) : 15 r e t u r n x [ 1 ] < DOLFIN EPS and on boundary 16 S l i p R a t e = E x p r e s s i o n ( ( (3+1.7 s i n (2 p i/%s x [ 0 ] ) ) \ 17 / %Le, 0. 0 ) ) 18 bcd = D i r i c h l e t B C (W. sub ( 0 ), S l i p R a t e, LowerBoundary )

30 Implementation II 19 #D e f i n e the p e r i o d i c c o n d i t i o n on the l a t e r a l s i d e s 20 c l a s s P e r i o d i c B o u n d a r y x ( SubDomain ) : 21 d e f i n s i d e ( s e l f, x, on boundary ) : 22 r e t u r n x [ 0 ] == 0 and on boundary 23 d e f map( s e l f, x, y ) : 24 y [ 0 ] = x [ 0 ] Le 25 y [ 1 ] = x [ 1 ] 26 pbc x = P e r i o d i c B o u n d a r y x ( ) 27 bcp = P eriodicbc (W. sub ( 0 ), pbc x ) 28 D e f i n e the v a r i a t i o n a l problem : a ( u, v ) = L ( v ) 29 ( v i, q i ) = T e s t F u n c t i o n s (W) 30 ( u i, p i ) = T r i a l F u n c t i o n s (W) 31 a = ( 0. 5 mu i n n e r ( grad ( v i )+grad ( v i ). T, grad ( u i ) \ 32 +grad ( u i ).T) d i v ( v i ) p i + q i d i v ( u i ) ) dx 33 L = i n n e r ( v i, G) dx 34 Matrix assembly and s o l u t i o n 35 U = F u n c t i o n (W) 36 s o l v e ( a==l, U, [ bcd, bcp ] ) 37 S p l i t the mixed s o l u t i o n to r e c o v e r u and p 38 ( u, p ) = U. s p l i t ( )

31 Convergence to Exact Solutions Errors in FEM velocity and pressure plotted against maximum element diameter, together with convergence rates m.

32 References I [Ach90] D. J. Acheson. Elementary Fluid Dynamics. Oxford University Press, New York, [AG99] Cleve Ashcraft and Roger Grimes. SPOOLES: an object-oriented sparse matrix library. In Proceedings of the Ninth SIAM Conference on Parallel Processing for Scientific Computing 1999 (San Antonio, TX), page 10, Philadelphia, PA, SIAM. [Bat00] G. K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, UK, [BR85] M. J. Balise and C. F. Raymond. Transfer of basal sliding variations to the surface of a linearly viscous glacier. Journal of Glaciology, 31(109), [CP10] K. M. Cuffey and W. S. B. Paterson. The Physics of Glaciers. Academic Press, Amsterdam, Boston, 4th edition, 2010.

33 References II [DH03] J. Donea and A. Huerta. Finite Element Methods for Flow Problems. Wiley, New York, 1st edition, [DM05] L. Debnath and P. Mikusinski. Introduction to Hilbert Spaces with Applications, Third Edition. Academic Press, 3rd edition, [Ern04] A. Ern. Theory and Practice of Finite Elements. Springer, Berlin, Heidelberg, [ESW05] H. C. Elman, D. J. Silvester, and A. J. Wathen. Finite Elements And Fast Iterative Solvers - With Applications in Incompressible Fluid Dynamics. Oxford University Press, New York, [GB09] R. Greve and H. Blatter. Dynamics of Ice Sheets and Glaciers (Advances in Geophysical and Environmental Mechanics and Mathematics). Springer, 1st edition. edition,

34 References III [Goo82] A. M. Goodbody. Cartesian tensors - with applications to mechanics, fluid mechanics and elasticity. E. Horwood, Chichester, [Jar08] A. H. Jarosch. Icetools: A full stokes finite element model for glaciers. Computers & Geosciences, 34: , [Jóh92] Tómas Jóhannesson. Landscape of Temperate Ice Caps. PhD thesis, University of Washington, [LJG + 12] W. Leng, L. Ju, M. Gunzberger, S. Price, and T. Ringler. A parallel high-order accurate finite element nonlinear stokes ice sheet model and benchmark experiments. Journal of Geophysical Research, 117, 2012.

35 References IV [LMW12] A. Logg, K. Mardal, and G. Wells, editors. Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book (Lecture Notes in Computational Science and Engineering). Springer, Berlin, Heidelberg, [Pir89] O. Pironneau. Finite element methods for fluids. Wiley, New York, [QV94] A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations. Springer, Berlin, Heidelberg, 1st edition, [TS63] A. N. Tikhonov and A. A. Samarskii. Equations of Mathematical Physics. Pergamon Press, Oxford, Translated by A.R.M. Robson and P. Basu and edited by D. M. Brink. [Wor09] G. Worster. Understanding Fluid Flow. Cambridge University Press, Cambridge, 1st edition, 2009.

Dynamics of Glaciers

Dynamics of Glaciers McCarthy Summer School 01 Andy Aschwanden Arctic Region Supercomputing Center University of Alaska Fairbanks, USA June 01 Note: This script is largely based on the Physics of Glaciers