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Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna 1 , Max Gunzburger 2 , Mauro Perego 1 , Konstantin Pieper 2 1 Sandia National Laboratories, 2 Florida State University Atlanta, March 2nd 2017 Sandia National Laboratories is a multi-mission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DEAC0494AL85000. Luca Bertagna (SNL) Ice-hydrology coupling Atlanta, March 2nd 2017 1 / 21 SAND2017-2256C

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Page 1: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Coupling Between Ice Sheets Movement and Subglacial Hydrology

Luca Bertagna1, Max Gunzburger2, Mauro Perego1, Konstantin Pieper2

1Sandia National Laboratories, 2Florida State University

Atlanta, March 2nd 2017

Sandia National Laboratories is a multi-mission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin

Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DEAC0494AL85000.

Luca Bertagna (SNL) Ice-hydrology coupling Atlanta, March 2nd 2017 1 / 21

SAND2017-2256C

Page 2: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Introduction

Introduction

Ice dynamics is complex:

nonlinear rheology

coupling with other components(atmosphere, ocean, land)

long time scales compared tosurrounding air/water

complex boundary conditions atbedrock interface

Our focus: boundary condition at the bedrock interface.

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Page 3: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Introduction

The governing equations

The velocity and pressure of the ice are widely assumed to satisfy the Stokes problem

−∇ ·[2µ(u)ε(u)

]+∇p = ρig

∇ · u = 0

with g = (0, 0,−g). The nonlinear viscosity is given by Glen’s law

µ(u) =1

2A−

1n (T )ε

1n−1

e

where, usually, n = 3, and

εij =1

2

(∂ui∂xj

+∂uj∂xi

), εe =

√1

2

∑ij

εijεij .

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Page 4: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Introduction

The First Order (FO) model

Under certain assumptions, a simplified model (Blatter 1995, Pattyn 2003) can bederived for the horizontal velocity u = (ux , uy ). In particular, the assumptions are

negligible bridging effects

vertical derivatives of horizontal velocities much larger than horizontal derivatives ofvertical velocity

Under these assumptions, εxz ' 12∂ux/∂z , and εyz ' 1

2∂uy/∂z , and the Stokes problem is

simplified to

−∇ · σ = −ρig∇s (1)

where s is the surface elevation, and

σ = 2µ(u)

[2εxx + εyy εxy εxz

εxy εxx + 2εyy εxz

]The constitutive law is still Glen’s law, where εe can be written as

2ε2e = ε2

xx + ε2yy + εxxεyy + ε2

xy + ε2xz + ε2

yz

Luca Bertagna (SNL) Ice-hydrology coupling Atlanta, March 2nd 2017 4 / 21

Page 5: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Introduction

Boundary conditions

For ice-air interface, we impose a stress-free condition

σn = 0.

For the ice-bedrock interface, we consider the ice-bedrock sliding condition

σn + βu = 0.

where β > 0 is a friction coefficient.

Obvious question: how to choose β?

Constant field β = β(x)

pro: linear BC

con: 2D field to estimate,problem dependent, does notevolve in time

Functional form β = β(u, p)

pro: potentially general (evolvesin time with u)

con: nonlinear BC, needs goodfunctional form guess

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Page 6: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Introduction

Sliding laws

One reasonable class of β is (Schoof 2004)

β = τc f (u)1

u(2)

where u = |u|. Some requirements on f :

continuous

non-decreasing

f (0) = 0

limu→∞ f (u) = 1

τc is the yield stress. Typically, τc = µfN, where N = ρigH + ρwgzb − pw is the effectivepressure and µf a scalar friction coefficient.

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Page 7: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Introduction

Examples:

f (u) = uqN r f (u) =

(u

u + λANn

)q

with q, r > 0, A as in Glen’s law, and λ geometric parameter for bed roughness (ratiobetween maximum bed bump length and maximum bed bump slope).

We consider the latter, which gives the boundary condition

σn + µfN

(u

u + λANn

)qu

u= 0 (3)

Note: Schoof suggests q = 1/n, with n as in Glen’s law.

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Page 8: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Introduction

Note: the field N (or, equivalently, pw ) is still unknown. Two possibilities:

Couple ice problem with a 2D subglacial hydrlogy model for N

Use a surrogate of field N.

For the latter, one can use the approximation (Bueler et al., 2008)

pw = αρigHh

hmax

where h is the thickness of water film between ice and bedrock.

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Page 9: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Introduction

Figure: Basal friction coefficient: 2D field estimation (left), parameter fitting (right).

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Page 10: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Distributed subglacial hydrology

Subglacial hydrology

In order to have a more realistic N, one needs to consider subglacial hydrology, and add a2D problem on Γb. For instance (Hewitt, Schoof, Werder, 2012)

∂h

∂t+∇ · q = ω

∂h

∂t=

hr − h

lr|ub| − AhNn

q = −k0h3

µw∇pw

pw = ρigH + ρwgzb − N := p0 − N

with h water layer thickness, k0 transmissivity constant, hr , lr bed bumps characteristiclength/height, ω water source, and n = 3 (as in Glen’s law).Assuming h known, one can derive an equation for N

−∇ ·(k0h

3

µw∇(N − p0)

)+ AhNn =

hr − h

lr|ub| − ω (4)

This is endowed with zero porewater pressure boundary condition, hence N = p0.

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Page 11: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Distributed subglacial hydrology

The well posedness of the hydrology follows from convex optimization. Define

J(N) =1

2

∫k0h

3

µw|∇N|2 +

1

4

∫AhN4 −

∫k0h

3

µw∇p0 · ∇N −

∫ (hr − h

lr|ub| − ω

)N.

Note: since dim(Γb) = 2, if we ask N ∈ H1(Γb), we have N ∈ Lq(Γb), ∀q ∈ [1,∞)(Sobolev embedding theorem), so all the integrals are well defined, provided h ∈ L∞(Γb),ω ∈ L2(Γb).

J is differentiable, so N is a weak solution iff ∂J(N) = 0. Since J is coercive and strictlyconvex, such point (the minimizer) exists and is unique, provided that h ≥ h0 > 0.Moreover,

‖N2 − N1‖H1 ≤ C(data)‖u1 − u2‖L

43

(5)

Note: With a little more work, using Stampacchia’s method, we can even show thatN ∈ L∞(Γb).

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Distributed subglacial hydrology

The coupled problem

The coupled ice-hydrology problem reads−∇ · σ = −ρig∇s

σn + µfN

(u

u + λANn

)qu

u= 0

−∇ ·(k0h

3

µw∇(N − p0)

)+ AhNn =

hr − h

lr|ub|+ ω

in Ω

on Γb

in Γb

(6)

existence of a solution?

uniqueness?

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Distributed subglacial hydrology

Existence/Uniqueness

Lemma

There are A,B,C > 0 independent on N, u such that

‖u‖W

1, 43 (Ω)≤ A

‖u1 − u2‖W

1, 43 (Ω)≤ B‖N1 − N2‖H1(Γb)

‖N2 − N1‖H1(Γb) ≤ C‖u1 − u2‖L

43 (Γb)

LetG : N → u(N), H : u→ N(u), T = G H.

The inequalities in the lemma ensure that

G is continuous

H is continuous

T is continuous (by composition)

v = T (u) is uniformly bounded

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Distributed subglacial hydrology

To use Schaefer’s theorem and prove existence, we need T to be compact. If

un u in W 1, 43 (Ω), letting vn, v be the trace of un, u on Γb, we have

un u in W 1, 43 (Ω)⇒ vn vn in W

14, 4

3 (Γb)⇒ vn → vn in L43 (Γb)

where the last implication follows from Sobolev’s embedding theorem. By virtue of the

previous lemma, we conclude T (un)→ T (u) in W 1, 43 (Ω). Therefore, using Schaefer’s

theorem,

Theorem

There is at least one solution to the coupled problem (6)

Combining the bounds in the previous lemma, one gets

‖T (u1)− T (u2)‖ ≤ γ‖u1 − u2‖.

If γ < 1, T is a contraction, so we have uniqueness.

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Distributed subglacial hydrology

Unsteady

We add the cavities equation to evolve h:

−∇ · σ = −ρig∇s

σn + µfN

(u

u + λANn

)qu

u= 0

−∇ ·(k0h

3

µw∇(N − p0)

)+ AhNn =

hr − h

lr|ub|+ ω

∂h

∂t=

hr − h

lr|ub| − AhNn

(7)

Using Stampacchia’s method, we can show that u and N are bounded in L∞. In particular

‖u‖∞ ≤ C1, ‖N‖∞ ≤ C2|h|h3

0

‖u‖∞.

so that |∂th| ≤ Q(|h|, h0), with Q rational function. If h0 + ε < h(x , 0) < M − ε, then,for t ∈ (0, δ) we have |h| ∈ (h0,M).

Question: how big can δ be?

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Distributed subglacial hydrology

A model with basal melting

We may consider also water formation due to melting:

m =1

L(G − τb · u) (8)

with L latent heat, G (net) geothermal flux and τb ice basal stress. Using sliding law,τb = −βu. The hydrology problem becomes

∂h

∂t+∇ · q = ω + 1

ρw L(G − β|ub|2)

∂h

∂t= 1

ρiL(G − β|ub|2) +

hr − h

lr|ub| − AhNn

q = −k0h3

µw∇pw = −k0h

3

µw∇(p0 − N)

The equation for N becomes

−∇ ·(k0h

3

µw∇(N − p0)

)+ AhNn − ρ

Lβ(N)|u|2 =

ρ

LG +

hr − h

lr|ub| − ω (9)

where ρ = 1/ρi − 1/ρw .

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Page 17: Coupling Between Ice Sheets Movement and Subglacial Hydrology · 2017-07-25 · Coupling Between Ice Sheets Movement and Subglacial Hydrology Luca Bertagna1, Max Gunzburger2, Mauro

Distributed subglacial hydrology

We can define the functional

J(N) =1

2

∫k0h

3

µw|∇N|2dx +

1

4

∫AhN4dx − ρ

L

∫ (∫ N

0

β(s)ds

)|u|2dx

−∫

k0h3

µw∇p0 · ∇Ndx −

∫ (ρ

LG +

hr − h

lr|ub| − ω

)Ndx .

Problem: J is not convex. The non convex term is continuous, and∣∣∣∣∫ (∫ N

0

β(s)ds

)|u|2dx

∣∣∣∣ ≤ ‖u‖L 43 (Γb)‖N‖2

L8(Γb) ≤ C‖u‖L

43 (Γb)‖N‖2

L2(Γb).

Therefore, J is still coercive, so that

Existence

The hydrology problem (9) has at least one solution

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Distributed subglacial hydrology

The lack of convexity does not allow to immediately get uniqueness. However, we canshow that any two solutions N1,N2 must satisfy

Ah0‖N2 − N1‖2L4(Γb) ≤

ρ

L‖u‖

L43 (Γb)

. (10)

Convexity is recovered for small data. In particular, we still have convexity and continuityof solution w.r.t. data if

k0θ3

µw≥ (1 + c2

p )C8,4ρ

L‖u‖

L43 (Γb)

which however is too strict of an assumption for real data.

Without well posedness (in particular, continuity of N w.r.t u), we cannot prove existenceof solutions for the coupled problem.

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Distributed subglacial hydrology

Steady hydrology

Recall the hydrology problem

∂h

∂t+∇ · q = ω +

m

ρw∂h

∂t=

m

ρi+

hr − h

lr|ub| − AhNn

q = −k0h3

µw∇pw

pw = ρigH − N := pi − N

If we assume ∂h∂t

= 0, then we have

h =hr/lr + G/(Lρi ) + β(N)|u|2/(Lρi )

|ub|/lr + ANn

And the equation for N becomes

−∇ ·(k0h(N)3

µw∇(N − pi )

)+β(N)|u|2

ρwL= −ω − G

ρwL(11)

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Distributed subglacial hydrology

pro: no need to assume h known (although we assumed ∂h/∂t = 0)

con: diffusion coefficient not guaranteed to be positive

Idea: add box constraint for N (Schoof, Hewitt, Werder, 2012). Physical argumentssuggest 0 ≤ N ≤ p0. We then turn (11) into a variational inequality. Wherever N = 0 orN = p0, (11) is regarded as an equation for h (not interesting for ice coupling).

Other issue: h = h(N, |ub|) is singular if N → 0 and |ub| → 0. However, if we assume|ub| ≥ um > 0, and we use box constraints for N, then the steady hydrology problem iswell posed.

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Conclusions

Looking ahead

sliding laws (e.g., regularized Coulomb) are appealing since they can evolve with thesolution

need to identify the parameters in the sliding law

regularized Coulomb requires to model effective pressure N

proved well posedness of quasi-static coupled FO-hydrology problem (at least forsmall data)

todo extend well-posedness do time-dependent. Under what conditions we have wellposedness for arbitrary large times?

todo Numerically solve the coupled FO-hydrology problem (Albany, ongoing).

doubt Is the case with melting intrinsically ill-posed?

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