Parameter sensitivity analysis of dynamic icesheet models ∗

Cheng Gong, Per LötstedtDepartment of Information Technology.

Uppsala University, SE-751 05 Uppsala, Sweden

January 27, 2020

Abstract

The velocity field and the height at the surface of a dynamic ice sheetare observed. The ice sheets are modeled by the full Stokes equations andshallow shelf/shelfy stream approximations. Time dependence is introducedby a kinematic free surface equation which updates the surface elevation us-ing the velocity solution. The sensitivity of the observed quantities at theice surface to parameters in the models, for example the basal topographyand friction coefficients, is analyzed by first deriving the time dependent ad-joint equations. Using the adjoint solutions, the effect of a perturbation in aparameter is obtained showing the importance of including the time depen-dence, in particular when the height is observed. The adjoint equations aresolved analytically and numerically and the sensitivity of the desired param-eters is determined in several examples in two dimensions. A closed formof the analytical solutions to the adjoint equations is given for a two dimen-sional grounding line migration problem in steady state under the shallowshelf approximation.

∗This work was funded by FORMAS 2017-00665

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1 Introduction

Large ice sheets cover Antarctica and Greenland, and glaciers are found in moun-tainous regions all over the world. The ice moves slowly to lower elevations onthe bedrock and it floats if it reaches the ocean. There is a need to predict whathappens to the ice due to the current and future climate change.

In computational models for the flow of ice in glaciers and continental ice sheets,it is necessary to choose models and equations and to supply parameters for thesliding between the ice base and the bedrock. Direct observations of the basalconditions by drilling holes in the ice are not feasible except for a few locations.Instead, the data for the sliding models are inferred from observations of the sur-face elevation and velocity of the ice from aircraft and satellites, see (Minchewet al. 2016 Sergienko and Hindmarsh 2013). How to do this is an important ques-tion because the basal sliding is a key uncertainty in the assessment of the futuresea level rise due to melting ice (Ritz et al. 2015).

We address here a related question how to determine the sensitivity of the obser-vations at the surface to the conditions at the base. The same solution technique isapplicable in sensitivity analysis, uncertainty quantification, and the inverse prob-lem. We extend the adjoint (or control) method by including the time evolution ofthe ice and its thickness.

The flow of ice is well modeled by the full Stokes (FS) equations, see (Greve andBlatter 2009). They form a system of partial differential equations (PDEs) for thestress and pressure in the ice with a nonlinear viscosity coefficient. The domainof the ice is confined by an upper surface and a base either resting on the bedrockor floating on sea water. The boundary conditions at the upper surface of the iceand at the floating part are well defined. For the ice in contact with the bedrock, afriction model with parameters determines the sliding force. The sliding dependson the topography at the ice base, the friction between the ice and the bedrock, andthe meltwater under the ice. The upper, free boundary of the ice and the interfacebetween the ice and the water are advected by equations for the height and theinterface.

The computational effort to solve the FS equations is quite large and there is oftena need for approximations. The FS equations are simplified by integrating inthe depth of the ice in the shallow shelf (or shelfy stream) approximation (SSA)(Greve and Blatter 2009 MacAyeal 1989). The spatial dimension of the problem

2

is reduced by one with SSA compared to FS. The pressure is also decoupled fromthe stress in the system.

The friction model is often of Weertman type (Weertman 1957) but other modelsare also considered (Tsai et al. 2015). The model for the relation between thesliding speed and the pressure and the friction at the bed is discussed in (Minchewet al. 2019 Stearn and van der Veen 2018). It is not clear how to formulate arelation that is generally applicable. When the parameters in the sliding model inthe forward equation are unknown in numerical simulations but data are availablesuch as the surface velocity and elevation of the ice, an inverse problem is solvedby minimizing the distance between the observations and the predictions of thenumerical PDE model with the parameters. The gradient of the objective functionfor the minimization is computed by solving an adjoint equation as in (Brondexet al. 2019 Yu et al. 2018). With a fixed thickness of the ice, the adjoint of the FSequations is derived in (Petra et al. 2012) and for SSA with a frozen viscosity in(MacAyeal 1993).

The basal parameters are estimated from uncertain observational data at the sur-face in (Gillet-Chaulet et al. 2016 Isaac et al. 2015) and initial data for ice sheetsimulations are found in (Perego et al. 2014) using the same technique as in (Petraet al. 2012). The sensitivity of the ice flow to the basal conditions is investigatedin (Heimbach and Losch 2012) with the adjoint solution. Part of the drag at thebase may be due to the resolution of the topography. The geometry at the ice baseis inferred by an inversion method in (van Pelt et al. 2013). The difficulty to sepa-rate the topography from the sliding properties at the base in the inversion is alsoaddressed in (Kyrke-Smith et al. 2018 Thorsteinsson et al. 2003). Considerabledifferences in the friction coefficient in the FS and SSA models are found afterinversion in (Schannwell et al. 2019). By linearization of the model equations, atransfer operator is derived in (Gudmundsson 2003; 2008) and it is shown in (Gud-mundsson and Raymond 2008) how the topography and the friction coefficientsare affected by measurement errors at the surface.

It is noted in (Vallot et al. 2017) that the friction coefficient varies in space andtime. The time scales of the variations are diurnal (Schoof 2010), seasonal (Soleet al. 2011), and decennial (Jay-Allemand et al. 2011). The time it takes for thesurface to respond to sudden changes in basal conditions are determined analyti-cally and numerically in (Gudmundsson 2008) with SSA and FS. Time dependentdata are used in (Goldberg et al. 2015) to infer time independent parameters inan ice model. By applying an inverse method in (Jay-Allemand et al. 2011), the

3

authors observe that a friction parameter varies several orders of magnitude in adecade at the bottom of a glacier. These papers indicate that it is not sufficientto infer the friction parameters from the time-independent adjoint to the FS stressequation but to include also the time dependent height advection equation in theinversion.

In this paper, we study how perturbations in the sliding conditions and the topog-raphy at the base of the ice affect observations of the height of the ice and thevelocity at the surface for the FS and SSA models. The friction law is due toWeertman (Weertman 1957). The sensitivity to perturbations of the parameters inFS and SSA is determined by solving an adjoint problem of the same kind as forinverse problems with a fixed ice thickness in (MacAyeal 1993 Petra et al. 2012).The difference here is that the adjoint of the time dependent advection equationfor the height is also solved allowing height observations and the influence isevaluated of the adjoint height in the adjoint solution. The adjoint equations fol-low from the Lagrangian of the forward equations after partial integration. Therelation is derived between the inverse problem to infer basal parameters fromsurface data and the sensitivity problem to observe changes in the velocity or theheight at the surface from perturbations of the basal parameters. The stationaryadjoint equations are solved analytically in two dimensions under simplifying butreasonable assumptions making the dependence of the parameters explicit. Thetime dependent adjoint equations are solved numerically for sensitivity analysisof perturbations varying in time. Conclusions are drawn in the final section. Ex-tensive numerical computations are reported in a companion paper (Cheng andLötstedt 2019) illustrating the accuracy of the analytical solutions and the adjointapproach.

2 Ice models

The full Stokes equations in glaciology are system of nonlinear PDEs which con-sists of conservation laws of mass, momentum and energy (Greve and Blatter2009). The nonlinearity is due to the viscosity of the ice according to Glen’s flowlaw (Glen 1955). The ice is assumed to be isothermal and obeys a friction lawat the contact surface between the ice and the bedrock. The upper surface of theice is a moving boundary and satisfies an advection equation for the height of theice. The aspect ratio of a continental ice-sheet and a floating ice-shelf, i.e. the

4

thickness scale V divided by the length scale L, is low V/L∼ 10−2 to 10−3. Thisscale difference is used in the SSA assumption to simplify the FS equations, see(Greve and Blatter 2009).

2.1 Full Stokes equation

Ice flux

h(x, y, t)

b(x, y)xGL

H

z

x

y

a∗∗∗∗

∗∗∗∗

∗∗∗

∗∗∗∗∗

∗∗

∗

∗

Γu ΓdγGL

Γb

Γw

y

xz

Figure 1: A schematic view of an ice sheet in the x− z (left panel) and x−y (rightpanel) plane.

Let u1,u2, and u3 be the velocity components of u = (u1,u2,u3)T in the x,y and zdirections and x = (x,y,z)T in three dimensions. Vectors and matrices are writtenin bold characters. The horizontal plane is spanned by the x and y coordinates andz is the coordinate in the vertical direction. Let the subscript x,y,z or t denote aderivative with respect to the variable. The height of the upper surface is h(x,y, t),the z coordinates of the bedrock and the floating interface are b(x,y) and zb(x,y, t),and the ice thickness is H = h−b, as shown in Fig. 1.The strain rate D and the viscosity η are given by

D(u) = 12(∇u+∇uT ), η(u) = 12A

− 1n (trD2(u))ν , ν = 1−n2n , (1)

where trD2 is the trace of D2. The rate factor A in (1) depends on the temperatureand Glen’s flow law determines n > 0, here taken to be n = 3. The stress tensor is

σ(u, p) = 2ηD(u)− Ip, (2)

where p is the pressure and I is the identity matrix. The domain occupied by theice is Ω with boundary Γ whose outward pointing normal is n. If x ∈ Ω then(x,y) ∈ ω in the x− y plane. The upper boundary is Γs at the ice surface. The

5

vertical, lateral boundary has an upstream part Γu where n · u ≤ 0 and a down-stream part Γd where n ·u > 0. The lower boundary at the bedrock is Γb and thefloating boundary is Γw. They are separated by the grounding line γGL defined by(xGL(y),y) by assuming that the ice mainly flows along the x-axis. If x ∈ Γu orx ∈ Γd then (x,y) ∈ γu or (x,y) ∈ γd where γ = γu∪ γd is the boundary of ω . Thedefinitions of these domains are

Ω = {x|(x,y) ∈ ω, b(x,y)≤ z≤ h(x,y, t)},Γs = {x|(x,y) ∈ ω, z = h(x,y, t)},Γb = {x|(x,y) ∈ ω, z = b(x,y), x < xGL(y)},Γw = {x|(x,y) ∈ ω, z = zb(x,y, t), x > xGL(y)},Γu = {x|(x,y) ∈ γu, b(x,y)≤ z≤ h(x,y, t)},Γd = {x|(x,y) ∈ γd, b(x,y)≤ z≤ h(x,y, t)},

(3)

and a schematic view in the x− y plane is shown in the right panel of Fig. 1.In two vertical dimensions as in the left panel of Fig. 1, x=(x,z)T , ω = [0,L], γu =0, and γd = L where L is the horizontal length of the domain.

The basal stress on Γb is related to the basal velocity using an empirical frictionlaw. The friction coefficient has a general form β (u,x, t) = C(x, t) f (u) wherethe coefficient C(x, t) is independent of the velocity u and f (u) represents somelinear or nonlinear function of u. For instance, f (u) = ‖u‖m−1 with the norm‖u‖ = (u ·u)1/2 introduces a Weertman type friction law (Weertman 1957) on ωwith a Weertman friction coefficient C(x, t)> 0 and an exponent parameter m> 0.Common choices of m are 1/3 and 1.

The density of the ice is denoted by ρ , the accumulation/ablation rate on Γs bya, and the gravitational acceleration by g. A projection (Petra et al. 2012) on thetangential plane of Γb is denoted by T = I− n⊗ n where the Kronecker outerproduct between two vectors a and c or two matrices A and C is defined by

(a⊗ c)i j = aic j, (A⊗C)i jkl = Ai jCkl. (4)

With h = (hx,hy,−1)T (in two dimensions h = (hx,−1)T ), the forward FS equa-tions for the height and velocity are

ht +h ·u = a, on Γs,h(x,0) = h0(x), x ∈ ω, h(x, t) = hγ(x, t), x ∈ γu,−∇ ·σ(u, p) =−∇ · (2η(u)D(u))+∇p = ρg, ∇ ·u = 0, in Ω(t),σn = 0, on Γs,Tσn =−C f (Tu)Tu, n ·u = 0, on Γb,

(5)

6

where h0(x) is the initial height and hγ(x, t) is a given height on the inflow bound-ary. The boundary conditions of the velocity on Γu and Γd are of Dirichlet typesuch that

u|Γu = uu, u|Γd = ud, (6)where uu and ud are some known velocities. In a special case where Γu is at theice divide, the horizontal velocity is set to u|Γu = 0, and the vertical component ofσn also vanishes on Γu.

2.2 Shallow shelf approximation

On the ice shelf and the fast flowing region, the basal shear stress is negligiblysmall and the horizontal velocity is almost constant in the z direction (Greve andBlatter 2009 MacAyeal 1989 Schoof 2007). The three dimensional FS problem(5) on Ω can be simplified to a two dimensional, horizontal problem with x =(x,y) ∈ ω by SSA, where only the horizontal velocity components u = (u1,u2)Tare considered. The viscosity for the SSA is

η(u) =12

A−1n

(u21x +u

22y +

14(u1y +u2x)2 +u1xu2y

)ν=

12

A−1n

(12

B : D)ν

, (7)

where B(u) = D(u)+∇ · uI with ∇ · u = trD(u). The Frobenius inner productbetween two matrices A and C is defined by

A : C = ∑i j

Ai jCi j. (8)

The vertically integrated stress tensor ς(u) is given by

ς(u) = 2HηB(u), (9)

Let n be the outward normal vector of the boundary γ = γu ∪ γd and t the tan-gential vector such that n · t = 0. The friction law is defined as in the FS casewhere the basal velocity is replaced by the horizontal velocity since the verti-cal variation is neglected in SSA. An example is Weertman’s law defined byβ (u,x, t) = C(x, t) f (u) = C(x, t)‖u‖m−1 with a friction coefficient C(x, t) ≥ 0.In Fig. 1, ω = Γb∪Γw and γu = Γu, γd = Γd .

7

The ice dynamics system is

ht +∇ · (uH) = a, 0≤ t ≤ T, x ∈ ω,h(x,0) = h0(x), x ∈ ω, h(x, t) = hγ(x, t), x ∈ γu,∇ · ς −C f (u)u = ρgH∇h, x ∈ ω,n ·u(x, t) = uu(x, t),x ∈ γu, n ·u(x, t) = ud(x, t),x ∈ γd,t · ςn =−Cγ fγ(t ·u)t ·u, x ∈ γg, t · ςn = 0, x ∈ γw.

(10)

The inflow and outflow normal velocities uu ≤ 0 and ud > 0 are specified on γuand γd . The lateral side of the ice γ is split into γg and γw with γ = γg∪γw. There isfriction in the tangential direction on γg which depends on the tangential velocityt ·u with the friction coefficient Cγ and friction function fγ . There is no frictionon γw. The structure of the SSA system (10) is similar to the FS equations in(5). However, the velocity u is not divergence free in SSA and B 6= D due to thecryostatic approximation of p.

In the case where an ice shelf or a grounding line exists, the floating ice is as-sumed to be at hydrostatic equilibrium in the seawater. A calving front boundarycondition (Schoof 2007 van der Veen 1996) is applied at γd by the depth integratedstress balance

ς(u) ·n = 12

ρgH2(

1− ρρw

)n, x ∈ γd, (11)

where ρw is the density of seawater. With this boundary condition, a calving rateuc can be determined at the ice front.

3 Adjoint equations

We wish to determine the sensitivity of a functional

F =∫ T

0

∫Γs

F(u,h)dxdt (12)

at Γs in the time interval [0,T ] to perturbations in the friction coefficient C(x, t) atthe base of the ice and the topography b(x) when u and h satisfy the FS equations(5) or the SSA equations (10). We introduce a Lagrangian L (u, p,h;v,q,ψ;b,C)for a given observation F with the forward solution (u, p,h) to (5) or (u,h) to (10)and the corresponding adjoint solutions (v,q,ψ) or (v,ψ). The adjoint solutions

8

solve the adjoint equations to the FS and SSA equations. These equations will bederived using the Lagrangian in this section and Appendix A.

The effect of the perturbations δC and δb in C and b on F is given by the pertur-bation δL in the Lagrangian

δF = δL= L (u+δu, p+δ p,h+δh;v+δv,q+δq,ψ +δψ;b+δb,C+δC)−L (u, p,h;v,q,ψ;b,C).

Examples of F(u,h) in (12) are ‖u− uobs‖2, |h− hobs|2 in an inverse problemto find b and C to match the observed data uobs and hobs at the surface Γs as in(Gillet-Chaulet et al. 2016 Isaac et al. 2015 Morlighem et al. 2013 Petra et al.2012), or F(u,h) = 1T u1(x, t)δ (x− x∗) with the Dirac delta at x∗ to measure thetime averaged deviation of the horizontal velocity u1 at x∗ on the ice surface Γswith

F =∫ T

0

∫Γs

F(u,h)dxdt =1T

∫ T0

u1(x∗, t)dt,

where T is the duration of the observation at Γs.

3.1 Full Stokes equation

The definition of the Lagrangian L for the FS equations is found in (58) in Ap-pendix A where (v,q,ψ) are the Lagrange multipliers corresponding to the for-ward equations for (u, p,h). In order to determine (v,q,ψ), the so-called adjointproblem is solved

ψt +∇ · (uψ)−h ·uzψ = Fh +Fu ·uz, on Γs,ψ(x,T ) = 0, ψ(x, t) = 0, on Γd,−∇ · σ̃(v,q) =−∇ · (2η̃(u)?D(v))+∇q = 0, ∇ ·v = 0, in Ω(t),σ̃(v,q)n =−(Fu +ψh), on Γs,Tσ̃(v,q)n =−C f (Tu)(I+Fb(Tu))Tv, on Γb,n ·v = 0, on Γb,

(13)

where the derivatives of F with respect to u and h are

Fu =(

∂F∂u1

,∂F∂u2

,∂F∂u3

)T, Fh =

∂F∂h

.

9

The adjoint viscosity and adjoint stress (Petra et al. 2012) are

η̃(u) = η(u)(I + 1−nnD(u):D(u)D(u)⊗D(u)

),

σ̃(v,q) = 2η̃(u)?D(v)−qI.(14)

The tensor I has four indices i jkl and Ii jkl = 1 only when i = j = k = l, oth-erwise Ii jkl = 0. In general, Fb(Tu) in (13) is a linearization of the friction lawrelation f (Tu) in (5) with respect to the variable Tu. For instance, with a Weert-man type friction law, f (Tu) = ‖Tu‖m−1, it is

Fb(Tu) =m−1

Tu ·Tu(Tu)⊗ (Tu). (15)

The ? operation in (14) between a four index tensor A and a two index tensor ormatrix C is defined by

(A ?C)i j = ∑kl

Ai jklCkl. (16)

The perturbation of the Lagrangian function with respect to a perturbation δC inthe slip coefficient C(x, t) is

δF = δL =∫ T

0

∫Γb

f (Tu)Tu ·Tv δC dxdt (17)

involving the tangential components of the forward and adjoint velocities Tu andTv at the ice base Γb.

The detailed derivation of the adjoint equations (13) and the perturbation of theLagrangian function (17) are given in Appendix A.3 from the weak form of the FSequations (5) on Ω, integration by parts, and by applying the boundary conditionsas in (Martin and Monnier 2014 Petra et al. 2012). The adjoint equations consistof the equations for the adjoint height ψ , the adjoint velocity v, and the adjointpressure q. Compared to the steady state adjoint equation for the FS equation in(Petra et al. 2012), an advection equation is added in (13) for the Lagrange multi-plier ψ(x, t) on Γs with a right hand side depending on the observation function Fand one term depending on ψ in the boundary condition on Γs. The adjoint heightequation of ψ can be solved independently of the adjoint stress equation sinceit is independent of v. If h is observed then the adjoint height equation must besolved together with the adjoint stress equation. Otherwise, the term ψh vanishesin the right hand side of the adjoint stress equation and the solution is v = 0 withδF = 0 in (17).

10

3.1.1 Time-dependent perturbations

Suppose that u1∗ = u1(x∗, t∗) is observed at (x∗, t∗) at the ice surface and thatt∗ < T , then

u1(x∗, t∗) = F =∫ T

0

∫Γs

F(u)dxdt,

with

F(u) = u1δ (x−x∗)δ (t− t∗), Fu1 = δ (x−x∗)δ (t− t∗), Fu2 = Fu3 = 0, Fh = 0.

The procedure to determine the sensitivity is as follows. First, the forward equa-tion (5) is solved for u(x, t) from t = 0 to t = T . Then, the adjoint equation (13)is solved backward in time for t ≤ T with ψ(x,T ) = 0 as the final condition. Ob-viously, the solution for t∗ < t ≤ T is ψ(x, t) = 0 and v(x, t) = 0. Denote the unitvector with 1 in the i:th component by ei. At t = t∗ we have

σ̃(v,q)n =−e1δ (x−x∗)δ (t− t∗)−ψh,

in the boundary condition in (13). For t < t∗, σ̃(v,q)n = −ψh. Since ψ is smallfor t < t∗ (see Sect. 3.1.3), the dominant part of the solution is v(x, t) = v0(x)δ (t−t∗) for some v0. To simplify the notation in the remainder of this paper, a variablewith the subscript ∗ is evaluated at (x∗, t∗) or if it is time independent at x∗.When the slip coefficient at the ice base is changed by δC, then the change in u1∗is by (17)

δu1∗ = δL =∫ T

0

∫Γb

f (Tu)Tu ·TvδC dxdt

≈∫

Γbf (Tu)Tu ·Tv0 δC(x, t∗)dx.

(18)

In this case, the perturbation δu1∗ mainly depends on δC at time t∗ and the con-tribution from previous δC(x, t), t < t∗, is small.

Let the height h∗ be measured at Γs. Then

F(h) = h(x, t)δ (x−x∗)δ (t− t∗), Fh = δ (x−x∗)δ (t− t∗), Fu = 0.

The solution of the adjoint equation (13) with σ̃(v,q)n = −ψh at Γs for v(x, t)is non-zero since ψ(x, t) 6= 0 for t < t∗. In a seasonal variation, there is a timedependent perturbation δC(x, t) = δC0(x)cos(2πt/τ) added to a stationary timeaverage C(x). The time constant τ could be for example 1 a (year). Assume

11

that f (Tu)Tu ·Tv is approximately constant in time (e.g. if u varies slowly, thenψ ≈ const and v≈ const for t < t∗. Then the observation at the ice surface variesas

δh∗ = δL =∫ T

0

∫Γb

f (Tu)Tu ·TvδC(x, t)dxdt

≈∫ t∗

0cos(2πt/τ)dt

∫Γb

f (Tu)Tu ·TvδC0 dx

=τ

2πsin(2πt∗/τ)

∫Γb

f (Tu)Tu ·TvδC0 dx.

(19)

When the friction perturbation δC is large at t∗ = 0,τ/2,τ . . . , the effect on h∗vanishes. If the middle of the winter is at nτ, n = 0,1,2, . . ., then the middle ofthe summer is at (n+1/2)τ . The friction is at its maximum in the winter and at itsminimum in the summer when the meltwater introduces lubrication. There is nochange of h∗ in the middle of the summer, δh∗= 0, but C+δC has its lowest valuethen. If h∗ is measured in the summer and compared to a mean value h(x), thenδh(x, t∗) = 0 and the wrong conclusion would be drawn that there is no change inC if the phase shift between δC and δh in (19) is not accounted for.

A two dimensional numerical example is shown in Fig. 2 with τ = 1 a and δC(x, t)=0.01C cos(2πt) in an interval x ∈ [0.9,1.0]×106 m where the ice sheet flows fromx = 0 to L = 1.6× 106 m. The grounding line is at xGL = 1.035× 106 m. Thedetails of the setup are found in the MISMIP (Pattyn et al. 2012) test case usedin (Cheng and Lötstedt 2019). The ice sheet is simulated by FS with Elmer/Ice(Gagliardini et al. 2013) for 10 years. The perturbations in δu1 and δh oscillateregularly with a period of 1 year after an initial transient and are small outside theinterval [0.9,1.0]× 106. An increase in the friction, δC > 0, leads to a decreasein the velocity and δC < 0 increases the velocity. There is a phase shift ∆φ intime by π/2 between δu and δh as predicted by (18) and (19). The weight in (19)for δC0 in the integral over x changes sign when the observation point is passingfrom x∗ = 0.9×106 to 1.0×106 explaining why the shift changes sign in the twolower panels.

The phase shift ∆φ between the surface observations and the basal perturbationsis investigated in (Gudmundsson 2003) with a linearized equation and Fourieranalysis. It is found that ∆φ = −π/2 between δC and δh for short perturbationwave lengths in the steady state as in Fig. 2.

12

0 2 4 6 8 10t (years)

−0.50.00.5

δu1

(m/a) ×10

−1 x∗ = 0.85 × 106 m

0 2 4 6 8 10t (years)

−20

2

δu1

(m/a)

x∗ = 0.90 × 106 m

0 2 4 6 8 10t (years)

−0.50.0

0.5

δu1

(m/a) ×10

1 x∗ = 0.95 × 106 m

0 2 4 6 8 10t (years)

−2.50.02.5

δu1

(m/a)

x∗ = 1.00 × 106 m

0 2 4 6 8 10t (years)

−101

δu1

(m/a) ×10

−1 x∗ = 1.02 × 106 m

−0.50.0

δh(m

)

×10−2

−0.50.00.5

δh(m

)

×10−1

−2.50.0

δh(m

)

×10−3

−10

1

δh(m

)

×10−1

0

1

δh(m

)

×10−2

Figure 2: Observations at x∗ = 0.85,0.9,0.95,1.0,1.02× 106 m with FS in timet ∈ [0,10] of δu1(blue) and δh(red) with perturbation δC(t) = 0.01C cos(2πt) forx ∈ [0.9,1.0]×106 m. Notice the different scales on the y-axes.

13

3.1.2 The sensitivity problem and the inverse problem

There is a relation between the sensitivity problem and the inverse problem toinfer parameters from data. Assume that (vi,qi,ψ i), i = 1, . . . ,d, solves (13) withFu = eiδ (x− x̄) or Fh = δ (x− x̄). With Fu 6= 0,Fh = 0 we have d = 2(or 3) intwo (three)-dimensions and with Fu = 0,Fh 6= 0 we have d = 1. Consider a targetfunctional F for the steady state solution with weights wi(x̄) multiplying δui inthe first variation of F . Using (17), δF is

δF =∫

ω

d

∑i=1

wi(x̄)δui dx̄ =∫

ω

d

∑i=1

wi(x̄)∫

Γbf (Tu)Tu ·Tvi δC dxdx̄

=∫

Γbf (Tu)Tu ·T

(∫ω

d

∑i=1

wi(x̄)vi dx̄

)δC dx.

(20)

It follows from (13) that (wi(x̄)vi(x),wi(x̄)qi(x),wi(x̄)ψ i(x)) is a solution withFu = wi(x̄)eiδ (x− x̄) or Fh = w(x̄)δ (x− x̄). Therefore, also(∫

ωwi(x̄)vi dx̄,

∫ω

wi(x̄)qi dx̄,∫

ωwi(x̄)ψ i dx̄

)is a solution with Fu =

∫ω wi(x̄)eiδ (x− x̄)dx̄ = wi(x)ei or Fh =

∫ω w(x̄)δ (x−

x̄)dx̄ = w(x).

In the inverse problem, F = 12∫

ω ‖u(x)−uobs(x)‖2 dx (Petra et al. 2012) and thefirst variation is δF =

∫ω(u(x)−uobs(x)) ·δu(x)dx. Let wi(x) = ui(x)−uobs,i(x)

in (20). Then we find that

δF =∫

Γbf (Tu)Tu ·Tṽ(x)δC dx, (21)

where

ṽ(x) =∫

ω

d

∑i=1

wi(x̄)vi(x)dx̄

is a solution to (13) with Fu = (w1,w2,w3)T = u−uobs or Fh = w = h−hobs.If we are interested in solving the inverse problem and determine δF in (20) toiteratively compute the optimal solution with a gradient method, then we solve(13) directly with Fu = u−uobs or Fh = h−hobs to obtain ṽ without computing vi.

14

3.1.3 Steady state solution to the adjoint height equation in two dimensions

In a two dimensional vertical ice, with u(x,z) = (u1,u3)T , the stationary equationfor ψ in (13) is

(u1ψ)x = Fh +(hψ +Fu) ·uz, z = h, 0≤ x≤ L. (22)When x > x∗, where Fh = 0 and Fu = 0, then ψ(x) = 0 since the right boundarycondition is ψ(L) = 0.

If u1 is observed at Γs then F(u,h) = u1(x)χ(x) and Fu = (χ(x),0)T and Fh = 0.The weight χ on u1 may be a Dirac delta, a Gaussian, or a constant in a limitedinterval. On the other hand, if F(u,h) = h(x)χ(x) then Fh = χ(x) and Fu = 0.

Let g(x) = u1z(x) when Fu 6= 0 and let g(x) = 1 when Fh 6= 0. Then by (22)(u1ψ)x−h ·uzψ = g(x)χ(x). (23)

The solution to (23) is

ψ(x) =− 1u1(x)

∫ x∗x

exp(−∫ ξ

x

h ·uz(y)u1(y)

dy)

g(ξ )χ(ξ )dξ , 0≤ x < x∗,ψ(x) = 0, x∗ < x≤ L.

(24)In particular, if χ(x) = δ (x−x∗) then F = u(x∗) or F = h(x∗) and the multiplieris

ψ(x) =−g(x∗)u1(x)

exp(−∫ x∗

x

h ·uz(y)u1(y)

dy), 0≤ x < x∗, (25)

which has a jump −g(x∗)/u1(x∗) at x∗.With a small h ·uz(y)≈ 0 above, an approximate solution is ψ(x)≈−g(x∗)/u1(x).Moreover, if u is observed with Fu 6= 0 and g(x)≈ 0, then ψ(x)≈ 0 and ψh≈ 0in (13). Consequently when u1 is observed, the effect on v of the solution of theadjoint advection equation is negligible. It is sufficient to solve only the adjointstress equation for v as in (Gillet-Chaulet et al. 2016 Isaac et al. 2015 Petra et al.2012), which may often be the case in FS.

3.2 Shallow shelf approximation

The adjoint equations for SSA are given and analyzed in this section. A La-grangian L of the SSA equations is defined with the same technique as in (Petra

15

et al. 2012) for the FS equations. By evaluating L at the forward solution (u,h)and the adjoint solution (v,ψ), the effect of perturbed data at the ice base canbe observed at the ice surface as a perturbation δL . The details of the deriva-tions are found in Appendix A.2. In a two dimensional vertical ice at the steadystate, the equations are simpler and analytical solutions for the forward and adjointequations are derived later in this section.

After insertion of the forward solution, partial integration in L , and applying theboundary conditions, the adjoint SSA equations are obtained as

ψt +u ·∇ψ +2ηB(u) : D(v)−ρgH∇ ·v+ρgv ·∇b = Fh, in ω,ψ(x,T ) = 0, in ω, ψ(x, t) = 0, on γw,∇ · ς̃(v)−C f (u)(I+Fω(u))v−H∇ψ =−Fu, in ω,t · ς̃(v)n =−Cγ fγ(t ·u)(1+Fγ(t ·u))t ·v, on γg, t · ς̃(v)n = 0, on γw,n ·v = 0, on γ,

(26)where the adjoint viscosity η̃ and adjoint stress ς̃ are defined by

η̃(u) = η(u)(I + 1−nnB(u):D(u)B(u)⊗D(u)

),

ς̃(v) = 2Hη̃(u)?B(v),(27)

cf. η̃ and σ̃ of FS in (14). The adjoint equation derived in (MacAyeal 1993) is thestress equation in (26) with a constant H, Fω = 0 and η̃(u) = η(u).

The adjoint SSA equations have the same structure as the adjoint FS equations(13). There is one stress equation for the adjoint velocity v and one equationfor the multiplier ψ corresponding to the height equation in (10). However, theadvection equation for ψ in (26) depends on the adjoint velocity v which leads toa fully coupled system for v and ψ .

The equations are solved backward in time with a final condition on ψ at t = T .As in (10), there is no time derivative in the stress equation. With a Weertmanfriction law, f (u) = ‖u‖m−1 and fγ(t ·u) = |t ·u|m−1, it is shown in Appendix A.1that

Fω(u) =m−1u ·u u⊗u, Fγ = m−1.

If the friction coefficient C at the ice base is changed by δC, the bottom topogra-phy is changed by δb, and the lateral friction coefficient Cγ is changed by δCγ ,

16

then it follows from Appendix A.2 that the Lagrangian is changed by

δL =∫ T

0

∫ω(2ηB(u) : D(v)+ρgv ·∇h+∇ψ ·u)δb− f (u)u ·vδC dxdt

−∫ T

0

∫γg

fγ(t ·u)t ·ut ·vδCγ dsdt.(28)

The weight in front of δC in (28) is actually the same as in (17).

Suppose that h is observed with Fu = 0 in (26). Then the adjoint height equationmust be solved for ψ 6= 0 to have a v 6= 0 in the adjoint stress equation and aperturbation in the Lagrangian in (28). The same conclusion followed from theadjoint FS equations.

The SSA model is obtained from the FS model in (Greve and Blatter 2009 MacAyeal1989) under some simplifying assumptions in the stress equation. An alternativederivation of the adjoint SSA would be to simplify the stress equation for v inthe adjoint FS equations (13) under the same assumptions. The resulting adjointequation would be different from (26) since the advection equation there dependson the adjoint velocity.

3.2.1 SSA in two dimensions

The forward and adjoint SSA equations in a two dimensional vertical ice are de-rived from (10) and (26) by letting H and u1 be independent of y and takingu2 = 0. Since there is no lateral force, Cγ = 0. The position of the groundingline, where the ice starts floating on water, is denoted by xGL as in Fig. 1 andΓb = [0,xGL], Γw = (xGL,L]. Let C be a positive constant on the bedrock andC = 0 on the water. Simplify the notation with u = u1 and v = v1. The forwardequation is

ht +(uH)x = a, 0≤ t ≤ T, 0≤ x≤ L,h(x,0) = h0(x), h(0, t) = hL(t),(Hηux)x−C f (u)u−ρgHhx = 0,u(0, t) = ul(t), u(L, t) = uc(t),

(29)

where ul is the speed of the ice flux at x = 0 and uc is the so-called calving rate atx = L. By the stress balance in (11), the calving front boundary satisfies

ux(L, t) = A[

ρgH(L, t)4

(1− ρρw

)

]n.

17

Assume that u > 0 and ux > 0, then η = 2A−1n u

1−nn

x and the friction term with aWeertman law is C f (u)u =Cum. The adjoint equations for v and ψ follow eitherfrom simplifying the adjoint equations (26) or deriving the adjoint equations fromthe two dimensional forward SSA (29)

ψt +uψx +(ηux−ρgH)vx +ρgbxv = Fh, 0≤ t ≤ T, 0≤ x≤ L,ψ(x,T ) = 0, ψ(L, t) = 0,(1nηHvx)x−Cm f (u)v−Hψx =−Fu,v(0, t) = 0, v(L, t) = 0.

(30)

The coefficient 1/n in front of η in the equation for v is a result of the adjointviscosity η̃ , which was 1 in the adjoint SSA formulation in (MacAyeal 1993).

The effect on the Lagrangian of perturbations δb and δC is obtained from (28)

δL =∫ T

0

∫ L0(ψxu+ vxηux + vρghx)δb− v f (u)uδC dxdt. (31)

The weights in the integral are denoted by wC =−v f (u)u and wb =ψxu+vxηux+vρghx.

3.2.2 The forward steady state solution

The viscosity terms in (29) and (30) are often small and the longitudinal stress canbe ignored in the steady state solution, see (Schoof 2007). The approximationsof both the forward and adjoint equations can then be solved analytically on areduced computational domain with x ∈ [0,xGL]. The simplified forward steadystate equation in (29) with f (u) = um−1 is written as

(uH)x = a, 0≤ x≤ xGL,H(0) = H0,−Cum−ρgHhx = 0,u(0) = 0,

(32)

and the adjoint equation in (30) is simplified when the gradient of the base topog-raphy bx is small

uψx−ρgHvx = Fh, 0≤ x≤ xGL,ψx(0) = 0, ψ(xGL) = 0,−Cmum−1v−Hψx =−Fu,v(0) = 0.

(33)

18

Numerical experiments in (Cheng and Lötstedt 2019) show that an accurate solu-tion compared to the FS and SSA solutions is obtained by calibration of H withHGL = H(xGL) in (32). All the assumptions made for the simplification in (32) arenot valid close to the ice divide at x = 0 and the ice dynamics in this area cannotbe captured accurately by SSA.

The solution to the forward equation (32) is determined when a and C are constantin (70) and (71) in Appendix B by integrating (69) from x to xGL

H(x) =(

Hm+2GL +m+2m+1

Cam

ρg(xm+1GL − xm+1)

) 1m+2

, 0≤ x≤ xGL,H(x) = HGL, xGL < x < L,u(x) =

axH, 0≤ x≤ xGL, u(x) =

axHGL

, xGL < x < L.

(34)

An example of the analytical solutions u(x) and H(x) for x ∈ [0,xGL] is given inFig. 3, where the ice rests on a downward sloping bedrock with the grounding lineposition at xGL as in the MISMIP test cases in (Pattyn et al. 2012). The detailsand specified data of this example are found in (Cheng and Lötstedt 2019). Whenx approaches xGL, then u increases and H decreases rapidly in the figure. Analternative solution to (29) when x > xGL is found in (Greve and Blatter 2009)where the assumption is that H(x) is linear in x.

0.0 0.2 0.4 0.6 0.8 1.0

x (m) ×106

0.0

0.5

1.0

u(m

/yea

r)

×103

0.0 0.2 0.4 0.6 0.8 1.0

x (m) ×106

1

2

3

4

H(m

)

×103

Figure 3: The analytical solutions u(x) and H(x) in (34) for a grounded ice in[0,xGL].

3.2.3 The adjoint steady state solution with Fu 6= 0

The analytical solution to (33) is derived in Appendices C to D assuming that bxis small such that bx� Hx and a and C are constants. The expressions for ψ and

19

v are found in (81)-(84). When u is observed at x∗, then F and F(u,h) satisfy

F =∫ L

0u(x)δ (x− x∗)dx = u∗, Fu = δ (x− x∗), Fh = 0,

and the adjoint solutions are

ψ(x) =Camx∗

ρgHm+3∗(xmGL− xm), x∗ < x≤ xGL,

ψ(x) =− 1H∗

+Camx∗

ρgHm+3∗(xmGL− xm∗ ), 0≤ x < x∗,

v(x) =ax∗

ρgHm+3∗Hm, x∗ < x≤ xGL,

v(x) = 0, 0≤ x < x∗,

(35)

with discontinuities at the observation point x∗ in ψ(x) and v(x). The analyticalsolutions ψ(x) and v(x) of the ice sheet in Fig. 3 at different x∗ positions are shownin the left panels of Fig. 4 and Fig. 5. In all figures, m = 1 in the friction model.In Fig. 4 and subsequent figures, a Dirac term αδ is plotted on a grid with gridsize ∆x as a triangle with base 2∆x and height α/∆x such that the integral over thetriangle is α . The derivative αδ ′ is plotted as a peak α/∆x2 followed by−α/∆x2.As in Sect. 3.1.3 for the FS model, ψ can be small in SSA when u is observedcompared to observations of h (cf. left and right panels of Fig. 4).

0.0 0.2 0.4 0.6 0.8 1.0

x (m) ×106

−3

−2

−1

0

1

ψ

×10−4 F (u, h) = uδ(x− x∗)

0.0 0.2 0.4 0.6 0.8 1.0

x (m) ×106

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

ψ

×10−2 F (u, h) = hδ(x− x∗)

Figure 4: The analytical solutions of ψ in (33) of the observations of u (left panel)and h (right panel) at different locations x∗ = 0.25×106,0.5×106,0.7×106 and0.9×106 m (blue, orange, green and red).

20

0.0 0.2 0.4 0.6 0.8 1.0

x (m) ×106

0

1

2

3

v

×10−9 F (u, h) = uδ(x− x∗)

0.0 0.2 0.4 0.6 0.8 1.0

x (m) ×106

−5

−4

−3

−2

−1

0

v

×10−8 F (u, h) = hδ(x− x∗)

Figure 5: The analytical solutions of v in (33) of the observations of u (left panel)and h (right panel) at different locations x∗ = 0.25×106,0.5×106,0.7×106 and0.9×106 m (blue, orange, green and red).

The perturbation of the Lagrangian in (31) is derived in (85)-(86) in Appendix D

δu∗ = δL =∫ xGL

0(ψxu+ vxηux + vρghx)δb− vum δC dx

=∫ xGL

x−∗

ax∗Hm

Hm+3∗[(m+1)HxH (x− x∗)+Hδ (x− x∗)] δb−

ax∗(ax)m

ρgHm+3∗δC dx

=u∗δb∗

H∗− u∗

ρgHm+2∗

∫ xGLx∗

C(ax)m((m+1)

δbH

+δCC

)dx,

(36)or scaled with u∗

δu∗u∗

=δb∗H∗− 1

ρgHm+2∗

∫ xGLx∗

C(ax)m((m+1)

δbH

+δCC

)dx. (37)

The Heaviside step function is denoted by H (x) in (36). The weights with respectto δC and δb in (36) are shown in the left panels of Fig. 6 and Fig. 7 for the sameice sheet geometry as in Fig. 3.

The following conclusions can be drawn from (36) and (37):

1. The weight in front of δC increases when x∗→ xGL. This is an effect of theincreasing velocity u∗ and the decreasing thickness H∗, see Fig. 3. There-fore, a perturbation δC with support in [x∗,xGL] will cause larger perturba-tions at the surface the closer x∗ is to xGL and the closer δC(x) is to xGL.

2. If δb∗= 0, i.e. the topography is unperturbed at the base below the observa-tion point, then the contribution of δb in (x∗,xGL] cannot be separated fromthe contribution of δC in (37).

21

0.0 0.2 0.4 0.6 0.8 1.0

x (m) ×106

−5

−4

−3

−2

−1

0

wC

×10−7 F (u, h) = uδ(x− x∗)

0.0 0.2 0.4 0.6 0.8 1.0

x (m) ×106

0

2

4

6

8

wC

×10−6 F (u, h) = hδ(x− x∗)

Figure 6: The analytical solution of the weights wC = −vum on δC in (31) for u(left panel) and h (right panel) observed at x∗ = 0.25× 106,0.5× 106,0.7× 106and 0.9×106 m (blue, orange, green and red).

0.0 0.2 0.4 0.6 0.8 1.0

x (m) ×106

−5.0

−2.5

0.0

2.5

5.0

wb

×10−7 F (u, h) = uδ(x− x∗)

0.0 0.2 0.4 0.6 0.8 1.0

x (m) ×106

−1.0

−0.5

0.0

0.5

1.0

wb

×10−5 F (u, h) = hδ(x− x∗)

Figure 7: The analytical solution of weights wb = ψxu+ vxηux + vρghx on δbin (31) for u (left panel) and h (right panel) observed at x∗ = 0.25× 106,0.5×106,0.7×106 and 0.9×106 m (blue, orange, green and red).

22

3. The change in u∗ due to δb in (36) is simplified if Hx � H. Then δu∗ ≈u∗δb∗/H∗ and the main effect on u∗ from the perturbation δb is localized ateach x∗.

4. The perturbation in u is very sensitive to H∗ due to the factor u∗/Hm+2∗ ∝1/Hm+3∗ in (36) when x∗ is moving downstream.

5. The relation between the two terms in (36) is estimated in a case when δC =0 and δb/H is constant in [x∗,xGL]. Let x∗ = 0.9 · 106 and xGL = 1.2 · 106.Then u∗ ≈ 130 in Fig. 3. The factor multiplying δb/H in the integral is bythe left panel in Fig. 7 approximately weight× interval length×H ≈ 150.The contribution by the two terms is about the same but with opposite signs.The integral term is reduced if the interval where δC 6= 0 is shorter.

6. With an unperturbed topography, let the perturbation of the friction coeffi-cient be a constant δC0 6= 0 in [x0,x1] ∈ [x∗,xGL] resulting in a perturbationof the velocity. Evaluate the integral in (36) to obtain

δu∗=−u∗

ρgHm+2∗

∫ x1x0

(ax)mδC0 dx =−amu∗

(m+1)ρgHm+2∗(xm+11 − xm+10 )δC0.

(38)The same δu∗ is observed with a constant perturbation in [x2,x3] ∈ [x∗,xGL]with the amplitude δC0(xm+11 − xm+10 )/(xm+13 − xm+12 ). Different δC cangive rise to the same observation δu∗. This holds true also for differentδb when δb∗ = 0. Inference of δC or δb from surface data requires moreobservation points than one.

7. Perturb C by δC = ε cos(kx/xGL) in (36) for some wave number k and am-plitude ε and let δb = 0 and m = 1. Then

δu∗ =−∫ xGL

x∗ε

a2x∗ρgH4∗

xcos(

kxxGL

)dx

=−ε a2x∗

ρgH4∗

x2GLk

(sin(k)− x∗

xGLsin(

kx∗xGL

)+

1k

(cos(k)− cos

(kx∗xGL

))).

(39)When k grows at the ice base, the amplitude of the perturbation at the icesurface decays as 1/k. The effect of high wave number perturbations of Cwill be difficult to observe at the top of the ice. When k vanishes, then δu∗tends to constant.

8. If δC = 0 and b is perturbed by δb = ε cos(kx/xGL), then any perturbation

23

at x∗ is propagated to the surface by the first term on the right hand side of(36). The integral term will behave in the same way as in (39).

Let the friction coefficient C be perturbed by a constant δC in [0,xGL] and takeδb = 0. Then it follows from (36) that

δu∗ = δC∫ xGL

x∗−vum dx = a

m+1x∗(xm+1∗ − xm+1GL )(m+1)ρgHm+3∗

δC. (40)

Computing the derivative of u(x,C) with respect to C in the explicit expression in(34) at x∗ yields the same result.

The sensitivity of surface data to changes in b and C is estimated in (Gudmundssonand Raymond 2008) with a linearized model and Fourier analysis. The conclu-sion is that differences of short wavelength in the bedrock topography cannot beobserved at the surface. Only differences with long wavelength in the friction co-efficient propagate to the top of the ice. This is in agreement with (36) and (37)and conclusions 7 and 8 above.

3.2.4 The adjoint steady state solution with Fh 6= 0

In the case when h is observed at x∗ and Fu = 0 and Fh = δ (x−x∗), the expressionsfor ψ and v are

ψ(x) =− Cam−1

ρgHm+1∗(xmGL− xm), x∗ < x≤ xGL,

ψ(x) =− Cam−1

ρgHm+1∗(xmGL− xm∗ ), 0≤ x < x∗,

v(x) =− Hm

ρgHm+1∗, x∗ < x≤ xGL,

v(x) = 0, 0≤ x < x∗.

(41)

There is a discontinuity at the observation point x∗ in v(x), see Fig. 5, but ψ(x)is continuous in the solution of (33). Actually, ψ(x) ∼ −δ (x− x∗) in the neigh-borhood of x∗ due to the second derivative term (1nηHvx)x which is neglectedin the simplified equation (33) but is of importance at x∗. A correction ψ̂ ofψ at x∗ is therefore introduced to satisfy

(1nηHvx

)x−Hψ̂x = 0. With vx(x∗) =

−δ (x− x∗)/(ρgH∗), the correction is ψ̂(x) =−δ (x− x∗)η∗/(nρgH∗). The solu-tion ψ is corrected at each x∗ in Fig. 4 with ψ̂ . The perturbation in h is as in (36)

24

with ψ and v in (41) and the additional term ψ̂

δh∗H∗

=∫ xGL

x−∗− uη∗

nρgH2∗δx(x− x∗)δb dx+

∫ xGLx∗

C(ax)m

ρgHm+2∗

((m+1)

δbH

+δCC

)dx

=aη∗

nρgH2∗

(x

δbH

)x(x∗)+

1ρgHm+2∗

∫ xGLx∗

C(ax)m((m+1)

δbH

+δCC

)dx,

(42)where a(xδb/H)x(x∗) = (uδb)x(x∗) represents the x-derivative of uδb evaluatedat x∗. When δb = 0 then δu∗ in (37) and δh∗ = δH∗ in (42) satisfy δu∗H∗ =−δH∗u∗ as in the integrated form of the advection equation in (32) and in (68).The contribution from the integrals in (37) and (42) is identical except for the sign(see also Fig. 6). The first term in (37) depends on δb/H and the first term in (42)depends on the derivative of xδb/H. Because of these similarities, the conclusions1, 2, 6, 7, and 8 from (36) and (37) for δu∗ are valid also for δh∗ in (42). Thechange in h caused by δC is less sensitive to H∗ than the change in u since thefactor multiplying the integral is proportional to 1/Hm+1∗ .

If uδb = axδb/H is constant, then δh is determined by the continuous weight inx ∈ [x∗,xGL] in Fig. 7 which has a shape similar to the δC weight in Fig. 6. Aperturbation of b at the base is directly visible locally in u at the surface while theeffect of δC is non-local in (42).

3.2.5 The time dependent adjoint solution

The time dependent adjoint equation (30) is solved numerically for the MIS-MIP test case in Sect. 3.2.2. Starting from the steady state solution, the fric-tion coefficient C has a seasonal variation in the forward equation (29), such thatC(x, t) = C0(1+κ cos(2πt)), 0 < κ < 1. Apparently, C has its highest value att = n, n = 0,1,2, . . ., i.e. the winter, and its lowest value at t = n+ 1/2, i.e. thesummer. The period of the seasonal variation is 1 a and the beginning of each yearis in the winter.

The amplitude of the pertubation is set to κ = 0.5 and the forward equation (29)is solved for 11 years. Observations on u and h are taken at x∗ = 9× 105 m for0.1 a in the four seasons of the tenth year, e.g., in the summer (t = 9.5), the fall(t∗ = 9.75), the winter (t = 10), and the spring (t = 10.25). The adjoint equations(30) are solved from the observation points backward in time, respectively, as in

25

Figs. 8 and 9. According to a convergence test, the time step is chosen to be 0.01 aand the spatial resolution is 103 m.

The adjoint weights wC and wb, as defined in (31), are shown in Fig. 8 for theobservations on u and h at x∗ = 9× 105 m in the four seasons. The time axisin the figure starts from the steady state solution and follows the time directionin the forward problem. As expected, most of the weights in space and time arenegligible. Therefore, we take a snapshot with the width of 105 m in space aroundx∗. Only δC and δb in a narrow interval around x∗ for t in [0, t∗] have an influenceon δu∗ and δh∗, see Fig. 8. A perturbation at the base is propagated to the x∗position on the surface but with a possible delay in time.

The temporal variations of the adjoint weights at x∗ in Fig. 8 are shown in Fig. 9for the four seasons with four different colors. As expected, the weights vanishwhen t > t∗. In the left panels of Fig. 9, the perturbations δC∗ and δb∗ have adirect effect on δu∗ at t∗, where wC(x∗) and wb(x∗) are both negative. The sameconclusion is valid for δu1∗ computed from the FS equations (18) in Sect. 3.1.1. Achange in δC∗ at the base is observed immediately as a change in u at the surface.The effect of δC on δu∗ for t < t∗ is weak in the upper left panel of Fig. 9. Thelargest effect of δC on δu∗ and δh∗ appears in the summer when C is small, asthe blue lines in the left panels.

However, when h is observed, the effect of δC∗ and δb∗ are not visible directlybecause wC∗ ≈ 0 and wb∗ ≈ 0 in the right panels of Fig. 9. Additionally, the effectof δC and δb is difficult to be separated, since the weight wb(x∗) has the similarshape as wC(x∗). The largest effect on δh∗ is from δC in the summer due to thepeaks as in the upper right panel. For the same δC, the largest δh∗ is observed inthe fall (orange), then the second largest δh∗ is in the winter (green) followed bythe spring observation (red). If δh∗ is observed in the fall and the time dependencyis ignored, then the wrong conclusion is drawn that δC in the fall has the strongesteffect. There is a delay in time between the perturbation and the observation ofthe effect.

A reference adjoint solution observed during the fall season (t∗= 9.75) with κ = 0in the forward equations is shown in the black dashed lines in Fig. 9. The weightwb for a constant b in time is well approximated by exp(−(T − t)/τ) with τ =1.4 a. For the weight wC, the same exponential function holds, but the constantτ = 1.8 a for the observation of h∗ and τ = 2.2 a for the u∗ case. Suppose thatthe temporal perturbation is oscillatory δC0 cos(2π f t) with frequency f . Then

26

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

summer

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

fall

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

winter

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

spring

−7.5 −5.0 −2.5 0.0 2.5 5.0 7.5×10−7

wC for F (u, h) = uδ(x− x∗)

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

summer

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

fall

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

winter

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

spring

−6 −4 −2 0 2 4 6×10−6

wb for F (u, h) = uδ(x− x∗)

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

summer

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

fall

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

winter

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

spring

−4 −2 0 2 4×10−7

wC for F (u, h) = hδ(x− x∗)

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

summer

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

fall

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

winter

8.5 9.0 9.5

x (m) ×1050.0

2.5

5.0

7.5

10.0

t(a

)

spring

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0×10−5

wb for F (u, h) = hδ(x− x∗)

Figure 8: The adjoint weights for the observations at x∗ = 9× 105 m of the fourseasons. The first row: wC for the observation of u. The second row: wb for theobservation of u. The third row: wC for the observation of h. The last row: wb forthe observation of h.

27

perturbation in h at t = t∗ is

δh =∫ T

0exp(−(T − t)/τ)δC0 cos(2π f t)dt

=

(cos(2π f T )+2πτ f sin(2π f T )− e−T/τ

4π2τ f 2 + τ−1

)δC0,

(43)

cf. (39). With a high frequency (diurnal), f � 1, then δh ∝ 1/ f and high fre-quency perturbations are damped efficiently. If the frequency is low (decennial),f � 1, then δh ∝ τ and the change in h is insensitive to the frequency. The sameconclusions hold true for δb where decennial variations seem more realistic.

0.0 0.2 0.4 0.6 0.8 1.0

t (a) ×101

−2

−1

0

wC

(x∗)

×10−5 F (u, h) = uδ(x− x∗)

0.0 0.2 0.4 0.6 0.8 1.0

t (a) ×101

0

1

2

3

4

5

wC

(x∗)

×10−7 F (u, h) = hδ(x− x∗)summer

fall

winter

spring

no seasonal

0.0 0.2 0.4 0.6 0.8 1.0

t (a) ×101

−5

0

5

wb(x∗)

×10−6 F (u, h) = uδ(x− x∗)

0.0 0.2 0.4 0.6 0.8 1.0

t (a) ×101

0.0

0.5

1.0

1.5

2.0

wb(x∗)

×10−5 F (u, h) = hδ(x− x∗)summer

fall

winter

spring

no seasonal

Figure 9: The adjoint weights for the observations of u and h at x∗ in the fourseasons of the tenth year with seasonal varying friction coefficient. The blackdashed line is a reference solution without seasonal variations which is observedat t = 9.75.

4 Conclusions

The sensitivity of the flow of ice to basal conditions is analyzed for time dependentand steady state solutions of the FS and SSA equations including the advectionequation for the height. Perturbations at the base of the ice are introduced in the

28

friction coefficient C and the topography b. The analysis relies on the adjointequations of the FS and SSA stress and advection equations. The adjoint FS andSSA equations follow from the Lagrangians of the forward FS and SSA equations.

The adjoint equations are derived for observations of the velocity u and heighth at the top surface of the ice. The adjoint height equation in the FS model issolved analytically for a two dimensional vertical ice. The relation between theinverse problem to find parameters from data and the sensitivity problem is es-tablished. The same adjoint equations are solved in the inverse problem but withother forcing functions.

If the perturbations in the basal conditions are time dependent then time cannotbe ignored in the inversion. It is necessary to include the adjoint height equationif h is observed. The wrong conclusions may be drawn with only static snapshotsfor both the FS and the SSA model. If u is observed then the contribution ofthe solution of the adjoint height equation is small and it is sufficient to solveonly the adjoint stress equation as is the case in many articles on inversion e.g.(Gillet-Chaulet et al. 2016 Isaac et al. 2015 Petra et al. 2012).

The adjoint equations of the FS and SSA models are similar and the analyticalsolutions based on the SSA equations for a two dimensional vertical ice show thatthe sensitivity grows as the observation point x∗ is approaching the grounding lineseparating the grounded and the floating parts of the ice. The reason is that thevelocity increases and the thickness of the ice decreases.

In the steady state solution of SSA, there is a non-local effect of a perturbation δCin C in the sense that δC(x) affects both u(x∗) and h(x∗) even if x 6= x∗, but δbhas a strong local effect concentrated at x∗. Nevertheless, the shapes of the twosensitivity functions for δb and δC are very similar except for the neighborhood ofx∗. It is possible to separate the effect of δb and δC in the steady state SSA modelthanks to the localized influence of δb in δu and δh. The same effect on δu andδh at one observation point can be achieved by different δC and δb. Perturbationsin C at the base observed in u at the surface are damped inversely proportional tothe wavenumber of δC thus making high wavenumber perturbations difficult toregister at the top.

In the time dependent solution of SSA, the strongest impact of a perturbation atx∗ is made at x∗ at the surface possibly with a time delay. There is a local effectof δC and δb at (x∗, t∗) on u∗ but not on h∗ where perturbations are caused byδC(x, t) and δb(x, t) with t < t∗. There is a time delay when a perturbation at the

29

ice base is visible at the surface in h but in u it is observed immediately. As in thesteady state, the same effect on δu and δh can be achieved by different δC andδb. The higher the frequency of the perturbation is the more it is damped at thesurface, but perturbations of low frequency reach the upper surface.

The numerical results in (Cheng and Lötstedt 2019) confirm the conclusions hereand are in good agreement with the analytical solutions.

A Derivation of the adjoint equations

A.1 Adjoint viscosity and friction in SSA

The adjoint viscosity η̃(u) in SSA in (14) is derived as follows. The SSA viscosityfor u and u+δu is

η(u+δu)

≈ η(u)(

1+ 1−n2n(2u1x+u2y)δu1x+ 12 (u1y+u2x)δu2x+(2u2y+u1x)δu2y+

12 (u1y+u2x)δu1y

η̂

).

(44)Determine B(u) such that

ρ(u,δu)B(u) = B(u)?B(δu).

First note that

B(u) : D(δu) = (D(u)+∇ ·uI) : D(δu) = D(u) : D(δu)+(∇ ·u)(∇ ·δu)= D(u) : (B(δu)−∇ ·δuI)+(∇ ·u)(∇ ·δu) = D(u) : B(δu).

Then use the ? operator in (16) to define B1−n2nη̂ ∑kl Bkl(u)Dkl(δu)Bi j(u) =

1−n2nη̂ ∑kl Dkl(u)Bkl(δu)Bi j(u)

= ∑kl Bi jkl(u)Dkl(δu) = (B ?D)i j.

Thus, let

Bi jkl =1−n2nη̂

Bi j(u)Dkl(u), η̃i jkl(u) = η(u)(Ii jkl +Bi jkl(u)),

or in tensor form

B =1−n

nB(u) : D(u)B(u)⊗D(u), η̃(u) = η(u)(I +B) . (45)

30

Replacing B in (45) by D we obtain the adjoint FS viscosity in (14).

The adjoint friction in SSA in ω and at γg in (26) with a Weertman law is derivedas in the adjoint FS equations (13) and (14). Then in ω with ξ = u,ζ = v,c =C,F = Fω , and at γg with ξ = t ·u,ζ = t · v,c = Cγ , f = fγ ,F = Fγ , we arrive atthe adjoint friction term c f (ξ )(I+F(ξ ))ζ where

F(ξ ) =m−1ξ ·ξ ξ ⊗ξ . (46)

A.2 Adjoint equations in SSA

The Lagrangian for the SSA equations is with the adjoint variables ψ,v,q

L (u,h;v,ψ;b,Cγ ,C) =∫ T

0∫

ω F(u,h)+ψ(ht +∇ · (uH)−a)dxdt+∫ T

0∫

ω v ·∇ · (2HηB(u))−C f (u)v ·u−ρgHv ·∇h dxdt=∫ T

0∫

ω F(u,h)+ψ(ht +∇ · (uH)−a)dxdt+∫ T

0∫

ω−2Hη(u)(D(v) : D(u)+∇ ·u∇ ·v)−C f (u)v ·u−ρgHv ·∇h dxdt− ∫ T0 ∫γg Cγ fγ(t ·u)t ·ut ·vdsdt

(47)

after partial integration and using the boundary conditions. The perturbed SSALagrangian is split into the unperturbed Lagrangian and three integrals

L (u+δu,h+δh;v+δv,ψ +δψ;b+δb,Cγ +δCγ ,C+δC)=∫ T

0∫

ω F(u+δu,h+δh)+∫ T

0∫

ω(ψ +δψ)(ht +δht +∇ · ((u+δu)(H +δH))−a)dxdt+∫ T

0∫

ω−2(H +δH)η(u+δu)D(v+δv) : B(u+δu)−(C+δC) f (u+δu)(u+δu) · (v+δv)−ρg(H +δH)∇(h+δh) · (v+δv) dxdt−∫ T0 ∫γg(Cγ +δCγ) fγ(t · (u+δu))t · (u+δu) t · (v+δv)dsdt= L (u,h;v,ψ;b,Cγ ,C)+ I1 + I2 + I3.

(48)

The perturbation in L isδL = I1 + I2 + I3. (49)

Terms of order two or more in δL are neglected. Then the first term in δLsatisfies

I1 =∫ T

0∫

ω F(u+δu,h+δh)−F(u,h)dxdt=∫ T

0∫

ω Fuδu+Fhδhdxdt.(50)

31

Using partial integration, Gauss’ formula, and the initial and boundary conditionson u and H and ψ(x,T ) = 0,x∈ω, and ψ(x, t) = 0, x∈ γw, in the second integralwe have

I2 =∫ T

0∫

ω δψ(ht +∇ · (uH)−a)+ψ(δht +∇ · (δuH)+∇ · (uδH))dxdt

=∫ T

0∫

ω δψ(ht +∇ · (uH)−a)dxdt+∫ T

0∫

ω−ψtδh−H∇ψ ·δu−∇ψ ·uδH dxdt.

(51)

The first integral after the second equality vanishes since h is a weak solution andI2 is

I2 =∫ T

0∫

ω−(ψt +u ·∇ψ)δh−H∇ψ ·δu+u ·∇ψδbdxdt. (52)

Using the weak solution of (10), the adjoint viscosity (27), (45), the friction coef-ficient (46), Gauss’ formula, the boundary conditions, and neglecting the secondorder terms, the third and fourth integrals in (48) are

I3 = I31 + I32,I31 =

∫ T0∫

ω−2(H +δH)η(u+δu)D(v+δv) : B(u+δu)−(C+δC) f (u+δu)(u+δu) · (v+δv)−ρg(H +δH)∇(h+δh) · (v+δv)) dxdt−∫ T0 ∫γ(Cγ +δCγ) fγ(t · (u+δu))t · (u+δu) t · (v+δv)dsdt

= I311 + I312− I313,

(53)

where

I311 =∫ T

0∫

ω−2HD(v) : (η(u+δu)B(u+δu))+2HD(v) : (η(u)B(u))dxdt

=∫ T

0∫

ω−2HD(v) : (η̃(u)?B(δu))dxdtI312 =

∫ T0∫

ωg−δC f (u)u ·vdxdt+∫ T

0∫

ωg−C( f (u+δu)v · (u+δu)− f (u)v ·u)dxdt=

∫ T0∫

ωg−δC f (u)u ·v+C f (u)(I+Fω(u))δu ·vdxdtI313 =

∫ T0∫

γg(Cγ +δCγ)( fγ(t · (u+δu))t ·vt · (u+δu)− fγ(t ·u)t ·vt ·u)dsdt

=∫ T

0∫

γg(Cγ +δCγ)( fγ(t ·u)t ·ut ·v+Cγ fγ(t ·u)(I+Fγ(t ·u))t ·δut ·vdsdt

I32 =∫ T

0∫

ω−ρgH∇h ·v−2ηD(v) : B(u)δH−ρg∇h ·vδH−ρgHv ·∇δhdxdt

=∫ T

0∫

ω−ρgH∇h ·v− (2ηD(v) : B(u)+ρg∇h ·v)δH+ρg∇ · (Hv)δhdxdt.

(54)

32

Collecting all the terms in (50), (52), and (53), the first variation of L is

δL = I1 + I2 + I3=

∫ T0∫

ω Fuδu−2HD(v) : (η̃(u)?B(δu))−H∇ψ ·δudxdt−∫ T0 ∫ωg C f (u)(I+Fω(u))v ·δudxdt−∫ T0 ∫γg Cγ fγ(t ·u)(I+Fγ(t ·u))t ·vt ·δudsdt−∫ T0 ∫γg δCγ fγ(t ·u)t ·ut ·vdsdt+∫ T

0∫

ω(Fh− (ψt +u ·∇ψ +2ηD(v) : B(u)−ρg∇b ·v+ρgH∇ ·v))δhdxdt+∫ T

0∫

ω−δC f (u)v ·u+(2ηD(v) : B(u)+ρg∇h ·v+u ·∇ψ)δbdxdt.

(55)

The forward solution (u∗, p∗,h∗) and adjoint solution (v∗,q∗,ψ∗) satisfying (10)and (26) are inserted into (47) resulting in

L (u∗, p∗;v∗,q∗;h∗,ψ∗;b,Cγ ,C) =∫ T

0∫

ω F(u∗,h∗)dxdt. (56)

Then (55) yields the variation in L in (56) with respect to perturbations δb,δCγ ,and δC in b,Cγ , and C

δL =∫ T

0∫

ω(2ηD(v∗) : B(u∗)+ρg∇h∗ ·v∗+u∗ ·∇ψ∗)δbdxdt−∫ T0 ∫γg δCγ fγ(t ·u∗)t ·u∗ t ·v∗ dsdt−∫ T0 ∫ω δC f (u∗)v∗ ·u∗ dxdt.

(57)

A.3 Adjoint equations in FS

The FS Lagrangian is

L (u, p,h;v,q,ψ;C) =∫ T

0∫

Γs F(u,h)+ψ(ht +h ·u−a)dxdt+∫ T

0∫

ω∫ h

b −v · (∇ ·σ(u, p))−q∇ ·u−ρg ·v dxdt=∫ T

0∫

Γs F(u,h)+ψ(ht +h ·u−a)dxdt+∫ T

0∫

ω∫ h

b 2η(u)D(v) : D(u)− p∇ ·v−q∇ ·u−ρg ·v dxdt+∫ T

0∫

Γb C f (Tu)Tu ·Tvdxdt.

(58)

In the same manner as in (48), the perturbed FS Lagrangian is

L (u+δu, p+δ p;v+δv,q+δq;h+δh,ψ +δψ;C+δC)= L (u, p,h;v,q,ψ;C)+ I1 + I2 + I3.

(59)

33

Terms of order two or more in δu,δv,δh are neglected. The first integral I1 in(59) is

I1 =∫ T

0∫

Γs F(u(x,h+δh, t)+δu,h+δh)−F(u(x,h, t),h)dxdt=∫ T

0∫

Γs Fu(δu+uzδh)+Fhδhdxdt.(60)

Partial integration, the conditions ψ(x,T ) = 0 and ψ(x, t) = 0 at Γs, and the factthat h is a weak solution simplify the second integral

I2 =∫ T

0∫

Γs δψ(ht +h ·u−a)+ψ(δht +u ·δh+uz ·hδh+h ·δu)dxdt

=∫ T

0∫

Γs δψ(ht +h ·u−a)dxdt+∫ T

0∫

Γs(−ψt−∇ · (uψ)+h ·uzψ)δh+h ·δuψ dxdt.

(61)

Define Ξ,ξ , and ϒ to be

Θ(u, p;v,q;C) = 2η(u)D(v) : D(u)− p∇ ·v−q∇ ·u−ρg ·v,θ(u;v;C) = C f (Tu)Tu ·Tv,

ϒ(u, p;v,q) = −v · (∇ ·σ(u, p))−q∇ ·u−ρg ·v.(62)

Then a weak solution, (u, p), for any (v,q) satisfying the boundary conditions,fulfills ∫ T

0

∫ω

∫ hb

Θ(u, p;v,q;C)dxdt−∫ T

0

∫Γb

θ(u;v;C)dxdt = 0. (63)

The third integral in (59) is

I3 = I31 + I32,I31 =

∫ T0∫

ω∫ h

b Θ(u+δu, p+δ p;v+δv,q+δq;C+δC)dxdt−∫ T0 ∫Γb θ(u+δu;v+δv;C+δC)dxdt,

I32 =∫ T

0∫

ω∫ h+δh

h ϒ(u, p;v,q)dxdt.

(64)

The integral I31 is expanded as in (53) and (54) or (Petra et al. 2012) using theweak solution, Gauss’ formula, and the definitions of the adjoint viscosity andadjoint friction coefficient in Sect. A.1. When b < z < h we have ϒ(u, p;v,q) = 0.If ϒ is extended smoothly in the positive z-direction from z = h, then with z ∈[h,h+δh] for some constant c > 0 we have |ϒ| ≤ cδh. Therefore,

|∫ h+δh(x,t)

hϒ(u, p;v,q)dz| ≤

∫ h+δh(x,t)h

sup |ϒ|dz≤ c|δh(x, t)2|,

34

and the bound on I32 in (64) is

|I32| ≤ ct|ω|max |δh(x, t)|2, (65)where |ω| is the area of ω . This term is a second variation in δh which is neglectedand I3 = I31.

The first variation of L is then

δL = I1 + I2 + I3=

∫ T0∫

Γs(Fu +ψh) ·δudxdt+∫ T

0∫

Γs(Fh +Fuuz− (ψt +∇ · (uψ)−h ·uψ))δhdxdt+∫ T

0∫

ω∫ h

b 2D(v) : (η̃(u)?D(δu))−δ p∇ ·v−q∇ ·δudxdt+∫ T

0∫

Γb C f (Tu)(I+Fb(u))Tv ·Tδudxdt+∫ T

0∫

Γb δC f (Tu)Tu ·Tvdxdt.

(66)

With the forward solution (u∗, p∗,h∗) and the adjoint solution (v∗,q∗,ψ∗) satis-fying (5) and (13), the first variation with respect to perturbations δC in C is (cf.(57))

δL =∫ T

0

∫Γb

f (Tu∗)Tu∗ ·Tv∗ δC dxdt. (67)

B Simplified SSA equations

The forward and adjoint SSA equations in (32) and (33) are solved analytically.The conclusion from the thickness equation in (32) is that

u(x)H(x) = u(0)H(0)+ax = ax, (68)

since u(0)= 0. Solve the second equation in (32) for u on the bedrock with x≤ xGLand insert into (68) using the assumptions for x > 0 that bx� Hx and hx ≈ Hx tohave ρg

CHm+1Hx =

ρgC(m+2)

(Hm+2)x =−(ax)m. (69)

The equation for Hm+2 for x≤ xGL is integrated from x to xGL such that

H(x) =(

Hm+2GL +m+2m+1

Cam

ρg(xm+1GL − xm+1)

) 1m+2

,

u(x) =axH, Hx =−

Cam

ρgxm

Hm+1.

(70)

35

For the floating ice at x> xGL, ρgHhx = 0 implying that hx = 0 and Hx = 0. Hence,H(x) = HGL. The velocity increases linearly beyond the grounding line

u(x) = ax/H(x) = ax/HGL, x > xGL. (71)

By including the viscosity term in (29) and assuming that H(x) is linear in x, amore accurate formula is obtained for u(x) on the floating ice in (6.77) of (Greveand Blatter 2009).

C Jumps in ψ and v in SSA

Multiply the first equation in (33) by H and the second equation by u to eliminateψx. We get

−Cmumv−ρgH2vx = HFh−uFu. (72)Use the expression for u and Hx in (70). Then

ρgH(mHxv−Hvx) = HFh−uFu, (73)

or equivalently ( vHm

)x=− 1

ρgHm+2(HFh−uFu). (74)

The solutions ψ(x) and v(x) of the adjoint SSA equation (30) have jumps at theobservation point x∗. For x close to x∗ in a short interval [x−∗ ,x

+∗ ] with x

−∗ < x∗ <

x+∗ , integrate (74) to receive∫ x+∗x−∗

( vHm

)x

dx =−∫ x+∗

x−∗

HFh−uFuρgHm+2

dx. (75)

Since H is continuous and u and v are bounded, when x−∗ → x+∗ , then

v(x+∗ )− v(x−∗ ) =−1

ρgH2∗

(H∗∫ x+∗

x−∗Fh dx−u∗

∫ x+∗x−∗

Fu dx). (76)

A similar relation for ψ can be derived

ψ(x+∗ )−ψ(x−∗ ) =1

H∗

∫ x+∗x−∗

Fu dx. (77)

36

With Fu = 0 and Fh = 0 for x < x∗ and v(0) = ψx(0) = 0, we find that

v(x) = ψx(x) = 0, ψ(x) = ψ(x−∗ ), 0≤ x < x∗. (78)

If F(u,h) = uδ (x− x∗), then by (76) and (77)

v(x+∗ ) =u∗

ρgH2∗, ψ(x+∗ )−ψ(x−∗ ) =

1H∗

, (79)

and if F(u,h) = hδ (x− x∗), then

v(x+∗ ) =−1

ρgH∗, ,ψ(x+∗ )−ψ(x−∗ ) = 0. (80)

D Analytical solutions in SSA

By Sect. C, v(x) = 0 for 0 ≤ x < x∗. Use equations in (33) with Hx in (70) forx∗ < x≤ xGL to have

vxv=−axCmu

m−1

ρgH3=−Cmu

m

ρgH2=

mHxH

.

Let H (x− x∗) =∫ x−x∗−∞ δ (s)ds be the Heaviside step function at x∗. Then

v(x) =CvH(x)mH (x− x∗), 0≤ x≤ xGL. (81)

To satisfy the jump condition in (79) and (80), the constant Cv is

Cv =

ax∗

ρgHm+3∗, F(u,h) = uδ (x− x∗),

− 1ρgHm+1∗

, F(u,h) = hδ (x− x∗).(82)

Combine (81) with the relation ψx = (Fu−Cmum−1v)/H and integrate from x toxGL to obtain

ψ(x) =Cvam−1C (xmGL− xm) , x∗ < x≤ xGL. (83)

37

With the jump condition in (79) and (80), ψ(x) at 0≤ x < x∗ is

ψ(x) =

− 1

H∗+

Camx∗ρgHm+3∗

(xmGL− xm∗ ), F(u,h) = uδ (x− x∗),

− Cam−1

ρgHm+1∗(xmGL− xm∗ ), F(u,h) = hδ (x− x∗).

(84)

The weight for δC in the functional δL in (31) is non-zero for x∗ < x≤ xGL

−vum =−Cv(ax)m. (85)

Use (81) and (33) in (31) to determine the weight for δb in δL ,

ψxu+ vxηux + vρghx = ρg(Hv)x+Fh=CvρgHm [(m+1)HxH (x− x∗)+Hδ (x− x∗)]+Fh. (86)

Acknowledgement

This work was supported by Nina Kirchner’s Formas grant 2017-00665. Lina vonSydow read a draft of the paper and helped us improve the presentation with hercomments.

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