A Mathematical Study of the Ice Flow Behavior in a Neighborhood of theGrounding Line
M
ARCO
A. F
ONTELOS
1
and A
NA
I. M
UN˜OZ
2
Abstract—
The description of the ice ﬂow in marine ice sheets is one of the problems that has attracted moreattention in the Scientiﬁc community interested in the motion of glaciers. It is widely assumed that the stabilityof the marine ice sheets, as in the West Antarctic Ice Sheet (WAIS), where ice shelves are formed, is mainlycontrolled by the dynamics of the grounding line. The grounding line is the line where transition between iceattached to the solid ground and ice ﬂoating over the sea takes place. In this paper, we present the analysis of amathematical model describing the behavior of the ice ﬂow in the neighborhood of the grounding line, whenconsidering the ice to be a ﬂuid with sheardependent viscosity of powerlaw type, including, as a particularcase, the Newtonian one. We prove the existence of solutions representing the transition from ice sheet to iceshelf and with ﬁnite viscous dissipation near the grounding line. The interface between the ice shelf and seawater is proved to be locally ﬂat near the grounding line.
Key words:
Free boundary problem, glaciology, Stokes’ ﬂow problem, NonNewtonian ﬂuid, mixed typeboundary conditions, convex analysis, existence and uniqueness of weak solutions, numerical resolution.
1. Introduction
Marine ice sheets are characterized by having much of the grounded ice lying on asubmarine bed and usually, at the continental margin, the ice ﬂow continues to ﬂoat onthe sea. Then, tongues of ice, which are called ice shelves, are formed. An example of thissituation can be found at the West Antarctic Ice Sheet (WAIS). The moving boundarythat localizes the transition from grounding ice to ﬂoating ice is termed grounding line. Itis widely accepted that the dynamics of the grounding line is of considerable importanceregarding the stability of the ice sheet, and therefore, of interest for predicting possiblesealevel rises. Some distance away from this grounding zone, either in the ice shelf or inthe ice sheet, there is some consensus as to which simpliﬁcations can be made in thestress equilibrium, either to have ﬂow dominated by shearing in horizontal planes, mostof it at the base as in the ice sheet, or to have the ﬂow dominated by lateral shearing and
1
Instituto de Matema´ticas y Fı´sica Fundamental, Consejo Superior de Investigaciones Cientı´ﬁcas,c/ Serrano, n 123, 28006 Madrid, Spain. Email: marco.fontelos@uam.es
2
Dpto. de Matema´tica Aplicada, ESCET, Univ. Rey Juan Carlos, c/ Tulipa´n s/n, 28933, Mo´stoles, Madrid,Spain. Email: anaisabel.munoz@urjc.esPure appl. geophys. 165 (2008) 1603–1618
Birkha¨user Verlag, Basel, 20080033–4553/08/081603–16
DOI 10.1007/s0002400403916
Pure and Applied Geophysics
longitudinal stretching, as in the ice shelf. The fundamental difference between the twoﬂow regimes seems to suggest the existence of a transition zone where all the stresscomponents are important and no simpliﬁcation can be made. The grounding linemigration and the coupling of ice sheet ﬂow with ice shelf ﬂow can rightly be consideredas one of those challenging modelling problems, which is nevertheless of primeimportance because, as we mentioned, it is the predominant mechanism by which theAntarctic ice sheet changes its dimensions. In the context of the study of ice ﬂow near thegrounding line, there are several recent contributions mainly due to H
INDMARSH
(1993),S
CHOOF
(2007a,b), V
IELI
and P
AYNE
(2005), W
ILCHINSKY
and C
HUGONOV
(2000), andN
OWICKI
and W
INGHAM
(2007). In these works various modelling approaches are proposedand the analysis of some stability hypothesis regarding the conﬁguration of the ice sheetice shelf junction is presented. One of the main challenges, both from the analysis and thesimulation point of view is the possible presence of singularities in the stresses close tomoving contact lines. Related to this fact, the analysis of the structure of the ﬂuid ﬂow inthe neighborhood of moving contact lines has been developed in relative to the spreadingof viscous ﬂuids over solid substrate (cf., F
RIEDMAN
and V
ELA´ZQUEZ
, 1995; F
ONTELOS
andV
ELA´ZQUEZ
, 1998). In this survey, we shall analyze a mathematical model describing theice ﬂow in a neighborhood of the grounding line. From the mathematical point of view,we are going to deal with a nonlinear free boundary problem, being the free boundary theone which corresponds to the part of the ice in contact with the sea. A more detailedtreatment of the theory presented here can be found in F
ONTELOS
and M
UN˜OZ
(2007),F
ONTELOS
et al
. (submitted). The aim of our analysis is to determine the velocity andpressure ﬁelds in the neighborhood of the grounding line and also the location andgeometry of the free boundary. In sections 2 and 3, we brieﬂy describe the setting of themathematical problem. In section 4, we shall prove the existence and uniqueness of asolution, with a moving grounding line, to the free boundary problem in the Newtoniancase. Moreover, we determine, analytically the ice ﬂow and the asymptotic behavior of the free boundary in the area of the grounding line. In section 5 we solve numerically theproblem for a general powerlaw rheology obtaining the asymptotic properties of the freeboundary and of the ﬂow when approaching the grounding line.
2. The Mathematical Problem
In the present work we aim to determine the ice ﬂow, together with the geometry andlocation of the free boundary near the grounding line. In order to do that we makemodeling simpliﬁcations. First, we consider a twodimensional domain by assumingindependence of all physical quantities in the third variable. This is a strong assumption,since it is known that remoteness from the grounding line ﬂow may be controlled bylateral stresses. Nevertheless, our results on the behavior of the ﬂow near the groundingline are essentially local and are expected to be valid even without the assumptionsconcerning symmetry. Changes due to temperature variations are neglected, therefore a
1604 M. A. Fontelos and A. I. Mun˜oz Pure appl. geophys.,
steady state and isothermal regime is considered. We assume that the contact line ismoving with constant velocity
U
in the main ﬂow direction (the xdirection, see Fig. 1).The ice is modelled as a ﬂuid with sheardependent viscosity of powerlaw type,including, as a particular case, the Newtonian one. In Figure 1, it is depicted the domain
D
:
¼ fð
x
;
z
Þ 2 ½
M
;
M
½
1
þ
b
ð
x
Þ
;
0
g
;
with
b
(
x
)
:
0 in
x
B
0, which represents avicinity of the grounding line. Two ﬂow regions appear where the ice ﬂow is wellunderstood. One is the inland ice sheet, where the ice ﬂow is dominated by shearing onhorizontal planes, most of it at the base, and the other one is the ice shelf, where the iceﬂow is dominated by lateral shearing and longitudinal stretching. Therefore one mightsuppose the existence of a transition region, near the grounding line, where all thecomponents of the stress should be of importance.The upper boundary of
D
;
denoted
C
0
, is the one corresponding to the ice in contactwith the air. We shall presume it to be ﬂat and located at
z
=
0. The part of the ice incontact with the solid substrate, denoted by
C
1
is also assumed to be ﬂat and located at
z
= 
1. The only free boundary we shall consider is
C
2
, the bottom of the ice ﬂoating onthe sea, described by
z
= 
1
?
b
(
x
). Note that at the grounding line, located at
x
=
0and
z
= 
1, different boundary conditions shall meet. This fact might give rise tosingularities in the solution. We shall consider a frame of reference attached to thegrounding line, which is equivalent to assume that the solid substrate is moving withconstant velocity

U
and the velocity ﬁeld is given by (

U
, 0)
?
(
u
,
w
). We shall useindistinctly the notation (
u
,
w
) and (
v
1
,
v
2
) when referring to the components of thevelocity vector. The strain tensor will be denoted by
D
, where
D
ij
¼
12
o
v
i
o
x
j
þ
o
v
j
o
x
i
;
x
1
¼
x
;
x
2
¼
z
;
i
;
j
¼
1
;
2
;
v
1
¼
u
;
v
2
¼
w
;
Figure 1Proximity of the grounding line. Flow domain
D
.Vol. 165, 2008 A Mathematical Study of the Ice Flow 1605
and the viscosity
l
depends on strain in the following way:
l
ðj
D
jÞ ¼
l
0
j
D
j
1
n
1
;
with
j
D
j
given by
j
D
j ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
D
:
D
p ¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
122
o
u
o
x
2
þ
122
o
w
o
z
2
þ
o
u
o
z
þ
o
w
o
x
2
s
ð
2
:
1
Þ
and
l
0
a constant. The exponent
n
is usually assumed for the ice ﬂow to be
n
=
3 (cf.,G
LEN
, 1958; L
LIBOUTRY
, 1964; H
UTTER
, 1981). Note that for
n
=
1, we are considering theice as a Newtonian ﬂuid. Due to the fact that in the Newtonian case, the problem issimpler because the viscosity is of constant value, we shall devote a subsection to itsstudy alone, since more regularity for the solution can be obtained in this case. Theequations to be considered are the equilibrium and mass conservation equations,
r
T
¼
0
;
r
v
!¼
0 in
D
;
ð
2
:
2
Þ
with
T
ij
= 
p
d
ij
?
l
D
ij
. The pressure
p
is in fact a reduced pressure, resulting fromthe substraction of
q
ice
g
(
z
?
1) from the real pressure, in order to absorb the gravityforces acting on ice (see, F
ONTELOS
and M
UN˜OZ
, 2007). In fact, it reduces to the actualpressure in the idealized case
q
ice
=
0. Concerning the boundary conditions at the upperpart of the sheet which is in contact with air, denoted by
C
0
, and assumed to be ﬂat, i.e.,
C
0
=
{(
x
, 0),
x
[
[

M
,
M
]}, we impose that both the normal velocity and the shear stressare zero there, i.e.,
v
~
n
~
¼
0
;
t
!
T n
!¼
0 at
C
0
;
ð
2
:
3
Þ
where
n
!
and
t
!
are the normal and tangent vectors to
C
0
, respectively. In the part of thebottom of the ice sheet which is in contact with the solid substrate we impose
v
~
¼
0
!
:
Inthe other part, which is in contact with the sea, we impose a balance between viscousstresses and the hydrostatic pressure. Therefore, at the base, depending on whether the iceis grounded or ﬂoating over the sea, we impose:
v
~
¼
0
!
on
C
1
;
ð
2
:
4
Þ
t
!
T n
!¼
0
;
T n
!¼
c
Ub
ð
x
Þ
n
!
on
C
2
;
ð
2
:
5
Þ
where
c
is a dimensionless parameter deﬁned by
c
¼
q
g
½
L
1
n
þ
1
=
U
l
0
;
based on thedensities of water and ice,
q
=
q
water

q
ice
, the velocity
U
, the typical longitudinallength scale and the factor
l
0
in the law of the viscosity (2.1). As inﬂow andoutﬂow conditions, we shall impose the values
v
!
in
ð
z
Þ
and
v
!
out
ð
z
Þ
of the velocity ﬁeld, at
x
?

M
and
x
?
?
M
, respectively, and assume, by compatibility with the boundaryconditions, that
v
~
in
ð
1
Þ ¼
0
!
;
v
~
in
ð
0
Þ
n
~
¼
0
;
and the shear stress
t
!
T n
!
is zero at(

M
, 0). We should choose
v
~
in
ð
z
Þ
to be of Poiseuille ﬂow type according to the physicsof the problem, that is
1606 M. A. Fontelos and A. I. Mun˜oz Pure appl. geophys.,
v
~
in
¼
r
2
ð
1
j
z
j
n
þ
1
Þ
;
0
:
ð
2
:
6
Þ
However, the study that we shall develop here is valid for more general inﬂowconditions. The outﬂow
v
~
out
ð
z
Þ
is such that
w
out
(0)
=
0 and the shear stress at(
M
, 0) and (
M
,

1) are zero. We may think, for instance, on a uniform velocityproﬁle, i.e.,
v
~
out
¼ ð
U
1
;
0
Þ
:
ð
2
:
7
Þ
Here
r
and
U
?
are two arbitrary positive parameters which are nevertheless linked by theconservation of mass constraint which implies
U
þ
Z
0
1
u
in
ð
z
Þ
dz
¼
U
ð
1
b
ð
M
ÞÞþ
Z
0
b
ð
M
Þ
u
out
ð
z
Þ
dz
;
hence, the velocity of the grounding line can then be determined from the givenphysically measurable entities
u
in
(
z
),
u
out
(
z
) and
b
(
M
):
U
¼
1
b
ð
M
Þ
Z
0
1
u
in
ð
z
Þ
dz
Z
0
b
ð
M
Þ
u
out
ð
z
Þ
dz
!
:
ð
2
:
8
Þ
Somewhat different outﬂow conditions appear for example in S
CHOOF
(2007) and inN
OWICKI
and W
INGHAM
(2007), where iceshelf stresses are prescribed, and the case inwhich the ice is allowed to slide over the bed is also considered. The boundary
C
2
mayevolve following the velocity ﬁeld, but in the stationary situation will be such that
½ð
U
;
0
Þþð
u
;
w
Þ
n
!¼
0
;
so that0
¼
Ub
x
þ
w
b
x
u
:
ð
2
:
9
Þ
The main results and conclusions of this study are stated in the following theorems. Webegin with the one referred to the Newtonian case with
n
=
1:
Theorem 2.1
.
Consider n
=
1,
then if v
!
in
and
v
!
out
are such that the norms
k
v
!
in
k
C
2
1
;
0
½
;
k
v
!
out
k
C
2
1
þ
b
ð
M
Þ
;
0
½
are small enough and b
(
M
)
is also small enough thenthere exists a unique (weak) solution v
!2 ½
H
1
ð
D
Þ
2
to the nonlinear free boundary problem given by
(2.2)–(2.7)
with U given by
(2.8).
Moreover, the free boundary b
(
x
)
[
C
1
?
d
[ 0,
M
]
and its asymptotic behavior near the grounding line located at
(
x
=
0,
z
= 
1)
is the following
:
b
ð
x
Þ ¼
Cx
32
þ
o
ð
x
32
Þ
:
Therefore, the velocity does not show any singularity near the grounding line,however, shear stress does, with a singularity of order
O
ð
x
12
Þ
:
Vol. 165, 2008 A Mathematical Study of the Ice Flow 1607