Abstract
In this article, we formulate a model describing the evolution of thickness of a grounded shallow ice sheet. The thickness of the ice sheet is constrained to be nonnegative. This renders the problem under consideration an obstacle problem. A rigorous analysis shows that the model is thus governed by a set of variational inequalities that involve nonlinearities in the time derivative and in the elliptic term, and that it admits solutions, whose existence is established by means of a semi-discrete scheme and the penalty method.
1 Introduction
The study of ice sheets melting and its correlation with the global warming problem has been attracting the interest of experts from all over the branches of science.
In this article, we propose and study a mathematical model describing the evolution of the thickness of a grounded shallow ice sheet (from now on, simply shallow ice sheet). The ice thickness of a shallow ice sheet evolves as a consequence of many factors like, for instance, the rate at which snow deposits, the rate at which melting occurs, as well as the velocity at which the glacier slides along the lithosphere. Since the ice thickness level is constrained to remain on or above the lithosphere at all times, the problem under consideration can be regarded as a time-dependent obstacle problem.
Grounded ice sheets and sea ice are two related but physically different problems. The mathematical literature related to ice sheets and glaciers and which deals with sea ice and glaciers is vast and abundant; in this direction, we mention, for instance, the previous articles [31,32,34,35,40,56,57] and the celebrated article by Hibler [29] to which we refer later on, and which deals with sea ice. The local-in-time well-posedness of Hibler’s model has recently been established in the article by Liu et al. [39]. The authors designed a regularizing scheme based on physical observations for deriving the sought local-in-time well-posedness.
Brandt et al. discussed in [7] the local-in-time well-posedness of Hibler’s model by means of a different regularization approach. The authors constructed the local strong solutions of the corresponding regularized system by investigating the analytic semi-group property of the parabolic operator of the regularized system in a bounded domain, and they also established the global well-posedness in time for initial data close to constant equilibria.
In this direction, we also mention the recently appeared pre-prints [2,3,8].
In the recent pre-print [24], Figalli et al. studied the phase transition of ice melting to water as a Stefan problem. The results established in this article provide a refined understanding of the Stefan problem’s singularities and answer some long-standing open questions in the field of free-boundary problems.
In 2002, Calvo et al. [13] published a paper discussing the evolution of the thickness of a shallow ice sheet as an obstacle problem. In their article, the authors only considered one spatial direction, they assumed the basal velocity to be smooth and the lithosphere to be flat. This seminal paper is, to our best knowledge, the very first record in the literature where a sound model describing the evolution of ice sheets thickness in the context of obstacle problems is discussed. This formulation was also exploited to justify the generation of fast ice streams in ice sheets flowing along soft and deformable beds [17]. In this direction, we also cite the related previous paper [18].
Ten years later, in 2012, Jouvet and Bueler [33] studied the steady (i.e., time-independent) version of the problem considered in [13], where, this time, two spatial directions and a more general lithosphere topography were taken into account.
It appears that the ice thickness evolution as a time-dependent obstacle problem over a two-dimensional spatial domain has not been addressed in the literature yet. The purpose of this article is exactly to address this problem under suitable assumptions.
This article is divided into four sections (including this one). In Section 2, we present the main notations we shall be using throughout the article, and we formally derive the governing equations for our model.
In Section 3, we formulate the “penalized” version corresponding to the obstacle problem introduced in Section 2, and we establish the existence of solutions to this model by resorting to a series of new preparatory results as well as the Dubinskiĭ’s compactness theorem.
Finally, in Section 4, we pass to the limit in the penalty parameter, and we recover the actual model corresponding to the model we formally derived in Section 2. This model, for which we also define the rigorous concept of solution, will take the form of a set of variational inequalities. The presence of the constraint, which adds a further nonlinearity to the two already considered (the first nonlinearity appears in the evolutionary term, while the second nonlinearity is the
2 Notations and formal derivation of the model
We denote by
A domain in
Let
and let the symbol
Let
The lithosphere elevation is described by the function
The elevation of the upper ice surface is described by the function
The ice thickness
This constraint implies the existence of a free boundary [10,12,23]. For all, or possibly almost every (a.e. in what follows)
The set
The variation of the ice thickness
In what follows, we assume that
and we recall that this function is assumed to be continuous in
Ice thickness is also influenced by the basal sliding velocity
and we recall that (cf., e.g., [28]), when the ice base is frozen, we have
The vector field
In the same spirit as Jouvet and Bueler [33], we assume that for each
In view of (2.5), we can naturally extend the volume flux
Besides, once again in the spirit of [33], we have that
The evolution of the ice thickness is governed by the ice thickness equation (cf., e.g., equation (5.55) in [28]), which we recall here:
The free boundary
By virtue of the fact that
To work out the following step of the boundary value problem recovery, let us recall that
On the one hand, in the region
On the other hand, specializing
Putting together (2.9)–(2.12) gives
The latter can be straightforwardly changed into:
Define
Multiplying the equation in the boundary value problem (2.15) by
Let
Combining (2.16) and (2.17) gives:
The arbitrariness of the test function
Find
for all test functions
and satisfying the following initial condition
where
Note that, at least for the time being, the rigorous concept of solution has not been defined yet as we did not specify the regularity of
Ice is viscous too; its viscosity is described in terms of the Glen power law (cf., e.g., equation (4.16) in [28]) with ice softness coefficient
By equation (5.84) in [28], the horizontal ice flow velocity
Observe that, not only does the formula (2.20) use Glen’s power law but also that it is only valid in a certain shallow limit derived from the Stokes model (cf., e.g., [55]).
Plugging formula (2.20) into (2.4) gives:
where the latter equality is obtained by an integration by parts with respect to the variable
The weak form (2.19) is a degenerate extension of the
For the sake of clarity, the magnitudes involved in the model are listed in Table 1.
Quantities entering the model
| Variable | Description |
|---|---|
|
|
Ice softness in the Glen power law |
|
|
Lithosphere elevation |
|
|
Gravitational acceleration |
|
|
Upper ice surface elevation |
|
|
Ice thickness |
|
|
Accumulation rate function |
|
|
Ablation rate function |
|
|
|
|
|
Basal sliding velocity |
|
|
Horizontal ice flow velocity |
|
|
Ice volume flux |
|
|
Ice density |
|
|
Index in the Glen power law |
As already mentioned by Jouvet and Bueler [33], the expression of
We also observe that the model we are considering follows the formulation originally proposed by Glen [26], according to which ice was regarded as a viscous fluid. The celebrated model proposed by Hibler, itself inspired by the article of Coon et al. [16], is on the one hand less precise than Glen’s model as the ice velocity
The operation of averaging considerably simplifies the expression of the ice volume flux
To overcome the difficulty arising as a result of the gradient degeneracy in the vicinity of the free boundary, we introduce the following transformation, originally suggested in [49] (see also [13]):
Observe that
second, we obtain
and, third, we obtain:
For all
Plugging (2.26) into (2.25) gives:
Finally, for all
and the function
Inserting (2.24)–(2.29) into (2.21) gives:
For the sake of brevity, we define the function
Observe that plugging (2.31) into (2.30) gives:
Plugging (2.23)–(2.29) and (2.32) into the boundary value problem (2.15) gives that the new unknown
Similarly to (2.19), the weak formulation associated with the boundary value problem (2.33) is expected to take the following form: Find
for all test functions
and satisfying the following initial condition
Once again, note that, at least for the time being, the rigorous concept of solution has not been defined yet. This task will be carried out in Section 3, in which the function spaces where the solutions are going to be sought will be defined as well as the requirements a function has to meet to be regarded as a solution.
3 Weak formulation of the unilateral boundary value problem
Let
and observe that
Define the set
The weak formulation of the unilateral boundary value problem and the definition of the concept of solution will be recovered as a result of a constructive proof, based on the penalty method. The idea of using the penalty method to derive the existence of solutions of time-dependent contact problems was first used by Bock and Jarušek in [4,5] in the context of dynamic contact problems and was further developed in the recent article [6]. A time-dependent obstacle problem in the case where the target space is finite-dimensional was also addressed in the context of the study of biological systems in [45,46]. The finiteness of the target space dimension played a fundamental role to improve the results obtained by Bock et al. in [6].
For the sake of simplicity, we make the following assumptions on the geometry of the problem:
The ice softness
The basal velocity
The function
Let us now establish some preparatory results. The first result collects some properties of the negative part operator.
Lemma 3.1
Let
is monotone, bounded, and Lipschitz continuous with Lipschitz constant equal to 1.
Proof
Let
Recall that the negative part of a function is also given by
We have that
and the monotonicity is thus established.
Let us show that the operator under consideration is bounded, in the sense that it maps bounded sets onto bounded sets. Let
and the boundedness of
Finally, to establish the Lipschitz continuity property, evaluate
where the inequality holds thanks to the Minkowski inequality. In conclusion, we have shown that
and the arbitrariness of
The following lemma presents an inequality that was originally proved in 1973 by Pedersen in the context of
Lemma 3.2
Let
Proof
If
Let us first show that
Since
thus proving the inequality:
An application of the triangle inequality to the right-hand side of the aforementioned estimate and the monotonicity of the mapping
We hereby provide a complementary result to Lemma 3.2.
Lemma 3.3
Let
Proof
Assume, without the loss of generality, that
Therefore, the sought inequality is equivalent to proving that
Consider the function
Observe that
Since
Thanks to Lemma 3.3, we can establish the following result, thanks to which we will be able to define a sound initial condition for the variational formulation we will be considering.
Lemma 3.4
Let
Then, we have that
Proof
Observe that
Since it was assumed that
To show that
Clearly, the fact that
It is actually possible to establish that
By Lemma 3.3, we have that the Cauchy-Schwarz inequality and the triangle inequality give
Observe that the continuity in
The next lemma establishes that the supremum can be interchanged with a monotonically increasing continuous extended-real-valued function.
Lemma 3.5
Let
Proof
By the definition of supremum, we can find a maximizing sequence
By the assumed continuity of
Since
Therefore, passing to the supremum on the left-hand side, we obtain that
and the proof is complete.□
The next lemma establishes a convergence property for sequences of functions enjoying the regularities announced in Lemma 3.4.
Lemma 3.6
Let
Then
Proof
Since the space
By using Lemma 3.3, Hölder’s inequality, and Lemma 3.5, we have that
The assumed strong convergence implies that the second factor is uniformly bounded with respect to
Since the second factor is uniformly bounded with respect to
The next lemma establishes an immersion that will be used in the proof of the existence of the solution.
Lemma 3.7
Let
The mapping
defined by
is linear, continuous, and injective. Besides, if
Proof
Let us check that
for all
Similarly, for each
for all
To show the continuity of the mapping
The boundedness of the mapping
To prove that
In particular, consider functions of the form
This in turn implies that
Since this conclusion is independent of the choice of
The conclusion of Lemma 3.7 is that the space
The proof of existence of solutions hinges on a compactness result proved by Dubinskiĭ [21] (see also [1] for some improvements and corrections), as well as other results proved by Raviart [50] that we recall in the following lemma.
Lemma 3.8
(Lemma 1.1 of [50]) Let
for all
The first inequality in this lemma is a direct consequence of Young’s inequality (cf., e.g., [61]), while the second inequality was proved by Simon [58].
The next preliminary result we recall is a generalized integration-by-parts formula, whose proof hinges on the Lebesgue theorem (cf., e.g., Theorem 2.11-3 of [15]).
Lemma 3.9
(Lemma 1.2 of [50]) Let
Then, the following formula holds
where
Finally, we recall Dubinskiĭ’s compactness lemma. The proof of Dubinskiĭ’s compactness lemma hinges on a preliminary lemma of independent importance, that we here recall for the sake of clarity. This lemma was originally proved by Dubinskiĭ [21] (cf. Lemma 2 of [21]) and was later improved by Barrett and Süli [1] (cf. Lemma 2.2 of [1]).
Lemma 3.10
Let
Assume that the set
Then, we have that
if
if
We are now ready to recall the well-known Dubinskiĭ compactness lemma (cf., e.g., Theorem 1 of [21] and Theorem 2.1 of [1]).
Lemma 3.11
Let
Assume that the set S is endowed with the semi-norm
Assume that the set
with
Then Y is relatively compact in
A discrete version of this compactness result has been established by Raviart [49].
Lemma 3.12
(Lemma 1.4 of [50]) Let
For each
and
where
Then, it is possible to extract a subsequence of the sequence
Let us also recall a result on vector-valued measures proved by Zinger [62] and later improved by Dinculeanu (see also, e.g., page 182 of [19], and page 380 of [20]).
Lemma 3.13
(Dinculeanu-Zinger theorem) Let
There exists an isomorphism between
for all
We now state the penalized problem, which is suggested by the formal model (2.33). Note that the initial condition makes sense thanks to Lemma 3.4.
Problem
and satisfying the following variational equations:
in the sense of distributions on
for some prescribed nonzero
4 Existence of weak solutions and variational formulation for Problem (2.33)
The recovery of the variational formulation for Problem (2.33) and the existence of solutions for this problem are broken into three parts. We first establish the existence of solutions for Problem
Problem
where
The following existence-and-uniqueness result can be established.
Theorem 4.1
Let
For each
Proof
For the sake of brevity, define
Consider the operator
Consider the operator
The operators
and the term on the right diverges as
Define the mapping
Next, we discuss the convergence of the sequence
Theorem 4.2
Let
Assume that the following decrease condition holds:
for all integers
Then, the following a priori estimates hold:
for some
Proof
Multiplying (4.1) by
By using the first estimate in Lemma 3.8 with
where
Multiplying by
and we finally obtain:
Therefore, there exists
so that the estimates (4.7) and (4.8) are proved.
By (4.7), for each
Since the left-hand side of the previous equation is bounded independently of
for some
To recover the estimates (4.9) and (4.11), let us multiply (4.1) by
For each
An application of Lebesgue’s inequality, the triangle inequality and Young’s inequality [61] gives
Since
Thus combining (4.16)–(4.18) gives
An application of Lemma 3.1 and (4.6) gives that for all
Thanks to (4.20) and the positiveness of
By summing (4.15) over
so that, in the end, we obtain the following estimate:
and the estimates (4.9) and (4.11) straightforwardly follow.
To establish the estimate (4.12), we exploit the boundedness of the
for all
By (4.7), for each
Since the first term is bounded independently of
for some
Remark 4.3
Observe that (4.21) suffices to derive the estimates (4.7)–(4.13). However, the decrease condition (4.6) is physically more realistic than (4.21), which could have very well been assumed in the statement of Theorem 4.2 in place of (4.6). The decrease condition (4.6) relates to a regime of pure melting, which is a common regime glaciers undergo during, for instance, the spring and the summer. Note also that assumption (4.6) means that the decrease of
Given
As a result of the estimates (4.7)–(4.13), we have that
Thanks to (4.24), we can establish the existence of solutions for Problem (
Theorem 4.4
Let
Besides, the weak-star limit
Proof
The convergence process (4.25) holds by virtue of an application of the Banach-Alaoglu-Bourbaki theorem (cf., e.g., Theorem 3.6 of [9]) to the estimates (4.24) and (4.22). The nontivial part of the proof amounts to identifying the weak-star limits
To begin with, we show that
Simple algebraic manipulations transform the latter into
and we observe that a sufficient condition insuring this is that
which is equivalent to writing
Let
is non-empty. Define the semi-norm
and define the set
An application of the Poincaré-Friedrichs inequality gives that there exists a constant
Let
Since
and observe that this mapping is hemi-continuous and monotone, being associated with the mapping
To show that
An application of (4.27) gives that the right-hand side of the latter term tends to zero as
Since the space
thus establishing the sought relative compactness.
The established relative compactness of the set
where, once again, the monotonicity of the mapping
Second, we show that
We observe that a sufficient condition insuring this is that
which is equivalent to writing
Let
is non-empty. Define the semi-norm
and define the set
An application of the Poincaré-Friedrichs inequality gives that there exists a constant
Let
Since
and observe that this mapping is hemi-continuous and monotone, since the mapping
To show that
Since
An application of (4.30) gives that the right-hand side of the latter term tends to zero as
Since the space
thus establishing the sought relative compactness.
The latter shows that
in turn implying that the set
where, once again, the monotonicity of the mapping
showing that the entire convergence process (4.25) holds.
We are left to show that the weak-star limit
By multiplying (4.32) by
We define
By letting
Let us rearrange the first term on the left-hand side of equation (4.33) as follows:
Therefore, equation (4.33) can be thoroughly re-arranged as follows:
Letting
Observe that an application of the Sobolev embedding theorem (cf., e.g., Theorem 6.6-1 of [14]) gives
Comparing equations (4.37) and (4.38) gives
Since
Since the duality in (4.40) is continuous with respect to
It is easy to observe that:
Let us now specialize
Since the integration in (4.42) is continuous with respect to
so that the injectivity of the monotone and hemi-continuous operator
The last thing to check is that
which in turn implies:
Passing to the
On the other hand, the specializations
Combining (4.45) and (4.46) gives:
We now exploit the method of Minty, known as Minty’s trick [41]. Let
Let
Dividing (4.49) by
Since
which finally implies that
The next step, which constitutes the final step of the existence result we want to prove, is the passage to the limit as
Based on the assumption (4.6), the estimates (4.7), (4.10), (4.12), and (4.13) in turn imply that there exists a constant
By the Banach-Alaoglu-Bourbaki theorem (cf., e.g., Theorem 3.6 of [9]), we infer that, up to passing to a subsequence still denoted by
Observe that the first convergence of (4.53), namely
The third estimate in (4.14) imply that
which means that
By using the same compactness argument as in Theorem 4.4, an application of Dubinskiĭ’s theorem (Lemma 3.11) with
Moreover, by Lemma 3.10(a), it can be established that:
The same monotonicity argument as in Theorem 4.4 and the third convergence in (4.53) in turn imply that the limit
The first convergence in (4.53) gives that
Note that, thanks to Stampacchia’s theorem [59], the announced convergence a priori makes sense. To prove that this convergence holds, fix any
since the first term converges to zero by the first convergence in (4.53) and the second term converges to zero thanks to (4.54). The convergence (4.56) is thus established. An application of Lemma 3.6 to the convergence (4.55) gives that:
and, so, it also strongly converges in
Define the linear and continuous operator
By (4.57) and the continuity of
Note that the variational equation in Problem
Integrating (4.58) in
where the last equality holds since
Let
Thanks to Lemma 3.6, the boundedness of
for some
for some
We now show that the inverse inequality holds true too. Observe that, by Stampacchia’s theorem, we have that
for a.e.
is attained for functions
so that the last two inequalities in (4.60) are, actually, equalities.
Let us now show that:
Let
for a.e.
Since the convergence in (4.62) holds for a.e.
and this proves (4.61). As a result of (4.60), (4.61), and the properties of the Bochner integral (cf., e.g., Theorem 8.13 of [36]), we have:
In conclusion, we have shown that:
By combining the latter with Bochner’s theorem (cf., e.g., Theorem 8.13 of [36]), we obtain:
By combining (4.59) and (4.64), we obtain
for some
Therefore, we can derive the following estimate plugging (4.65) into (4.58):
By applying the boundedness of
for some
Therefore, by the Dinculeanu-Zinger theorem (Lemma 3.13), there exists a vector-valued measure
It can be observed that the vector-valued measure
To see this, let
Keeping the convergence (4.68) in mind, as well as the fourth convergence in (4.53), we are in position to carry out the same compactness argument as in Theorem 4.4 to apply Dubinskiĭ’s theorem (Lemma 3.11) with
and
Thanks to Lemma 3.7 and the Dinculeanu-Zinger theorem (Lemma 3.13), we have that the following chain of embeddings holds:
Thanks to Lemmas 3.6 and 3.9, we have:
By specializing
In the special case where
By combining (4.71) and (4.73) and the first convergence in (4.53), we obtain that (4.72) gives:
Since the
for all
Let us now consider an arbitrary function
Let
In the end, we are in position to write down the model governing the evolution of the thickness of a shallow ice sheet, and to assert that this model admits at least one solution. This is the main result of this article, the proof of which has been given earlier.
Theorem 4.5
Let
Then, the family
Moreover, if the following regularity condition holds
then
Problem
satisfying the following variational inequality:
for all
for some prescribed
We conclude the article with a remark where we propose an alternative regularity condition, which is more physically realistic than the abstract regularity condition (4.77).
Remark 4.6
Observe that the following condition implies (4.77), namely:
There exists a constant
This regularity condition, which relates to the melting regime of the grounded ice sheet, is more physically realistic than (4.77) and saves us the effort of resorting to vector-valued measures. The condition (4.78) means that the melting regime is at some point overcome by the accumulation regime. One can expect this type of behavior by virtue of the cyclic accumulation and ablation phases ice sheets undergo. In particular, the regularity condition (4.78) lets us obtain the following sharper result:
Starting from the latter, the rest of the proof hinges on classical arguments and we derive that the weak-star limit
for all
for some prescribed
5 Conclusion
In this article, we formulated a time-dependent model governing the evolution of the thickness of a shallow grounded ice sheet undergoing regimes of melting and ablation. The shallow ice sheet relative height constitutes the main unknown of the problem we considered in this paper. Note that the modelling imposes the shallow ice sheet relative height to be nonnegative. This means that the shallow ice sheet height cannot be smaller than the height of the bedrock. For this reason, the problem under consideration can be regarded as an obstacle problem.
First, we recovered the formal model, based on Glen’s power law. This passage was inspired by the modeling for the static case of Jouvet and Bueler [33].
Second, we incorporated the constraint, in the form of a monotone term, in the model. By so doing, it was possible to write down the variational formulation of the model under consideration in terms of a variational equation posed over a vector space. The existence of solutions for the penalized model was established thanks to the Dubinskiĭ compactness theorem and a series of preliminary results, which we stated in the form of lemmas.
Third, we let the penalty parameter approach zero and we recovered the actual model, which takes the form of a variational inequality tested over a nonempty, closed, and convex subset of the space
It is worth noticing that the proofs hinge on a suitable decrease condition, i.e., (4.6), and a regularity condition, i.e., (4.77), that, in the same spirit as [49], we can replace by the corresponding more realistic one (4.78).
At a first glance, the time-dependent model we studied appears to be very challenging, and it is not clear whether it is possible to establish the existence of solutions to it without assuming that (4.6) and (4.77) (or the more physically realistic regularity condition (4.78)) hold.
It is worth mentioning that the concept of solution we are considering in this article is different from the one discussed in Definition 3.1 on page 688 of [13]. In our context, the modeling did not lead us to consider the analogue of the function
In the forthcoming article [48], we plan on investigating the uniqueness of the solution, its continuous dependence on the data, and its regularity.
Acknowledgments
The authors would like to express their most sincere gratitude to Professor Ed Bueler (University of Alaska, Fairbanks) for the insightful comments on the physics of the problem and for the suggested improvements. The authors would like to express their most sincere gratitude to the Anonymous Referees for the insightful comments and the suggested improvements. This article was partly supported by the Research Fund of Indiana University.
-
Funding information: This article was partly supported by the Research Fund of Indiana University.
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Author contributions: Both the authors equally contributed to each part of the manuscript.
-
Conflict of interest: Authors state no conflict of interest.
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Statements and declarations: All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or nonfinancial interest in the subject matter or materials discussed in this article.
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Data availability statement: No data was generated or analysed during this research.
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