Notations

Letter subscripts denote derivatives. Scales are denoted by brackets.

3.1. Scaling analysis

If the sliding velocity has order not much higher than the component velocity due to shearing deformation, then scales of fields describing shearing flow in glaciers (subscript o) are well-known (Reference Salamatin and MazoSalamatin and Mazo, 1984):

Here [x] and [a] can be determined from observations. Physically, this relation, between ε₀, [a] and the other parameters, is determined by equality of the scales of the pressure longitudinal drop and the vertical gradient of the shear stress. From observations it is known that ε≤ ε0 ∼ 10"² - 10⁻³.

The case when the shear stress has an order not less than the longitudinal deviatoric stress in ice streams ([T1]>[τ₂]) was considered by Reference FowlerFowler (1982) and that of slow varying traction was considered by Reference Hindmarsh and PeltierHindmarsh (1993). Here we consider all the situations when the shear stress can be much less or much higher than the longitudinal deviatoric stress.

Here we assume [x] to be prescribed and known from observations. Along the ice stream, the ice thickness is assumed to decrease by a value comparable with the thickness itself, therefore [Ux] = [u]/[x]. Evidently [u] = [q]/[z] and w = eu from mass conservation. Because the ice stream can have a large catchment area, we generally do not assume [q] = [a][x]. Scale for the pressure is [z]p(i)g. It should be noted that if the upper surface slope is much less than the ice-stream aspect ratio, then [px] ≪ p(i)ge. Again, if the ice-thickness gradient is much less than the aspect ratio, then ux ] ≪ [u]/[x]. These cases require more subtle scaling analysis which is not considered here. However, when the ice-thickness gradient is comparable with the upper surface slope and both are less than the aspect ratio, then in scaling analysis we can assume the typical ice thickness [z] (instead of [x]) to be prescribed and find the typical slope of the upper surface from scaling analysis, which will determine [x] much higher than the ice-stream length.

To find [Uz] we decompose u = U(sl) + vu(sh), where U(sl) is the sliding velocity and Vu(sh) is the shearing component velocity. The dimensionless parameter v is to be chosen so that [U(sh)] = [u] and is a typical ratio of the shearing component of velocity to the velocity u. Hence [Uz] = v[u]/[z] and from Equations (4) and (5) we have [τ2]/[τ1] ∼ ejv if ε2 < v determining uz. On the other hand, from the boundary conditions at the upper surface we have hence at the upper surface T ∼ er₂, therefore in the ice stream. This means that */>£² always.

For the stresses we have

where when (the typical shear stress is not less then the typical longitudinal deviatoric stress) and when (the typical longitudinial stress is not less then the typical shear stress). From the sliding law in Equations (10) we derive

Determination of v using the sliding law is correct only if the shear stress has an order at the bed not less than at the upper surface (shearing is determined mainly by the traction at the bed and τ1(x,b)∼[τ1]), when the order of the sliding velocity is not less than the order of the shearing component velocity. This is because this determination corresponds to finding the scale of the shear stress [n] from the sliding law. If then this determination is not valid. On the other hand, if we derive v » 1, then it means that sliding is negligible and we must put v = 1 in further analysis.

When the shear stress is maximal at the upper surface, then, as it was shown above, v ∼ ε². Therefore, in order for Equations (16) and (17) to be correct, it is sufficient that ε² ≪ v, which is, as will be seen later (Equation (18)), equivalent to the confirmation that a value of the second term of (1) is negligible in comparison with the other terms.

3.2. Governing equation in dimensionless form

Denoting dimensionless variables by capital letters, for the ice stream we obtain

where

and

Boundary conditions:

3.3. Determination of dependent scales

From Equation (18) we infer that A must be of order 1. Letting A = 1, we can find the typical ice thickness or the ice-stream aspect ratio

When v derived from Equations (16) and (17) is not higher than 1, we can write these formulas as one via [k]:

This equation does not include information about rheology Comparing aspect ratios of glaciers of the same typical length with sliding ε and without it ε₀, we can write

From Equations (27)–(30) we have

We can find typical value [k] (τ) k](r) of the sliding coefficient determining equality of the scales of the shear and longitudinal deviatoric stresses. Equating ε = v yields

Putting v = 1, we can find typical value [k](v) of the sliding coefficient determining equality of the scales of the sliding velocity and the shearing component velocity:

3.4. Shelf-like dynamics of ice streams

The parameter ε²/v in (18) determines the importance of the longitudinal stress deviator in the momentum conservation equations. If ε² /v ∼ 1, then we expect an analogy with shelf dynamics. Reference Hindmarsh and PeltierHindmarsh (1993) calls such streams "shelfy streams". In this case U = U_(sl) + O(e²) and the equation for the ice thickness will be (Reference Muszynski and BirchfieldMuszynski and Birchfield, 1987)

with the boundary condition determining the ice thickness at the grounding line to be found.

3.5. Plug and shearing flow

Let us consider the situation when ε²/v ≪ 1. In this case to order O(e² /v) + O(ε²) from Equations (18)–(26) we derive

Mass balance:

Boundary conditions which have not yet been taken into account:

If v ≪ 1, then Q = KH(-S_xH)^m + O(y). If v ∼ 1, then Q = KH(-S_xH)^m + |S_x|ⁿH_n+ ²/(n+2). In the first case (plug flow) we have an equation for the ice thickness (Reference FowlerFowler, 1982)

In the second case we derive (e.g. Reference Salamatin and MazoSalamatin and Mazo, 1984)

Equations (42) and (43) require boundary conditions at the grounding line to be found. One can note that (42) and (43), contrary to (34), do not require the value of the ice thickness at the begining of the ice stream to be imposed.

For ice streams we expect v ≪ 1, while for glaciers with no-slip condition: v ∼ 1. Therefore it is necessary to estimate value ε²/ v to determine what model to apply: Equation (34) or (42) and (43). Evidently, if ε ≤ v (order of the shear stress is not less than the order of the longitudinal deviatoric stress), then and (42) and (43) are to be applied. If v<e, then from Equations (30) we find . For Ice Stream E we have [x] = 300 km, [z]₀ = 2.4 km, [z] = 1.1km (e.g. Reference LingleLingle, 1984), therefore for n = 3 we obtain ε²/*/ ∼ 0.005 and we do not expect "shelf-like" dynamics of this ice stream. For ice streams of Vilcheck Land (Franz-Josef Land) ε = 0.01, ε₀ =0.02, therefore ε²/v ∼ 0.01 for n = 3 and we arrive at the same conclusion. On the other hand, putting v = ε², we find and typical ice thickness is of the order of only several metres, which is not observed.

5.1. Revealing singularity near the grounding line

Differential equations (34), (42) and (43) determining the ice-stream thickness require the boundary condition at the grounding line. Solution (46), determining the ice-shelf thickness, has one free parameter determining the ice-shelf thickness at the grounding line. Evidently, if the shear stress in the ice stream is much larger than the longitudinal stress deviator, then all the stresses in the transition zone are of the same order and the transition zone is singular (e.g. Reference Chugunov and WilchinskyChugunov and Wilchinsky, 1996). Let us then consider the situation when the value of the parameter v, describing the importance of the shearing component velocity, is small near the grounding line and presume that the (shallow-ice-approximation) solution for the ice shelf found (46) is valid up to the grounding line. In this case the solution of the problem of marine-ice-stream dynamics determines the ice thickness at hydrostatic equilibrium and continuous stresses at the grounding line. This means that strain rate is also continuous at the grounding line. On the other hand, due to the small value of v, to order of O(y) we have Hence, because Q and H are continuous at the grounding line, we obtain continuity of the ice-thickness gradient at the grounding line.

First, we consider the situation when the upper surface slope in the ice stream has order not less than the ice-thickness gradient S× > Hx. At the same time H_× ∼ S_x/6 in the ice shelf, due to the condition of hydrostatic equilibrium. This yields in other words, such a solution determines a ten-time break of the upper surface slope and Sxx much higher near the grounding line, than it is in the ice stream or in the ice shelf. However, from physical considerations we expect the smooth upper surface profile.

Second, we consider the situation when the upper surface slope in the ice stream is much less than the ice-thickness gradient. Evidently, such a situation can occur only if B× > 0 when X ≤ Xg. In this case we start from investigation of the solution for the ice shelf (46). In deriving this solution we assumed that S× ≤ 0, which is physically consistent. Therefore, this determines Hx(X(g)+) ≤ 0 and Bx(X(g)+) ≥ 0 at the grounding line. Evidently, if Bx(X{g)-) ≠ Bx(X{g)+) then matching the strain rate also determines a break of the upper surface. Although investigation of Equation (34), written for the ice stream and for the ice shelf, does not allow us to conclude that if Bx(X(g)-) = Bx(X(g)+) then the strain-rate is continuous at the grounding line, one could assume it. However, in this case the condition Bx(X{g)-) =Bx(X{g)+) implies that H(w)(X(g)) = (1–δ)H(X(g)+) and H(w)× (X(g))=(1–δ)Hx(X(g)+), where H(w) is the water depth. Therefore, due to (44) the water depth H M at the grounding line must satisfy the equation

which is assumed to be a very strong restriction on the bed profile determining the existing of the solution of the marine-ice-stream dynamics.

Eventually, we conclude that in any case, even if we consider "shelf-like" dynamics of the ice stream, the reduced models found adequate in the ice stream or in the ice shelf are not adequate near the grounding line and the transition zone is singular. This can be caused not only by the change of flow style in the transition zone, but also by the change of boundary condition at the base from the absence of the Archimedes force to the presence of it.

5.2. Scaling analysis

Let us determine the scales of the transition zone. For this purpose we consider Equation (1).

First, in the transition zone we expect that all the stresses important in both the ice stream and the ice shelf are important in the transition zone. Therefore, all terms in (1), describing the influence of the stresses, must have the same order (as in the ice shelf), because all these terms determine the change in flow style in the xz plane.Footnote ^*

Second, if the longitudinal deviatoric stress in the ice stream is much larger than the shear stress and this holds in the transition zone, then the boundary condition at the lower surface yields [τ1]/[τ2] ∼ ε(s) ≪ 1 and the shallow-ice approximation is valid in the transition zone, which contradicts the conclusion made above. Hence, in any case in the transition zone all the stresses are of the same order.

On the other hand, both statements are possible only when [τ1] = [τ2], [a;] = [z] = [h] (generally [u] + [w]). This means that the full system of Stokes equations must be solved in the ice-stream-ice-shelf transition zone.

Denoting h _(g) h(x_(g)) the ice thickness at the grounding line, we introduce new variables, pointed by bars with the same (yet unknown) scale [τ] for the stresses and the excess pressure; and the scale h(g) for the coordinates and the ice thickness. We also move the origin of the coordinate system to the bed at the grounding line.

Integral equations (12) and (13) for X > 0 (in the ice shelf) take the forms

where

Eliminating from Equations (47) and (48), we derive

where

From this equation it can be seen that 0 ∼ 1, otherwise either the first term on the righthand side (which describes the driving force in the ice shelf) is not balanced or is not important which contradicts the transitional behaviour of the flow. Hence, Equation (47) means that to order O(S) the ice thickness can be found from the condition of hydrostatic equilibrium and the upper surface slope is of order O(S). However, for high sliding velocities the latter estimate can be improved in the ice stream near the transition zone, where the shallow-ice approximation is still valid.

First, for determining [τ] we use Equation (11) taking into account (48):

Here we assumed that B× ∼ ε when X ≤ 0. This equation is written at X ∼ 1, where the shallow-ice approximation is valid (near the transition zone, but not in it). Due to the above estimate on the upper surface slope, we have . The scales of the shear stress and longitudinal stress deviator at so chosen X (near the transition zone, but where the shallow-ice approximation is valid) are denoted by [τ1]0, [τ2]0 and are generally different from those in the ice stream. Evidently, the scale of the shear stress, τ1 b in the zone of integration is the maximum of the typical shear stresses near the transition zone [τ1]0 and in it [r].

Equating the typical values of lefthand-side and right-hand-side terms yields the physically reasonable relation

The scales [τ1]0, [τ2]0 can be found from Equation (15) using (16), (17) and (29) to determine the typical ice-surface slope of the ice stream particularly near the grounding line as a function of h(g) (in this case) when the shallow-ice approximation is valid (one should not forget that these formulas were derived with the assumption s_× ∼ h_x). On the other hand, from (49) we derive [r] = h_(g)P&amp;gS. Elimination of h_(g) yields

These estimates are valid when the value, v_(t), of the parameter v near (not in) the transition zone is no higher than 1 (_v(t) is not equal to the value of v in the ice stream because the scales of the flow near the the transition zone and in the ice stream are generally different). Equating v(t) = 1 in (16) and using (15) and (55) we derive [τ] and the corresponding value [k](v,t) of the sliding coefficient making v(t) ∼ 1:

For the typical ice-stream surface slope near (not in) the transition zone ε(t) from (29) we obtain

Although these relations determine the typical upper surface slope near the transition zone, because they were derived using estimates for the shallow-ice approximation, we expect that these estimates are also valid in the transition zone. Otherwise, if the surface slope in the transition zone is much higher, this could lead to much higher longitudinal deviatoric stress, which contradicts (52).

IF V(t) ∼ 1 (sliding is negligible), then substituting Equation (55) in (45) for [z]_s and using (56) we find that in this case the typical slope of the ice shelf surface near the transition zone is 2⁻2^n–1^.

In the derivation of Equations (53)–(59) we used formulas (16) and (17), valid when the shear stress in the ice stream near (but not in) the transition zone reaches its maximum at the lower surface. It is true particularly when the gradient of the longitudinal deviatoric stress in (1) is much less than the other terms. Evidently, if in the ice stream the shear stress is much larger than the longitudinal deviatoric stress then this is true. If the situation is the contrary and we assume that all the terms in (1) have the same order, then equating [p_x] = P_(i)g[h_(g)]/[x] = [T_2x} = [τ]/[x] we find [T]=_P(i)g[h_(g)], which contradicts (55). Hence our scaling is adequate.

For the use of formulas (53)–(59) we can find the typical value [k]_(t) of the sliding coefficient, which determines [τ1]₀ ∼ [τ2]0. Equating [_τ1]₀ = [τ₂]₀ yields

Evidently [τ1]_{c τ2} when [k] (t) [k] and vice versa.

To compare the typical values of the sliding coefficient determining different flow styles, we find the following ratios:

Using these formulas we derive for m == n and 6 == 0.1:

It can be seen that the value of the sliding coefficient determining the equality of the typical sliding velocity and the shearing component velocity in the ice stream [k]_M also determines the equality of the typical longitudinal deviatoric stress and the shear stress near the grounding line, where the shallow-ice approximation is still valid. This value is much less than that determining the equality of the typical longitudinal deviatoric stress and the shear stress in the ice stream [k] _(τ) This is caused by the higher surface slope near the grounding line than that in the ice stream.

5.3. Problem for the transition zone

In order to find the boundary condition at the grounding line relating the ice thickness and horizontal mass flux at it, we scale values in the transition zone by the scales of flow without sliding (shearing flow):

Here q_(g) is the (yet unknown as well as h_(g)) horizontal mass flux at the grounding line. In the transition zone the full system of Stokes equations (considered as the inner problem) must be solved (new variables are pointed by hats):

where

Boundary conditions:

where

For determination of the suface profiles, instead of the boundary conditions at the free surfaces, we use the integral equations:

where

The initial condition is β, In these variables ff _(w is unknown.

The parameter (3, as it will be seen later, plays an important role and is proportional to the ratio of the longitudinal stress deviator found with the shallow-shelf approximation to the shear stress found for the Pouiselle flow at the grounding line.

Solution of the problem (65)-(78) and matching the solutions found for the ice stream, ice shelf and transition zone is beyond the scope of this work. However, it should be noted that values in (65)-(78) are not of order 1. If expansion on 6 is used, then, proper scaling must first be performed. For example, near the grounding line we cannot neglect the upper surface slope (as it was done in the case of the no-slip condition at the bed; see Reference Chugunov and WilchinskyChugunov and Wilchinsky, 1996), because the longitudinal deviatoric stress, which plays a significant role in the transition zone, is determined by it. We can assume that the upper surface is plane and horizontal (to order O(δ)\ but the value of must enter the kinematic boundary condition at the upper surface (70), which in case of no-slip condition at the base yields and does not include information about the surface slope.

5.4. Boundary condition at the grounding line

It is physically consistent to presume that when the bed profile and horizontal mass flux are monotone functions of x, then a unique solution of the complete problem describing the ice flow in the ice stream, transition zone and ice shelf exists. In this case, because the (outer) solutions found for the ice stream and ice shelf have free parameters (their thicknesses at the grounding line) to be determined from matching with the solution of the (inner) problem (65)(78), problem (65)-(78) has a unique solution not including free parameters. On the other hand, problem (65)-(78) includes only five dimensionless parameters n, m, β, γ, δ which determine similar flows in the transition zone. Influence of the outer flow is "hidden" in /3 and γ, where the parameter /3 includes the unknown ice thickness at the grounding line h(g), and parameter 7 is a known function of x(g). Presumption of uniqeness of the solution leads to the relation /3 = β(n, m, S, 7 ). That is for certain choices of n, m,γ,δ, only the one value of /3 determines continuous fields at the grounding line. If sliding is negligible, then /3 depends only n and 6. As can be seen from (49) and (78), the typical value of β is [τ]/[r](t)

Finally, the boundary condition at the grounding line (78) relating the ice thickness and horizontal mass flux can be written as

which, when dependence β(n, m, 6, 7 γ and Hw(n, m, 6, 7 γ is known, lets us determine x(g) and h(g), because h(g) = h(w) (x)/H{w)(n, m, S, 7 ) , and the lefthand as well as righthand parts of (79) are known functions of x(g) It should be noted, that the form of (79) generally implies that several solutions x(g) can exist if b(x) or q(x) are not monotone.

In variables scaled by the scales of flow without sliding we derive

where Q_(g) = Q(X_(g)) and £ is the aspect ratio of the glacier without sliding (14).