1. Introduction

The mechanics of a bounded ice sheet with steady free surface has been considered by Reference NyeNye (1959), Reference WeertmanWeertman (1961), and more recently by Reference Morland and JohnsonMorland and Johnson (1980). Nye considered both plane and axisymmetric flow in order to estimate the effects of a third dimension in comparison with the plane-flow restriction. Morland and Johnson’s analysis was for an ice sheet deforming in plane flow only, solving the full momentum equations by a regular perturbation technique. The equivalent axisymmetric solution is now presented. There are large similarities between the plane and axisymmetric analyses and details which apply to both problems are not repeated. Henceforth Reference Morland and JohnsonMorland and Johnson (1980) will be referred to as M and J.

Consider an ice sheet shown by the cross-section in Figure 1, with horizontal bed z = 0 and surface z = h(r) in cylindrical polar coordinates (r, θ, z), and all physical variables independent of θ. The velocity components are (u, o, w) and g is the constant acceleration due to gravity. τ is the Cauchy stress tensor with σ_rθ = σ _rz = 0, and (t_n, t_s) denote normal and tangential tractions on the surface. The ice sheet is maintained in steady flow by surface accumulation and ablation and basal drainage, with zero net flux.

Fig. 1.

Ice-sheet cross-section.

By considering constant surface accumulation Reference NyeNye (1959) obtained the solution

(1)

for the profile, where

and A and m are constants appearing in the basal sliding law

q is the constant accumulation, ρ is the uniform density of ice, h_c is the height at the centre of the ice sheet, and l the radius.

As for the plane-flow profile, the condition of small surface slope is violated in the axisymmetric profile except in some central zone. Nye’s analysis is independent of the flow law and any temperature dependence. He has, nevertheless, deduced that observed temperature profiles give rise to an approximately uniform horizontal velocity profile above a thin bottom layer, and used this as the basic kinematic approximation. The analysis in this paper does not include any temperature variation so that Nye’s deduction cannot be tested, nor can any effect of temperature variation be predicted.

Following the approach of M and J, for the axisymmetric problem the boundary conditions are

(2)

(3)

where the basal drainage b and the sliding coefficient λ depend on the overlying height of ice. If q is the accumulation (volume flux per unit horizontal area) referred to the horizontal cross-section, then

(4)

The free-surface conditions are

(5)

(6)

and symmetry requires

(7)

Formulation of the full problem in dimensionless variables introduces a small parameter v. However, the approximation v = 0 does not permit a bounded ice sheet. Rescaling the horizontal coordinate by a small factor allows a regular perturbation solution in ε. v and ε are identical to the parameters which arose in the plane-flow analysis, and again ε defines the small magnitude of the surface slope. The small-slope solution is valid up to the margin (h = 0) provided the sliding coefficient as for the plane flow case.

The results obtained are compared with those for the plane-flow solution (M and J) for the case m = 1. In all illustrations it is found that the aspect ratio decreases (or is unchanged), as predicted by Reference NyeNye (1959), but here the results demonstrate the dependence of the decrease on the accumulation rate and the sliding law.

2. Balance laws and constitutive equations

The ice is assumed to be incompressible, so mass balance requires

(8)

Inertia terms are negligible in this slow viscous flow so momentum balance requires

(9)

Following M and J, the ice is assumed to be an incompressible non-linear viscous fluid with a temperature-dependent rate factor (Reference MorlandMorland, 1979), so

(10)

where

(11)

and The non-zero components of D are

(12)

T denotes temperature, and a(T) is the rate factor, normalized by a(T ₀) = 1 for some temperature T ₀, with σ₀ and D ₀ denote a constant stress magnitude and a constant strain-rate magnitude respectively, so that , the invariants Î₂, Î₃, and the response functions ø₁, ø₂ are dimensionless.

The more commonly adopted laws for D give the simpler form

(13)

which implies

(14)

With

(15)

Glen’s law gives

(16)

and the Colbeck and Evans polynomial law gives

(17)

In the following analysis it is assumed that in the general law (10) the stress contribution from a non-zero ø₂ term is not of greater magnitude than that of the ø₁ term. It is also assumed that but the singularity in ø₁, arising from Glen’s law (Equation (16)) will be considered.

3. Dimensionless formulation and the small parameter

Introduce dimensionless variables by

(18)

where h₀ is a magnitude of the maximum ice thickness and q_m is a magnitude of maximum accumulation density. Take q _m = q(h ₀) on the assumption that ablation at lower heights does not significantly exceed this value. It is supposed that the drainage magnitude is not greater than that of Q. Define

(19)

Setting

(20)

in terms of a dimensionless stream function Ψ(r, z) satisfies the mass-balance equation(8). The momentum equations(9) become

(21)

Boundary conditions (2)–(7) become, for z = 0:

(22)

(23)

for z = H(R):

(24)

(25)

(26)

and for R = 0:

(27)

The tensors have non-zero components

(28)

where, under typical conditions (M and J),

(29)

To deal with the singularity in ø₁ at Î₂ = 0 for Glen’s law (see M and J), let

(30)

so that

(31)

is bounded, and for Glen’s law

(32)

and ø₁ is constant, while the finite viscosity case is obtained by setting α = 0, with ø₁ = O(1). Hence Equations (30) and (31) cover both the cases with a power-law singularity and those which are non-singular. Equation(10) is now

(33)

where and

(34)

The parameters δ, v, and s are identical to those of the plane-flow analysis and typically

(35)

(see table 1, M and J), which is the condition adopted. Note also that

(36)

for practical conditions.

Table I.

Radius ρ _m⋆ and height η _c for linear accumulation Q⋆(η) and linear sliding law (m = 1) with linear coefficient Λ(η). Solutions for Colbeck and Evans’ law (CE) and Glen’s law with n = 1, 3

By setting v = 0 in the momentum equations a uniform parallel slab of infinite extent is predicted with

(37)

excluding a margin at finite radius.

4. Scaled variables and slope magnitude

For a non-zero surface slope, P, to leading order, must not be independent of R, so the horizontal momentum balance given by the first of Equations(21) must involve the shear stress gradient in Z. This suggests a horizontal coordinate contraction

(38)

(cf. M and J), where ρ, γ = O(I) and Γ has magnitude ε. To retain the surface accumulation balance given by Equation(24), a stream-function scaling

(39)

is required, so Equation (24) becomes

(40)

Note that for plane flow Ψ was scaled by ε but the relations between the velocity components and the axisymmetric stream function are different.

As before, a balance in the momentum equations requires and only leading-order terms are presented for brevity. Thus, by Equations(28) and (33)

(41)

(42)

The balance now requires

(43)

and hence

(44)

This restriction is identical to that in the plane-flow case so the ø₂ term does not contribute to ∑ to leading order with the assumption For most, if not all, practical conditions

(45)

which is assumed here.

5. Leading-order approximation

A power-series expansion in ε is again appropriate, so let and

(46)

Denote all leading-order quantities by a subscript or superscript 0, so

(47)

Proceeding with the solution as before gives rise to the ordinary differential equation

(48)

for the free surface H = η(ρ), where

(49)

Λ is normalized on the scale ε_I = ε (n = I) and so is independent of n. The g functions arising from the constitutive laws are unaltered, and so for Glen’s law and the Colbeck and Evans polynomial laws respectively

(50)

and

(51)

Here the argument —η′η is always positive. Ψ₀ is given by

(52)

where

(53)

The leading order stress components are

(54)

and these have the same form as for plane flow.

6. General properties and validity

Equation(48) is analogous to equation (68) in M and J. It differs in the appearance of the ρ term on both sides of the equation which leads to a stricter restriction on Λ(η) and also requires a slightly different numerical approach. Rewrite Equation(48) as a first-order ordinary differential equation for η′ = γ₀(η):

(55)

For Glen’s law and Colbeck and Evans’ law respectively

(56)

Zero mass flux requires that, integrating over the surface and the bed,

(57)

which implies for η → 0:

(58)

As η → η_c = η(0), Λ and Q^⋆ are finite and γ₀, ρ → 0, so the essential behaviour of Equation(55) is

(59)

where l = min (m, n). Hence

(60)

As the dominant terms of Ψ₀ , in both limits, from Equation(52), are

(61)

Thus, as γ₀ → 0, η → η_c

(62)

Direct differentiation of both terms in (61) show that bounded requires (i) if l= n then l = n = I and m = 1 or 2 or m ≥ 3; (ii) if l = m then l = m= I and n = 1 or 2 or n ≥ 3. These are the same restrictions as for plane flow.

It now remains to look at the behaviour of γ₀ as η→0 to find any restrictions on t and m to ensure that β ≥ 0. Let

(63)

Since

(64)

Substituting these asymptotic values in Equation (55) and balancing both sides of the equation leads to

(65)

and

(66)

where ρ_m = ρ (η = 0).

The restriction β = 0 is the same as for the plane-flow problem.

The appearance of the ρ term in Equation(55) means that the problem cannot be converted to an initial-value problem as before, but must be solved as a two-point boundary-value problem. Equation(55) can be written as two first-order ordinary differential equations for η, ρ with γ₀ as the independent variable:

(67)

where

and the prime denotes the derivatives with respect to the argument. In the centre, as ρ, γ₀ → 0

(68)

since l = 1. Also

(69)

where The limit of F ₂, Equation(69), is bounded for l ≥ 1 in the light of the second of Equations(55) and Equation(60). However, since l = 1 the limit must be evaluated explicitly in order to determine As ρ, γ₀ → 0 the essential behaviour of Equation(48) is

(70)

which integrated gives

(71)

and so

(72)

So finally, Equations(67) are integrated numerically using the explicit end-point values

(73)

(74)

using the first of Equations (58) and Equation (65), where γ_m and γ₁ are given by Equation (66), and Ω₀| _n=1 for Glen’s law and Colbeck and Evans’ law respectively is

(75)

The boundary conditions are:

(76)

Solutions for various values of the physical parameters were found using a shooting technique. It was found that numerical convergence hinged on good initial choices of ρ_m and η_c.

7. Illustrations

For comparison between laws with different exponents n, all results are expressed in terms of the same dimensionless horizontal coordinate ρ^⋆ , based on the scale factor ε₁ = ε (n = 1) and so, from Equation(44)

(77)

All calculations adopt the values in Equations(15) with a = 0.1, q_m = 3 X 10^–9 m s^–1, so that θ = 0.09. The real slope magnitude ε for different n and h₀ is shown in table 1, M and J. Illustrations are shown for linear Λ and Q^⋆ , namely

(78)

For comparison with Reference NyeNye’s (1959) profile, the latter is rewritten in terms of the dimensionless coordinates (ρ, η), taking q = q _m, λ = λ (h₀) giving

(79)

where η_c or ρ_m has to be prescribed.

Comparisons between Nye’s solution as given by Equation(79) and the complete small-slope solution are shown in Figure 2 with Q _o = λ ₀ = 1, where the value of η_c in Equation(79) has been set to that obtained by the analysis presented here.

Fig. 2.

Comparison of Nye’s solution with the complete small-slope profile for Glen’s law with linear Λ and Q⋆, with λ₀= Q₀ (i)

m = 1, n = 1, (ii) – – – – – m = 2, n = 3.

Figure 3 shows the profiles obtained for four sets of (λ₀, Q₀, m) using the Colbeck and Evans polynomial law. These profiles are analogous to those shown in figure 3, M and J, and it is clear that the effects of changing the parameters (λ₀, Q₀, m) are similar for the plane flow and axisymmetric cases. Values of ρ_m ^⋆ and η_c for (λ ₀, Q ₀) = 1, 5, 10 with m = 1, are shown in Table 1 (cf. M and J, table II).

Fig. 3.

Comparison of profiles for Colbeck and Evans’ law with linear Λ and Q⋆; (i)

λ₀ = 1, Q₀= 1, m = 1, (ii)– – – – – λ ₀ = 1, Q₀ = 5, m = 1, (iii)     λ₀ = 5, Q₀ = 1, m = 1, (iv)–  –  –  – λ₀ = 1, Q₀= 1, m = 3.

Table II.

Percentage decrease in the aspect ratio from plane to axisymmetric flow

It now remains to compare the plane-flow and axisymmetric solutions. Plane-flow analysis is simpler and other physical solutions, such as a non-symmetric ice sheet, may be considered. However, by looking at the axisymmetric case, some insight into the effect of a third dimension can be obtained. Reference NyeNye (1959) predicted a decrease in the aspect ratio of

The analysis presented here and in M and J does not yield a simple formula for the change in the aspect ratio, but a comparison between table II (M and J) and Table I confirms a decrease (or no change) in all cases considered, as shown by Table II. These values range from 0–17% compared with Nye’s prediction of 21%. It is, however, apparent that changes in the accumulation rate and the sliding law affect this decrease.

Finally, Figure 4 shows both the plane-flow and axisymmetric profiles on the same horizontal axis for the Colbeck and Evans polynomial law, with Q₀ = λ₀ = 1, and m = 1.

Fig. 4.

Comparison between plane and axisymmetric flow for Colbeck and Evans’ law with linear Λ, Q⋆, where λ₀ = Q₀ = 1, and m = 1; (i)

plane flow, (ii)– – – – – axisymmetric flow.