1. The Flow Law
Laboratory studies (Reference SteinemannSteinemann, [1956]; Reference GlenGlen, 1955; Reference Tabor and WalkerTabor and Walker, 1970) of the steady-state creep of polycrystalline ice in uniaxial compression gave a power law relating strain-rate 
and applied stress σ:
where K is a temperature-dependent constant of the form A e−Q/RT and n is approximately constant ( ≈ 3) over the stress range 0.1 to 2 MN m−2.
Neglecting elastic strains, Odqvist’s (1934; 1966, p. 21) generalization to three dimensions of the uniaxial law can be written:
where 
and 2τ
0
2/3 are the second invariants of the strain-rate and stress deviator tensors respectively:
where x, y and z are mutually perpendicular axes, 
is the strain-rate in the x direction and σ′
xx
is the deviatoric stress in the x direction. 
is the effective shear strain-rate and τ
0 is the effective shear stress. Reference NyeNye (1953) proposed alternative definitions with 
and 2τ
0
2/3 replaced by 
and 2τ
2 so that in a pure shear experiment the shear stress is τ and the shear strain-rate is 
. This slightly modified form of Odqvist’s generalization has since been accepted in glaciological work. The constant B
0 is replaced by B which can be expressed in terms of K and n. The strain-rate in any direction x then becomes:
2. The Creep of Ice Shelves
2.1. Unconfined ice shelf
Floating ice shelves represent perhaps the simplest natural ice forms. They rest on a frictionless bed, stress conditions are uniform over large distances, and boundary conditions at the upper and lower surfaces are known. Consequently analysis of their creep behaviour avoids many of the problems associated with land glaciers. For an unconfined ice shelf of constant thickness H we have zero shear stresses of the type σ ij if we choose x- and y-axes horizontal at sea-level and z-axis vertical (Fig. 1). By balancing stresses in the x direction summed over depth against sea-water pressure at the ice front Reference WeertmanWeertman (1957) showed that, for an ice shelf of uniform density ρ and zero transverse creep
where h is the elevation above sea-level of the ice-shelf top surface and B is the value of B averaged over depth.
2.2. Bounded ice shelf
Weertman’s analysis predicts that the longitudinal strain-rate 
will increase rapidly with increasing ice thickness. But Reference BuddBudd (1966) reported observations on the Amery Ice Shelf showing the opposite trend with 
rapidly decreasing almost to zero as the ice thickness (and distance from the ice front) increased. The Amery Ice Shelf lies between two almost parallel flanks of land ice and Budd suggested that under these conditions Weertman’s expression for an unrestricted ice shelf should not be expected to apply. Instead, part of the driving force due to the weight of ice above sea-level is used in overcoming the restraining effect of the ice-shelf margins. Consequently the ice thickness increases rapidly away from the ice front, and this feature was considered by Reference CraryCrary (1966) to be responsible for overdeepening at the inland end of fjords.
Nye’s velocity solution for a flat ice shelf held at its sides and deforming by laminar flow (Reference NyeNye, 1952) can be written
where V c is the velocity at the centre line of the ice shelf and V y is the velocity of the ice at a distance y from the centre line of the ice shelf. ∂p/∂x is the pressure gradient (assumed to be independent of y) along the ice shelf. Budd substituted
(where z is measured in the vertical direction, b and s refer to the lower and upper surfaces of the ice shelf, and bars denote values averaged over depth) for ∂p/∂x in Equation (5) to obtain an expression for the strain-rate gradient along the centre line. However, in this situation ∂p/∂x is the stress gradient averaged over depth
so that Budd’s expression applies only to ice shelves of constant thickness. Furthermore Equation (5) is based on the assumption that 
is zero whereas Budd reported large values of longitudinal strain-rate on the Amery Ice Shelf.
In Reference BuddBudd (1969, p. 115 and 123) a general expression was derived for 
where 
is the longitudinal deviatoric stress and is 
for zero transverse strain. With the assumption that the effective shear stress can be written as 
Budd then obtained an expression involving the longitudinal strain-rate gradient:
where ω is the surface slope and f is the boundary friction coefficient defined in terms of the shear stress at the perimeter of a cross-section and a shape factor. Here the bars denote averages taken over the section.
For ice shelves, Reference BuddBudd (1969, p. 138) reduced Equation (6) to:
with the implicit assumption that terms involving ∂H/∂x on the left-hand side of Equation (6) are insignificant, or
Using data for g1 on the Amery Ice Shelf given in Reference ThomasThomas (1973) we find ∂H/∂x ≈ 2×10−3and
and in this instance the assumption is certainly not warranted.
Furthermore for “Weertman” creep, Equation (7) reduces to
whereas differentiation of Equation (4) gives
Thus Equation (7) is limited in its application to the case where
which in general implies ∂H/∂x → 0 and the ice shelf is of almost uniform thickness.
In the next section we shall derive a general expression for creep in an ice shelf where the sole restriction is that of zero shear stresses in the xz and yz planes.
3. General Expression for Ice-Shelf Creep
We choose rectangular axes with the x-axis at sea-level in the direction of ice movement and z-axis upwards (Fig. 1). The symbols to be used are listed below:
-
σ jk j, k = x, y, z; j = k: direct stress; j ≠ k: shear stress
-
σ′ jk stress deviator = σ jk:−δp; p = hydrostatic pressure; δ = 1 when j = k; δ = 0 when j ≠ k
-
strain-rate; u = velocity -
H ice-shelf thickness
-
h ice-shelf surface elevation
-
ρ density.
strain-rate
; u = velocityThe subscripts i and w refer to ice and sea-water respectively. All densities are assumed independent of x, y and z in order to simplify the equations. However, at appropriate points, equations in squared brackets are included to incorporate ice density as a function of z.
For quasi-static creep the equilibrium conditions are:
We make the following assumptions:
-
(a) Ice is incompressible and hence
-
(b) The ice shelf is in hydrostatic equilibrium so that
-
(c) Zero shear strains in the xy and yz planes. This means that velocity and strain-rates are independent of z.
-
(d) The generalized flow law (Equation (3)) holds for ice.
Together with assumptions (a) and (c) this means that
Equation (10) now becomes:
Field results give values of 
and 
at some point P(x, y, z), on the ice-shelf surface, and we can express each in terms of 
:
In general α and β are functions of x and y but, from assumption (c), they are independent of z. Except in the special cases summarized at the end of section 3.1, α and β are deduced from measurements of the strain-rate components.
From the flow law we also have that the ratio 
is constant at any point, so
and
or
Using Equation (11) and neglecting atmospheric pressure we also have
So, integrating Equation (12) over z gives:
where
is the total force opposing the movement of a unit vertical section at P.
In order to obtain a solution for the strain-rate we next express the effective shear stress τ in terms of σ′ xx .
By definition:
and
The flow law
now becomes:
with the sign of 
determined by that of σ′
xx
.
Thus:
and using the assumption (c) we can write:
Comparing Equations (13) and (15):Footnote *
which, for convenience, we write:
where
and the sign of 
is determined by that of
Note that with β = 0 the term (2 + α) θ/|2+α| reduces to
which can be shown to be equal to 1/ϕ, where ϕ is Budd’s “transverse strain function” (Reference BuddBudd, 1969, p. 126).
Equation (16) is of little use unless we can evaluate F, so we shall consider two special cases which approximate conditions in actual ice shelves.
3.1. Ice-shelf movement restricted in at least one direction by sea-water pressure only
By choosing the x-axis to coincide with this direction, F becomes the total force exerted by the sea on a unit vertical section of ice shelf:
Equation (16) becomes:
With α = β = 0, Equations (17) and (18) reduce to Reference WeertmanWeertman’s (1957) expressions (Equation (4) is one of these) for an ice shelf free to creep in one horizontal direction only. The expression for an ice shelf (or iceberg) uniformly spreading in all directions is obtained by setting α = 1 and β = 0, giving:
These equations will not apply very near the ice front where shear stresses are induced in the xz plane by unbalanced hydrostatic pressure at the ice cliff.
3.2. Ice shelf flowing between approximately parallel sides
Consider a vertical element of unit width taken parallel to the x-axis at a distance y from the central line of the ice shelf (Fig. 2). In this case F is the force due to water pressure F w plus that due to shear past the sides Fs. So Equation (16) becomes:
The net shear stress acting up-stream on the element at some point (x, y, z) is − (∂σ xy /∂y) and the total up-stream force due to shear on the entire section between x = x and x = X (the point at which the ice shelf leaves its protective margins) becomes:
To proceed further we make the additional assumptions
-
(e) ∂σ xy /∂y is independent of y,
then
-
(f) σ xy at the sides of the ice shelf averaged over z reaches some limiting value which is independent of x and equal to τ s so that
where a is the half-width of the ice shelf.
Equation (21) then becomes:
and
With τ s equal to some appropriate “yield stress”, say 105 N m−2 we can solve this equation using values of H, a and h measured at different points along the ice shelf.
We can also obtain a solution in terms of the shear strain-rate 
:
and with
Substituting for τ in Equation (24) gives:
From assumption (c), 
is independent of z, so we can write:
and, with
If values of the strain-rate components are measured at several different values of y, Equation (25) can be used to find n. The correct value of n is that which gives the best straight line for a plot of 
versus
assuming B is approximately independent of x and y.
We can write Equation (25)
Thus, measurements of strain-rate at some point at distance y from the centre line can be used to evaluate τ s in terms of B if we know n. This value can then be substituted for τ s in Equation (23) which then becomes:
where 
, α1, and β
1, need not necessarily be measured at the same point as 
, α and β. The sign of 
is decided by that of 
so Equation (27) can be written
If 
and 
are measured at the same point this reduces to:
This expression can be used at any point on the ice shelf, except on the centre line where both β and y are zero.
In a companion paper (Reference ThomasThomas, 1973) the equations derived above will be used to Interpret available ice-shelf data.
Acknowledgements
I thank the British Antarctic Survey for sponsoring this work, Dr W. F. Budd for helpful discussions, and members of the Scott Polar Research Institute for reading and improving the manuscript.
and 
