2.1.Integration of the equations of motion

Two-dimensional motion or “plane strain-rate” only is considered.

We adopt a right-hand orthogonal system of axes, x, z such that the horizontal is inclined at an arbitrary angle χ to the positive x direction. All angles will be taken positive for an anticlockwise rotation from the. x-axis (see Fig. 1).

Fig. 1.

Coordinate system mid definition of quantities.

Let −α of be the surface slope of the ice mass at position x, −β be the basal slope of the ice mass at position x, Z ₁ be the ordinate of the surface at position x, Z ₂ be the ordinate of the base at position x, write Z = Z ₁−Z ₂ for the ice thickness at x and −θ = −α + χ, −ϕ = −β + χ, for the slopes of the surface and base, with respect to the x-axis. The stress components at (x, z) are denoted by (σ_x, τ_xz σ_z). We consider an ice mass of constant density p Let g be the gravitational acceleration, and write g_x = +g sin χ, g_s = −g cos χ for its components in the directions of the axes.

The equations of equilibrium for slow steady motion may then be written as

(1)

(2)

These equations are true everywhere in the ice mass for any such system of axes so defined.

We require an expression for the longitudinal stress deviator σ′_x = ½(σ_x−σ_z). Hence we differentiate Equation (1) with respect to z and Equation (2) with respect to x and subtract to yield

(3)

This equation also does not depend on the choice of the axis orientation. It is only when this equation is integrated that it becomes necessary to specify the boundary conditions in terms of (he axis direction.

We integrate Equation (3) with respect to z from z ₁ to z

(4)

From Equation (1) we note at the surface that

(5)

Hence, using this, Equation (4) may be written

(6)

Now integrating again with respect to z this time from z ₁, to z ₂ noting the first two terms on the right are constant with z and also Z = z₁ − z ₂ gives

(7)

This equation is exact and expresses the mean longitudinal stress deviator gradient in terms of the boundary conditions al the surface and base of a column of ice. We shall shortly examine the surface and base boundary conditions in detail. But first it is often required to integrate this equation with respect to x. To do this we note that the left-hand side may be written

(8)

2.2 Boundary conditions

At the surface we make the assumption that the shear stress parallel to the surface is zero, and that the normal stress is the atmospheric pressure p.

If θ is the angle between the. x-axis and the surface, then using the standard formula for the rotation of axes, the normal and shear stresses are related to the components in the x, z directions at the surface by (cf. e.g. Reference JaegerJaeger (1962))

(9)

(10)

For Equation (8) we require only Equation (10) in the form (dividing by cos²θ)

(11)

Similarly at the base, if −τ _b is the basal shear stress parallel to the bed, where −ϕ is the angle between the base and the x-axis, then

and therefore

(12)

Since and , using Equations (11), (12) and (8) in Equation (7) and writing

(13)

The term is zero for points where the surface is parallel to the x-axis. For other slopes however this term depends on the longitudinal stress and stress-gradients and the curvature of the surface. We now evaluate this term in full to show under what conditions it may be approximated by ρg_z tan θ.

If s denotes the distance along the curved surface,

(14)

Using Equation (2) this may be written

Similarly expanding and using Equation (1) gives

(15)

Now at the surface if σ_n, σ₁ are the normal and longitudinal stresses and σ′₁ = ½(σ₁ − σ_n)

Hence

and

Substituting in Equation (15) and taking surface values

(16)

Hence to take as the first approximation

it is necessary (and sufficient provided the longitudinal stress and stress gradients are not too large) that not only the slope θ be small but also the slope gradient, i.e or less. Since the longitudinal stresses vary with the surface slope this is usually the case. However abrupt changes of surface slope or stress (e.g. surface crevasses) will not be covered by the approximate formula.

Finally substituting Equation (16) inEquation (13) and writing the components of g in full gives

(17)

This equation is exact, it has arbitrary orientation χ for the x-axis, and applies everywhere along the ice mass with the same rectilinear coordinates, and hence it may be directly integrated with respect to x, without resorting to curvilinear coordinates. From this equation the conditions required for various simplified forms may be determined.

2.3 Special cases of longitudinal axis inclination

(1) Small slopes.

For small slopes χ, θ, ϕ and slope gradients (i.e. neglecting second and higher orders). all except the first two and last two terms on the right of Equation (17) are negligible so that it reduces to the form given by Reference BuddBudd (1968)

(18)

where In this case only the surface slope is the surface slope is relevant and it is therefore immaterial whether the axes are taken horizontal, parallel to the surface or parallel to the base. The senses of the signs in Equation (18) are such that an increasing tension in the direction of motion is required to balance either a smaller negative slope downwards in the direction of motion) or a greater basal stress against the direction of motion.

Other simplified forms of Equation (17) may be readily obtained by choosing the longitudinal axis horizontal (χ = o), parallel to the surface (θ = o, χ = a), or parallel to the base (φ = 0, χ = β), at some particular position.

(2) For the longitudinal axis x parallel to the surface at some position we have χ = α, θ = 0, φ = α –β and Equation (17) reduces to

(19)

An alternative form may be obtained directly from Equation (7)

(20)

These results correspond to the form derived by Reference NyeNye (1969) except that here Z is not necessarily constant, and the no-slip condition at the base is not required (i.e. (σ_x – σ_z) is not necessarily zero, and τ _b is given in general by Equation (12)).

For small (α−β) these results all reduce to Equation (18) above. However, since the x-axis here is parallel to the surface, whose slope varies along the ice mass, this equation cannot be readily integrated with respect to X without resort to curvilinear coordinates.

(3) For the X-axis parallel to the base, χ = β, φ = o, θ = α – β. Equation (17) becomes. for small (x − β),

(21)

For both small (α – β) and β this immediately reduces to the form of Equation (18). This equation is also useful in the case of small (α – β) but moderate values of β and if the, x-axis is chosen parallel to the average basal slope over the length, then it may be readily integrated with respect to x.

(4) For the x-axis horizontal χ = o, θ = α, ϕ = β. This form may he useful for considering a wedge-type profile say near a terminus, with small β and ∂α/∂x but moderate α. In this ease if the longitudinal surface stress and slope gradients are not large Equation 17) reduces to

(22)

Again even for moderate θ this equation differs only slightly from Equation (18).

3.1 Preliminary comments

The theory so far, for stresses, is exact and applies generally (for slow creep) whatever the How law. The application of this equation to the study of flow properlies of ice masses from elevation and bedrock profiles and surface strain-rates, requires some additional less substantiated assumptions.

Before examining the flow law, it. is necessary to make a comment concerning the second term on the right of Equation (18) viz.

This term will be discussed in a separate paper concerning ice flow over undulations (Reference BuddBudd, 1970), but it has already been indicated (Reference BuddBudd. 1968, 1969) that T is important for small wavelength undulations (λ ≈ 3Z). T is zero of Y_xz constant or increasing linearly with X but for flow over undulations it enhances the relative maximum extension on crests and compression in troughs, such that Equation (18) may be written

Where for long waves, α_s is the local surface slope and α is the mean slope over c. 10 to 202.

The assumption of replacing τ _b by is largely empirical (ef. Reference BuddBudd, 1968) but it is a good approximation provided α is calculated over such a distance that the longitudinal stress is unimportant (x > c 10Z). Several earlier workers (Reference Robin DeRobin, 1967; Reference BuddBudd. 1968: Reference CollinsCollins, 1968) have used a power law for flow, say

and replaced σ_x in Equation (18) by This is in many instances unsatisfactory since it neglects the effect of the vertical shear. The question arises: is the longitudinal strain-rate much higher in regions of high vertical shear for the same values of longitudinal stress devialor and ice temperatures? For Newtonian How it would not be, but for a high power law for flow it would be.

The following analysis aims at devising a method to answer questions such as this by determining the flow law from measurements of longitudinal stress and strain-rates in regions with differing degrees of vertical shear.

3.2 Derivation of the flow law of ice from the longitudinal stress and strain-rate gradients

To convert the equation for longitudinal stress gradient to an equation in strain-rate gradient it is necessary to re-examine the flow law of ice.

With a power law for flow it is only satisfactory to replace by if or n ≈ 1. In practice (Reference McLarenMcLaren, 1968) τ_xz is often much larger than . The major problem then is to consider the relation between and when τ_xz is not necessarily small.

We abandon the power-law formulation of the stress strain-rate relation for ice

(23)

where is the stress deviator and τ the octahedral shear stress, since n and B are both found to vary with stress. Instead we adopt a single-parameter “generalized viscosity” relationship of the form

(24)

Where η(τ, θ) is a function of both stress and temperature.

For the octahedral shear values of stress τ, and strain-rate Equation (24) gives (cf. Reference NyeNye, 1953)

(25)

Hence

(26)

This may be regarded as an alternative definition of η. Equation (25) may be regarded as the flow law of ice, and for each constant temperature represents a single curve on the versus τ diagram. It is these curves we wish to determine. The important result is that for a given stress state the longitudinal stress and strain-rate have the same ratio as the octahedral values.

So, substituting for the average longitudinal stress deviator through the ice column in the equation for stress gradient (18)

(27)

We obtain

(28)

(29)

or

(30)

We now define a weighted mean (low parameter η* through a vertical column by

Then by integrating Equation (30) (for fluctuations around a mean value, i.e. taking for δα* = 0)

(31)

or

(32)

Where

(33)

Hence from the ratio of the longitudinal stress deviator to the longitudinal strain-rate we obtain η* for that τ and θ. We now use these values of η* and τ to determine the How law in terms of octahedral values.

Since in terms of the octahedral values we can now obtain the flow law of ice by calculating from η* and τ for each value and illustrate this by then plotting τ against τ.

For two-dimensional flow the octahedral shear stress τ is calculated from

(34)

taking

(35)

Here α is taken as the mean surface slope over a distance x about ten times the ice thickness. The approximation in Equation {34) is expected to be close since τ_xz increases linearly with depth and only varies slightly in the upper layers.

These values of stress are derived directly from the measured elevation and ice thickness profiles. The surface strain rate can be measured, but to obtain the average strain-rate through the column something must be known about the ratio say.

This normally requires information on the velocity-depth profile. However if the ice is not slipping at the base then we may expect the strain-rates to vary with depth in a similar wav to the velocity V, i.e.

(36)

For cold ice caps the velocity profile depends on the temperature profile and estimates cart be calculated (cf. Reference BuddBudd, 1969).

In the absence of a velocity profile the value of λ can be assumed to lie between ⅔ (for viscous flow with no slip) and 1 (for block sliding) being near 90% for typical ice cap temperature profiles. In terms of the measured surface strain-rates, then, we may write

(37)

From the measured variations in surface strain rates , and surface slope α, the generalized viscosity can be determined from Equation (37) and then using the values of mean octahedral shear stress from Equations (34) and (35), and for the mean temperature for the column at that position, a point on the stress strain-rate relationship can be established by plotting

(38)

versus τ.

Strictly is not a unique single-valued function of τ and θ because it depends not only on the mean values but also on the vertical distributions of θ and τ. However until more is known about the vertical distribution of longitudinal strain-rate this method provides a useful means of analysing measured surface longitudinal strain-rates in regions of different mean temperature and stress.

To obtain the complete set of curves for many values of covering a wide range of shear stress τ and température θ are required. For a typical cold ice cap the ice temperature θ and the shear stress τ both generally increase from the inland towards the coast and hence each contribute to higher 1/η* values (i.e. ratios) near the coast. For temperate ice masses at pressure melting point throughout) we may expect the variations in the ratio of the longitudinal strain-rate to longitudinal stress to depend just on the variations in the magnitude of the octahedral shear stress.