Ice movement

The only fixed distance on solid ground in the network, common lo both the 1965 and 1968 surveys, lies between points 11 and 13. A comparison of the adjusted values obtained from both surveys shows a difference of 0.028 m (1 951.270 m in 1965, 1 951.242 m in 1968), which is most likely due to random measurement errors as well as systematic scale differences between the two different geodimeters used. Before the coordinates of both surveys can be compared, this difference must be distributed over both surveys, i.e. the scale of each survey must be changed so that the average distance (11-13) becomes (1 951.270+1 951.242)/2 == 1 951.256 m. This means a slight scale change for both surveys. For the 1965 survey data, the scale must be reduced by about 7 p.p.m.; for the 1968 data, the scale must be increased by the same amount. The resulting coordinates, listed in Table II, are likely to provide a better comparison than the originally adjusted coordinates, although they only differ by at most 3 cm (point 16). In Table II, the coordinate differences Δx, Δy (displacements) and the actual velocity in m year^-1 are listed for all points on the ice. The time difference between both campaigns was 2 years and 11 months.

The displacements of points 12 and 14 on the grounded ice rise (Fig. 1), compared with those on the floating ice shelf, are very small. Any movements on this ice rise are only of local origin and are mainly dependent on the surface slope. Due to the various kinds of observational errors and possible changes in inclination of the survey markers, these determined displacements may not be of great significance.

There is an obvious movement of the ice shelf of about 0.5 m year^-1 towards the north. Assuming this value as constant across the eastern part of the Ward Hunt Ice Shelf (east of Ward Hunt Island), the rate of discharge can be estimated. With the present distance of about 13 km between Ward Hunt Island and Cape Albert Edward, an ice thickness of 40 m (Reference CraryCrary, 1958) would give a discharge of 0.26 X 10⁶ m³ year^-1. Neglecting continuous small-scale ice calving, an area of approximately 6.5 km² of ice will be formed each year. Despite this low value and the fact that the present ice front is fixed tightly at both ends, it seems rather difficult to predict another extensive break-up of the eastern ice shelf in the near future. Massive ice calving also depends to a great extent on océanographie effects (e.g. tide and swell).

According toReference NyeNye (1959), an average deformation or strain-rate tensor within a quadrilateral can be determined, if all of its sides were measured twice at different instants of time. This is the result of the transformation properties of strain-rates. Assuming a steady-state ice flow, i.e. the velocity components u, v s are functions x,y only (independent of time), the Jacobian matrix

expresses velocity changes due to small point displacements. The deformation itself is usually given by the components of the strain-rate tensor, defined by a symmetric matrix

i.e.

The strain-rate tensor generally changes its components if the coordinate system changes its orientation. Denning an orthogonal rotation matrix by

where α is the angle between the original x-axis and the rotated x’-axis, both the Jacobian F as well as the strain-rate tensor E are transformed into the coordinate system x’, y’ by the expressions

One essential result is that the strain-rate έ_α in a certain direction (e.g. under an angle α against the original x-axis: έ_α = έ’_xx) can be expressed in terms of a and the three independent components of E:

The six sides of the quadrilateral (15-16-17-18) therefore give rise to six equations in three unknowns έ_xx, έ_xy, έ_yy . These equations can be treated as observation equations from which best values of the strain-rate components are reduced according to the principle of least squares.Footnote * Table III gives the quantities and their values required for computation of the strain-rate tensor of quadrilateral 15-16-17-18.

Table III.

Strain-Rates 4_a Of The Quadrilateral (15-16-17-18)

After solving the normal equations, the following components are obtained

The relatively large r.m.s. errors are caused partly by systematic velocity changes within the quadrilateral and partly by unavoidable measurement errors. Using 0.015 m as standard error for one distance measurement (Table I), the value increases for the difference of two measurements to 0.015x2½ = 0.023 m. This yields an a priori standard error for the strain-rate (S = 1000 m, t = 2.9 year) of 0.8 X 10^-5 year^-1, which part can be considered as accidental.

The directions of the principal strain-rates are parallel to the principal axes of the tensor ellipse. Whereas these directions are given by the eigenvectors of the strain-rate tensor, the principal strain-rates are identical with its eigenvalues. The eigenvalue problem leads to a solution of the quadratic equation

resulting in the two eigenvalues

The first value is orientated within the coordinate system according to its eigenvector (ξ₁ ; = 1; η₁= 11.1) as the result of either of the two equations

As shown in Figure 4, an obvious expansion occurs parallel to the main ice How and compression perpendicular to it. Since the ridges and troughs (Fig. 1) are orientated parallel to the main ice compression, they cannot be originated by internal ice forces. This would be in accordance with Reference SmithHattersley-Smith’s (1957) theory.

Fig. 4.

Velocity vectors and principal axes of strain-rate tensor.

Ice stresses

The determined strain-rates can be used to obtain some indication of the stress conditions on the ice shelf in general and within the measured strain figure in particular. Based on Glen’s power flow law of the form έ — (τ/έ)^η, where έ and τ are the second invariants of the strain-rate and stress-deviator tensors, respectively,Footnote *defined by

some dynamic properties of the ice shelf can be estimated in a way similar to Reference BuddBudd’s (1966) investigations on the Amery Ice Shelf, Antarctica. Application of theoretically derived equations by Reference WeertmanWeertman (1957) and Reference NyeNye (1965) lead to good estimations for the flow parameters B and n of Glen’s equation.

According to Reference NyeNye (1965), the velocity v in an infinitely deep channel of uniform width 2a is given by

y being the distance lateral to the flow, measured from the center to either of the two channel walls(y _max = a). For purely laminar flow, the velocity at the center is expressed as

where dp/dx is a uniform longitudinal pressure gradient. The flow through the channel between Ward Hunt Island and Cape Albert Edward, however, is somewhat complicated by the presence of the Marvin Islands (Fig. 5) and a number of smaller ice rises not taking part in the general flow. As indicated in Figure 5, two main ice streams may be distinguished, in which the orographically left one obviously has a higher velocity. An idealized velocity profile across the ice shelf, though difficult to determine, may be assumed to lie somewhere between the two profiles, corresponding to the entire width 2a = 11.6 km, and to the partial width 2a’ = 5.4 km of the left stream only. The maximum velocities at the centers of two such channels can be determined from Equation (1), since v = 0.53 m year^-1 is the known velocity within the strain figure situated about 1.4 km apart from the Ward Hunt ice rise. For different values of n, these center velocities are listed in Table IV. Also listed are the lateral velocity gradients dv/dy, dv/dy’, respectively, obtained simply by differentiation of Equation (1). An average lateral velocity gradient estimated from the corner velocities of strain figure 15-16-17-18 yields a value of —0.026 m year^-1 km^-1 (+0.007 between 16 and 15, —0.059 between 17 and 18). This value is significant only with respect to its order of magnitude, the corner velocities already having standard errors of ±0.02 m year^-1. The most probable exponent (Table IV) seems to be very close to n = 4.

Fig. 5.

Simplified flow of the Ward Hunt Ice Shelf between Ward Hunt Island (W.H.I.) and Cape Albert Edward (C.A.E.).

Table IV.

Centrai, Velocities And Their Lateral Gradients For Entire And Partial Ice-Shllf Width

Fig. 6.

Estimation of an idealized velocity profile.

In Figure 6 the two velocity profiles are plotted. An average curve with v"_mix = 0.62 m year^-1and a fictitious width 2a" = 7.4 km are assumed in the following considerations.

For a determination of B, Reference BuddBudd (1966) gave the equation

which relates the velocity v and its longitudinal gradient dv/dx to the boundary dimensions of the ice shelf (thickness H, width 2a), and the flow parameters n and B. The gradient dv/dx could be obtained similar to dv/dy from the strain-figure data, but it is not needed since its gradient d² v/dx² , actually not known from the available observations, may be neglected because of the slow flow of the Ward Hunt Ice Shelf. The density of ice (p = 0.92 Mg m^-3) and an average thickness gradient (dH/ds = 3.0 m km^-1) were compiled from Reference CraryCrary’s (1958) data. With the above values and ·2α = 7.4 km, we obtain B = 2.8 X 10⁷ N m^-2 s^¼ (3.7 bar year¼).

Fig. 7.

Relationship between B and n as deduced from Weertman’s equation and Measured creep rate. Comparison with previous determination using Nye’s equation (◬) and experimental data (◯,[inlineeq]).

The flow parameter B is also a function of temperature T of the form

where T₀ is the temperature (in K) of the melting point of ice, B(T₀) = B₀ is a constant equal to 2.67 bar year¼ (2.0 X 10⁷ N m^-2 s^¼),Footnote * Q is an activation energy for creep and is equal to 68 kJ mole^-1 (Reference Mellor and TestaMellor and Testa, 1969[a], [b]), R is the gas constant 8.32 J K^-l mole^-1, and n is the second flow parameter, taken here as 4. To find B for any temperature T = T₀+t (t in °C), the above equation can be transformed into

The mean annual surface temperature of the ice shelf is — 18.5°C (Reference Lyons and RagleLyons and Ragle, 1962), while the temperature at the base is — 1. 8°C, the freezing 7f sea-water. Reference Lyons and RagleLyons and Ragle (1962) showed several graphs of temperature versus depth from different parts of the ice shelf, although none of these extended to the base. Temperature appears to increase linearly with depth, at least approximately. The mean annual ice temperature therefore is equal to (—18.5—1.8)/2 = —10.2°C. This gives a value for B of 2.7X 10⁷ N m^-2 s¼, which corresponds very well to the previous estimate of 2.8 X 10⁸. In Figure 7, the point corresponding to (B = 2.75 X 10⁸; n = 4) is plotted as a triangle.

By taking Nye’s relation for steady-state creep for the case of zero transverse creep, and taking advantage of the fact that the ice shelf is floating, Weertman (1957) obtained for the creep rate

where Δp = p_w-p_i is the difference between the densities of water and ice, H = ice thickness, and

Weertman’s equation for the extension rate provides the means of establishing a relation between the two flow parameters n and B for different values of H, the other quantities having known values. K can be set equal to the measured longitudinal surface extension rate 1.17 X 10^-4 year^-1 (3.26 X 10^-12 s^-1), an assumption that is right insofar as K is constant with depth, but is valid only near the center flow line of the ice shelf. However, Weertman’s creep rate is still a good approximation to the extensional rate elsewhere. According to height measurements by H. Serson,Footnote * the ice shelf around the deformation figure has a freeboard of h = 4.39 m. The entire thickness of the ice shelf is given by

where p_w = 1.03 Mg m^-3 is the density of sea-water. The mean density p _i of the ice shelf lies somewhat between 0.90 and 0.92 Mg m^-3 (Grary, 1958), which results in an ice thickness Hbetween 35 and 41 m.

Taking p_i as parameter, [inlineeq] can be expressed in terms of n:

In order to obtain a linear graph, [inlineeq] is represented in logarithmic and n in hyperbolic scale in Figure 7.

In terms of principal strain-rates, the effective shear strain-rate έ is given by the expression

where έ₃ is determined through the continuity equation (no volume change)

With έ₁ = 1.17 X 10^-4 year^-1, έ₂ = —0.56 X 10^-4 year^-1 (Fig. 3) one obtains έ₃ = —0.61 X 10^-4 year^-1. This yields an effective shear strain-rate of έ = 1.01 X 10^-4 year^-1. The effective shear stress is determined by [inlineeq], and the principal deviatoric stresses by

The result is τ = 0.37 bar, σ’₁ = 0.43 bar, σ’₂ = —0.20 bar, σ’₃, = —0,22 bar. Since the vertical stress σ₃ = 0 on the ice surface, the actual horizontal principal stresses σ₃, given by the relationship

are

Elevations and atmospheric refraction

Though primarily measured to reduce slope distances to horizontal distances, the vertical angles may be used for determining the elevations of all stations. They can also reveal some information about the refractive index of air during the period of observation. It is known that in the lower atmospheric layers above sea ice, shelf ice or inland ice (see e.g. Reference LijequistLiljequist, 1964) the gradient of the refractive index, as a direct function of the temperature gradient, can be extremely high. The reason lies in the radiation losses from the surface which may develop temperature inversions. Any ray of light extending between the surface and such an inversion is subject to a more or less large path curvature, which, under certain circumstances, may give rise to mirages. The effect forbids any precise trigonometric height measurements. Only if vertical angles and temperature gradients are measured simultaneously at both stations, more reliable heights may be obtained. If the distance between two such stations is not too long, the path of the light ray can be assumed as part of a vertical circle. Reference BrocksBrocks (1954), amongst others, showed that, along a horizontal surface, the curvature of a light path is directly proportional to the air-temperature gradient:

if influences of air and vapor pressure are neglected. In surveying, a so-called “local coefficient of refraction" (or relative curvature of light) is used, defined as

with R = radius of the Earth. In practice, this quantity is taken as a constant (e.g. k = 0.13), if the light path is sufficiently far away from the surface. In polar regions, however, k may assume values higher than 3, and even negative values are common.

At all stations on the Ward Hunt Ice Shelf, vertical angles were measured. Therefore, both angles are given between two neighbouring stations, although these were not measured at the same instant of time. By means of given instrument and target heights above corresponding stations and the distance between instrument and target, the actual light path can be approximated by a circular are, the radius of which is equal to the inverse of an average path curvature K. Thus all elevation differences along the lines of the network (Fig. 1) can be determined. Since more observations are available than required, a height adjustment of the entire network can be carried out.

The result is shown in Table V, where the elevation of station 18 is taken from a geometric level survey run by H. Serson in 1969. All elevations can be considered to have a standard error of 0.2 m over the comparatively short distances involved.

Table V.

Elevations or Ward Hunt Ice SheIf Survey Stations. Reference Point For Ice Stations Is Top Of Marker, For Ground Stations Bench Mark

Table VI shows refraction conditions of the lower atmosphere for various lines of sight (Fig. 1), together with the resulting angles of refraction (i.e. vertical angle on a station between circular light path and straight line to opposite station), local refraction coefficients and computed temperature gradients. Of particular interest are observation periods 1 and 2 (column 1), because of equal elevation of all stations under consideration. Whereas typical white-out conditions yield a refraction coefficient of 0.6, the “mixed" values during the second period show a rather high influence of the pure radiation condition (clear sky), resulting in a somewhat extrapolated coefficient of 1.5 + (1.5—0.6) = 2.4; note the extremely high temperature gradient of 0.3 deg m^-1. For observation periods 3 and 4, a proper analysis is virtually impossible with the sparse data available, due to asymmetric vertical profiles with height differences up to 50 m. The negative coefficient of refraction along line 11-13 is a result of the dark, snow-free and heated terrain surface with associated heat convection into the surface layers of air.

Table VI.

Refraction Of Air During Observation Periods