Introduction
The Erebus Glacier Tongue (EGT) flows from the western side of Mount Erebus into McMurdo Sound, about 20 km north of New Zealand’s Scott Base and the nearby United States base, McMurdo ( Fig. 1). Its close proximity to excellent logistical support has meant that the EGT has been the focus of a number of experimental and theoretical studies to map its geometry and structure (Reference HoldsworthHoldsworth, 1982), and to understand its dynamics (Reference HoldsworthHoldsworth, 1969, Reference Holdsworth1974; Reference Goodman and HoldsworthGoodman and Holdsworth, 1978; Reference Holdsworth and HoldsworthHoldsworth and Holdsworth, 1978; Reference Holdsworth and GlynnHoldsworth and Glynn, 1981; Reference Vinogradov and HoldsworthVinogradov and Holdsworth, 1985; Reference Gui and SquireGui and Squire, 1989).
Fig. 1. Map showing Erebus Glacier Tongue in relation to New Zealand−s Scott Base and the U.S. McMurdo Station.
The EGT is known to have calved at least three times since the turn of the century: in March 1911 members of Scott’s British Antarctic (Terra Nova) Expedition observed a major calving event and the formation of an approximately 4 km2 tabular iceberg weighing more than 100 × 10° tonnes; a similar calving was deduced by Reference HoldsworthHoldsworth (1974) to have occurred in the 1940s and he suggested another calving was imminent. The last calving occurred on 1 March 1990 (Reference Robinson and HaskellRobinson and Haskell, 1990), 3 months after the measurements that are reported in this paper were collected.
These calvings and many others in similar ice tongues took place at some distance from the grounding line, where one might expect the bending moments and stresses to be largest. This has led to the suggestion that the calving is due to the build-up of standing waves in the tongue and that these are of sufficient amplitude to produce fatigue failure of the ice after many cycles. All theoretical studies have assumed that the waves are standing. Various groups have measured strain or tilt on the EGT (e.g. Reference Goodman and HoldsworthGoodman and Holdsworth, 1978; Holds-worth and Holdsworth, 1978) and have attributed the observed oscillations in strain to standing flexural waves.
In 1984, the New Zealand Department of Scientific and Industrial Research (DSIR) Physical Sciences buried sensitive strain gauges in the EGT to monitor oscillations and to observe a calving event, noting that one was overdue. Strains were measured daily from October 1984 to 1989, and were telemetered to Scott Base where they were recorded on floppy disk. Initially, two strain gauges were buried lengthways either side of the tongue, 3.5 km from the snout. The strains from these two instruments were in phase and of similar amplitudes, which is consistent with a lengthwise flexure of the tongue (Reference Robinson and HaskellRobinson and Haskell, 1992). A rosette of three strain gauges was used to investigate further the mode of vibration. The major principal strain was found to lie along the length of the tongue and the minor principal strain was found to be approximately of the major principal strain (Reference Robinson and HaskellRobinson and Haskell, 1992). This was additional evidence that the strains were due to lengthwise flexure of the ice tongue. Since, at this time, it was not possible to distinguish between standing and travelling waves, two more strain gauges were placed at distances of 180 and 1700 m towards the snout. Results from the three strain gauges suggested that flexural waves were travelling from the snout towards the grounding line at a phase speed of approximately 70 ± 10 ms−1 (Reference Robinson and HaskellRobinson and Haskell, 1992).
To clarify further these observations, which were quite sparse, the present intensive study was staged during November 1989. It is the analysis of these 1989 data that is the subject of this paper. The long data set telemetered from Scott Base during 198-89 is deficient in several respects, mainly because of limitations in the band width of the telemetry system. The 1984-89 data set cannot, for example, be used for spectral analysis because the length of the data records transmitted is too short for meaningful statistical interpretation. It was intended for, and is therefore more suited to, climatological work; for instance, looking for long-term changes preceding a calving event. The November 1989 experiment, on the other hand, only took place over a few days but the records were collected with the aim of interpreting the data fully. Unfortunately, the long-awaited EGT calving event of March 1990 took place after all the DSIR strain gauges had been uplifted.
The most recent measurements of the dimensions of the EGT were made in December 1985 (Reference DeLisle, Chinn, Karlen and WintersDeLisle and others, 1989). Its length was 15 km, of which the last 13.5 km were floating, and its thickness varied from 50 m at the snout to 300 m at the grounding line. Its width varied from 0.5 to 2 km with many large bays along its sides. The depth of McMurdo Sound is about 300 m at the EGT’s snout. The EGT grows at the rate of about 150 m year−1 (Reference DeLisle, Chinn, Karlen and WintersDeLisle and others, 1989). Most of the year it is surrounded by sea ice of 1-3 m thickness and it is only for a short time at the height of summer that the sea ice melts around the tongue (Reference Robinson and HaskellRobinson and Haskell, 1992).
The results of Reference Robinson and HaskellRobinson and Haskell (1992) suggest that the strains in the EGT are due to length-wise flexural waves, and in the rest of this paper we shall assume that this is the case.
Flexural Waves
A commonly used model for a floating ice sheet which is flexing under the action of small-amplitude waves is that of a perfectly elastic, thin plate on a fluid foundation. The alternative use of a thick-plate model, which includes the effect of rotatory inertia and transverse shear, has been investigated by Reference Fox and SquireFox and Squire (1991a) and it has been shown to be unnecessary except in the case of very short waves or very thick ice. For example, Reference Fox and SquireFox and Squire (1991a) found the thin-and thick-plate models agreed for periods greater than 20 s with 200 m thick ice. Thus, the present geometries for either the EGT or the sea ice of McMurdo Sound do not call for this more complicated theory; a thin-plate theory is quite adequate.
The ice-plate model and dispersion relation
The dispersion relation for a thin sea-ice plate of thickness h and density ρ′ is derived by assuming an homogeneous ice sheet occupying-∞ < x ∞,-∞ < y ∞, 0 < z ∞ h floating on inviscid, irrotational water of depth H and density ρ ( Fig. 2). Then, for no downward load, it is straightforward to derive a linearized surface-boundary condition for the ice sheet as follows (Reference Fox and SquireFox and Squire, 1991b):
where η(x, y, t) represents a small vertical displacement of the ice sheet, and compressive stresses have been neglected. The effective flexural rigidity of the ice L = Eh 3/12(1 − υ2), where E is the effective Young’s modulus and υ is Poisson’s ratio. The dispersion relation for uniform plane waves of wave number k and angular frequency ω travelling in the positive x-direction is found by assuming that the surface displacement η is everywhere proportional to ei(kx-wt). Then, using the kinematic boundary condition
we substitute Equation (2) into Equation (1), to get
where h′ = ρ′h/ρ. This expression has been used many times to represent waves beneath a sea-ice plate (e.g. Reference SquireSquire, 1984a, Reference Squireb; Reference Fox and SquireFox and Squire, 1990, Reference Fox and Squire1991b; Reference Davys, Hosking and SneydDavys and others, 1985; Reference Wadhams and UntersteinerWadhams, 1986), where the thickness h is small compared to the wavelength (i.e. h << 2π/k). In contrast to Equation (3), the dispersion relation most appropriate to a glacier tongue derives from consideration of a floating beam rather than a plate. The only difference, however, is that the flexural rigidity for a beam is defined by L = EI, where I is the moment of area of the beam about the xz-axis (Reference Landau and LifshiuLandau and Lifshitz, 1970, p. 89). In the case of a beam with a rectangular cross-section, which is basically a narrow plate, L = Eh 3/12. Then, the dispersion relation is identical to Equation (3), except for the factor (1-v 2) which appears in the denominator of L.
Fig. 2. Schematic showing ice sheet or beam of thickness h floating on water depth H.
Surface strain
Anticipating the data analysis to follow, the strain amplitude at the ice surface must now be related to wave amplitude. For an infinite, homogeneous plate of thickness h bent infinitesimally, the strains are due to a simple change in geometry. Therefore the strains ∈xx and ∈yy in the x-and y-directions, respectively, are given by
For a wave of amplitude η0 in the sea ice, the surface displacement will be η(x,t) = η0ei(k,x-wt). Thus, writing the vector wave number k = (k cos θ, sin θ), where θ is the angle between k and the x axis, the surface-strain amplitudes for the sea ice are
together with a shear-strain term ∈12.
For an ice tongue of finite width, there will also be a surface strain perpendicular to the direction of the major principal strain. This strain, i.e. the minor principal strain, was measured by Reference Robinson and HaskellRobinson and Haskell (1992). Assuming that the wave is propagating in the x-direction, i.e. along the beam, the principal strain ∈0 will also be in the x direction and
An independent issue of particular concern to the EGT relates to the assumption in the derivation of Equation (4) that the neutral plane occurs half-way through the ice-tongue’s thickness. This is clearly an approximation, as the ice properties are known to vary markedly throughout the thickness.
For both the sea ice and the EGT, we may write the displacement amplitude in terms of the principal surface strains using Expressions (5) and (6):
where
Angle θ may also be determined to the extent of four possibilities using orthogonal strain gauges located on the sea ice, since
Note that, to find the displacement amplitude η 0 it is necessary to determine first the wave number k, which implies that the dispersion relation in Equation (3) must be used.
Energy density
The mean energy density is the sum of the energy density in the water column and that in the beam or plate. This will be used later to compare the energies of the surface gravity waves in the sea ice with those in the EGT. For a wave of amplitude η 0and wave number k, the mean energy density E is given by
or, in terms of the strain amplitude using Equation (7), by
The Wave Data
The raw data comprise a series of surface-strain measurements made on the EGT and on the surrounding sea ice, using strain gauges sensitive to nano-strain. Four sets of strain measurements were made simultaneously at six locations for a period of 4 h each. Sampling was carried out at 2 Hz, so the Nyquist frequency was 1 Hz. The strain-data stream was first low-pass-analogue filtered at a cut-off frequency of approximately 0.5 Hz to remove any high frequencies, and was then recorded via a Maclab Α-D system on to the hard disk of a Macintosh computer. Four experiments were done on 19, 20, 21 and 22 November 1989. Three of the strain gauges were mounted on the EGT with their axes pointing along the tongue and three were located on the sea ice. The configuration is shown in Figure 3, with the distances between the instruments having been obtained by an accurate survey of the site. The numbers which distinguish the strain gauges in Figure 3 will be referred to as “channels” later in the text.
Fig. 3. Schematic showing positions and separation of the strain gauges on the ice tongue in relation to those on the sea ice. The numbers 1-6 attached to the strain gauges correspond to the channel numbers referred to in the text.
Pre-processing
The raw data are unsuitable for immediate spectral analysis for two reasons. First, there are long-term effects such as the action of the Sun on the ice and ice creep; both lead to a slow mean drift or trend in the strain readings which is not usually linear. This must be removed before spectral analysis to avoid contamination from irrelevant d.c. or near-d.c. energy. Secondly, there are steps in the data: to ensure that the instruments are always on scale and to maximize dynamic range, the strain gauges have a stepping motor that fires when the data drift out of range. The effect is to pull the data back on to scale and this appears as a near-instantaneous step from full scale to zero. The number of re-zeroes in a particular time series depends on the selected gain and on the degree to which the instruments are drifting due to natural effects. The steps are easily removed in the computer by setting to zero any changes greater than some threshold. Long-term trends in the data were removed by means of a Butterworth digital filter with a cut-off frequency of 0.0077 Hz (approximately 130 s) (Reference Oppenhem and SchaferOppenhem and Schäfer, 1975, chapter 5). Each data set was first low-pass filtered so that the drift could be examined without the distraction of oscillating waves. The drift was then subtracted from the original data. This method was chosen rather than direct application of a high-pass filter, so that the optimum cut-off frequency could be chosen by eye. Filtering was done twice, from left-to-right and from right-to-left, to avoid phase distortion. Once the data had been filtered, they were ready for spectral analysis. Sections of time series from the three strain gauges on the sea ice and the three on the ice tongue are shown in Figure 4a and b, respectively. The records suggest that the ice-tongue strain data were consistently of longer period than the data from the sea ice; counting cycles, the ice-tongue data appear to oscillate at around 50 s, whereas the principal period for the sea-ice data is about 12-13 s. The ice-tongue data are also of a consistently higher amplitude than the sea-ice data and less “noisy”.
Some very narrow band-width, high-frequency noise was present in channels 4 and 6, and in some of the records from the strain gauges on the sea ice at precisely the same frequency. The noise is believed to be due to transmissions from the U.S. or N.Z. bases about 20 km down McMurdo Sound; the long lengths of unshielded wire used in recording the data acted as antennae. The effect may be associated with the pulse-repetition frequency of a typical radar. Fortunately, the noise was easily removed by notch-filtering the data.
Fig. 4. Strain time series from (a) instruments on the sea ice and (b) instruments on the EGT. The channel numbers correspond to those of Figure 3.
Spectral analysis
Spectral analysis was performed on pairs of sequences using the Welch method of power-spectrum estimation. The two sequences of length 28 800 data points were divided into sub-sections of length 512 points, giving 112 degrees of freedom. Successive sub-sections were Hanning-windowed and then transformed with a 512-point fast Fourier-transform algorithm. With 112 degrees of freedom, the 95% confidence interval is approximately ±26% (Reference Bendat and PiersolBendat and Piersol, 1986, p. 286). We denote the estimate of the auto-power spectrum of channels 1 and 2 by G 1(f) and G 2(f), respectively, where f is the frequency (ω = 2π f) and their cross-power spectrum estimate by G 12(f) An estimate of the transfer function H(f) between sequences 1 and 2 is therefore
(Reference Bendat and PiersolBendat and Piersol, 1986, p. 166), so that the phase function β(f) = arg(H). The coherence function, defined
(Reference Bendat and PiersolBendat and Piersol, 1986, p. 172), expresses the degree of statistical dependence between sequences.
A 95% confidence interval for the transfer function H(f) may be found. For 112 degrees of freedom, the squared difference between our estimate and the true transfer function is less than r 2(f), where
Thus, the error in the phase is Δβ(f) ≈ sin−1 ∣r(f)/H(f)∣ (Reference Bendat and PiersolBendat and Piersol, 1986, p. 317).
EGT power spectra
Power spectra derived from the channels on the ice tongue were consistent from day to day. The peak strains occurred at a frequency of 0.0195 ± 0.004 Hz (i.e. at about 50 s). The power spectrum of channel 4, located nearest the end of the ice tongue on 20 November 1989, is shown together with its confidence interval in Figure 5. Although a clear peak is apparent between about 1.8 × 10−2Hz and 2.2× 10−2Hz (45-55s), taking into account the resolution band width of 3.9 × 10−3Hz, there is still a spread of significant energy from 0.01 to 0.035 Hz (28-100 s). Subsequently, we refer to the peak as being at 50 s, noting that there is some band width associated with this estimate. The power spectra for channels 4-6 on 19 November 1989 are shown in Figure 6. Channel 5 generally had larger power values than channels 4 and 6, and this is believed to be due to a difference in the geometry of the sites (calibration errors have been dismissed). If the strain gauge recorded as channel 5 were located, for instance, in an area where the EGT was thinner or above a heavily crevassed zone or even above an ice cave (of which there are many), then we would expect to see greater strain readings. This does not, of course, indicate that more energy is present. Unfortunately, the geometry of a site could not be ascertained as radio-echo sounding gear was not available. This highlights the problem of attempting to compare direcdy the amplitudes of time series recorded by different strain gauges; the power spectrum is not a measure of energy itself but of the square of strain amplitude, which is proportional to energy. Energy is also a function of frequency, ice thickness, etc. Thus, the constant of proportionality which relates the area beneath the power spectrum to energy varies between sites on the EGT.
Fig. 5. Power spectrum (linear scale) of channel 4 together with its 95% confidence interval for 20 November.
Fig. 6. Power spectra (logarithmic scale) for channels 4 (solid line), 5 (dashed line) and 6 (dotted line) on 19 November.
EGT transfer functions
Estimates of the transfer functions and coherence between channels on the EGT showed a number of salient features. Although there was some variation in the quality of the results obtained on each of the 4d, the results were, in general, compatible. There was consistently a peak in the coherence of around 0.5. The most striking features were non-zero coherences in the range of frequencies from about 0.01 to 0.05 Hz, despite there being little energy present at the higher frequencies. This can be seen in Figure 7 which shows coherence for channels 4-5, 4-6 and 5-6 for data collected on 21 November. The broad span of non-zero coherence allows us to estimate the transfer function for a range of frequencies. Since the flexure detected by a strain gauge depends on the nature of a particular site as well as on the amplitude of the wave, the gain of the transfer function is of little interest and it is the phase function β(f) which is of greatest value. The phase function between channels 4 and 5 is shown in Figure 8 together with its 95% confidence interval. This shows a number of features that were common to all of the phase functions, and the interpretation of this figure is central to the conclusions of this paper.
Fig. 7. Coherence functions for channels 4-5 (solid line), 4-6 (dashed line) and 5-6 (dotted line) for data collected on 21 November.
Fig. 8. Phase function between channels 4 and 6 for 19 November, with 95% confidence intervals.
We consider the possibility that at any given frequency f
0 there are waves travelling in both directions, and reference our calculations to the first of two physically separated sites on the ice tongue. The strain at the first site will be (A + B)ei2πf
0
t
, where A and B are the complex amplitudes of the waves travelling in each direction. The strain at the second site will be where ifo is the ratio of the amplitudes at either site, and ξ
0 is the phase difference between the sites at frequency f
0 The transfer function at f
0 is therefore given by
In general, there is insufficient information to calculate the wave amplitudes A and B. However, we may consider two possibilities. The first occurs when the wave amplitudes have equal magnitude. Then, B = Ae2iα, where 2α is the arbitrary difference in the arguments of A and B, and the two waves interfere to create a standing pattern. The phase function in this case is given by
The second possibility is that the waves are travelling in only one direction. Then, B = 0, say, and the phase function becomes
In this case, the phase difference between any two sites separated by a distance d for a wave propagating with velocity C 0(= 20/k) is
Thus, if the wave is non-dispersive, the phase function will be linear in frequency f with a slope given by
We now return to Figure 8. The low-frequency limit of the phase function is approximately 180°, suggesting that at low frequencies the waves are travelling in both directions. From about 0.01 to 0.03 Hz, however, the phase function is approximately linear with a slope which gives a phase speed of 67 m s−1 using Equation (17). Equation (16) may also be used to obtain a velocity estimate at each frequency ( Fig. 9). This figure shows a distinct region of constant velocity of approximately 65 m s−1. Noting that these two estimates are independent, this suggests that the band of waves centred around 50 s period is travelling with a phase velocity of approximately 65 m s−1. The apparent decrease of phase speed to zero at low frequencies in Figure 8 is an artifact of applying Equation (16), which assumes the waves are travelling in only one direction, to frequencies at which waves are travelling in both directions.
Fig. 9. Phase velocity c as a function of frequency, computed for channels 4 and 6 on 19 November with 95% confidence intervals.
Sea-ice power spectra
Strain measurements on the sea ice near to the EGT were inferior to those on the EGT itself. The strain gauges on the sea ice were not buried like those on the tongue and the measured strains were generally smaller (see Fig. 4). It proved possible to estimate power spectra for the sea-ice channels, but it was not possible to estimate transfer functions with any statistical accuracy. Sea-ice strain-power spectra (channels 1, 2 and 3) on 20 November are shown in Figure 10. Note that the power spectra for channels 1 and 2 are very similar, which is what would be expected from parallel but laterally displaced instruments. There are two peaks apparent at 0.15 ± 1.95 × 10−3Hz (~67s) and 0.075 ± 1.95 × 10−3Hz (~13s). Like the EGT, there is also significant energy present in the sea ice at a frequency of 0.02 ± 1.95 × 10−3 Hz (~50s). The low values of coherence between the sea-ice channels makes transfer-function estimates unreliable, and thus phase speeds of the waves in the sea ice cannot be determined. It is assumed that the strains in the sea ice are due to flexural-gravity waves (Reference Fox and SquireFox and Squire, 1991b) and are therefore governed by the dispersion Equation (3). The two strain gauges at right-angles do show a consistent phase change of zero, as expected.
Fig. 10. Power spectra for sea ice on 20 November; channel 1 (solid line), channel 2 (dashed line) and channel 3 (dotted line).
The direction of the incoming flexural waves may be found from the two perpendicular strain gauges using Equation (8). This has been done for channels 2 and 3, which give a result of θ = 22° + 3° to 95% confidence. In choosing θ from its four possible values, we have assumed that the waves are propagating down McMurdo Sound, and have chosen an angle between the axis of strain gauge 2 and the axis of the EGT.
Dispersion: theory and experiment
Equation (3) describes the behaviour of flexural-gravity waves in the sea ice adequately but there are problems in applying it to the EGT over and above those discussed earlier. The ice tongue is quite irregular in shape and its cross-section is only approximated by a rectangle. Further, its cross-section changes along its length. Another problem is that the EGT is heavily crevassed and has many ice caves. These will reduce both its moment of area and the value of E. The value of the EGT’s thickness h is only known approximately. The last radio-echo sounding was done about 4 years before the experiments reported in this paper (Reference DeLisle, Chinn, Karlen and WintersDeLisle and others, 1989), when it was found that the ice-tongue’s thickness at approximately 1.5 and 3 km from the snout was 68.0 and 103.8 m, respectively. It is therefore clear that estimating the flexural rigidity L accurately is not possible. Finally, the water depth under the EGT is also not known accurately. Inevitably, then, values for the velocity of flexural waves in the EGT will be at best approximate.
It is now assumed that the most significant coupling between the flexural-gravity waves beneath the sea ice and those under the EGT occurs when the phase velocities, as given by Equation (3), are equal. We must allow for the fact that the velocity of the waves in the sea ice is at some angle ψ to the axis of the EGT. Thus, the phase-velocity component c ψ of the waves down the EGT is related to the velocity in the sea ice ( Fig. 11) by c ψ = Csea ice/cos ψ. The theoretical dispersion curve for the sea ice with flexural rigidity L = 11.3 GN m, water density ρ = 1025 kg m−3, ice density ρ′ = 922.5 kg m−3 and the corresponding curves for the EGT with various values of L (L = 1 × 105, 2 × 105, 4 × 105 GN m, chosen because of our uncertainty about its true value), h = 80 m and the same densities as for the sea-ice plate are shown in Figure 12. All curves are for a water depth of 300 m. Sea-sice thickness is not critical for low-frequency waves where the dominant restoring force is gravity. The intersections of the dispersion curve for sea ice with those for the EGT certainly lie in the vicinity of the peak frequencies seen in the power spectra.
Fig. 11. Schematic showing the angle the waves are believed to be striking the EGT.
Fig. 12. Theoretical dispersion curves for the ice tongue with three values of flexural rigidity: L = 1 × 105 GN m (solidline), 2 × 105 GN m (dashedline),4 × 105 GN m (dotted line) and for the sea ice (chained line).
Fig. 13. The dispersion curve for the ice tongue (L = 2 × 105 GN m) plotted together with phase velocities computed from channels 4-6, 19 November.
Assuming L = 2 GN m, the dispersion curve for the EGT is plotted in Figure 13, together with phase velocities computed from the data. The figure shows a reasonable agreement between theory and the measured velocities except at low frequencies (f ≤ 0.01 Hz), where we believe the velocity estimates are invalid as explained previously.
Energy density
If coupling is occurring between the waves under the sea ice and those under the EGT at 50 s, then it is of interest to obtain an order-of-magnitude estimate of the respective energy densities. To calculate this, the dispersion equation is required and it will be assumed that our theoretical dispersion Equation (3) is valid. To simplify the problem, we assume that all the energy in the EGT is at 50 s and that all the energy in the sea ice in the band 0.0156-0.0234Hz is at 0.02 Hz (50s).
To find the energy density, we use Equation (10). For the EGT, the square of the strain amplitude is calculated from the integral over all frequencies up to Nyquist. (In fact, this is no different from integrating over the first ten frequency bins, since there is virtually no energy at higher frequencies.) For the sea ice, the square amplitude of the strain at 50 s is found by integrating strain spectral-density values from 0.0156 to 0.0234 Hz, including the peak value at 0.0195 Hz. From Equation (7), the strains for channels 2 and 3 must be summed to get the principal strain amplitude in the sea ice. An average value for the strain amplitude of the sea ice is 8 nano-strain and for the EGT is 20 nano-strain. Choosing a phase speed c = 65 m s−1 based on Figure 9, Equation (3) gives a wavelength of 3250 m. This gives a displacement amplitude for the EGT of η0 = 0.1 mm, and for the sea ice of η0 = 1 mm. Using Equation (10), the energy density of the flexural-gravity wave under 3 m thick sea ice is approximately 5 × 10−3 J m−2, whereas under the EGT, assuming a thickness of 80 m, it is about 5 × 10−5 J m−2. Thus, the wave-induced energy density of the EGT is about one-hundredth of that in the sea ice.
Conclusions
Like many geophysical measurements, there was undoubtedly significant noise with some unknown statistical distribution present in the strain measurements. This is demonstrated in the coherence function which reached a maximum of only 0.5, thereby indicating that only half the energy is accounted for by a linear-transfer function. However, the data sets used in the study were all very long and thus we have some confidence that statistically significant information has been extracted.
Our first conclusion is that, for the duration of the experiment, flexural waves of a period centred on approximately 50 s were travelling up the tongue with a phase speed of roughly 65 m s−1. At low frequencies, there appears to be evidence of standing waves, or at least evidence that a component of the wave is propagating down the EGT from the grounding line. In this context, it is worthwhile attempting to compute the eigenfrequencies corresponding to the normal flexural modes of oscillation of the ice tongue, invoking a simple fixed-free cantilever beam theory on a foundation which responds by buoyancy forces alone. Because the EGT tapers from its grounding line to snout and because of the many irregularities along its length, including variations in the bulk properties of the ice structure due to crevasses and ice caves, for example, the values obtained should be treated with caution. The first five non-dimensionalized modal wave numbers kl, where l is the length of the tongue, are 1.875, 4.694, 7.855, 10.996 and 14.137. These all correspond to eigenfrequencies of about 0.059 Hz (~17s). Although kl is unaffected by the foundation on which the tongue rests, conversion to the eigenfrequencies at these wave numbers is dominated by buoyancy forces, and hence a full analysis which models the water beneath the ice tongue completely is probably necessary. Holds-worth and Glynn (1981) have used finite differences to solve the equations for a tapered beam on a fluid foundation. Their work shows that very long-period modes of standing oscillation are indeed possible in the tongue. Holdsworth (personal communication, 1993) also noted that a low-order harmonic may occur at about 50 s, and suggested that we may be observing a flexural wave propagating along the tongue which later is dissipated due to attenuation between the terminal area and the coastline, reflection at the coast and attenuation on its return journey. This is very plausible as the propagating strains detected are small.
It is also apparent that significant energy is present in the local sea ice, similarly in a band centred at approximately 50 s. If the EGT is radiating energy into the sea ice, then it would have to be taking energy input from some source other than the sea ice. Although there are candidates for this, such as the wind or tidal surges, this still poses a problem. Thus, if significant coupling between the EGT and the surrounding sea ice exists, it is most likely that it is the flexural-gravity waves in the sea ice which are driving the EGT. Indeed, the data suggest that the energy density at 50 s is much greater in the sea ice than in the EGT, which adds conviction to the notion that the waves in the sea ice are driving the EGT.
It has been assumed that optimum coupling occurs when the phase speed of the waves in the sea ice is equal to that in the EGT. This is obviously an approximation. More importantly, above 0.02 Hz, the phase speeds in the EGT and in the sea ice diverge, and coupling is unlikely. The sea-ice wave spectra peak between 59-77 s, yet the EGT spectra peak from 45 to 55 s. Why the EGT favours a band centred on 50 s period oscillation is unclear but it is probably associated with the geometry of the ice tongue.
Another difficult question to answer is the source of the peak in the sea-ice spectra at 67 s. A very long-period swell (infra-gravity wave) or a tidal seiche are both possibilities (Reference Robinson and HaskellRobinson and Haskell, 1992) but it is impossible to establish definitively the causal mechanism behind the long-period observations. Since the seiche fundamental in McMurdo Sound is not close to the observed period, non-linear energy-transfer processes would need to be invoked, or coupling at some harmonic; there are insufficient auxilliary data to proceed with either hypothesis. Infra-gravity waves of periods greater than 30 s, on the other hand, are certainly an ubiquitous feature in pressure spectra taken in the deep ocean. Again, they are believed to originate from a non-linear transfer of energy from shorter-period waves, in this case surface gravity waves (Reference Webb, Zhang and CrawfordWebb and others, 1991). Their presence in McMurdo Sound, which would appear very shallow to waves of this period, has not been established independently. The 67 s peak thus remains an open question.
The measurements in this experiment were made under conditions of solid-ice cover surrounding the EGT; conditions unlike those when the ice tongue calved when there was significant open water present. Therefore, it is difficult to draw conclusions about the mechanism of ice-tongue calving. What is apparent is that the EGT can be modelled as a beam floating on water. It is also apparent that the EGT is capable of being excited by water waves. Finally, it appears that there is some reflection of low-frequency waves at the grounding line to create standing waves. Further measurements are called for to reject or to confirm the standing-wave calving theory, and to qualify further the way in which the EGT is vibrating.
Acknowledgements
The authors are indebted to the New Zealand Foundation for Research, Science and Technology, the University of Otago, and the staff of the New Zealand Institute for Industrial Research and Development for their support. M.M. is a Ph.D. student supported on a Duffus Lubecki Postgraduate Scholarship in Applied Science. The authors are grateful to Dr G. Holdsworth of the Arctic Institute of North America for his constructive comments and suggestions.
The accuracy of references in the text and in this list is the responsibility of the authors, to whom queries should be addressed.
