Hydrodynamic preliminaries

Surface gravity waves treated in this study are considered to propagate in both ice-free and ice-covered water. In ice-covered water, the ice cover can consist of broken sea ice, ranging up to several meters thick, to iceberg and ice-shelf cover, with thickness ranging up to several hundred meters. In all cases studied, wave amplitude, η ₀, is small compared to water-column thickness (water depth minus ice thickness if ice-covered), h, which is generally large compared to wavelength, λ = 2π/k. Throughout the analysis that follows, wave trains produced from glacial sources will be assumed to be planar, being described by a simple sinusoidal profile of the sea surface (or ice surface, in ice-covered water):

where η(x, t ) is the free-surface displacement relative to the mean sea level, k = 2π/λ is the wavenumber, x is the direction of propagation, ω = 2π/T is wave frequency, with T being wave period, and t is time. Water motion below the surface of the ice-free ocean, and below the basal surface of any ice cover, is irrotational and restricted to the x−z plane (Earth rotation effects are assumed to be negligible, provided T ≪ 2π/Ω, where Ω is the angular frequency of Earth rotation), where z is the vertical coordinate, positive upward. These irrotational motions decay with increasing z. In circumstances where λ is small compared to h (i.e. the deep-water limit) vertical decay is exponential with an e-folding scale equal to the horizontal wavelength. In the shallow-water limit, when λ ≫ h, horizontal motion in the water associated with the wave is independent of z. The water is assumed inviscid.

In the absence of flexural effects introduced by ice cover, the equations of hydrodynamics predict that waves of the type described above propagate according to the dispersion formula (Reference Munk, Miller, Snodgrass and BarberMunk and others, 1963):

where g = 9.81 m s⁻² is the acceleration due to gravity. Conventional, earthquake and submarine-landslide triggered tsunamis are observed in the low-frequency (or shallow-water) limit for which kh → 0 leading to the well-known undispersed relation:

which corresponds to a velocity (both phase, C, and group, U) of

However, records of the 2004 Indian Ocean tsunami on hydrophone, island seismometers, as well as on our iceberg-deployed seismometers, have detected its dispersion according to Equation (2) throughout the frequency spectrum (Reference OkalOkal, 2005; Reference Okal and MacAyealOkal and MacAyeal, 2006; Reference Okal, Talandier and ReymondOkal and others, 2007). In particular, in the high-frequency (or deep-water) limit where kh → ∞, Equation (2) becomes

leading to a phase velocity C = ω/k = g/ω and a group velocity U = ∂ω/∂k = g/(2ω) = C/2 (Reference Munk, Miller, Snodgrass and BarberMunk and others, 1963; Reference Bromirski and DuennebierBromirski and Duennebier, 2002). In turn, U ≈ 1/ω predicts an arrival time linearly dispersed with frequency. This range of frequencies is the traditional domain of sea swell and also of our ‘micro-tsunami’ observations.

Although of considerable interest, the effects of ice cover on the dispersion and attenuation of waves on the ocean surface (waves in the ice-covered ocean) will not be considered in detail in this study because the limited knowledge of ice-thickness conditions (i.e. thickness maps of icebergs on which seismometers were deployed do not exist) prohibits analysis using results of previous studies (e.g. Reference Williams and RobinsonWilliams and Robinson, 1981; Reference Squire, Robinson, Meylan and HaskellSquire and others, 1994, Reference Squire, Dugan, Wadhams, Rottier and Liu1995; Reference SquireSquire, 2007). In most cases, our observations suggest that glacial micro-tsunamis are generated by iceberg and ice-shelf fragments that fall into, or otherwise disturb, an ocean that is ice-free or covered by thin, broken sea ice. In this case, propagation of the long-wavelength components of excitation (e.g. with wavelengths >100 m) is negligibly influenced by sea-ice flexure and attenuation, and analysis of micro-tsunami signals can be performed using the dispersion relation given by Equation (2). In cases where micro-tsunami generation is within the interior of an iceberg or ice shelf, where ice thickness is orders of magnitude larger than for sea ice, ice-flexure effects and attenuation are likely to be significant for even the longest wavelengths generated. We acknowledge this complexity, but refrain from addressing it due to the fact (discussed below) that our observation design did not permit satisfactory determination of micro-tsunami source locations. Without source location, treatment of flexure and attenuation is moot. In spite of our disregard of the flexure and attenuation effects of ice cover, we find that the majority of events witnessed in our observation program were consistent with sources in the open or sea-ice-covered waters of the Ross Sea that allow analysis using the simple dispersion relation given by Equation (2).

One of the important utilities of the dispersion described in Equation (5) is that it allows source-to-receiver distance to be obtained from seismometer observations. Defining the cyclic frequency, f = 2π/ω, and using the concept of group velocity, U = dω/dk, a relation between f and wave travel distance, Δ, from the source to the receiver is obtained:

where t in the above context is the observed time at which energy with frequency f arrives at the receiver. The above formula is used to determine Δ from measurements of df /dt that are obtained on signal spectrograms (described below).

Field deployment

We deployed seismometers on icebergs in the Ross Sea (C16, B15A, B15K) and on the McMurdo Ice Shelf, the Ross Ice Shelf and on sea ice in McMurdo Sound (Fig. 1). The instruments were Guralp 40T seismometers provided by the Incorporated Research Institutions for Seismology (IRIS) Program for the Array Seismic Studies of the Continental Lithosphere (PASSCAL) instrument pool, and are sensitive to signals in the 0.01–50 Hz range on three axes of motion. Signals were sampled at both 1 and 100 Hz, digitized by Quanterra Q330 equipment and stored in Quanterra Bailers. The seismometers were deployed on ceramic flooring tiles at the bottom of snow pits ∼2 m deep, that were covered with plywood and snow-block roofs. Solar panels provided power to a 12 V battery system. As discussed by Reference Okal and MacAyealOkal and MacAyeal (2006), these seismometers are able to sense sea-surface motion in the low-frequency range, below ∼0.15Hz (corresponding to a period of 6.7 s), and microseisms (e.g. Reference GutenbergGutenberg, 1947; Reference Longuet-HigginsLonguet-Higgins, 1950) at frequencies above this level. Such fidelity to the recording of sea-surface wave conditions is rarely achieved, if at all, in seismometers deployed on grounded ice sheets or on land. The typical frequency content and amplitude of signals recorded by the seismometers is associated with transoceanic propagation of sea swell coming from storms in the distant Pacific and Indian Oceans. These signals are described at length by Cathles and others (in press).

An initial deployment, in October 2003, focused all seismometer placement on iceberg C16, where four sites were operated for a period of ∼3 months (Fig. 1b). This array provided the opportunity to examine the nature of iceberg motions excited by ocean waves incident on the iceberg’s seaward-facing edges. This four-site array was replaced in January 2004 with a single instrument at the C16 center site (site A in Fig. 1b), and was maintained during the 2004 and 2005 field seasons. In January 2006, C16 drifted away from the range of logistical operations, and the iceberg and seismometer (with data collected since early November, 2005) are currently located off the coast of East Antarctica.

From January 2004 onward, single-seismometer stations were deployed on icebergs C16, B15A, B15K, the McMurdo Ice Shelf (near the US and New Zealand scientific bases), on landfast, multi-year sea ice in McMurdo Sound, and on the Ross Ice Shelf (at a site called ‘Nascent Iceberg’ where a large-scale tabular iceberg is expected to calve in the next decade; Fig. 1a). Table 1 summarizes the periods of deployment and data collection for these stations. Generally speaking, the instruments operated throughout the period solar charging was possible, and did not operate during the period from roughly the end of May to mid-October of each year of the field program. All data are in the process of being archived at the IRIS Data Management Center.

Background noise

For events to be added to the catalogue, their signal had to be distinguishable from background noise in the 0.01–0.15Hz band and had to have frequency dispersion implying a local (<500 km) source, so as to rule out signals originating from storms and other events beyond the glacial environment of Antarctica. Waves with frequencies higher than ∼0.15 Hz were typically not observed due to attenuation by sea ice surrounding the icebergs or ice shelf, and by reflection at the cliff-like seaward-facing edges of the iceberg or ice shelf. Waves with frequencies lower than ∼0.05 Hz were typically not observed due to the constant bobbing motion of the icebergs and ice shelf at this frequency. Floating ice of thickness h _i and density ρ _i bobs at a period T given by (Reference SchwerdtfegerSchwerdtfeger, 1980)

where ρ _w is the density of sea water and g = 9.81 m s⁻². For h _i ∼ 250 m, the approximate thickness of the Ross Ice Shelf at the Nascent Iceberg site, this period is ∼30 s, giving a frequency of maximum noise ∼0.03Hz. For h _i ∼ 75 m, the approximate average thickness of iceberg C16 and of the McMurdo Ice Shelf, this period is ∼16 s, giving a frequency of maximum noise ∼0.06Hz. Background noise recorded by the seismometers on iceberg C16, and on the McMurdo and Ross Ice Shelves, conforms to the frequency band of bobbing motions (pitching and rolling motions are also possible, at slightly higher frequency (Reference Okal and MacAyealOkal and MacAyeal, 2006)) and are probably excited by passing meteorological conditions and by ocean swell propagating into the Antarctic from afar (e.g. Reference Williams and RobinsonWilliams and Robinson, 1981; Reference MacAyealMacAyeal and others, 2006b). The band of noise in the 0.01–0.06Hz range is well displayed by the spectrograms of all example events shown in Figure 5a.

Fig. 5. Background noise and other seismic signals to be excluded from the event catalogue. (a) Earthquake arrival recorded at Nascent Iceberg (NIB); top panel: vertical ground motion (LHZ) seismogram filtered (four-stage Butterworth filter) to 0.05–0.15 Hz frequency band; middle panel: spectrogram of signal (dB of log₁₀ of counts²Hz⁻¹; color bar at bottom); bottom panel: unfiltered vertical (LHZ) seismogram. Inset: map showing source and receiver for M = 6.7 earthquake on Pacific coast of North America. Ocean noise due to ice-shelf buoyancy oscillations (bobbing) indicated by bright signal strength in spectrogram between two horizontal gray lines. P-wave and surface-wave (Rayleigh wave) arrival of the earthquake indicated by vertical gray lines. (b) Iceberg tremor and micro-tsunami recorded on C16, site B (Fig. 1).

Glacial source identification criteria

Once candidate events for the catalogue are differentiated from noise in the above manner, their likelihood of being generated by glacial mechanisms is determined using the following criteria:

1. 1. Their duration within the seismogram or spectrogram record, including the effect of dispersion, is short (i.e. less than ∼3 hours). This distinguishes the glaciogenic ocean-wave signals from those that are generated by meteorological events (e.g. lasting days, as described by Cathles and others, in press).

2. 2. They often display quasi-periodicity (e.g. arriving at the same time of day over a number of days) at timescales associated with the dominant diurnal tide of the Ross Sea. This implies a source mechanism that is sensitive to ocean tide. Ocean tide is seen to be an important driver of iceberg motion (Reference MacAyeal, Okal, Thom, Brunt, Kim and BlissMacAyeal and others, 2008b) and modulates ice-shelf and ice-stream flow (Reference Bindschadler, King, Alley, Anandakrishnan and PadmanBindschadler and others, 2003; Reference BruntBrunt, 2008; Reference Wiens, Anandakrishnan, Wineberry and KingWiens and others, 2008).

3. 3. Frequency dispersion implies a source within the local glaciated environment of the Ross Sea (e.g. distances inferred from dispersion range from ∼1 to ∼500 km). This range is consistent with sources along the calving fronts of the ice shelf, the edges of the icebergs, outlet glaciers along the coastlines and rifts and other features within the ice shelf, both proximal to the ice front and proximal to the grounding lines.

4. 4. Efforts were made to rule out other impulsive arrivals such as those associated with teleseismic earthquake signals and local iceberg harmonic tremor (IHT) signals.

The above criteria for discriminating glaciogenic sources from others are admittedly ad hoc, and room exists for occasional misidentification of signals that neither have glacial origin nor propagate as ocean surface waves. We were particularly attentive to two classes of signals that could be prone to misidentification. The first of these two classes concerns signals arriving from teleseismic earthquake sources: typically a weak P-wave arrival followed by a high-amplitude dispersed surface- or Rayleigh-wave arrival. Teleseismic earthquake signals were received on the icebergs and the Ross Ice Shelf, despite the fact that they are disconnected from the solid Earth by an intervening sea-water layer. The large length scale of teleseismic waves means that the water and ice layers move in concert with the seabed (Reference Okal and MacAyealOkal and MacAyeal, 2006). An example of a teleseismic earthquake signal is shown in Figure 5a, and results from a magnitude 6.7 earthquake that occurred in the Pacific, northwest of North America, at a distance of ∼131^°. The characteristic upper-right verging ‘hook’ of the solid-Earth surface-wave arrival in the spectrogram is used to identify the event as a teleseismic earthquake signal. Note, in particular, that its dispersion is not linear in the 0.05–0.07 Hz range. Also seen in Figure 5a is a faint P-wave arrival as a broad, undispersed vertical band of barely discernible energy in the spectrogram ∼45 min prior to the hook-like surface-wave arrival. (S-wave arrivals were typically not observed at our seismometer sites, due to their low expected amplitude exacerbated by losses in the conversion of shear motions to vertical motions at the sea bottom.) Our heuristic signal-identification technique involved looking for signs of P-wave arrival and the tight, hook-like curvature of the spectrogram energy swath signifying the arrival of Rayleigh waves.

The second class of signal excluded from the catalogue arises from IHT described by Reference MacAyeal, Okal, Aster and BassisMacAyeal and others (2008a). IHT is generated by stick–slip motion along the edge-on-edge contact between two tabular icebergs that are slowly sliding past each other. Typically there are thousands of discrete stick-slip events over a period of hours, and their inter-event time spacing (typically 0.1–10 s) leads to a striated, or ‘chevron’, pattern in signal spectrograms. An example of IHT in its low-frequency range that overlaps with the <0.15 Hz signals of glaciogenic wave-making events is shown in Figure 5b. To discriminate between ocean wave trains associated with glaciogenic sources and short bursts of IHT, we looked for (1) the classic pattern of integer-multiple harmonics that exist at frequencies higher than the fundamental sub-0.15 Hz vibration of the IHT and (2) the possible presence of high-frequency (above ∼5Hz) hydro-acoustic signals also generated by stick–slip motions at the edges of colliding icebergs and observed as seismo-acoustic phases at our stations (Reference MacAyeal, Okal, Aster and BassisMacAyeal and others, 2008a).

Ice-flexure effects

An initially perplexing feature of the ocean-wave signals in our catalogue is that they appear very simple, showing little immediately recognizable effect of the flexural rigidity of the ice cover on which the seismometers were deployed. To illustrate an event where flexural rigidity effects are important, we show the signal generated on the sea ice of McMurdo Sound resulting from the landing of a 100 000 kg aircraft on a 5 m thick sea-ice runway overlying water (>100 m deep) near McMurdo Station (Fig. 6). The flexural-gravity-wave signal from the aircraft landing was observed at a seismometer located ∼1800 m from the point where the aircraft touched down and ∼1350 m from the point where the maximum reverse thrust was applied on the aircraft engines. The seismometer was located ∼100 m from the side of the runway, and so was in the expected maximum radiation pattern for flexural-gravity waves of a load moving at high speed across the runway surface.

Fig. 6. Flexural-gravity wave on a sea-ice covered ocean surface recorded by a seismometer on landfast, multi-year sea ice (∼5 m thick) in McMurdo Sound (Fig. 1) generated by the landing of a large cargo aircraft (Boeing C17). Top panel: Seismogram of vertical channel (LHZ) filtered to 0.05–0.15 Hz frequency band (four-stage Butterworth filter); middle panel: LHZ spectrogram displaying right-upward concave pattern characteristic of flexural-gravity wave propagation (dB of log₁₀ of counts² Hz⁻¹; color bar at bottom); bottom panel: unfiltered HHZ (100 Hz sample rate vertical channel) seismogram of same event. Initial aircraft wheel-on-runway touchdown and secondary reverse thrust application are apparent in both the spectrogram and HHZ seismogram.

The seismograms and spectrogram associated with the aircraft landing are shown in Figure 6. The top seismogram displays the 1 s sample rate LHZ signal filtered into the 0.01–0.5Hz band. At face value, this signal would appear to be a compact packet of surface gravity waves generated by some impulsive event in the ocean surrounding the seismometer site. What distinguishes it as a packet of flexural gravity waves involving the elastic flexure mechanics of the sea ice is the fact that short wavelengths (high frequencies) arrive at the seismometer before long wavelengths (low frequencies). This dispersion pattern is the reverse of the pattern of dispersion for surface waves traveling on deep water in the absence of a flexurally rigid ice layer. This pattern is also opposite to the normal dispersion characteristics for seismic surface waves. Ice-flexural dispersion is seen in the bottom seismogram of Figure 6, which shows the 0.01s sample rate vertical ground-motion seismogram signal. In the spectrogram, the arrival of the wave packet generated by the aircraft landing shows a characteristic upper-left verging curvature indicative of the effects of elastic flexure on short-wavelength waves. First arrivals are of the highest-frequency energy; later arrivals involve the low-frequency waves that are not as strongly influenced by elastic flexure of the ice. Two other features in the spectrogram of the aircraft landing are interesting: the initial touchdown of the aircraft on the ice surface appears to be followed by a second series of waves generated by the application of reverse thrust on its jet engines (the energy >1.5 Hz that appears after 01:11) and there is a high-frequency tremor appearing after 01:14 of unknown origin.

By far the majority of surface-wave phenomena observed on the icebergs and the Ross Ice Shelf lacked the tell-tale signs of elastic-flexure dispersion described above. An example of this is given in Figure 7, showing C16 when four seismometers were operating. The signal is observed at all stations at roughly the same time (exact arrival times could not be determined because the signals were too noisy) with identical frequency dispersion, i.e. identical df /dt, where f is the cyclic frequency and t is the time of observation. Site B (Fig. 1b) displays the largest signal in both the LHZ spectrogram (shown) and seismogram (not shown), with the brightest spectral-energy swath showing the distinctive linearity signifying a source-to-receiver distance of ∼15km (as discussed below). Comparing the spectrograms from the other three sites (A, C and D) with that of site B reveals that the slope of the signal is the same for all sites, implying an identical source-to-receiver distance of ∼15 km, despite the intervening distance between site B and the other sites. This uniformity of df /dt implies that travel of the wave train across the iceberg from the station closest to the source to the station most distant from the source produces little discernible dispersion associated with flexure effects. (A thorough analysis of how such effects would be manifest is left for future theoretical treatments.)

Fig. 7. LHZ spectrograms of micro-tsunami arrival (16 December 2003 at approximately 16:00) at four seismometer stations on C16 (Fig. 1b) (dB of log₁₀ of counts² Hz⁻¹; color bars at bottom). Except for signal amplitude and noise level, the signatures of the micro-tsunami represented by the sloping signal density swaths in the spectrograms are virtually identical at all four sites. The slope (black line) at site A is identical to those of the three other sites (black lines), and indicates that signal dispersion along paths that traverse the iceberg is negligible. This suggests that wave dispersion is primarily a function of the open-water leg of wave-train travel to the iceberg, and that the signal, once it reaches the iceberg, is transmitted to sites within the iceberg through a combination of flexural and rigid-body motions of the iceberg. The lack of significant dispersion within the iceberg, and the lack of easily identified arrival times associated with the emergent signals, made it impossible to determine micro-tsunami source locations from the C16 seismometer array.

The absence of the characteristic signature of flexural-gravity-wave dispersion in the spectrograms of events in our catalogue (e.g. Figs 3 and 7) initially surprised us, and this inspired a detailed analysis of several of the strongest events recorded by the four-station seismometer array on C16 to determine why. Investigation of the iceberg-surface (ground) motion revealed that the initial vibration of the iceberg excited by wave trains incident somewhere on its seaward-facing edge was almost immediately transmitted (e.g. within several wave periods) to all other parts of the iceberg. This is consistent with the expected high propagation velocity of flexural-gravity waves (e.g. Reference Holdsworth and GlynnHoldsworth and Glynn, 1978; Reference Goodman, Wadhams and SquireGoodman and others, 1980; Reference Williams and RobinsonWilliams and Robinson, 1981; Reference Kristensen, Squire and MooreKristensen and others, 1982) for ice in the 50–150 m thickness range measured on C16 at the four seismic stations, using a hand-held pulse radar. The extreme rapidity of flexural-gravity-wave transmission resulted in short travel times between the site of initial excitation (i.e. a seaward-facing edge onto which an ocean wave train is incident) and distant sites elsewhere on the iceberg. Dispersion by flexural effects within the iceberg leg of wave propagation was thus insignificant (owing to the short travel times). Dispersion effects associated with surface-gravity-wave propagation in ice-free (or thinly covered) ocean conditions evident from a given wave train’s propagation across an open-water leg leading from the source to the iceberg’s seaward-facing edge was preserved without distortion, and recorded at all points on the iceberg uniformly. All seismometers on the iceberg record motion at the same frequency, because flexural effects transmit frequency changes observed at the initial point of excitation across the iceberg with a timescale shorter than that of frequency change associated with the incident wave energy.

To illustrate the above explanation for the lack of flexural-gravity-wave dispersion patterns in catalogue event seismograms, a schematic diagram of elastic-flexural effects on the iceberg is shown in Figure 8. An initial glacial wave-making event, such as the collapse of a seaward-facing ice cliff (Fig. 8, right), generates an impulsive wave that subsequently propagates a distance through open water (or through water covered by sea ice of insufficient thickness to introduce elastic-flexure effects). During the open-water propagation leg, the initially impulsive wave front is dispersed into a wave train led by long-wavelength/low-frequency energy and trailed by short-wavelength/high-frequency energy. Eventually, the dispersed wave train impinges on the edge of the iceberg (Fig. 8, on the right of the tabular iceberg). Relative to the timescale necessary to transmit the edge excitation across the length of the iceberg, the edge excitation of the impinging wave appears monochromatic, i.e. the time evolution of the frequency of the impinging waves determined by dispersion in the open-ocean environment is slow compared to the transmission time of waves at a given frequency across the iceberg by elastic-flexure effects. Seismometers at locations spread across the iceberg thus sense the frequency information of the impinging wave, but not the change in this frequency information associated with the short-timescale dispersion effects of flexural rigidity.

Fig. 8. Schematic diagram showing (1) micro-tsunami generation by small-scale edge wasting of seaward-facing ice cliff, (2) micro-tsunami dispersion on deep water during travel across ice-free ocean surface, (3) excitation of vibration at the seaward-facing ice cliff of an iceberg’s edge or ice shelf’s ice front and (4) subsequent fast propagation as flexural-gravity wave through iceberg/ice-shelf interior to various receiving stations. The waveform depicted in flight between the source and the receiver is adopted from the observed micro-tsunami signal shown in the upper-right panels of Figure 3.

Flexural-gravity resonance modes of iceberg

The considerations in the preceding subsection explain the observed fact that flexural effects are generally absent from the glaciogenic wave-making events recorded in our catalogue. The same considerations can be used to suggest that the icebergs (and likely the Ross Ice Shelf as well) are subject to flexural oscillations excited by the waves that impinge on them (e.g. Reference Goodman, Wadhams and SquireGoodman and others, 1980; Reference Wadhams, Kristensen and OrheimWadhams and others, 1983; Reference SquireSquire and others, 1994; Reference SquireSquire, 2007). To investigate this possibility, we again examined the four-station seismometer motions on iceberg C16.

As shown in Figure 9, the kinematics of horizontal and vertical ice-surface movement on C16 is complex. The horizontal polarization of iceberg-surface (ground) motion and relative phasing between the sites changes as the frequency of an arriving event changes according to its spectrogram. As the maps of horizontal ice-surface motion recorded by the seismometers suggest (Fig. 9), the horizontal polarity of seismometer motion at each of the four sites (represented by the elliptical patterns of point locations superimposed on the map using an exaggerated horizontal scale) changes as the frequency of the event evolves (as shown on the spectrogram of Fig. 9). The vertical motions at each of the four sites also change, progressing from uncorrelated ‘noisy’ motions prior to the event’s arrival, to phase-coordinated sinusoidal vibrations as the frequency of the arriving wave train passes through specific ranges (e.g. ∼0.065 Hz for the map labeled 3 in Fig. 9 and its associated sub-seismogram). Gentle undulations of the amplitude envelope of the vertical ice-surface motion seismogram recorded at site B, on the northeast corner of C16, suggest that different modes of iceberg vibration are being excited in sequence.

Fig. 9. Iceberg (ground) displacements during the arrival of a micro-tsunami recorded on 16 December 2003. (a) A seismogram (upper graph) and a spectrogram (lower graph) of C16 B site’s vertical displacement during a 60 min period of micro-tsunami arrival. The seismogram signal was filtered to the 0.05–0.15 Hz frequency band using a four-stage Butterworth filter. The vertical displacement seismogram is in units of dB of log₁₀ of m² s² Hz⁻¹; red color denotes high signal strength. The vertical displacement seismogram (upper graph) is divided into four segments to display stages of signal evolution (labeled 1–4). Stage 1 represents pre-event noise; stage 2 represents event onset; stage 3 represents event development as a flexural/elastic mode of iceberg vibration (resonance); stage 4 represents event decay. (b) Maps of exaggerated horizontal position (left panels) and graphs of vertical motion (right panels) for each of the four stages of evolution designated in (a).

The vertical and horizontal motions observed at the seismometer sites during four sub-intervals of the evolution of a single micro-tsunami arrival are displayed in Figure 9b. These motions suggest that there are four distinct stages in the response of the iceberg to the excitation induced by incoming waves in the ocean. Stages 2–4 suggest the progressive excitation of successive eigenmodes of the iceberg’s flexure, such as those modes illustrated in Figures 10 and 11. We relegate to future study the question of the identification and further analysis of such modes, which will likely require a considerable quantitative effort to compare our observations with simulations of iceberg/ocean modes. In Figure 10, the two most distinguishable vertical modes of vibration are depicted (we do not know how these modes relate to the spectrum of other eigenmodes (e.g. whether they are most grave or not)). The lowest-frequency vibration is a ‘see-saw’ motion of the two corners defining the northern end of C16, with relatively little movement in the rest of the iceberg. The next-highest-frequency mode consists of a ‘trampoline’ motion involving oscillation between the center of the iceberg and its two ends. The northern end of C16 was originally part of the Ross Ice Shelf closest to the shear zone where the ice shelf flows past the eastern end of Ross Island (Fig. 1), defining the western end of the ice-shelf extent. Airborne radar surveys of C16 conducted in the early and mid-1970s, when it was still part of the Ross Ice Shelf (Reference Bentley, Clough, Jezek and ShabtaieBentley and others, 1979), suggest that this end of the iceberg (constituting the northern end during the seismometer observation campaign) was thin, with ice thicknesses ranging below 100 m. This is confirmed by point radar measurements at sites B and C when the seismometers were deployed, giving thicknesses of 53 ± 5 and 40 ± 5 m, respectively. It is likely that this thin area of the iceberg would vibrate with the slowest-frequency see-saw mode due to its reduced flexural rigidity (which depends on the cube of ice thickness). The higher-frequency vibration of the trampoline mode may be explained by the greater thickness, and hence flexural rigidity, of the rest of the iceberg relative to the area of the northern end. Ice-thickness measurements at sites A and D are 145 ± 20 and 150 ± 20 m, respectively; thus, motions involving the central, elastically stiffer, part of the iceberg should have higher frequency.

Fig. 10. Two vertical modes of iceberg vibration determined from vertical displacements of the seismometers. Station displacements are shown schematically as cylinders (height of cylinder denotes position relative to undisturbed location) at various phases of the oscillation. (a) Iceberg vibration involving ‘see-saw’ of the northern end (top in figure) of the iceberg, with relatively little motion in other parts of the iceberg. (b) ‘Trampoline’ mode of iceberg vibration where the center of the iceberg goes up and down, out of phase with the edges of the iceberg.

Fig. 11. Two horizontal modes of iceberg vibration determined from horizontal displacements of the seismometers. Station displacements are shown schematically as stick figures at various phases of the oscillation (dots denote station location; sticks are line segments connecting stations). (a) Iceberg vibration involving dilatation of the northern end (top in figure) of the iceberg, with relatively little motion in other parts of the iceberg. (b) Iceberg vibration involving twisting of the northern end of the iceberg relative to the center of the iceberg.

Horizontal modes of oscillation are depicted in Figure 11, where the two most distinguishable horizontal modes of vibration visible in animations made using seismometer horizontal displacements are (1) a stretching mode of the line segment formed by the two corners defining the northern end of C16 (sites B and C) where the iceberg is relatively thin and (2) a torsional mode involving twisting oscillation of the northern end of the iceberg relative to its center. Consideration of the ice-thickness and flexural-rigidity variations on the iceberg again suggests that the stretching mode should be lower-frequency than the torsional mode, which involves torsion of the thicker ‘waist’ of the iceberg near site A.

Event periodicity and source location

The wave-train travel-distance information derived above supports the notion that the majority of the glaciogenic sources are associated with icebergs that were located north of Ross Island during the field campaign. Collisions between these icebergs driven by the gyrations of B15A and B15J associated with ocean tides were noted as a major kinematic feature of these icebergs (Reference MacAyeal, Okal, Thom, Brunt, Kim and BlissMacAyeal and others, 2008b). It is thus reasonable to inquire whether events of the catalogue occur on a quasi-periodic, diurnal basis given the strong dominance of the diurnal tide in the southwestern Ross Sea (Reference MacAyealMacAyeal, 1984; Reference Robertson, Padman, Egbert, Jacobs and WeissRobertson and others, 1998; Reference Padman, Erofeeva and JoughinPadman and others, 2003). Examination of events observed on C16 reveals that, indeed, a great many tend to occur at the same time of day for several days in a row. These periodic events are also often associated with IHT episodes which are generated by stick–slip cycles produced by tangential motions in edge-on-edge contact zones between two icebergs (Reference MacAyeal, Okal, Aster and BassisMacAyeal and others, 2008a). These associations imply that a majority of the events are associated with iceberg collisions that are paced by the ocean tide.

Micro-tsunamis as informative signals

The most significant conclusion drawn from our observations is the realization that micro-tsunamic signals generated by floating portions of the Antarctic ice sheet are common-place and readily observed. This result invites speculation about whether micro-tsunami signals might inform study of iceberg-calving and ice-shelf-disintegration processes; and about whether glaciogenic tsunamis (close to source, where amplitude can be in the meters range) may themselves play a mechanical role in stimulating iceberg calving and ice-shelf disintegration. A disappointing aspect of our field campaign was that the design of our seismometer array was inadequate to pin down the source locations and focal mechanisms of micro-tsunamis. Our failure in this regard does not, however, rule out the possibility of gathering this important information if the seismometer array design is judiciously improved.

Assuming that this technical improvement is feasible, we suggest that observations of micro-tsunamis on ice shelves prone to ‘Larsen-style’ disintegration may help in understanding the dynamics of explosive ice-shelf disintegration. Recent study of the explosive collapse of parts of the Wilkins Ice Shelf in February and May 2008 (Reference Braun, Humbert and MollBraun and others, 2008; Reference Braun and HumbertBraun and Humbert, 2009; Scambos and others, in press) suggests that the disintegration was due to a calving and mass-movement process concentrated on the calving edge rather than due to a process that simultaneously affected large areas of the ice-shelf interior. Satellite imagery (Scambos and others, 2009) suggests that the disintegration was an edge-wasting process that initially produced many small, MacAyeal uncapsized ice-shelf fragments of various sizes, and that a large population of these fragments (notably with width smaller than thickness) subsequently capsized and fragmented into small, irregularly shaped pieces. To further differentiate between ‘edge wasting’ and ‘ice-shelf interior’ processes at play in ice-shelf disintegration, a study of micro-tsunamis generated by iceberg calving, capsize and fragmentation would be warranted. By deploying seismometers on a disintegrating ice shelf (e.g. in the rear of a disintegrating ice shelf which may be expected to survive the disintegration intact), the locus of micro-tsunamic signal sources could potentially distinguish between edge and interior source mechanisms.

Glaciogenic tsunamis as a forcing mechanism

The extreme rapidity of the break-up of portions of the Wilkins Ice Shelf in February 2008 (Reference Braun, Humbert and MollBraun and others, 2008; Reference Braun and HumbertBraun and Humbert, 2009) suggests that the requisite calving, capsizing and break-up of ice-shelf fragments is a self-stimulating, chain-reaction-type process that requires mechanical processes capable of acting over large distances with short, sub-diurnal timescales (e.g. following the ideas of R.A. Massom and others (unpublished information)). This suggestion is exemplified best by the comparison of MODIS (moderate-resolution imaging spectroradiometer) imagery between 28 and 29 February 2008, when ∼150 km² of unfragmented ice-shelf area converted into a densely packed mass of small icebergs that expanded seaward from the original edge of the ice shelf at a rate of ∼0.3 m s⁻¹ (http://nsidc.org/news/press/20080325 Wilkins.html). Simple, rough consideration of the energetics associated with ice and sea-water movement during the fragmentation and capsize processes indicates that the gravitational potential energy, ΔE, released by the ice-shelf disintegration process is:

where H is the original thickness of ice shelf, H − ΔH is the thickness of idealized ‘ice rubble’ occupying the ocean surface after disintegration, A is the surface area of ice shelf involved in disintegration, and ρ _i and ρ _w are the densities of ice and sea water, respectively. We note that the above expression is positive definite. Evaluating the above formula for conditions associated with the disintegration of the Wilkins Ice Shelf in February 2008, where initial ice-shelf thickness is ∼180 m, the final ice-rubble thickness is ∼50 m, and where a 10 km by 35 km area of the original ice shelf is involved, gives:

If all this energy is converted into kinetic energy, T = 1/2ρ _i AHV ², the velocity, V, of the seaward-moving mass of ice rubble (typically seen as a blue bubble in the wake of an outer edge of uncapsized ice-shelf fragment ice-bergs in the MODIS imagery) following disintegration will be ∼11.7 m s⁻¹. This velocity is far above that implied by the expansion rate of the ice-rubble mass in the MODIS imagery of 28 and 29 February 2008 (i.e. ∼0.3 m s⁻¹). The implication is that a large fraction of potential energy released by the disintegration is radiated hydrodynamically as traveling tsunamis. These tsunamis may subsequently interact with unbroken, uncapsized icebergs and the remaining intact, undisintegrated ice shelf to help sustain the disintegration process. Break-up of one small segment of the ice shelf could produce tsunamis that cause other parts of the ice shelf that have been preconditioned by environmental and climate-change effects (e.g. surface melting (Scambos and others, in press)) to disintegrate. This idea is admittedly speculative, but it serves to illustrate how further seismological observations of glaciogenic micro-tsunami signals may inform understanding of the ice-shelf disintegration process.