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  • View in gallery

    (top left corner) Helheim Glacier and Sermilik Fjord with bathymetry contours: the map is reprinted with permission from Schjøth et al. (2012). (top middle) Fenris and (top right corner) Midgard are two additional calving glaciers. Moorings (black stars) were placed along deeper side of Sermilik Fjord. A weather station is located at the Helheim front (red star).

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    The deviation from mean surface height η during the month of June is plotted for unfiltered data, T_TIDE filter, and high-passband filter with a 1-h wave period cutoff (offset by −0.5 m for clarity). Calving on Helheim occurred on 9, 11, and 29 June. The high-pass filter performs the best for the purpose of identifying calving-generated waves.

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    The average of magnitude squared coherence between each pair of events is plotted with 95% confidence levels (dashed line) for (a) class 1 events, which are among confirmed calving events on Helheim, (b) events from class 2.

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    The deviation from mean surface height η for 16 confirmed calving events on Helheim. The vertical lines indicate the beginning of the signal on the respective tsunameter; the distance between the lines is the computed time lag. Propagation of the initial sharp peak is from TSU1 (dark blue) toward TSU5 (red) in all cases, when the data are available. The vertical offset is proportional to the relative distance between the tsunameter locations.

  • View in gallery

    Examples of class 2 type events (legend is the same as in Fig. 4, and the signal from different tsunameters is offset for clarity): (a) Disturbance originates near TSU1 (dark blue) or at the glacier front and propagates toward the fjord mouth. (b) Disturbance originates near TSU3 (green) and propagates in both directions. (c) Disturbance originates near TSU2 (light blue) and propagates in both directions. The vertical offset is proportional to the relative distance between the tsunameter locations.

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    Calving catalog. (a) Potential calving events (black solid line) preselected from TSU1 record: 13 of these corresponded to calving on Helheim (shown in gray dotted lines overlaying the respective black lines from preselected events), and 3 additional calving events were detected on TSU4 and TSU5 after TSU1 stopped recording. (b) The first row shows the record from the camera located at Helheim front: calving events are shown in black, and gray indicates time interval during which Helheim calving front advanced out of the view of the camera. The next three rows show the record from satellite data. The width of each square indicates the time interval during which calving occurred on the respective glacier: red indicates calving on Helheim, orange indicates calving on Fenris, and yellow indicates calving on Midgard.

  • View in gallery

    Power spectral density from TSU5 with 90% confidence intervals and labeled tidal constituents. Locations of two lowest modeled resonant modes (52- and 32.5-min periods) are marked with dashed lines.

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    Power spectral density of response of the fjord system when forced at different wave periods; each line corresponds to an individual run with monochromatic forcing as labeled.

  • View in gallery

    Spectrogram from each tsunameter for first three calving events. Vertical axis shows the time since onset of the calving event, and the plots are normalized. The 4 h that precede calving are included as well for comparison with neutral state. High-frequency content associated with calving is highest on TSU1 and decays as the tsunameter position moves southward.

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    Normalized power spectral density over 7-h-long interval for the first eight calving events as seen on (left) TSU1 and (right) TSU5. For TSU1, wave periods of 21 min together with 2–10-min short periods dominate. For TSU5, the 21- and 32.5-min wave periods are strongest. Wave periods of dominant peaks are consistent among the events for most cases.

  • View in gallery

    A normalized Hovmöller diagram showing the deviation from the mean surface height η for the first four calving events. The vertical axis represents the distance along the fjord from the calving front.

  • View in gallery

    Normalized power spectral density over 6-h-long interval for different forcings near the glacier boundary: (a) at vTSU1 and (b) at vTSU5.

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    A normalized Hovmöller diagram showing the modeled deviation from mean surface height η for different forcings near the glacier boundary. The vertical axis represents the distance along the fjord from the calving front. (a) Damped oscillator with 40-min period. (b) Damped oscillator with 10-min period. c) Damped oscillator with 5-min period. (d) Free evolution following an initial displacement of water surface at the glacier boundary.

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    (a) Wind speed and direction (2-h running mean) as recorded at Helheim Glacier front. (b) Class 3 event associated with change of wind properties.

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    Tidal phase and distribution of calving events. The large black circle is oriented counterclockwise and represents the tidal phase: (left) semimonthly and (right) semidiurnal. Calving events are shown as circles centered at the point of the tidal phase when they occurred on TSU1. The radius of each circle indicates the relative amplitude of the event. The events are chronologically placed on the color bar and assigned a color to indicate time relative to each other.

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    Cumulative positive degree-days and calving events (after a calving event occurred, the count is reset). The location of events is represented with a line; its lower portion colored in black indicates the relative amplitude of the event. After a long winter break, calving season starts in the spring, when the calving event follows only a few days after the rise of average air temperature above the melting point.

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Calving Signature in Ocean Waves at Helheim Glacier and Sermilik Fjord, East Greenland

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  • 1 Courant Institute of Mathematical Sciences, New York University, New York, New York
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Abstract

When glaciers calve icebergs, a fraction of the released potential energy is radiated away via gravity waves. The characteristics of such waves, caused by iceberg calving on Helheim Glacier in east Greenland, are investigated. Observations were collected from an array of five high-frequency bottom pressure meters placed along Sermilik Fjord. Calving-generated tsunami waves were identified and used to construct a calving event catalog. Calving events are observed to cluster around high and low semidiurnal tides and around high and prior-to-low semimonthly tides. In the postcalving ocean state, discrete spectral peaks associated with calving events are observed, and they are consistent among all the events. A numerical model is used to compute the resonant modes of the fjord and to simulate calving-generated ocean waves. Damped oscillator boundary forcing with 5- to 10-min periods is found to reproduce well the observed properties of calving waves. These observations and modeling are relevant for better understanding of wave dynamics in glacier fjords.

Denotes Open Access content.

Corresponding author address: Irena Vaňková, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185. E-mail: vankova@cims.nyu.edu

Abstract

When glaciers calve icebergs, a fraction of the released potential energy is radiated away via gravity waves. The characteristics of such waves, caused by iceberg calving on Helheim Glacier in east Greenland, are investigated. Observations were collected from an array of five high-frequency bottom pressure meters placed along Sermilik Fjord. Calving-generated tsunami waves were identified and used to construct a calving event catalog. Calving events are observed to cluster around high and low semidiurnal tides and around high and prior-to-low semimonthly tides. In the postcalving ocean state, discrete spectral peaks associated with calving events are observed, and they are consistent among all the events. A numerical model is used to compute the resonant modes of the fjord and to simulate calving-generated ocean waves. Damped oscillator boundary forcing with 5- to 10-min periods is found to reproduce well the observed properties of calving waves. These observations and modeling are relevant for better understanding of wave dynamics in glacier fjords.

Denotes Open Access content.

Corresponding author address: Irena Vaňková, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185. E-mail: vankova@cims.nyu.edu

1. Introduction

Calving-generated ocean waves are tsunami-like waves, which have the potential to cause sudden and large mixing events and affect melt rates at the glacier ocean interface. Furthermore, waves generated at the open ocean have been hypothesized to have an influence on calving. Tides or other long-period nontidal components of sea level (i.e., surges) put the glacier out of buoyant equilibrium and increase stresses, both of which can trigger calving. Bending forces due to oscillating tidal motion can deepen preexisting crevasses or create new ones. Tides have been therefore linked to an increase in seismic activity of glaciers (Hammer et al. 2015). Tsunami waves generated during capsizing of large icebergs in Antarctica were suggested to cause further calving of ice shelves and floating tongues (MacAyeal et al. 2009). It is therefore important to gain better understanding of wave dynamics in the vicinity of calving glaciers. Nevertheless, waves in fjords with glaciers have not been systematically investigated. This is partly because of the difficulties in accessibility and recovery of moorings in these fjords, which are densely packed with icebergs and sea ice (i.e., ice mélange).

Tide gauges (Nettles et al. 2008) or single moorings (Amundson et al. 2010) have been previously used to detect ocean waves associated with calving, but their sampling rate was too low or their positioning did not allow one to determine the direction of propagation or other properties that evolve during the lifetime of these waves. Marchenko et al. (2012) recorded calving-generated ocean waves in the vicinity of a glacier front and reported spectral peak at the 93-s wave period. Amundson et al. (2010) presented results from a calving event at Jakobshavn Isbræ recorded on a pressure sensor loosely attached to the seafloor. The dominant wave periods were at 30–60 s, and the calving waves were suggested to have exceeded 1-m amplitude at 3-km distance from the glacier. It has also been suggested that calving excites seiches in fjords (Amundson et al. 2012).

A few theoretical and experimental studies have been dedicated to the topic of calving-generated ocean waves and suggested useful bounds or phenomena to consider in the context of glacier fjords. MacAyeal et al. (2011) proposed a theoretical estimate for an upper bound of iceberg capsize–generated tsunami wave amplitude to be 1% of the iceberg height. Burton et al. (2012) came to a similar result in an experimental setup of capsizing of plastic icebergs in an unstratified fluid. Massel and Przyborska (2013) studied an idealized problem of wave propagation caused by an object falling or sliding into a body of incompressible irrotational fluid flow and obtained slightly different wave patterns for different generating mechanisms. MacAyeal et al. (2012) used a simulation to model a channel with and without 20 equally spaced icebergs, concluding that the presence of icebergs lengthens the period of low-frequency modes and introduces band gaps.

Currently, seismic techniques and semiautomated algorithms are being used to detect calving and construct calving catalogs based on icequakes on seismic networks (Nettles and Ekström 2010). Walter et al. (2013) developed a related method that detects calving with the use of seismic signal induced by long-period ocean waves, which were observed to remain in the fjord after calving. Parallel to this work on calving ocean waves, Mei et al. (2016, manuscript submitted to Cryosphere) use seismic techniques to detect and locate icequakes at Helheim.

While many different calving mechanisms have been proposed (Benn et al. 2007), it remains unclear which calving factors dominate and how much they vary from glacier to glacier or seasonally. For the case of Helheim, James et al. (2014) and Murray et al. (2015) inferred from observations that the glacier often undergoes slow bottom up rotation for many days prior to breaking, which then occurs via buoyant flexure. On the other hand, a modeling study in a Helheim-like two-dimensional setup by Cook et al. (2014) concluded that meltwater in crevasses and increased basal water pressure are significant calving factors, while submarine melting and back stress by mélange are not.

In this work a new year-long dataset from an array of moorings in a Greenlandic glacier fjord is presented and analyzed. The moorings were equipped with high-frequency bottom pressure meters (i.e., tsunameters) capable of detecting waves in a fjord in high temporal resolution. This setup made it possible to determine the direction of propagation of waves in this fjord and allowed to distinguish between glacier- and ocean-generated signal. Different types of high-frequency waves are identified, and calving-generated waves are characterized. A numerical model is used to reproduce the observed properties of calving-generated waves and a damped oscillator forcing with a period between 5 and 10 min is suggested to represent well the effect of calving on the fjord. A model with this forcing and a realistic geometry reproduces well the spectrum of calving waves as well as their propagating nature that is not observed to settle to a pure standing wave.

A calving catalog is constructed and used to discuss influences of external factors on calving on Helheim Glacier. It is observed that calving is clustered around spring tide and prior to neap tide. In the case of semidiurnal tide, larger events are clustered around both extremas of the tide but not at the transition from one to the other. Further, the calving catalog shows two different modes: long winter break with a single calving event and a calving season whose beginning seems to be onset by the rise of average daily temperatures above the melting point in early spring. This points to the importance of previously suggested seasonal drivers such as meltwater in crevasses or back stress exerted by mélange on calving dynamics at Helheim (Howat et al. 2010).

It is shown that an array of tsunameters in a fjord can provide a new, although perhaps more challenging, alternative to tracking glacier activity. In principal, this approach should be capable of detecting all types of calving events, given the release of potential energy is enough to radiate away detectable wave amplitudes (~1 mm). Ocean waves carry additional information about calving, and therefore to understand the mechanisms involved it may be useful to combine both seismic and ocean wave analysis and detection techniques.

In section 2, the area of interest and the details about data collection are presented. The methods for the construction of the calving catalog are described in section 3. Resonant modes of the fjord system are computed in section 4. Properties of the calving-generated waves are analyzed and modeled in section 5. Other types of detected high-frequency waves are briefly discussed in section 6. The role of ocean waves coming from the open ocean and the role of seasonal variations in temperatures on calving are discussed in section 7. Concluding remarks and further suggestions are finalized in section 8.

2. Observational setup

Helheim Glacier is connected to the open ocean via a 100-km-long and 600–900-m-deep fjord system. Helheim Fjord runs west to east and after ~20 km merges almost orthogonally into the north to south running Sermilik Fjord. There are two additional calving glaciers, Fenris and Midgard, letting out into Sermilik Fjord. A map is shown in Fig. 1. Fenris and Midgard are smaller and shallower than Helheim; they drain only a small part of the ice sheet and therefore have received much less attention than the well-studied Helheim Glacier (Nettles et al. 2008; Murray et al. 2015). The circulation and water properties in Sermilik Fjord have been extensively studied in recent years (Straneo et al. 2010; Sutherland et al. 2013).

Fig. 1.
Fig. 1.

(top left corner) Helheim Glacier and Sermilik Fjord with bathymetry contours: the map is reprinted with permission from Schjøth et al. (2012). (top middle) Fenris and (top right corner) Midgard are two additional calving glaciers. Moorings (black stars) were placed along deeper side of Sermilik Fjord. A weather station is located at the Helheim front (red star).

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

Helheim Fjord is generally densely packed with mélange and is difficult to recover moorings from; therefore, moorings were placed in the adjacent Sermilik Fjord. Five moorings were deployed in August 2013 and recovered in August 2014. The first mooring was placed at the northern end of the fjord and the remaining four were spaced in intervals of 10.5, 12.1, 17.8, and 13.8 km, respectively, to cover the full length of the fjord. The moorings were placed along the deeper western side of the fjord at depths of 595, 638, 683, 880, and 908 m, respectively. Positions of the moorings are shown in Fig. 1. The moorings were recovered within 100 m of the location where they were deployed the year before.

Each mooring was designed as a large, circular float, with a hole through the middle in which an acoustic release was attached. The release held on to a metal bar that passed through opposite walls of a ~20-cm-wide and ~1-m-long rigid pipe. The bottom of the float was in direct contact with the top end of the pipe. The pipe was made from hard plastic, and its bottom end was attached to a heavy rectangular metal anchor, which secured the mooring at a fixed position at the seafloor. The advantage of this rigid setup as opposed to a chain is that it does not allow for movements of the float with bottom currents, for example. The instruments were attached through additional holes in the body of the float.

The moorings were equipped with a CTD (model SBE 37 MicroCAT) and a high sampling frequency pressure meter (model SBE 54), that is, a tsunameter (TSU1 is first tsunameter from the glacier, TSU5 is the last one). The CTDs sampled every 10 min, and the tsunameters sampled every 4 s. The resolution of the tsunameter depends on the sampling rate and depth; the worst case scenario of 4-s sampling rate at 900 m of depth gives a resolution of 0.4 mm. The device could be sensitive to temperature gradients but is well insulated, and the temperature on the ocean floor did not vary by more than 1 K throughout the year. The high sampling frequency of the tsunameters was crucial for calving wave detection; since a barotropic wave travels the length of the fjord in less than 20 min, the sampling rate of the pressure measurement from the CTD does not give sufficient temporal resolution. TSU1, TSU4, and TSU5 sampled almost until the recovery time: 8, 16, and 18 August, respectively. TSU2 and TSU3 stopped recording on 23 June and 30 May, respectively.

A weather station overlooking Helheim Glacier is located ~500 m above sea level near the average terminus position of the last years. Atmospheric data were recorded at 10-min intervals. A camera (model CC5MPX) was additionally installed on the weather station in order to visually confirm calving events detected on tsunameters. The images were recorded at 1-h intervals. Further, MODIS satellite imagery (Herried et al. 2013) was used to identify time intervals during which calving events occurred. The satellite image spatial resolution is 250 m, and at most one image per day is available.

3. Calving detection method

a. Identification of calving waves

The recorded signal from the tsunameters shows strong semidiurnal tidal signal with amplitude varying from 0.5 to 1.8 m depending on the spring–neap cycle. The tidal signal was filtered out using the T_TIDE package (Pawlowicz et al. 2002). Although this removed a significant portion of the high-amplitude low frequencies attributed to tides, it was not sufficient to clearly see the calving signal, mainly due to the residual low-frequency oscillations (surges and remaining part of different tidal components), which still had significant amplitudes. Therefore, additionally a high-pass band filter with a cutoff wave period of 1 h was used. This specific choice of cutoff period was determined empirically, such that it was the highest-frequency filter that kept all the qualitative features of sudden high-amplitude events superimposed on the tide and often directly observable on the raw signal. It is further justified by the fact that observed calving events last on the order of tens of minutes. At this point, there is no need to use the tidal filter prior to the high-pass filter since the cutoff period is much shorter than the shortest period tidal component removed by T_TIDE. A comparison of the two different filters is shown in Fig. 2.

Fig. 2.
Fig. 2.

The deviation from mean surface height η during the month of June is plotted for unfiltered data, T_TIDE filter, and high-passband filter with a 1-h wave period cutoff (offset by −0.5 m for clarity). Calving on Helheim occurred on 9, 11, and 29 June. The high-pass filter performs the best for the purpose of identifying calving-generated waves.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

Data from TSU1 were used for calving identification since there the noise from short deep-water waves is the lowest and amplitude from tsunami signal is the largest. Events whose amplitude exceeded the threshold of three standard deviations (7 mm) were preselected and then manually inspected and different characteristic waveforms were identified. Most of these preselected events start with a sudden sharp peak that then slowly decays (class 1 and 2). However, this method also picked up a few long-lasting events, which gradually increase and decrease in amplitude (class 3).

Classes 1 and 2 both show abrupt change in amplitude, but they differ by maximum amplitude reached, duration of the event, and mainly by visibly distinct dominant wave period after the onset. This is quantified by spectral coherence analysis in sections 5 and 6 and shown in Fig. 3. A necessary characteristic of a calving wave is its propagation from the glacier front toward the fjord mouth. Waveforms that do not consistently propagate in this direction can thus be rejected. The maximum value of cross correlation between signals on neighboring tsunameters over 1-h-long segment was used to determine the lag. All class 1 events propagated from TSU1 to TSU5 at realistic barotropic speeds and were therefore included as candidates for the calving signal. Figure 4 shows initial propagation of the signal southward and properties of these waves are discussed further in section 5. Class 2 events do not always originate at the glacier front. In Fig. 5, different examples are shown, depicting where these events originate before TSU1, near TSU2, and near TSU3 and then propagating in all directions. Based on this observation, class 2 is rejected from the preselected events.

Fig. 3.
Fig. 3.

The average of magnitude squared coherence between each pair of events is plotted with 95% confidence levels (dashed line) for (a) class 1 events, which are among confirmed calving events on Helheim, (b) events from class 2.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

Fig. 4.
Fig. 4.

The deviation from mean surface height η for 16 confirmed calving events on Helheim. The vertical lines indicate the beginning of the signal on the respective tsunameter; the distance between the lines is the computed time lag. Propagation of the initial sharp peak is from TSU1 (dark blue) toward TSU5 (red) in all cases, when the data are available. The vertical offset is proportional to the relative distance between the tsunameter locations.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

Fig. 5.
Fig. 5.

Examples of class 2 type events (legend is the same as in Fig. 4, and the signal from different tsunameters is offset for clarity): (a) Disturbance originates near TSU1 (dark blue) or at the glacier front and propagates toward the fjord mouth. (b) Disturbance originates near TSU3 (green) and propagates in both directions. (c) Disturbance originates near TSU2 (light blue) and propagates in both directions. The vertical offset is proportional to the relative distance between the tsunameter locations.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

Class 3 events do not show a clear onset; they last for many hours to days, slowly increase, and then decrease in amplitude. Cross correlation between adjacent tsunameters does not show consistent direction of propagation from one end of the fjord to the other and the lags obtained are highly sensitive in both magnitude and direction to the time interval around the higher-amplitude event considered and may give unrealistically high propagation speeds, which indicate a standing wave component. Based on the long duration of the buildup of the event and lack of propagation, class 3 is also rejected from preselected events. Possible causes of class 2 and 3 events are discussed further in section 6.

b. Confirmation of calving on Helheim

Camera and satellite imagery, when available, served to confirm whether calving on Helheim occurred. From 28 November to 18 December, the calving front moved out of the scope of the camera view; however, no sharp peaks were observed during this time, and no retreat was seen on satellite images. Therefore, it is likely the first calving event of the record occurred on 18 December. The calving front was again out of the view of the camera between 17 January and 4 April. Although satellite images were available from 22 January on, many days of thick cloud cover obscures the view, and usually fresh snow cover makes it difficult to distinguish the glacier from the mélange. A small event detected on the tsunameter on 25 February coincides with slight movement of mélange out of the Helheim Fjord between 24 and 27 February. Similarly between 22 and 23 March a large piece of mélange broke off, but the remaining piece stayed rigid and attached. A larger event was detected on the tsunameters, but also strong winds of almost 20 m s−1 from the southwest were present between 21 and 23 March. While this could have been a Helheim calving event, it is possible that the signal on the tsunameters was entirely due to the breaking of this piece of mélange as a result of long-lasting unidirectional strong winds. With this limitation, and excluding the above mentioned two occurrences, 16 calving events were confirmed to occur on Helheim, 13 of which were among the preselected events from class 1 and the remaining 3 occurred after TSU1 stopped recording. The Helheim calving catalog is shown in Fig. 6, and it summarizes the tsunameter, camera, and satellite information as well as the final confirmed Helheim events.

Fig. 6.
Fig. 6.

Calving catalog. (a) Potential calving events (black solid line) preselected from TSU1 record: 13 of these corresponded to calving on Helheim (shown in gray dotted lines overlaying the respective black lines from preselected events), and 3 additional calving events were detected on TSU4 and TSU5 after TSU1 stopped recording. (b) The first row shows the record from the camera located at Helheim front: calving events are shown in black, and gray indicates time interval during which Helheim calving front advanced out of the view of the camera. The next three rows show the record from satellite data. The width of each square indicates the time interval during which calving occurred on the respective glacier: red indicates calving on Helheim, orange indicates calving on Fenris, and yellow indicates calving on Midgard.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

Most but not all detected large-amplitude sharp peaks correspond to calving on Helheim. Many smaller-amplitude events remain unidentified for a variety of reasons. First, there were no cameras on Fenris and Midgard Glaciers, and satellite imagery with the given resolution is only capable of indicating bigger changes on the glacier front. However, even when a camera is present as is the case of Helheim, fog, cloud, and limited hours of sunlight during winter months can cause the calving front to be undetectable for hours to days, and small events can easily go unnoticed.

4. Spectral characteristics of the fjord

a. Observed spectrum

The power spectral density from TSU5 is shown in Fig. 7. The year-long record was divided into six segments and averaged to obtain confidence intervals; 90% confidence levels are shown. The peaks corresponding to semidiurnal and diurnal tidal frequencies dominate the signal. Further, sharp narrow peaks correspond to higher tidal harmonics, namely, M3, M4, and M6. At neither TSU does the spectrum show convincing sharp narrow peaks that would be characteristic of natural modes of a simple system (Rabinovich 2009), although there are regions of enhanced wave response in the shorter periods, corresponding to computed modes. The first two computed modes are marked with dotted lines at the observed spectrum at TSU5.

Fig. 7.
Fig. 7.

Power spectral density from TSU5 with 90% confidence intervals and labeled tidal constituents. Locations of two lowest modeled resonant modes (52- and 32.5-min periods) are marked with dashed lines.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

b. Computed resonant modes

The motivation for computing the resonant modes of the fjord comes from the need to interpret discrete sharp peaks in the spectrum of calving waves in section 5. The forced response of the fjord system was assessed with a numerical model by forcing the system at different wave periods following Sammartino et al. (2014). The MITgcm (Marshall et al. 1997a,b) numerical model was set up with observed geometry and bathymetry from Schjøth et al. (2012). The horizontal resolution was 500 m, and the bathymetry was binned to 5 vertical levels of 180 m each. The model was set to solve the hydrostatic primitive equations on an f plane with a linear drag of 0.0002 m s−1 and with constant density and salinity. No-slip boundary conditions were prescribed at solid boundaries. Along the partially opened eastern boundary a 20-km-thick sponge layer with enhanced viscosity was imposed to damp the waves constantly input in the system. The model was forced by prescribed monochromatic periodic oscillations in the meridional velocity field imposed at the southern boundary. The forcing periods of 5, 10, 20, 30, 40, and 60 min were tested and each run lasted 20 h.

Resulting spectrum from different forcing periods on virtual tsunameters in the model (vTSU1 is the model analog of TSU1, etc.) is shown in Fig. 8. For each run the highest peak corresponds to the forcing period. The first three resonant modes are found at forcings with wave periods of 52, 32.5, and 25 min. A large number of small peaks then follow in shorter periods.

Fig. 8.
Fig. 8.

Power spectral density of response of the fjord system when forced at different wave periods; each line corresponds to an individual run with monochromatic forcing as labeled.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

A significant challenge when modeling the natural modes of a proglacial fjord is the effect of highly variable mélange and large icebergs in the fjord that can alter the propagation properties of shallow-water waves. For example, MacAyeal et al. (2012) suggested that periodically spaced icebergs can introduce band gaps in the seiche spectrum and lengthen the period of low-frequency modes. Similarly, sea ice cover is known to damp ocean waves and can possibly change their dispersion as well (Squire et al. 1995). None of these effects were considered in the presented model.

5. Results, modeling, and discussion of calving waves

a. Calving wave properties

Calving-generated waves (class 1 events) are characterized by a sharp sudden onset of high-amplitude oscillations reaching up to 24 cm at 600-m depth 30 km away from the glacier front. The initial disturbance propagates from the glacier front southward to the fjord mouth, as is presented in Fig. 4. The average speed of propagation of this initial peak between each pair of tsunameters, determined by cross correlation of the neighboring signals, was 75 ± 8, 69 ± 11, 67 ± 2, and 81 ± 6 m s−1 in each section, going from north to south. The shallow-water barotropic speed estimate , where g is gravity and H is the averaged depth at the two neighboring mooring locations, would be 78, 80, 88, and 94 m s−1, respectively. The first baroclinic mode of a two-layer system, which Sermilik Fjord can be approximated as (thin layer of cold, fresh layer overlaying thick layer of salty, warm water), would propagate at only 1–4 m s−1. The large difference between the propagation speed of the barotropic and baroclinic modes lets us conclude that the detected calving-generated waves are barotropic waves. In this dataset, no calving-generated internal waves were detected, although the sensitivity of the instrument would allow for detection of large internal waves of a few meters in amplitude.

The degree of similarity between the different calving events on Helheim was evaluated using coherence analysis for a time period of 10 h around the onset of the event. Magnitude squared coherence was computed between each pair of these events on TSU1, and the calving signals were found to be coherent with 95% confidence level in the interval of 1–8-min periods, as shown in Fig. 3a. The same result is obtained when the computation is repeated for TSU2 and TSU3. For TSU4, the 1–3-min part of the peak drops below the given confidence level, and for TSU5, the peaks are only in the 3–8-min interval.

The time evolution of the spectral properties of calving waves is captured on a spectrogram in Fig. 9, showing the first three events. The logarithmic distribution of power spectral density from each tsunameter was computed with 30-min-long segments using a 15-min overlap, and the plots were normalized. High-frequency content of the calving event is in the wave periods of 30–120 s; it is the highest on TSU1, and it decays as the tsunameter position moves southward. On TSU5, which is closest to the mouth of the fjord, only the lower-frequency part of the spectrum persists. Further, these high frequencies decay within an hour or two, and their source may lie in the adjustments in the glacier and mélange following up a calving event. A similar range of high-frequency content as on TSU1 and TSU2 was reported by Amundson et al. (2010) from ocean pressure measurements of a calving event at Jakobshavn Isbræ. There, however, these wave periods of 30–60 s dominated the signal, possibly because the device was placed much closer to the calving front, 3 km away.

Fig. 9.
Fig. 9.

Spectrogram from each tsunameter for first three calving events. Vertical axis shows the time since onset of the calving event, and the plots are normalized. The 4 h that precede calving are included as well for comparison with neutral state. High-frequency content associated with calving is highest on TSU1 and decays as the tsunameter position moves southward.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

Both coherence analysis and the spectrogram show that the high-frequency waves (wave periods below 3 min) associated with calving are dissipated before they arrive at the fjord mouth.

A dominant feature seen from the spectrogram is the concentration of energy in discrete peaks whose locations are consistent among the events, although the relative intensity varies and not all peaks are present at every event. This is further shown in Fig. 10, where the normalized power spectral density over 7 h since the beginning of a calving event is shown for the first few events. While at TSU1, calving events tend to excite the mode of the 21-min period; at the TSU5 location, both the 21- and the 32.5-min periods dominate the spectrum. The latter is consistent with the second lowest resonant mode computed numerically.

Fig. 10.
Fig. 10.

Normalized power spectral density over 7-h-long interval for the first eight calving events as seen on (left) TSU1 and (right) TSU5. For TSU1, wave periods of 21 min together with 2–10-min short periods dominate. For TSU5, the 21- and 32.5-min wave periods are strongest. Wave periods of dominant peaks are consistent among the events for most cases.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

Although much of the calving energy input into the fjord is projected onto resonant modes, the fjord after calving is not observed to settle to a pure standing wave. Rather, the calving waves manifest as traveling waves that remain in the fjord for over 10 h following the event and slowly radiate out until the fjord settles to the preevent state. A Hovmöller diagram in Fig. 11 shows how these calving waves propagate back and forth while reflecting from the fjord mouth and from the glacier end of the fjord. In the case of a typical calving event, for an hour or two following the initial peak, there is a strong presence of shorter higher-frequency waves originating at the glacier front and reflecting from the side fjords. Later, the fjord settles to a dominant lower-frequency wave of a 20–40-min periods (corresponding to 1 to 2 times the length of the fjord), which, however, still shows propagation. This indicates that these modes are not associated with pure standing waves but rather that they may be quasi-normal modes associated with bay openings allowing for energy radiation to a shelf described by Bellotti et al. (2012).

Fig. 11.
Fig. 11.

A normalized Hovmöller diagram showing the deviation from the mean surface height η for the first four calving events. The vertical axis represents the distance along the fjord from the calving front.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

b. Modeled calving waves

This section is aimed at modeling and reproducing observed properties of calving-generated ocean waves with proposed forcings near the glacier boundary imitating the actual calving process. A similar setup of the MITgcm model was used for the computation of resonant modes, and the following modifications were made: In the southern boundary and open part of the eastern boundary, a radiation condition was applied with phase speed fixed and set to shallow-water speed at the given depth that ensures minimal reflection of waves from this boundary (Durran et al. 1993). The disturbance at the Helheim front was prescribed in two different ways:

  1. At the line of points adjacent to the boundary representing Helheim Glacier front, the initial surface of the water level was uplifted by 2 m and let to evolve freely afterward. This setup may simulate sudden displacement of water column by the glacier.
  2. Both velocity components were prescribed at six grid cells at the northern boundary adjacent to the calving front of Helheim. The velocity u of boundary points evolved in time t as a damped oscillator:
    eq1
    with the damping time scale Td fixed at 10 min and amplitude A fixed at 5 cm s−1. The choice of Td was made such that the initial vigorous motion is confined to the first few minutes and the forcing decays within an hour. With Td of 10 min, the surface fluctuations decay to 60% of the original amplitude within the first 5 min and to 0.25% within an hour. Three different runs were performed with period To set to 5, 10, and 40 min. This forcing setup may simulate a calving situation of a block of ice undergoing a slow rotation while breaking.

Again, the normalized power spectral density over the duration of the run is plotted and shown in Fig. 12. Similarly as for observed calving events, energy is concentrated in discrete peaks. Further, these peaks agree well with the observed events in Fig. 10.

Fig. 12.
Fig. 12.

Normalized power spectral density over 6-h-long interval for different forcings near the glacier boundary: (a) at vTSU1 and (b) at vTSU5.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

There is a slight shift from 32.5 to 30 min and from 20 to 19 min in the resonant modes of damped oscillator-type forcing, which is an artifact of the way the forcing was prescribed. There the top 12 grid cells of the domain were removed, which brought the northern edge of the Helheim front to the northern boundary of the domain and allowed to prescribe velocity there, but as a side effect this resulted in shortening the side fjord next to Midgard by a few kilometers.

At vTSU1, the prescribed forcings have high energy content in shorter periods, and they tend to excite the mode of the 20-min period as well as the 30–33-min mode. The damped oscillator forcing with the oscillation time scale To of 40 min is too slow to show some of the high-frequency peaks observed during calving. The To of both 5 and 10 min as well as free-evolving displacement give sufficient high-frequency content required for a calving wave as well as noticeable peaks at the 20-min mode.

The relative size of the 30–33-min mode peak tends to be exaggerated in the modeled events compared to the observed ones. At the vTSU5 location, most energy is contained in the 30–33-min mode and this does not vary much as a function of the forcing that indicates that most of the information about the forcing simulating a calving process has been lost by this point.

For further comparison with reality, a Hovmöller diagram of the modeled waves is shown in Fig. 13. With the exception of the first panel showing the waves generated by the 40-min damped oscillator, the main characteristics observed in Fig. 11 are well represented in the remaining three cases. Initially higher-frequency waves and lower-frequency traveling waves at later times are present. Reflections back and forth between Helheim and Midgard maintaining high amplitudes on vTSU1 are apparent, and traveling waves are present in the fjord for many hours following an event.

Fig. 13.
Fig. 13.

A normalized Hovmöller diagram showing the modeled deviation from mean surface height η for different forcings near the glacier boundary. The vertical axis represents the distance along the fjord from the calving front. (a) Damped oscillator with 40-min period. (b) Damped oscillator with 10-min period. c) Damped oscillator with 5-min period. (d) Free evolution following an initial displacement of water surface at the glacier boundary.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

6. Discussion of class 2 and 3 events

Class 2 events are characterized by sudden onset of high-frequency propagating oscillations that decay within an hour. Magnitude squared coherence between each pair of these events was computed, and the resulting averaged value is shown in Fig. 3b. There is one strong peak above the 95% confidence level at the 30–60-s period. This is much different from the calving waves coherent in the interval of 1–8-min periods. Further, class 2 waves are of much lower amplitude, only 1–2 cm. Because of their small amplitude and high frequency, these waves often do not propagate to all the tsunameters and dissipate quickly. Given that class 2 events can originate anywhere in the fjord, as described in section 3 and shown in Fig. 5, it is possible they may be associated with the capsizing of large icebergs; however, there is no direct confirmation of this. It is also possible that those class 2 events propagating from TSU5 to TSU1 are associated with a different calving mode or with adjustments in the mélange.

Class 3 events are long-lasting occurrences of enhanced ocean swell with no clear beginning or direction of propagation. They may be associated with strengthening wind speed or changes of the wind direction, and one such example is shown in Fig. 14. Here, the amplitude of oscillations increases over a few days during which the wind speed stayed tripled and direction was reversed compared to both previous and following days. Nevertheless, these events are not always correlated with distinct events on the weather station at the glacier front.

Fig. 14.
Fig. 14.

(a) Wind speed and direction (2-h running mean) as recorded at Helheim Glacier front. (b) Class 3 event associated with change of wind properties.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

7. External calving factors

a. Waves from open ocean

While previous sections were aimed at the influence of calving-generated waves on the fjord, the focus here is on the potential influence of waves coming from the ocean on the calving of the glacier.

First, the two main features that visibly dominate the unfiltered data, the semidiurnal tide and its envelope of the period 14–15 days and the semimonthly tide, are considered. Figure 15 shows phase diagrams of calving event distribution over these semidiurnal and spring–neap tides. Calving events are represented as circles centered at the point of the tidal phase when they occurred on TSU1, and their radius is the relative amplitude. It is possible to make the observation that in the case of semimonthly tide two clusters of points appear: first around spring tide and second prior to neap tide. In the case of semidiurnal tide, larger events are clustered around both high and low tide but not in the transition from one extrema to the other. However, since there are only 16 data points, further observations would be necessary to make a statistically significant conclusion.

Fig. 15.
Fig. 15.

Tidal phase and distribution of calving events. The large black circle is oriented counterclockwise and represents the tidal phase: (left) semimonthly and (right) semidiurnal. Calving events are shown as circles centered at the point of the tidal phase when they occurred on TSU1. The radius of each circle indicates the relative amplitude of the event. The events are chronologically placed on the color bar and assigned a color to indicate time relative to each other.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

Surges were calculated by subtracting the tidal prediction by T_TIDE from the observation. While tides reach amplitudes between 0.5 and 1.8 m, the maximum surge amplitude was only 0.25 m but usually much less. Much of this component of the signal is coherent with the atmospheric pressure, and there is no clear consistent propagation of the signal either in or out of the fjord. This suggests that the observed surges are mainly a consequence of local atmospheric conditions rather than a signature of distant storms reaching the fjord. No correlation between calving events and nontidal ocean waves was observed. Additionally, no tsunamis propagating northward were recorded. Furthermore, given the thick mélange cover and the low amplitude (less than a centimeter) of high-frequency wind-generated waves on TSU1, the influence of these waves at the glacier front is negligible.

b. Calving frequency distribution

Frequency of calving on Helheim was highly variable throughout the year, as seen from the constructed calving catalog. Only a single calving event occurred during fall and winter. The first large spring event occurred in early April (although unclassified, smaller calving events could have occurred given that the glacier front moved out of the scope of the camera). After the first event, calving season was initiated, and the glacier calved regularly every 1 to 3 weeks. This is illustrated in Fig. 16, which shows cumulative positive degree-days (cumulative days when average daily temperature was above 0°C). The first spring calving event follows directly after the increase of average air temperature above the melting point. This was the first time since October that the temperature stayed above freezing for more than a day. A similar behavior of freezing for several days, warming up, and staying above freezing for couple of days followed by calving can be seen for the next two events as well.

Fig. 16.
Fig. 16.

Cumulative positive degree-days and calving events (after a calving event occurred, the count is reset). The location of events is represented with a line; its lower portion colored in black indicates the relative amplitude of the event. After a long winter break, calving season starts in the spring, when the calving event follows only a few days after the rise of average air temperature above the melting point.

Citation: Journal of Physical Oceanography 46, 10; 10.1175/JPO-D-15-0236.1

Two important factors in proposed calving mechanisms that vary seasonally with air temperature are the weight of water in filled crevasses (moulins) and back stress from mélange (Howat et al. 2010). However, rearrangements in subglacial water storage and drainage is another seasonally varying cause of glacier speed up, often independent of meltwater production (Benn et al. 2007). The relative importance of these factors is not settled yet and it may vary between different glaciers. For example, on Helheim, Foga et al. (2014) studied the relation of the calving rate and mélange properties based on satellite imagery and concluded that mélange has some role on calving rate variation and that its rigidity is due to air temperature rather than due to iceberg content. Cassotto et al. (2015) used MODIS surface temperature data to study the seasonal variability of mélange and its relation to the changes in the terminus position and glacier velocities of Jakobshavn Isbræ. From the satellite data complemented with time-lapse photography and seismically detected calving record, it was shown that calving was often preceded by an increase mobility of the mélange. On the other hand, Cook et al. (2014) modeled a Helheim-like, two-dimensional glacier to evaluate the importance of calving factors and concluded that meltwater in crevasses was a significant factor and back stress from mélange was not. The studies of mélange generally depend on satellite imagery (MODIS), which gives an average sampling frequency 4.75–6 days (Seale et al. 2011; Foga et al. 2014). This is likely not sufficient temporal resolution for a reliable estimate of the time scale on which the transition from rigid to weak mélange occurs. Current available estimates suggest this weakening to occur on a scale of days to weeks (Krug et al. 2015). While some studies based on satellite observations are available for mélange rigidity–calving rate relation, no analogous observations for the time scale of filling and drainage of moulins are available. The presented observations suggest that whichever is the responsible seasonal driver of calving, it needs to be able to act on the scale of days.

8. Conclusions

In this study, a new dataset was presented and the analysis and modeling focused on better understanding of wave dynamics in fjords with calving glaciers. During 1 year, starting on August 2013, an array of five tsunameters moored along Sermilik Fjord recorded waves propagating between the open ocean and the Helheim Glacier front. Different types of high-frequency waves were classified from the tsunameter data, and an empirical procedure for selecting calving-generated ocean waves was outlined. Southward-propagating waves were preselected as the signature of calving events, and all Helheim calving events identified with a camera belonged to it. In the future, this method, which includes high-pass filtering, preselection of events based on amplitude threshold, and classification and rejection of class 2 and class 3 type events, could be semiautomated.

The Helheim calving catalog resulted in 16 confirmed calving events during the year, whose characteristics were further analyzed. The amplitude of waves caused by calving varied between 1 and 24 cm at 600-m depth, and the waves propagated at the barotropic wave speed. The three closest tsunameters to the calving front show significant high-frequency content, which, however, dissipated well before the signal arrived at the mouth of the fjord, 70 km away. The calving-generated waves persist through partial reflection at the fjord mouth and travel in the fjord for several hours before they fully radiate away. The ocean spectrum following a calving event was dominated by discrete peaks corresponding to different resonant periods of the fjord, which were computed numerically. At TSU1, peaks at the 2–10-min periods and a peak at the 21-min period dominated, while at TSU5, the 32.5-min period was the most prominent.

Calving waves were simulated with a numerical model. Forcing was imposed at the northern boundary of Helheim Fjord as either a free oscillation following a displacement of a water level in the vicinity of the glacier boundary or a damped oscillator with prescribed damping time scale of 10 min. Both the free oscillation and prescribed damped oscillator with 5- and 10-min periods reproduced well the observed calving events, producing similarly distributed spectral content as well as their propagation characteristics.

Furthermore, the influence on calving by waves coming from the open ocean was considered. Surges reached only small amplitudes compared to the tide and did not show any relation with calving. Calving events clustered around high and low semidiurnal tides and around high and prior-to-low semimonthly tides. Last, calving on Helheim was very seasonal, only a single calving event occurred during the fall and the winter. Calving resumed after a first multiple-day sequence of positive degrees-days occurred. Past then Helheim calved regularly every 1 to 3 weeks. This observation suggests that the seasonal driver that modulates calving on Helheim Glacier acts on the scale of days, which provides a constraint for future consideration of relative importance of seasonal calving factors.

Acknowledgments

Field work at Sermilik Fjord and Helheim Glacier and the work of both authors was supported by NSF Office of Polar Program Grants ARC-0806393 and ARC-1304137, NASA Polar Programs Grant NNX08AN52G, and the NYU Abu Dhabi Center for Global Sea Level Change Grant G1204. Moorings in the fjord were deployed by Carl Gladish in August 2013. NYU High-Performance Computing clusters in New York were used to run MITgcm simulations. We thank the reviewers, Mark Inall and Jason Amundson, for useful suggestions and insights that significantly improved the quality of this article.

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