1. Introduction
Recent years have seen an increased interest in the fate of icebergs shed from high-latitude glaciers and ice shelves. They remain a threat to shipping as well as offshore oil and gas exploration. This is of particular relevance as retreating Arctic sea ice and increasing hydrocarbon demands have garnered the attention of industrial developers interested in both shipping and drilling in the Arctic Ocean (Pizzolato et al. 2014; Unger 2014; Henderson and Loe 2016). Concurrently, ongoing global climate change is being held responsible for an observed increase in calving fluxes from Antarctic and Greenland glaciers, an increase that is projected to accelerate during the coming decades (e.g., Rignot and Kanagaratnam 2006; Copland et al. 2007; Rignot et al. 2011; Joughin et al. 2014) and that is expected to impact regional ecosystems and oceanographic conditions (e.g., Vernet et al. 2012; Smith et al. 2013; Stern et al. 2015; Duprat et al. 2016). Furthermore, rapid shedding of icebergs from Northern Hemisphere ice sheets during the Heinrich events of the last glacial period are believed to have affected oceanic and atmospheric conditions on a global scale [see reviews in Hemming (2004) and Stokes et al. (2015)].
In light of these factors, icebergs have recently begun to be implemented in state-of-the-art global climate models (GCMs; e.g., Martin and Adcroft 2010; Hunke and Comeau 2011; Stern et al. 2016) and Earth System Models of Intermediate Complexity (EMICs; e.g., Jongma et al. 2009, 2013; Bügelmayer et al. 2015a,b). An improved physical understanding of iceberg dynamics is important for this model development and will aid in the interpretation of the model simulation results.
Previous iceberg drift studies have often focused on the ability to (i) reproduce individual iceberg trajectories using comprehensive dynamic hindcast models (Smith and Banke 1983; Lichey and Hellmer 2001; Keghouche et al. 2009; Turnbull et al. 2015) or (ii) predict trajectories using statistical relationships derived from observed trajectories. A well-known feature of the latter approach is the empirical rule of thumb that icebergs move at approximately 2% of the wind velocity relative to the ocean current (e.g., Garrett et al. 1985; Smith and Donaldson 1987; Smith 1993; Bigg et al. 1997). Other studies have focused on large-scale freshwater release from icebergs into the high-latitude oceans (e.g., Bigg et al. 1997; Death et al. 2006; Martin and Adcroft 2010; Jongma et al. 2013; Roberts et al. 2014). These studies typically use a representation of iceberg drift that is based on the model introduced by Bigg et al. (1997).
Here, we examine the salient characteristics of how iceberg trajectories are determined. We develop an idealized iceberg drift model, which allows an analytical solution of the iceberg velocity as a function of the local water and air surface velocities. Because the iceberg trajectories are found to depend on iceberg size, we couple the drift model to an idealized decay model. This is adapted from the decay representation in the comprehensive iceberg model of Bigg et al. (1997). The Lagrangian iceberg model presented here is computationally inexpensive and requires only three input fields to simulate iceberg trajectories: ocean and atmosphere surface velocities and sea surface temperature (SST). Furthermore, the somewhat idealized formulation of the model facilitates detailed physical interpretation and helps build understanding of the processes that determine iceberg drift.
This article is structured as follows: Section 2 introduces the iceberg drift and decay representations as well as the analytical solution for the drift velocity. Section 3 presents iceberg trajectories that are computed from the analytical solution, with surface conditions taken from an observational state estimate. Section 4 discusses the role winds and currents play in determining iceberg trajectories, focusing on the limits of small icebergs (Arctic) and large icebergs (Antarctic). Concluding remarks are given in section 5.
2. Iceberg drift model
a. Governing equation for iceberg drift
The model developed in this study retains the main components of previous formulations. It is, however, somewhat idealized, with the central approximations being as follows:
- The acceleration term
- The pressure gradient force is approximated from the ocean velocity by assuming geostrophic ocean currents.
- The iceberg speed |vi| is taken to be much smaller than the surface wind speed |va|.
- The drag forces from sea ice and wave radiation are neglected based on the expectation that they are a small component of the momentum balance.
- The water drag is approximated based on the ocean current velocity at the surface alone, neglecting vertical variations in the current over the depth of the iceberg. Similarly, the wind felt by the iceberg at any height is approximated to be equal to the surface wind.
Approximation 1: This approximation is best satisfied for small icebergs. Using the model simulations of section 3 (below), we find that this approximation is typically fairly well satisfied for Arctic icebergs. When the iceberg length is less than 1.5 km, we find that 
Approximation 2: The pressure gradient force has previously been argued to be well approximated by assuming a geostrophic ocean velocity (Smith and Banke 1983; Gladstone et al. 2001; Stern et al. 2016). It should be noted, however, that Bigg et al. (1996) found that this approximation may introduce substantial errors in regions where the ageostrophic component of the ocean velocity is large.
Approximation 3: While the typical velocity scale of surface winds is |va| ~ 10 m s−1, icebergs tend to travel at speeds |vi| ~ 0.1 m s−1 (Robe 1980), such that typically |vi|/|va| ~ 0.01.
Approximation 4: Previous studies have either modeled wave radiation explicitly (e.g., Bigg et al. 1996, 1997; Death et al. 2006; Jongma et al. 2009) or included it in the wind drag term where it alters the drag coefficient (Smith 1993; Keghouche et al. 2009). However, a number of studies have found that the contribution of wave radiation is typically small compared to the air and water drag terms (Bigg et al. 1997; Gladstone et al. 2001). Hence, for simplicity, we neglect the wave radiation term. We also neglect the sea ice drag term, which allows us to model the motion of icebergs outside of the central sea ice pack. Icebergs can spend long periods of time locked-in by sea ice in regions with heavy sea ice, such as north of Greenland and in the Weddell Sea (Lichey and Hellmer 2001; Schodlok et al. 2006). Neglecting sea ice drag also omits the seasonality in iceberg drift and decay associated with the large seasonal cycle of the polar sea ice cover. The drift model developed here will hence have limited applicability for regions and periods with high sea ice concentration.
Approximation 5: Vertical variations of the currents and winds near the surface are neglected for simplicity, as in most previous studies (e.g., Bigg et al. 1997; Jongma et al. 2009; Martin and Adcroft 2010). This approximation limits the applicability of the model for predicting the trajectories of icebergs in regions with large vertical shear in the ocean currents or winds. We note that some iceberg models include an explicit representation of this vertical structure (e.g., Kubat et al. 2005; Turnbull et al. 2015; Merino et al. 2016; FitzMaurice et al. 2016).
Bird’s eye view schematic of the horizontal surface air velocity (green arrow) and water velocity (blue arrow) driving the iceberg motion (red arrow). The wind-driven component of the iceberg velocity 
Citation: Journal of Physical Oceanography 47, 7; 10.1175/JPO-D-16-0262.1
The pressure gradient force is defined as 
b. Analytical solution
From Eq. (6) we see that the wind drives icebergs at an angle 
c. Iceberg decay model
Iceberg motion is affected by the decay of the iceberg, since the coefficients α and β depend on the iceberg length S. Iceberg decay is modeled here using a modified version of the thermodynamic decay model developed by Bigg et al. (1997). This model accounts for three melt processes: (i) wind-driven wave erosion, (ii) turbulent basal melt, and (iii) sidewall erosion from buoyant convection. The main difference between the present model and that of Bigg et al. (1997) is that we use a different iceberg rollover criterion because the rollover criterion used by Bigg et al. (1997), which was adopted from Weeks and Mellor (1978), has been found to contain several errors Wagner et al. 2017, manuscript submitted to Ocean Modell.). The decay model is described in more detail in the appendix.
3. Model validation using ECCO2 output
The model is forced using the NASA ECCO2 product, a global ocean state estimate of the period 1992–2012 that is obtained using satellite and in situ data in concert with an ocean general circulation model (Menemenlis et al. 2008). The surface wind forcing in ECCO2 is taken from the Japanese 25-year Reanalysis Project (JRA-25; Onogi et al. 2007). For simplicity we idealize the icebergs to be noninteracting passive Lagrangian particles. This allows the efficient computation of large numbers of iceberg trajectories.
We consider two scenarios:
- small icebergs (L < 1.5 km) released from three main outlet glaciers in Greenland; and
- large tabular icebergs (L > 15 km) released off the coast of the Antarctic Peninsula.
The upper size limit of the Greenland icebergs was chosen to correspond to that used by Bigg et al. (1997) and subsequent studies. The size limit of the Antarctic icebergs is similar to the lower bound of icebergs tracked by the National Ice Service [which is 10 nautical miles (n mi; 1 n mi = 1.852 km)], and it is 10 times the size of the largest Arctic icebergs simulated in scenario a.
The ECCO2 dataset, which is used for both scenarios, consists of output fields averaged over 3-day intervals. We compute iceberg trajectories using Eq. (6) coupled to the decay model described in section 2c. This requires as input the ECCO2 surface water velocities and JRA-25 surface wind velocities as well as the sea surface temperature. The wind velocities, which are given on a 1° × 1° horizontal grid, are interpolated onto the 0.25° × 0.25° grid of the ocean fields.
We integrate the iceberg trajectories using a forward Euler time-stepping scheme with 1-day temporal resolution. The 3-day ECCO2 and JRA-25 fields are linearly interpolated from time interval centers onto a 1-day time resolution to match the time stepping. Iceberg velocities are computed at each time step using currents and winds from the spatial grid box that is centered nearest to the iceberg location.
Grounding events are not resolved explicitly; instead we set iceberg velocities to zero when icebergs come within one grid box of land until the surface circulation moves them away from the coast, akin to schemes used previously (e.g., Wiersma and Jongma 2010).
a. Arctic iceberg simulations
We release Arctic icebergs near the outlets of three main Greenland ice discharge glaciers: Helheim, Kangerlussuaq, and Jakobshavn (Fig. 2). For each iceberg, the release location is chosen randomly from the centers of a 4 × 4 grid of ECCO2 grid points near each fjord outlet. Icebergs are released at random times within the first 3 years (1992–94) of the ECCO2 dataset and advected until they melt completely. This time period is sufficiently long to produce a relatively converged freshwater distribution. We consider 10 initial iceberg sizes (Table S1), with dimensions ranging from 100 × 67 × 67 to 1500 × 1000 × 300 m3, similar to the classification of Bigg et al. (1997). A total of 500 icebergs are released for each size class from each glacier.
Arctic iceberg trajectories. (a) Drift trajectories for 50 icebergs released at each of three Greenland outlet glaciers: Kangerlussuaq (K; green), Helheim (H; red), and Jakobshavn (J; blue). Iceberg release locations are indicated by yellow stars. The dashed line indicates the observed “normal” range of icebergs as given by the International Ice Patrol (2009). (b) Freshwater flux [centimeters per 250 km3 ice volume released, which corresponds to approximately half the total yearly ice volume released from calving around all of Greenland (Enderlin et al. 2014)].
Citation: Journal of Physical Oceanography 47, 7; 10.1175/JPO-D-16-0262.1
Figure 2a shows 50 iceberg trajectories for each of the three glaciers. Size classes and release dates for these trajectories are chosen at random. We find that most icebergs from Kangerlussuaq Glacier drift close to the coast southward in the East Greenland Current, with frequent groundings. Small icebergs from Helheim Glacier are commonly deflected eastward by winds once they are subject to the strong westerlies near the southern tip of Greenland. The larger Helheim icebergs remain more commonly in the coastal current, drift around the tip of Greenland, and subsequently make their way north (sizes are not indicated in Fig. 2). Icebergs from Jakobshavn, on the other hand, quickly make their way across Baffin Bay and follow the “Iceberg Alley” south along the Labrador coast toward Newfoundland. They mostly melt completely by the time they reach the Grand Banks (approximately 45°N, 50°W). However, some simulated icebergs survive substantially longer and drift beyond the commonly observed iceberg boundary as estimated by the International Ice Patrol (2009), which is indicated by a dashed line in Fig. 2. This may be partially due to the decay model not accounting for the breakup of large icebergs, which is likely a dominant driver of large iceberg deterioration (e.g., Wagner et al. 2014; Tournadre et al. 2016), but for which an adequate model representation is still missing. Furthermore, some studies have added a temperature dependence to the parameterization of wind-driven wave erosion (e.g., Gladstone et al. 2001; Martin and Adcroft 2010), which causes faster decay in warmer waters, although others have not (e.g., Jongma et al. 2009; Bügelmayer et al. 2015a).
Figure 2b shows the simulated freshwater input distribution due to iceberg melt. This is computed by averaging over all icebergs within each size class and then weighting each iceberg size class field according to the lognormal distribution used in Bigg et al. (1997, their Table 1) and subsequent studies. Considering the substantial simplifications of the present simulations, the resulting meltwater distribution is in fairly good agreement with those of Martin and Adcroft (2010, their Fig. 2b) and Marsh et al. (2015, their Fig. 3), although the eastward transport of freshwater from east Greenland icebergs is somewhat exaggerated here.
b. Antarctic iceberg simulations
We qualitatively validate the model against Antarctic tabular icebergs using the observed trajectories of large icebergs as catalogued in the Antarctic Iceberg Tracking Database. Figure 3a shows QuikSCAT/SeaWinds scatterometer observations (Ballantyne and Long 2002), the full dataset of which tracked 352 icebergs, mostly of diameter > 10 nm, over the years 1999–2009.
Antarctic iceberg trajectories. (a) Observed iceberg trajectories from the Antarctic Iceberg Tracking Database, which is derived from QuikSCAT/SeaWinds scatterometer data (Ballantyne and Long 2002). Here, only the trajectories of icebergs with inception points around the Antarctic Peninsula are included. (b) Simulated trajectories during 1 year for 200 large icebergs (L = 15–20 km), computed using Eq. (6) with ECCO2 input fields. Icebergs are released from random locations within the dashed polygon region.
Citation: Journal of Physical Oceanography 47, 7; 10.1175/JPO-D-16-0262.1
Using the model described in Eq. (6), we release 200 large icebergs of lengths between 15 and 20 km in the ECCO2 fields off the east coast of the Antarctic Peninsula and locations in the Weddell Sea and track these icebergs for 1 year (Fig. 3b).
Note that we ignore the drag and reduced melting effects of sea ice in these simulations (approximation 4). The large uncertainties inherent in this comparison should be emphasized. Iceberg release locations, iceberg dimensions, and drift periods are among the unconstrained factors that make a direct comparison between model output and satellite observations difficult. Considering these uncertainties, the simulated trajectories show fairly good agreement with observations, accurately capturing the general drift pattern along the east coast of the Antarctic Peninsula and into the Antarctic Circumpolar Current. Furthermore, the corresponding simulated meltwater distribution (not shown) compares reasonably well to that derived from observations by Silva et al. (2006). It should be noted by caveat that the iceberg trajectories shown in Fig. 3b do not terminate with the disappearance of the icebergs but rather with the end of the 1-yr time window. Large tabular icebergs in these simulations would survive considerably longer, which is expected to be mainly an artifact of the model not accounting for breakup processes as well as the omission of an SST dependence in the wave erosion term, as discussed above.
4. The role of winds and currents
Here, we address the roles that the three terms in Eq. (6) play in determining iceberg trajectories. Specifically, we focus on the following questions:
- Are icebergs primarily driven by winds or by currents? To what degree does this depend on iceberg size and on the magnitude of the air and water velocities?
- What determines how much the wind drives icebergs in along-wind versus across-wind directions? In other words, how does the angle θ depend on the surface velocities and iceberg size?
To address these questions, we consider first the analytical solution (section 4a) and subsequently the iceberg trajectories and velocities that were numerically computed using ECCO2 (section 4b). Note that other factors, including iceberg shape and tangentially acting drag (Crepon et al. 1988), are also expected to affect the answers to the questions above, but these factors are beyond the scope of the present study.
a. Winds and currents in the analytical solution
1) Direction of wind-driven motion
(a) Coefficients α and β, as well as 
Citation: Journal of Physical Oceanography 47, 7; 10.1175/JPO-D-16-0262.1
The direction in which the wind drives icebergs θ as a function of the dimensionless wind speed Λ (solid). The dashed curves show the asymptotic approximations for small and large Λ, which are given in Eq. (11). The point where these asymptotics meet, 
Citation: Journal of Physical Oceanography 47, 7; 10.1175/JPO-D-16-0262.1
(i) 
This is in close agreement with previous observational estimates, which have found the empirical rule of thumb that icebergs typically drift at approximately 2% of the wind velocity relative to the water. Such empirical estimates include γ = 0.018 ± 0.7 (Garrett et al. 1985) and γ = 0.017 (Smith 1993), translating the previous results into the formalism of the present study. Here, by contrast, γ = 0.018 is derived analytically from physical first principles.
(ii) 
In this limit, the icebergs move in the direction perpendicular to the wind velocity, relative to the ocean current. The reason for this is contained in Eq. (6): the across-wind term is dominant for large icebergs since in this limit, the Coriolis force, which gives rise to the across-wind component and scales with mass M, is large relative to the drag forces, which scale with cross-sectional area.
2) Surface winds versus currents
In the limit of small Λ, both α and β decrease when Λ decreases. This implies that the total wind contribution to iceberg motion drops off for low winds or large icebergs. This is a further consequence of the Coriolis term growing large, since it depends only on 
b. Wind, current, and iceberg velocities from simulations with ECCO2
1) Arctic iceberg simulations
A single 3-day snapshot (1–3 Jan 1993) of the ECCO2 velocities is shown in Fig. 6. Also shown is the wind-driven component of iceberg velocities 
Velocities in the North Atlantic from ECCO2 output averaged over the 3-day period 1–3 Jan 1993. (a) Surface wind. (b) Surface ocean current. (c) Wind-driven component of the iceberg velocity field for small (class 1) icebergs. (d) As in (c), but for large (class 10) icebergs. (e) Iceberg velocity field for small icebergs, equal to the sum of (b) and (c). (f) Iceberg velocity field for large icebergs, equal to the sum of (b) and (d). In all panels, arrows indicate the velocity field, and shading indicates the associated speed. Note that the wind field features the signature of a winter storm.
Citation: Journal of Physical Oceanography 47, 7; 10.1175/JPO-D-16-0262.1
In agreement with the discussion above, we find that (i) the wind-driven component is stronger for smaller icebergs and (ii) that its direction is closely aligned with that of va for small icebergs but not for large icebergs. Figure 6 shows that the iceberg velocity field of the small size class is largely determined by the wind field, while the larger icebergs move primarily with the ocean.
The relative importance of each of the three terms in Eq. (6) for the motion of Arctic icebergs can be quantified using the mean current and wind speeds experienced by the icebergs along their trajectory. These wind speeds are shown in Fig. S3. For a given glacier, the wind speed is found to be approximately constant, with the two east Greenland glaciers (Helheim and Kangerlussuaq) experiencing higher wind speeds (≃6.5 m s−1) and Jakobshavn having lower wind speeds (≃4 m s−1).
First, we consider the relative importance of the two wind-forced terms in Eq. (6). Inserting the average value of 
Next, we consider the relative importance of the current- and wind-driven terms of Eq. (6). The right vertical axis of Fig. 4 shows the coefficient R [Eq. (14)] using the mean simulated velocity ratio 
2) Antarctic iceberg simulations
The results above suggest that, for large tabular icebergs as observed in Antarctica, the wind drag can be neglected to a good approximation and Eq. (6) reduces to the relation 
Simulated Antarctic iceberg trajectories. Shown are simulations of the full analytical model (red), iceberg motion that follows the ocean currents (blue, overlaying most of the red trajectories), and iceberg motion that follows the 2% rule (green). Icebergs are released from random locations within the region indicated by the dashed polygon.
Citation: Journal of Physical Oceanography 47, 7; 10.1175/JPO-D-16-0262.1
5. Conclusions
We have presented an idealized iceberg drift model with an analytical solution for the velocity of icebergs as a function of the ocean current and wind. This solution facilitates
- an improved understanding of the underlying mechanisms determining iceberg drift; and
- computationally inexpensive simulations of large numbers of iceberg trajectories.
In these simulations, the wind-driven iceberg motion dominates the ocean-driven motion approximately threefold for small icebergs (L < 200 m), and it becomes less than 10% of the ocean-driven motion for large icebergs with L > 12 km.
We use the model to demonstrate that in the limit of small icebergs or strong winds, icebergs drift at ~2% of the surface winds relative to the water. This asymptotic result of the analytical model agrees with an empirical rule of thumb used in previous studies. However, the results highlight the limitations of this 2% rule: in the limit of large icebergs or weak winds, the wind contribution to driving the icebergs becomes negligible, and the icebergs drift with the surface ocean current. By considering trajectories computed from ECCO2 output fields, we find that these two limits approximately correspond to typical (small) Arctic icebergs and typical (large) Antarctic icebergs, respectively.
The dependence of the drift velocity regime on iceberg size can be explained through the relative importance of the drag terms compared with the Coriolis term (which includes the pressure gradient force in this representation). Since the drag terms scale with surface area 
We are grateful to Jeff Severinghaus and Ralph Keeling for helpful discussions during the development of this work. This work was supported by National Science Foundation Grant OCE-1357078. MATLAB code for the iceberg drift and decay model presented here is available online (at http://eisenman.ucsd.edu/code).
APPENDIX
Iceberg Decay Representation
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