## Abstract

The fate of icebergs in the polar oceans plays an important role in Earth’s climate system, yet a detailed understanding of iceberg dynamics has remained elusive. Here, the central physical processes that determine iceberg motion are investigated. This is done through the development and analysis of an idealized model of iceberg drift. The model is forced with high-resolution surface velocity and temperature data from an observational state estimate. It retains much of the most salient physics, while remaining sufficiently simple to allow insight into the details of how icebergs drift. An analytical solution of the model is derived, which highlights how iceberg drift patterns depend on iceberg size, ocean current velocity, and wind velocity. A long-standing rule of thumb for Arctic icebergs estimates their drift velocity to be 2% of the wind velocity relative to the ocean current. Here, this relationship is derived from first principles, and it is shown that the relationship holds in the limit of small icebergs or strong winds, which applies for typical Arctic icebergs. For the opposite limit of large icebergs (length > 12 km) or weak winds, which applies for typical Antarctic tabular icebergs, it is shown that this relationship is not applicable and icebergs move with the ocean current, unaffected by the wind. The latter regime is confirmed through comparisons with observed iceberg trajectories near the Antarctic Peninsula.

## 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

We develop an iceberg drift model that is adapted from the canonical family of drift models introduced by Bigg et al. (1997) and used by Gladstone et al. (2001), Martin and Adcroft (2010), Marsh et al. (2015), and other studies. These models mostly differ only in minor details, and the momentum equation is typically written in the form

where *M* is the mass of the iceberg, **v**_{i} is the iceberg velocity, and *f* is the Coriolis parameter, which varies with latitude. The terms on the right-hand side represent Coriolis force , pressure gradient force *F*_{p}, water drag *F*_{w}, air drag *F*_{a}, wave radiation force *F*_{r}, and sea ice drag *F*_{i}.

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 is neglected in Eq. (1) based on the expectation that it is much smaller than other terms in the momentum balance (e.g., Crepon et al. 1988).

The pressure gradient force is approximated from the ocean velocity by assuming geostrophic ocean currents.

The iceberg speed |

**v**_{i}| is taken to be much smaller than the surface wind speed |**v**_{a}|.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.

The justification for each approximation is given below.

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 estimated from the simulated velocities is typically less than 10% of the air drag term. For very large icebergs, such as those in the Antarctic, this approximation begins to break down; for lengths of about 20 km, the magnitude of the acceleration term is typically similar to the air drag term. However, when the model is solved numerically both with and without the inclusion of the acceleration term, we find that approximation 1 ultimately has relatively little influence on typical iceberg trajectories for both small and large icebergs (supplemental material Figs. S1 and S2).

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 |**v**_{a}| ~ 10 m s^{−1}, icebergs tend to travel at speeds |**v**_{i}| ~ 0.1 m s^{−1} (Robe 1980), such that typically |**v**_{i}|/|**v**_{a}| ~ 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).

The water and air drag terms are given by

where and . Here, *C*_{w} and *C*_{a} are bulk drag coefficients of water and air, *ρ*_{w} and *ρ*_{a} are water and air densities, and *A*_{w} and *A*_{a} are the cross-sectional areas on which the water and air velocities act, respectively. We take the icebergs to be cuboids of height *H*, length *L*, and width *W* (Fig. 1), and we only consider drag exerted on the vertical surfaces of the icebergs. Previous studies have typically assumed a fixed orientation of the iceberg relative to the wind and current (e.g., Bigg et al. 1997). We instead adopt the approximation that icebergs are oriented at a random angle *ϕ* relative to the wind and current. In this case, the long-term mean horizontal length of the vertical working surface area for both drag terms is such that

The pressure gradient force is defined as where ∇*P* is the horizontal pressure gradient due primarily to sea surface slope. Approximating that the pressure gradient force acting on the icebergs arises only from the geostrophic component of ocean flow (approximation 2) allows us to write the pressure force as The first two terms in Eq. (1) can then be combined.

Making use of approximations 1–5 leads to

where we have used that (approximation 3) and defined . Here, and are approximated by the surface velocities (approximation 5).

The importance of the terms in Eq. (3) can be quantified by introducing dimensionless quantities

which describe the magnitudes of the water and air drag terms relative to the Coriolis term, respectively. Note that these quantities are both analogous to the Ekman number for fluid flows.

Equation (3) can then be simplified to

with unit vectors and .

### b. Analytical solution

Equation (5) contains **v**_{i} in the quantities , Λ_{w}, and Λ_{a}. Note that this vector equation can be written as a set of two coupled scalar algebraic equations, which are nonlinear in the components of **v**_{i}. Equation (5) can be solved analytically for **v**_{i}, giving as a solution

Here, *γ* is a dimensionless parameter, which describes the relative importance of water drag versus air drag:

The dimensionless parameters *α* and *β* in Eq. (6) are

where

with being the harmonic mean horizontal length of the iceberg. Since the Coriolis parameter does not vary substantially along the simulated iceberg trajectories, the variable Λ can be approximately interpreted as the ratio of wind speed to iceberg size scaled by several constants.

From Eq. (6) we see that the wind drives icebergs at an angle . How *θ* varies with Λ will be discussed further in section 4. Note that the iceberg velocity is independent of iceberg height *H* since the drag terms and the Coriolis term both scale linearly with *H*, and only the ratios of these terms feature in Eq. (5).

### 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; andlarge 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 m^{3}, similar to the classification of Bigg et al. (1997). A total of 500 icebergs are released for each size class from each glacier.

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.

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

The Coriolis term in Eq. (3) causes some of the wind drag to project onto the direction perpendicular to the wind velocity, giving rise to the cross-product term in Eq. (6), which is somewhat analogous to Ekman transport. The importance of this term relative to the along-wind term can be assessed by considering the coefficients *α* and *β* (Fig. 4). We compute first-order series expansions of *α* and *β* for small and large Λ:

These asymptotics are included in Fig. 4. Note that since is a measure of wind speed relative to iceberg size, these limits can arise in two different ways: Λ ≫ 1 can apply to strong winds or alternatively to small icebergs. Similarly, Λ ≪ 1 applies to weak winds or large icebergs.

The angle *θ*, at which the surface winds drive the iceberg (relative to the direction of the wind), is indicated schematically in Fig. 1, and its dependence on Λ is shown in Fig. 5. From Eq. (10), we obtain the asymptotic limits for the ratio *α*/*β* = tan *θ*:

Both the angle *θ* and the ratio *α*/*β* decrease monotonically with increasing Λ: the stronger the wind blows, the more it drives the iceberg in the direction of the wind. The two asymptotic regimes meet when *α* = *β*, in which case Λ = 1 and *θ* = 45°. Figure 5 shows that these asymptotic solutions provide a reasonably close approximation to the exact solution in the full range of Λ. For strong winds, *α*/*β* ≪ 1 (or equivalently Λ ≫ 1), and we find that *θ* → 0; that is, the wind drives icebergs in the direction that it blows. For weak winds, on the other hand, *α*/*β* ≪ 1, and the icebergs move at *θ* ≃ 90° to the wind. We conclude that icebergs drift primarily in the along-wind direction relative to the ocean currents when Λ > 1 and across wind when Λ < 1. Next, we consider these limits in more detail.

##### (i) (strong winds, small icebergs)

From Eq. (10), we find that for strong winds or small icebergs the across-wind component of iceberg drift approaches zero, since *α* → 0, and the along-wind component approaches a constant value, with *β* → 1 (Fig. 4). Equation (6) therefore reduces to

Using density values *ρ*_{a} = 1.2 kg m^{−3}, *ρ*_{w} = 1027 kg m^{−3}, and *ρ*_{i} = 850 kg m^{−3} and taking bulk coefficients *C*_{a} = 0.9 and *C*_{w} = 1.3 (Bigg et al. 1997), we find from Eq. (7) that *γ* = 0.018.

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.

These results not only shed light on the origin of this empirical rule of thumb, but they also provide constraints on its validity; icebergs satisfy the 2% rule only when Λ ≫ 1. We can interpret this dimensionally in terms of iceberg size. Assuming a constant midlatitude Coriolis parameter *f* = 10^{−4} s^{−1} and a typical aspect ratio *L*/*W* = 1.5 (Bigg et al. 1997), Eq. (9) becomes

where *c* = 130 s. This means that the limit Λ ≫ 1 is satisfied when

We conclude that the 2% rule is a good approximation for iceberg drift *only* when the length of the iceberg is sufficiently small or the surface wind speed is sufficiently large or the length of the iceberg is sufficiently small. In section 4b, we will consider how this relates to the simulated icebergs.

##### (ii) (weak winds, large icebergs)

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 (and not ). Large tabular icebergs (as occur mostly in Antarctica) can therefore be approximated to be driven by the ocean currents alone, as we confirm in section 4b below.

The relative importance of wind and ocean currents can be quantified in terms of the ratio of the associated speeds. Previous studies have stated either that the water drag dominates (Matsumoto 1996) or that water and air drags are of similar magnitude (Gladstone et al. 2001). Here, we establish this relative importance quantitatively. We define the ratio of the magnitudes of the wind-driven and ocean-driven velocity components in Eq. (6) as

We will see below that the ratio is approximately constant for icebergs from a given region. This implies that small icebergs (with ) move predominantly with the wind (i.e., *R* > 1) when . For very large icebergs, the water velocities (i.e., the denominator in *R*) will always be dominant, since in that case (Fig. 4). In the following, we compare these limits to the actual velocity ratios experienced by icebergs simulated with the ECCO2 surface conditions.

### 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 and its magnitude for iceberg size classes 1 and 10 (second row). The third row of Fig. 6 represents the corresponding iceberg velocity fields.

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 **v**_{a} 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 m s^{−1} for all Arctic icebergs into Eq. (13) gives *L* = 770 m/Λ. This length scale, indicated as the top horizontal axis of Fig. 4, acts as a measure of the relative strength of the across-wind and along-wind terms in Eq. (6). The critical length corresponding to Λ = 1, m, separates the regimes where *α* and *β* dominate. This means that Arctic icebergs will be driven mostly along wind if and mostly across wind if . Alternatively, an iceberg of size *L* = 770 m will move primarily across wind for m s^{−1} and along wind for m s^{−1}, as illustrated in the top horizontal axis of Fig. 5.

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 . This means that the role of wind drag dominates that of water drag by a factor of *R* ≈ 3 for small icebergs and that wind drag becomes negligible compared to water drag (*R* < 0.1) for icebergs larger than *L* ≈ 12 km.

#### 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 , that is, large icebergs move with the surface ocean current. To demonstrate the accuracy of this approximation, we perform two more sets of simulations using the same initial conditions as those for the Antarctic simulations discussed in section 3b (above). First, we approximate icebergs to move at the water velocity (blue trajectories in Fig. 7). Next, we instead set icebergs to move according to the 2% rule (green trajectories in Fig. 7). Whereas the icebergs with drift in close agreement with those using the full solution (red trajectories in Fig. 7), the icebergs following the 2% rule show a substantially different drift pattern because they are influenced by the strong prevailing winds around the Antarctic Peninsula. This supports our conclusion that large icebergs move approximately with the ocean currents. It should be noted that the empirical 2% rule was originally introduced based on Northern Hemisphere iceberg observations (e.g., Garrett et al. 1985; Smith and Donaldson 1987; Smith 1993), and it has typically been applied around Greenland rather than in the Antarctic.

## 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 , and the Coriolis term scales with volume (assuming similar horizontal dimensions, ), the Coriolis term dominates the momentum balance [Eq. (3)] in the limit of large icebergs. This implies that : icebergs move with the water. On the other hand, the drag terms dominate the momentum balance [Eq. (3)] in the limit of small icebergs. In this case, the solution is : icebergs move at 2% of the wind velocity relative to the water.

## Acknowledgments

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

Here, further details are given regarding the iceberg decay representation, which is summarized in section 2c. This is based on the thermodynamic decay model of Bigg et al. (1997), using a representation adapted from Martin and Adcroft (2010). The three dominant melt processes, (i) wind-driven wave erosion *M*_{e}, (ii) turbulent basal melt *M*_{b}, and (iii) thermal sidewall erosion from buoyant convection *M*_{υ}, are represented as follows:

where is the sea state, m^{1/2} s^{−1/2}, , m s^{−1} °C^{−1}, m s^{−1} °C^{−2}, and m^{−2/5} s^{−1/5} °C^{−1}. Here, is the sea surface temperature, and is the temperature of the ice, taken to be fixed at C (El-Tahan et al. 1987). Other processes, such as surface melt, have been found to be small compared to these terms (Savage 2001) and are neglected here. The iceberg dimensions evolve as and . Finally, we impose that an iceberg capsizes when its width-to-height ratio falls below a critical value (MacAyeal et al. 2003; Wagner et al. 2017, manuscript submitted to *Ocean Modell.*), where

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## Footnotes

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JPO-D-16-0262.s1.

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