Glaciology - Calving

Calving comprises the mechanisms by which icebergs and fragments of ice break off glaciers terminating in the sea (tidewater glaciers) or in lakes (freshwater glaciers). Calving in tidewater is more common and the flowing speed of which is generally faster than the freshwater glaciers.

1. Water-depth law

Brown et al., (1982) noted a strong (empirical) correlation between calving rates and water depth based on the study of 12 Alaskan tidewater glaciers, in the form of $U_{\mathrm{calv}} = a + bD_{\mathrm{water}}$ (with $b \sim 17\ \mathrm{yr}^{-1}$ in the original paper), where $D_{\mathrm{water}}$ is the width-averaged water depth and $a$ and $b$ are empirically determined coefficients.

This law is further developed by Pelto and Warren (1991) using a larger dataset ($n = 22$) from Alaska, Greenland and Svalbard, for which the empirical constant $a$ and $b$ was constrained to be $70$ and $8.33$ respectively. It was also applied to freshwater glaciers, while a much smaller constant sets were derived ($a = 17.4$, $b = 2.3$, based on 21 freshwater glaciers studied by Warren and Kirbride, 2003). This difference between tidewater and freshwater glacier can be explained by 1. the water depth at tidewater glacier terminus is often greater than that of the freshwater glacier; 2. greater density of seawater than fresh water means that tidewater glaciers will flow more rapidly for the same depth of water at the terminus (Benn et al., 2007). A global data set compiled by Haresign (2004) shows that relationships between calving rate and water depth vary between regions, and Van der Veen, 1996, Van der Veen, 2002 has shown that it can also change through time for a single glacier.

Despite proved useful in multiple cases, it is notable that this law relies on an empirical rather than physical relationship between water depth and calving rate. Indeed, water depth is correlated with the glacier velocity directly: For a constant ice thickness (and therefore constant $P_{\mathrm{ice}}$), greater water depth leads to greater basal water pressure ($P_{\mathrm{water}} = \rho_{\mathrm{water}}gD_{\mathrm{water}}$), which reduces effective pressure ($N = P_{\mathrm{ice}} - P_{\mathrm{water}}$). Since sliding velocity depends inversely on effective pressure ($u_b = k\tau_b^3N^{-1}$), deeper water ultimately promotes faster glacier flow. However, this does not imply a direct causation between water depth and calving rate: Interpreting ice velocity as calving rate requires assuming that the frontal position is stationary ($dL/dt \approx 0$), an assumption that often fails during dynamic glacier retreat (van der Veen, 1996).

2. Buoyancy

Van der Veen (1996)’s height-above-buoyancy law focused on factors controlling the position of a calving terminus rather than calving rate. By examining the data series collected by USGS on Columbia Glacier, van der Veen noted that the calving front tends to be located where the ice is approximately 50 m greater than the flotation thickness. If the glacier thinned (due to either melting or longitudinal extension), the calving front tended to retreat to a position where this height-above-buoyancy calving criterion was again satisfied. A more general form of the buoyancy criterion was introduced by Vieli et al. (2000), replacing the fixed value of 50 m with a fraction of the flotation thickness. As revealed by Choi et al. (2018) using the Ice Sheet System Model, this law reproduced the magnitude of observed retreat along 53% of flowlines across nine Greenland tidewater glaciers, but failed to capture the timing or continuity of retreat. An important shortcoming of this calving criterion is that it cannot explain why some glaciers can thin past the flotation and form ice shelves before calving (the height above buoyancy of ice shelves is zero rather than 50 m). This limits its application in predicting the behaviour of Antarctic glaciers or ice streams that flow into ice shelves, and the evolution of marine ice sheets from floating ice. (Benn et al., 2007)

3. Crevasses depth

From inner glacier to the outlet, basal drag decreases with high water pressure, and lateral drag also decreases with thinning and widening of fjord walls. As a result, the glacier accelerates down-fjord ($\partial u/\partial x > 0$). This velocity structure results in a longitudinal strain rate ($\dot{\varepsilon} = \partial u/\partial x$). The depth of the crevasses ($d$) equals to the point when the crevasse opening rate caused by this longitudinal strain rate is balanced by the creep closure rate due to overburden pressure ($\dot{\varepsilon} = A(\frac{1}{2}\rho_i g d)^n$). Therefore, the depth can be determined by $d_{\mathrm{dry}} = \frac{2}{\rho_i g}\left(\frac{\dot{\varepsilon}}{A}\right)^{1/n}$, and the depth of water-filled crevasse equals: $d_{\mathrm{wet}} = \frac{2}{\rho_i g}\left[\left(\frac{\dot{\varepsilon}}{A}\right)^{1/n} + \rho_w g d_w\right]$.

Based on this, Benn et al. (2007) determined the calving front position as where $d$ reaches the water line. This law worked well in explaining the behaviour of Columbia glaciers. Nick et al. (2010) further developed this by taking bottom crevasses into consideration, suggesting that calving occurs where combined surface and basal crevasses span the full thickness ($d_{\mathrm{surf}} + d_{\mathrm{basal}} = H_{\mathrm{ice}}$, $d_{\mathrm{basal}} = \frac{2}{\rho_i g}\left[\left(\frac{\dot{\varepsilon}}{A}\right)^{1/n} + P_w\right]$). This law was applied to the seasonal flotation tongues of Helheim, and successfully produced $\pm 10%$ of the frontal position.

This law has been successfully applied to Sermeq Kujalleq (Store Glacier), where both Elmer/Ice and HiDEM models reproduced its observed stable terminus as a stress-governed critical point (Benn et al., 2023). Among the six calving laws tested by Amaral et al., 2020, the crevasse depth model provides the best balance of high accuracy and low sensitivity to imperfect parameter calibration. Therefore they argue that while the crevasse depth model appears unlikely to capture the true controls on crevasse penetration, numerically, it reproduces observed terminus dynamics with high fidelity.

Although the classical calving laws treat iceberg detachment as an intrinsic response to the glacier’s own geometry and stress state, recent studies highlight the importance of external controls such as the back-pressure supplied by iceberg mélange and land-fast sea ice. Mélange refers to the dense, jumbled mass of calved icebergs, smaller debris, and slush that crowds the fjord in front of tidewater glaciers. Land-fast sea ice, in turn, is rigid sea ice anchored to coastal features that can freeze around and stabilize the mélange. Together, they act as a floating buttress, resisting glacier motion and suppressing calving. Recent 3D modelling at Thwaites Glacier (Crawford et al., 2024) shows that the same internal stress field can produce either large calving events or front stability, depending solely on mélange strength. Similarly, at Store Glacier, Benn et al. (2023) find that fracture propagation halts when mélange back-pressure is high. These findings challenge the adequacy of stress-based or position-based laws alone, suggesting that calving cannot be fully captured without a dynamic representation of the downstream force balance.

4. First principle

The model treat glacier ice as a granular material made of interacting boulders of ice that are bonded together under local stress, such as frictional forces, damped elastic collisions and elastic bonds. When all bonds are intact, the model represents a crude discretization of an elastic body. Bonds would break when either the tensile or shear stress exceeds its respective yield strength. As bonds fail, the ice becomes increasingly fragmented and the behaviour begins to resemble a granular material, an analogy previously noted for the Columbia Glacier based on field observations (Post, 1997).

First-principles (FP) calving laws outperform traditional geometric rules by integrating multiple stressors into a single physical framework. While CD law mainly focuses on extensional forces, FP also accounts for mélange buttressing, lateral pinning, undercutting, and wave-induced bending. FP laws also require less empirical tuning. Slater and Wagner (2024) reproduced calving thresholds across 12 Greenland glaciers using a single physically derived criterion, whereas traditional laws needed site-specific adjustment. This parsimony enables FP models to apply across diverse settings, from tidewater glaciers to Antarctic shelves (Bassis and Jacobs, 2013). Together, these strengths mark FP laws as a necessary step beyond purely geometric approaches.