Calving

Yelmo has two calving paths. By default (ycalv.use_lsf = True), the calving front is a level set that moves with the ice velocity plus a calving velocity. With use_lsf = False, calving is a mass-balance rate applied to front cells. The allowed calving laws depend on the path. Frontal melt is a separate term (see Mass conservation).

Default: level-set front with von Mises calving

The defaults (input/yelmo_defaults.nml) use a level-set front (ycalv.use_lsf = True), the von Mises law vm-m16 for floating and marine-grounded fronts (calv_flt_method = calv_grnd_method = "vm-m16") and the subgrid front (ytopo.front_subgrid = "marine"). This path is computed in calc_ytopo_calving_lsf (src/yelmo_topography.f90).

Calving rate

vm-m16 follows Morlighem et al. (2016, Eq. 4). On each velocity face the calving rate is

c = u \, \max\left(0, \frac{\tau_1}{\tau_{\rm ice}}\right)

directed against the ice velocity u, where \tau_1 is the first principal value of the depth-averaged deviatoric stress (mean of the two cells, or the ice cell’s value at a face between ice and an ice-free cell) and \tau_{\rm ice} is the ice strength: ycalv.tau_ice_flt (default 250 kPa) on floating faces (f_grnd_ac = 0) and ycalv.tau_ice_grnd (default 1 MPa) on the other marine faces. The floating law is used on faces with a grounded fraction of zero, the grounded law on the other faces. On faces between two land cells (bed at or above sea level), c = -u, so the level set does not move there. The front is stationary where \tau_1 = \tau_{\rm ice}.

The stresses come from the last velocity solution. A cell that received ice since then takes the mean of its edge neighbours that were part of that solution (fill_stress_new_ice).

Other laws on this path:

• "zero" / "none": no calving;
• "equil": the front is held in place (c = -u);
• "threshold": the front retreats where the ice is thinner than Hc_ref_flt (floating) or Hc_ref_grnd (grounded);
• "exp1"–"exp5" (floating): the CalvingMIP experiments;
• "ismip7" (grounded): frontal retreat from the frontal melt of Rignot et al. (2016), with the subglacial discharge bnd%Qd and the thermal forcing bnd%T_shlf minus the seawater freezing point at the water depth (Jenkins, 1991; ISMIP7 protocol). It does not use bnd%tf_shlf.

Level-set front

The level set \varphi is negative in the ice domain and positive in the ocean; its zero contour is the calving front (src/physics/calving/lsf_module.f90). It is advected with the front velocity w = u + c (with c against u),

\frac{\partial \varphi}{\partial t} + w \cdot \nabla \varphi = 0,

with first-order upwinding on the faces and explicit sub-steps (LSFupdate). w is extended into the ocean from the faces next to ice, and \varphi is kept in [-1, 1]. Cells with the bed at or above sea level are set to \varphi = -1. With ycalv.lsf_method = "snap" (default), cells more than two cells from the front are reset to \pm 1 (with dt_lsf > 0, the level set is also reset to \pm 1 from the ice mask every dt_lsf years); "redist" uses Sussman/Osher redistancing instead (lsf_redist_n_iter iterations). Marine cells where bnd%mask_ice is NONE are set to \varphi = 1.

Ice in cells with \varphi > 0 is removed in one step, except in cells that can be partial front cells (below). Ice-free marine cells behind the front without an ice-covered edge neighbour are returned to the ocean (\varphi = 1), except with calv_flt_method = "equil".

Subgrid front

With ytopo.front_subgrid = "marine", ice cells with the bed below sea level ("floating": floating cells only) can be partially ice covered. A front cell (such a cell with an ice-free ocean edge neighbour) has an effective thickness H_eff, taken from its thickest interior edge neighbour (or diagonal neighbour, if there is none) minus ytopo.front_dHdx·distance (at least ytopo.front_H_eff_min, with a limit on the rise of the effective surface), and the ice fraction f_ice = H_ice/H_eff. Front cells without an interior neighbour keep H_eff = H_ice, and so do front cells entirely behind the level-set front (a_lsf = 1). Ice thinner than 1 mm (H_ice_eps) is ice free (f_ice = 0) and is never a front cell.

With the level set, the thickness of these cells follows the front (calc_G_lsf_front): a_lsf, the area fraction of the cell behind the zero contour, is computed from the cell’s centre, edge and corner values of \varphi. Ice-free land neighbours, where \varphi is held at −1, take the cell’s own value, so they do not count as ice. Cells with a_lsf < 0.1 are emptied. With the level set, eligible cells cut by the front (a_lsf < 1) that touch the ocean only at a corner are front cells too; the trim and f_ice use this one front/interior classification. Front cells with a_lsf < 1 are trimmed to a_lsf·H_ref, with H_ref the reference thickness from the interior neighbours, so f_ice is approximately a_lsf. In the momentum balance, partial front cells use H_eff as their thickness and the front boundary condition is applied on their ocean faces. Cells with f_ice < A_FRONT_MIN = 0.1 keep their ice and fill by transport, but are ice free in the momentum balance until they reach 0.1 (H_ice_dyn = 0); otherwise a film of a few millimetres would be a full H_eff column and turn the ocean face of the neighbouring front cell into an interior face.

Mass-balance calving path

With use_lsf = False, the calving laws give a calving mass balance cmb in front cells:

• floating ice (calv_flt_method):
  • "zero" / "none": no calving;
  • "threshold": ice thinner than a critical thickness calves with the time scale calv_tau. The critical thickness changes from Hc_ref_flt to Hc_deep as the bed deepens from zb_deep_0 to zb_deep_1 (bed smoothed with zb_sigma);
  • "vm-l19": von Mises effective stress (Lipscomb et al., 2019), below;
  • "eigen": eigen calving (Levermann et al., 2012), scaling factor k2;
  • "kill": all floating ice is removed;
  • "kill-pos": floating ice is removed where bnd%calv_mask is set;
  • both kill methods act at the end of the calving step, after the front advance, so no floating ice remains in the kill region after each step;
• grounded ice (calv_grnd_method): "zero" / "none" or "stress-b12" (Bassis and Walker, 2012). Any grounded law also adds calving of grounded ice where the sub-grid bed roughness z_bed_sd is large, rising from zero at sd_min to calv_grnd_max at sd_max.

With the subgrid front (ytopo.front_subgrid ≠ "none"), the calving demand is applied along the front and the front can advance into neighbouring cells. Without it, thin ice is calved at the rate calv_thin below Hc_ref_thin (vm-l19, eigen), and thin floating tongues are removed.

Lipscomb et al. (2019)

c = k_\tau\, \tau_{\rm ec}

where k_\tau [m yr^{-1} Pa^{-1}] is an empirical constant (kt_ref, default 0.0025, changing to kt_deep as the bed deepens from zb_deep_0 to zb_deep_1) and \tau_{\rm ec} [Pa] is the effective calving stress,

\tau_{\rm ec}^2 = \max(\tau_1,0)^2 + \omega_2 \max(\tau_2,0)^2.

\tau_1 and \tau_2 are the eigenvalues of the depth-averaged horizontal deviatoric stress tensor and \omega_2 is an empirical weighting constant (ycalv.w2, default 0; Lipscomb et al., 2019, use 25). The calving rate is applied to floating cells with an ice-free ocean neighbour, as the mass balance -\min(H\,c/\Delta x, 2000\ \mathrm{m\,a^{-1}}).

The stress tensor is \tau_{ij} = 2\bar\mu\,\dot\varepsilon_{ij} (calc_stress_tensor_2D), and its eigenvalues are the roots of

\lambda^2 - (\tau_{xx} + \tau_{yy})\,\lambda + \tau_{xx}\tau_{yy} - \tau_{xy}^2 = 0

(calc_2D_eigen_values).

References

• Bassis, J. N. and Walker, C. C. (2012). Upper and lower limits on the stability of calving glaciers from the yield strength envelope of ice. Proc. R. Soc. A, 468, 913–931.
• Jenkins, A. (1991). A one-dimensional model of ice shelf–ocean interaction. J. Geophys. Res., 96(C11), 20671–20677.
• Levermann, A., et al. (2012). Kinematic first-order calving law implies potential for abrupt ice-shelf retreat. The Cryosphere, 6, 273–286.
• Lipscomb, W. H., et al. (2019). Description and evaluation of the Community Ice Sheet Model (CISM) v2.1. Geosci. Model Dev., 12, 387–424.
• Morlighem, M., et al. (2016). Modeling of Store Gletscher’s calving dynamics, West Greenland, in response to ocean thermal forcing. Geophys. Res. Lett., 43, 2659–2666.
• Rignot, E., et al. (2016). Modeling of ocean-induced ice melt rates of five West Greenland glaciers over the past two decades. Geophys. Res. Lett., 43, 6374–6382.