Physics

This section describes the continuum equations and numerical methods that Yelmo uses to solve the coupled momentum balance, mass conservation, and (in time) thermodynamics of the ice sheet. The goal is to document what the code actually implements, with the same variable names used in the source so that the equations here can be traced back to specific routines. The notation follows Robinson et al. (2020) for mass conservation and Robinson, Goldberg, and Lipscomb (2022) for the momentum balance.

Momentum balance

The ice flow is described as a slow, incompressible, gravity-driven flow of a power-law (Glen) fluid. The full Stokes problem is never solved. Instead Yelmo offers three approximations, listed in order of decreasing fidelity to Stokes:

Approximation	Membrane stresses	Vertical shear	Use case
DIVA	yes	yes (closure)	first-class momentum balance, valid from slow interior to streaming flow and shelves
SSA	yes	no (plug flow)	floating shelves, fast-streaming grounded ice
SIA	no	yes (closure)	slow grounded interior, used as a complement to DIVA in hybrid mode

DIVA is the recommended choice in Yelmo and is solved as a single 2D problem for the depth-averaged horizontal velocity. SSA is recovered from DIVA in the limit of vanishing vertical shear, and SIA is recovered in the limit of vanishing membrane stresses and large basal drag. All three formulations share the same driving stress, the same Glen viscosity, and (where applicable) the same family of friction laws — see Parameters for the parameter choices.

The fourth page in this subsection documents how the DIVA / SSA momentum balance is discretised and solved, including the two solvers currently supported: the residual assembler inherited from Yelmo v1, and the energy assembler that minimises a discrete energy functional.

Mass conservation

The continuity equation evolves the ice thickness $H$ in time given the depth-averaged velocity from the momentum balance and a set of mass-balance inputs (surface, basal, frontal, calving). Yelmo offers several discretisations of the resulting hyperbolic transport problem; the two recommended choices — an explicit donor-cell upwind scheme and an implicit upwind scheme solved with LIS — are documented in Numerical solution.

Notation used throughout

• $u, v$: horizontal velocity components $[\mathrm{m\,a^{-1}}]$.
• $\bar u, \bar v$: depth-averaged horizontal velocity (the DIVA/SSA unknowns and the advecting field in the continuity equation).
• $u_b, v_b$: basal horizontal velocity.
• $H$: ice thickness $[\mathrm{m}]$; $s$ and $b$: ice surface and basal elevations.
• $z$: vertical Cartesian coordinate, $b \le z \le s$.
• $\rho_i$, $g$: ice density and gravitational acceleration.
• $A(T')$: Glen rate factor, depending on the pressure-corrected temperature $T'$ and (where relevant) the water content $\omega$. $n = n_\mathrm{glen}$ is the Glen exponent (typically 3).
• $\dot\varepsilon_{ij}$: components of the horizontal strain-rate tensor.
• $\dot\varepsilon_e$: effective strain rate, the second invariant of $\dot\varepsilon_{ij}$.
• $\mu$: effective viscosity; $\bar\mu = \frac{1}{H}\int_b^s \mu \,\mathrm dz$: depth-averaged viscosity.
• $\boldsymbol\tau_d = (\tau_{d,x},\tau_{d,y})$: gravitational driving stress.
• $\boldsymbol\tau_b = (\tau_{b,x},\tau_{b,y})$: basal shear stress.
• $\beta$: basal friction coefficient, defined by $\boldsymbol\tau_b = \beta\,\mathbf u_b$.
• $\dot a$: surface mass balance (smb).
• $\dot b_g, \dot b_f$: basal mass balance on grounded and floating ice (bmb).
• $\dot c$: calving rate (cmb; mass loss, $\dot c \ge 0$).

All horizontal fields live on the Arakawa C-grid (called the ac-grid in Yelmo): scalars at cell centres (aa-nodes), $u$ on the right face of each cell (acx-nodes), $v$ on the top face (acy-nodes), and cross-derivative quantities at the cell corners (ab-nodes).