SSA — Shallow Shelf Approximation

The SSA is the membrane-stress balance that DIVA reduces to in the limit of vanishing vertical shear. It treats the ice column as a vertical plug: the horizontal velocity is independent of z, so \bar{\mathbf u} is the basal velocity, \bar{\mathbf u} = \mathbf u_b. The implementation is in src/physics/velocity_ssa.f90.

The SSA is solved by its own Picard loop (calc_velocity_ssa) with the same linear-system assemblers as DIVA (ydyn.ssa_solver, see Numerical solution); the difference is in the closure. SSA is the appropriate model for floating ice shelves (no basal drag, no vertical shear) and the physical limit in fast-streaming grounded ice. It is used with ydyn.solver = "ssa", where \bar{\mathbf u} = \mathbf u_b, and in the hybrid mode ("hybrid"), where the SSA basal velocity is added to the SIA shear velocity, \bar{\mathbf u} = \bar{\mathbf u}^{(\mathrm{SIA})} + \mathbf u_b.

Continuum equations

The depth-integrated stress balance has the same shape as the DIVA equation but with the friction coefficient \beta instead of \beta_\mathrm{eff}:

\frac{\partial}{\partial x}\!\left[\, 2\,\bar\mu\,H\,(2\,\bar\varepsilon_{xx} + \bar\varepsilon_{yy})\,\right] + \frac{\partial}{\partial y}\!\left[\, 2\,\bar\mu\,H\,\bar\varepsilon_{xy}\,\right] - \beta\,\bar u \;=\; \tau_{d,x},

\frac{\partial}{\partial y}\!\left[\, 2\,\bar\mu\,H\,(2\,\bar\varepsilon_{yy} + \bar\varepsilon_{xx})\,\right] + \frac{\partial}{\partial x}\!\left[\, 2\,\bar\mu\,H\,\bar\varepsilon_{xy}\,\right] - \beta\,\bar v \;=\; \tau_{d,y},

with depth-averaged viscosity \bar\mu, driving stress \boldsymbol\tau_d and friction law \boldsymbol\tau_b = \beta\,\mathbf u_b defined exactly as in DIVA.

What is dropped relative to DIVA

The effective strain rate omits the \frac{\partial u}{\partial z} and \frac{\partial v}{\partial z} terms:

\dot\varepsilon_e^{\,2} \;=\; \dot\varepsilon_{xx}^{\,2} + \dot\varepsilon_{yy}^{\,2} + \dot\varepsilon_{xx}\,\dot\varepsilon_{yy} + \dot\varepsilon_{xy}^{\,2} + \varepsilon_0^{\,2},

so the Glen viscosity \mu depends only on the horizontal strain rates (and, through A(T'(z)), still varies with depth before it is depth-integrated). There is no F-integral closure: F_2 \to 0, so

\beta_\mathrm{eff} \to \beta, \qquad u_b \to \bar u.

The basal stress diagnosed after the solve is simply \boldsymbol\tau_b = \beta\,\bar{\mathbf u}. \beta is the friction coefficient after the grounding-line scaling, staggering and beta_min limit (see Basal friction).

For ice shelves \beta = 0 identically and only the membrane terms and \boldsymbol\tau_d remain — this is the SSA in its purest form.