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.

Yelmo uses the same depth-integrated 2D solver as DIVA — the difference is purely in the closure. SSA is the appropriate model for floating ice shelves (no basal drag, no vertical shear) and the right physical limit in fast-streaming grounded ice.

Continuum equations

The depth-integrated stress balance has the same shape as the DIVA equation but with the raw friction $\beta$ 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:

$$\dot\varepsilon_e^{\,2} \;=\; \dot\varepsilon_{xx}^{\,2} + \dot\varepsilon_{yy}^{\,2} + \dot\varepsilon_{xx}\,\dot\varepsilon_{yy} + \tfrac{1}{4}\,\dot\varepsilon_{xy}^{\,2} + \varepsilon_0^{\,2},$$

so the Glen viscosity $\mu$ depends only on horizontal strain rates. 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 $\boldsymbol\tau_b = \beta\,\bar{\mathbf u}$.

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