DIVA — Depth-Integrated Viscosity Approximation

DIVA is the first-class momentum balance solved in Yelmo. It is a 2D (depth-integrated) problem for the horizontal velocity, augmented with a 1D vertical closure that recovers the full 3D velocity field. The formulation follows Goldberg (2011), Arthern et al. (2015), Lipscomb et al. (2019), and Robinson et al. (2022), whose notation we use here.

DIVA is valid uniformly from slow grounded ice (where vertical shear dominates, the SIA limit) to fast streaming ice and floating shelves (where membrane stresses dominate, the SSA limit). It does so by retaining the depth-integrated membrane stress balance of SSA while reintroducing vertical shear through a closure on $\frac{\partial u}{\partial z}$ that is consistent with the Glen flow law.

The implementation is in src/physics/velocity_diva.f90.

Continuum equations

Depth-integrated stress balance

The unknown is the depth-averaged horizontal velocity $\bar{\mathbf u} = (\bar u, \bar v)$. The momentum balance is the depth integral of the horizontal Stokes equations, with the assumption that the longitudinal stresses do not vary in the vertical ($\frac{\partial u}{\partial z}$ contributes only to the strain-rate invariant, not as an independent unknown):

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

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

with the depth-averaged effective viscosity

$$\bar\mu \;=\; \frac{1}{H}\int_b^s \mu \,\mathrm dz.$$

(The product $\bar\mu\,H$ is stored in the code as visc_eff_int.) The horizontal strain rates are

$$\bar\varepsilon_{xx} \;=\; \frac{\partial \bar u}{\partial x}, \qquad \bar\varepsilon_{yy} \;=\; \frac{\partial \bar v}{\partial y}, \qquad \bar\varepsilon_{xy} \;=\; \tfrac{1}{2}\!\left(\frac{\partial \bar u}{\partial y} + \frac{\partial \bar v}{\partial x}\right).$$

The driving stress on the right-hand side is the depth-integrated gravitational force,

$$\tau_{d,x} \;=\; \rho_i\, g\, H\, \frac{\partial s}{\partial x}, \qquad \tau_{d,y} \;=\; \rho_i\, g\, H\, \frac{\partial s}{\partial y},$$

evaluated on ac-faces by calc_driving_stress in velocity_general.f90. A grounding-line variant (calc_driving_stress_gl) replaces $H$ with a flotation-corrected thickness so that no spurious driving stress is generated at the flotation contact. At calving fronts the lateral boundary contributes a depth-integrated water-pressure imbalance $\tau_{l,\mathrm{int}}$ that enters as a Neumann condition.

Effective viscosity from Glen's flow law

The effective viscosity follows Glen's law,

$$\mu \;=\; \tfrac{1}{2}\, A^{-1/n}\, \dot\varepsilon_e^{\,(1-n)/n},$$

(routine calc_viscosity_glen in deformation.f90). The effective strain rate $\dot\varepsilon_e$ in DIVA uses the horizontal strain rates plus a contribution from the (closure-defined) vertical shear, regularised by $\varepsilon_0$ to avoid singular viscosity in stagnant ice:

$$\dot\varepsilon_e^{\,2} \;=\; \dot\varepsilon_{xx}^{\,2} + \dot\varepsilon_{yy}^{\,2} + \dot\varepsilon_{xx}\,\dot\varepsilon_{yy} + \tfrac{1}{4}\,\dot\varepsilon_{xy}^{\,2} + \tfrac{1}{4}\!\left(\frac{\partial u}{\partial z}\right)^{\!2} + \tfrac{1}{4}\!\left(\frac{\partial v}{\partial z}\right)^{\!2} + \varepsilon_0^{\,2}.$$

The two $\left(\frac{\partial u}{\partial z}\right)^{\!2}, \left(\frac{\partial v}{\partial z}\right)^{\!2}$ terms are what make DIVA different from SSA: they reintroduce the vertical-shear contribution to the deformation rate, and hence soften the ice in places where vertical shear is large (slow, grounded interior).

Vertical-shear closure

Because the depth-integrated equation does not solve for the full vertical velocity profile, $\frac{\partial u}{\partial z}$ and $\frac{\partial v}{\partial z}$ are obtained from a closure that is consistent with the Stokes balance under DIVA's assumptions. With the basal shear stress $\boldsymbol\tau_b$ acting at $z = b$ and vanishing shear stress at the surface $z = s$,

$$\frac{\partial u}{\partial z}(z) \;=\; \frac{\tau_{b,x}\,(s-z)}{\mu(z)\,H}, \qquad \frac{\partial v}{\partial z}(z) \;=\; \frac{\tau_{b,y}\,(s-z)}{\mu(z)\,H}$$

(Robinson et al. 2022 Eq. 21; routine calc_vertical_shear_3D). Define the F-integrals

$$F_m \;\equiv\; \int_b^s \frac{1}{\mu(z)}\!\left(\frac{s-z}{H}\right)^{\!m}\,\mathrm dz$$

(Robinson et al. 2022 Eq. 15; routine calc_F_integral). These are precisely the integrals needed to relate the depth-averaged velocity to the basal velocity:

$$\bar u \;=\; u_b \,+\, \tau_{b,x}\,F_2, \qquad \bar v \;=\; v_b \,+\, \tau_{b,y}\,F_2.$$

Basal stress closure and effective drag

With a basal friction law of the form $\boldsymbol\tau_b = \beta\,\mathbf u_b$ (see Parameters for the supported choices of $\beta$), the relation $\bar u = u_b + \tau_{b,x}\,F_2 = u_b\,(1 + \beta\,F_2)$ inverts to express $\boldsymbol\tau_b$ directly in terms of the depth-averaged velocity:

$$\boldsymbol\tau_b \;=\; \beta_\mathrm{eff}\,\bar{\mathbf u}, \qquad \beta_\mathrm{eff} \;\equiv\; \frac{\beta}{1 + \beta\,F_2}$$

(Robinson et al. 2022 Eq. 19; routine calc_beta_eff). In the no-slip limit ($\beta \to \infty$) this reduces to $\beta_\mathrm{eff} = 1/F_2$ (Robinson et al. 2022 Eq. 20), which is the form used when the user requests no_slip = .TRUE.. The effective drag $\beta_\mathrm{eff}$ — not the raw $\beta$ — is what appears in the depth-integrated momentum equation above.

Picard iteration

The viscosity $\mu(\bar{\mathbf u})$ and the effective drag $\beta_\mathrm{eff}(\bar{\mathbf u})$ both depend on the solution, so the momentum equation is nonlinear. Yelmo solves it by Picard iteration:

1. Compute $\mu$ from the current $\bar{\mathbf u}$ (and the current 3D viscosity $\mu(z)$).
2. Relax $\mu$ in log-space against the previous iterate (Sandip et al. 2023): picard_relax_visc.
3. Compute $\bar\mu\,H$, the F-integrals $F_m$, and $\beta_\mathrm{eff}$.
4. Assemble and solve the linear system for $\bar{\mathbf u}$ (see Numerical solution).
5. Relax $\bar{\mathbf u}$ against the previous iterate (picard_relax_vel) and test an L2-relative convergence criterion.

When the iteration converges, the basal stress $\boldsymbol\tau_b = \beta_\mathrm{eff}\bar{\mathbf u}$ and basal velocity $u_b = \bar u - \tau_{b,x}\,F_2$ are diagnosed, and the 3D horizontal velocity field is reconstructed by integrating the vertical-shear closure from the bed upward:

$$u(z) \;=\; u_b \,+\, \tau_{b,x}\, \tilde F_1(z), \qquad \tilde F_1(z) \;\equiv\; \int_b^z \frac{1}{\mu(z')}\!\left(\frac{s-z'}{H}\right)\,\mathrm dz'$$

(routine calc_vel_horizontal_3D).

Limits

• SSA limit: as $\frac{\partial u}{\partial z}, \frac{\partial v}{\partial z} \to 0$, the F-integrals enter only through $\beta_\mathrm{eff} \to \beta$ (no vertical-shear enhancement of $\dot\varepsilon_e$, no basal/depth-averaged offset). The depth-integrated stress balance reduces to the SSA equation — see SSA.
• SIA limit: as membrane gradients become negligible relative to vertical-shear gradients, the balance collapses to $\boldsymbol\tau_b \approx \boldsymbol\tau_d$, and the closure on $\frac{\partial u}{\partial z}$ reproduces the SIA velocity profile — see SIA.