Modeling melange back-pressure

Equation (33) above, describing the stress boundary condition for ice shelves, can be written in terms of velocity components:

(38)$$\begin{array}{r}\begin{array}{rl}2\nu H(2u_x+u_y){\mathit{n}}_x+2\nu H(u_y+v_x){\mathit{n}}_y& ={\displaystyle {\int }_b^h(p_{\text{ice}}(z)-p_{\text{water}}(z))dz{\mathit{n}}_x,}\\ 2\nu H(u_y+v_x){\mathit{n}}_x+2\nu H(2v_y+u_x){\mathit{n}}_y& ={\displaystyle {\int }_b^h(p_{\text{ice}}(z)-p_{\text{water}}(z))dz{\mathit{n}}_y.}\end{array}\end{array}$$

Here $\nu$ is the vertically-averaged ice viscosity, $H$ is the ice thickness, $b$ is the elevation of the bottom and $h$ of the top ice surface, $p_{\text{water}}$ and $p_{\text{ice}}$ are pressures of the column of ice and water, respectively:

(39)$$\begin{array}{r}\begin{array}{rl}{p}_{\text{ice}}& ={\rho }_{\text{ice}}g(h-z),\\ p_{\text{water}}& ={\rho }_{\text{water}}gmax(z_{\text{sea level}}-z,0).\end{array}\end{array}$$

We call the integral on the right hand side of (38) the pressure difference term.

It can be re-written as

(40)$$\begin{array}{r}\begin{array}{rl}{\int }_b^h{p}_{\text{ice}}(z)-p_{\text{water}}(z)dz& =H({\overline{p}}_{\text{ice}}-{\overline{p}}_{\text{water}}),\text{where}\\ {\overline{p}}_{\text{ice}}& =\frac{1}{2}{\rho }_{\text{ice}}gH,\\ {\overline{p}}_{\text{water}}& =\frac{1}{2}{\rho }_{\text{water}}g\frac{max(z_s-b,0{)}^2}{H}.\end{array}\end{array}$$

PISM’s ocean model components provide ${\overline{p}}_{\text{water}}$, the vertically-averaged pressure of the water column adjacent to an ice margin.

To model the effect of melange [105] on the stress boundary condition we modify the pressure difference term in (38), adding ${\overline{p}}_{\text{melange}}$, the vertically-averaged melange back pressure:

(41)$${\int }_b^h(p_{\text{ice}}-(p_{\text{water}}+{\overline{p}}_{\text{melange}}))dz.$$

By default, ${\overline{p}}_{\text{melange}}$ is zero, but PISM implements two ocean model components to support scalar time-dependent melange pressure forcing. Please see the Climate Forcing Manual for details.

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