# Modeling melange back-pressure¶

Equation (32) above, describing the stress boundary condition for ice shelves, can be written in terms of velocity components:

(37)\begin{align}\begin{aligned}2 \nu H (2u_x + u_y) \nx + 2 \nu H (u_y + v_x) \ny &= \displaystyle \int_{b}^{h}(\pice(z) - \psw(z)) dz\, \nx,\\2 \nu H (u_y + v_x) \nx + 2 \nu H (2v_y + u_x) \ny &= \displaystyle \int_{b}^{h}(\pice(z) - \psw(z)) dz\, \ny.\end{aligned}\end{align}

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, $$\psw$$ and $$\pice$$ are pressures of the column of ice and water, respectively:

(38)\begin{align}\begin{aligned}\pice &= \rhoi\, g (h - z),\\\psw &= \rhow\, g\, \max(\zs - z,\, 0).\end{aligned}\end{align}

We call the integral on the right hand side of (37) the pressure difference term.

It can be re-written as

(39)\begin{align}\begin{aligned}\int_b^h \pice(z) - \psw(z) dz &= H (\bar p_{\text{ice}} - \bar p_{\text{water}}),\, \text{where}\\\bar p_{\text{ice}} &= \frac12\, \rho_{\text{ice}}\, g\, H,\\\bar p_{\text{water}} &= \frac12\, \rho_{\text{water}}\, g\, \frac{\max(z_s - b, 0)^2}{H}.\end{aligned}\end{align}

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

To model the effect of melange [100] on the stress boundary condition we modify the pressure difference term in (37), adding $$\pmelange$$, the vertically-averaged melange back pressure:

(40)$\int_{b}^{h}(\pice - (\psw + \pmelange))\, dz.$

By default, $$\pmelange$$ 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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