# 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  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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