Modeling melange back-pressure

Equation (17) above, describing the stress boundary condition for ice shelves, can be written in terms of velocity components:

\[\begin{split}\newcommand{\Diff}[2]{ \frac{\mathrm{d}#1}{\mathrm{d}#2} } \newcommand{\diff}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\var}[2]{ {#1}_{\text{#2}} } \newcommand{\h}[1]{ \var{h}{#1} } \newcommand{\T}[1]{ \var{T}{#1} } \newcommand{\m}[1]{ \var{m}{#1} } \newcommand{\ms}[1]{ \var{m^{*}}{#1} } \newcommand{\psw}{p_{\text{ocean}}} \newcommand{\pice}{p_{\text{ice}}} \newcommand{\pmelange}{p_{\text{melange}}} \newcommand{\n}{\mathbf{n}} \newcommand{\nx}{\n_{x}} \newcommand{\ny}{\n_{y}} \newcommand{\phimin}{\phi_{\mathrm{min}}} \newcommand{\phimax}{\phi_{\mathrm{max}}} \newcommand{\bmin}{b_{\mathrm{min}}} \newcommand{\bmax}{b_{\mathrm{max}}} \newcommand{\bq}{\mathbf{q}} \newcommand{\Up}[2]{\operatorname{Up}\left(#1\big|#2\right)} \newcommand{\uppair}[2]{\left\{\begin{matrix} #1 \\ #2 \end{matrix}\right\}}\end{split}\]
(22)\[ \begin{align}\begin{aligned}2 \nu H (2u_x + u_y) \nx + 2 \nu H (u_y + v_x) \ny &= \displaystyle \int_{b}^{h}(\pice - \psw) dz\, \nx,\\2 \nu H (u_y + v_x) \nx + 2 \nu H (2v_y + u_x) \ny &= \displaystyle \int_{b}^{h}(\pice - \psw) dz\, \ny.\end{aligned}\end{align} \]

Here \(\nu\) is the vertically-averaged ice viscosity, \(b\) is the ice base elevation, \(h\) is the ice top surface elevation, and \(\psw\) and \(\pice\) are pressures of the column of sea water and ice, respectively.

We call the integral on the right hand side of (22) the “pressure difference term”. To model the effect of melange [77] on the stress boundary condition, we assume that the melange back-pressure \(\pmelange\) does not exceed \(\pice - \psw\). Therefore we introduce \(\lambda \in [0,1]\) (the melange back pressure fraction) such that

\[\pmelange = \lambda (\pice - \psw).\]

Then melange pressure is added to the ordinary ocean pressure so that the pressure difference term scales with \(\lambda\):

(23)\[ \begin{align}\begin{aligned}\int_{b}^{h}(\pice - (\psw + \pmelange))\, dz &= \int_{b}^{h}(\pice - (\psw + \lambda(\pice - \psw)))\, dz\\&= (1 - \lambda) \int_{b}^{h} (\pice - \psw)\, dz.\end{aligned}\end{align} \]

This formula replaces the integral on the right hand side of (22).

The resulting stress boundary condition at the shelf front is

(24)\[ \begin{align}\begin{aligned}2 \nu H (2u_x + u_y) \nx + 2 \nu H (u_y + v_x) \ny &= \displaystyle (1 - \lambda) \int_{b}^{h}(\pice - \psw) dz\, \nx,\\2 \nu H (u_y + v_x) \nx + 2 \nu H (2v_y + u_x) \ny &= \displaystyle (1 - \lambda) \int_{b}^{h}(\pice - \psw) dz\, \ny.\end{aligned}\end{align} \]

By default, \(\lambda\) is set to zero, but PISM implements a scalar time-dependent “melange back pressure fraction offset” forcing in which \(\lambda\) can be read from a file. Please see the Climate Forcing Manual for details.


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