tmelange.rst - pism - [fork] customized build of PISM, the parallel ice sheet model (tillflux branch)
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       tmelange.rst (2186B)
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            1 .. include:: ../../../global.txt
            2 
            3 .. _sec-model-melange-pressure:
            4 
            5 Modeling melange back-pressure
            6 ------------------------------
            7 
            8 Equation :eq:`eq-cfbc` above, describing the stress boundary condition for ice shelves,
            9 can be written in terms of velocity components:
           10 
           11 .. include:: ../../../math-definitions.txt
           12 
           13 .. math::
           14    :label: eq-cfbc-uv
           15 
           16    2 \nu H (2u_x + u_y) \nx + 2 \nu H (u_y + v_x)  \ny &= \displaystyle \int_{b}^{h}(\pice - \psw) dz\, \nx,
           17 
           18    2 \nu H (u_y + v_x)  \nx + 2 \nu H (2v_y + u_x) \ny &= \displaystyle \int_{b}^{h}(\pice - \psw) dz\, \ny.
           19 
           20 Here `\nu` is the vertically-averaged ice viscosity, `b` is the ice base elevation, `h` is
           21 the ice top surface elevation, and `\psw` and `\pice` are pressures of the column of sea
           22 water and ice, respectively.
           23 
           24 We call the integral on the right hand side of :eq:`eq-cfbc-uv` the "pressure difference
           25 term". To model the effect of melange :cite:`Amundsonetal2010` on the stress boundary
           26 condition, we assume that the melange back-pressure `\pmelange` does not exceed `\pice -
           27 \psw`. Therefore we introduce `\lambda \in [0,1]` (the melange back pressure fraction)
           28 such that
           29 
           30 .. math::
           31 
           32    \pmelange = \lambda (\pice - \psw).
           33 
           34 Then melange pressure is added to the ordinary ocean pressure so that the pressure
           35 difference term scales with `\lambda`:
           36 
           37 .. math::
           38    :label: eq-cfbc-3
           39 
           40    \int_{b}^{h}(\pice - (\psw + \pmelange))\, dz &= \int_{b}^{h}(\pice - (\psw + \lambda(\pice - \psw)))\, dz
           41 
           42    &= (1 - \lambda) \int_{b}^{h} (\pice - \psw)\, dz.
           43 
           44 This formula replaces the integral on the right hand side of :eq:`eq-cfbc-uv`.
           45 
           46 The resulting stress boundary condition at the shelf front is
           47 
           48 .. math::
           49    :label: eq-cfbc-mbp
           50 
           51    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,
           52 
           53    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.
           54 
           55 By default, `\lambda` is set to zero, but PISM implements a scalar time-dependent "melange
           56 back pressure fraction offset" forcing in which `\lambda` can be read from a file. Please
           57 see the :ref:`Climate Forcing Manual <sec-ocean-frac-mbp>` for details.