refreezing

cosipy.modules.refreezing.refreezing(GRID)

Refreeze water in layers.

This approach is adapted from Bartelt & Lehning (2002).

\[ \begin{align}\begin{aligned}\Delta\theta_{i} &= -\Delta\theta_{w}\frac{ \rho_{w} }{ \rho_{i} }\\\Delta T &= \frac{ c_{w} \rho_{w} \Delta\theta_{w} }{ c_{i} \rho_{i} \theta_{i} + c_{w} \rho_{w} \theta_{w} }\\\theta_{refrozen} &= \Delta\theta_{w} h\end{aligned}\end{align} \]

For the maximum water available for refreezing, the latent energy release from refreezing water equals the layer warming of the layer’s ice content, the newly frozen water, and the remaining water that cannot be refrozen:

\[ \begin{align}\begin{aligned}Q_{refreeze} &= \theta_{i} + \Delta\theta_{i} + (\theta_{w} - \Delta\theta_{w})\\\Delta\theta_{w} L_{f} \rho_{w} &= \Delta T_{max} \left [ (c_{i} \rho_{i}(\theta_{i} + \Delta\theta_{i})) + (c_{w} \rho_{w} (\theta_{w} - \Delta\theta_{w})) \right ]\\\Delta\theta_{w} L_{f} \rho_{w} &= \Delta T_{max} \left [ \left ( c_{i} \rho_{i} \left ( \theta_{i} + \frac{ \rho_{w} }{ \rho_{i} }\Delta\theta_{w} \right ) \right ) + (c_{i} \rho_{w}(\theta_{w} - \Delta\theta_{w})) \right ]\end{aligned}\end{align} \]

Re-arranged in terms of \(\Delta\theta_{w}\), limited by the maximum cold content:

\[\Delta\theta_{{w}_{max}} = \frac{ -\Delta T_{max}(\rho_{i}c_{i}\theta_{i} + \rho_{w}c_{w}\theta_{w}) }{ \rho_{w}(L_{f}-\Delta T_{max}(c_{i}-c_{w})) }\]

Note

The units for \(\Delta\theta_{w} h\) cancel out to m w.e. as long as the density of waer is set to 1000 kg m^-3. Note that \(\Delta\theta_{i} h\) is in m ice equivalent, but both the refreeze parameter and returned refrozen water are in m w.e.

Parameters:

GRID (Grid) – Glacier data structure.

Returns:

Refrozen water, [m w.e.].

Return type:

float