PISM, A Parallel Ice Sheet Model
stable v2.11g6902d5502 committed by Ed Bueler on 20231220 08:38:27 0800

◆ L()
Latent heat of fusion of water as a function of pressure melting temperature. Following a reinterpretation of [AschwandenBuelerKhroulevBlatter], we require that \( \Diff{H}{p} = 0 \): \[ \Diff{H}{p} = \diff{H_w}{p} + \diff{H_w}{p}\Diff{T}{p} \] We assume that water is incompressible, so \( \Diff{T}{p} = 0 \) and the second term vanishes. As for the first term, equation (5) of [AschwandenBuelerKhroulevBlatter] defines \( H_w \) as follows: \[ H_w = \int_{T_0}^{T_m(p)} C_i(t) dt + L + \int_{T_m(p)}^T C_w(t)dt \] Using the fundamental theorem of Calculus, we get \[ \diff{H_w}{p} = (C_i(T_m(p))  C_w(T_m(p))) \diff{T_m(p)}{p} + \diff{L}{p} \] Assuming that \( C_i(T) = c_i \) and \( C_w(T) = c_w \) (i.e. specific heat capacities of ice and water do not depend on temperature) and using the ClausiusClapeyron relation \[ T_m(p) = T_m(p_{\text{air}})  \beta p, \] we get \begin{align} \Diff{H}{p} &= (c_i  c_w)\diff{T_m(p)}{p} + \diff{L}{p}\\ &= \beta(c_w  c_i) + \diff{L}{p}\\ \end{align} Requiring \( \Diff{H}{p} = 0 \) implies \[ \diff{L}{p} = \beta(c_w  c_i), \] and so \begin{align} L(p) &= \beta p (c_w  c_i) + C\\ &= (T_m(p)  T_m(p_{\text{air}})) (c_w  c_i) + C. \end{align} Letting \( p = p_{\text{air}} \) we find \( C = L(p_\text{air}) = L_0 \), so \[ L(p) = (T_m(p)  T_m(p_{\text{air}})) (c_w  c_i) + L_0, \] where \( L_0 \) is the latent heat of fusion of water at atmospheric pressure. Therefore a consistent interpretation of [AschwandenBuelerKhroulevBlatter] requires the temperaturedependent approximation of the latent heat of fusion of water given above. Note that this form of \( L(p) \) also follows from Kirchhoff's law of thermochemistry. Definition at line 358 of file EnthalpyConverter.cc. References m_c_i, m_c_w, and m_L. Referenced by enthalpy(), enthalpy_liquid(), and water_fraction(). 