c enthalpy enthalpy_cold enthalpy_cts enthalpy_liquid enthalpy_permissive EnthalpyConverter is_temperate is_temperate_relaxed L m_beta m_c_i m_c_w m_do_cold_ice_methods m_g m_L m_p_air m_rho_i m_T_0 m_T_melting m_T_tolerance melting_temperature pressure pressure pressure_adjusted_temperature Ptr temperature temperature_cold validate_E_P validate_T_omega_P water_fraction ~EnthalpyConverter

L() double pism::EnthalpyConverter::L ( double  T_pm ) const Latent heat of fusion of water as a function of pressure melting temperature. Following a re-interpretation 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 Clausius-Clapeyron 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 temperature-dependent 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().