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_cold_mode 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 364 of file EnthalpyConverter.cc. References m_c_i, m_c_w, and m_L.