PISM, A Parallel Ice Sheet Model  stable v2.1-1-g6902d5502 committed by Ed Bueler on 2023-12-20 08:38:27 -0800

## ◆ 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().