Snow

Snow layers are simulated following the idea of Riley and Bonesmo (2005). Below 1.8°C air temperature, an increasing fraction of snow is assumed in the falling rain, which adds to a snow layer.

\[n_l = \begin{cases} 1 & T_a \geq T_{as} \\ \frac{T_a - T_{ls}}{T_{as} - T_{ls}} & T_{ls} < T_a < T_{as} \\ 0 & T_a \leq T_{ls} \end{cases}\]

\(n_l\) Fraction of liquid precipitation \([mm \, mm^{-1}]\)
\(T_a\) Daily mean air temperature \([^{\circ} C]\)
\(T_{ls}\) Threshold temperature for liquid water in snow \([^{\circ} C]\)
\(T_{as}\) Threshold temperature for snow accumulation \([^{\circ} C]\)

The amounts of liquid and snow precipitation are derived from:

\[N_l = n_l \cdot k_l\]

\(N_l\) Liquid precipitation \([mm]\)
\(n_l\) Fraction of liquid precipitation \([mm \, mm^{-1}]\)
\(k_l\) Correction factor liquid precipitation \([mm]\)

and

\[N_s = (1-n_l) \cdot k_s\]

\(N_s\) Snow precipitation \([mm]\)
\(n_l\) Fraction of liquid precipitation \([mm \, mm^{-1}]\)
\(k_s\) Correction factor snow precipitation \([mm]\)

The snow layer density is calculated as follows:

\[\rho_{sn} = \rho_{sn\,min} + \rho_{sn\, max} \cdot \frac{T_a - T_{ls}}{T_{as} - T_{ls}}\]

\(\rho_{sn}\) Density of fresh snow \([kg \, dm^{-3}]\)
\(\rho_{sn \, min}\) Minimum density of fresh snow \([kg \, dm^{-3}]\)
\(\rho_{sn \, max}\) Maximum density of fresh snow \([kg \, dm^{-3}]\)
\(T_a\) Daily mean air temperature \([^{\circ} C]\)
\(T_{ls}\) Threshold temperature for liquid water in snow \([^{\circ} C]\)
\(T_{as}\) Threshold temperature for snow accumulation \([^{\circ} C]\)

Snow starts thawing above 0.31 °C and increases its density:

\[W_{sm} = \begin{cases} 0 & T_a < T_{sm} \\ a_{sm} \cdot ( T_a - T_{sm}) & T_a \geq T_{sm} \end{cases}\]

\(W_{sm}\) Water from melting snow \([mm]\)
\(a_{sm}\) Snow ageing (limited to 4.7)
\(T_a\) Daily mean air temperature \([^{\circ} C]\)
\(T_{sm}\) Base temperature snow melt \([^{\circ} C]\)

where

\[a_{sm} = 1.4 \cdot \frac{\rho_s}{0.1}\]

\(a_{sm}\) Snow ageing (limited to 4.7) \([kg \, dm^{-3}]\)
\(\rho_s\) Snow density \([kg \, dm^{-3}]\)

Liquid water in the snow layer re-freezes below –1.7 °C:

\[W_{sf} = 1.5 \cdot (T_{sm} - T_a)^{0.36}\]

\(W_{sf}\) Re-freezing water in the snow layer \([mm]\)
\(T_a\) Daily mean air temperature \([^{\circ} C]\)
\(T_{sm}\) Base temperature snow melt \([^{\circ} C]\)

The water-holding capacity of snow is calculated within given boundaries as:

\[C_s = \frac{0.1 \cdot C_{smax}}{\rho_s}\]

\(C_s\) Water-holding capacity of snow \([mm]\)
\(C_{smax}\) Maximum water-holding capacity of snow \([mm]\)
\(\rho_s\) Snow density \([kg \, dm^{-3}]\)

From this, the amount of water held back in the snow layer can be calculated:

\[W_{sr} = C_s \cdot W_s\]

\(W_{sr}\) Liquid water in the snow layer \([mm]\)
\(C_s\) Water-holding capacity of snow \([mm]\)
\(W_s\) Water equivalent in the snow layer \([mm]\)

where

\[W_s = W_f \cdot W_l\]

\(W_s\) Water equivalent in the snow layer \([mm]\)
\(W_f\) Water equivalent of the frozen water \([mm]\)
\(W_l\) Liquid water in the snow layer \([mm]\)

\[W_f(t) = W_f(t-\Delta t) + N_s - W_{sm} + W_{sf}\]

\(W_f(t)\) Water equivalent frozen water at time \(t\) \([mm]\)
\(W_f(t-\Delta t)\) Water equivalent frozen water at preceding time step \([mm]\)
\(W_{sm}\) Water from melting snow \([mm]\)
\(W_{sf}\) Re-freezing water in the snow layer \([mm]\)

and

\[W_l(t) = W_l(t-\Delta t) + N_l + W_{sm} - W_{sf}\]

\(W_l(t)\) Liquid water in the snow layer at time \(t\) \([mm]\)
\(W_l(t-\Delta t)\) Liquid water in the snow layer at preceding time step \([mm]\)
\(W_{sm}\) Water from melting snow \([mm]\)
\(W_{sf}\) Re-freezing water in the snow layer \([mm]\)

The amount of liquid water flowing from the snow layer onto the soil surface is calculated for the case \(W_l\) > \(W_{sr}\) \(W_l\) > \(W_{sr}\) as

\[W_i = W_l - W_{sr}\]

\(W_i\) Liquid water flowing from the snow layer \([mm]\)
\(W_l\) Liquid water in the snow layer \([mm]\)
\(W_{sr}\) Liquid water in the snow layer \([mm]\)

Finally, snow height can be calculated as:

\[S = \frac{W_s \cdot \rho_W}{\rho_s}\]

\(S\) Snow height \([mm]\)
\(W_s\) Water equivalent of snow \([mm]\)
\(\rho_W\) Snow density \([kg \, dm^{-3}]\)
\(\rho_s\) Density of water \([kg \, dm^{-3}]\)