# Snow processes

For each cell, a dynamic snow storage is simulated at a daily time step, adopted from the model presented by Kokkonen et al. (2006). The model keeps track of a snow storage, which is fed by precipitation and generates runoff from snowmelt. Refreezing of snowmelt and rainfall within the snowpack are simulated as well.

#### Snow and rainfall <a href="#toc534724165" id="toc534724165"></a>

Depending on a temperature threshold, precipitation is defined as falling in either solid or liquid form. Daily snow accumulation, which is defined as solid precipitation, is calculated as:

*Equation 13*

$$
P\_{s,t} = \begin{cases}
Pe\_{t} &\text{if } \space T\_{avg,t}\le T\_{crit} \\
0 &\text{if } \space T\_{avg,t} > T\_{crit}
\end{cases}
$$

with $$P\_{s,t} (mm)$$ the snowfall on day $$t$$,   $$Pe\_{t} (mm)$$the effective precipitation on day $$t$$, $$T\_{avg,t} (\degree C)$$ the mean air temperature on day $$t$$, and $$T\_{crit} (\degree C)$$ a calibrated  temperature threshold for precipitation to fall as snow. The precipitation that falls as rain is defined as liquid precipitation, and is calculated as:

*Equation 14*

$$
P\_{l,t} = \begin{cases}
Pe\_{t} &\text{if } T\_{avg,t}>T\_{crit} \\
0 &\text{if } T\_{avg,t} \le T\_{crit}
\end{cases}
$$

with $$P\_{l,t} (mm)$$ being the amount of rainfall on day $$t$$.

#### Snowmelt, refreezing, and storage <a href="#toc534724166" id="toc534724166"></a>

To simulate snowmelt, the well-established and widely used degree-day melt modeling approach is used (Hock, 2003). The application of degree-day models is widespread in cryospheric models and is based on an empirical relationship between melt and air temperature. Degree-day models are easier to set up compared to energy-balance models, and only require air temperature, which is mostly available and relatively easy to interpolate (Hock, 2005). Using a degree-day modeling approach, the daily potential snowmelt is calculated as follows:

*Equation 15*

$$
A\_{pot,t} = \begin{Bmatrix}
HT\_{t}\*DDF\_{s} &\text{if } T\_{max,t} >0\\
0 &\text{if } T\_{max,t} \le 0
\end{Bmatrix}
$$

*Equation 16*

$$
HT\_t=Max \begin{Bmatrix} 0, \Big( \sum\_{i=1}^{1=24}T\_{avg,t}+\big(T\_{avg,t}+(T\_{avg,t}-T\_{max,t})cos(\pi\*i/12)\big)\Big)/24\end{Bmatrix}
$$

with $$A\_{pot,t} (mm)$$ the potential snowmelt on day $$t$$,  $$T\_{max,t} \space (\degree C)$$ the max air temperature on day $$t$$ and $$DDF\_s (mm \degree C^{-1}d^{-1})$$ a calibrated degree-day factor for snow. $$HT\_t$$ calculates the fraction of day where the hourly temperature is above melting point, while the average temperature is below melting point. The actual snow melt is limited by the snow store at the end of the previous day, and is calculated as:

*Equation 17*

$$
A\_{act,t} = min(A\_{pot,t},SS\_{t-1})
$$

with $$A\_{act,t} (mm)$$ the actual snowmelt on day $$t$$, and $$SS\_{t-1} (mm)$$ the snow storage on day $$t-1$$. The snow storage from day $$t-1$$is then updated to the current day $$t$$, using the actual snowmelt $$(A\_{act,t})$$ and the solid precipitation$$(P\_{s,t})$$. Part of the actual snowmelt freezes within the snowpack and thus does not run off immediately. When temperature is below the melting point, meltwater that has frozen in the snowpack during $$t-1$$ is added to the snow storage as:

*Equation 18*

$$
SS\_t = \begin{cases}
SS\_{t-1}+P\_{s,t}+SSW\_{t-1} &\text{if } T\_{avg,t}<0 \\
SS\_{t-1}+P\_{s,t}-A\_{act,t} &\text{if } T\_{avg,t} \ge 0
\end{cases}
$$

with  $$SS\_t$$ the snow storage on day t,  $$SS\_{t-1}$$the snow storage on day  $$t-1$$,  $$P\_{s,t}$$ the solid precipitation on day t,  $$A\_{act,t}$$ the actual snowmelt on day t, and  $$SSW\_{t-1}$$ the amount of frozen meltwater on day  $$t-1$$. The units for all terms are mm.

The capacity of the snowpack to freeze snowmelt is characterized by introducing a calibrated water storage capacity$$(SSC (mm \cdot mm^{-1}))$$, which is the total water equivalent of snowmelt (mm) that can freeze per mm water equivalent of snow in the snow storage. The maximum of meltwater that can freeze$$(SSW\_{max}(mm))$$ is thus limited by the thickness of the snow storage:

*Equation 19*

$$
SSW\_{max,t}=SSC\*SS\_t
$$

Then the amount of meltwater stored in the snowpack, and that can freeze in the next time step, is calculated as:

*Equation 20*

$$
SSW\_{t} = \begin{cases}
0 &\text{if } T\_{avg,t} < 0\\
min(SSW\_{max,t},SSW\_{t-1}+P\_{l,t}+A\_{act,t}), &\text{if } T\_{avg,t} \ge 0
\end{cases}
$$

with $$SSW\_t$$ the amount of meltwater in the snowpack on day $$t$$, $$SSW\_{max,t}$$ the maximum of meltwater that can freeze on day $$t-1$$, $$SSW\_{t-1}$$ the amount of frozen meltwater on day $$t-1$$, $$P\_{l,t}$$ the amount of rainfall on day $$t$$, and $$A\_{act,t}$$  the actual snowmelt on day $$t$$. The units of all terms are in mm.

The total snow storage, $$SST \space (mm)$$, consists of the snow storage and the meltwater that can freeze within it, according to:

*Equation 21*

$$
SST\_t =(SS\_t +SSW\_t)\*(1-GlacF)
$$

with  $$(1-GlacF)(-)$$ the grid-cell fraction not covered with glaciers. In SPHY it is therefore assumed that snow accumulation and snowmelt can only occur on the grid-cell fraction determined as land surface. Snow falling on glaciers is incorporated in the glacier module.

#### Snow runoff <a href="#toc147754710" id="toc147754710"></a>

Runoff from snow, $$SRo \space (mm)$$, is generated when the air temperature is above melting point and no more meltwater can be frozen within the snowpack, according to:

*Equation 22*

$$
SRo\_t = \begin{cases}
A\_{act,t}+P\_{l,t} - \Delta SSW&\text{if } T\_{avg,t} > 0\\
0 &\text{if } T\_{avg,t}\le 0
\end{cases}
$$

with $$\Delta SSW (mm)$$ the change in meltwater stored in the snowpack according to:

*Equation 23*

$$
\Delta SSW=SSW\_t-SSW\_{t-1}
$$
