# Glacier processes

Since the SPHY model usually operates at a spatial resolution between 250m and 1km, the dynamics of glaciers such as ice flow cannot be resolved explicitly. However, SPHY has a mass-conserving glacier evolution algorithm to represent changes in glacier cover through time.

#### Glacier melt <a href="#toc534724169" id="toc534724169"></a>

Glacier melt is calculated with a degree-day modeling approach as well (Hock 2005). Because glaciers that are covered with debris melt at different rates than debris-free glaciers (Reid et al. 2012), a distinction can be made between different degree-day factors for both types. The daily melt from debris-free glaciers $$(A\_{ci} (mm))$$ is calculated as:

*Equation 23*

$$
A\_{CI,t}\begin{Bmatrix}
T\_{avg,t}\*DDF\_{CI}\*F\_{CI} &\text{if } & T\_{avg,t}>0 \\
0 &\text{if } & T\_{avg,t}\le0
\end{Bmatrix}
$$

with $$DDF\_{ci} (mm \degree C^{-1}d^{-1})$$ a calibrated degree-day factor for debris-free glaciers and $$F\_{ci} (-)$$ the fraction of debris-free glaciers within the fractional glacier cover (GlacF) of a grid cell. The daily melt from debris-covered glaciers $$(A\_{DC} (mm))$$ is calculated in a similar way, but with a different degree-day factor:

*Equation 24*

$$
A\_{DC,t}\begin{Bmatrix}
T\_{avg,t}\*DDF\_{DC}\*F\_{DC} &\text{if } & T\_{avg,t}>0 \\
0 &\text{if } & T\_{avg,t}\le0
\end{Bmatrix}
$$

where $$DDF\_{DC} (mm \degree C^{-1}d^{-1})$$  is a degree-day factor for debris-covered glaciers and $$F\_{DC} (-)$$ is the fraction of debris-covered glaciers within the fractional glacier cover of a grid cell. The total glacier melt per grid cell$$(A\_{GLAC} (mm))$$ is then calculated by summing the melt from the debris-covered and debris-free glacier types and multiplying by the fractional glacier cover, according to:

*Equation 25*

$$
A\_{GLAC,t}=(A\_{CI,t}+A\_{DC,t})\cdot GlacF
$$

#### Glacier runoff <a href="#toc534724170" id="toc534724170"></a>

In SPHY, a fraction of the glacier melt percolates to the groundwater while the remaining fraction runs off. The distribution of both is defined by a calibrated glacier melt runoff factor (GlacROF (–)) that can have any value ranging from 0 to 1. Thus, the generated runoff GRo (mm) from glacier melt is defined as:

*Equation 26*

$$
GRo\_{t}=A\_{GLAC,t} \cdot GlacROF
$$

#### Glacier percolation <a href="#toc534724171" id="toc534724171"></a>

The percolation from glacier melt to the groundwater $$(G\_{perc,t} (mm))$$ is defined as:

*Equation 27*

$$
G\_{perc,t}=A\_{GLAC,t} \cdot (1-GlacROF)
$$

The percolated glacier water is added to the water that percolates from the soil layers of the non-glacierized part of the grid cell (Section 2.7.1 and 2.7.7), which eventually recharges the groundwater.

**Glacier ice redistribution**

The model takes sub-grid variability into account by calculating the snow and glacier melt runoff from glaciers. By intersecting the glacier outlines with the coarse model grid, the glaciers or parts thereof (fraction) that lie within each model grid cell can be identified. Future changes in (parts of) glaciers in response to the precipitation and temperature are taken into account by using a mass-conserving ice redistribution approach (Khanal et al 2021). The ice redistribution is done once per year at the end of the hydrological year, which is also the end of the melting season (October 1st). At that moment the accumulated snow in the accumulation zone is transformed into ice and distributed downwards to the ablation area. The net imbalance (I), that is, the difference in the volume of total snow accumulated (SnowS) and total volume of melt generated from the glaciers (GM), forms the basis of ice redistribution:

*Equation 28*

$$
I\_{n,j}=SnowS\_{n,j}-GM\_{n,j}
$$

where the subscript n is the glacier id, and j is a unique-id. Only when the net imbalance is negative, the volume of ice is redistributed (Vred) over the ablation zone according to:

*Equation 29*

$$
Vred\_{n,j}\begin{Bmatrix}
0 &\text{,} & j\isin B\_{n,j} \\
\sum I\_{n,j} \times \frac{Vini\_{n,j}}{\sum\_{j \isin A\_{n,j}}Vini\_{n,j}} &\text{,} & j\isin A\_{n,j}
\end{Bmatrix}
$$

Where Aj is the part of the glacier with a negative imbalance. Bj is the part of the glacier with a positive imbalance in any glacier-id. The redistribution is propotional to the initial volume of ice (Vini), that is, glacier partswith a larger initial ice volume will receive a large volume of accumulated ice from the accumulation zone to the ablation zone
