SEBM formulation and upgrades

A.1. GRENBLS formulation

GRENBLS is a physically based SEBM that requires atmospheric hourly atmospheric input variables to be run adequately (Keller and Goyette, 2005). It is therefore driven by air temperature, T [K], dew point temperature, T_d [K], anemometer-level horizontal wind speed, v [m s^-1], precipitation rate, P [mm s^-1], surface pressure, p_sfc [hPa] and net incoming solar radiation,

R

s↓ [W m^-2]. The model computes the remaining reflected solar and infrared radiative fluxes, the surface turbulent sensible and latent fluxes, and the heat flux in the ground and in the snow pack. Surface temperature, soil wetness and snow mass are prognostic variables in the model. The net radiation at the surface,

R

N [W m^-2], can be expressed as follows,:

T R R R

R

N= s^↓− s^↑+ L^↓−

ε σ

_sfc4 ^(A.1)

where

R

s↑ is the reflected shortwave radiation at the surface,

R

_L^↓ the downward atmospheric long-wave radiation,

ε

the surface emissivity and

σ

T_sfc⁴ the outgoing long-wave radiation from the surface where

σ

is the Stefan-Boltzmann constant. T_sfc is the computed snow temperature, T_snow, if the snow covers entirely the surface and the top ground temperature,

Tg, if there is no snow. The snow pack and the ground each have their own temperature. The snow fraction on the ground,

δ

_snow, is computed according to Roesch et al. (2001):

 quantity of snow required for the surface to be entirely covered. The snow fraction equals to 1 when

δ

_snow exceeds 10 kg m^-2 and to 0 when there is no snow.

ρ

snow). The surface energy budget is computed as follows:

 penetrating the snow pack. The last term in the equation is the energy change associated with the melting rates of frozen soil moisture, latent heat of fusion. The temperature of the snow pack is computed prognostically through the heat storage using a force-restore method as:



diurnal frequency and C_* the effective heat capacity of the surface. The surface sensible, latent and snow heat fluxes are computed as follows:

) Ch is the bulk heat transfer coefficient and V

r

The water budget at the surface is calculated as follows:

off bottom drainage. M_S represents the melting rate of snow, P_S the solid precipitation rate and E_s the snow evaporation rate. The partitioning of W into E_L, the liquid, and E_F, the solid part of the soil moisture, depends on T_air and is estimated as follows:

0

Precipitation is considered as solid if T_air is less than that of the triple point of water. Liquid precipitation on a snow pack induces snow melt. Melted snow goes directly into the soil as liquid moisture, latent heat is released and energy is transferred to the surface. The surface energy budget is computed at each model time step over snow cover. The radiative and turbulent fluxes are first computed. Then, the heat storage in the snow pack is calculated. If the heat storage is positive and the snow temperature below the melting point, the excess energy is first used to raise the temperature of the pack. Once its temperature reaches the melting point, any additional excess energy is used to melt the snow. The temperature of the snow is held below the melting point until the snow has melted. The melted snow goes directly into the ground as liquid moisture.

A.2. GRENBLS upgrade

The snow density parameterization of the surface energy balance model GRENBLS has been extended from an earlier version that has been developed at the University of Fribourg by Keller and Goyette (2005). Among several formulations of snow density that have been tested¹, the one of Verseghy (1991) has been selected as it has shown the best results.

The snow heights are not always proportional to the mass because the density also grows namely with the increasing quantity of snow on the ground and as a function of temperature and solar radiation. In the former version of GRENBLS, the density was fixed at 270 kg m^-3 when the snow pack was over 10 kg m^-2 and equalled 220 kg m^-3 otherwise. A new formulation, taking into account different aspects such as temperature with melting and re-freezing of snow crystals, aging of the snow pack, liquid and solid precipitation and solar income has been developed. Snow density is assumed to be constant with depth to avoid undue mathematical complexity or the use of a multi-layer snow pack parameterization.

Verseghy (1991) proposed the following expression that incorporates an asymptotic value of the density due to aging:

1 A special thank you goes to Gregory Lucato for having helped me implement the new parameterization

where

ρ

₋₆ is the density from the previous 6 hours in kg m^-3 ,

ρ

_a is the asymptotic value of density due to aging in kg m^-3.

τ

is the e-folding time in hour representing the time interval within which density, exponentially growing, increases by an e factor. In order to use it in GRENBLS, the formulation has been further adapted:

a a

old

snow

ρ ρ

t

τ ρ

ρ

=( − )exp(−1*∆ /3600* )+ (A.13)

where

ρ

_old is the oldest density, ∆t is the time step used in the SEBM and

τ

^is

expressed in seconds. According to the values used by Verseghy (1991) in his formulation, a range of possible estimates for

τ

^and

ρ

a has been tested and respectively the 500 h and 320 kg m^-3 values were finally selected. Following every snowfall, the updated density is processed using a mass weighting of the density of the previous snow pack and that of the new snow.

A.3. References

Keller B. and Goyette S. (2005): Snow melt under the different temperature increase scenarios in the Swiss Alps. In Climate and Hydrology in Mountain Areas (eds C. de Jong, D. Collins and R. Ranzi), John Wiley & Sons, Ltd, Chichester UK: Chapter 19.

Roesch A., Wild M., Gilgen H. and Ohmura A. (2001): A new snow cover fraction parameterization for the ECHAM4 GCM. Climate Dynamics 17(12): 933-946.

Verseghi D. L. (1991): CLASS – A Canadian land surface scheme for GCMS, Part I: Soil model. International Journal of Climatology 11: 111-133.

Dans le document Modelling snow and runoff patterns in Swiss mountain environments under conditions of climate change (Page 96-101)