This article presents a comprehensive description of the 3.0.2 stable release of the Crocus snowpack model in the SURFEX modelling platform. It synthesizes and harmonizes a number of equations disseminated in various previous publications, introduces a number of unpublished parameterizations and includes new developments implemented since 2012. Among the novelties, an explicit representation of the evolution of impurity mass in snow (e.g. black carbon, mineral dust) allows representing their impact on solar radiation absorption in the snowpack at different wavelengths and their feedback on all snowpack properties. The model also allows the formation of surface ice layers due to freezing rain. In addition, Crocus is coupled to the MEB “big-leaf” vegetation scheme and can therefore be applied in forested areas. A module for snow management can also be optionally activated to simulate the snowpack on ski slopes in ski resorts. The model can be coupled with various blowing snow schemes. The MEPRA expert system which analyses the mechanical stability of the simulated snowpack has been implemented directly within SURFEX. For each physical process represented by empirical parameterizations, several new parameterizations from the literature were implemented. The different combinations of these parameterizations constitute the ESCROC multiphysics ensemble model. It allows the quantification of simulations uncertainty for various applications. Finally, a technical solution was proposed for externalized applications allowing the use of the scheme in other Land Surface Models. The paper also reviews the available scientific evaluations and applications of the model. It describes its numerical efficiency and the main scientific and technical challenges providing guidance for the future of snow modelling.
European Research Council949516Introduction
A large variety of snowpack modelling systems e.g. have been developed for several decades for various applications : the computation of energy fluxes at the atmosphere-cryosphere interface in climate modelling and numerical weather prediction; hydrological simulations for discharge forecasting, water resources and hydropower management; physical process studies in the snowpack; avalanche hazard forecasting; glacier mass balance assessment; sea ice modelling; etc. The most detailed snowpack models include a detailed representation of the snowpack stratigraphy as well as an explicit representation of some microstructural properties of the snow layers through the implementation of empirical parameterization of snow metamorphism. There is a relatively limited number of snowpack models with such a level of detail: mainly the “Swiss” SNOWPACK model , the “American” SNTHERM model , the “Japanese” SMAP model and the “French” Crocus model. The latter has been initially developed by and was implemented in the early 2010s in the SURFEX platform of surface modelling to facilitate coupling with atmospheric models and the other components of surface modelling, especially soil and vegetation schemes of the ISBA Land Surface Model. The last description of the model was published by after this major evolution.
However, numerous evolutions of the model have been implemented after . Some of them are only partly described in dedicated scientific publications . These articles were generally published before the merging of all new developments in a unique and stable code version. A comprehensive and accurate description of the state of the last official model release is therefore missing. This lack of documentation has been identified by as one of the main factors of human errors in numerical simulations. The purpose of this paper is to provide an updated reference with all the functionalities available in the latest stable release of the model. To help the users to find any information they may need to understand the model implementation, all the equations disseminated in the various papers are reported here either in the main text or in Appendices, including the equations already published in previous references of and in order to provide a self-sufficient reference describing the whole model. Following the recommendations of , a major effort has been dedicated to check the consistency and comprehensiveness between the code and the equations. The scope of this paper is limited to the snowpack on the ground. The coupled modules representing soil, vegetation and blowing snow remain beyond the scope of this paper and only the terms involved in the coupling are mentioned. More details can be found in the reference publications of the coupled models: for soil and vegetation; and for blowing snow, cf. Sect. . Note that the standalone Crocus model was last designated as version 2.4. This versioning was discontinued by in favor of the SURFEX versioning system. However, due to the impossibility to synchronize Crocus and SURFEX main stable releases and the integration of Crocus in other Land Surface Models, a dedicated versioning is necessary for Crocus. This paper thus describes version 3.0.2 of Crocus. Section refmodel presents all scientific equations necessary to evolve the state variables of the model. Section presents complementary diagnoses computed as output of the model. Section presents technical features associated with running the model (simulation geometries, numerical efficiency, and associated tools to facilitate running and visualization). Finally, Sect. reviews the available scientific evaluations of the model and provides an overview of its applications. In light of these elements, we discuss the confidence that can be placed in the simulation results and the main remaining challenges for the future, including perspectives in terms of data assimilation.
Physical model
Compared to , the model has become highly modular with the extension of applications and the development of the ESCROC multiphysics framework by . The common modelling structure includes the discretization procedure and the solving of the heat diffusion equation which is the core of the model. Figure presents an overview of the sequence of routines and summarizes the main changes (new routines and updated routines) which are detailed in this section. Table summarizes the different available options for each process and associated requirements. The details of each option are described in the subsection dedicated to the corresponding process. The ESCROC multiphysics ensemble system consists in the combination of these differents physical options and different values of some key parameters such as τa (Sect. ), Ril (Sect. ).
Sequence of Crocus subroutines for each time step without (blue) or with (green) coupling with MEB. Optional subroutines in light blue. Compared to the version of , new subroutines are labelled “NEW”. The label “UPDATED” refers to subroutines in which changes were applied due to the new state variables for microstructure and where new physical options were implemented. The heat diffusion, melting and sublimation routines were only concerned by minor changes or technical optimizations.
Available physical options in Crocus. Default options appear in bold. The last column shows options which are not available in the externalized release but only within the SURFEX implementation of Crocus. Although the combination of all options is possible in theory, the options for snow management in ski resorts were only tested with the default options of the other processes and the blowing snow schemes were never tested together with the MEB vegetation scheme.
The model has to be forced with subdaily time series of air temperature Ta (K) and specific humidity qa (kg kg−1) at a known level za (m), wind speed U (m s−1) at a known level zu (m), incoming longwave radiation LW↓ (W m−2), incoming shortwave direct and diffuse radiation SWDIR↓ and SWDIF↓ (W m−2), rainfall Pr and snowfall Ps (kg m−2 s−1) and surface pressure Ps (Pa). The very low sensitivity of model results to Ps allows the user to provide constant Ps values when a time series is not available. The split of global shortwave radiation SW↓ between direct and diffuse components is only used with specific model options (coupling with TARTES optical scheme, Sect. , and/or coupling with MEB big-leaf vegetation scheme, Sect. ). When none of these options are activated, the split between both components is not necessary as only their sum is considered. When TARTES or MEB are activated, an accurate forcing of SWDIR↓ and SWDIF↓ should always be preferred. Nevertheless, for users interested in these options without available data to separate both components, the following parameterization as a function of the cosine of the solar zenithal angle μ and derived from SBDART clear-sky modelling at Col de Porte is available for mid-latitude areas. Following , the estimated error in cloudy conditions for the simulated broadband albedo is lower than 0.1. The parameterization is applied by default in the code if the diffuse component is set to zero:
SWDIFSWDIR+SWDIF=minexp-1.549919μ3+3.735357μ2-3.524211μ+0.029911,1..
Optionally, dry and wet deposition fluxes of Light-Absorbing Particles (LAP) Wd,j and Ww,j (kg m−2 s−1) can be included for each type j depending on the physical option selected for radiative transfer (Sect. ).
State variables
The prognostic variables (i.e. transmitted from a given time step to the next one) describing each snow layer i are its mass mi (kg m−2), density ρi (kg m−3), enthalpy Hi (J m−2) defined as the energy required to melt the snow layer i, age Ai (days since snowfall), and complementary variables for snow microstructure. The state variables initially used for snow microstructure as described in (dendricity, sphericity, grain size) were replaced by optical diameter di (m) and sphericity Si∈[0,1]. After several years of coexistence, the formulation from was removed from the code in order to improve its readability but also its efficiency as numerous expensive conditional statements could be removed. This change of state variables does not only affect metamorphism evolution laws but also various equations based on microstructure properties in other simulated processes. All modified equations are provided in the following section or in an appendix. In addition, a historical tracker hi is used as a last state variable with its meaning summarized in Table .
Possible values of the historical tracker hi.
Liquid water at any time before tFaceted crystals at any time before tMore than one freezing-melting cycle before t0NoNoNo1NoYesNo2YesNoNo3YesYesNo4YesNoYes5YesYesYes
Light Absorbing Particles (LAP) are organic or mineral substances able to increase radiation absorption in snow. added LAP mass contents Mi,j (kg m−2) as new optional state variables for each layer i and LAP type j. Although the code is designed to deal with any number of LAP types, only the parameters corresponding to black carbon and mineral dust are implemented in version 3.0.2. LAP interact with the other variable states of the model through absorption of solar radiation when the TARTES optical scheme is activated (Sect. ).
Note that the layer thickness zi (m), the layer temperature Ti (K) and the layer mass of liquid water li (kg m−2) can be directly derived from the state variables and are used in many parts of the model.
2zi=miρi3Ti=minT0,Ti*4li=max0,Ti*-T0micILm
where
Ti*=T0+HimicI+LmcI.T0 (K) is the water melting point temperature which in SURFEX is assumed to be equal to the triple point value. Lm (J kg−1) is the latent heat of fusion, and cI (J kg−1 K−1) the ice thermal capacity. Note also that the temperature θi in °C is also used in some parameterizations:
θi=Ti-T0.
LayeringVertical discretization
One of the main original feature of Crocus compared to the majority of snowpack schemes available in the literature is its Lagrangian vertical discretization based on a dynamical evolution of the number and thicknesses of the numerical snow layers. The number of active layers N varies between Nmin=3 and a user-defined maximum Nmax (the default value is 50). Each layer is referenced by the index i which varies from 1 for the surface layer to N for the deepest layer. The main discretization principles described in are still valid in the current version but they have been a bit complexified to solve numerical issues in specific applications. The current article provides the first comprehensive formulation of the algorithm.
First, a similarity criterion D(i,i+1) between adjacent layers i and i+1 is defined. The initial formulation was a Manhattan distance of weighted values of dendricity, sphericity and grain size but it has never been published. The strict variable transformation of led to a unjustified complexity in the formulation of this distance. Therefore, this distance was simplified in the same spirit by:
D(i,i+1)=50|Si+1-Si|+2000|di+1-di|+δh(i,i+1)ifAi-Ai+1<300D(i,i+1)=200otherwise
where
δh(i,i+1)=1ifhi>1andhi+1≤1orhi≤1andhi+1>1δh(i,i+1)=0otherwise.
The 2000 coefficient is chosen for normalization considering the typical range of 5×10-4 m between highest and lowest values of di and the range of 1 between highest and lowest values of Si. The δh function avoids the aggregation between cold snow and wet/refrozen snow. The specific case of significant age difference was introduced to avoid the aggregation of a recent snow layer with layers describing permanent snow or glacier. This distance is noted D(n,1) when it is applied between a new snowfall and the surface layer.
When it is necessary to choose two layers to aggregate, a penalty criteria P(i,i+1) is defined by:
P(i,i+1)=D(i,i+1)+25zizi*+zi+1zi+1*
where zi* (m) is the layer thickness of the optimal attractor profile defined in Appendix .
Layers lsup and linf are the adjacent layers which satisfy:
Plsup,linf=mini∈[1,N-1](P(i,i+1)).
The algorithm modifies the thicknesses at the previous time step zit-Δt for i∈[1,Nt-Δt] towards a new profile zit for i∈[1,Nt] following Eq. () by a succession of exclusive conditional statements depending on the total initial depth Zt-Δt=∑i=1Nzit-Δt, the initial thickness of the first layer z1t-Δt, the thickness of new snowfall to add zn=mnρn (m), and the initial number of layers Nt-Δt. The computations of the mass mn (kg m−2) and density of new snowfall ρn (kg m−3) are described later in Sect. .
Ifzn>0IfZt-Δt<ZminorNt-Δt<NminNt=maxNmin,minNmax,⌊100zn⌋zit=znNt∀i∈1,NtelseIfz1t-Δt<z1*andD(n,1)<0.2ormnΔt<3×10-5kgm-2s-1andz1<2z1*orz1t-Δt<10-4Nt=Nt-Δtz1t=z1t-Δt+znzit=zit-Δt∀i∈2,NtelseIfNt-Δt<NmaxNt=Nt-Δt+1z1t=znzit=zi-1t-Δt∀i∈2,NtelseNt=Nmaxz1t=znzit=zi-1t-Δt∀i∈2,lsupzlinft=zlsup+zlinfzit=zit-Δt∀i∈linf+1,NmaxelseIfz1t-Δt<5×10-3Nt=Nt-Δt-1z1t=z1t-Δt+z2t-Δtzit=zi+1t-Δt∀i∈2,NtelseIfzNt-Δtt-Δt<10-2Nt=Nt-Δt-1zit=zit-Δt∀i∈1,Nt-1zNtt=zNtt-Δt+zNt+1t-ΔtelseIf∃j∈1,minNt-1,Nmax-4|zjt-Δt>8-Nmax-Nt-ΔtNt-Δt-Nminzj*Nt=Nt-Δt+1zit=zit-Δt∀i∈[1,j-1]zit=zjt-Δt2∀i∈[j,j+1]zit=zi-1t-Δt∀i∈j+2,NtelseIf∃j∈2,Nt-1|zjt-Δt<5zj*D(i,i+1)andzjt-Δt+zj+1t-Δt<4.5maxzj*,zj+1*Nt=Nt-Δt-1zit=zit-Δt∀i∈[1,j-1]zjt=zjt-Δt+zj+1t-Δtzit=zi+1t-Δt∀i∈j+1,NtelseNt=Nt-Δtzit=zit-Δt∀i∈1,Nt.
In the definition of the initial number of layers (first line of Eq. ()), ⌊.⌋ designates the floor operator. The literal translation of Eq. () is as follows. A uniform layering is applied for new snowfall on the ground while the total snow depth does not reach Zmin=0.03 m. In other cases, snowfall is aggregated to the surface layer when it is similar to a sufficiently thin surface layer or when it is very low. When new snowfall is too thick or too different from the surface layer, a new layer is created. However, if Nmax was already reached, this layer creation is done after the preliminary aggregation of the two closest layers of the profile lsup and linf (as defined by Eq. ). When there is no snowfall, the algorithm applies only one of the following modification by order of priority: aggregation of surface layer or bottom layer when too thin, split of internal layer when too thick, aggregation of internal layer when too thin, relative to the optimal attractor profile. Note than only one modification by point is allowed at each time step. Thus, when the conditions for an internal split or aggregation are obtained for several layers (i.e. several values of j) at the same time, it is only applied for the uppermost layer min(j). For glacier applications, aggregation is strictly forbidden between snow and ice layers (i.e. when ρi<ρG and ρi+1>ρG where the threshold density for glacier ρG can be adjusted by the user (default 850 kg m−3).
This algorithm is applied independently at each simulation point. Therefore, the vertical layering differs between points in terms of number and thickness of snow layers. This raises a number of vectorization issues in the management of loops which were the topic of recent investigations detailed in Appendix .
Aggregation of layers
When two layers i and i+1 are aggregated (zit=zit-Δt+zi+1t-Δt from Eq. ) the state variables are modified in order to conserve mass and enthalpy. The optical diameter is updated in order to obtain a mass-weighted average of the corresponding albedo. A mass-weighted average is also applied for sphericity and age.
12mit=mit-Δt+mi+1t-Δt13ρit=mitzit14Hit=Hit-Δt+Hi+1t-Δt15dit=0.9-mit-Δt0.9-15.4dit-Δt+mi+1t-Δt0.9-15.4di+1t-Δtmi+mi+115.4216Sit=mit-ΔtSit-Δt+mi+1t-ΔtSi+1t-Δtmi+mi+117Ait=mit-ΔtAit-Δt+mi+1t-ΔtAi+1t-Δtmi+mi+118hit=hit-Δtifmit-Δt≥mi+1t-Δthi+1t-Δtotherwise19Wd,it=Wd,it-Δt+Wd,i+1t-Δt20Ww,it=Ww,it-Δt+Ww,i+1t-Δt
The same equations apply to aggregate falling snow with the surface layer replacing xit, xit-Δt and xi+1t-Δt by respectively x1t, x1t-Δt and xn for each x representing all state variables m, ρ, H, d, S, A, h, Wd and Ww in Eqs. ()–().
Splitting of layers
When a layer i is split into two layers i and i+1 (zit=zi+1t=zit-Δt2 from Eq. ), mass and enthalpy are divided into equal parts and microstructure properties are not modified:
21mit=mi+1t=mit-Δt222ρit=mitzit23Hit=Hi+1t=Hit-Δt224dit=di+1t=dit-Δt25Sit=Si+1t=Sit-Δt26Ait=Ai+1t=Ait-Δt27hit=hi+1t=hit-Δt28Wd,it=Wd,i+1t=Ww,it-Δt229Ww,it=Ww,i+1t=Ww,it-Δt2.
Evolution equationsSnowfall
The new snow amount is the result of solid precipitation, but also deposition of blowing snow and snowmaking, when the corresponding modules are activated. These three mass sources are respectively referred by the subscripts SP, BS and SM in the following. In case of simultaneous occurrence, only one additional layer is created with weighted physical properties. Thus, the mass of the new layer is defined by:
mn=mSP+mBS+mSM.
The density of new snow ρn, optical diameter dn, and sphericity Sn are expressed by:
31ρn=ρSPmSP+ρBSmBS+ρSMmSMmn32dn=dSPmSP+dBSmBS+dSMmSMmn33Sn=SSPmSP+SBSmBS+SSMmSMmn
The mass of solid precipitation during the time step Δt is directly provided by the forcing:
mSP=PsΔt.
Several empirical expressions of the density of natural falling snow ρSP are implemented , as a function of 2 m air temperature Ta* (°C) =Ta-T0 and 10 m wind speed U10 (m s−1). They can be activated through the SNOWFALL physical option following Eq. (). An illustration comparing these formulations is provided in , Fig. 1 in the corrigendum. The parameters are listed in Appendix . When the forcing wind speed is not available at a 10 m height, a logarithmic adjustment is applied following Appendix .
IfSNOWFALL=V12:ρSP=maxρmin,aρ+bρTa*+cρU10IfSNOWFALL=S14:log10ρSP=eρ+fρTa*+gρ+hρsin-1iρ+jρlog10maxU10,2ifTa*≥-14°Clog10ρSP=eρ+fρTa*+hρsin-1iρ+jρlog10maxU10,2otherwiseIfSNOWFALL=A76:ρSP=ρmin+maxkρTa*+lρ1.5,0.
Adjustments of parameter cρ were proposed by to account for the uncertainties associated with the impact of wind speed on Arctic snowfall density. Although hard-coded in version 3.0.2, this parameter will be tunable in a future version.
Several formulations of microstructure properties of new snow were implemented by depending on SNOWDRIFT option. Again, Appendix is used to assess the 5 m wind speed U5 from the forcing wind speed U at height zu. SNOWDRIFT options are reformulated here with the new microstructure variables now used in the model:
IfSNOWDRIFT=NONESSP=minmax0.0795U5+0.38,0.5,0.9dSP=10-4δNONE+1-δNONE4-SSPwhereδNONE=minmax1.29-0.173U5,0.2,1IfSNOWDRIFT=DFLTSSP=0.5dSP=10-4IfSNOWDRIFT=VI13SSP=minmax0.035U5+0.43,0.5,0.9dSP=10-4δVI13+1-δVI134-SSPwhereδVI13=minmax1.14-0.07U5,0.2,1IfSNOWDRIFT=GA01SSP=1-0.49δGA01dSP=10-4δGA01+1-δGA014-SSPwhereδGA01=minmax2.868exp-0.085U5-1,0,1.
VI13 refers to the approach of and GA01 refers to the approach of . The initial value of the historical tracker is hn=0 and the initial value of snow age is An=0. At each following time step, this variable remembers the duration from the snowfall event by the simple evolution:
Ait+Δt=Ait+Δt86400.
Blowing snow
The mass, density, optical diameter and sphericity of blowing snow (mBS, ρBS, dBS, SBS) can be provided by an external snow transport module. Within SURFEX, three different snow transport modules can be coupled to Crocus. The SYTRON module is designed to simulate erosion and accumulation from the windward to the leeward side of the crests in French operational simulations based on an idealized topography. The SnowPappus module comprehensively described by is designed for gridded simulations at horizontal resolutions between 30 and 250 m . Finally, snow transport can be solved by an explicit coupling with the MESO-NH Large Eddy Simulation model for process studies on small study domains and at the event scale .
Machine-made snow
A representation of machine-made snow can optionally be provided by a dedicated module activated by the logical option SNOWMAK_BOOL. This enables the model to compute the mass, density, optical diameter and sphericity of machine-made snow (snowmaking) (mSM, ρSM, dSM, SSM) and its interaction with the rest of the snowpack. First, the wet bulb temperature Tw (expressed in K) is computed following Eq. () in Appendix . The machine-made production is allowed if both the wet bulb temperature and the wind speed are lower than the respective user-defined thresholds Tlim and Ulim. When the meteorological conditions are satisfied, snowmaking is activated only during two periods over the winter season: the “base-layer generation” production period and the “reinforcement” production period . The dates of beginning and end of each perio day1 to day3 and t1 to t4 described in Appendix . Two different strategies can be adopted depending on the SELF_PROD logical option. When it is set to True, the production follows a pre-defined set of rules. In this case, during the base-layer generation period the production is allowed until a given amount of water plim (kg m−2) is used. During the reinforcement period, instead, the production is allowed if the total snow depth Z is lower than a threshold Zlim. If SELF_PROD is False, the production is forced to match a water use target (kg m−2) defined by the user for each simulated point, regardless of any meteorological or timing condition.
The mass of machine-made snow is then obtained by:
mSM=1-LSMISMASMaSMTw-T0+bSMΔt
where ISM=1 when production is allowed, and 0 otherwise, ASM is the surface area covered by a snow gun set to 3300 m2, and LSM the loss factor set to 0.4 by . aSM (kg K−1 s−1) and bSM (kg s−1) are regression coefficients of the parameterization of the potential mass produced by a snowgun. They can be adjusted by the user or derivated from .
The density ρSM of machine-made snow can be adjusted by the user whereas the snow microstructure properties dSM and SSM are fixed following .
For more information about the implementation of the snowmaking practices in Crocus, please refer to and .
Freezing rain
When liquid precipitation occurs at negative temperatures, the model can simulate the formation of an ice layer at the top of the surface following the work of . In a first step, the whole precipitation mass is assumed to form a new ice layer at T0, only aggregated to the previous surface layer if a thin ice layer is already present (Eq. 39)
The energy associated with the phase change EFRZ (W m−2) is computed by Eq. () and accounted for further as an additional energy source in the surface energy balance (Eq. ), able to partly melt this new ice layer:
IfPr>0andTa<T0:EFRZ=1-ϕFRZLmPrΔtwhereϕFRZ=cWLmT0-Taelse:EFRZ=0.
The assumption behind Eq. () is that a fraction ϕFRZ of the latent heat release due to refreezing is consumed by the increase of temperature from Ta to T0 and the remaining part is fully stored by the surface layer and can either be available for melting or be partly dissipated through diffusion or heat exchanges with the atmosphere, as solved later in Sect. and .
Note that some other implementations of this process in other models consider than while all the precipitation eventually freezes and contributes to the formation of an ice layer at the surface, only a fraction of the associated latent heat is kept in the ice layer, while the remaining latent heat is transferred to the atmosphere at a shorter time scale than the model time step .
Metamorphism
The prognostic equations of microstructure variables di and Si are still fully empirical in the absence of physical evolution laws. They depend on conditions of the vertical temperature gradient Gi estimated by:
G1=2|T2-T1|z1+z2Gi=2|Ti+1-Ti-1|zi-1+2zi+zi+1∀i∈[2,N-1]GN=2|TN-TN-1|zN-1+zN.
Following and , the evolution of sphericity ΔSi during a time step Δt is defined by Eq. () (dry metamorphism) and Eq. () (wet metamorphism):
42Ifli=0:IfGi<5Km-1IfSi<1ΔSi=sph1e-6000/Tifcorrdi,Si,hiΔtelseΔSi=0elseIfSi>0ΔSi=-sph2e-6000/TiGi0.4ΔtelseΔSi=043Ifli>0:IfSi<1:ΔSi=maxsph3100liρizi3,sph2e-6000/T0ΔtIfSi=1:ΔSi=0.
Parameters sph1, sph2 and sph3 are provided in Appendix . fcorr(di,Si,hi) is an unpublished parameterization in the original code of which modifies the general behaviour of Eq. () by reducing the sphericity increase of snow layers with microstructure properties typical of depth hoar or large faceted crystals and submitted to low thermal gradients. This prevents the formation of rounded grains from depth hoar or large faceted crystals. Indeed, the persistence of anisotropy in this case was obtained with a phase-field numerical model applied to snow microstructure and recently confirmed by unpublished tomography observations.
IfSi≥0.5;hi=1andgsidi,Si>5×10-4mfcorrdi,Si,hi=0IfSi<0.5;hi=1andgsidi,Si>5×10-4mfcorrdi,Si,hi=exp3×10-4-gsidi,Si10-4otherwisefcorrdi,Si,hi=1
hi is a tracker of the snow layer history (Table ) for which odd values (1, 3, 5) corresponds to the occurrence of depth hoar at any time since the layer creation. gsi(di,Si) is a variable originally used to describe snow microstructure by (grain size). Its retrieval from the current state variables di and Si is described in Appendix . It must be noticed that this parameterization was forgotten by because it was unpublished and was only recently restored in the code.
Several prognostic evolutions of di during dry metamorphism were implemented by and , following the works from , and . Furthermore, several issues were found by in the translation of the formalism (using dendricity and grain size) in terms of optical diameter evolution by (details in Appendix ). A new parameterization (B21) was therefore recently implemented to solve the associated issues and now replaces the original implementation of now removed from the code. The resulting available evolution laws of di during dry metamorphism are given by Eqs. ()–().
45IfSNOWMETAMO=B21andli=0:Ifdi≥10-44-Si:IfGi>15Km-1andSi=0:Δdi=12fθihρigGiΦΔtelse:Δdi=di-4×10-41+SiΔSielse:Δdi=10-4-sph2e-6000/TiJiSi-3+ΔSiΔt104di-1Si-3ΔtwhereJi=Gi0.4ifGi≥5Ji=1ifGi<546IfSNOWMETAMO=F06andli=0:Δdi=2r˙0ρi,Ti,Giτρi,Ti,Gidi2-r0+τρi,Ti,Gi1κρi,Ti,GiΔt47IfSNOWMETAMO=S–Bandli=0:IfAi≤2dΔdi=asρidi26+bsTi6ρims-1dims-2else:Eq.(46)48IfSNOWMETAMO=S–Fandli=0:IfAi≤2dEq.(48)else:Eq.(47).
Functions f(θi), h(ρi), g(Gi) and parameter Φ used in Eq. () for the growth of faceted crystals in the case of high gradient metamorphism follow and are defined in Appendix . In Eq. () from , coefficients r˙0(ρi,Ti,Gi), τ(ρi,Ti,Gi) and κ(ρi,Ti,Gi) are retrieved from look-up tables provided in a parameters file (cf. section on Data and Code availability) and ro=5×10-5 m. Experimental parameters as, bs, ms in Eq. () are defined in Appendix from .
Regardless of the SNOWMETAMO option, wet metamorphism is always parameterized with the laws published by , rewritten here with the new microstructure prognostic variables and the methodology described in Appendix :
Ifli>0Ifdi<10-44-Si:IfSi<1:Δdi=10-4104di-1Si-3-(Si-3)ΔSiIfSi=1:Δdi=2.10-4maxsph3100liρizi3,sph2e-6000/T0Δtelse:IfSi<1:Δdi=di-4.10-4Si+1ΔSiIfSi=1:Δdi=2πv0+v1100liρizi3di2Δt.
To follow its definition in Table , the historical tracker hi is updated at the end of this subroutine by:
Ifdi≥10-44-Si:IfSi=0andhit=0:hit+Δt=1else:IfSi=1andliρizi>0.005:Ifhit=0:hit+Δt=2Ifhit=1:hit+Δt=3else:hit+Δt=hitelse:IfTi<T0:Ifhit=2:hit+Δt=4Ifhit=3:hit+Δt=5else:hit+Δt=hitelse:hit+Δt=hitelsehit+Δt=hit.
Natural compaction
For a given layer of density ρi the mechanical settling under the over burden σi (Pa) is expressed with a visco-elastic model :
IfSNOWCOMP∈[B92,T11]:Δρi=ρiσiηiΔt
where
σi=gcosΘ∑j=1i-1ρjzj∀i∈[2,N]σ1=gcosΘ×0.5ρ1z1.g is the gravitational acceleration and Θ the slope angle from horizontal.
The viscosity ηi is a function of density ρi and temperature θi (°C) depending on the SNOWCOMP option :
IfSNOWCOMP=B92:ηi=f1wif2di,Siη0ρicηeaη-θi+bηρiIfSNOWCOMP=T11:ηi=f1wif2di,Si⋅0.05ρi-0.0371θi+4.410-4e0.018ρi+1.
The parameters for case B92 are defined in Appendix . The multiplicative function f1 accelerates the settlement of wet snow as a function of the volumetric liquid water content wi. The multiplicative function f2 reduces the settlement of layers consisting of faceted snow types as a function of microstructure properties di and Si (translation of the formula from in the new formalism).
54f1wi=11+60wiρw55f2di,Si=min4,expmin4,gsidi,Si×104-2ifdi>4-Si×10-4andSi≤0.51otherwise
where gsi(di,Si) is defined by Eq. (). The compaction rate however has a complex dependence on snow microstructure which cannot be described by the representation of snow microstructure in Crocus. Alternatively to Eq. (), it is possible to use a parameterization from derived from tomographic observations and representing a non-linear relationship between settlement, the stress σi (Pa) and the optical diameter increase for the first 48 hours after snowfall:
IfSNOWCOMP=S14:Δρi=BSΔdidi2σikSifAi≤2dEq.(53)ifAi>2days
with BS=3.96×10-2 and kS=0.18. The current Crocus parameterization is applied when the snow layer age exceeds 2 d.
Alternative parameterizations reducing compaction in the presence of low vegetation were proposed by combining earlier works from and . Although not available in version 3.0.2, they will be implemented in a future version.
Grooming
The effects of grooming machines (snowcats) on the snow physical properties include the compaction induced by their overburden weight and the mixing of surface layers produced by the tiller. Both these effects can optionally be simulated in Crocus .
The compaction effect is activated using the logical switch SNOWCOMPACT and only applies if the total mass M=∑i=1Nmi is higher than 20 kg m−2 (a minimum value required by the grooming machines) and between 20:00 and 21:00 LT (and also 06:00–09:00 LT in case of snowfall during the night). Grooming starts on 1 November and continues until the date dayEND chosen by the user. Its frequency fGRO (number of grooming sessions per day) is also user-defined. It is worth noticing that it is possible to activate grooming without activating snowmaking, but the opposite would not be realistic. The static stress due to the weight of the snowcat itself σGROi (Pa) is expressed for layer i by:
IfSNOWCOMPACTIf∑j=1imj<50kgm-2:σGROi=gcosΘ×500If50kgm-2≤∑j=1imj<150kgm-2:σGROi=gcosΘ×150-∑j=1imj⋅5If∑j=1imj≥150kgm-2:σGROi=0
and the corresponding density change is obtained by replacing σi by σGROi in Eq. ().
The tilling effect is activated using the logical switch SNOWTILLER, applies down to 35 kg m−2 below the surface and only if SNOWCOMPACT = True. The tiller mounted at the rear of snowcats produces two main effects: it further increases the density of the snow by loading the snowpack with extra pressure and it modifies the snow microstructure by creating smaller, rounded grains. As a result, all impacted layers are mixed together, their properties are homogenized and some of them are modified . These effects are simulated in Crocus by directly modifying the density, optical diameter and sphericity of the impacted layers. The density reached by the snowpack after grooming is parameterized as:
IfSNOWTILLER:ρi=maxρ‾,2ρ‾+3ρGRO5∀i∈[1,k]|∑i=1kmi≤35kgm-2
where ρ‾=∑i=1kρimi∑i=1kmi is the weighted average density of the k impacted layers and ρGRO is the target density that should eventually be reached by the grooming process . The optical diameter and sphericity of snow are altered analogously, using the respective target values dGRO and SGROAppendix ,.
Drifting snow
Even if the Crocus model is not coupled with a dedicated module able to transport snow from one simulation point to another, it is possible to activate a parameterization from simulating the impact of drifting snow on compaction and metamorphism, but with mass conservation. This parameterization relies on two possible definitions of a mobility index MOBi:
IfSNOWMOB=GM98Ifdi<10-44-Si:MOBi=0.51-Si+0.75δielse:MOBi=0.8331-Si-0.583gsiIfSNOWMOB=VI12Ifdi<10-44-Si:MOBi=0.34×0.51-Si+0.75δi+0.66Fρielse:MOBi=0.34×0.8331-Si-0.583gsi+0.66Fρi
where
F(ρ)=1.25-0.004maxρmin,ρ-ρminandρmin=50kgm-3.
In addition, a threshold is applied on the mobility index in case liquid water had been present at any time in the snow layer:
Ifhi≥2:MOBi=min-0.0583,MOBifromEq.61.
The mobility index was expressed with the original formalism of metamorphism. The conversion functions δi(di,Si) and gsi(di,Si) are defined in Appendix refAmetamob21 by Eqs. () and (). Redefining this index as a function of the current state variables (di and Si) would be more consistent and flexible but it would require a significant new effort of evaluation. It highlights the issues involved by changes in the variable states in a model which have been underestimated in the work of .
A driftability index Di is then obtained by combining the mobility index and the 5 m wind speed U5:
Di=maxMOBi-2.868⋅exp-0.085×U5-1.,0.
We then introduce:
DEFFi=Di×exp-10×∑j=1i-1zj3.25-Dj+zi23.25-Di
to formulate the variation of density due to snow drift by:
Δρi=DEFFimaxρMAX-ρi,0)ΔtτDRIFT
with ρMAX=350 kg m−3 and τDRIFT=172800 s (2 d). Adjustments of parameters ρMAX and τDRIFT were proposed by several authors for Arctic snow as summarized by . Although hard-coded in version 3.0.2, they will be adjustable by the user in a future version. Then, the variation of microstructure properties due to fragmentation during snow transport are obtained by Eq. (). The origin of this expression is provided in Appendix .
65Ifdi<10-44-Si:Δdi=DEFFi×10-42.5-1.5Siδi-1+SiΔtτDRIFTelseΔdi=DEFFi-5×10-41+Si2+di-4×10-41-Si1+SiΔtτDRIFT66ΔSi=DEFFi1-SiΔtτDRIFT
Optionally, a mass loss due to blowing snow sublimation can be estimated and removed from the surface layer. The parameterization is inspired from with a modification of a threshold wind speed Ut to account for the microstructure-related mobility index. This option is not activated by default due to the large associated uncertainties and lack of evaluation but is considered to be necessary in polar environments . The mass reduction of the surface layer due to sublimation Δm1 is obtained by:
Δm1=min0.5m1,aSUBLT0TaγSUBLUtρaqsatTa,Ps1-qaqsatTa,PsU5UtbSUBLΔt
where
Ut=-logMOB1+1cSUBLdSUBL.
The air density ρa (kg m−3) and the saturation specific humidity qsat(Ta,Ps) (kg kg−1) are computed following Appendix . aSUBL, bSUBL, γSUBL, cSUBL, dSUBL are dimensionless parameters defined in Appendix .
Absorption of solar radiation
Two schemes of different complexities are currently implemented and can be activated through the SNOWRAD option. When SNOWRAD=B92, the initial 3-band scheme of is applied, inspired from . The incoming solar radiation at the interface between layers i and i+1 is defined by:
Ri=∑k=131-αkγkSW↓exp-∑j=1iβkjzj∀i∈[0,N]ifSNOWRAD=B92.
The spectral partitioning of incoming radiation is fixed by parameters γk (Appendix , with ∑k=13γk=1).
Spectral albedo values αk are parameterized by:
αk=χαk1+(1-χ)αk2∀k∈[1,3]
where
71χ=0.8min1,z10.02+0.2min1,max0,z1-0.020.0172α1i=max0.6,min0.92,0.96-1.58di-0.2Aiτa×minmax0.5,PsPCDP,1.5ifρi<ρGα1i=α1Gifρi≥ρGfori∈[1,2]73α2i=max0.3,0.9-15.4diifρi<ρGα2i=α2Gifρi≥ρGfori∈[1,2]74α3i=346.3di′-32.31di′+0.88ifρi<ρGwheredi′=mindi,0.0023α3i=α3Gifρi≥ρG.
Note that compared to , the consideration of optical diameters of the first two layers, already in the code but not documented, is now made explicit in Eq. (). The time constant τa in Eq. () is the main control of the parameterization reducing snow albedo in the visible band as a function of the age of the layer Ai in order to mimic the effect of Light-Absorbing Particles (LAP). Its default value is set to 60 d with an elevation-dependent multiplicative correction factor (function of Ps) assuming that LAP deposition decreases with elevation. However, it is recommended to adjust this parameter depending on the expected amount of LAP in the target region , to consider calibration against observed albedo time series when possible, or to apply several values of this parameter in multiphysics applications . The last modification about absorption of solar radiation consists in applying constant glacier spectral albedo values αkG on snow-free glacier surfaces. Default values (Appendix ) are taken from but must be adjusted to each specific glacier e.g.
Absorption coefficients βki (m−1) for band k and layer i are parameterized by:
75β1i=max40,0.00192ρidi∀i∈[1,N]76β2i=max100,0.01098ρidi∀i∈[1,N]77β31=+∞.
Alternatively to Eq. (), an option is available for solar radiative transfer calculation in the snowpack (SNOWRAD = T17) combining the TARTES radiative scheme Two-streAm Radiative TransfEr in Snow model, and an explicit modelling of LAP . TARTES is a two-stream radiative transfer scheme based on an analytical formulation of radiative transfer in snow . The scheme is applied separately for the direct (DIR) and diffuse (DIF) components of solar radiation. For both components, TARTES computes spectral solar absorption within each layer EkiEqs. 59–63 in the comprehensive scientific documentation of TARTES, and the spectral albedo αkEq. 63 in. Equation () is used to come back to the profile of Ri used in Crocus:
Ri=∑k=1Nk1-αkDIRγkDIRSWDIR↓-∑j=1iEkjDIR+1-αkDIFγkDIFSWDIF↓-∑j=1iEkjDIF∀i∈[0,N]ifSNOWRAD=T17.
The default spectral resolution is 20 nm for Nk=111 spectral bands in the interval [300–2500 nm]. Coefficients γkDIR and γkDIF to split input direct and diffuse broadband radiation in spectral solar irradiance are currently provided as fixed parameters derived from SBDART under the conditions encountered at Col de Porte.
To compute Eki, TARTES accounts for the effect of snow physical properties (SSA and density) and Light-Absorbing Particles using the mass absorption efficiency of each LAP type and its mass content vertical profile. For dust, the mass absorption efficiency is defined following Eq. (83) of with parameters λ0=400 nm, MAE(λ0)=110 m2 kg−1 and AAE=4.1values for dust PM2.5 from Libya in Table 4 of. For black carbon, the mass absorption efficiency is defined following Eq. (82) of with a constant density ρBC=1270 kg m−3 and a constant refractive index mBC=1.95-0.79i from. The MAE is then scaled with a multiplicative factor fBC=1.638 to obtain an MAE value at 550 nm of 1.125×104 m2 kg−1 consistently with measurements of . The scaling makes it possible to implicitly account for the potential absorption enhancement due to internal particle mixing or particle coating. The implementation of new types of LAP would require the implementation of the description of their associated mass absorption efficiency. The vertical profile of LAP mass content is obtained following Sect. .
Light-absorbing particles
From , the evolution law of the mass Mi,j of LAP j in layer i is expressed as follows:
79M1,j+=M1,j-+Ww,jΔt+Wd,jΔtexp-z1h∑k=1Nexp-Zkh80Mi,j+=Mi,j-+Wd,jΔtexp-Zih∑k=1Nexp-Zkh∀i∈[2,N]
where Zi represents the depth of layer i from the surface Zi=∑j=1izj and h the e-folding depth characterizing the decrease rate with depth of the impact of the dry deposition flux Wd,j. The wet deposition flux Ww,j only affects the uppermost layer.
However, in case of thin layers at the surface, a homogeneous repartition is applied on the uppermost N10 layers gathering the first 10 kg m−2 of snow. This limits artificial albedo variations due to vertical regridding :
Mi,j+=mi∑k=1N10mk×∑k=1N10Mk,j+∀i∈1,N10
where N10 is the highest integer such that ∑k=1N10-1mk<10 kg m−2.
Note than the wet deposition flux should be consistent with the occurrence of precipitation in the model input data. If not, any positive wet deposition flux without precipitation is not incorporated in the snowpack LAP mass content .
Turbulent fluxes
The sensible and latent turbulent heat fluxes are expressed by:
82H=ρacPCHU>1T1Πs-TaΠa83LE=ρaLsCHU>1qsatT1,Ps-qa.ρa is the air volumetric mass (kg m−3), Πs and Πa the Exner functions at the surface and at the level of the atmospheric forcing and qsat(T1,Ps) the saturation specific humidity at surface temperature T1 (kg kg−1), cf. Appendix for their computations. A minimum value of 1 m s−1 is applied to wind speed (U>1=max(U,1 m s-1)) to maintain minimal fluxes even with very low wind speeds. Note that a simplification has been introduced compared to by considering only surface sublimation and not evaporation of liquid water. The exchange coefficient CH depends on the stability of the atmosphere through the Richardson Number Ri (Appendix ) following :
CH=κVK2lnzuz0lnzaz0h1-15Ri1+Ch|Ri|ifRi<=0CH=κVK2lnzuz0lnzaz0h×11+15Ri*1+5Ri*ifRi>0whereRi*=minRi,Ril
where κVK is the Von Karman constant. Coefficient Ch in the unstable case is defined by:
Ch=153.2165+4.3431lnz0z0h+0.5360lnz0z0h2-0.0781lnz0z0h3k2lnzuz0lnzaz0hzaz0hp
where
p=0.5802-0.1571lnz0z0h+0.0327lnz0z0h2-0.0026lnz0z0h3.
In the stable case, an adjustable threshold Ril is applied on the Richardson number . Considering the high uncertainty of these parameterizations of turbulent processes, sensitivity analyses to momentum and thermodynamic roughness lengths z0 and z0h m) and to parameter Ril are recommended for robust applications of the model .
Heat diffusion and energy balance
The formalism of this section is largely inspired by the documentation of the ISBA-ES (Explicit Snow) snowpack scheme from . The model solves the heat diffusion in the stratified snowpack using an implicit time integration scheme:
cIρiziTi+-Ti-Δt+Ei=Gi-1+-Gi+∀i∈[1,N]
where Gi+ (W m−2) represents the heat flux between layers i and i+1∀i∈[1,N-1] at the end of the time step. G0+ and GN+ are the heat fluxes at the interfaces with atmosphere and soil at the end of the time step and Ei (W m−2) the energy of phase change for layer i. The main assumption of the model is that it is possible to separate heat diffusion and phase changes. Thus, the diffusion is solved assuming Ei=0∀i∈[1,N].
The heat flux between two snow layers i and i+1 is the sum between the radiative flux Ri and the heat conduction Ki+ (W m−2) between these layers:
Gi+=Ki++Ri∀i∈[1,N-1].
The conduction flux Ki is expressed at the end of the time step by:
Ki+=2λi‾Ti+-Ti+1+zi+zi+1∀i∈[1,N-1]
where λi‾ is the harmonic weighted mean of the thermal conductivities of layers i and i+1:
λi‾=zi+zi+1ziλi+zi+1λi+1.
Equation () replaces the arithmetic mean used in previous versions of Crocus . Although there is not a clear agreement in the literature that harmonic means outperform the modelling of heat diffusion , this solution is more consistent with the other components of SURFEX and more commonly used today in snow models .
The thermal conductivity λi of layer i is parameterized as a function of density ρi following parameterizations of , or :
IfSNOWCOND=Y81:λi=maxaλρiρw1.88;λminIfSNOWCOND=C11:λi=bλρi2+cλρi+dλIfSNOWCOND=I02:λi=eλ+fλρi2+gλ+hλTi-+iλP0P.
All empirical parameters are provided in Appendix . Alternatives to Eq. () were proposed by for Arctic snow. Although not available in version 3.0.2, they will be implemented in a future version.
The heat flux between the atmosphere and the surface is the sum of all energy fluxes at the surface:
G0=R0+ϵLW↓-σT14-H-LE+PrΔt×cWTa-T0+EFRZ.
All fluxes in Eq. () are expressed at time t+Δt with the following approximation:
F+=F-+∂F∂T1T1+-T1-.
The computation of ∂qsat∂T1 is detailed in Appendix . The heat flux between the bottom layer and the ground GN+ is expressed with a semi-implicit coupling (i.e. considering the ground surface temperature TG1- at time step t) in order to solve separately the thermal diffusion in the soil, outside the snowpack model:
GN+=2λN‾TN+-TG1-zN+zG1+RN.
where λN‾ is the harmonic mean between the thermal conductivity of the bottom snow layer λN and the thermal conductivity of the first soil layer λG1:
λN‾=zN+zG1zNλN+zG1λG1/
Combining and rearranging Eqs. ()–(), () and () the system to solve becomes:
cIρ1z1Δt+2λ1‾z1+z2+4ϵσT1-3+ρaCHU>1cPΠs+Ls∂qsat∂T1T1+-2λ1‾z1+z2T2+=cIρ1z1ΔtT1-+R0-R1+ϵLW↓+3σT1-4+ρaCHU>1cPTaΠa+Lsqa-qsatT1-,Ps+∂qsat∂T1T1-+PrΔt×cWTa-T0+EFRZ-2λi-1‾zi-1+ziTi-1++cIρiziΔt+2λi‾zi+zi+1+2λi-1‾zi-1+ziTi+-2λi‾zi+zi+1Ti+1+=cIρiziΔtTi-+Ri-1-Ri∀i∈[2,N-1]-2λN-1‾zN-1+zNTN-1++cIρNzNΔt+2λN‾zN+zG1+2λN-1‾zN-1+zNTN+=2λN‾zN+zG1TG1-+cIρNzNΔtTN-+RN-1-RN.
The first line of Eq. () is rewritten in the form:
1CTΔtT1+-ζ2CTΔtT2+=ζ1CTΔt
where
99CT=1cIρ1z1100ζ2=2CTλ1‾Az1+z2101A=1Δt+CT2λ1‾z1+z2+4ϵσT1-3+ρaCHU>1cPΠs+Ls∂qsat∂T1102ζ1=BT1-+CA103B=1Δt+CT3ϵσT1-3+ρaCHU>1∂qsat∂T1104C=CTR0-R1+ϵLW↓+ρaCHU>1cPTaΠa+Lsqa-qsatT1-,Ps+PrΔt×cWTa-T0+EFRZ.
Therefore, the temperature profile at time t+Δt is obtained by:
T1+T2+…Ti+……TN+=B1C100……0A2B2C20……00A3……………00…BiCi…0………Ai……0………………CN-100…00ANBN-1Y1Y2…Yi……YN
where vectors A=(A2,…,Ai,…AN), B=(B1,…,Bi,…BN), C=(C1,…,Ci,…CN-1), Y=(Y1,…,Yi,…YN) are defined by:
Ai=-2λi-1‾zi-1+zi∀i∈[2,N]B1=1CTΔtBi=cIρiziΔt+2λi‾zi+zi+1+2λi-1‾zi-1+zi∀i∈[2,N-1]BN=cIρNzNΔt+2λN‾zN+zG1+2λN-1‾zN-1+zNC1=-ζ2CTΔtCi=-2λi‾zi+zi+1∀i∈[2,N-1]Y1=ζ1CTΔtYi=cIρiziΔtTi-+Ri-1-Ri∀i∈[2,N-1]YN=2λN‾zN+zG1TG1-+cIρNzNΔtTN-+RN-1-RN.
Adjustments in case of surface melting
The possibility for T1+ to exceed the freezing point in the solving of Eq. () can lead to overestimate the surface energy fluxes that depend on T1+ and to overestimate the heat conduction K1 below surface. To avoid this numerical artefact, Crocus distinguishes the case of first melting (T1-<T0 and T1+>T0), and ongoing melting (T1-≥T0 and T1+>T0).
In the case of a first melting (T1-<T0 and T1+>T0), the temperature profile for layers i∈[2,N] is not updated, so only the surface fluxes are adjusted replacing T1+ by T0 in Eqs. (), () and (). The new temporary surface temperature (before melting) is obtained by converting the difference in both consecutive estimates of the surface energy flux G0 in terms of temperature change:
IfT1-<T0andT1+>T0:G0+=R0+ϵLW↓-σT1-34T0-3T1--ρacPCHU>1T0-Ta-ρaLsCHU>1qsatT1-,Ps-qa+∂qsat∂T0T0-T1-+PrΔt×cWTa-T0T1+=T1first+ΔtcIρiziG0+-G0first
where T1first is the solution of Eq. (), and G0first is the flux obtained applying Eq. () with T1first. This ensures that the energy budget over the snowpack is closed.
In the case of an ongoing melting (i.e. the solution of Eq. () provides a temperature above freezing point for the surface layer at two consecutive time steps: T1-≥T0 and T1+>T0), the system is solved a second time for layers i∈[2,N] by constraining T1+=T0:
T2+…Ti+……TN+=B2C20……0A3……………0…BiCi…0……Ai……0……………CN-10…00ANBN-1Y2+2λ1‾z1+z2T0…Yi……YN.
Then, the surface energy fluxes are updated replacing T1+ and T1- by T0 in Eqs. (), (), () and the conduction flux between the two first layers is updated with the solution of Eq. (). A new estimate of the surface layer T1+ is temporarily obtained from Eq. () (i=1) consistently with these updated fluxes, before the transfer of exceeding energy in phase change (E1, see Sect. ):
IfT1-≥T0andT1+>T0:G0+=R0+ϵLW↓-σT04-ρacPCHU>1T0-Ta--ρaLsCHU>1qsatT0,Ps-qa+PrΔt×cWTa-T0K1+=2λi‾T0-T2+zi+zi+1T1+=T0+ΔtcIρiziG0+-K1+-R1.
This also guarantees the closure of the energy budget in that case. Note that recently proposed alternative model formulations to compute a more stable surface energy balance with a better coupling between surface melting and heat transfers. This should be explored in the future to avoid the need of such numerical adjustments.
Coupling with MEB
MEB Multiple Energy Balance, is a variant of the ISBA land surface scheme in which the soil-vegetation system is no longer described by a composite approach but by an explicit representation of vegetation with a big-leaf approach. This allows representation of the main physical processes involved in forest-snow interactions including snowfall interception and radiative impacts of the trees. An extensive description of this implementation is beyond the scope of this paper but available in . However, as coupled processes require a coupled solving of snow surface and vegetation temperatures T1 and Tv, the solving of heat diffusion is modified when Crocus is coupled to the MEB scheme. In that case, the values of the surface energy fluxes are no longer obtained through the implicit solving of Eqs. ()–() but are imposed as a boundary condition to conserve energy. Equation () is modified for the surface layer to maintain the fluxes obtained from MEB:
cIρiziT1+-T1-Δt+E1=G0MEB-R1-K1+
where
G0MEB=R0+ϵLW↓-σT1MEB4-HMEB-LEMEB+PrΔt×cWTa-T0.G0MEB represents the total surface energy flux in agreement with the first estimate of surface temperature from MEB T1MEB and the associated turbulent fluxes HMEB and LEMEB. LW↓ accounts for vegetation radiation and is therefore linked to the solution obtained for vegetation temperature TvMEB. It must be noted that as R0 and R1 are already needed in MEB, they are computed with the same routine as described in Secti. but earlier in the time step (before the MEB solving) to guarantee identical values at both steps (MEB and Crocus), and accounting for shading by trees in SW↓.
Consistently with Eq. (), the first line of Eq. () is replaced by:
cIρ1z1Δt+2λ1‾z1+z2T1+-2λ1‾z1+z2T2+=cIρ1z1ΔtT1-+G0MEB-R1
Rewriting Eq. () with the same form as Eq. () is equivalent to replacing Eqs. (), () and () by Eqs. (), (), () and solving the linear system of Eq. () with coefficients C1 and Y1 of Eq. () modified according to the new values of ζ2 and ζ1 from Eqs. () and ().
113A=1Δt+CT2λ1‾z1+z2114B=1Δt115C=CTG0MEB-R1
The adjustments of Sect. are no longer required as G0 is imposed by MEB and phase change was already accounted for to compute this flux.
The validity of Eq. () would require that no modification of the state variables of the snowpack has occurred between the computation of G0MEB and the solving of Eq. () because G0MEB depends on snow properties through Eqs. (48) and (49) in and the associated Appendix I4. In practice, the SURFEX code structure makes this constraint especially challenging. The violation of this assumption can generate numerical instabilities especially with thin surface layers due to the violation of the second principle of thermodynamics . To reduce as much as possible the occurrence of this problem, the sequence of routines is modified following Fig. . However, there is currently no solution to avoid modifications due to snow interception by vegetation because this term cannot be computed before solving MEB (the mass balance depends on latent heat terms known only after the solving). Numerical instabilities are therefore still possible and further investigations are in progress to safely allow large scale applications of MEB-Crocus. This issue is more challenging than in the case of the coupling of MEB with ES snow scheme which has already been successfully applied in large scale simulations. This is probably due to the possible occurrence of thinner surface snow layers with Crocus.
Total melting or sublimation
The snowpack is assumed to totally disappear when
G0+-GN+≥-∑i=1NtHi++HsublΔtormax0,LE+ΔtLs≥∑i=1Ntmi.
The first condition corresponds to a total melt of the snowpack (energy gain during the time step exceeds available internal energy for melting). It requires to remove the enthalpy of surface snow potentially sublimated Hsubl from the total enthalpy. The second condition corresponds to a total sublimation of the snowpack, which is a very unusual case of snow disappearance but can appear especially when a snow transport module provide a very low amount of snow on bare ground in cold and windy conditions.
In both cases, for mass conservation, the runoff at the bottom of the snowpack is set to the total snow mass and the energy associated with LE+ is transmitted to the soil scheme as a correction term. If Eq. () is not verified, the procedure follows Sect. to .
Melting
Let us define the mass of solid and liquid fractions of the snow before any phase change si- and li- (kg m−2):
si-=mi-li-.
When the heat diffusion solving provides a temperature Ti+ above the melting point T0 for layer i, the model simulates melting. The energy available for fusion Efi (J m−2) on layer i can be constrained either by the available heating energy or by the available mass of the solid fraction of snow before melting si-. Thus, the energy and mass fi (kg m−2) of melting during the time step are computed by:
118Efi=mincIρiTi+-T0,Lmsi-119fi=EfiLm.
The mass of solid and liquid fractions after melting are:
120si+=si--fi121li+=li-+fi
The corresponding updates of depth and total density can be expressed by:
122zi+=zi-×mi-li+mi-li-123ρi+=mizi+.
In case of melting, the melting point temperature is attributed to the layer temperature:
Ti++=minT0,Ti+.
In practice, a first evaluation of Eqs. ()–() is computed for all layers to identify cases where a numerical layer fully melts out (cIρi(Ti+-T0)≥Lmsi-). In such a case, the numerical layer i is aggregated with the numerical layer i+1 for i∈[1,N-1], following Eqs. ()–(). If the bottom layer N is concerned, it is agregated with the above layer N-1. Several iterations can be done in case of melting of multiple consecutive layers. Then, the melting is computed again with this updated discretization with the guarantee that all numerical layers remain defined.
Refreezing
When the heat diffusion solving provides a temperature Ti+ below the melting point T0 whereas liquid water is present, the model simulates refreezing. The energy available for refreezing Eri (J m−2) can be constrained either by the layer cooling after diffusion or by the maximum available liquid water content before refreezing li-. Thus, the energy and mass ri (kg m−2) of refreezing during the time step are computed by:
125Eri=mincIρiT0-Ti+,Lmli-126ri=EriLm.
The mass of solid and liquid fractions after refreezing are:
127si+=si-+ri128li+=li--ri.
The energy conservation during the refreezing process is expressed by:
cIρiTi+-Ti++si++Eri+cIρiT0-Ti+ri=0
where Ti+ and Ti++ (K) are the layer temperature of the solid fraction before and after refreezing. In Eq. (), the first term corresponds to the heating of the solid fraction after refreezing, the second term to the latent heat release due to refreezing and the third term to the cooling of the refrozen part necessary for the thermal equilibrium of the solid phase.
By combining Eqs. () and (), the evolution of the layer temperature due to refreezing is computed by:
Ti++=T0+Ti+-T0si-si++EricIρisi+.
Equations ()–() are not applied independently but jointly with liquid water percolation as described in Sect. within Algorithm 1.
Buckets algorithm for liquid water percolation.
F0=PrΔt
fori∈[1,N]do
Compute ri, si, wi with Eqs. (127)–(133)
wi*=wi+Fi-1+fi-rizi
Fi=max(0,wi*-wimax)zi
wi=wi*-Fizi
ρi+=ρi-(Fi-Fi-1)ρWzi
ifρi+>ρIthen
zi=ρi+ρIzi
ρi+=ρI
endif
endfor
Liquid water percolation
Let us define the volumetric liquid water content wi (kg m−3):
wi=lizi
and the snow porosity:
ϕi=1-ρi--wiρI
where ρI is the volumetric mass of pure ice. The liquid water flow Fi (kg m−2) between layers i and i+1 is computed by a simple and conceptual bucket approach where the layers are seen as superposed water reservoirs with a maximum liquid water holding capacity wimax. The excess water drains to the underlying layer when wi exceeds wimax following Algorithm 1. Snow layer densities ρi are updated with the resulting net mass flux Fi-Fi-1. However, in the limit case of ice layers (ρ=ρI) and refreezing (ri>0), the excess liquid water mass increases the ice layer depth at constant density. Note that another choice would be possible by considering ice layers impermeable (i.e. computing Fi before any refreezing and ri after percolation) with potential impacts on glaciers simulations.
Several formulations of wimax were implemented by :
IfSNOWLIQ=B92:wimax=0.05ρwϕiIfSNOWLIQ=SPK:wimax=ρw0.08-0.10230.97-ϕiifϕi≥0.77wimax=ρw0.0264+0.0099ϕi1-ϕiotherwiseIfSNOWLIQ=B02:wimax=ρirmin+rmax-rminmax0,ρr-ρiρr.
Parameters rmin, rmaxρr are defined in Appendix .
The resolution of Richards equations might help improve the realism of this process in a detailed snowpack model . However, the developments of are not sufficiently robust to be available in this official release. More stable and numerically efficient alternatives are emerging and might be preferred in the future following the recommendations of .
Scavenging of LAP
When LAP are activated (Sect. ), liquid water percolation may carry a fraction of LAP mass :
Mi,j+=Mi,j--Fi×Cscav,j×Mi,j-mi.
The scavenging coefficient Cscav,j can be adjusted by the user for each LAP j (default values are set to Cscav,j=0 i.e. no scavenging).
Sublimation and deposition
Sublimation and deposition are accounted for by adding or removing mass to the surface layer accordingly with the surface latent heat flux (Eq. where T1 is the solution of Eqs. (), () or () depending on the occurrence of melting). The corresponding evolution of the surface layer properties is described by Eqs. ()–().
135z1+=maxz1+LEΔtLsρ1--l1z1-,0136ρ1+=ρ1-+l1z1+.
Microstructure properties are not modified. Therefore, in case of deposition, the current Crocus snow model only increases the mass of the surface snow layer but it does not allow the formation of a dedicated layer with microstructure properties typical of surface hoar. As a result, it is neither possible to track the burying of surface hoar in the snowpack. In the very unusual cases when the mass of the surface layer is insufficient for sublimation -LEΔtLs>ziρi--lizi- which only occurs for extremely thin snowpacks), the quantity -LEΔtLs-ziρi--lizi- is later extracted from the first soil layer in the ISBA-DIF soil scheme to conserve energy, and the remaining liquid water, if any, is transferred to the next layer. A homogeneous regridding is applied before the next time step.
Unloading from vegetation
In case of unloading from vegetation, the initial implementation of MEB consisted in adding the unloaded mass to the mass of solid precipitation mSP, following Eqs. () and () for snow density and microstructure. This could lead to unrealistic surface density when unloading occurred in cold conditions. A recent new parameterization instead attributes a fixed density ρUN and microstructure properties dUN and SUN to unloaded snow. The associated snow mass is either aggregated to the surface snow layer following Eqs. ()–() or associated to a new snow layer depending on the properties of the surface snow layer.
Diagnoses
This section describes the complementary diagnoses of a Crocus simulation provided in addition to the state variables of the model. These diagnoses can be useful for model evaluation or interpretation in research or operational applications.
Diagnoses of recent, wet or refrozen snow
We define nX as the number of snow layers more recent than X days, i.e. satisfying the following condition:
Ai<X∀i∈1,nX.
Thus, thickness ZX and mass MX of snow more recent than X days are defined by:
138ZX=∑i=1nXzi139MX=∑i=1nXmi.
These diagnoses are provided for X∈[0.5,1,3,5,7]. Similarly, the number of wet and refrozen snow layers from the surface nw and nr are defined by:
140li>0∀i∈1,nw141li<0∀i∈1,nr.
The thickness of wet and refrozen snow Zw and Zr are diagnosed by:
142Zw=∑i=1nwzi143Zr=∑i=1nrzi.
Grain type classification
For each snow layer, a diagnosis of grain type Ψi is derived from values of optical diameter and sphericity through the following classification (Eq. ) in which δi(di,Si) and gsi(di,Si) are defined respectively by Eqs. () and (). The values taken by Ψi are taken from the International Snow Classification and include PP (Precipitation Particles), D (Depth Hoars), MF (Melt Forms) and combinations of these types.
Ifdi<10-44-SiIfδi∈[0;0.3[IfSi<0.55Ψi=DF+FCelseΨi=DF+RGIfδi∈[0.3;0.6[Ψi=DFIfδi∈[0.6;0.8[Ψi=PP+DFIfδi∈[0.8;1.0]Ψi=PPelseIfhi=0IfSi∈[0;0.2[Ψi=FCIfSi∈[0.2;0.8[Ψi=RG+FCIfSi∈[0.8;1.0]Ψi=RGIfhi=1Ifgsi∈[0,0.55[IfSi∈[0;0.2[Ψi=FCIfSi∈[0.2;0.8[Ψi=RG+CIfSi∈[0.8;1.0]Ψi=RGIfgsi∈[0.55,1.05[IfSi<0.55Ψi=FC+DHelseΨi=MF+DHIfgsi≥1.05IfSi<0.55Ψi=DHelseΨi=MF+DHIfhi∈(2;4)Ifgsi∈[0,0.55[IfSi<0.55Ψi=MF+FCelseΨi=RG+MFIfgsi≥0.55IfSi<0.55Ψi=MF+FCelseΨi=MFIfhi∈(3;5)IfSi<0.55Ψi=MF+FCelseΨi=MF+DH.
Snowmaking diagnoses
The water use for snowmaking is obtained by:
VSM=mSMASM1-LSMρw.
It is cumulated since the beginning of the season in the model diagnoses.
Optical diagnoses
The specific surface area SSAi (m2 kg−1) of each snow layer i is directly diagnosed from the optical diameter by:
SSAi=6diρI.
When CSNOWRAD = T17, spectral albedo values are also provided from an integration of incoming and absorbed radiations over user-defined spectral bands.
Mechanical diagnosesPenetration resistance
A penetration resistance is computed for each layer. It is designed to represent the measurement obtained by the rammsonde commonly used in field snowpack observations . The value of penetration resistance Rpi is inferred for each layer i∈[1,N] by Eq. (147) from microstructure properties (di, Si and the transformation gsi(di,Si) following Eq. ), density ρi, liquid water content wi (kg m−3) and temperature θi (°C). The result is expressed in kgf (1 kgf ≃ 9.81 N).
Shear strength
To be able to compute stability indexes of the snowpack, a shear resistance Rsi is diagnosed for each layer i. It is also computed from microstructure properties, density, liquid water content and temperature through Eq. (). Rsi is expressed in kgf dm−2 (1 kgf dm−2≃ 0.981 kPa).
Ifwi≥5andΨi∈RG,MF,RG+FC,RG+MF,DF+RG,MF+FC,MF+DHIfρi<200:Rsi=0.1If200≤ρi<320:Rsi=0.02ρi-3.9Ifρi≥320:Rsi=0.068ρi-18.64else:Rsi=max0.05,C1Si,hi×C2di,Si×C3di,Si,hi×C4wi,ρi×C5di,Si,hi,wi,ρi×ρi2⋅10-4-0.6+0.12.
C1, C2, C3, C4 and C5 multiplicative functions are defined in Appendix .
MEPRA
MEPRA was a standalone module designed to estimate the avalanche hazard from the snowpack stratigraphy simulated by Crocus, from mechanical diagnosis and expert rules . It has been fully implemented in the SURFEX platform, and its output are now available with the other diagnostic variables. The general idea is to compare the shear strength Rsi as defined previously, to the shear stress in the layer. For natural release, only the weight of overlying layers is taken into account while the load related to the presence of a skier at the snowpack surface is added to represent the accidental triggering. Expert rules are then defined to determine a hazard index from these mechanical stability indicators, both for natural release and accidental triggering. These expert rules were initially implemented by but remained largely unpublished and have evolved to answer to feedbacks of operational avalanche forecasters, without detailed documentation. Equations implemented in the current version of SURFEX are described below. They are only valid for slope angle Θ=40°.
Mechanical stability of snowpack layers
A simple mechanical diagnosis for stability is computed by dividing the shear resistance Rsi by the shear in the layer. Two values are computed, to discriminate between natural avalanche activity and human triggering.
Natural release
The stability index Snati for natural release is defined in each layer i as follows:
Snati=Rsiσi+1
where σi+1 is the weight of overlying layers (including considered layers), projected on the slope-parallel axis, as defined by Eq. ().
Accidental triggering
The stability index for accidental triggering Sacci is similar to Eq. (), with a supplementary term for shear stress to represent the additional load on the snowpack:
Sacci=Rsiσi+1+Ξiσacci
where σacci is designed to represent the shear load induced by a skier, defined as a piecewise linear decreasing function of depth Zi=∑j=1izj:
σacci=4-15ZiifZi<0.12.5-10Zi-0.1if0.1≤Zi<0.152-8Zi-0.15if0.15≤Zi<0.21.6-4Zi-0.2if0.2≤Zi<0.351-2Zi-0.35if0.35≤Zi<0.50.7-1.5Zi-0.5if0.5≤Zi<0.80if0.8≤Zi
and Ξi is designed to represent the bonding effect reducing the shear stress in the layers, defined by:
Ξi=∑j=1imjξj∑j=1imjwhereξi:IfΨi∈MF,RG+MF,MF+FC,MF+DFIfθi<-0.2,ξi=0.5else:ξi=1.1else:IfRsi>1.5:ξi=1.0else:ξi=1.2.
Hazard indexes
MEPRA analyses the profiles of Snati and Sacci mechanical indexes with other parameters (mainly grain type, temperature and liquid water content and snow heights), to assess a natural avalanche hazard index Hnat, on a scale of 0–5 and accidental hazard index Hacc on a scale 0–3, with a set of expert rules presented below. These hazard indexes are associated with levels of instability (one for high instability, Zh, and one for moderate instability, Zm, at most). For natural release, a classification between 5 avalanche types is also provided.
Natural release
The natural release analysis relies on a classification of the upper layers of the snowpack (referred as “superior profile”) in 4 classes: NEW (new snow), WET (wet snowpack), FRO (refrozen snowpack) or NAN (when it could not be classified in other class), based on the conditions listed in Table . A height of this superior profile ZSUP is also defined below which the snowpack is significantly different, and called inferior profile. The latter is also classified into three classes : SOF (soft), HAR (hard) and NAN (undefined) as a function of the maximum penetration resistance, following Table .
Levels of high and moderate instabilities are looked for in the superior profile, as the uppermost buried layer where Snati is below a threshold Sh=2 or Sm=3 respectively (or Sm=3.05 for a NEW type of superior profile):
153If∃j|Zj>0.1andZj≤ZSUPandSnatj≤Sh:Zh=Zj-zj2154If∃k|Zk>0.1andZj≤ZSUPandSnatj≤Smandj≠k:Zm=Zk-zk2.
A first natural avalanche hazard index Hnat is defined depending on depths of instability level Zh and Zm, superior profile height ZSUP and superior profile type with the following expert rules:
Hnat1=Zh>0.8NEW→5WET→5FRO→3Zh>0.4NEW→4WET→4FRO→3Zh>0.2NEW→2WET→2FRO→3Zh>0NEW→1WET→1FRO→1Zhnotdefined→0Hnat2=Zm>0.8NEW→3WET→3FRO→1Zm>0.4NEW→3WET→3FRO→1Zm>0.2NEW→3WET→3FRO→1Zm>0NEW→1WET→1FRO→1Zmnotdefined→0Hnat3=ZSUP>0.8NEW→1WET→1FRO→0ZSUP>0.4NEW→1WET→1FRO→0ZSUP>0.2NEW→1WET→1FRO→0ZSUP>0NEW→0WET→1FRO→0ZSUPnotdefined→0NEW,WET,orFRO→Hnat=2ifHnat1=2maxHnat1,Hnat2,Hnat3otherwiseNAN→Hnat=0.
The avalanche situation is then classified in 6 classes following Appendix . Finally, Hnat is updated by expert rules accounting from the temporal evolution between times t-ΔtM and t following Appendix . Note that these rules are sensitive to the MEPRA time step ΔtM, set to 3 h by .
Accidental triggering
Hacc is based on the identification of a slab structure in the snowpack including a slab (layer i and possibly layers above) over a weak layer (layer i+1). This structure is identified through the following conditions:
MF∉Ψi-1Rsi>1.3Ψi∈[DF+RG,RG,RG+FC,DF]0.01m≤Zi<1mΨi+1∈[FC,DH,FC+DH,PP,PP+DF,DF].
Similarly to the natural instability levels (Eqs. and ), levels of high accidental instability and moderate accidental instability are looked for among the identified weak layers satisfying Eq. ():
157If∃i|Ψi+1∈[FC,DH,FC+DH]andSacci+1<1.5:Zh=Zi+1-zi+12158If∃i|Ψi+1∈[PP,PP+DF,DF]orΨi+1∈[FC,DH,FC+DH]and1.5≤Sacci+1<2.5:Zm=Zi+1-zi+12.
The accidental hazard index is finally defined with the following expert rules:
If∃i|Eq.(159):IfZc<0.01:Hacc=max3,Heqelse:Hacc=max1,HeqIf∃i|Eq.(160):IfZc<0.01:Hacc=max2,Heqelse:Hacc=max1,HeqIfi|Eqs.(159)or(160):Hacc=max0,Heq
where Heq s a function of the natural index Hnat following Appendix and Zc is the cumulated depth of crusts above a layer:
160Zch=∑izifori|θi<0.2andMF∈ΨiandZi<ZhandPP∉ΨjandDF∉Ψj∀j∈[1,i-1]161Zcm=∑izifori|θi<0.2andMF∈ΨiandZi<ZmandPP∉ΨjandDF∉Ψj∀j∈[1,i-1])
Note that only one value for Zh and Zm is provided in the diagnoses output file. If Hacc=Heq in Eq. (), they correspond to Eqs. () and (), otherwise they correspond to Eqs. () and ().
All variables in the code are defined as vectors including at least a spatial dimension, and when necessary the snow layers as second dimension. The implication in terms of numerical efficiency of loops in the code is described in detail in Appendix . This 1D spatial dimension allows to use either discontinuous collections of points or regular grids for simulations. A typical example of a collection of points is the semi-distributed geometry based on homogeneous massifs and topographic classes which has been used for more than 30 years for operational simulations in French mountains and in the associated 66-year reanalysis . Gridded experiments can also be easily defined through the standard SURFEX tools which include regular latitude-longitude coordinates or various conformal projections. Over the French territory, the Lambert 93 projection is recommended as a national standard for gridded simulations i.e..
Parallelization and numerical efficiency
In the general case where Crocus is not coupled with a snow transport model, the processes currently represented by the model do not involve any mass or energy exchange between the snowpacks of the different spatial units. Therefore, parallelization can be very efficiently applied without any need of communication between processors in the physical model. A distributed memory parallelism has be chosen for that purpose in the offline driver of the SURFEX platform based on MPI libraries. However, the standard SURFEX Input-Output routines are not currently parallelized. Therefore, all input and output data are forwarded to a single processor. As a result, over large domains, the efficiency of parallel computing is currently limited by the saturation of the IO processor, in all cases but especially when the user chooses to output a large number of diagnoses at high temporal resolution. The XIOS library implemented in SURFEX is designed to deal with this issue but the implementation of its compatibility with Crocus variables is still in progress. The numerical cost of the model itself depends on the number of layers created by the model. Therefore, it can vary significantly from one domain to another and from a season to another, depending on the number of layers of each simulated snowpack which highly depends on total snow depth.
To give the magnitude of the numerical cost, Table presents the computing time of a 1-year simulation over 4471 simulation points in the French Alps. It must be noticed that the numerical cost of Crocus is very low compared to the cost of IO, providing a clear guidance for priorities in future optimizations. Crocus also only represents 26 % of the computation time of the ISBA land surface model itself. Even in an alpine region, the variability of snow cover in time and space makes the snow component relatively cheap with a similar level of complexity compared to the 20-layer soil model running all year over all points. As a result, the numerical cost of Crocus (0.3 cores × s per year and simulation point for this example) can not be considered as a valid argument to prefer simpler models in large scale applications of Land Surface Models or coupled applications.
Time (cores × s)Ratio with Crocus time (%)SURFEX run359642480IO reading and communications293902026ISBA land surface model5513380Crocus snow model1450100Snowfall and vertical discretization49133.4Metamorphism17311.9Absorption of solar radiation16211.2Compaction956.6Regridding (aggregation/dissociation)775.3Heat diffusion (Eqs. 107 and 108)644.4Thermal conductivity614.2Percolation and refreezing584.0Melting302.1Aggregation of vanishing layers271.9Drift251.7Diagnostic of energy fluxes231.6Energy balance (Eqs. 102–106)201.4
Considering the partitioning of numerical costs between the different subroutines within Crocus, it appears clearly that the complexity of the discretization rules emphasized by Eq. () has a significant impact as this routine represents 33 % of the whole model cost. The metamorphism routine is the second most contributing routine although its cost has been considerably reduced compared to previous versions . The numerical core of the model solving heat diffusion and energy balance is very efficient contributing to less than 6 % of the cost. Numerical optimizations are still possible in some routines representing an unjustified contribution compared to their low complexity (e.g. thermal conductivity). Possible optimizations may concern the management of loops (cf. Appendix ) or some iterative calls to scalar functions in external modules.
The memory consumption of the model is relatively low with the standard physical options. In the experiment described above, the maximum Resident Set Size (RSS) is lower than 17 GB. However, the high spectral resolution of the TARTES optical scheme slightly increases the memory consumption (25 GB) and highly increases the total numerical cost when this option is activated (28 588 cores × s for the TARTES module in an experiment identical to Table but with SNOWRAD = T17, i.e. about 20 times the reference numerical cost of Crocus).
Running environment and visualization
Beyond the FORTRAN code itself, most offline applications of the model have to deal with the management of input and output files to perform various experiments with different forcing files, namelists, or binaries, different setting of initial and final simulation dates, standard initialization procedures (soil spinup, etc.). Therefore, a common running environment of the model in Python is provided in an independent package called snowtools with full user documentation and an interface for technical support (cf. Code availability section).
Crocus scalar diagnoses can be easily processed by any scientific plotting software supporting netcdf format as input. However, the irregular vertical discretization of the snowpack model complexifies the visualization of the simulated profiles. A simple software provided in the snowtools package is able to combine the variables to plot with the depth of each layer to produce detailed instantaneous stratigraphies or temporal evolution of a given stratified variable (as in Appendix ). For spatial simulations, plotting the spatial variability of the vertical structure of the snowpack is still a challenge.
Externalization
Although the reference implementation of Crocus is within the SURFEX land surface modelling platform , there is an increasing need of being able to couple Crocus with other land surface schemes. For that purpose, an externalized version of the source codes is now available as an independent Fortran library that can be compiled alone and called by other land surface models (cf. Code availability section). It includes all the processes described in this paper except the coupling with external components (snow transport modules and MEB vegetation module). Thus, Crocus is also now fully integrated within the SVS2 land surface system . It was also recently implemented within the Flexible Snow Model FSM2, allowing the coupling with its more detailed vegetation model . SURFEX, SVS2 and FSM2 rely on a unique code repository of Crocus that guarantees the long-term maintenance and convergence of the code and therefore facilitates the contributions of different research groups to the model developments. For instance, as mentioned for several processes in Sect. , various new parameterizations better suited for Arctic snow were recently proposed within the SVS2 implementation . Thanks to this method, these developments will integrate soon a future release and be beneficial for SURFEX and FSM2 applications. Therefore, we strongly encourage other groups that have copied the code within their specific applications to try to converge towards this unique code version. This includes the implementation of Crocus within WRF-Hydro , the coupling of Crocus with Noah Land Surface Model , the MAR regional climate model largely used for polar regions and from which the Crocus version has diverged for a long time , and even applications which have only extracted specific routines such as the CryoGrid permafrost model .
Review of evaluations and scientific applicationsWith a local scale meteorological forcing
The most direct evaluations of the model are performed on well-instrumented sites allowing to minimize errors in the meteorological forcing and in the observations used for evaluation. Model skills have been documented in detail at the Col de Porte experimental site , a mid-elevation meadow in French Alps. The very first evaluations of present qualitative evaluations of surface and internal temperature, snow depth, and basal runoff during short periods of the winter 1986–1987. extended the evaluations to the whole winter 1988–1989 on the same variables as well as a subjective comparison between the simulated stratigraphies and weekly observed profiles. Extensions of the evaluation period were successively published by , , , and in the context of model intercomparisons, and also by . These papers also included evaluations of albedo and Snow Water Equivalent. extended these evaluations to all multiphysics options and accounted for observation uncertainties usually ignored in previous papers. The accuracy of the model on these variables has not significantly changed since the first years of development. Complementary evaluations of density and microstructure profiles are provided by and . However, quantitative evaluation of internal snow properties is still a methodological challenge due to frequent discrepancies between numerical and observed snow layers that can easily lead to double penalty issues. This can only be partly solved with vertical adjustments algorithms or with an expert layer tracking .
More challenging evaluations include a variety of environmental and climate conditions. Evaluations driven by local meteorological observations were performed in Svalbard , at the high elevation site of Weissflujoch, Switzerland , over tropical glaciers and moraines in Bolivia and Ecuador , at Sherbrooke University, Quebec , at Torgnon, Italy . The most comprehensive evaluations in terms of number of sites and years were performed on 10 contrasted sites through the Earth System Model-Snow Model Intercomparison Project ESM-SnowMIP, and with a few more sites for albedo evaluations by . Typical errors on snow depth, SWE, albedo, surface temperature and ground temperature were in the range of state-of-the art snow models as illustrated with up-to-date simulations with this model version in Appendix . A similar overall skill can be obtained by simpler models on these variables. The cold bias in surface temperature identified by may be attributed to the parameterization of turbulent fluxes which are suspected to be underestimated with the previously standard configuration (Ril=0.2) but this bias is removed with the new default value Ril=0.026,. This assumption was also recently supported by more detailed evaluations of all components of the energy balance including eddy-covariance observations of turbulent fluxes in Québec and in Finnish peatlands . However, it is especially important to be aware of the equifinality between the different empirical parameterizations (i.e. different model configurations providing similar model results) and to be cautious with the complex compromises in multivariate evaluations the optimal model configuration can change depending on the evaluated variable,. These difficulties should encourage future attempts to improve processes representations to be tested robustly with ensemble multiphysics simulations and multivariate evaluations e.g..
With a regional scale meteorological forcing
In many other applications, the model was forced by meteorological reanalyses e.g. or short-term forecasts e.g. or a combination of both . Although these studies often include snow depth evaluations on a large range of stations, in this case the resulting modelling errors of any snow model are dominated by errors in the meteorological forcing and Crocus does not make an exception . Thus, the attribution of some limitations of the simulation results to the snowpack model itself is difficult. However, this is sometimes the only possible method for the assessment of some specific processes. For instance, the simulated concentrations of Light-Absorbing Particles , the spectral reflectances from the TARTES optical scheme , the blowing snow fluxes were only evaluated in such context. The same applies to the ability of the model to reproduce the properties of polar snow e.g., the spatial distribution of snow conditions depending on topography , or complex remote sensing signals . Similar simulation frameworks are used to investigate the suitability of the model for hydrological diagnoses e.g., glacier mass balance e.g. or avalanche activity .
Based on the confidence provided by these available evaluations, the model is used for various purposes including the understanding of internal physical processes e.g., the quantification of contributions to the energy balance e.g.for light-absorbing particles, the monitoring of the long-term climatology of extreme snow loads or avalanche activity , the investigation of the links between snow cover and alpine ecosystems evolutions , and climate projections of natural snow conditions , avalanche hazard and ski resorts operating conditions . The development efforts were originally dedicated to operational avalanche hazard forecasting . However, after about 30 years of operation, the model did not become the main tool of the forecasters because this application is especially sensitive to uncertainties among numerous other challenges as reviewed by . The statistical post-processing of simulations through various techniques of artifical intelligence might provide guidance to improve the practical interest of such simulations for operational applications including avalanche forecasting.
Towards data assimilation
The long memory of the snowpack on the past meteorological conditions and past snow processes imply that all sources of modelling errors tend to cumulate all along the season. Therefore, there is a large avenue for data assimilation algorithms in order to improve the initial states of the simulations . However, the variable dimension size of the Crocus state vector (due to the variable number of layers) and the high non-linearities in the simulated processes make the application of a number of data assimilation algorithms to this model challenging. Therefore, most recent efforts intended to apply different variants of the Particle Filter to weight the members of an ensemble of simulations according to their distance to some observations . The different options of the algorithm are described in and the implemented variables are snow depth and optical reflectance, for which the observation operator is simply the identity function as these variables are direct diagnoses of the model. The assimilation of optical reflectance is however constrained by the retrieval errors of this variable in complex terrain that still exceed the requirements for an efficient data assimilation . Large efforts are planned in a near future to extend these possibilities. For some common satellite observations (e.g. snow cover fraction, wet snow fraction), this will require the development of appropriate observation operators from the simulated state variables.
Conclusion
This article provides a comprehensive description of all equations implemented in version 3.0.2 of the Crocus snow model. It gathers the recent developments of the last 13 years in a unique publication: (i) modelling of Light-Absorbing Particles, (ii) coupling with the TARTES optical scheme, (iii) modelling of ice layers due to freezing rain, (iv) coupling with the MEB big-leaf vegetation scheme, (v) snow management practices on ski slopes, (vi) coupling with blowing snow schemes, (vii) multiphysics parameterizations for most processes and (viii) diagnosis of the snowpack mechanical stability. In addition, this article documents a number of equations implemented in previous code versions but never published in the literature. It must be mentioned that during the preparation of this publication, a considerable number of errors or discrepancies between the code and previous publications were identified and corrected. This is in full agreement with the main conclusions of suggesting that insufficient model documentation is a key factor for the difficulty to improve snow modelling in the last decades. This comprehensive documentation is expected to help snow scientists to better interpret results based on this model. This is especially important in the context of an in-progress extension of Crocus applications in several land surface schemes. This documentation effort is also expected to help the snow modelling community improve numerical models in the future thanks to an accurate knowledge of the existing parameterizations and numerical difficulties. Despite our best efforts to minimize errors, previous literature and experience suggest that some errors may still remain in this publication. In such a case, corrigenda will be associated to this publication in due course.
Attractor profile in layering
The attractor profile zi* used in Eq. () depends only on the total snow depth Z and number of layers N. Let us define N* and Z* by:
ifZ>20andN>Nmax3+2N*=N-N+Nmax6Z*=3elseN*=NZ*=Z.
Then, the attractor profile is defined by:
z1*=min0.01,Z*N*z2*=min0.0125,Z*N*z3*=min0.03,Z*N*ifN*>3z4*=min0.04,Z*N*ifN*>4z5*=min0.05,Z*N*ifN*>5z6*=max0.07,minZ*N*,0.5ifN*>6z7*=max0.07,minZ*N*,1.ifN*>7z8*=max0.07,minZ*N*,2.ifN*>8z9*=max0.07,minZ*N*,4.ifN*>9z10*=max0.07,minZ*N*,10.ifN*>10zi*=max0.07,minZN,∀i∈11,N*ifN*>11zi*=i-N*2(Z-3)N-N*N-N*+1∀i∈N*+1,NifZ>20zN*=min0.02,ZN*ifZ>3andN*>10zN-1*=0.66zN*+0.34zN-3*zN-2*=0.34zN*+0.66zN-3*insteadofpreviousdefinitions.
It extends the definition of in order to converge towards a profile allowing a numerically stable resolution of heat diffusion for thick snowpacks and glacier applications.
Adjustment of wind speed
Several parameterizations of the model are formulated with a wind speed at a specific height corresponding to the experimental conditions but might not correspond to the reference height of the forcing variable. In these cases, the wind speed Uz (m s−1) at height z (m) is adjusted assuming a logarithmic profile in the surface boundary layer based following:
Uz=Ulnzz0lnzuz0
where z0 is the surface roughness length (m).
Growth of faceted crystals in B21 metamorphism parameterizations
The growth of faceted crystals in B21 parameterization (Eq. ) is based on cold room experiments from . These functions already published by are repeated here for the comprehensiveness of this paper. f, g, h and Φ are dimensionless functions from 0 to 1 given by:
C1fθi=0ifθi<-40°C0.011×θi+40if-40≤θi<-22°C0.2+0.05×θi+22if-22≤θi<-6°C0.7-0.05θiotherwiseC2hρi=1.ifρi<150kgm-31-0.004×ρi-150if150<ρi<400kgm-30.otherwiseC3gGi=0.ifGi<15Km-10.01×Gi-15if15≤Gi<25Km-10.1+0.037×Gi-25if25≤Gi<40Km-10.65+0.02×Gi-40if40≤Gi<50Km-10.85+0.0075×Gi-50if50≤Gi<70Km-11.otherwiseC4Φ=1.0417.10-9ms-1.
New formalism of metamorphism
The translation of the original metamorphism parameterizations in terms of dendricity, sphericity and size from to the new formalism of in terms of optical diameter and sphericity, was based on the original expression of the optical diameter di as a function of dendricity δi, sphericity Si and grain size gsi (as already published in Eq. 13 of ):
di=10-4δi+1-δi4-Siifδi>0(dendriticcase)gsi×Si+1-Si×max4.10-4,gsi2ifδi=0(non-dendriticcase).
This relationship has always been required to compute the absorption of solar radiation from the equations of Sect. .
In the dendritic case, the transformation (Si,δi)⇒(Si,di) is bijective. The inversion of this relationship lead to:
δi=104×di-4+SiSi-3ifdi<10-44-Si
and the new evolution law of di was obtained by using
ddidt=∂di∂δidδidt+∂di∂SidSidt.
Thus, the combination of Eq. () with the equations from Table 1 of provides the second line of Eq. () in this paper (which is close to the right column of Table 2 in after typo corrections).
In the non-dendritic case, the transformation (Si,gsi)⇒(Si,di) is not bijective. When Si=0, Eq. () gives di=4×10-4∀gsi∈[3×10-4,8×10-4]. As a result, the metamorphism functions from Table 1 of cannot be reproduced in the space (Si, di) when Si=0 (e.g. depth hoar). Then, Eq. () is discontinuous at the limit between dendritic and non-dendritic cases. Indeed, the limit of the dendritic case gives limδi→0di=10-4(4-Si). When combined with the non dendritic case, this would result in gsi=3×10-4 while the initial value of gsi was actually higher for non-spheric particles in : gsi=10-4(4-si). In the other cases, the inversion of Eq. () can provide a relationship for gsi as a function of di and Si:
gsi=2diSi+1ifdi≥4×10-4Si+1di-4×10-41-SiSiifdi<4×10-4Si+1.
Equation () is actually more complex than Eq. (3) of which was an incorrect simplification corresponding only to the initialization of grain size at the dendritic - non-dendritic transition. It is also not defined when Si=0 and di<4×10-4(Si+1). Last, the derivation of an evolution law for optical diameter was obtained by in the non-dendritic case (left column of their Table 2) by considering only ddidt=∂di∂SidSidt. This is actually inconsistent with the original formalism in the non-dendritic case as it ignores the term ∂di∂gsidgsidt. Ignoring this term may be convenient to avoid the problems mentioned above but it affects the metamorphism of faceted crystals and depth hoar without any scientific justification whereas they are the most critical snow types for further analyses in terms of mechanical stability. Furthermore, a number of parameterizations of other processes and diagnoses in the code were also affected by the incorrect simplification of Eq. (). The original difficulty of this translation of formalisms comes from the fact that Eq. () is neither bijective nor continuous. However, we considered that it is better to adapt this unpublished formula and preserve as much as possible the metamorphism laws, the parameterizations of other processes and the diagnoses relying on microstructure properties. A new metamorphism option (B21) has therefore been defined, replacing the expression from by:
di=gsi×Si+1-Si4×10-4+gsi2ifδi=0(non-dendriticcase).
This allows to replace Eq. () by:
gsi=2di-2×10-41-Si1+Si.
This way, the evolution of optical diameter in the non dendritic case can be obtained by:
D7ddidt=∂di∂gsidgsidt+∂di∂SidSidtD8=1+Si2dgsidt+gsi2-2×10-4dSidtD9=1+Si2dgsidt+di-4×10-41+SidSidt.
The main advantage of this new formalism is that it respects the original evolution law of dgsidt as published by . As a result, all parameterizations and diagnostics calibrated in the original formalism can still considered to be valid in this new formalism. Finally, the first line of Eq. () corresponds to the combination of Eq. () and Table 1 of . Complementary analyses by have shown that the obtained microstructure properties of B21 option are closer to the original formalism of than the implementation of (also referred as C13 in . This is especially true when this parameterization is combined with the snow drift options. More details are available in .
Similarly for wet metamorphism, the combination of Eq. () and Table of provides Eq. () of this paper. The transformation of wet metamorphism laws in this new formalism were neither provided in nor correctly implemented.
Evolution of optical diameter during snow drift
The evolution of microstructure properties was parameterized in terms of dendricity δi, sphericity Si and grain size gsi from simple evolution laws provided in Table 3 of in which sign errors must be accounted for in the evolution of δi and gsi:
E1dδidt=-δi2τDRIFTifδi>0(dendriticcase)E2dgsidt=-5×10-4τDRIFTifδi=0(non-dendriticcase)E3dSidt=1-SiτDRIFT.
In the dendritic case, the evolution of the optical diameter di is obtained by the introduction of Eqs. () and () in Eq. () and computing ∂di∂δi and ∂di∂Si from the first line of Eq. (). In the non-dendritic case, the evolution of the optical diameter di is obtained by the introduction of Eqs. () and () in Eq. () and computing ∂di∂gsi and ∂di∂Si from Eq. (). Finally:
ddidt=10-4Si-3×-δi2τDRIFT+10-4δi-1×1-SiτDRIFTifdi<10-44-Siddidt=Si+12×-5×10-4τDRIFT+gsi-4×10-42×1-SiτDRIFTifdi≥10-44-Si.
The first line can be slightly simplified and Eq. () introduced in the second line to finally obtain the equivalent discrete formulation of Eq. ().
Thermodynamical functions
The air volumetric mass ρa is obtained by:
ρa=PsRaTa1+RvRa-1qa+g×za.
The Exner functions at surface and at the forcing level are defined by:
F2Πa=PaP0RacPF3Πs=PsP0RacP
where P0=105 Pa and the atmospheric pressure at forcing level Pa is obtained from hydrostatism:
Pa=Ps-ρagza.
The saturation specific humidity at temperature T is obtained by:
qsatT,Ps=RaRv×esat(T)Ps1+RaRv-1esat(T)Ps
where the water vapor partial pressure at saturation esat(T) is obtained from Eq. () which is an approximate integral of the Clausius–Clapeyron formula:
IfT≥T0esat(T)=expαw-βwT-γwln(T)IfT<T0esat(T)=expαI-βIT-γIln(T)
where
F7γw=cW-cPvRvF8βw=LvRv+γwT0F9αw=lnesatT0+βwT0+γwlnT0F10γI=cI-cPvRvF11βI=LsRv+γIT0F12αI=lnesatT0+βIT0+γIlnT0F13esatT0=611.14Pa.
The derivative ∂qsat(T,Ps)∂T is obtained by:
∂qsatT,Ps∂T=∂esat(T)∂TqsatT,Ps×11+RaRv-11+RaRv1qsatT,Ps-1
where
∂esat(T)∂T=βwT2-γwT.
The Richardson number is computed by:
Ri=gcosΘzu2θva-θvs0.5θva+θvsmaxU,Uth2za
where
F17θva=TaΠa1+RvRa-1qaF18θvs=T1Πs1+RvRa-1qsatT1,Ps.
The wet bulb temperature Tw* (°C) is computed by Eq. ():
Tw*=γ⋅Ta*+∂esat(T)∂T⋅Td*γ+∂esat(T)∂T
where the slope of the saturation vapor pressure curve ∂esat(T)∂T is given by Eq. (); the dew point temperature Td* (°C) is parameterized by
Td*=[116.9+237.3ln(e)]/[16.78-ln(e)].
and the psychrometric constant γ (in kPa K−1) is obtained by:
γ=cPm⋅Ps0.622⋅Lv
where cPm is the specific heat capacity of moist air at constant pressure (in kJ kg−1 °C−1).
Functions in the parameterization of shear resistance
G1IfSi>0.8andhi∈[3,5]:C1di,Si=1.05else:C1di,Si=0.45+0.7SiifSi<0.250.625+1.0⋅Si-0.25if0.25≤Si<0.50.875+0.6⋅Si-0.5if0.5≤Si<0.751.025+0.5⋅Si-0.75if0.75≤SiG2C2di,Si=1-0.4δiifδi<0.250.9-0.4⋅δi-0.25if0.25≤δi<0.50.8-0.8⋅δi-0.5if0.5≤δi<0.750.6-0.6⋅δi-0.75if0.75≤δi
where δi=di×104-4+SiSi-3Ifdi≤4-Si×10-4:C3di,Si=1else:Ifgsidi,Sidi,Si≤4×10-4-10-4Si:C3di,Si=1else:C3di,Si=1-530⋅(0.8-0.2s)⋅-4×10-4+gsidi,Si+10-4Si
where gsi(di,Si) is defined by Eq. ().
Ifwiwimax<0.9:C4wi,ρi=1+wiwimaxifwiwimax<0.11.1-2.35wiwimax-0.1if0.1≤wiwimax<0.30.63-0.4wiwimax-0.3if0.3≤wiwimax<0.9else:C4wi,ρi=max0.15,min0.35,ρi-wi×10-4
where wimax(ρi,wi) is defined by Eq. ().
Ifhi∈[0,1]orwiwimax>0.5:C5=1else:Ifwi=0:C5=1.5⋅1.15+0.21-Si1.15⋅1+0.2C3di,Sielse:Ifhi∈[2,3],C5=1else:C5=1.5-2wiwimaxifwiwimax<0.11.3-0.75wiwimax-0.1ifwiwimax≥0.1.
ConditionTypeRpi≤8∀i∈[k,l]wherel:∑i=klzi≤ZSUP4<∑i=kl+1ziSOF∃i∈[k,l]:Rpi>8and∑i=klzi≤ZSUP4<∑i=kl+1ziHARZ<1.25ZSUPNANClassification of avalanche situations
The avalanche situation is classified in 6 typical classes: NEW_DRY (new snow, dry), NEW_WET (new snow, wet), NEW_MIX (new snow, mixed type), MEL_SUR (melting at surface), MEL_GRO (melting, not mainly at surface) and AVA_NAN if could not identify to an other type. The expert rules determining avalanche type from superior profile, inferior profile, temperature and liquid water content of layers of superior profile are described in following equation:
NEWIfwi<5∀i|Zi≤ZSUP→NEW_DRYIfwi≥5∀i|Zi≤ZSUP→NEW_WETelse→NEW_MIXWETHAR→MEL_SURSOF→MEL_GROelseIfwi≥5∀i|0.1<Zi≤ZSUP→MEL_GROelse→MEL_SURFROIfHnat=0→AVA_NANelseHAR→MEL_SURSOF→MEL_GROelseIfwk≥5fork|Zk=ZSUPand∑i=jkzi>ZSUP3wherej|Zj-1<0.1<Zj→MEL_GROelse→MEL_SURNAN→AVA_NAN.
Accounting for the temporal evolution in natural hazard index
If at time t, superior profile is NEW but avalanche type is not NEW_MIX and if between t-ΔtM and t, avalanche type has remained unchanged and Z and ZSUP have decreased, then Hnat is updated from Table . The same rule is applied if avalanche type is NEW_MIX but only if the continuous wet thickness ZW=∑i=0nWzi where nW|wi>5∀i∈[1,nW] has decreased or remained constant since t-ΔtM.
If superior profile is WET or FRO at t and at t-ΔtM and Z-ZSUP has not reduced by more than 0.05 m between t-ΔtM and t:
If Hnat=3 then it is reduced to 2
If superior profile is WET and Hnat(t)∈[4,5] then:
If Hnat(t-ΔtM)∈[3,4,5], then Hnat(t)=3
If Hnat(t-ΔtM)=1, then Hnat(t)=1.
Update of Hnat depending on Hnat(t) as assessed from Eq. (157) and from its value at previous output time step Hnat(t-ΔtM). – represents an undefined value.
Hnat(t-ΔtM)Hnat(t)012345–0012345–1011145–2011145–3011134–4012334–5012334––012345–Equivalent natural hazard index
In Eq. (), Heq is designed to account for natural hazards in the assessment of the accidental hazard index. It is defined by:
ifZ≥0.2mIfHnat≥4:Heq=3If2≤Hnat<4:Heq=2IfHnat=1:Heq=1else:Heq=1.
Overview of simulation results with the ESM-SnowMIP dataset
This appendix illustrates the application of Crocus 3.0.2 at the 10 ESM-SnowMIP reference sites. In-situ meteorological forcing and evaluation dataset are available from . Simulations are performed with default physical options as defined in Table except for the 3 BERMS forest sites (Saskatechewan, Canada) where the coupling with MEB is activated. Default values are used for all adjustable parameters as defined in Appendix except for za and zu which are adjusted following the site-specific metadata . Simulations are compared with observations of snow depth, snow water equivalent, albedo and surface temperature at two alpine sites where all observations are available (Col de Porte, 1325 m, France, Fig. , and Weissfluhjoch, 2540 m, Switzerland, Fig. ) and only with snow depth for the 8 other sites (Figs. –). The illustrations show the ability of the model to simulate contrasted stratigraphies between wet sites dominated by Melt Forms (Figs. refcdp and ), cold sites dominated by Faceted Crystals or Depth Hoar (Figs. – and ), and high-elevation sites with thick and often dry snowpacks dominated by Rounded Grains with only thin layers of Faceted Crystals (Figs. , and ). Table summarizes the model skill with scores computed following on the 7-year periods selected for illustration. They do not exhibit any systematic bias on the evaluated variables and errors in the magnitude of other snow models . These results can be considered as a benchmark of the model skill in various environment and climate conditions. This benchmark can be useful to assess the impact of future model developments.
Scores of Crocus 3.0.2 simulations forced and evaluated by the ESM-SnowMIP dataset. Missing cells correspond to unavailable observations in the dataset. N is the number of available data considered to compute the scores. 7-year evaluation periods for all sites: 2007–2014 for Col de Porte (cdp), Weissfluhjoch (wfj), Sapporo (sap), Senator Beck (snb), Sodankyla (sod), Swamp Angel (swa); 2003–2010 for BERMS old aspen (oas), old black spruce (obs) and old jack pine (ojp); 2001–2008 for Reynolds Mountain East (rme), as in the associated illustrations (Figs. –).
Illustration of Crocus simulation with ESM-SnowMIP dataset. Snow grain type, snow depth, snow water equivalent, snow surface albedo, snow surface temperature. Col de Porte, France, 2007–2014.
Illustration of Crocus simulation with ESM-SnowMIP dataset. Snow grain type, snow depth, snow water equivalent, snow surface albedo, snow surface temperature. Weissfluhjoch, Switzerland, 2007–2014.
Illustration of Crocus simulation with ESM-SnowMIP dataset. Snow grain type, snow depth. BERMS Old Aspen, Saskatchewan, Canada, 2007–2014.
Illustration of Crocus simulation with ESM-SnowMIP dataset. Snow grain type, snow depth. BERMS Old Black Spruce, Saskatchewan, Canada, 2007–2014.
Illustration of Crocus simulation with ESM-SnowMIP dataset. Snow grain type, snow depth. BERMS Old Jack Pine, Saskatchewan, Canada, 2007–2014.
Illustration of Crocus simulation with ESM-SnowMIP dataset. Snow grain type, snow depth. Reynolds Mountain East, Idaho, USA, 2007–2014.
Illustration of Crocus simulation with ESM-SnowMIP dataset. Snow grain type, snow depth. Sapporo, Japan, 2007–2014.
Illustration of Crocus simulation with ESM-SnowMIP dataset. Snow grain type, snow depth. Senator Beck, Colorado, USA, 2007–2014.
Illustration of Crocus simulation with ESM-SnowMIP dataset. Snow grain type, snow depth. Sodankylä, Finland, 2007–2014.
Illustration of Crocus simulation with ESM-SnowMIP dataset. Snow grain type, snow depth. Swamp Angel, Colorado, USA, 2007–2014.
Numerical efficiency of loops
All equations of this paper must be applied iteratively over all simulation points and often over all snow layers. The variable number of active snow layers N between points introduces a spatial dependence of the boundary of the loop iterator operating on snow layers. Furthermore, the maximum number of snow layers also depends on time. Several options are possible to implement these specificities, in the context where in SURFEX the leftmost dimension of arrays represent the spatial dimension, which is also the fastest varying dimension in Fortran with continous memory storage. Without extended analyses of their numerical impact, two options were implemented in previous versions of the code. A compressed-index form (CINDX, Algorithm 1) where the loop over snow layers is inside the loop over points had been chosen by in most parts of the code. This option minimizes the number of iterations and operations as the boundary of the snow layer iterator can vary between points and dates. However, the successive accesses to the values of the arrays are not performed continuously relatively to the memory storage involving more expensive memory accesses and preventing the vectorization of computations. A full iteration with condition (FCOND, Algorithm 2) had been chosen in other parts including the heat diffusion and the TARTES optical scheme. This option does not minimize the number of iterations but still minimizes the number of operations and allows continuous memory accesses. However, it adds a potentially expensive conditional statement which still prevents vectorization. Other options could be considered. For instance, a full iteration with a predefined mask (FMASK, Algorithm 3) allows continuous memory access and vectorization. However, it adds operations on empty layers so that the interest of this approach is expected to increase with the matrices density. Note also that some operations such as divisions must be secured with this approach to avoid floating point exceptions on empty layers.
Loops with compressed index (CINDX): illustration for the computation of vertical gradient.
forp∈[1,Np]do
fori∈[1,N(p)-1]do
G(p,i)=2T(p,i)-T(p,i+1)z(p,i)+z(p,i+1)
endfor
endfor
Loops with full iteration with condition (FCOND): illustration for the computation of vertical gradient.
fori∈1,Nmax-1do
forp∈1,Npdo
ifi≤N(p)-1then
G(p,i)=2T(p,i)-T(p,i+1)z(p,i)+z(p,i+1)
endif
endfor
endfor
Loops with full iteration with mask (FMASK): illustration for the computation of vertical gradient. MASK is a precomputed array with 1 values for defined snow layers (i≤N) and 0 for undefined layers (i>N).
fori∈1,Nmax-1do
forp∈1,Npdo
G(p,i)=2T(p,i)-T(p,i+1)z(p,i)+z(p,i+1)×MASK(p,i)
endfor
endfor
Comparing the numerical efficiency of the whole model with these different options would represent a considerable amount of work because it would need to code all the model loops with these options. Therefore, the efficiency of these different options were only compared on simple test cases with random initialization values: the sum of a quantity over the vertical dimension (i.e. computation of total snow depth Z), the computation of the vertical gradient of a quantity (i.e. Eq. ), the solving of a linear system with a tridiagonal matrix (i.e. Eq. ), and finally the series of Equations to represent metamorphism (Sect. ). The obtained results are presented in Table and exhibit a large variability depending on the operations and matrix filling. Although, the initially implemented compressed-index method was found to be more efficient for all operations in the case of sparse matrices (N=12 layers), the efficiency is highly deteriorated in case of dense matrices (N closer to 50) probably due to the discontinuous accesses to memory. The full iteration with condition has the best efficiency for simple operations. The full iteration with mask improves the efficiency of the inversion of tridiagonal matrix compared to other methods but deteriorates the efficiency of the complex metamorphism routine with numerous conditional statements. Finally, the whole code was homogenized using a full iteration with condition (FCOND) allowing continous memory accesses and presenting more stable computing times in the most common cases. We recommend that future developments follow the same approach, unless a dedicated efficiency performance test is able to demonstrate that an added value is obtained with another method for a specific and expensive algorithm.
SymbolDescriptionisnow layer, i=1 refers to the surface layer, increasing indexes going downjtype of Light-Absorbing Particleskspectral band for solar radiationpsimulation point (spatial dimension)
VariablesSymbolUnitsDescriptionAidaysAge of snow layer iCH–Exchange coefficient for turbulent fluxesdimOptical diameter of layer idnmOptical diameter of new snowdSPmOptical diameter of solid precipitationdBSmOptical diameter of blowing snowD(i,i+1)–Similarity criteria between layers i and i+1Di–Driftability index of layer ifikg m−2Mass of melting for layer i during a time stepEFRZW m−2Energy released at the surface by freezing of supercooled rainEiW m−2Energy of phase change for layer iEfiW m−2Energy available for fusion in layer iEriW m−2Energy available for refreezing in layer iEkiW m−2Absorbed solar radiation by layer i for spectral band kFikg m−2Liquid water flow between layer i and i+1 during a time stepGiK m−1Vertical temperature gradient in layer iGiW m−2Global heat flux between layers i and i-1hi–Historical tracker of layer iHW m−2Surface turbulent sensible heat fluxHiJ m−2Enthalpy of layer iHnat0–5Natural avalanche hazard index of the simulated snow profileHacc0–3Accidental avalanche hazard index of the simulated snow profilelikg m−2Mass of liquid water of layer iKiW m−2Conduction heat flux between layers i and i-1LEW m−2Surface turbulent latent heat fluxlinf, lsup–Adjacent layers minimizing the penalty criteriaLW↓W m−2Incoming atmospheric longwave radiationmikg m−2Total mass of layer imnkg m−2Mass of new snowmSPkg m−2Mass of new solid precipitationmBSkg m−2Mass of new blowing snowmSMkg m−2Mass of new machine-made snowMXkg m−2Mass of snow more recent than X daysMi,jkg m−2Mass of Light-Absorbing Particles of type j in layer iMOBi–Mobility index of layer iN–Number of active layersNp–Number of simulation pointsnX–Number of layers more recent than X daysnr–Number of refrozen layers at the surfacenw–Number of wet layers at the surfacePrkg m−2 s−1Rainfall fluxPskg m−2 s−1Snowfall fluxPsPaAtmospheric pressure at surfaceP(i,i+1)–Penalty criteria for aggregation between layers i and i+1qakg kg−1Air specific humidity at reference height zaqsat(T,Ps)kg kg−1Saturation specific humidity for temperature T and pressure Psrikg m−2Mass of refrozen water for layer i during a time stepRiW m−2Shortwave radiative flux between at the interface between layers i and i+1Ri–Richardson numberRpikgfPenetration resistance of layer iRsikgf dm−2Shear resistance of layer isikg m−2Mass of solid phase of layer iSi–Sphericity of layer iSn–Sphericity of new snowSSP-Sphericity of solid precipitationSBS–Sphericity of blowing snowSnati–Stability index of layer i for natural avalanche releaseSacci–Stability index of layer i for accidental avalanche releaseSSAim2 kg−1Specific Surface Area of snow layer iSymbolUnitsDescriptionSWDIR↓W m−2Incoming direct solar shortwave radiationSWDIF↓W m−2Incoming diffuse solar shortwave radiationSW↓W m−2Incoming total solar shortwave radiationSWk↓W m−2Incoming spectral shortwave radiation for band kTaKAir temperature at reference height zaTa*°CAir temperature at reference height zaTw*°CWet bulb air temperatureTd*°CDew point air temperatureTiKTemperature of layer iTG1KTemperature of surface soil layerUm s−1Wind speed at reference height zuU>1m s−1Wind speed at reference height zu with a minimum threshold of 1 m s−1 for the computation of turbulent fluxesUzm s−1Wind speed at height zVSMkg m−2 s−1Water consumption for snow makingwikg m−3Volumetric liquid water content of layer iwimaxkg m−3Maximum liquid water holding capacity of layer iWd,jkg m−2 s−1Dry deposition flux for Light-Absorbing Particles of type jWw,jkg m−2 s−1Wet deposition flux for Light-Absorbing Particles of type jzimThickness of layer iznmThickness of new snowfallZimDepth of the bottom of layer i from the surfaceZXmThickness of snow more recent than X daysZrmThickness of refrozen snow at the surfaceZwmThickness of wet snow at the surfaceαk–Spectral surface albedo for band kβkim−1Absorption coefficient of solar radiation for band k and layer iηikg s−1 m−1Viscosity of layer iθi°CTemperature of layer iΨi–Grain type of layer i in the International Snow ClassificationλiW m−1 K−1Thermal conductivity of layer iλG1W m−1 K−1Thermal conductivity of surface soil layerλi‾W m−1 K−1Integrated thermal conductivity for layers i and i+1μ–Cosine of solar zenithal angleΠa–Exner function at height zaΠs–Exner function at the surfaceρakg m−3air volumetric massρikg m−3Density of layer iρnkg m−3Density of new snowρSPkg m−3Density of natural snowfallρBSkg m−3Density of blowing snowσiPaPressure of over burden snow for layer iσGROiPaStatic stress due to snowcat for layer iϕi–Porosity of snow layer iϕFRZ–Fraction of latent heat release due to the freezing of supercooled rain consumed by the heating up to T0Fixed parameters
SymbolValues and unitsDescriptionThermodynamical and physical parameters cI2.106×103 J kg−1 K−1Ice specific heat capacitycW4.218×103 J kg−1 K−1Liquid water specific heat capacitycP1.0047×103 J kg−1 K−1Dry air specific heat capacitycPv1.8461×103 J kg−1 K−1Vapor specific heat capacitycPm1.013×103 J kg−1 K−1Typical moist air specific heat capacityκVK0.4Von Karman constantLm3.337×105 J kg−1Latent heat of ice fusionLv2.5008×106 J kg−1Latent heat of vaporization of liquid waterLs2.8345×106 J kg−1Latent heat of ice sublimationρI917 kg m−3Volumetric mass of pure iceρw1000 kg m−3Volumetric mass of liquid waterRa287.05967 J kg−1 K−1Specific gas constant for dry airRv461.52499 J kg−1 K−1Specific gas constant for water vaporT0273.16 Kwater melting point temperature assumed to be equal to the triple point valueσ5.6705×10-8 W m−2 K−4Stefan–Boltzmann parameterg9.80665 m s−2Gravitational accelerationLayering parameters Nmin3Minimum number of snow layersZmin0.03 mThreshold for uniform layeringParameters for density of new snow when SNOWFALL = V12 ρmin50 kg m−3Minimum thresholdaρ109 kg m−3Regression coefficientbρ6 kg m−3 K−1Regression coefficientcρ26 kg m−7∕2 s+1∕2Regression coefficientParameters for density of new snow when SNOWFALL = S14 eρ3.28Empirical parameterfρ0.03Empirical parametergρ-0.36Empirical parameterhρ-0.75Empirical parameteriρ0.8Empirical parameterjρ0.3Empirical parameterParameters for density of new snow when SNOWFALL = A76 kρ1.7 kg m−3 K−1.5Regression coefficientlρ15 KRegression coefficientParameters for metamorphism sph111 574.07 s−1Empirical parametersph22314.81 s−1Empirical parametersph37.2337×10-7 s−1Empirical parameteras1.1×10-6 m2 kg−1 s−1Empirical parameterbs3.1×10-8Empirical parameterms3.1Empirical parameter
SymbolValues and unitsDescriptionParameters for thermal conductivity when SNOWCOND = Y81 aλ2.22 W m−1 K−1Empirical parameterλmin4×10-2 W m−1 K−1Empirical parameterParameters for thermal conductivity when SNOWCOND = C11 bλ2.5×10-6 W m5 K−1 kg−2Empirical parametercλ-1.23×10-4 W m2 K−1 kg−1Empirical parameterdλ2.4×10-2 W m−1 K−1Empirical parameterParameters for thermal conductivity when SNOWCOND = I02 eλ2.0×10-2 W m−1 K−1Empirical parameterfλ2.5×10-6 W m5 K−1 kg−2Empirical parametergλ-6.023×10-2 W m−1 K−1Empirical parameterhλ-2.5425 W m−1Empirical parameteriλ-289.99 KEmpirical parameterP0105 PaEmpirical parameterParameters for solar radiation absorption when SNOWRAD = B92 γ10.71Fraction of shortwave radiation in band [0.3–0.8 µm]γ20.21Fraction of shortwave radiation in band [0.8–1.5 µm]γ30.08Fraction of shortwave radiation in band [1.5–2.8 µm]PCDP8.7×105 PaReference atmospheric pressure at Col de PorteParameters for solar radiation absorption when SNOWRAD = T17 λ0400 nmReference wavelength for dust MAEMAE(λ0)110 m2 kg−1Mass Absorption Efficiency of dust at wavelength λ0AAE4.1Angström absorption exponent for dustρBC1270 kg m−3Density of black carbonmBC1.95–0.79 iRefractive index of black carbonfBC1.638Multiplicative factor to compute black carbon MAEParameters for Light-Absorbing Particles h0.005 mE-folding depth of the exponential decay rate for dry depositionParameters for compaction when SNOWCOMP = B92 η07.62237×106 kg s−1 m−1Empirical parameteraη0.1 K−1Empirical parameterbη0.023 m3 kg−1Empirical parametercη250 kg m−3Empirical parameterParameters for compaction when SNOWCOMP = S14 BS3.96×10-2Empirical parameterkS0.18Empirical parameterParameters for snow drift aSUBL1.8×10-3Empirical parameterbSUBL4Empirical parametercSUBL2.868Empirical parameterdSUBL0.085Empirical parameterγSUBL3.6Empirical parameter
SymbolValues and unitsDescriptionParameters for percolation when SNOWLIQ = B02 rmin0.03Mass of liquid fraction parameter in B02 parameterizationrmax0.1Mass of liquid fraction parameter in B02 parameterizationρr200 kg m−3Density parameter in B02 parameterizationParameters for snowmaking and grooming ASM3300 m2Surface area covered by a snowgunLSM0.4Loss factor during snowmakingdSM2.8×10-4 mOptical diameter of machine-made snowSSM0.9Sphericity of machine-made snowρGRO450 kg m−3Target density of groomed snowdGRO2.6×10-4 mOptical diameter of groomed snowSGRO0.9Sphericity of groomed snowParameters for unloading ρUN200 kg m−3Density of unloaded snowdUN6×10-4 mOptical diameter of unloaded snowSUN0.9Sphericity of unloaded snow
Physiographic parameters
SymbolUnitsDescriptionzGjmDepth of soil layer jΘradSlope angle
Parameters adjustable in namelistSymbolDefault values and unitsDescriptionNmax50Maximum number of layersΔt900 smodel time stepϵ0.99snow emissivityRil0.026Threshold on the Richardson number for the exchange coefficient parameterizationz010−3 msnow roughness for momentumz0h10−4 msnow roughness for heatza2 mReference height for air temperaturezu10 mReference height for wind speedα1G0.38Glacier albedo in band [0.3–0.8 µm]α2G0.23Glacier albedo in band [0.8–1.5 µm]α3G0.08Glacier albedo in band [1.5–2.8 µm]ρG850 kg m−3Density threshold to separate snow and ice on glaciersτa60 dTime constant in visible albedo parameterizationCscav,j0Scavenging coefficient for Light-Absorbing Particles of type jTlim269.15 KSnowmaking: wet but temperature thresholdUlim4.2 m s−1Snowmaking: wind speed thresholdday11 NovemberSnowmaking: day of beginning of the base-layer generation production periodday215 DecemberSnowmaking: day of end of the base-layer generation production periodday331 MarchSnowmaking: day of end of the reinforcement production periodt10 s (= 00:00 LT)Snowmaking: time of beginning of the base-layer generation production periodt286 400 s (= 12:00 LT)Snowmaking: time of end of the base-layer generation production periodt364 800 s (= 18:00 LT)Snowmaking: time of beginning of the reinforcement production periodt428 800 s (= 08:00 LT)Snowmaking: time of end of the reinforcement production periodplim150 kg m−2Snowmaking: water use allowance for the base-layer generation production periodZlim0.6 mSnowmaking: total (natural + machine-made) snow depth threshold for the reinforcement production periodρSM500 kg m−3Snowmaking: machine-made snow densityaSM-0.4377 kg K−1 s−1Snowmaking: coefficient to compute the production potential mass of lance gunsbSM-0.47 kg s−1Snowmaking: coefficient to compute the production potential mass of lance gunsdayEND4, 30, 4, 30Grooming: month and day at which grooming is stopped (without and with snowmaking, respectively)fGRO1 d−1Grooming: daily frequency of groomingCode and data availability
The Crocus snowpack model is developed within the opensource SURFEX project within CeCILL-C 1.0 license (https://cecill.info/licences/Licence_CeCILL-C_V1-en.html, last access: 30 March 2026). The source code of the version referred in this work can be accessed freely on 10.5281/zenodo.20493693. Obviously, the code will evolve after submission and publication. The most up-to-date stable version of Crocus can be accessed through the branch cen of the SURFEX_CEN git repository, currently hosted on https://github.com/UMR-CNRM/ (last access: 30 March 2026). All informations to access to this repository are available on https://umr-cnrm.github.io/snowtools-doc/surfex/surfex-index.html (last access: 30 March 2026). Latest developments not yet stabilized are in branch cen_dev. For reproductibility of results, providing a git tag is recommended in any publication based on Crocus simulations with any modification of the source code compared to the version associated with this paper. The version described in this work is tagged as crocus3.0.2.
The SURFEX Land Surface Model comes with a comprehensive documentation including user's guide, technical and scientific documentation available at https://www.umr-cnrm.fr/surfex (last access: 30 March 2026). Nevertheless, we recommend to combine the use of SURFEX-Crocus with the snowtools_git Python3 package (Sect. ) which includes pre and post-processing tools. Version 2.0.3 of the snowtools package was fully tested with Crocus3.0.2 and archived on 10.5281/zenodo.17122726. The installation of the most up-to-date version and execution procedures are described in https://umr-cnrm.github.io/snowtools-doc/ (last access: 30 March 2026). It also includes informations to request technical support on registration. This documention also includes a summarized documentation to install SURFEX-Crocus and run a first test case (https://umr-cnrm.github.io/snowtools-doc/surfex/surfex-index.html, last access: 30 March 2026).
The externalized version of Crocus is also available on 10.5281/zenodo.20493693. However, model developers who intent to couple their Land Surface Model with Crocus are encouraged to access the code through the SURFEX git repository following the procedure described at https://umr-cnrm.github.io/snowtools-doc/surfex/surfex-extern.html (last access: 30 March 2026). This will highly facilitate future updates of the code coming from the SURFEX implementation and allow to provide new contributions potentially useful for the whole Crocus community.
Author contributions
ML wrote the paper with contribution of all authors and has led the Crocus model development since 2012 with support of MD and SM. BDF (with major contribution), JB, KF, MB and LR read and help to fix typos in the equations and their consistency with the code. LVG wrote the description of the mechanical diagnoses and MEPRA. We only mention here the contributions to the code from the previous reference paper (2012). RN and MF undertook various technical improvements, merging and optimizations. BC and ML implemented the different multiphysics options. ML implemented TARTES within Crocus, the management of glacier configurations, and model diagnoses. ML coupled Crocus with MEB with the help of ABoo. ABou implemented unloading from vegetation with the scientific supervision of ABoo. FT implemented light-absorbing particles with the scientific supervision of MD. PS and CC implemented machine made snow and snow grooming with the scientific supervision of SM. CC initiated the implementation of the new formalism of metamorphism with the scientific supervision of SM. MB fixed this implementation of metamorphism as described in this paper. MB and AH coupled Crocus with Snowpappus and adjusted the drift parameterizations. LQ implemented the management of ice layers after freezing rain with the scientific supervision of VV. PH implemented the MEPRA module within SURFEX. VV coupled Crocus with the SYTRON blowing snow module within SURFEX. VV coupled the externalized version of Crocus with the SVS2 platform and has frequently provided contributions to the code. GM coupled Crocus with the FSM2 platform. MM has supervised the different updates of Crocus within SURFEX. SM, MD and ML led the snow modelling team of CNRM/CEN respectively from 2009 to 2015, 2015 to 2020, and 2021 to today.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. The authors bear the ultimate responsibility for providing appropriate place names. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.
Acknowledgements
Authors acknowledge the pioneering work of Eric Brun, now retired, who initially developed the model in the late 1980s and implemented a large part of the current version within the SURFEX platform in 2010. They also acknowledge the numerous contributions of students and permanent staff that contributed to the model development, evaluations and applications during 35 years. CNRM/CEN is part of LabEX OSUG@2020.
Financial support
This paper received financial support and contributions from authors (Marie Dumont, Basile De Fleurian, Kévin Fourteau, Julien Brondex, Léon Roussel, François Tuzet) funded by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (IVORI; grant no. 949516).
Review statement
This paper was edited by Yuanchao Fan and reviewed by Richard L. H. Essery and one anonymous referee.
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