## 2.8.2. Ice Content[¶](#ice-content "Permalink to this headline")
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The conservation equation for mass of ice in snow layers is

(2.8.4)[¶](#equation-8-17 "Permalink to this equation")\\\[\\begin{split}\\frac{\\partial w\_{ice,\\, i} }{\\partial t} = \\left\\{\\begin{array}{lr} f\_{sno} \\ q\_{ice,\\, i-1} -\\frac{\\left(\\Delta w\_{ice,\\, i} \\right)\_{p} }{\\Delta t} & \\qquad i=snl+1 \\\\ -\\frac{\\left(\\Delta w\_{ice,\\, i} \\right)\_{p} }{\\Delta t} & \\qquad i=snl+2,\\ldots ,0 \\end{array}\\right\\}\\end{split}\\\]

where \\(q\_{ice,\\, i-1}\\) is the rate of ice accumulation from precipitation or frost or the rate of ice loss from sublimation (kg m\-2 s\-1) in the top layer and \\({\\left(\\Delta w\_{ice,\\, i} \\right)\_{p} \\mathord{\\left/ {\\vphantom {\\left(\\Delta w\_{ice,\\, i} \\right)\_{p} \\Delta t}} \\right.} \\Delta t}\\) is the change in ice due to phase change (melting rate) (section [2.6.2](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Soil_Snow_Temperatures/CLM50_Tech_Note_Soil_Snow_Temperatures.html#phase-change)). The term \\(q\_{ice,\\, i-1}\\) is computed in two steps as

(2.8.5)[¶](#equation-8-18 "Permalink to this equation")\\\[q\_{ice,\\, i-1} =q\_{grnd,\\, ice} +\\left(q\_{frost} -q\_{subl} \\right)\\\]

where \\(q\_{grnd,\\, ice}\\) is the rate of solid precipitation reaching the ground (section [2.7.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#canopy-water)) and \\(q\_{frost}\\) and \\(q\_{subl}\\) are gains due to frost and losses due to sublimation, respectively (sectio [2.5.4](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Fluxes/CLM50_Tech_Note_Fluxes.html#update-of-ground-sensible-and-latent-heat-fluxes)). In the first step, immediately after \\(q\_{grnd,\\, ice}\\) has been determined after accounting for interception (section [2.7.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#canopy-water)), a new snow depth \\(z\_{sno}\\) (m) is calculated from

(2.8.6)[¶](#equation-8-19 "Permalink to this equation")\\\[z\_{sno}^{n+1} =z\_{sno}^{n} +\\Delta z\_{sno}\\\]

where

(2.8.7)[¶](#equation-8-20 "Permalink to this equation")\\\[\\Delta z\_{sno} =\\frac{q\_{grnd,\\, ice} \\Delta t}{f\_{sno} \\rho \_{sno} }\\\]

and \\(\\rho \_{sno}\\) is the bulk density of newly fallen snow (kg m\-3), which parameterized by a temperature-dependent and a wind-dependent term:

(2.8.8)[¶](#equation-8-21a "Permalink to this equation")\\\[\\rho\_{sno} = \\rho\_{T} + \\rho\_{w}.\\\]

The temperature dependent term is given by ([van Kampenhout et al. (2017)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#vankampenhoutetal2017))

(2.8.9)[¶](#equation-8-21b "Permalink to this equation")\\\[\\begin{split}\\rho\_{T} = \\left\\{\\begin{array}{lr} 50 + 1.7 \\left(17\\right)^{1.5} & \\qquad T\_{atm} >T\_{f} +2 \\ \\\\ 50+1.7 \\left(T\_{atm} -T\_{f} + 15\\right)^{1.5} & \\qquad T\_{f} - 15 < T\_{atm} \\le T\_{f} + 2 \\ \\\\ -3.833 \\ \\left( T\_{atm} -T\_{f} \\right) - 0.0333 \\ \\left( T\_{atm} -T\_{f} \\right)^{2} &\\qquad T\_{atm} \\le T\_{f} - 15 \\end{array}\\right\\}\\end{split}\\\]

where \\(T\_{atm}\\) is the atmospheric temperature (K), and \\(T\_{f}\\) is the freezing temperature of water (K) ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)). When 10 m wind speed \\(W\_{atm}\\) is greater than 0.1 m\-1, snow density increases due to wind-driven compaction according to [van Kampenhout et al. 2017](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#vankampenhoutetal2017)

(2.8.10)[¶](#equation-8-21c "Permalink to this equation")\\\[\\rho\_{w} = 266.861 \\left(\\frac{1 + \\tanh(\\frac{W\_{atm}}{5})}{2}\\right)^{8.8}\\\]

which is added to the temperature-dependent term (cf. equation [(2.8.8)](#equation-8-21a)).

The mass of snow \\(W\_{sno}\\) is

(2.8.11)[¶](#equation-8-22 "Permalink to this equation")\\\[W\_{sno}^{n+1} =W\_{sno}^{n} +q\_{grnd,\\, ice} \\Delta t.\\\]

The ice content of the top layer and the layer thickness are updated as

(2.8.12)[¶](#equation-8-23 "Permalink to this equation")\\\[w\_{ice,\\, snl+1}^{n+1} =w\_{ice,\\, snl+1}^{n} +q\_{grnd,\\, ice} \\Delta t\\\]

(2.8.13)[¶](#equation-8-24 "Permalink to this equation")\\\[\\Delta z\_{snl+1}^{n+1} =\\Delta z\_{snl+1}^{n} +\\Delta z\_{sno} .\\\]

In the second step, after surface fluxes and snow/soil temperatures have been determined (Chapters [2.5](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Fluxes/CLM50_Tech_Note_Fluxes.html#rst-momentum-sensible-heat-and-latent-heat-fluxes) and [2.6](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Soil_Snow_Temperatures/CLM50_Tech_Note_Soil_Snow_Temperatures.html#rst-soil-and-snow-temperatures)), \\(w\_{ice,\\, snl+1}\\) is updated for frost or sublimation as

(2.8.14)[¶](#equation-8-25 "Permalink to this equation")\\\[w\_{ice,\\, snl+1}^{n+1} =w\_{ice,\\, snl+1}^{n} +f\_{sno} \\left(q\_{frost} -q\_{subl} \\right)\\Delta t.\\\]

If \\(w\_{ice,\\, snl+1}^{n+1} <0\\) upon solution of equation [(2.8.14)](#equation-8-25), the ice content is reset to zero and the liquid water content \\(w\_{liq,\\, snl+1}\\) is reduced by the amount required to bring \\(w\_{ice,\\, snl+1}^{n+1}\\) up to zero.

The snow water equivalent \\(W\_{sno}\\) is capped to not exceed 10,000 kg m\-2. If the addition of \\(q\_{frost}\\) were to result in \\(W\_{sno} > 10,000\\) kg m\-2, the frost term \\(q\_{frost}\\) is instead added to the ice runoff term \\(q\_{snwcp,\\, ice}\\) (section [2.7.6](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#runoff-from-glaciers-and-snow-capped-surfaces)).

