## 2.6.3. Soil and Snow Thermal Properties[¶](#soil-and-snow-thermal-properties "Permalink to this headline")
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The thermal properties of the soil are assumed to be a weighted combination of the mineral and organic properties of the soil ([Lawrence and Slater 2008](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#lawrenceslater2008)). The soil layer organic matter fraction \\(f\_{om,i}\\) is

(2.6.75)[¶](#equation-6-77 "Permalink to this equation")\\\[f\_{om,i} =\\rho \_{om,i} /\\rho \_{om,\\max } .\\\]

Soil thermal conductivity \\(\\lambda \_{i}\\) (W m\-1 K\-1) is from [Farouki (1981)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#farouki1981)

(2.6.76)[¶](#equation-6-78 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{lr} \\lambda \_{i} = \\left\\{ \\begin{array}{lr} K\_{e,\\, i} \\lambda \_{sat,\\, i} +\\left(1-K\_{e,\\, i} \\right)\\lambda \_{dry,\\, i} &\\qquad S\_{r,\\, i} > 1\\times 10^{-7} \\\\ \\lambda \_{dry,\\, i} &\\qquad S\_{r,\\, i} \\le 1\\times 10^{-7} \\end{array}\\right\\} &\\qquad i=1,\\ldots ,N\_{levsoi} \\\\ \\lambda \_{i} =\\lambda \_{bedrock} &\\qquad i=N\_{levsoi} +1,\\ldots N\_{levgrnd} \\end{array}\\end{split}\\\]

where \\(\\lambda \_{sat,\\, i}\\) is the saturated thermal conductivity, \\(\\lambda \_{dry,\\, i}\\) is the dry thermal conductivity, \\(K\_{e,\\, i}\\) is the Kersten number, \\(S\_{r,\\, i}\\) is the wetness of the soil with respect to saturation, and \\(\\lambda \_{bedrock} =3\\) W m\-1 K\-1 is the thermal conductivity assumed for the deep ground layers (typical of saturated granitic rock; [Clauser and Huenges 1995](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#clauserhuenges1995)). For glaciers,

(2.6.77)[¶](#equation-6-79 "Permalink to this equation")\\\[\\begin{split}\\lambda \_{i} =\\left\\{\\begin{array}{l} {\\lambda \_{liq,\\, i} \\qquad T\_{i} \\ge T\_{f} } \\\\ {\\lambda \_{ice,\\, i} \\qquad T\_{i} <T\_{f} } \\end{array}\\right\\}\\end{split}\\\]

where \\(\\lambda \_{liq}\\) and \\(\\lambda \_{ice}\\) are the thermal conductivities of liquid water and ice, respectively ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)). The saturated thermal conductivity \\(\\lambda \_{sat,\\, i}\\) (W m\-1 K\-1) depends on the thermal conductivities of the soil solid, liquid water, and ice constituents

(2.6.78)[¶](#equation-6-80 "Permalink to this equation")\\\[\\lambda \_{sat} =\\lambda \_{s}^{1-\\theta \_{sat} } \\lambda \_{liq}^{\\frac{\\theta \_{liq} }{\\theta \_{liq} +\\theta \_{ice} } \\theta \_{sat} } \\lambda \_{ice}^{\\theta \_{sat} \\left(1-\\frac{\\theta \_{liq} }{\\theta \_{liq} +\\theta \_{ice} } \\right)}\\\]

where the thermal conductivity of soil solids \\(\\lambda \_{s,\\, i}\\) varies with the sand, clay, and organic matter content

(2.6.79)[¶](#equation-6-81 "Permalink to this equation")\\\[\\lambda \_{s,i} =(1-f\_{om,i} )\\lambda \_{s,\\min ,i} +f\_{om,i} \\lambda \_{s,om}\\\]

where the mineral soil solid thermal conductivity \\(\\lambda \_{s,\\min,i}\\) is

(2.6.80)[¶](#equation-6-82 "Permalink to this equation")\\\[\\lambda \_{s,\\, \\min ,i} =\\frac{8.80{\\rm \\; }\\left(\\% sand\\right)\_{i} +{\\rm 2.92\\; }\\left(\\% clay\\right)\_{i} }{\\left(\\% sand\\right)\_{i} +\\left(\\% clay\\right)\_{i} } ,\\\]

and \\(\\lambda \_{s,om} =0.25\\)W m\-1 K\-1 ([Farouki 1981](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#farouki1981)). \\(\\theta \_{sat,\\, i}\\) is the volumetric water content at saturation (porosity) (section [2.7.3.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#hydraulic-properties)).

The thermal conductivity of dry soil is

(2.6.81)[¶](#equation-6-83 "Permalink to this equation")\\\[\\lambda \_{dry,i} =(1-f\_{om,i} )\\lambda \_{dry,\\min ,i} +f\_{om,i} \\lambda \_{dry,om}\\\]

where the thermal conductivity of dry mineral soil \\(\\lambda \_{dry,\\min,i}\\) (W m\-1 K\-1) depends on the bulk density \\(\\rho \_{d,\\, i} =2700\\left(1-\\theta \_{sat,\\, i} \\right)\\) (kg m\-3) as

(2.6.82)[¶](#equation-6-84 "Permalink to this equation")\\\[\\lambda \_{dry,\\, \\min ,i} =\\frac{0.135\\rho \_{d,\\, i} +64.7}{2700-0.947\\rho \_{d,\\, i} }\\\]

and \\(\\lambda \_{dry,om} =0.05\\) W m\-1 K\-1 ([Farouki 1981](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#farouki1981)) is the dry thermal conductivity of organic matter. The Kersten number \\(K\_{e,\\, i}\\) is a function of the degree of saturation \\(S\_{r}\\) and phase of water

(2.6.83)[¶](#equation-6-85 "Permalink to this equation")\\\[\\begin{split}K\_{e,\\, i} = \\left\\{ \\begin{array}{lr} \\log \\left(S\_{r,\\, i} \\right)+1\\ge 0 &\\qquad T\_{i} \\ge T\_{f} \\\\ S\_{r,\\, i} &\\qquad T\_{i} <T\_{f} \\end{array}\\right\\}\\end{split}\\\]

where

(2.6.84)[¶](#equation-6-86 "Permalink to this equation")\\\[S\_{r,\\, i} =\\left(\\frac{w\_{liq,\\, i} }{\\rho \_{liq} \\Delta z\_{i} } +\\frac{w\_{ice,\\, i} }{\\rho \_{ice} \\Delta z\_{i} } \\right)\\frac{1}{\\theta \_{sat,\\, i} } =\\frac{\\theta \_{liq,\\, i} +\\theta \_{ice,\\, i} }{\\theta \_{sat,\\, i} } \\le 1.\\\]

Thermal conductivity \\(\\lambda \_{i}\\) (W m\-1 K\-1) for snow is from [Jordan (1991)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#jordan1991)

(2.6.85)[¶](#equation-6-87 "Permalink to this equation")\\\[\\lambda \_{i} =\\lambda \_{air} +\\left(7.75\\times 10^{-5} \\rho \_{sno,\\, i} +1.105\\times 10^{-6} \\rho \_{sno,\\, i}^{2} \\right)\\left(\\lambda \_{ice} -\\lambda \_{air} \\right)\\\]

where \\(\\lambda \_{air}\\) is the thermal conductivity of air ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)) and \\(\\rho \_{sno,\\, i}\\) is the bulk density of snow (kg m\-3)

(2.6.86)[¶](#equation-6-88 "Permalink to this equation")\\\[\\rho \_{sno,\\, i} =\\frac{w\_{ice,\\, i} +w\_{liq,\\, i} }{\\Delta z\_{i} } .\\\]

The volumetric heat capacity \\(c\_{i}\\) (J m\-3 K\-1) for soil is from [de Vries (1963)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#devries1963) and depends on the heat capacities of the soil solid, liquid water, and ice constituents

(2.6.87)[¶](#equation-6-89 "Permalink to this equation")\\\[c\_{i} =c\_{s,\\, i} \\left(1-\\theta \_{sat,\\, i} \\right)+\\frac{w\_{ice,\\, i} }{\\Delta z\_{i} } C\_{ice} +\\frac{w\_{liq,\\, i} }{\\Delta z\_{i} } C\_{liq}\\\]

where \\(C\_{liq}\\) and \\(C\_{ice}\\) are the specific heat capacities (J kg\-1 K\-1) of liquid water and ice, respectively ([Table 2.2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#table-physical-constants)). The heat capacity of soil solids \\(c\_{s,i}\\) (J m\-3 K\-1) is

(2.6.88)[¶](#equation-6-90 "Permalink to this equation")\\\[c\_{s,i} =(1-f\_{om,i} )c\_{s,\\min ,i} +f\_{om,i} c\_{s,om}\\\]

where the heat capacity of mineral soil solids \\(c\_{s,\\min,\\, i}\\) (J m\-3 K\-1) is

(2.6.89)[¶](#equation-6-91 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{lr} c\_{s,\\min ,\\, i} =\\left(\\frac{2.128{\\rm \\; }\\left(\\% sand\\right)\_{i} +{\\rm 2.385\\; }\\left(\\% clay\\right)\_{i} }{\\left(\\% sand\\right)\_{i} +\\left(\\% clay\\right)\_{i} } \\right)\\times 10^{6} &\\qquad i=1,\\ldots ,N\_{levsoi} \\\\ c\_{s,\\, \\min ,i} =c\_{s,\\, bedrock} &\\qquad i=N\_{levsoi} +1,\\ldots ,N\_{levgrnd} \\end{array}\\end{split}\\\]

where \\(c\_{s,bedrock} =2\\times 10^{6}\\) J m\-3 K\-1 is the heat capacity of bedrock and \\(c\_{s,om} =2.5\\times 10^{6}\\) J m\-3 K\-1 ([Farouki 1981](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#farouki1981)) is the heat capacity of organic matter. For glaciers and snow

(2.6.90)[¶](#equation-6-92 "Permalink to this equation")\\\[c\_{i} =\\frac{w\_{ice,\\, i} }{\\Delta z\_{i} } C\_{ice} +\\frac{w\_{liq,\\, i} }{\\Delta z\_{i} } C\_{liq} .\\\]

For the special case when snow is present (\\(W\_{sno} >0\\)) but there are no explicit snow layers (\\(snl=0\\)), the heat capacity of the top layer is a blend of ice and soil heat capacity

(2.6.91)[¶](#equation-6-93 "Permalink to this equation")\\\[c\_{1} =c\_{1}^{\*} +\\frac{C\_{ice} W\_{sno} }{\\Delta z\_{1} }\\\]

where \\(c\_{1}^{\*}\\) is calculated from [(2.6.87)](#equation-6-89) or [(2.6.90)](#equation-6-92).

