Title: 2.6. Soil and Snow Temperatures — ctsm CTSM master documentation

URL Source: https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Soil_Snow_Temperatures/CLM50_Tech_Note_Soil_Snow_Temperatures.html

Markdown Content:
The first law of heat conduction is

(2.6.1)[¶](#equation-6-1 "Permalink to this equation")\\\[F=-\\lambda \\nabla T\\\]

where \\(F\\) is the amount of heat conducted across a unit cross-sectional area in unit time (W m\-2), \\(\\lambda\\) is thermal conductivity (W m\-1 K\-1), and \\(\\nabla T\\) is the spatial gradient of temperature (K m\-1). In one-dimensional form

(2.6.2)[¶](#equation-6-2 "Permalink to this equation")\\\[F\_{z} =-\\lambda \\frac{\\partial T}{\\partial z}\\\]

where \\(z\\) is in the vertical direction (m) and is positive downward and \\(F\_{z}\\) is positive upward. To account for non-steady or transient conditions, the principle of energy conservation in the form of the continuity equation is invoked as

(2.6.3)[¶](#equation-6-3 "Permalink to this equation")\\\[c\\frac{\\partial T}{\\partial t} =-\\frac{\\partial F\_{z} }{\\partial z}\\\]

where \\(c\\) is the volumetric snow/soil heat capacity (J m\-3 K\-1) and \\(t\\) is time (s). Combining equations and yields the second law of heat conduction in one-dimensional form

(2.6.4)[¶](#equation-6-4 "Permalink to this equation")\\\[c\\frac{\\partial T}{\\partial t} =\\frac{\\partial }{\\partial z} \\left\[\\lambda \\frac{\\partial T}{\\partial z} \\right\].\\\]

This equation is solved numerically to calculate the soil, snow, and surface water temperatures for a 25-layer soil column with up to twelve overlying layers of snow and a single surface water layer with the boundary conditions of \\(h\\) as the heat flux into the top soil, snow, and surface water layers from the overlying atmosphere (section [2.6.1](#numerical-solution-temperature)) and zero heat flux at the bottom of the soil column. The temperature profile is calculated first without phase change and then readjusted for phase change (section [2.6.2](#phase-change)).

