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LPJmL has an energy balance model, including one-dimensional heat conduction, convection of latent heat. Freezing and thawing has been added to better account for soil ice dynamics. The one-dimensional heat conduction equation is:

where:

α is thermal diffusivity, λ thermal conductivity, c heat capacity. Soil temperature T at position x and time t is solved in its finite difference form following Bayazıtoğlu, Yıldız and Özişik, M. Necati. (1988):

for soil layers m, including a snow layer, and time step i with the following boundary conditions:

We assume a heatflux of zero below the lowest soil layer. The largest possible, numerically still stable time step Δt is calculated depending on Δx, which is given by soil layering and soil thermal diffusivity α (Bayazıtoğlu, Yıldız and Özişik, M. Necati. 1988): with

equation for temp_change becomes:

For numerical stability, (1-2r) needs to be > 0, so that r ≤ 0.5 as Δx is given from soil depth and α can be calculated from soil properties. The maximum stable Δt can be calculated:

For the diurnal temperature range after Parton and Logan, 1981, at least 4 time steps per day are calculated and the maximum time steps are defined to be 40 per day.
Heat capacity (c in J/K) of the soil is calculated as the sum of the volumetric-specific heat capacities (in J/(m^3 \cdot K)) of soil minerals c_m=1.9259 x 10⁶, soil water content c_w=4.2 x 10^6 and soil ice content c_i=2.1 x 10⁶ and their corresponding shares (in m³) of the soil bucket:

The heat capacity of air is neglected because of its comparatively low value. Thermal conductivity λ is calculated following Johansen, 1977. Sensible and latent heat fluxes are calculated explicitly for the snow layer by assuming a constant snow density of 0.3 t/m³ and the resulting thermal diffusivity of 3.17 x 10^-7 m²/s. Sublimation is assumed to be 0.1 mm/d. The active layer thickness represents the depth of maximum thawing of the year. Freezing depth is calculated by assuming that the fraction of frozen water is congruent with the frozen soil bucket. The 0°C-isotherme within a layer is estimated by assuming a linear temperature gradient within the layer and this fraction of heat is assumed to be used for the thawing respectively freezing process. Temperature represents the amount of thermal energy available, whereas heat transport represents the movement of thermal energy into the soil by rain and melt water. Precipitation and percolation energy and energy which is determined by the temperature difference of the temperature of the above layer or the air temperature for the upper layer and the temperature of the below layer is assumed to be used for conversion of latent heat fluxes first and the residual energy is used to increase soil temperature. The soil temperature is initialized at the beginning of the spinup simulation by the mean annual temperature.

source:trunk/src/soil/soiltemp.c

Parameters are defined in source:trunk/par/soil_new.par

Bayazıtoğlu, Yıldız and Özişik, M. Necati. (1988):, ‘Elements of heat transfer’, McGraw-Hill, New York, 1988. ISBN 0070041547.

Parton, W.~J. & Logan, J.~A., 1981, `A model for diurnal variation in soil and air temperature’, Agricultural Meteorology, 23(0),~205—216.

Johansen, O.. 1977, ‘Thermal conductivity of soils’, PhD thesis, University of Trondheim, Trondheim, Norway.