The soil temperature at the current time step is a function of the past soil temperature, soil

moisture state, and the surface forcing. The latter depends on the time of day, wind,

The temperature gradient in a non-uniform soil layer can be described by the one-

dimensional heat flow equation:

∂*T * ∂ ⎛ ∂*T *⎞

- *v *p,*w*

= ⎜ *k*th

+

⎟

∂*t * ∂*z *⎝ ∂*z *⎠

(4.1)

∂*k *∂*T*

∂*T*

2

= th

+ *k*th 2 - *v*

+

∂*z *∂*z*

∂*z*

where *T *is the temperature (*K*), *t *is time (*s*), *k*th is the thermal diffusivity (*m*2/s), *c*p,w is the

specific heat of water (*J/kg*⋅K), *c*p is the specific heat of the soil (*J/kg*⋅K), *v *is the vertical

rate of water flow (*m/s*), *l*fus is the latent heat of fusion (*J/kg*), θi is the volumetric ice

content (*cm*3/cm3), ρi is the density of ice (*kg/m*3), ρw is the density of water (*kg/m*3), and *z*

is depth (*m*) measured positive downward from the surface. Further discussions on the

water flow rate and the change in ice content are found in Chapter 5 and Chapter 6,

respectively. The temperature is subject to the following boundary condition:

ρ ∂θi

(4.2)

∂*T*

κ

+ *l * fus i

∆*z *- *vc * pT = 0

@ *z *= 0*m*

ρw ∂*t*

∂*z*

radiation (*W/m *) both reflected and emitted, *H *is the sensible heat (*W/m*2), *L *is the latent

conductivity (*W/mK*). In Equation (4.2), heat that is transferred to the surface is

considered positive, as is shown in Figure 4.1. Energy that can be transferred either

to/from the surface depending on the gradient is shown as double-headed arrows.

The first term in Equation (4.2) represents the amount of solar, or shortwave radiation,

absorbed by the surface. The second term is the incoming longwave radiation while the

third term is the outgoing longwave radiation, which is composed of radiation emitted

from the surface and incoming radiation reflected from the surface. The sensible and

latent heat fluxes together are called the turbulent heat fluxes and have non-zero values in

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