Values from Table
m-3
Other values….
Thermal admittance of dry soil ~ 102 J m-2 s-1/2 K-1
Thermal admittance of wet saturated soil ~ 103 J m-2 s-1/2 K-1
Soil density, thermal conductivity, thermal
admittance.
Elevated %
of quartz and
clay minerals
Water content
High values
Sandy
Clay
Peat
Low
values
Elevated %
of organic
matter
(this is only qualitative
the relations are non
linear)
Amplitude of the temperature wave at the surface
DT.
Elevated %
of quartz and
clay minerals
Water content
Low
values
Sandy
Clay
Peat
High values
Elevated %
of organic
matter
(this is only qualitative
the relations are non
linear)
Specific heat
Elevated %
of quartz and
clay minerals
Sandy
Water content
Low values
Clay
Peat
High
values
Elevated %
of organic
matter
(this is only qualitative
the relations are non
linear)
Thermal diffusivity.
Elevated %
of quartz and
clay minerals
Water content
High values
Sandy
Clay
Peat
Low values
Low values
Elevated %
of organic
matter
(this is only qualitative
the relations are non
linear)
Examples:
Dry Sandy Soil (40% pore space)
soil density   s  1.6  103 kg m -3
specific heat  cs  0.8  103 J kg 1 K - 1
thermal conductivity  k  0 .3 W m - 1 K - 1
Heat Capacity  C s   s c s  1.28  106 J m - 3 K - 1
k
Thermal diffusivity   
 0.24  10  6 m 2 s - 1
scs
Thermal admittance   C s k  620 J m - 2 s - 1/2 K - 1
Damping depth (daily cycle)  z D 
2
 0.08 m

2
 1. 5 m

For a maximum Ground Heat Flux of 200W/m2 the
Damping depth (annual cycle)  z D 
temperature variation between night and day is
DT 
QG
 38o C
 
Saturated Sandy Soil (40% pore space)
soil density   s  2.0  103 kg m -3
specific heat  cs  1.48  103 J kg 1 K - 1
thermal conductivity  k  2 .2 W m - 1 K - 1
Heat Capacity  C s   s c s  2.96  106 J m - 3 K - 1
k
Thermal diffusivity   
 0.74  10  6 m 2 s - 1
scs
Thermal admittance  C s k  2550 J m - 2 s - 1/2 K - 1
Damping depth (daily cycle)  z D 
2
 0.14 m

2
 2.7 m

For a maximum Ground Heat Flux of 200W/m2 the
Damping depth (annual cycle)  z D 
temperature variation between night and day is
DT 
QG
9o C
 
Limitations of the previous approach:
•Measurements show that the ground heat flux is not
sinusoidal in time. In particular during night-time is
more uniform and much flatter.
•The assumed sinusoidal variation of the surface
temperature may be not realistic.
•The simplifying assumption of the homogeneity of the
submedium is often not realized.
max
min
9 hrs
1st approach:Statistical parameterizations
Reasonable expectation that QG is a fraction of
Q* forcing. The surface QG leads the Q*
forcing by about 3 hours. Therefore a daily plot
of QG vs Q* results in a hysteresis loop
This loop can be modeled as
 Q * 
QG  aQ * b
c
 t 
Where a, b, c are deduced from measurements.
Ex. For bare soil (Novak, 1981):
a=0.38,b=0.56 hrs, and c=-27.3 W m-2
This approach ignores the role of wind
(Convection) in heat sharing at the surface
2nd approach: physically based models
They take into account net radiation, latent and
sensible heat fluxes at the surface
The Force-Restore method (Deardorff, 1978)
Two layer approximation
A shallow thermally active layer near the surface, and
a thicker layer below.
Energy budget of the shallow layer
TG
1  *

 Q  Q H  Q E  QG d 

t
cd 
Q*
QH
QE
d
QG ( d )
Q*=net radiation
QE=Latent Heat Flux
QH=Sensible Heat Flux
QG =Ground Heat Flux
TG=ground temperature
of the shallow layer
d= depth of the shallow
layer
C= specific heat
soil density
N.B. Non radiative
positive fluxes are
directed away from the
surface. QH and QE are
positive when upward, QG
when downward. Q*
(radiative flux) is positive
when downward.
c=Cs is the heat
capacity of the
soil, function of
the water content.
Dz
Assuming that
T T
QG ( d )  k m G
Dz
with Tm temperature of the
thick layer
TG
1  *
k TG  Tm

 Q  Q H  Q E  
 C sd
t
C sd 
Dz
from the definition of damping depth
and thermal diffusivity
1/ 2
k
 2 
zD   
, 
Cs
 
2 C
z
2
k
s
2 
zD
k D
C s
2
CG C s d
is the heat capacity per unit area
of the near ground soil layer
2
z
TG
T  Tm
1  *

 Q  Q H  Q E   D G
 2d
t
CG 
Dz
z
assuming d  D and Dz  z D
2
TG
1  *

 Q  Q H  Q E   TG  Tm 

t
CG 
Surface forcing term
Restoring term
If the surface forcing term is removed, the restoring
term will cause TG to move exponentially towards Tm
To estimate Tm two possibilities:
•Constant (equal to the mean air temperature of the
previous 24hrs)
•Computed assuming that the ground heat flux at the
bottom of the thicker layer is zero.
Tm
 Tg  Tm 
t
Multi-Layer Soil Models (Tremback and
Kessler, 1985)
Q*
QH
QE
QG ( zi )
Compute the soil temperature in several layers in the soil
solving numerically:
TG
  T 
  G 
t
z  z 
The thermal diffusivity  is computed as a function of
the soil heat capacity and soil moisture potential
The soil moisture potential  is an indirect
measure of water content. It is the energy
necessary to extract water from the soil matrix.
Units are those of a negative pressure (Pa).
The forces which bind soil water are related to the soil
porosity and the soil water content (S, volume of water
per volume of soil). The forces are weakest for open
textured, wet soils and greatest for a clay soil
For a given soil, the potential increases as S
decreases. It is relatively easy to extract moisture
from a wet soil but as it dries out it becomes
increasingly difficult to remove additional units
Vertical flux of liquid water in soil (in absence
of percolating rain) is result of:
• Gravity
•Vertical water potential gradient (flux
gradient relationship as for heat). Darcy’s Law

z
K f  hydraulic conductivity
J  K f
g
The effect of evapotranspiration is to create a vertical
positive potential gradient which becomes greater
than the opposing gravitational gradient and
encourage the upward movement of water.
Soil heat flux measurements (Oke, 374-5)
In theory QG can be calculated from TG profiles and
knowledge of k or  – in practice this is not really possible,
since the values of k and  are variable and very difficult to
measure.
Most use soil heat flux plats (similar idea to net
radiometer thermopile)
Plates should be inserted in un-disturbed soil (few cm
depth), and not right at the surface. The depth depends on
the nature of the soil and the presence of roots.
Need to consider energy budget between plate and surface
z
Plate
DT
1
QG ( 0 )  QG ( z )

Dt C S Dz
 DT 
QG ( 0 )  QG ( z )  
C S Dz
D
t


measured
measured
Soil heat capacity estimated from
volume fraction of mineral, organic
matters and water
CS =Cm qm + Co qo + Cw qw + Ca qa