Outline
• Announcements
• Soil Thermal Regime
Soil Physics 2010
Announcements
• Homework 6 due now
• Review sessions next week
Quiz!
Soil Physics 2010
Quiz, question 1
DT
(L2/T)
The diagram below shows how a specific soil thermal property
varies as a function of some other soil property. What are
those 2 properties? Label the axes, including units.
q (L3/L3, or unitless)
Soil Physics 2010
Quiz, question 2
Rank the following in order (most, middle, least) according to the
heat energy required to raise the temperature by 1 K of a liter
of:
Most
a) water
Middle
b) saturated soil
Least
c) dry soil
Soil Physics 2010
Quiz, question 2
Two 1 L cubes of ice, temperature -10 °C, are surrounded by air
that has a temperature of 40 °C. The ice cubes are not in direct
sunlight, and are exposed to the same windspeed, but cube A is
in the Amazon rain forest, and cube S is in the Sahara desert.
Which cube melts faster, and why?
The major difference in the
cubes’ environments is the
humidity of the air. Air in the Amazon has much
more water vapor in it. When the water vapor
touches the ice cube, it condenses, releasing
its latent heat. So cube A receives much more
heat from the air, and melts faster.
Soil Physics 2010
Soil thermal regime
Steady-state
T
qh  
z
Transient
T
T
  DT 2
t
z
2
Instantaneous change
(like infiltration)
Cyclical change
New! Improved! Dynamic!
Soil Physics 2010
Simple soil heat model
T
 2T
Given:
  DT 2
t
z
Assume:
1-D soil
DT constant
Constant temperature Ta at z = ∞
Sinusoidal temperature at z = 0,
with amplitude A0
Then:

T z, t   Ta  A0 sin t  0  z
Soil Physics 2010

e
d
z
d
Some new terms

T z, t   Ta  A0 sin t  0  z
•
•
•
•

e
d
z
d
1-D soil (depth z, positive down)
DT constant in z and t
Constant temperature Ta at z = ∞
Sinusoidal temperature at z = 0, with amplitude A0
 = 2p / period (say, 24 hours): normalizes the
“clock time” t to the 2p sine wave period.
d = depth z at which thermal amplitude is
A0/e: normalizes “physical depth” z to exponential
function depth. Specifically,
2 DT
d

Soil Physics 2010
The sine part
T 0, t   sint 
This is about the soil
surface warming
during the day, and
cooling at night.
Soil Physics 2010
More sine stuff

T z, t   A0 sin t  0  z

d
Clock time at
the surface
Phase shift
with depth
Phase constant:
adjust so peak is at
the right time of day
1
6:00 am
1:00 pm
1
3
0
0
midnight
-1
Soil Physics 2010
2
noon
4
5
6
For a period of 24 hours, and
a peak at the surface at 1:00
pm (the 13th hour),
midnight
7
 0  2p
24
The e -z part
T  z   Ta  A0 e
exponential decay,
half-lives, etc.
Soil Physics 2010
1.0
z
ye
0.8
d
0.6
 kz
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Summary
• Thermal properties (specifically DT)
appear only in the definition of
damping depth:
2 DT
d

• Phase shifts (delays) as sine wave
propagates downward
• Amplitude decreases as the wave
propagates downward
• Temperature constant at infinite
depth
Soil Physics 2010
Applications

T z, t   Ta  A0 sin t  0  z
The questions we ask this equation
are usually about either
• timing and phase shift, or
• amplitude
but not both.
When it’s a timing question, you
only need the sin() part
When it’s about amplitude, you
only need the e-z/d part
Soil Physics 2010

e
d
z
d
Example application
On the coldest day of the year, at what depth is the warmest
soil found?
z

T z, t   Ta  A0 sin t  0  z
Translation: what depth z is ½ cycle
(i.e., p) later than the surface?
½ cycle delay requires that z d  p ,
where d 
so z  p
Soil Physics 2010
2 DT

2p
and  
,
period
2DT 365days
2p

e
d
d