Transcript d = “damping depth” - Soil Physics, Iowa State University

Outline
• Announcements
• Soil Thermal Regime
• Evaporation
Soil Physics 2010
Announcements
• Review sessions this week:
• Today, 11-1 in G217
• Wednesday, 11-1, 1581
• No quiz today!
Soil Physics 2010
Homework 6, question 3a
You are monitoring soil temperature, wetness, and CO2 concentrations, and want to
calculate the CO2 efflux. A spreadsheet of your data (SP_HW6_gasdiff.xls) is
available on the course website. Plot the calculated (a) CO2 diffusivity as a function
of depth and time. (see pdf online for more details)
0.05
0 - 6 mm
6 - 12 mm
0.04
12 - 18 mm

0.5   
D , T   0.03
DCO T 
2.5
18 - 24 mm
24 - 30 mm
0.5
2
0.02
0.01
0.00
Soil Physics 2010
196
200
204
208
212
216
Homework 6, question 3b
Plot the calculated (b) CO2 efflux (diffusion across the soil surface) as a function of
time for the period given. State your assumptions.
Temperature
What is D(,T) for
How do you average
diffusivity (or
conductivity)?
Conductors in series use a
harmonic mean (in parallel,
they use the arithmetic mean)
Soil Physics 2010

CO2
10
?
0
3
0–6
9
6 – 12
15
12 – 18
21
18 – 24
27
24 – 30
36
30 – 36
42
36 – 42
3
15
27
45
60
100
75
Homework 6, question 3b
Plot the calculated (b) CO2 efflux (diffusion across the soil surface) as a function of
time for the period given. State your assumptions.
series: harmonic mean
=harmean(1,2,3…) in excel
y-axis units are ppm / cm2 s
100
3 mm
15 mm
27 mm
75
parallel: arithmetic mean
50
=average(1,2,3…) in excel
25
0
Soil Physics 2010
196
200
204
208
212
216
Back to the soil thermal regime

T z, t   Ta  A0 sin t  0  z

e
d
z
Ta = Average Temperature
A0 = amplitude of temperature at the surface
 = 2p / period (say, 24 hours): normalizes the
“clock time” t to the 2p sine wave period.
d = “damping depth”: depth z at which
thermal amplitude is A0/e: normalizes
“physical depth” z to exponential function depth.
Specifically,
Soil Physics 2010
d
2 DT

d
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
Clock time at
the surface,
normalized
to 2p

d
Phase shift
with depth
Phase constant:
adjust so peak is at
the right time of day
1
6:00 am
3:00 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 3:00
pm (the 13th hour),
midnight
9
 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 is 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, focus
on the sin() part
When it’s about amplitude,
concentrate on 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  
,
365 d
2DT 365d
 p DT 365d
2p

e
d
d
Example application
Around June 20, the soil surface temperature may
have an amplitude of 15 °C in one day. At what
depth is the amplitude only 2.5 °C in one day?

T z, t   Ta  A0 sin t  0  z

e
d
z
d
This is an amplitude problem, so we are only concerned
with the e-z/d part.
e
z
d
1
6
z
d
 6
 ln 1
 6
z   ln 1
Soil Physics 2010
2DT

Evaporation
Soil wets: Infiltration
Soil dries: Drainage
Transpiration
Evaporation
Soil Physics 2010
– John?