Proc. Indian Acad. Sci. (Earth Planet. Sci.), Vol. 104, No. 2, June 1995, pp. 257-271.
9 Printed in India.
Estimation of surface temperature from MONTBLEX data
K NARAHARI RAO
Centre for Atmospheric Sciences, Indian Institute of Science, Bangalore 560012, India
Abstract. It is observed that the daily mean temperature of the soil is linear with depth and
the variation of the temperature is sinusoidal with a period of a day. Based on these
observations the one-dimensional heat conduction equation for the soil can be solved which
gives the amplitude and phase variation of the temperature wave with depth. Given the
temperature data at three levels below the surface, the amplitude and phase variation and
hence the surface temperatttre variation over the day are estimated. The daily mean temperature of the surface is estimated from linear extrapolation of the daily means at the three levels
below the surface. Estimated values of soil thermal diffusivity show a subtantial change after
sudden and heavy rains.
Keywords. Soiland surface temperature; diurnal variation; phase and amplitude of temperature waves; thermal diffusivity.
1. Introduction
Because of the importance of the surface layer in determining the eddy fluxes of mass,
m o m e n t u m and energy, the M o n s o o n T r o u g h B o u n d a r y Layer Experiment ( M O N T BLEX) conducted during the m o n s o o n of 1990 gave particular importance to the
surface layer measurements. The locations chosen for such observations lie along the
trough, and are:
9
9
9
9
l i T K h a r a g p u r (KGP),
Banaras H i n d u University at Varanasi (BHU),
l i T Delhi (DEL), and
C A Z R I at J o d h p u r (JDP).
30 m masts were erected at each site with the required instrumentation at six levels (1, 2,
4, 8, 15 and 30 m) above g r o u n d level to measure vertical and horizontal velocities, wind
direction, temperature and humidity (Rudra K u m a r et al 1991).
The soil" temperature was measured at three levels (0.1, 0-2 and 0.3 m) below the
g r o u n d using resistance thermometers of 12.5 microns diameter platinum wire encapsulated in ceramic. These were sampled at 1 Hz, averaged over one minute or three
minutes and stored on audio cassette tape t h r o u g h a Campbell data logger. F o r
a proper analysis of these data, it is important to have precise information a b o u t surface
parameters such as roughness length, temperature, moisture etc. T o estimate the
surface temperature, which is the purpose of the present analysis, we use 30 minute
averages of soil temperature obtained from the logged data..
257
258
K Narahari Rao
2. Equations for the surface layer
The wind and temperature profiles in the surface layer are governed by the following
equations:
u(z) = ( u , / x ) ( l n ( z / z o ) - ~kM(z) ),
(1)
0(z) -- 0 o = (0"74 0,/x)(ln (z/z o) -- @~(z)),
(2)
where u(z) = mean wind speed at height z, m/s
u,
= characteristic (friction) velocity, m/s
z
= vertical height, metres
zo
= roughness length, metres
r
= v o n Karman constant = 0.41
O(z)
= temperature at height z, degrees
0,
= characteristic temperature, degrees
0o
= surface temperature, degrees
ffM, ~bu = correction factors for u and 0 in the log law.
The surface temperature, which is not an easy quantity to measure, was not directly
measured during MONTBLEX. However, since the soil temperatures have been
recorded at three levels below the ground, Oo can in principle be inferred from this data.
3. Evaluation of surface temperature
We first assume that temperature is governed by the one-dimensional heat conduction
equation in the soil,
O0/Ot = D O20/Oz 2,
(3)
where
D = K / p C , thermal diffusivity of the soil, m2/s
p = density of soil, kg/m 3
C = specific heat of soil, cal k g - 1 deg C - a
K = conductivity of the soil, c a l m - 1 s - 1 deg C - 1
We can in principle obtain the thermal diffusivity using equation (3). One approach
is the following: The time derivative of temperature at the middle level can be obtained
from
OO/Ot = (O(t + 1) - O(t - 1))/2At.
(4)
The second derivative at the middle level is obtained from the three temperature
measurements a.t zl, z2 and z 3 as:
O20/Oz 2 = (O(z3) - 20(zz) + O(z t ))/(Az) z,
(5)
Surface temperature from Montblex data
50,0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
JDP: Julian day 1 9 4
~40. O"
259
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/"
"x.
. ........
":"
-.=
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"""
~30.0
;:...
II
- - -i
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. ~
~0.0
~
. . . . . . .
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] " . . . . . .
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4
. . . .
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,
~ I . . . . . . .
f2
8
l
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, '
I '
1=1 1 ' ~
1'~'
20
f6
24
T~w~e ( ~ H r a
50.0
JDP; Julian day 195
~ 40.0
pr
.~ ....
-.
'r
~.
@0
=: :, :, :, ,:o, ,, *, i l
|ll,lL'~ ""
=='it*lit
ao~*m~
1o~
n
swx
1 ; : : ' : ' " i ] i i l " * ,. a. .' .~.'.*.8 " ' ~"' s. ,. .. . .".` "
i"
~[_30.0
20,0
O
4
Ttw~e ~,~ Itrs
8
t2
16
20
24
50,0
JDP: Julian day 196
I
~ 40.0.
%
.
::::
30,0.
20,0
,,,,,,,|,,
0
W l l l l " l | l l l l l l l ' | l I ] l l l l | l l l l l l
4
8
12
18
'~lllllll
24
Figure I. Variation of temperature over the day (* - 0"1 m, x 0.2 m, A - 0"3m and - - - - -surface).
and substituting from equations (4) and (5), D can be evaluated from equation (3). Once
D is known, applying the equations to the first level it is possible to evaluate the surface
temperature 0 o.
Computations however show that, many times during the day, the sign of dO/dt
evaluated as above does not agree with that of (~20/~z2, implying a negative coefficient
of diffusion! The derivatives depend heavily on the accuracy of the temperatures
measured, which in turn depends on the calibration constants and drifts. The difficulties of obtaining derivatives numerically are well known, and it was felt that this
method is not the appropriate one to use here in the present situation.
Measurements show that the temperature variation of the soil is approximately
a sine wave with a period of one day at all three levels below the surface, with a phase
K Narahari Rao
260
0.1
JDP days 194.19,5 & 196
-~ - 0 . 0
~ -0.1
~ -
~
~
~'5
'
-0.2
-0.3
-0.4
32
"--'
33
'
34
'
-'
3'6
'
J7
'
38
Daily Mean Terftp (Deg C)
Figure 2. Variation of daily average temperature with depth at JDP. (A - day 194; * - day
195 and [ ] - d a y 196).
0.1
Jodhpur Julian Day 194
./~"
~ -O.f
-0.3
-0.4
~5
-7 '
Pha~e(radia~)
'
~3
~1
0.1
Jodhpur Julian Day 195
--~
0.0
~e
~ -0. f -0.2-
r
-0.3-0.4
-?
-5
-
-f
t
P h ~ e ( ro.dic~uO
0.!
Jodhpur Julian Day 196
0.0~ -0. I-0.2-0.3 -
/f
-0.4
'
'
'26
'
'
23'
-t
f
Figure 3. Variation of the phase with depth ( [ ] - beginning, * - m i d , o
A - logarithm of the amplitude).
maximum and
261
Surface temperature f r o m M o n t b l e x data
60
84
0
r~
0
0/.,"
40
ooO.
..---
~
ooo
-.
0 o
O/
0~,
"~,
0
~.
O0
000~
9/
~
oo oo
,,e"
20
~ 000
o ~ 0 o 0 o o.,o/0
|
0
I
i
'
4
/e
'
/8
'
e'o
'
e4
Figure 4. Comparison of computed and measured surface temperatures- data from Pune,
4th-6th May 1992.
Table 1. Variation of alpha.
No of pts:
Mean alpha:
Std. Dev.:
Limit for 95% confidence:
BHU
DEL
JDP
KGP
60
3.54
1.31
0.34
43
6.82
1-90
0.59
64
6.63
1.77
0.44
54
4.98
1.68
0.46
difference between different levels (figure 1). With this condition, the solution of
equation (3) can be obtained as (Kirkham and Powers (1972)):
O(z, t) = O(z) + 0"e-~ sin(~ot - ~z),
(6)
where Ois the mean temperature at depth z over the day, 0 is the amplitude of the sine
wave at the surface,
= ~-(-~/2D) and has the units metre- 1,
~o = 2rc/P in radians/hr, and
P = 24 hrs, the period of the sine wave.
It is also observed that the mean temperature over the day varies linearly with depth
(figure 2). Hence, a value of Oat z = 0 can be obtained from extrapolation, and is shown
there in figure 2.
In equation (6), one finds that the amplitude decreases like e - ~ and the phase of the
sine wave shifts by e z with depth z. Therefore, from the measurements, ~ can be
determined in two ways - one from the amplitude and the other from the phase.
In the first method the maximum and minimum of the temperature wave, the
difference between which gives twice the amplitude, is determined. With temperature
measurements at three levels below the ground level, knowing that the amplitude varies
like e -'z, we can evaluate 7. Figure 3 shows the variation of the logarithm of the
amplitude with depth; the slope of the straight line in these plots gives ~. The value
shown at z = 0 is from extrapolation. It may be noted that the amplitude at the lowest
262
K Narahari Rao
II
I I
I I
I I
~1~
~ ~ 1
I I ~
I I
8
=.
1 I~ ' ~ ~1
~
I I I I I I I I
Surface temperature from Montblex data
263
.=r
I J
I'N
oJ
J I ~
~
I I
~
I I
~
§
r~I
I
~ ~ Z
r~ ~
II
I I~
I I I I I I I I I I I I I~
I~
I I I
mmmmmmm
I I I I I I ~
I ~ ~ - ~ - ~ - ~ ~ ~ 1 7 6 1 7 6
i i i i i i ~
i ~ ~
~
~
~
~
m ~ m m m m ~ m m m m ~ m m m m m m ~ m m ~ m m ~ m m m ~
264
K Narahari Rao
illl
ii
iiiiiiiirlll
Ii
/rll
rl
rllflrllrlll
/r
iJii
rr
iffrllllJiIi
ir
~eh
I
~
~
6
~
6
~
II
I
r l l l l l l l l l l l l l
II
I
IJPIrlllllllll
II
I
IJIIIrlllllllr
~
~
~
6
~
o
~
6
~
6
.
"
~
.
~
.
.
.
.
Surface temperature from Montblex data
265
Ill"
Ill~l
o
8.
.<
r
.<
o
~
I I I I I I I / I I J I I I i I I
~
I I I
r~
I I I I I I I I I I I I I
I
[-
~
I I I
r162
r162162
I I I I I I I I I I I I I
r162
r162162162162162
t~
266
K Narahari Rao
level does not fit into the trend. This may be due to difficulty in accurate evaluation of
a small amplitude or in the calibration constant for that sensor.
The phase differences between the three waves below the ground can be found in four
different ways. Since the sine wave has a period of a day, we can determine the phase
from the point where the wave crosses the mean. This happens at two points during
every cycle. However, it is found from experience that the crossings around midnight
are more difficult to determine than the middle one, which is simply due to the fact that
the wave crosses the mean with a distinct positive slope around midday. Figure 3 shows
the variation of phase determined from both the midnight and midday crossings with
depth. The fourth method is to determine the time when during the day the temperature
4a
(a)
Mean Surf Temp -
0 o
VARANAS.
.o*
tJ
0
4O
0
0 0
0
o
0
G
0 r
0 o
OooO
o
0
0
~ 0 ~ ~
~
o
35
~
~
oO~
d~
0
o
0
30
.
,
i
140
I
!
!
!
I
~
I
180
160
I
I
!
w
2,00
I
|
I
!
I
220
I
i
I
240
280
du~rt DcV
I0
(b)
VARANASI
O
5
0
o*~
~ == .
0
0
0
o
O~
0
o ~176176
:
.
o
o~
0% 0 % ~
.
0
0
i
~
=
I
160
140
dul~n
i
i
!
i
I80
~
I
=
200
!
i
I
i
i
i
2,2,0
I
|
w
v
2,60
2,40
Do,l~
160
(c)
VARANASI
120
8O
4O
!
140
I60
Ju, l ~
Figure
fBO
2.00
|
J
220
Da.tt
5. (a) Surface mean temperature; (b) ~c;and (e) rain at BHU.
m.
240
n-
__.J
260
Surface temperature from Montblex data
267
43
Mean Surf Temp
0
(a)
DELHI
-
O
0
0
L~ 3B
%
0
r :L~
o
0%~ %0
0
g**
0
~~
0
33
0
oo
,~8
,
,
I
I
i
!
i
|
140
160
,lumen DaV
.--%
I0
~
!
I
180
I
i
!
i
200
I
I
|
i
220
I
i
I
260
(b)
DELHJ o
O0
0000
0
I
240
d)
o~
0
0
0
0
0
0 o
0
o
0
0
0
5
D
~
0
2
0
0
!
i
=
I
~
140
160
Jutic~n D a y
:
1
1
!
1
180
i
I
i
i
i
200
I
i
i
r
!
i
z
2,40
220
160
i
260
DELHI
(c)
I20
80
40
T~
I
140
Figure 6.
160
MO
200
'
'
!
'
Z20
!
240
,
260
(a) Surface mean temperature; (b) ct; and (e) rain at DEL.
waves have a maximum, which is also a measure of the phase of the wave. In principle,
this is more difficult as the time at which the maximum occurs is not sharply defined,
since a sine wave has zero slope at this point. However, it can be seen from figure 3 that
this method also gives a reasonable estimate of ~.
It may be noted here that the value of the temperature measured within the soil
depends on the calibration constant and drift (if any) of the instrument; so does the
amplitude. On the other hand, the phase does not depend on either of the above, and
hence should actually give the most reliable information about ~.
It.is possible that ct cannot be estimated by all the four methods mentioned above,
mainly because of the discontinuity in data at some stretches during the day. ~ obtained
268
K Narahari Rao
from different methods is averaged for further analysis. If~ cannot be estimated by two
methods at least, such data has been discarded.
Knowing the mean temperature at three levels below the ground a straight line can
be fitted through those three points and extrapolated to get the mean temperature at
the surface (figure 2). The phase difference and the amplitude of the surface temperature
can be computed from ~ and the temperature data at the first level below the surface.
The temperature of the surface for the whole day can be computed. The dotted line in
figure 1 shows the result. Taking into account the longer day at Jodhpur during that
time, this temperature wave crosses the mean around the same time delay after sunrise
and sunset.
(a)
4a
.
40
oo
Meon Surf "[ernp - JODHPUR
oo
0
~
0
0
o
0
o
~
oo
30
~
o
0
%
000
0
~,%. 0 o==o
~o
~
0
!
i
160
140
I
!
I
!
180
I
'
i
i
200
I
+
i
,
220
I
+
i
,
240
260
Ju,5o,~ DcV
I0
00 0
0
!+
~ o~; o oo
**%
0
O ~
0
0
0
(b)
JODHPUR o
0 o
0
0
o
0
~o
o
0
0
0
o
0 0
o%O
00
0
0
0
0
0
0
0
i
i
i
140
I
I
1
,
160
I
i
I80
|
!
I
i
i
I
=
I
2,20
200
=
I
i
w
w
240
260
Jt4,L~.~1. Da~
160
(c)
JODHPUR
120
8O
40
0
'
140
Ju3~
Figure
'
'
I
f 60
n,
,
'
I
fSO
,
!
gO0
,
,
!la
220
DeW
7. (a) Surfacemean temperature;(b) ~; and (c) rain at JDP.
o
240
i
260
Surface temperature from Montblex data
35
269
Mean ~;UH Ternp -
(a)
KHARACPUR
0
=.
o,e%p
~
0
0
0
o
25
ooD
~90
,
,
,
I
,
,
w
140
160
J~zLi~zzL D " t i
'
'
I
180
'
'
'
~00
I
'
'
I
~gO
260
,?,40
10
(b)
KHARAGPUR
,h
|
0
~0
o
2:
9
0
~
cP o
aO
O o
o
..~
o
o
o
o
0
.
|
'
i
I
140
IBO
J~im~ Day
9
1
I
IBO
,
1
,
200
!
i
,
220
!
o
,
260
240
160
(c)
KHARAGPUR
f20
q
0
,
I40
'
'
I
160
'
~
"
!
180
!
In]
l
_~1_
|
200
n
!
l
w
I
'
'
220
'
I
240
t
,
,
260
Julizz~ Do.~l
Figure 8. (a) Surfacemean temperature;(b) ~; and (c) rain at KGP.
Figure 4 gives the observed surface temperature along with the surface temperature
computed as explained above from the data at three levels below the ground during 4th to
6th May 1992 at Pune (kindly supplied by Dr. Verneker, IITM). The agreement is very
good, giving us confidence in the present procedures for obtaining the surface temperature.
4. Conclusions and recommendations
Table 1 gives the mean, standard deviation and 95% confidence levels for the mean of
~t for all the four stations. The values are seen to be different for different stations. The
spread seems to be large (about 25 to 30% of the mean). The values of the thermal
K NarahariRao
270
(a)
45
,
o
o
Meon Surf Temp - JODHPUR
0
o
0
o
0
40
o
o
*
o
r,J
0
0
0
'|
I
o
0
I
!
,
0
30
, o
0
0
I
I
0
0
I
!
o
25
I
I
!
I
t '70
I
I
I
!
I
I'75
|
I
!
I
I
180
I
!
I
185
!
I
I
190
''!
200
195
Julian DaU
10
(b)
JODHPUR
0
0
0
0
0
0
0
0
0
I
~
0
9
0
0
O
0
I
I
O
O
I
I
0
0
,q
I
I
I
I
'70
I
a
I
~
o
I'75
!
i
I
I
I
I
180
i
I
I85
I
I
I
190
i
i
i
200
I95
Jul{a,n D(Z~l
160
(c)
JODHPUR
120
i
8O
i
i
~4
!
40
i
i
!
0
!
170
!
!
!
I
175
!
i
i
i
I
I80
!
!
I
I!
!
185
!
!
1
190
!
!
!
!
I
196
!
!
!
1
200
Juli=n nay
Figure 9,
(a) Surface mean temperature; (b) ~; and (c) rain at JDP around a heavy rainfall.
Surface temperature from Montblex data
271
diffusivity calculated from 9 are in the range (about 0.1 x 10 -6 to 1-3 x 10 -6 mZ/s)
reported by Kirkham and Powers (1972).
The mean surface temperature over the day, the amplitude of the temperature wave
at the surface and ct are computed over the period of the experiment and are given in
table 2 which can be used for getting the surface temperature at any time of the day. The
daily mean surface temperature, ct and the rainfall at all the four stations are presented
in figures 5 to 8.
It can be seen from the Jodhpur data (figure 7) that the ~ and the mean surface
temperature change appreciably following heavy rains. To illustrate this more clearly,
figure 9 shows the period of such an event at Jodhpur. At this station, the mean 0t over
the full duration of the experiment is 6.63 rad/m with a standard deviation of 1.77
rad/m. Following rain on day 183, the value of~ drops to about 1.7 rad/m on day 184
and the mean surface temperature over the day falls drastically from 39.5~ on day 183
to 28"1~ on day 184. It is observed during the experiment that the air temperature
during the early hours of day 184 was around 25~ From the data, the estimated
probability of ~ attaining this low value is less than 1%. A similar trend was observed
around day 220. It is therefore clear that the thermal diffusivity and the surface
temperature of the soil must have changed substantially after sudden, very heavy rains.
At other sites, the rainfall was almost uniformly spread over the period of experiment; so no sudden changes can be observed. However, as long as the temperature
waves are sinusoidal and the mean temperatures are linear with depth, the present
analysis should still hold good.
References
Kirkham D and Powers W L 1972AdvancedSoil Physics(WileyInterscience)
Rudra Kumar S, Srinivasan H P, Srikrishna R, Ameenulla S, Prabhu A (1991)Available tower data from
MONTBLEX 1990,R~p 91 M D 1 CAS, IISc, Bangalore 12.