Meteorologists are interested in modeling the vertical flow of heat and moisture through the soil in order to better simulate the vertical and temporal variations of the atmospheric boundary layer. The one dimensional planetary boundary layer model of is modified by the addition of transport equations to be solved by a finite difference technique to predict soil moisture

Bornstein, R. D.

Santhanam, K.

English

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ONE-DIMENSIONAL SIMULATION OF TEMPERATURE AND
MOISTURE IN ATMOSPHERIC AND SOIL BOUNDARY LAYERS
by
Robert D. Bornstein
San Jose State University
San Jose, California
and
Krishnamurthy Santhanam
Informatics Inc.
Pairs Alto, California
FINAL REPORT
NASA - Ames
Grant No. NSG - 2184
	
January 1981
	
'	 J
(NASA-CR- 1b3841) ONE-LIHENSICNAL SIMULATION
	
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UP TEMPERATU&E AND MOISTUhl IN ATMOSPHERIC
AND SOIL BOUNDARY LAYERS rinai Report (San
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7.5
ONE-DIMENSIONAL SIMULATION OF TEMPERATURE AND
MOISTURE IN ATMOSPHERIC AND SOIL BOUNDARY LAYERS
Krishnamurthy Santhanam
Informatics Inc.
Palo Alto, California
and
Robert D. Bornstein
San Jose Sta gs University
San Jose, California
L. 410
1. INTRODUCTION
Meteorologists are inter-
ested in modeling the vertical flow of
heat and moisture through the soil in
order to better simulate the vertical
and temporal variations of the atmos-
pheric boundary layer. In the present
study the one-dimensional PBL model of
Diertele (1979) is modified by the
addition of transport equations to be
solved by a finite difference technique
to predict soil moisture. The model of
Diertele is a refinement of the two-
dimensional model of Bornstein (1975)
and Bornstein and Robock (1976) in which
a one-dimensional time-dependent model
of the planetary boundary layer simu-
lates the vertical distribution of wind,
temperature, and water vapor.
2. FORMULATION
Time dependent solutions for
the vertical distribution of wind,
temperature, and water vapor in the
planetary boundary layer (PBL) can be
obtained from equations used by
Bornstein (1975) in his URBMET (for ur-
ban meteorology) model. This system of
equations describes a two-dimensional
(x-z plane) boundary layer model, with
a lower, analytical. "constant flux"
layer of 25 meters, and an upper, numer-
ical transition layer of 1900 meters.
A one-dimensional version of
the model valid over homogeneous terrain
was developed by Diertele (1979) to in-
clude the effects of: 1) heating due to
divergence of the long and short wave
radiative fluxes, and 2) an energy con-
servation equations for the air-ground
interface, including radiative, sensible,
and latent heat fluxes. Use of this
equation requires inclusion in the
model of a subsurface layer for predic-
tion of soil sensible heat flux. In the
current formulation, the differential
equations of soil heat and moisture
fluxes are solved using in explicit
fin i te difference technique to obtain
soil temperature and moisture respec-
tively. instead of using a parametri-
zation for solving surface latent heat
flux, it is Gbtained more accurately
from the calcula+.ions of surface mois-
ture and temperature.
a.	 Transition Layer
The equations governing at-
mospheric variables in a one-dimensional,
hydrostatic, Boussinesq transition layer
were given by Bornstein (1975). With
the additional assumptions discussed
above, the equations become
d= = f dz * dz7(KMC)	 (11
a f (ug-u) * 6z (KM^)
	
(2)
60	 d	 dQ	 1	 8ON	 (3)
ZE 6z (KHdz )
 - pmcp W z
6t =6z (Kjap	 (4)
U
	 (5)
	dz "	 (6)
where all symbols are defined in Appendix.
The verticity and stream function
approach, used in the original model, is
retained in the present one-dimensional
study for convenience. The eddy diffu-
sivities KM , KH , and KV (set equal to KH)
are specified using the interpolation
formula of O'Brien (1970). Details of
the formulation of radiative flux diver-
gence which enters into the thermal
energy equation (2-3) for PBL, can be
found in Santhanam (1980) as can the
94
details of the equations for the constant
flux surface boundary layer. Only a
cloudless and pollutant free atmosphere
is considered in the present study, which
therefore restricts the generality of the
model.
In order to obtain the follow-
ing similar equation for surface moisture,
a balance is assumed between the atmos-
pheric surface water vapor flux and soil
surface moisture flux:
.4^
b.	 Soil Laver
The flow of moisture through
soil can be written as
where the capillary action term (first
term on the right hand side of the equa-
tion) is generally larger than the con-
vection term (middle term) and the
gravity term (last term). The flow of
heat through the soil can be written as
6Ts 6 A' r	 6	 6
6t I-z (c Ras) + LEpwU(	 a	 (e)
where the first term on the right hand
side of the equation represents sensible
heat flux and the second term represents
latent heat flux. All values of the
moisture dependent diffusivities (DN , DNV,
and DT), as well as the hydraulic conduc-
tivity (KW) and the modified thermal
conductivity (A') were obtained from
Sasamori (1970). One difference between
current 'formulation and that of Sasamori
(1970) is the present use of moisture de-
pendent values of soil heat capacity and
9ensity.
C.	 Grid and Boundary Conditions
The one-dimensional version
of the interlaced grid of Fromm (1964)
computes vorticity and stream function
values at grid locations vertically dis-
placed by one -half of a grid interval
from locations of the velocity components.
Studies by Clark ( 2970) and Taylor and
Delage ( 1971) discuss the superiority of
variable grid spacing in achieving high
resolution near the surface, and the grid
network used in the present model pos-
sesses such resolution. A complete list
of boundary conditions used + n the pres-
ent simulation can be found in Santhanam
(1980).
In order to obtain the fol-
lowing equation for the time-varying
surface temperature, a balance is assumed
at the surface between the computed net
radiant flux and convection in the atmos-
phere, conduction into the soil, and
evaporation:
QR + QLi - QL1 - esOT94 + pmcpu*8*
IT
+ pmLEu *q * + a' jz - + LED D rava = 0. (9)
95
pmu *q. + pwDg6 A + pwDT6 + p V . 0.(10)
Equations (9) and (10) are partial dif-
ferential equations with two unknowns,
i.e., moisture and temperature. Solu-
tions of these two equations are obtained
by using a Newton -Ralphson double itera-
tive technique (Carnahan at al., 1969).
The specific humidity at
ground can be obtained from
qg
 - qs x hg ,	 (11)
where qs is obtained from an expression
given by Murray (1967) and hg is obtained
from equations from Philip ( 1957) and
Nappo (1975).
3. SOLUTION
At each time step, the
following procedure is performed:
determination of soil layer profiles;
construction of constant flux profiles;
determination of the new values of 0, q,
n, and U at the surface and at the lowest
finite determination of the new 0, q, ;,
7, u, v, •
 and U p,-+files using newest
available values c: all parameters. De-
tails can be found in Santhanam (1980).
4. RESULTS
This simulation attempts to
reproduce the temperature and moisture
fields in the atmospheric and soil boun-
dary layers. Initial conditions for
these simulations were taken as those
existing at the beginning of the 5th
general observational period at O'Neill.
Nebraska, 1230 LST on August 24, 1953
(Lettau and Davidson, 1957). Numerical
integration was carried out for a 48
hour period.
Computed soil moisture values
near the surface (approximately 0.25 cm
below surface) are compared with those
of Sasamori (1970) in Fig. 1. Soil
moisture displays a minimum value in the
afternoon as the surface dries because
evaporation is greater than the moisture
flux from below. A maximum value is
found after midnight. Wetness values
decrease from 5.4 % at 1200 LST on the
first day to 3.1• at 1200 LST on the
next day, indicating a continuous deple-
tion of soil moisture. The depletion
arises as the moisture flux from below
the surface is less than surface evapo-
ration and as nighttime surface recon-
densation is also less than daytime
evaporation. In general, the present
results are similar to those of Sasamori
ta
a
r.r
results are similar to those of Sasamori
(1970).
Computed surface relative
humidity values (Fig. 2) compare well
with those of Happo (1975), who obtained
his values by dividing observed mixing
ratio values at 10 cm into the atmos-
phere (which he assumed equal to that at
the surface) by the saturation mixing
ratios. However, his specific humidity
values which were obtained by multiply-
ing relative humidity by saturation
specific humidity (which was not explic-
itly presented in his original paper) do
not n.;ateh the present predictions (Fig.
3) in amplitude and phase. However, the
present calculations do agree somewhat
better with those of Sasamori (1970).
Estimated Bowen ratios (of
sensible flux to latent heat flux) from
various models are shown in Fig. 4.
Those of Suomi (computed from observa-
tions of net radiation and soil heat
flux) show an unrealistic jump at 1500
LST due to a too small latent heat flux.
In addition, his values beyond 18 hours
of simulated time, as well as those of
Lettau (Lettau and Davidson, 1957),
which are obtained in a similar manner
to those of Suomi, do not agree with the
present results and those of Sasamori
(1970). These later results as well as
the present ones, are from complete PBL
formulations, while those of Suomi and
Lettau are associated with simple para-
metrizations.
5.	 CONCLUSION
The two dimensional URBMET
urban planetary boundary layer model has
been simplified to one dimension and
modified to perform a detailed analysis
of the flow of soil moisture into the
atmosphere. Model calculations of tem-
perr.ture and moisture in the soil and
atmospheric boundary .layers were vali-
dated against existing observations at
O'Neill, Nebraska.
Results have indicated that
simple models with Parametrizat ions of
the surface latent heat flux do well in
predicting surface relative humidity but
do not do well for specific humidity,
because they do well in predicting sur-
face temperature. Calculations show
that models using empirical techniques
to evaluate the surface energy balance
do not predict Bowen ratios as well as
those incorporating a soil layer.
ACKNOWLEDGEMENT
This research was cerried
out at Nasa-Ames Research Center and
San Jose State University under NASA
grant NSG 2184. Thanks are extended to
John Millard of Nasa/Ames for his many
suggestions concerning this research.
The first author is greatly indebted to
Dr. Louis Capone of San Jose State Uni-
versity for his many helpful discussions.
REFERENCES
Bornstein, R. D., 1975: The two dimen-
sional URBMET urban boundary layer model.
J. Appl. Meteor., 14(8), 1459-1477.
Bornstein, R. D. and A. D. Robock, 1976:
Effects of variable and unequal time
steps in the simulation of the urban
boundary layer using the two dimensional
URBMET model. Mon. Weather Rev., 104,
260-7
Carnahan, E., Luther, H. A., and J. O.
Wilkes, 1969: Applied Numerical Methods.
John Wiley i Sons, Inc., New York, 	 pp.
Diertele, D., 1979: Simulation of urban
surface energy balance, including the
effects of anthropogenic heat production.
M.S. Thesis, Dept. of Meteorology, San
Jose State University.
Fromm, J., 1964: The time dependent
flow of an incompressible viscous fluid.
J. Comp. Phys., 3, 345-382.
Lettau, H. H., and B. Davidson, 1957:
Exploring the Atmosphere's First Mile.,
Vols. I and 2, Pergamon Press, 578 pp.
Murray, F. W., 1967: On the computation
of saturated vapor pressure. J. Appl.
Meteor., 6, 203-204.
Nappo, C. J., 1975: Parametrization of
surface moisture and evaporation rate in
a planetary boundary layer model. J.
Appl. Meteor., 14, 289-296. 	 —`
O'Brien, J., 1970: On the vertical
structure of the eddy exchange coeffi-
cient in the planetary boundary layer.
J. Atmos._Sci., 27, 1213-1215.
Philip, J. R., 1957: Evaporation, and
moisture and heat field in the soil.
J. Appl. Meteor., 14, 354-366.
Santhanam, K., 1980: One-dimensional
simulation of temperature and moisture
in atmospheric and soil boundary layers.
M.S. Thesis, Dept. of Meteorology, San
Jose State University.
Sasamori, T., 1970: A numerical study
of atmospheric and soil boundary layers.
J. Atmos Sci., 27, 1122-1137.
Taylor, P. A. and Y. Delage , 1971: A
note on finite difference schemes for
the surface and planetary boundary lay-
ers. Boundary Layer Meteor., 2, 108-121.
4
	 96
ice. i
aA
APPENDIX
List of Symbols
Variables and Constants - Roman Alphabet
c soil heat capacity
c 
specific heat of dry air
Dn
total soil moisture dif-
fusivity
Dnv vapor moisture diffusiv-
ity
DT total soil thermal dif-
fusivity
f Coriolis parameter
hg surface relative humidity
KH, !, Kq eddy exchange coef f i.: ienls
for heat, momentum, and
water vapor
% hydraulic condictivity
LE latent heat of vaporiza-
tion
q # qg. qs. 9r	 specific humidity, sur-
face q, saturated q,
and friction q
QE' OH vertical flvx of latentand sensible heats
QLf'QL+ upward and downward di-rected long-wave radia-
tion
ON' QR net all-wave and solarraRiation at the
surface
C time
TV
 To surface and soil tempera-ture
u, u horizontal component of
g the wind and geostropin
wind in x-direction
U, friction velocity
v horizontal component of
the wind in the y-
direction
X, y,	 z horizontal coordinate iii
direction of geostropic
wind, perpendicular to
geostropic wind and
vertical coordinate
Variables and Constants - Creek Alphabet
to wave length integrated
surface emissivity
C modified y-component of
sorticity
nS soil wetness
of
	
0t potential temperature and
friction i
U' modified soil, thermal
conductivity
on constant space averaged
atmospheric density
Ow density of water
o Stefan-Boltzman constant
T stream function
K
E
T MRS)
Figure 1.
Predicted soil moisture (by volume) at
0.25 cm below surface, where 12, the
'.nitial time, corresponds to 1200 LST.
97
r f
^---- ►^[piCT[0
•
E
^ r
r
>r
•
n	 1•	 >H	 a	 N
T MRS)
Figure 2.
Predicted onrface relative humitidy (1)
I•
». 1
Ie
o Ir.^s
a.
v
A 1!
li. l
11
t l IRS 1
Figure 3.
Predicted surface specific humidity.
sso
------ PREOICTEO
SASAMORI (19701
SUOMI (19571
-"'—• l[TTAU(19571
• 1^
10D
Z
i
	
•90 -F	 r----^ --r- --•
	
12	 1•	 71
T ( MRS 1
Figure 4.
Predicted Bower, ratios.
a
x
98
v.


One-dimensional simulation of temperature and moisture in atmospheric and soil boundary layers

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