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RD-RiSi 286 THE BEHAVIOR OF THE ATMOSPHERE IN THE DESERT PLANETARY I/iBOUNDARY LAYER U BEN-GURION UNIV OF THE NEGEY SEDEBOOER (ISRAEL) JACOB BLAUST.. L BERKOFSKY 14 OCT 84
UNCLASSIFIED RFOSR-TR-85-0227 AFOSR-84-0036 F/G 4/i NL
IllllllllllIIIIIIIIIIIIIIfllfll~fIIIIIIIIIIIIIIlfflf~fllllllllllhllI
1.0 2.
111U. 1U 11E1U1 .251 1111.6I
MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANIDAROS-1963 A
07
~~* ~ ~ ~ W V - . C i. ?" lWV . .
AFOSR.TR g- 0 P2 7
FINAL REPORT
co LOUIS BERKOF SKY
cv THE BEHAVIOR OF THE ATMOSPHERE IN THE DESERT PLANETARY BOUNDARY LAYER
In
AFOSR-84-0036
THE JACOB BLAUSTEIN INSTITUTE FOR DESERT RESEARCH
BEN-GURION UNIVERSITY OF THE NEGEV
SEDE BOQER CAMPUS 84990, ISRAEL
15 OCTOBER 1983 - 14 OCTOBER 1984
85- 02 27 0374
p-1
o.- . • , . ° o ... ,t• , . ... . .... . .• . , . . .... . . . .. . . .
SECUIITY CLASSIFICATION OF THIS PAGE (When DetsEntered) _ _ _._
REPORT DOCUMENTATION PAGE READ INSTRUCTIONS .REORDCUENAO PBEFORE COMPLETING FORM
I. REPORT NUMBER 12. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER -.
AFOSR-TR- 85-0227 JAL,__ _ _ __ _ _
4. TITLE (-nd Subtitle) S. TYPE OF REPORT & PERIOD COVERED F
THE BEHAVIOR OF THE ATMOSPHERE IN THE DESERT Final Scientific Report15 October 1983-14 October 984
PLANETARY BOUNDARY LAYER. 6. PERFORMING 01G. REPORT NUMBER
7. AUTHOR(s) 3. CONTRACT OR GRANT NUMBER(*)
Louis Berkofsky AFOSR-84-0036
9. PERFORMING ORGANIZATION NAME AND ADDRESS i0. PROGRAM ELEMENT. PROJECT, TASK
AREA & WORK UNIT NUMBERSThe Jacob Blaustein Institute for Desert ResearchBen-Gurion University of the Negev " V "--=Sede Boqer Campus 84990, Israel
I I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATEXL 14 October 1984U.S. Air Force Office of Scientific Research/IJC 14 O 1984Bolling AFB, D.C. 2 03 3 2 -(4c' I. NUMBERFPAGES.
14. MONITORING AGENCY NAME & AODRESS(II different from Controlling Office) IS. SECURITY CLASS. (of this report)
Unclassified
15. DECLASSIFICATION/DOWNGRADINGSCHEDULE
16. DISTRIBUTION STATEMENT (of this Report)
Approved for public release, distribution unlimited
17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, If different from Report)
18. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on reverse side if necessary and identify by block number)
Atmosphere, Desert, Boundary Layer, Numerical Model, Inversion, Transport,
Mesoscale, Dust, Limited Area Model
20. ABSTRACT (Continue on reveree side If necessary and Identify by block number)
One of the aims of this investigation was to develop a limited areaplanetary boundary layer desert model for computers of limited power. Wederived a general system of vertically integrated equations, including a
4 dust concentration equation and an inversion height equation. The boundarylayer was divided into a constant flux layer, a transition layer, and aninversion layer. The model equations predict the mean (vertically averaged)winds in the transition layer, the potential temperature at the top of the
DD IJAN73 1473 EDITION OF I NOV 6S IS OBSOLETE Q j r .N
SECURITY CLASSIFICATION OF THIS PAGE (When DOe Entere)
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unclassifiedSECURITY CLASSIFICATION OF THIS PAGE(rhe, Date Entered)
surface layer, the potential temperature at the ground, the height of theinversion layer, the dust concentration at the top of the surface layer,the moisture at the top of the surface layer, and the soil moisture at theground. The radiation flux is also calculated as a function of time.
The equations were programmed for solution on a 300 x 600 km grid, 2with 10 km grid spacing, on a grid staggered in both time and space.The lateral boundary conditions were of the radiation type.
Initially, the one-dimensional version was tested (no horizontaladvection). All fields showed reasonable evolution for a twenty-four hourpredicti66". Data (dust concentration, inversion height) are now beinggathered for verification.
The two-dimensional version was first run with a time step of twominutes and boundary conditions held fixed in time. Although the interiorfields, starting from artificial initial data, developed reasonably, thecalculations blew up after 4 hours, probably due to the restrictive boundaryconditions.When the radiation boundary conditions were used, the modelran for 6 hours, did not blow up, but developed unrealistically near theboundaries. Efforts are continuing to improve the model by introducingappropriate smoothing. Upon checkout, real data including topography, forIsrael will be introduced as initial conditions.
Accession For
NTIS GRA&IDTIC TABUnannounced [Justificatio
By -Distribution/_"Availability Codes
Avail and/orDist 'pecialI I
C
C . O T-
SECURITY CLASSIFICATIOIR OF ?U*, PAOE(Whon Di~ta En/eated).,-',
AFOSR -84 -0036 -0001
Grant Number: AFOSR 0036
THE BEHAVIOR OF THE ATMOSPHERE IN THE DESERT PLANETARY BOUNDARY LAYER
Louis BerkofskyThe Jacob Blaustein Institute for Desert ResearchBen-Gurion University of the NegevSede Boqer Campus 84990, Israel
14 October 1984
Final Scientific Report, 15 October 1983 -14 October 1984Approved for public release; distribution unlimited
Prepared for:
AIR FORCE OFFICE OF SCIENTIFIC RESEARCHBOLLING AFB, D.C.
and
EUROPEAN OFFICE OF AEROSPACE RESEARCH AND DEVELOPMN
LONqDON, ENG LAND
Tht3
MA'T?:4-' J-
r'r.w• w a-a - -. ,-'
>-v
TABLE OF CONTE"TS
Page
INTRODUCTION ................................................ 1
THE MODEL AND MODEL EQUATIONS ........................... ..... 2
THE FINITE-DIFFEREN1CE SCHJ'IE ........................ 29
LATERAL BOUNDARY CONDITIONS .................................. 31
EXPERIMENTS . ................................ ........... 36
One-Dimensional ................................... 36
Two-Dimensional ......................................... 41
CNCLUSIONS .............................................. 43
RECOMM ENDATIONS................ .. ...... 4
DUST CONCENTRATION DATA ..................... ............... 48
BIBLIOGRAPHY ........... ....... ..... 5
ACKNOWLEDGMENTS .............. o.......*............*............. 53
FIGURES .. .............. ... 54--,
4
4 eo o•o o o 4 o o oo o•o o ao o o o o o o o
.4 -
Q.T E W
PREFACE
The research described in this report was conducted by personnel of the
Jacob Blaustein Institute for Desert Research, Ben-Gurion University of the
Negev, Sede Boqer, Israel from 15 October 1983 to 14 October 1984 under Grant
No. AFOSR-0036 to the Ben-Gurion University of the Negev, Research and
Development Authority, P.O. Box 1025, Beersheva, Israel.
Participating personnel concerned with the tasks described in this report
include Prof. Louis Berkofsky, Principal Investigator, Dr. Avraham Zangvil,
Research Associate, Ms. Andrea Molod, Meteorologist-Programmer, Ms. Perla
Druian, Meteorologist-Programmer, and Mr. Tapani Koskela,
Meteorologist-Programmer.
Observational data used in this study were obtained from the Institute's
meteorological (4 m) tower, and from the radiation center, collected on a data
logger and analyzed in the laboratory. Dust data were obained from the
Institute's Size Selective Inlet High Volume Sampler.
The Director of the Institute during the conduct of this study was Prof.
Joseph Gale.
This report should be cited as follows:
Berkofsky, L., 1984: "The Behavior of the Atmosphere in the Desert PlanetaryBoundary Layer," Final Scientific Report, prepared by the DesertMeteorology Unit of the Jacob Blaustein Institute for Desert Research forthe U.S. Air Force Office of Scientific Research, AFSC, Bolling AFB, D.C..20332. -
I-. .2
r ,-. .
INTRODUCTION-
There exist a large number of planetary boundary layer models each designed
for specific purposes, (Deardorff, .1974; Mahrt and Lenschow, 1976; Stull, 1976;
Heidt, 1977; Yamada, 1979; Tennekes and Driedonks, 1981; Chen and Cotton,
1983). Many of these consider the top of the bourary layer a material
surface. Some consider the top of the boundary layer to be coincident with
the inversion, and consider entrainment across its interface. Some are multi -
level, some are bulk, single - level models. The various models are of one,
two and three dimensions. The greater the number of dimensions, the greater
the computational complexity
It is possible to reduce the computational complexity, and still not
eliminate the three - dimensionality completely by using a variation of the - -
"momentum integral" method (Schlichting, 1968). By means of this approach, the
vertical structure of several of the variables is specified and incorporated
into the vertically integrated equations. In this way, the model becomes two -
dimensional in the horizontal, and vertical variations are incorporated into
various coefficients. The method was introduced into meteorology by Charney
and Eliassen (1979), who derived the highly successful "equivalent barotropic
model". The method has also been used in studying nocturnal drainage flow
(Manins and Sawford, 1979).
In this study. one of the guiding principles was that computers of limited
power would be available to us. Therefore we decided at the outset to attempt
to model the atmospheric circulation in the desert planetary boundary layer by
means of a vertically - integrated, parameterized model.
-1--
THE MODEL AND MODEL EQUATIONS
We consider a model of the planetary boundary layer (depth approximately
1km). This layer is itself divided into a surface layer (20m), a transition
layer, which at certain times and places becomes very well mixed (and is
frequently called the "mixed layer") and an inversion layer. (See Figure 1).
We shall not assume that the transition layer is always well mixed.
Very often, the top of the planetary boundary layer is capped by an
inversion. This is particularly true in many desert and semi - desert regions.
In Israel for example, there are 222 days per year, on the average, of mid-day
inversions just about 100 km north of the beginning of the Negev desert. (Shaia
and Jaffe 1976) As we go farther south, closer to the center of the Hadley
cell, the frequency of occurrence of inversions is probably even greater. When
the capping inversion exists, and when convection occurs below the inversion,
the inversion changes height due to upward and corresponding downward fluxes
through it by turbulence. The inversion height changes affect the dust
concentration. These processes have to be modelled. Further, the processes
which we wish to model are on such a scale that fairly high resolution is
needed on the order of 10-20 km in the horizontal, over a region approximately
300x600 km. If then the vertical resolution above the surface layer is to be
very fine - say 100m - the computation time for solution of only the boundary
layer mesoscale equations may become prohibitive. Thus, in spite of the fact
that the optimum mesoscale model must be three-dimensional, we expect to gain
valuable insights with a vertically parameterized two-dimensional nonsteady
model.
4 -2- %, q:.i
We shall concern ourselves with a form of the primitive equations derived
by ensemble averaging over a horizontal area 4x4y which is large enough to
contain the sub-grid scale phenomena, but small enough to be a fraction of the
mesoscale system. We define
( O ) (1)
where o(is any scalar variable.
With the above definitions, the appropriate equations are, approximately
6 LA (2)
-(3)
+ (4)|
- .-
*• -, |-~~~, - --- (ur 5
i pcP "It at:-
~C ~ (7)
In the above set of equations, the unbarred .,ariables
u,v,w,u ,v Oe,F,P,q,c, are all mean values according to Equation (1).
We have used the Boussinesq assumnption, and the variables are defined as
follows:
u'v'w components of the wind vector
u ',v 1w' turbulent components of the wind vector
uglv'g ,geostrophic wind components
f = Coriolis parameter
n a e potential temperature
p = air density
uv ,,, F a net radiation flux
S q = specific humidity
c dust concentration
e' ,' ,c' = turbulent fluctuations of Oqc
xyZft = space variables and time
e ptetalt-4raur
=6i est
c = specific heat of air at constant pressure
lp
L~r) = fall velocity of particle of radius r
Equations (2) and (3) are the horizontal equations of motion, Equation (4) is
the continuity equation, Equation (5) is the thermodynamic energy equation,
Equation (6) is the moisture equation, Equation (7) is the dust concentration
equation.
The lower boundary condition is
w WT~ =YV Ta z =z T terrain height (8)
Here t~ u + v =horizontal wind vector, ~.and are unit vectors in the E
and N directions respectively and V is the gradient operator in 2 -dimensions,
In the present investigation, we assume that any condensed moisture stays
in the air. Thus we do not treat clouds or precipitation explicitly in this
model (but implicit predictions are possible).
In order to expedite deriving appropriate expressions for predicting
inversion height, we first derive inversion "interface" conditions. The
results are essentially the upper boundary conditions .Let h(x,y,t) be the
* inversion height. Let 9be a small layer of constant thickness above the -
inversion level. In Mahrt and Lenschow (1976) this is called the turbulent
inversion layer, which is sufficiently thin so that terms of O(S) may be
-5--."
neglected in the integrated equations. -
Definej
dz (9)
Let
=jump in q'across inversion
* By Leibniz's rule,
where x (x,y,t) (1
We apply the operator Equation (9) to Equations (2), (3), (5), (6), (7),
allow to approach zero, and obtain.
Lk AL 'A-I'L/ (12)
4 7- (L4 (3
-.-
Ago"
and -(
- (U + 4 = r (15)
XL11
In the above derivations, we have assumed that iii
Each of the above equations can be viewed as equations for the vertical
eddy fluxes, or as prediction equations for h if these eddy fluxes are known.
The quantity ( - Wh), which represents the motion of the air
relative to that of the inversion, is called the "entrainment velocity", we
wh is the larger scale vertical velocity at the interface. The terms -
involving lh/ax and '4h/*Oy are usually omitted in derivations of these interface
7 I
....-7-lh
conditions, since most inversion height models assume horizontal homogeneity.
Our approach will be to integrate the system of Equations (2)-(7)
inclusive, with respect to z from z = k = constant = height of surface layer,
to z = h = inversion height, then to introduce modelling assumptions for all
the variables, i., e., to specify their variations with height. If we do this,
it becomes possible to express all of the jump quantities in Equations
(12)-(16) inclusive in terms of their values at specific levels.
Parameter izat ions
Winds
Surface Layer
We assume that the surface layer is neutral, so that we may use a constant
flux profile
where k= von Karman constant = 0.4,V* = friction velocity,
zO roughness parameter. Here V, (u2 +v, 2 ) 1 2
*-8- -
where u*and v* are defined below.
- Transition Layer:
We assume
* (xyzt) = A(z) .(xyt).
v (x,,fzot) = B(z) v (x,ylt)
where
(0e) (19)
At the level z =k, the two expressions (17), (18) must match
* Therefore
-~ (i.±)(20)
- -9-
Potential Temperature
Surface Layer
Assuming that the turbulent flux of heat is also constant with height in the
surface layer, we find
e (x,y,z,t) = OGR + ("k- GR) (21)
We derive a modelling approximation for potential temperature in regions
where a nighttime inversion exists, and where surface heating during the day
leads to convection and turbulent mixing. In Figure 1, an inversion exists in ,
the early morning. The potential temperature at z = k is eki = e(x,y,k,o).
The potential temperature increases linearly with height according to
e(x,y,z,t) = +kI + Y(O)(z-k) (22)Ik
up to z=h, and linearly from there up to z = h+4, with a lapse rate60) within
the inversion layer. Here Y(O) is also the lapse rate above the inversion
layer.
Thus
I-10-
&(x,y,h+i&t) e = +(0) (z-k) + V'in (0)& (23)
It is assumned that heating destabilizes the lapse rates, so that, at some later
time
e(x,y,z,t) = e + Irlt) (z-k)
(24)
B(x,y,h,t) =eh =ek +rV(t) (h-k)
@(X,y,,h4- t) O h +j ek + fl t) (h-k) + rim, (t)g (25)
It is assumed that Equations (23) and (25) will yield the same result for
GS.In that case,
4 9 (ekl-&k) + ((0O)-rC(t)] (h-k) + 'r (0) (26)
From Equation (26), we see that
-6t
and, since
from first of Equations (25), we have
2L 0 (27)
65
This is similar to the result obtained by Tennekes and Driedonks (1981) for
a well-mixed, horizontally homogeneous layer, i.e., "the magnitude of the
temperature jump 0 increases in two ways: it increases as the height of the
mixed layer increases, and decreases if the boundary layer warms up". Here we
have not assumed well-mixedness (eh #am) or horizontal homogeneity, so
that Equation (27) is more general than was realized.
In the case of flow over water, the lapse rate '((t) is replaced by the
appropriate expression for a marine environment, say l(t).
Moisture
we follow the same formulation as for potential temperature,
q(x,y,z,t) = qk(x,yt) + (t) (z-k) (28)
q(x,y,z,O) = () +( 0)(z-k) (29)
4q =q lkI- qk) + [(O) - (t)] (h-k) + inv(0) (30)
-12 -
In the case of flow over water, the lapse rate of moisture 3(t) is replaced
by its appropriate value over water, say
Dust
We assume that the dust concentration is given by
C(x,y,z,t) : D(z) Ck(X,y,t) (31)
So that
C = [D(h+%) - D(h) ] Ck(x,yt) (32)
If we make the further assumption that there is no dust at the top of the
inversion layer, i.e., at z = h +, then D(h+ ) = 0, (See Carlson and
Prospero, 1972, concerning top of Saharan dust layer), and
C = - D (h) Ck (x,y,t) (33) -
I
- 13 - g
I.
-~- -.--- ~ -~-S - ~ -. ~- .- -. ~ - -. -. --- - . - . -A
Surface Parameterizations
For the momentum, we use
(u'w')k = Cd (A(k)l II
(34)
(vw')k = ( B(k)] r1i j
where Cd is the turbulent transfer coefficient for momentumn
d
For the convective heating,
w')K =HO A()u (4GR 9k (35)
where eGR is the potential temperature at the ground, C(o is the
For the moisture,
(w q t) CH A(k) u qq '
Cat r eR k (36)
* -14 -
- . - V C r r -- r r-..
where qs means saturation mixing ratio at temperature
'eGR, WGR = ground soil moisture, Wk potential saturation
value of W.
For the dust,
(w'c')k = Cd A(k) u (CGR Ck) (37)
where CGR is the dust concentration in a thin layer near the ground.
Ground Albedo
Ck a +b - ,b<0 (38)
gr W it
a,b constants
In the above, WGR, 8GR, CGR must be predicted.
Radiation -
The formulation of short and longwave energy exchange in the air and at the .
ground is based on Gates et al (1971). The system is applied at three levels: -)
1- 15 - ,.i-
I.
07 F - -- =
ground, top of surface layer, and top of transition layer: Thus the temperatures
at these levels (z = 0, z = k, z = h) are needed, as is the temperature at 2m.
All of these are obtainable within the framework of the model by converting
potential temperatures as given to .temperatures using T = e( - )
Longwave radiation balance ( positive if upwards) at each of the three
levels is defined by Rh, Rk and RGR as follows:
o + 6 I .~ ) V T.-+OA r
4, + -( U
where raur-a -,|
*o"S
-16- -;]
(40)----7 temperature at 2m.--
Stefan-Bolt~mann constant
U i is the effective water vapor content between ground level and-. !
i, and is obtained from-'
I _ . !i16
~ (41)
dt|
where p pressure at the ground pi = pressure at level i.* Pgr
o-.--
(42)
where gr is the mixing ratio at the ground and > = constant = 2.92
(Smith, 1966).
For levels k, h,oa, we obtain
*~LP (43)
Tf I N
Note: if we assume uk _0,(uk*)= 1 '(u-uk) (u.).
Also, if we assume Tk >> (T 4 Tk), we obtain ja simple form for Rk. We have not made use of these.
The effect of CO2 absorption is taken into account by the
coefficients 0.736 and the term 0.6 - 0.1. The former value
actually applies at 600 mb in the Mintz - Arakawa model, but we use this
for h and k, which are
- 17 -
very much lower than 600 mb. The effect of the error is not known.
Shortwave radiation (positive if downwards) is given by
S< +~ <,,(44)
whrSa part of the solar radiation subject to absorption = - -S0 x 0.651 cos Z
S0 = part of the solar radiation subject to scattering =
S0 x 0.-9 cos Z
= ground albedo,."gr,-- ,O = i ,005 027lg1 o ]
= sky albedo, (45)
- = zenith angle of the sun
S0 = solar constant60
The radiation balance = outgoing - incoming, i.e. negative
radiation balance at the ground means input of radiative heat to the
surface.
-18-
6 '
Closures
At this point, we define several closure formulas, and will indicate
later where and why they are used.
w' ' ).. A')4 (46)
We are now in a position to derive the final form of the equations.
The procedure is to apply the averaging Equation (19) to Equations (2)-(7)
inclusive and use the interface conditions Equations (12)-(16) inclusive to
eliminate the fluxes at inversion height, thus obtaining the transition
layer equations. If we do this in all of the equations, we are left
without a prediction equation for - . However if we use the first of
Equations (44), which states that the turbulent flux of heat at the
interface is a fraction of, and opposite in sign, to that at the top of the
surface layer, in the first law of thermodynamics, this latter equation is
then closed. We may then use this same (well-known, see Carson, 1973)
closure in the interface Equation (14), together with the various modelling
0 assumptions, to derive a prediction equation for
The ground temperature and soil moisture equations are adapted from
Deardorff (1978).
The final form of the equations, in which we have used Leibniz's rule
in the form
- 19 -
r~Q +,.-'
+ -. -- €
is
First Equation of Motion
* --
A (A.. +9) -~-(') 1 A [ r-A& ) rg
A4.
Pt (A L) Cy
Second Equation of Motion
+A t (A) +' )' g) (49)
4-4 4 t-
(*50)c.+ Oro, _k + +DMR. 4,"First Law of Thermod ynamics ;
+4 _
-. A"
L..-20-
Inversion Height Equation
u A 41 +4, A )- (AP4f+40 q)vo•-.-
A BG A(Si) &j - 04A 5
Dust Concentration Equation
4- 2L ) 4 (kr (52)D -+ + A b A SID V-
Moisture Equation
+ -A- 4- . +-3 WI& +rAjoi -Nix -) k____A+_
4 ' ( l~) (53)-A.
zCwo V* ll (- )1
Ground Temperature Equation
-21-
A
Here = net radiation balance at the ground
Soil Moisture Equation
- - _______(55)
C1 and c2 are constants Ps T,
-w density of liquid water
W2 bulk moisture (analogous to 02 )
P precipitation rate (prescribed
Fs soil density
cs = soil specific heat
dl (ks~l) 1/2
ks = soil thermal diffusivity
t = period of 1 day
92= deep soil potential temperature
In Equations (50) and (53) above, there appear the lapse rates of
- potential temperature and mixing ratio, r(t), and 3 (t), and their
derivatives and In this type of model, it is not
possible to derive an equation for prediction uf these quantities. Thus we
have specified V(t) and 3(t). See Fig. 2 for the form of f(t).
In addition to the prediction scheme, Equations (48)-(55) inclusive,
there exist three diagnostic equations, two for the geostrophic wind
components, and the equation of continuity for the vertical velocity.
I2
*- 22- 6
Geostrophic Wind Equations
(56)
Continuity Equation
The expression for wk is derived in the following way:
Substituting Eq. (20) into Eq. (17), we have
(58)
Then, integrating the continuity equation
-'-3(59)
* -23 -
.i.e.
~+
(60)
wT is given by Eq. 8, and in that equation is evaluated from
(61)6i
Given initial values of 'U',t, 'k,h,C,q, the
system can be solved. If we are concerned with a limited area, lateral
boundary conditions are important. These will be discussed later. The upper
,- and lower boundary conditions have already been prescribed.
If we assume no change of magnitude or direction of horizontal wind with
height within the transition layer only, then A(z) = B(z) = 1. This situation
freguently exists when the transition layer is well mixed. The equations then
become:
- First Equation of Motion
(62)
-24 -
z- .1 T 71 . - .1
Second Equation of Motion
'b c + +4 4 Wit*W U& )
-~ - ~ -~ (i-A.)(63)R -4
First Law of Thermodynamics
£'\. ,_A\
(64) '
L +A C,. V.-AR(4A>
Inversion Height Equation
Dust Concentration Equation
-"
-25-
Moisture Equation
r' {--&C- i ( ,4) -
The Ground Teperature Equation (54), SoilMoisture Equation (55), and
* Geostrophic Wind Equations (56) are unchanged.
The expressions for u and v in the surface layer become
( -~ (68)
C~ V
-| .
Euaosof( Coinuit a
(69)
From q. (6), wededuc
(70)
4 -26 -
This expression is useful in a model in which wh is prescribed, as in
the one - dimensional model to be described below. For a range of values of h
(300mihA2000m), k = 20m, Z= 0.7cn, and with wT = 0, we find
-.06 wh wk e-.Olwh
Thus, in this model wk is very small, and is essentially zero, unless
WT 0.
Finally, we write the equations for a model in which horizontal gradients
are absent.
First Equation of Motion
Second Equation of Motion
b -27-
G - ~27 -.- _
I-:
First Law of Thermodynamics
-~ ~ ( -0 _ __ _
_ _ _ _ _ _ _ _ _ _ (73)
* Inversion Height Equation
Dust Concentration Equation
DutCnetaionEuton!!i
Moisture Equation
(0) (76)3 to
4 28- 28 - -
The Ground Temperature Equation (54) and Soil Moisture Equation (55) are
unchanged. The geostrophic wind Equations (56) cannot be applied,
Aso~ U vg must be specified. Similarly, the Equations of Continuity
(69) cannot be applied, so that, if an estimate of the effect of wh ish ,is-
desired, wh must be specified. Then, wkcan be-deduced from Eq. (70).
THE FINITE - DIFFERENCE SCHEME6!
We have used a domain 300km x 60km, with 4x = 10km. Fig. 3 shows the
Eliassen grid (Mesinger and Arakawa, 1976), which is a space - time grid
staggered in both space and time, convenient for the leapfrog scheme associated
with centered space differencing.
Two and four point averages of variable quantities were taken as needed to
provide values at the grid point under consideration.
Thus averages are defined either asII
(4 .) (77)
or
= ± ~ O z, (78)-J
Centered differences were taken over one or two grid intervals as dictated
-29-
by locations of variables on the grid. For example, when predicting uja
term like (Zh/'x) is approximated using a one-grid interval centered
difference
A-k (79)
n~n
When predicting 8 j ,a term like (6h/x)n is approximated
using a two-grid interval centered difference
( (80)
When a two or four point average was necessary for squared terms, the
averaging operator was performed first, then the average was squared.
When a choice was necessary between one or two grid - interval differences
for nonlinear terms, such as (uO), the one grid interval difference was
used. All products, such as uO, were formed after appropriate averaging to
provide values at the same point.
The leap-frog time differencing scheme is
1 0 (81)
* - 30-
-d
- '~XI* P -- -
--I
LATERAL BOUNDARY CONDITIONS
We use the radiation condition proposed by Orlanski (1976). This makes use
of the Sommerfeld radiation condition.
L 0c - 4 -- o (82)
L4
where C is constant.
The finite - difference leap-frog representation of this equation is
uti) 3-M)(83)
Here JM refers to the boundary point.
The essence of Orlanski's (1976) technique is: instead of fixing a
constant value of the phase velocity, we numerically calculate a propagation
* velocity from the neighboring grid points, using the same Equation (83) for
each variable to calculate C. Thus we find
I c#_- a-,-- (Th-
(84
-31 -
* To determine the boundary grid point, C+ from Equation (84) is substituted for
C in Equation (83), in which the time index is increased by 1. We find
aHA
-~i~j(85)
if° 0
N th rmior uato e re quinter lCti w/atin C4) 0so wesmut
No information has cm from the bin exr scr ed ion w pr evs we muste
4•32-
Finally, we have:
Calculate C4 from Equation (84).
If C + > Ax/4t (limiting outflow), set C+ =4xAt
If 0 < C < AX/At (outflow) , use C1 as is (88)
If C+ < 0 (inflow), set C#= 0
Use C from Equation (88) to calculate tn+l (JM) from Equation (85).
In our initial tests, we shall use only the condition corresponding to
* C = 0, i.e., Equation (87), on all boundaries, since this is an easy
condition to apply for testing the model equations themselves.
In discussing the above, we have tacitly assumed that we are dealing
with the rigLt and bottom boundaries respectively. To derive the
appropriate equations for the left and top boundaries , we must
We have
-' (-L (m) + (89)
where, due to the method of indexing, JM+l now refers to the first grid
point in from the left and top boundaries. Applying Equation (89) to
neighboring grid points,
when applied to the top and bottom boundaries, Ax is replaced by4y
in the set of equations.
- 33 -
(TMmt +41, (90)
We now substitute Equation (90) into Equation (89), using C, for C, and
increase the time index by 1.
We obtain
In this formulation, we require -A x/At < C, < 0
For the limiting outflow condition, C+ = -A x/lAt,
(92)
as before.
if
- - (93)
6 -34-
Q I W W"°
This is again regarded as inflow information. .-..-
Finally, we have:
Calculate C4 from Equation (90).
If C4) < -A x/At, (limiting outflow), set C = -AX/At
If -Ax/At < C+,< 0, (outflow), use C4 as i . (94)
If Cp > 0 (inflow), set C4 = 0
Use C from Equation (94) to calculate n+l (JM) from Equation (91).
In the above Equations (84) and (90), it is conceivable that C+ may
become infinite or indeterminate.
Case 1 - C. infinite
Equation (85) may be written
2~~t t1) A-__ 4~-~ (95)
+ (96)I-.
Similarly, Equation (91) becomes
-35-
-. - - -. -. ' -- -. - ~ . - - -.-.
IC 7 7
.o.
Case 2 - C, indeterminate
This situation arises when, in addition to the fact that the
denominator of Equation (84) is zero, the numerator is also. then
n-n-2
n ( )(98)
If applied to the boundary point (JM), this is essentially the
statement Equation (87) corresponding to C4 = 0. Thus, in case the
numerator of Equation (84) or Equation (90) is zero, we use the condition
corresponding to C = 0, i.e., Equation (87) or Equation (93).
4
EXPERIMENTS
One - Dimensional
Our initial experiments are all concerned with a dry atmosphere. We
first ran the one - dimensional model, given by Equations (71)-(75)
inclusive plus Equation (54), with the following conditions:
A (h+g) = B (h+j) = 1 (no wind-shear across inversion)p. ii.
Iu = v g=0 (no geostrophic wind)
a.) wh =0; w, =0
b.) wh - (h-k) +Z=-(h-k) B (B = constant); wk=O
-36-
S.-
-5 -IZ
B -5xlO 5(0900-1500)
-5xlO-S " + 5xl0-('1500-2100)
5xl0-5(2100-0300)
5-5-.5xl 5 - - 5x10 50300-0900)A ~ -
C.) u V = 100 cm s- .
d.) ek = 303 OK, eGR= 303.2°k
-2e.) Cho= 4 x 1f.) h =300 m
0
g.) Ck 100gm-3-3
h.) CGR = 10014gm = constant;
i.) CGR = 10,000 /Agm = constant
The results of the experiment with wh = 0 are shown in Figures 4, 5,
and 6.
Figure 4 shows the evolution of the fields of u and 'v (the caps are
omitted from the figure). These fields evolve smoothly. The absolute value of
the wind reaches a maximum at about 1300 LST and a minimum at about 0100 LST.
Fig. 5 shows the evolution of the ground temperature 6.Q and the temperature at .
the top of the surface layer 0 k" G&p reaches its maximum value at about
1400, while 0k reaches its maximum at about 1600 local time. Both of these
are reasonable. The ground temperature reaches its minimum at about 0600,
while the air temperature reaches its minimum at about 0800, which is somewhat
late. The maximum and minimum of % f both exceed the corresponding values for,
Ok which is realistic. The range of both values is somewhat less than
should be expected on a summer day with light winds.
Figure 6 shows the evolution of the inversion height (H in the figure) and
dust concentration C for the two cases,
k0 -37-
-3 -3
CGR= 100 / m 3 , CGR - 10,000 •gm.
The former of these corresponds to an average air value for Saharan dust. It
turned out, after 6 months of observation at Sde Boqer, that the average value
is closer to 60g However, the use of 100 will not alter these
-3results in any substantial way. The value 10,0001Agm corresponds to that
within a strong dust storm. The first thing that can be seen from this figure
is that the inversion height rises steadily throughout the day. This may be
indicative of the fact that a convective boundary layer model should apply only
during daylight hours - or it may indicate that subsidence at inversion level
is required to bring the inversion down at night.
The curve C = 100 M gm-3 shows the trend of Ck (dust
concentration at 20 meters). This quantity decreases steadily with time. This
is not surprising, since, according to Equation (75), the tendency of Ck is -.1inversely proportional to the inversion height. It also indicates that the
turbulent transport of dust was too weak to overcome the thinning out of Ck
-3as the inversion rose. With CGR = 10,000/4gm - , there is continuousasR
intense turbulent transport, so that the dust concentration at z = k is kept -
high throughout the period.
It should be recalled that both ek and h are dependent upon the assumed
form of '(t) (Figure 2). Thus, it is possible to "tune" the model by U
experimenting with other curves of r(t).
Figures 7, 8, 9 shows the results for the same quantities when wh, the
vertical velocity at inversion height, is not zero. In actual fact, this
quantity should be obtained by solving the continuity equation, which is not
available in a model without horizontal transport. Thus we assume values of
the horizontal divergence B (given in the list of data), which vary with time a
throughout the day. Again, this quantity is "tunable", but we wish to
-38-_
highlight the effect of subsidence on inversion height.
Figure 7 shows the evolution of 't and ' when wh 1 0. The results are
indistinguishable from those of Figure 4. This is simply because the Wh
terms vanish from Equations (71) and (72) under the conditions we have assumed.
Figure 8 shows the evolution of GR and ek. There are some small
differences between these results and those with wh = 0, but the evolution
is more or less the same in both cases.
The major differences in the case wh # 0 can be seen in Figure 9, when
compared with Figure 6. Figure 9 shows the evolution of inversion height, h,
and of dust concentration, Ck. The inversion height now reaches a maximum
of about 1400 m at about 1800 LST, and then falls due to subsidence. It reaches
a minimum of about 600 m at about 0500, and then starts up again. Even though
the value itself at the minimum may not be realistic, this experiment very -.
strongly highlights the effect of the vertical velocity at inversion height
level on the inversion height itself.
The same type of result is true for Ck . For CGR= 100<gm- 3 -
there is little change in C,, except that it does not decrease as much as
it did in the case wh = 0. This is so because the subsidence during the
latter part of the period tends to increase the concentration at lower levels.
This effect is even more marked in the case of CGR = 10,000pgm- .
There is intense turbulent transport upward during the entire period. During
the second half of the day, the upward transport to level z=k is intensified by
subsidence to that level, so that Ck continues to increase.
It should be realized that the assumption of CGR = constant for 24
hours is rather weak, in the sense that wind erosion, which causes variations
in CGR, occurs in bursts. We shall discuss this point later.
The legends at the top of all the figures contain the heading "NCN -
-39-
PARAMETERIZED DUST EQ." We have distinguished these results from results with
"PARAMETERIZED DUST EQ", to be shown below, for the following reasons.
We have used the interface Equations (12)-(16) inclusive , to eliminate the
turbulent fluxes, at z=h except for Equation (14). It is also possible to
parameterize the other fluxes, for example (w'c')h in Equation (16). If
we use the flux - gradient relation
* "-C-w'c' =A (z) (99)
then -(w'c')h A (h) D'(h)ck (100)
h k
where A (z), the coefficient of eddy diffusion , is given by Equation (46).
In that case, Equation (16) becomes
*")(101)
and the dust concentration equation becomes
(102)$j V I,- V11,-A) CCe):.
instead of Equation (75). We now see that the model in this form contains two
-40-
* .-.
inversion height equations, Equation (74) due to convection of heat and
Equation (102) due to turbulent diffusion of dust. We carried out experiments
with the same initial data , but in which the inversion height was calculated
as an average of the results from Equation (74) and (102). The results are
shown in Figures 10-15 inclusive. The major differences between these results
and the non-parameterized results are in the inversion heght and dust
calculations, seen in Figures 12 and 15. These shbuld be compared with Figures
6 and 9 respectively. For the wh = 0 case, the results show similar trends,
but the non - parameterized inversion height went much higher, and the curve is
less smooth. This is undoubtedly due to the averaging involved in obtaining
the parameterized height. The variation of Ck is quite similar in both
.-3cases, although when CG = 10,000 Mgm , Ck does not drop down, inGRk
the parameterized case as it did in the non-parameterized case. When wh f
0 the curves look quite similar, except that the maxima and minima of inversion
height occur about an hour later in the parameterized case than in the
non-parameterized case.
The conclusion to be drawn from these comparisons is that the
non-parameterized system, which is theoretically more correct, should be used.
S4
Two -Dimensional Experiments
We used the system of Equations (62) - (66) inclusive, Equations (68),
(69), (70). We have not assumed A (h+4), B (h+S) = O,i.e., we have allowed
wind shear through the inversion.
41
* -41-° 6
a). Constant Boundary Conditions
We consider a region 300km x 600km, with a grid spacing 4 x = Ay = lOkm. In
this experiment, we used the lateral boundary condition corresponding to C+ =
0, i.e., ....
-~ 4)(87)
Thus we expect that this fixed boundary condition will eventually lead to
instability due to reflection at the boundaries.
As this experiment was essentially a check of the program, we used very
simple initial conditions.
u = 100 cm s-I everywhere
v =0
h =650 m
ek =303'k
6GR = 303. 2k
'(t) as in Figure 2.
Ck = 100) gm
CGR = l00pm3
A(h+) = B(h+i) = 1.01 - corresponding to 1ms- 1 (10m)- 1
u was taken as 1 msgV = 0g
We expect changes in the variables due to changes in the radiation fluxes.
-42-
We used a time step of4t = 1 minute. When the proqram ran smoothly
for 33 time steps, we tried 4t = 2 minutes. with identical results. WI
then ran for a lonqer period, but the calculations blew up at time step 322
= 4.07 hours, primarily due to boundary instability. The evolution of thf -.
interior fields seemed reasonable.-
b.) Radiation Boundary Conditions
Startinq from the same initial fields, we applied the radiation
* conditions Equations (821 - (98) inclusive. The experiments have been run
-o far for s hours, with one minute time steps. The evolution of the
interior fields seemed reasonable. Stronm qradients did not develop in all
"he interiors as they did in the constant boundary condition case.
c.) Comparison of Results of the Two Experiments
In the fiqures to be discussed below, we have displayed the printoutc
for the various fields, only for rows 1,2,3,4.5,58,59,60,61. The remaininq
fields are smooth transitions. Due to the printinq limitations, each row
nf the qrid is represented by almost two full rows on the printout, so that
columns 1 throuqh 17 qo from left to riqht, while columns 18 throuqh 31,
iust beneath them, qo from riqht to left. We have since modified the print
routine to qive a clear rectanqular array, and future results will b-
printed in the manner.
4 Piaure 16 shows the ,u-field at 180 minutes. with constant boundary
conditions. Fiq. 17 shows the ,O-field with radiation boundary conditions.
Althouqh the numbers on the bottom boundary of the latter are larqer than
in the constant B.C. case, the fields everywhere else are quite uniform and
-43-
smooth. Fiqures 18 and 19 show the '-fields in the constant and radiation
cases. Aqain, the interior fields in the radiation case are much smoother,
and less subject to qradients created by boundary reflection. The
inversion heiqhts are shown in Fiqures 20 (constant) and 21 (radiation).
There is no question of the superiority of the radiation condition for this
field, even thouqh the left and upper boundaries are not sufficiently
smooth. Fiqures 22 and 23 show the fields-in the constant andakr
radiation cases. Except for the bottom row. the radiation condition aive5
smoother field. Fiqures 24 and 25 show s)GR in the constant and
radiation cases. Aqain. the radiation case is clearly superior. Figures
S6 and 27 show the Ck (dust) fields in the constant and radiation
cases. In this field, the interior for the constant case alredy shows
larae, unrealistic values, while these values for the radiation case are
very smooth. Finally, Fiqures 29 and 29 show the wh values andw.
radiation cases. These values should all be zero, as both u and v should
be developinq uniformly. While all are very small. the fields in the
constant case are already beqinninq to develop, while they are almost all
z ero in the radiation case.
These results clearly illustrate the efficacy and suverioritv of the
radiation boundary condition over the constant boundary condition. Yet the
boundaries still require additional treatment. We plan to experiment with
suitable filters (Shapiro, 1975) to control spurious hiqh frequency6
os cillations on the boundaries themselves. There are also suqqestions
(Miller and Thorpe, 1981) for improved versions of the radiation boundary
condition.
As noted earlier, the radiation boundary condition experiments have
been continued for 6 hours, with one minute time steps. All of the field.q
developed reasonably, with little encroachment from the boundaries, except
44 -
the inversion heiqht field. While the calculations still did not blow up
at 6 hours, spurious oscillations were developinq in this field. It is
clear that some kind of smoothinq is required. This will be attempted in
future experiments.
Cc(NCLUSICNS
The one - and two - dimensional versions of the desert planetary
boundary layer model appear to be checked out, in the sense that they qive
reasonable metereoloqical results.
The one - dimensional version runs well for 24-hours, and can certainly
be run for lonqer periods. It is now rossible to use this model for a
number of sensitivity experiments, such as prescription of dust
concentration at the qround as a function of time, variable vertical
velocity at inversion height, variable surface albedo.
The two - dimensional version runs well for up to 6 hours before
fluctuations in the inversion heiqht field set in. Thus, more numerical
investiqation is needed in order to carry the experiments further. A3
already stated above, the investiqations will take the form of smoothino
and filterinq operators, and possible improved forms of the radiation
houndary conditions.
We have already prepared a set of base data of all fields for Israel,
and will carry out experiments with these data, over Irreqular terrain,
following adoption of more suitable numerical techniques.
-45- 6
RECOMMENDATIONS
There are a number of steps which can be taken which may lead to 9improvement of the present model. Although the model appears to be checked
out, in the sense that it does not blow up by 6 hours, and gives reasonable
evolution of the fields during that time it is desirable to focus on a
model which may be useful operationally.
There are still several steps which may be taken to improve the
radiation boundary conditions. Among them are the application of a
suitable filter (Shapiro,1975), which was successfully applied to this type
of problem by Eliassen and Thorsteinsson (1984). Another is the
possibility of applying an improved version of the radiation boundary
condition according to the method suggested by Miller and Thorpe (1981).
There are also several ways of improving the physics in the dry
model. We could try a possibly more realistic version of the stability
Y.t). We could also try to improve the assumption that the dust4
concentration near the ground, C GR, is constant. To do so requires a
method for predicting erosion of soil by wind . A first step in this
direction has been taken by Berkofsky (1984) who developed an equation
describing the processes of detachment, transport and deposition which make
up wind erosion.
When all of the above has been incorporated we can then include
moisture and a thermally active surface layer. Real data (already compiled
for Israel for this model) can then be used as input, together with
topography, for operational testing.
-46- • -S
Finally, it is possible to combine this model with that of the free
atmosphere for a simplified version of a tropical operational model. Such
a model has been suggested by Berkofsky (1983).
I 7
I
---
4 -47-
j.i
W ~ ~ M W-
DUST CONCENTRATION DATA
In Table 1 we present dust concentration data for Sede Boqer, compiled
simce June 1984. For a variety of technical reasons, the data are not
continuous. These dust concentration data were obtained with an ultra -
high volune sampler (Sierra InstruLnents) with cascade impactor, with a
constant flow meter operating at 40 cubic feet per minute. These
categories are:
C 1 : 7.2 - 00 /4.
C2 : 3.0 - 7.2 ,,.<
C3 : 1.5 - 3.0 A
C4 : 0.95- 1.5,M
C5 : 0.49 -0.95 A
C6 : < 0.49 p_
The averages inMgm,-3 are:
Total C1 C2 C3 C4 C5 C6
59.81 11.24 15.23 12.13 10.26 5.98 18.71
The minimun value was 10.47Agm on 11/11/84. The maximun value
was 133. 20Agm - on 1/8/84. The wind direction is given as a sort of
twenty - four hour average. It is seen that westerly winds dominate.
These data will have to be analyzed with respect to the prevailing
synoptic situation.
A48 -:.,
w
Ci) NZ Z U) ZU) ZU)N
b~U N. xN N N N N N
to . Z Z .7 W Z .X Z~ .Z Z Z .W .Z U .Z . . . . . . . . . . . . . . . . . . . . . .w -r or -N . NNN Z. NN NN -Z NN NZ N -rNZ ZN V-e - NoN-Z'Zr-N-%0W o' rC. ~,wMM4ro rw rM r
M in NN NN NZ ZZ NN ZNN N NZZ.Z,1 ,0 -- O4100 r -0NK ' )- N - 4 b N r ,NZ 0 0 NVIC V)- ZK KM0 CZZZZ0Z.ZZZZC -N '4 .,ZZZ - u 0%Z M D LO-Z r )ZsZZZ DZZZ0ZZZKZZZt 4 - - MZ D0-ZrZZ KC- O MZr -Z
10 . . . . . . . . . . . . . . . . . . . . . . . . .L) Au CV D0 OC4 0 -U)q 0-00 ,0NO W 0MM0-Wr O ' D0h OU )- r U
N ,a -MM0M0NI , rMW0 0 0-
C N K -0 0MM0MM0M% oN 0 D ,o ,W0N 0K0. r0. .
L)f f '0ON '0K 0L ' )- m >C400 D0 rU)*NNC O l v rC)V V(4eV
C.). Cl 0C)W0 DO - rr,0V -b)O 0mNWW -W K00- - CWW ,iR & N 0- J M It) - n ' 4 C eJNC,- -a K-0Nr C40 D N-r- r J-CJCDJ' '4D4 M -4, 0N . - O
.L) C1 ' o(1a r )i Ib)0P D9 I)K I - OM4 rW I rNKN
Cl . 0 -c . ,I )tn N4 r1.- r - l nNh ,NN -c)N0 , C)C-L D 0W.............................................................................................................................................
C-4 N 144
C.)j 4r0 0 K P.C
8 K~ P.' . -CJ-- -- 0 0w ww KKNK.- M 04N4 M-444 - 0 -4CI' K 0 0w 00 -4-4
4r fI*owI rw rwq V rW .W
C')~~~~~~~~~~~~ M fl C''C.C ObdbC.SOCW' MN-hhN '0 1 a, a, a 0 0 '-4NN'0If
.4..
Bibliography
1. Deardorff, J., 1974: Three-Dimensional Numerical Study of the
Height and Mean Structure of a Heated Planetary Boundary
Layer, Boundary Layer Meteor., 7, 81-106.
2. Mahrt, L. and D.H. Lenschow, 1976: Growth Dypamics of the
Convectively Mixed Layer, J. Atmos. Sci. , 33, 41-51.
3. Stull, R.B.,1976: The Energetics of Entrainment Across a
Density Interface, J. Atmos. Sci. , 33, 1260-1267.
4. Heidt, F.D., 1977: The Growth of the Mixed Layer in a
Stratified Fluid Due to Penetrative Convection, Boundary
Layer Meteorology , 12, 439-461.
5. Yamada, T., 1979: Prediction of the Nocturnal Surface
Inversion Height, J. Appl. Meteor. 18, 526-531.
6. Tennekes, H. and A.G.M. Dreidonks, 1981: Basic Entrairnment
Equations for the Atmospheric Boundary Layer, Boundary
Layer Meteorology , 20, 515-532.
7. Chen, C. and W.R. Cotton, 1983: Numerical Experiments With a
One-Dimensional Higher Order Turbulence Model:
Simulation of a Wangara Day 33 Case, Boundary Layer
Meteorology , 25, 375-404.
8. Schlichting, H., 1968. Boundary layer Theory , 747 pp.,
McGraw-Hill, New York.
9. Charney, J.G. and A. Eliassen, 1949: A Numerical Method for
Predicting the Perturbations of the Middle Latitude
Westerlies, TELLUS , 1, 38-54.
-50 -
"-'9
• mm imm mmmmmnmmmimmmmmmmmmi i mm mmmnmiimtmmim mm '-.n
10. Manins, P.C. and B.L. Sawford, 1979: A Model of Katabatic Winds. J.
Atmos. Sci. , 36, 619-630.
11. Shaia, J.S. and S. Jaffe, 1976: Midday Inversion Over Bet-Dagan,
Israel Meteorological Service, Bet Dagan, Israel, Series S
(Meteorological Notes) No. 33.
12. Carlson, T.N. and J.M. Prospero, 1972: The Large-Scale Movement of
Saharan Air Outbreaks Over the Northern Equatorial Atlantic, JAM,
Vol. lli, No. 2, 283-297.
13. Berkofsky, L., 1982: A Heuristic Investigation to Evaluate the
Feasibility of Developing a Desert Dust Prediction Model. Mon.
Wea. Rev. , 110, 2055-2062.
14. Gates, W.L. E.S. Batten, A.B. Kahle and A.B. Nelson, 1971: A
Documentation of the Mintz - Arakawa Two-Level Atmospheric General
Circulation Model: A Report Prepared for Advanced Research
Projects Agency, Rand Corporation, Santa Monica, California,
U.S.A., 408 pp.(ARPA order No. 189-1), R-877-ARPA.
15. Smith, W.L. 1966: Note on the Relationship Between Total Precipitable
Water and Surface Dew Point, JAM , V.5, No. 5, 726-727.
16. Carson, D.J., 1973: The Development of a Dry Inversion - Capped
Convectively Unstable Boundary layer, Ouart. J.R. Met. Soc. , 99,
450-467.
17. Deardorff, J.W. 1978: Efficient Prediction of Ground Surface
Temperature and Moisture With Inclusion of a Layer of Vegetation,
J. Geophys. Res. , 1889-1902.
18. Mesinger, F. and A. Arakawa, 1976: Numerical Methods Used in
Atmospheric Models, GARP PUBLICATIONS SERIES No. 17, V.1, pp.
53.
-51-
19. Orlanski, I., 1976: A Simple Boundary Condition for Unbounded
Hyperbolic Flows, J. Comp. Phys. , 21, 251-269.
20. Shapiro, R. 1975: Linear Filtering, Mathematics of Computation , 29,
1094-1097.
21. Eliassen, A. and S. Thorsteinsson, 1984: Numerical Studies of
Stratified Air Flow Over A Mountain Ridge on the Rotating Earth.
TELLUS , 36, 173-187. -
22. Miller, M.J. and A.J. Thorpe, 1981: Radiation Conditions for the
Lateral Boundaries of Limited Area Numerical Models, Quart J.R.
Meteorology Soc. , 107, 615-628.
23. Berkofsky, L., 1984: Wind Erosion, paper presented at Ben-Gurion
University - U.C.L.A. Conference on Desert Research held at Sede
Boqer, Israel, 16-21 Sept. 1984, 19 pp.
24. Berkofsky, L. 1983: Tropical Limited Area Modelling: A Background
Study, World Meteorological Organization , Geneva, 75 pp.
Io
41
526
-52- ?:-*
ACKNOWLEDGMENTS
The program was prepared by Andrea Molod, assisted by Perla Druian.
The final checkout was accomplished by Tapani Koskela, a visitor from
University of Helsinki. The suggestions and discussions with all of the
above contributed substantially to the development of the model in its
present form The typing was done by Dina Feingold. The assistance of
all of these people is gratefully acknowledged.
53
I I
.'."
U
-53- o
7 Z:.*- & '; C -1 -mc W .7- - -1 -J
I-'I
LIL
-F 0
%ago
I-. A
0*5-
E 4
* 03 -
0-
8:00 12:00 20:00 4:00TIME (hour of day)
Fig. 2. - The Lapse-Rate ~()
-55-
0 -0..: .
Fi .3 p c -i eG i tg ee oh nS aead T m ,C n ei n o h
Lepro chm AscatdwihCetre pceDffrncn (fe
Elasn 1956).v
56
x No V INI
IDI
N. -4
N.CI
cr..
o 0c
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op 4.1
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03AI-1L:)-3 N
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0 lO 10 90-0 10~ 10 1 T: IG 331 100 10 100 13U0 1~ 100 10 100
17 97 97 97 ;7 07 q7 97 97 97 97 97 97 97
I r. A r' '4 i-'J 6 j ~ 6 9^~ ~'~ 6 6 9^
1L tJ il 1~ 1 103 1 00 1301001 .
Fig. 16. - G field at 180 minutes, constant B.C.
46 6 96 96 9t 96 96 9o 96 96 9 6 9 6 9602 S'& 96 96 9.i 911, 96 96 9A 9t 96 96 96 96 96 96 96
9 96 96 76 0 96 9 i 96 96 96 96 96 96 96
Y.6 96 96 7", 96 9696 9-59696969694'9'699^o; ~ 91 96 96 9 t 96 9S 96 96 96 96 96 96 9
58 99 96 7 ?7 6 , 9 6 96 96 969 ?6 69696 96 996
4
-69-
V VELOCITY -FIELD "N *.1 CI/SXC -3T E P 1 3y6 Cjl 7- C3 0~ C.5 'S 57 0S ' 12 11 12 13 14 15 16 17
03 0 -73 -7! -7 -73 -75 -757 -73 -7 1 -75 -75 -75 -75 -75 -75 -75*: -'s 7 ' 7 -7z; -75 -7T -75 -75 -75 -"5 -75 -75 -7S
7 7 I. v i 5 i f 1
* l7~7 r 7 T~ Z -- 7 ? 7- 7 - 7 ~
-7 7 -7-7 -7-7 7 7 - 7 7-77 -77 -77-- 47 -77 - 7- - 7-7 -77 -7 -77 -7
Fig. 18. v~ f ield at 180 minutes, constant B.C.
I: 'a1 Z) L 7 03 oo 10 o 1 12 13 14 15 16 17
7-71 -75'76 -71-7 ? -7Q -73 -78~3' - ~ -7 ; 7 -72 -7t 7 7~~ 7 7 -78 -78 7 -7-3 7? 7- 7- -7- - 7 4-7,5 - 7 S -70i -7 -7~ -7! -78-7 -7 -8
- - -73 -7o - 7. -7P - 73'. -7R -78 -73 -78 -78 -78 -78
5, -70-r
1: 01 02 03 04 C5 0 1 f7 '5 0,; 10 11 12 13 14 15 16 1731 3 29 Z 2 7 2o 25 24 23 22 21 20 19 1 8
Ij043 At. bi 1 1 1M~7474 7 14 71T1 1 1 1 1 1
-- 5 3 1571 714' 71~ '714 714 714 714 714 714 714 714
6 7 , 71 71 7 71 7 1. 714' 717 71:' 714 727 7277i Qi .74
7 7 -- 1 1 1 91 P1 6lp'1 '0 691 6914~~~ A 01 6. . -- ~ 9 1~ 1 691 691
Fig. 20. -Inversion height at 180 minutes, constant B.C.
6. b. ~ Fi r, :T.i 1 z ZI: 0 01 04 Ox0 07 O0 00 10 11 12 13 14 15 16. 17
t1 2? 2 27 Z6. 25 24 2! 22 21 20 19 71a
i t
-u o;3 3 ~ ~ U' 1I f ~ IH Zj 7 1I Z:o 7i' 1l S ilb 1 T i -. ( 0 Q f1719 719 713 713 71a 712 717 718 718 717 718 717 718 719
04 7 04 717 718 71 7 19 719 719 711 710 710 718 719 719 718 719 719
713 719 711 719 7 13' 7193 713e 713 719 718 718 719 718 1
1 1 1
4p 6 pq0 :
:1 71
.. .I --
• .a
01 30 Z 3J3 3303 303 303 33~ 303 3! 3~ 0 C3 303 303 3 03 303 303 303 - 4!03- 303 303 3 C 30 0 T 3 7-730 3 0 - 3Q '03 303 303 303 303
4 4 1 .1 *J 1 1 44 _-
'II
35 7 '33 V 3 0 2 30816 &.5 6. - .. .-. - 3.. - 0.7 e ; Q! "
.1~r - , -,
J 1
-- 4 3 C±~ 3-Z4
Fig. 22. -eat 180 minutes, constant B.C.
-~ :3 Fm I-
01 309 306 ,,3C7 3 30 3 70 353 30"'_ 3G7 3j7 3'3 307 307 307 307 307 307307 307 303 3J- 327 30 ,7 307 307 30? 303 307 307 303
_-,- -.-_i i . - - - - - - - - - - - - -----i- i " i 31 i 3ii -! 11 ! "
-t - i . % VI j .. 7.; 7 - - ..6; 'j 1 14 1. d J 4 j d .... .. J j Q
2> 307 327 3'7 3.2 7 3 0 9 308 308 303 308 303 30SS307 33 7 7S 30' 33 0 C 3 00; 2 30.3 305 30 3 30. 30 3-
Y.. "7 37 7 Q 7 I 3 3 3 30, 3 0 S 30 30 3 03 303
O=.Z'7T q.C71q7 ; ir'v u"i .'. -' -- "',J. " I I " -- : i"m t' 'l I
.5 37 333 Q , - 3 . 3- . . 3 ' 30. 3T 308 3 C 3 03 08".07)330 7 3 +d.-07-3507 307 30. 3C7 303 30 303 308 3C8 0
.10 ., .13.. 07 307 307 303 303 303 308 30 307 307 307 307 30860 307 307 30" "07 3C7 337 307 307 307 307 307 3 7 307 307 307 307 307
Fig. 23. - Oat 180 minutes, radiation B.C.
7
- 72 -
I-.
01 303 303 303 03 1 - . 3 3 - 303 303 31 3 303 303 303 3033 303 7Q7 "03 303 3"3 303 303 303 303 303 303 303 303"
02 303 303 309 3 309 3019 301 "30,9 3090O 309 30 9 309 309 309 309
- 1 1ii 1 1 1 11 '11 311 311 311 311
05 303 3 311 11 -11 11 11 11 11 7.11 311 Z11 311 311 311 311 3115, 3 3 1 j1 11 11 L11 11 A .-11 11 3,11 311 311 311 311
', . -- 1- 3 -.1 ( 4 . - - _.,
, --____.:zzI- 1 132-" ..--. .- 2-.- ,,. - I "" - " " ' -_", 1-- 1 1 313 3--313--1-3 313 . -
I 303 303 303 .. 3t3 , 3 .3 303 3303 3- 33- ' 3 f) 303 3 3 '33 303 303
Fig. 24. - t at 180 minutes, constant B.C.
01 311 311 311 311 311 311 311 311 311 .311 311 311 311 311 311 311 311L1 311 31 111 311 311 311 ,11 311 311 311 311 311 3110 311 311 311 "11 311 11 311 311 l11 311 311 311 311 311 311
01 311 311 310 311 .511 317 5!1 311 311 311 311 311 311 316 311 '311 011 "
011 311 311 '11 11 31i5 ,i1 1 311 6)11 311 311V 311 "311 311"
Z 11Q30 311 311 !31~.1 311 11 '4 311 311 311 311 311 311
5d 311 311 311 311 11 -11l 311 311 311 211 111 7'11 311 311 311 311 311--
11 3 C# 501 l 1 30 131Z 1 ' 713,3 6 1 j 3114
Fig. 25. - 9 ,aat 180 minutes, radiation B.C. 1
-73-
DUST CONC, -PIELD IN U1 G f-'3 STiP li;C
02 1 919 23 19 19 1 19 10 19 19 19 19 19 19 1903 10 19 1~ 1 1 10 19 19 19 19 19 19 19
101 1 1 1 1 1 1~ 10 110 1 1919 191
10 11 10 1~ 1~ 11
11
Fig. 26. C Ck at 180 minutes, constant B.C.
!DUST CONCP -FIELD -*T ' ?P STED 1 :IJ - __
402 10 1 0 130 10 0 1 3 1 10 1 10 1 0 1 0 1 0 1 0 1 0 1 013 1 10 1 0 10 10 10 10 10 10 10 i
- 0 1 310 10 10
53 13 10 13 J41l 10 13 1 Q ) 10-i 1:1 1 k 10 10 1C 10 10 16iC 1 0 1 0 1- 13 10 1'3 10 1 10 10 1059 10 10 13 13aQ 1 10 1) 10 10 1 10 10 10 10IL 10 10
Fig. 27. C Ck at 180 minutes, radiation B.C.
-74-
0~ .-.-.~---.-
W AT- H -FIELD IN MM/ S C STEP 1 ..1: 01 02 03 C4 05 06 07 03 09 10 11 12 13 14 15 16 17" "
31 30 29 2U 27 26 25 24 23 22 21 20 19 1-8i J ; .__w
*. _ o - - - I - .. 4 - "
03 0 2 1 0 o 0 0 0 0 0 0 C 0 0 0 0 0_0 -1 1 1 0 C 0 0 0 0 0 0 G.- - .
104 0 1 C . 0 3 0 0 0 0 0 0 0 00115- 971 779 ' llV , 0 .M w 0 o . 6
-, -1 ,- 0 ; 0 0 0 ,0 03.3 0 . U J L' ' C 0 0 C - - 0 3
1 r 0 0
1- 1.- 1-- -- -.---- -- 1-- " "-"-3 - . . - ' - - - -- ' -. -" --- .- - -L---- -:' 2 - 2"
" 0 0 0 0 0 0
Fig. 28. - Wh at 180 minutes, constant B.C.
, W AT M' -FIELD :u 1,;k: / S;:C -STEP 123II 01l02 0 0z CS 06 07 0309 111112 13 1415 1 -
31 30 29 2; 27 26 5 24 23 22 21 20 19 18
-, ;" _ " - z!_ " L . B _________ ______..
03 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 G 0 0 0 0 0 0 0 0 0
-4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0O3 99j9779t.~ 7 W Ynl 2 ;b d. 0 0 0 0 0 0 0 0 0 0
0 0 0 C 0 0 00 0 0 0 00 58 0 0 0 0 0 0 0 0 0 0 :0 0 0 0 0 0__
0 0 0 0 0 0 -10 0 0 0 0 0
-3 .0 - - - - - " ' - C .. .3 '.. '2 - r - - -._____
~~~~ 0*.0 0 0 0 0 0 .0 .. *;O - W
Fig. 29. -Wh at 180 minutes, radiation B.C... .
0
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