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

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NATIONAL BUREAU OF STANIDAROS-1963 A

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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

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THE BEHAVIOR OF THE ATMOSPHERE IN THE DESERT Final Scientific Report15 October 1983-14 October 984

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Louis Berkofsky AFOSR-84-0036

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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.

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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

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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.

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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

-4

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op 4.1

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03AI-1L:)-3 N

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0 901MI atJ )I131 a

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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

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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

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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... .

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