-------
l.n combination wltri the energy equation reducing IT, to
aq, ( i 2} . As In the formulations of refs. i and 7 We will
/-A
add a term proportional to iA/q ) g_. u.' fcTr/To LO permit the
direct effect of stratification,,
It is immediately obvious that trie scale equation contains
much more arbitrariness than the Reynolds stress equations
where many of the terms were determined precisely without
recourse to modeling or coefficients. With such a large
number of coefficients in the scale equation, a correspond-
ingly large number of different experiments must be matched
concurrently if the resulting coefficients are to have any
invariant validity.
The coefficient s^ ^ay ^e estimated I'rom trie decay of homo-
geneous grid turbulence. If homogeneous turbulence is
assume a to decay as
q2 ~ *'"n (14)
then from the q equation
(15)
(16)
re,,t.nt, review cf gi lu t ui bw..i.ci.Co tx^ei^inicnt & cy Go.d-el-Hak
h.nri
u
shows values of n predonilnantiy ostweeri 1 and
I,-; W:.lh more of the values lying near ,: , 25 , This value of
To reduce the number- cf diffusion coefficients it is desir-
able to keep the coefficient of the direct diffusion term, SQ,
the same as in the Reynolds stress equation. This calls for
SQ - 1 , It is compatible with this assumption to take So = 0,
since with SQ - 1 any combination of m and n for an expression
of the form
. v
.
Dt e
would lead to an equation for iJA/Dt wltli s,~ = 0,
A simple relationship between ?>r>) s-( , Sp and Sh may be obtained
by looking at the reduced form of the scale equation In the
steady, neutral, constant-shear-stress layer near a solid
9
-------
boundary where we know that
A = a z (18)
with z the distance from the wall and a a constant set equal
to approximately 0.6 in ref. 3. In this region the scale
equation reduces to
which, with the aid of the energy equation, may be reduced
to
s,, = sAV + (b/a2)(sn - s0) (20)
An estimate for s,-, the coefficient of the stratification
term, can be obtained from the stable, constant flux layer
for Richardson number equal to its critical value, where both
A and q become constant. Under these conditions the scale
equation reduces to
0 = - s - THwT - s bq + s A JL. ^TJT (2i)
which, with the aid of the simplified Reynolds stress equa-
tions for this region^, reduces to
s = s + -£3-. (3 - s } (22)
^ u^A *
VHrien the values of the parameters used in ref. 3 are chosen,
this yields
s5 = 5.6 s1 - 4.6 s2 (23)
This leaves s,, sg, and s7 to be determined by computer
optimization. Prior to doing this we must determine boundary
conditions for the scale.
The boundary conditions on the scale equation are not as
straightforward as those on the Reynolds stress equations,
since the scale need not go to zero at the free boundary
of a region of turbulence. In fact, the eddies extending
the farthest are expected to be the largest eddies present
in the region. At a free boundary we therefore set A
equal to some fraction of the spread of the region of
turbulence. For axisymmetric problems a good choice appears
10
-------
he region oi turbulence
as measured by trie r ad! ..."-; oi In-.- regir.n w>. -re q^ equals 1/4
of Its maximum value. Tills oo^nri&ry condition Is 2 times the
algebraic approximation used everywhere for A in ref. 4. For
a two-dimensional region of turbulence a value uf A-,,j^e = 0.6
times the spread of the region of turbulence se^m-s to be
appropriate. The otner boundary ;oiiditlon is that near the
surface of a solid ooundary A she Li Id go to zero like a times
the distance normal to the* s^rfaci-.-, consistent with eq. (18).
The edge boundary condition .appears simpler if one chooses
to use the dissipation equation, since t - bq-V'A clearly
approaches zero. However, this je;-:1. L-S A 1 ree to approach
any value from 0 to «> a,;:> q -+ i; :nd no Independent information
is gained.
Since the production term is small for a fnomirntumless wake,
this provides a flow for estimating sg and s,v to lie between
0,3 and 1. To r e d u c e t h e un c -o i' t a i r \ y i r: t he s e c o e f f i c i e n t s
we have rather- ari?!tr'j.ril.y si'4" bi, --. j»^_ - ^//2J The last
coefficient Ok^y now be determined by computer fit with the
we- have some
; iycfiij Ibly due
Ii t^rms; but
. ron,iie. This
.c-a by Roal° who
'oei'flcient from
,rc - 1
Wltn one abo-c- cr.'Oic=.-h lor r -,- - c> ''i viexiib^ o^r proposed
scale tquat i>.-n i..'c.c; !(K;I:<
DA A
0,375
, '' ^.A ... L_z C?4^
dx. / ? T v '
Q O
As noted earlier, we cannot assign ctt> riigti a confidence level
to this equation as to our modeled Reynolds stress equations.
In fact, it appears that unless the turbulence is far out of
equilibrium, our previous approach-*-1 of simply, limiting the
A to be equal to the .lowest, of the three hounds
11
-------
i
11) A < C.2 6rjl
ill) A < C.
1 /?
rn i -*" / '
o
gdF/dz !
(25;
is quite satisfactory for the planetary boundary layer
problem.
The scale variation with height for the neutral planetary
boundary layer is shown in fig, 1 using both eqs. (24) and
(25), The upper value was given whatever small value permit-
ted a smooth tail to the curve,, rather than a fixed boundary
value for this flow.
In summary, eq. (24) appears to provide a reasonable estimate
of the macroscale for a number of different flow geometries;
i.e., for homogeneous turbulence behind a grid; for flat
plate boundary layer flow; for a free jet; for self -preserving
drag wake; for a moment umlesj; wakej arid for the atmospheric
boundary layer under both neutral and stable conditions.
Thus we will tentatively use this equation, but may anticipate
modifications ^^ farther flows art considered.
B, TRANSFORMATION TO STREAM FUNOTIGIJ COORDINATES TO
PERMIT SPATIAL VARIATION IN ROUGHNESS
Tr.e planetary boundary layer £,r^gra::i for- ref, 1 set the mean
vertical velocity, w> zero everywhere £s seen by eq. (2),
this is appropriate as long as the other velocity components
have no horizontal spatial variation. To permit spatial
inhomogeneity , we may transform eqs, (l) arid (4) to stream
function coordinates without adding any further equations.
This may be done by defining a stream function -^ such that
so that eq. (2) is satisfied automatically as long as v is
independent of y, Then the convective derivatives in eqs. (l)
and (4) reduce to ud/dx for steady flow and d/dz =
For steady flow, our transformed equations, as programmed in
coordinates x and ^ now may be written as
12
-------
Figure 1:
A, gross features
A, dynamic
equation
40
60
80 100 120
A, meters
140
160
Macroscale variation with height for a neutral
planetary boundary layer as given by the
dynamic equation, eq. (24); and by the gross
features of the flow, eq. (25).
13
-------
a'"1"*
- i an sir, * - . (281
\ u
2
I ~\ T I {*"1 1 1 A \ __ ^ I 1 i 1 1 1 ! ^*.
_____ ^ ^/ -. I L[ L>1 /V """ - i
4 v
c
|vwi \ _ ju
01^ / uA \
40
| ^ /
Jff (uivi sin 0 - u'w' cos 0} (29)
w'
, x dv'w' \ q
( quA -%-. - j - ~- v
\ ^ o^ / uA
: t- TT , x : :
-^ - w'w' -T-T + V -3-7 ( quA -%-. - j - ~- v'w'
c ^
u Tw ' sin 0L u'v1 cos 0]
u'v' sin 0 (30
(31)
5r ^u ,, 5 /,, du 'w' \ qu'W
- W ' W ' -r-r- -I- '«/ --c-r I QUA -T-r 1 - -*-r
01// C OT/' \ OT^ / UA
-^ [v'wr sin 0 - w'w1 CQS 0 f u'u1 cos 0] (32j
-------
-*r i; ' \fu ' ~^~- ; -^" - i '-J Uit "^ -* \ ~ rjp. |J *\J
dd- ' oy ' f ci-^ \ * of / UA
v' - v'wr cos ^ - u'u' sin v1! (34)
L*
n t1-;-,- irquan io:j.s gi
-------
foi'T; a set ol"' as ge m^ 1 o reiat j.onaai us Detwea.a crie Individual
Correlations and ':rrj vie an fK-w derl v -. '. ives ^ bat r'et.aln.Ing
jo.Mveotion and lif^.^ioi': of the t ^fb-.-l^-ace oy oar-ryir.g tht
full equation for- q1-' ; In t>:is ;,/.;-:.,, eqs, ' ;I4 ) cr.i^ugh (1J)
a r e r e p 1 a c e d c y
OL, . a /
2
Dt
,
A.
(30;
I u.'u,' ,
q I i 11 ox
g. -
o - rp
,
O --J
ol Iniinated tfjy .., -i
ecuations, base-."i or;
the planetary bo'.'p. '-
.. ., . i-.:J :.o .-, . ! 37 ) to allow
...;- ;- : ^y^teL'i c .'c-i-de termined .
lh--, ., ', ^o influerice of the
: ^ 'i '3-; 'ire distributed
", C--TJ ''- - of q-': , We have also
t '. ' :\e Reynolds stress
, i ..-rj r'^it, SiA./q « 1 within
r ; :-.>,,/, '. f i t^ o .:.-;, i -eq..ii j.i briuni results
.,- ;-,.. ,-, f,.-..^.ic ca icuia" Ion of the
^c'bi llbrl urn dr st. r \ b-.-\ ">,.,\*-. i., i,r-:- \ larvtary boundary layer
!'?BLj« The I'igar-^ .SIIOIA/L rii.rJ in- aLt;.:<',.xirru.te model gl -/es
\;. tnr: fui 1 ;-..?:- for this steady flow,
,.L^I i/^/,^.L;L. ^ v.ae dynamic response
- v>r; ', ;. irr,-.-, c f f " j surface condition
!'-< nq i 1 lii-i .;.; Jist r-ib\if. ioris obtained
-------
2400
2100
U,V,10*QQ
Figure 2: Velocity distributions in the neutral planetary
boundary layer (ZQ = .0774 m, f = 10"^ sec"1,
Up. = 10 m/sec) computed from the quasi-
o
equilibrium approximation () and from the full
set ( ).
17
-------
l.6r-
o I 2
-------
by the full equations ^ ::.s s. ov/-. I': ;">>. 2 ,> are taken as initial
condi'.. jr,s ru,t Surface neat flu< 1 o . rl*, l wir:_ tiT~ aa given
later ^n eq, ^j), Figure i >;:iows v,.-. re^po-if.: jf cue maximam
normal .'Cicc.jty v and the maximum q^ o,a a function cf time for
3 hourt; beginning at sunrise, !vlax 5 :r u": deviat! cu between the
two solutions is 10% during this p;.rr jf the -itay w.en turbu-
lence is undergoing its most rapid .-uanges , Alt/.ough the
quasi -eqaili brium solution u^s so:;ic piobiems Ir. t: acking
the turbulence, the overall '.greem-u'.1 is accept:aoie.
Comparisons cf the two metho'ic '..*.-: , \ sr, b-^ei^ rn=.ae ir; calcula-
tions of an axisymmetri c , moriontu^ie-LU:, wake 12 wnich show
approximately a 10^' difference j n tne resulting oi stributions.
The great simplification rr^ct'-j pcfc^JLl.; by the QE approximation
appears Ir, t^ aniplc Juct M'i v -r Ir,n ^ :- '.cce pi 1-.£ ^ he slight
loss in sc:.'.":-..cy exnibitcd ir ". ue.u; :-o':.parisons . Encouraged
' ^ ^ -" f^1 <~ - r^-t ; ~' r ^ 1,1;:. t-j r r> 1 "i *:"- H f \" f~ ~'-'r~ : ' '*( V ~\ ^'''" ^ ^ "\ '" ^ C*- Q C* £* ~\ C*M ~°
~~ j '' ' . ^ *- ^ J ~ > *- ) '* * i '/< . ^ - L !--' J. J '- -* ^ *-'^~ ( ' ° - - ^ ^- A. J LI L<. .,- ^ 1 _ i. w CX ^, >-^. J- *-* vA
lation of tr,;.. diurnal vari',,ticr in iha V31 (ser; .Section Vi ;,
a calculation which would require excessive time on our _ .
small computer with thv~ f'.ll iv-t of
-------
SECTION V
MODIFICATIONS TO THE POLLUTANT DISPERSAL KGLE..
A. INCLUSION OF THE CAPABILITY TO COMPUTE MEANDER
A.I Background
The diffusion of a pollutant; plume released into a turbulent
field strongly depends on the scale length of t,\c plume aa
well as the scale of the turbulence. La the £,R /-?, invari-
ant model, turbulent diffusion is proportional to t;;e product
of the plume scale arid trie root mean square of the 'curbulent
kinetic energy in the plume, Much, of the work described below
deals with improving the computational technique^ for both the
scale and energy during the early .stages of pi ;,ae growth
before the plume scale nas spread t:,- the scale of tr?; turbu
-
- , i L.
X
w , i. ^ ...
ri.-.tt :'-~kr
,^. j.oii _
This is :n.-.-^"sv-idenr,
t)l .'Ties are reie-Hed into t-.striblo 'i.e.
j , the piurrie
;.,ca.le 1 L, proi't-f,tv -u' s,ju;,od \nd !.''.' : o ^n^rgy ^;;5ociated with
die scale of ~:i. -: j. ,. a/,e :_,:, oo/u. i ;]<,/
-------
ciC '£"
;^'ogj:-a'ii re,4^i:^i:3 less
,. - 'wr /i -.. ,M'W i 4'-'- ; . -, .';, j -i-
r./ ,*'/ ;:: . - ~o :3z
M , . Jill1, co:.Ci -
<:-.: In 1 ^ti-j. ted In
L a .::.a -__dl_n:_ens ionai
11 r, p -. >^ \ --^ r* 1 £1 ^ "t ni f^ (1 g 1
. ^'~..- :'cr- the
(40)
-------
^C ' w
lr, eqs , (40) - (4'i) we assume that i;^ Re^iio^'j^
iriigh e 'lough so t hat \7iscou3 dlf'fubloc leftr,^ -'fi-y
cciiipai'ed co the turbulent d"! f'f 'jslc/: te^Tis. i'r =;
of these equations is 'ieRcriC^J lr '.'L-i. \ "or
reader ,
The values for ^n/.- various triodel pdratneters ar..,-., -,s f.-xpla
in Section lvr, vV - 0.3. A.! - 0,7^, - - l.ri. --~na :, -- 0,i2!
'er-tns in t:>.e u>S~ and 6 ' 9 ' eqaatiott,,
j c o ! iic s w ICL ^ ~. nc . u i * ^ u ju v^ n L o ^ ^ i [T11, o L» 11 u..rry o1 L i ^ . o
turbulent Pr^adtl 'L'Tber under neutril scifcll: ;._
arid that the spectra of concentration f 1 jc t ua\ i'
tc that of t-'-oerat'urr n^r:4;.': v :.:..-. jOi-ot-^,
iiOcij" Concentration
- ' . ^'' -s -1 t ~ , , -? ", ^T.
. _ - A : i ti,
The constrain! simply e ^,''-,; .hat '/'.^ .aixing scale cannot be
larger than tu. largest . LLI-L'^ ie./. j ^:.;, . The pr-,-^,T" lonal ity
-------
,j»J /, ^ .
''.:' ' -j >i. , ,' , '.(. to'j.o j.-j t'l;.. ' J J~
- [j
I.'-,.- :,; v iraj,e Kinetic a>:v:, :gy coij'Cd.
r.', e^u:;',:S with. ,: scale Is.ng* L A<- ,
, Is : . ;; o?.s-] s t en"; w:1": T, j -x:,cr-i-
'=*->''>-,,>-'.>: . -f '- > -; ",-)'- ' . ' - ' -- ~(\ : '
* > .^ r ' r- -,'--"- s
the p I ;; "
start
-------
w ' w
(' C, I
)i . . ' -j ', r'fan 10:
a^.nt..
he plum
- ai.d
-------
the v.. a 1*"?'^ of ::;a-> niauerieal technique used v; -;a' t . : " '-.
*2J;JiJL2a2iiL2±i£2. P-^tion of the plume history was lava:. . ;,%; -
by .".onslderlng a plume with an initial Gaussian 0:0!.-' ..ate
defined by a standard deviation of 1 m and a noi'Tiai iaaa
:-eiittrrilne concentration of 1.0 released into an :GoirapJ.'.
vurbulcnca field with FH/P" -.- 1 r^/sec?. The q0 ~* a i:.;.ar
wao aenievad by setting iaentic'^Ily ic a the oiii'as aa a.:,
reiidenoy-co- ! sotropy terirs in eq. (^1), Ac wae. 'lo^r, . , "-ov-.-
iaie amlyr.ic sol.:;lio;j showa tnat tae plune splits !aio ,aa
Identical w;vc-b, ajco, having a, aaxifiiurr concentr1: _ _.an of t, ,
ana ..r'avelln^ iWa,}/ r i-oa the je.ater cp the plun.e t::1:'1 -.'.
y^-iC'' i.i ,y j: I ia 'ii-1 /.
'i;i-...^ -« oij^':.'.", " tf-: .'c-./^^ala-a "> i A ' /a uc "- ic%a .In t "a . ... ;a "
a/a :-! 'jownwlna ;jf plurutr rsleast-;, The ayrneaiaii -r - .'._.-:-
'.vat; a f ully-ii'Lpl i'.1 i.t tridiagonal algorithm1'1 wl-r .-j
aei,',!t ; iritr ^ jiaximurn local concenti'af ion i-hanp>' a-
>i .-t -; nl," .-'.:*,'. '." ;i.' '. . ;_-- i '/. '. fjc1 -";! .j i .-.o;-i'. whit, L'J it ;j£ or, i ; i. t 1 7"j l ^;' the lar^';-
s'r'aj; i-jr t,i:.(p .<:'.::- _-;-quI re
,,. i .
fiold, :':<& ]! ;ri£- scale wa-5 iuv1,i cor;-,tant <.,l ^.v,'- " ., , ;
:,uu';,l r a !\j,niuL.t ;':.' "'.] c:ia! V(, Wf se .-idj'jL. V".">. t.,.. 1.0 ur.1, <
.f;^! ^ ..-I,; v'0.1.7 ? '' o'rd--r' to coincide w;1"1"; c};e e.x af. ?a "
a,lv."^ U,/ ens , (bt "> c-iid ', ;;1^, rlgui'-:a a and o on:
;'or;par.f:ai ^.f'.d axact ?orioenira* iaa pi'aflies a1, ai,i4";;aa .a
-------
CONCENTRATION PROFILE
IN WAVE-DOMINATED REGION
x = IOOm
JOr
z, m
^-Numerical solution
Exact solution
Figure 4: Comparison of numerical and analytical
predictions for one-dimensional wave
propagation.
26
-------
a
o
a: LU
a. H-
^ o
2 o
O T
? *
O CO
O U.
O U.
Q
o:
o
L_
O
in
CVJ
O
O
CVJ
o
cd
£
O
H
ra
d
a>
H
T3
I
0)
ra
C
O
O
H
T3
0)
tn
a
05
O
cd
d
cd
ri
£
cd
cd
o
H
fn
CU
O
£
O £
ra o
H -H
^ ra
cd d
O
in
O
O
O
m
S
O -H
O 73
0
PH
M
H
27
-------
CONCENTRATION PROFILE
FOR DIFFUSION-DOMINATED REGION
x = 5000m
lOOOr
800
600
z,m
400
200
10'
Numerical solution
Exact solution
10
-2
10
-i
Figure 6: Comparison of numerical and analytical
predictions for one-dimensional diffusion
far downstream of the release point.
28
-------
'".r-gfi ;.>K-J -^A-<,.
, ex^(-t value .>,^L :.t.- '; > , ~. : . jt
t,- vocation at v-;;iicti crie :>ac t :'ri,Llc
"s wit?. in 1'? )f : " ;. rvv;x!mum value is ^.ecura'o : ^"r._-,t
'!1v:e if' :'-:;rp.:t :; : nio -vere made wi^h tt fi^xi'nirr. S".^L .. va.,^/-. of
to insure ^°( -ice ^-'i-y : a c
doQir;ato:j pi re , This co'noinaf- ion ie^dsJ to ;>irc,e:' errors
during the c^r'y w.j /-- ioa:i-i ;,ted r-eg'ou, but 8l/;,--_j tfi- extent
of* tf-iis regior. i:; -\ smj,ii porvt:ja of ~ru- plunie '. i r..-t : :nj, it
IB c:msid?r-d to b - ,;t- ,v^'-pfai?] e [-.-- -.> r-y , " i-rTiaf^::
corripvit 3 t lor. * -rr^ -^.d riu.L'.er ica..- ictao i. .. Lt y ,
t arbu-
'',.. (52)
u re U'L-,- -«x t f-r.^e ioca -
2^^ oi" t be aiaxiraum
(4^; itnii has jeen
(ie. ?(.>r 1 tj-;- ["'...upe tu,a.lfo I'1 th>; wave-
','^r, '",-" "''- p-Tfoses or deitcnsti'at ing
;i. ; ' -v.;in !',.,.if 'o thie pi ar^e scale.
-------
!';' [5
-------
OJ
CD
10
-P
ra
£
O
o
05
G
O
H
-P
k
O
a
o
SH
CX
0)
x:
4^
o
H
P
O
ra
aj
0)
r-i
cd
o
CQ
E
«
-------
cr
LU
Q
LU
2
LU
2
Z>
_J
Q.
O
<&
LL
O
I-
O
LU
Li_
LL.
LU
^f
O T3 r-f
C a; to
o -P >
o -P
O m
OJ iH 3
Sao
a o oJ
H >
0 -P
C! cd fn
H PH O
rH ^J CH
PH fl
0 0 -P
0 O O
o o a
CO
0
Pn
O
O
E
-------
z,m
EFFECT OF PLUME SCALE FACTOR
ON CONCENTRATION PROFILES
x = IOO m
0
.02
.04
.06
.10
.12
Figure 9: Mean concentration profiles 100 m downstream of a
Gaussian plume release for different values of
scale factor p. (u = 10 m/sec, w'w' = 1 m^/sec,
q2 = 3 m2/sec2, neutral stability, Oj_ = 1 m,
Cmax, = -1-)-
1 33
-------
I40r
120
100
Z, m
80
60
40
20
EFFECT OF PLUME SCALE FACTOR
ON CONCENTRATION PROFILES
x = IOOO m
.005
.010
.015
.020
Figure 10:
Mean concentration profiles at 1 km for the
same conditions as fig. 9.
34
-------
800
700
600
500
2, m
400
300
200
100
EFFECT OF PLUME SCALE FACTOR
ON CONCENTRATION PROFILES
x=IO,OOOm
.001
.002
c
.003
.004
Figure 11
Mean concentration profiles at 10 km for
the same conditions as fig. 9.
35
-------
r
-------
fO
O
CD
1 01
O
LU
Q_
U_ ^
O 5E
o
LU
o
00
O
ii
ISI
o
x
O
E
o
d
o
H
-P
^
O
-P
ra
H
-P
a
dl
0
CO
co
o
o
ra
0) CO
3 CD
iH rH
cd -H
> CM
O
rH L<
Cd CX
H
-P C
H O
d -H
H -P
cd
CM in
O -P
a
CD CD
O O
d d
CD o
2 o
d x!
H -P
OJ
rH
CD
bO
H
Pn
37
-------
"/'f.'.'in *. few hiind"ed rue t era neight variation. Trier1; 1.-: r
' ".ds "'..;.>': a 1 difficulty in including tnis effect in tne
7:'o?i",H'1 since we are computing variations in y and z -3s
'.u''C - ion of x} but !.t does require the computation of tn
pj,.j';ie in. the full (y, z) plane since this will destroy t'
^ymmeiry of trie plume in the y direction. This
require a doubling of computer time.
-------
SECTION VI
SAMPLE CALCULATIONS AND MODEL VERIFICATION
The computer model is now capable of making a wide range
of calculations of practical interest. The following
examples have been chosen for demonstrating its capabil-
ities and for their own inherent Interest,,
A. STEADY STATE, NEUTRAL, PLANETARY BOUNDARY LAYER
The mean wind and Reynolds stress components existing in
the steady state, neutral planetary boundary layer have been,
computed and reported on in ref. 11 which Is included in
this report as the Appendix. The macroscale used In those
computations was determined by one of the bounds given in
eq. (25), The results were compared (in fig. A3V with the
wind and turf,u,1 e.nce profiles computed by Deai'dcri'-"-1-^ using
his three-d iver-oion-ii, unsteady oa ; ^-,1;^ ions , ..^r c.d.i,3u-
latio.n rias D- er. !;,uedted u.sing the 1;, r.amic sea.;; /q^ation
given i n. en , ' ''} , Thoo- re;'; Its d"-- ,-,- ' so cciLC-ir^c; with
Deardorff's "cr/lt,; (appropriately un^ormaiizen'' jr< fig, 13-
There is relhi^ly little di. f ferer cr: between tne results
for the deter-ifl rut. loj. of n, v, arid q 'tsing the two approaches
to the deter-" irv.t i or: of the u-.-aie. In either c-n-:e ; the wind
profiler: agr"o<: r-,r- >oit-'h iy we]' with Deordorff's but the
turbulence i;s a I ^: i f 1 rant ly lower, P.trt of the discrepancy
is probably t au.seci cy the low riJtliu.i^ at which Deardorff
matches with the geost ropfiic wind, but it may also indicate
that our predictions for the herizontnL wind variance perpen-
dicular to the ..surface wind are low. This component is
difficult to .determine empirically. Measurements as reported
by Panofsky1^ contain an uncertainty of a factor of 2 in what
this component should be. For the present, however} we will
accept this discrepancy and use these resulting equilibrium
profiles as a starting point for a more Interesting result -
the dynamic response of the earth's boundary layer to chang-
ing surface heat flux,
B. DIURNAL VARIATIONS IN THE PLANETARY BOUNDARY LAYKR
In the Appendix we discuss the detailed calculation of the
response of the planetary boundary layer to trie changing
diurnal conditions embodied in the rising and set.!lr;t; of the
-------
2000,
17SO
-2
U.V.lOad CMPSD
Figure 13:
Comparison of velocity distributions in the
neutral planetary boundary layer (ZQ = 0.008 m,
f = 10"^ sec"1, ug = 10 m/sec) as predicted by
the model with dynamic scale equation with
Deardorff's model predictions ( )(ref. 15).
-------
sun. Tne present discussion will briefly summarize the input
conditions for this calculation and highlight the results.
A further calculation made with the scale equation -.f, its
dynamic operation will also be touched upon,
For the actual calculation of the diurnal variation of the
planetary boundary Layer, we program, arid solve eqs. (l) - (11',.
subject to the quasi-equilibrium approximation, Geostrophic
boundary conditions were applied at altitude while at the
surface layer, conditions were matched with the results of
the Monir.-Obukhov similarity analysis, A temperature inver-
sion lid was initially established at 2 km and permitted to
drift up or down through the course of the three-day calcula-
tion as the computation deemed necessary. The inltid.1 profiles
(with d - Q everywhere below ? km) -; re as give;. 1: fig, A3
and discussed in Sect ion VI A.. To simulate the effect of the
sun, we set the surface heat flux varint ion as
(w'e') =,0.25 sin (0.00007275t) > 0.025°C-n./st-c (53'
o
as suggested DJ,
Kansas, Rewova
cons tent ra;; : ->'
and '-> -;r;
reaul; ,.% o f : ,
C-f* r* ^- >"'' i ' i . i
s ' ' . f. '
The pro gnat.. - : ;
dot a o I'
re
of
P.fc'ai so;j.i
At th-? mor". "
(in fig. A 9 ,; i
the surface an
This strongly
'. rig
ea anro of
-. t u a , :u, and
y n a IT: i C S O 1U -
8 a.-A. and 8 p.m.
. r- :?-: ress u*
:v.ii pc;int, while
a 'l,i fig. AiOi i,ih.c;ws the rtvarse trend.
,ests i'.it r -in-ersion lid near 2,5 km.
The "breaking throigh" of the early morning boundary layer
is particuLi": v /ivlc1 t:^re wh.-r-' '-y 11 a.m. the character
This effect is
of the q:^
carried
loop predicted
;I:±,J8 ,iK WV
as a i'./.c
f 1 c a n t d 1 f f e r e n c e
-------
scale jquj'ion ('--+, . ^ atv <--v!.-
repeated UK ing r.r,e £a>i:t- -t ft' "','--" i -
with two varia^ :o/;B, K Sce-qs eu-cir
during the day Iw prouat-Iy x^-:r,o.- 1 d
hours, Thi...-3, in ch^- t-ai here q^x I (-.-,;.
implemented oetweer, i,ue hoir.'-t-. wi - t'
the dynamic scale equation wbs loeii;
Figure 14 gives t'v~ surface M res-/. -,
for the three days of c^ Ic^Iatlo.n ,
shows a close sim ! lai-i ly exc^r/L fV r
11 a,m. where fig, 14 shows i.nr-et ,l;i
tions. This efl'e,..t apparently rn.:v K rr'rMjir^d Ar :.he
dynamics of the scale nation, ;d \
inversion "i la ib t- ;-.g clown : ., r : u
fig. 15 for trie kinetic- energy q1-'.
reveals thai, trjt- '^s \ --.-^lor; }"n now
intensity of i nc turcalen^t --1" t rr,,
increased by l-'O^. 'firie .rise 1;t q-- 1:
dramatic; ita rallof* ;,;.r-:t !-:
to the prco^'<; ,.-f' !a.'g'-. _.;".,-, -
The dvaarni--: - ,-. '..: = :,
the ^-o.^-, ;
C. TURBTFLFiNCM 1^1: i'KllrVJT I'VJ " , ':';:;<- I;1' ,
ABRUPT C.lAlk.i, IK CUl'KACK J{01.: dl"!" ;"S
The boundary ,i.ay,-r equaLi,-).ti& it!
(eqs. (27) - (?4)} may i:e ust d U
the- near-wai'1 i'cur'd'xi^. I.,.; " < o -
r o ugh n e s s z,... k -. i-,! c ^ ] ;: i o ^ w
a t mo s p he r i c ,; ; r f ,-. c e d .! ) t' K- i '."i d i
shear, and velocities a-. -.1 .. ^'.le
Transitions w>-re made rV.^" :\, >.gv
rough where th.e hycir'.;.i./Jia''1' '. t,j.-!^
of the rough.iese. '' ; >,:<:: ? :,'.; f ';.;'-.-,
consisted o! octad t'd ?'.-. ;. 1 j-bbi
r cj . ^
\ -,rii oles
s " -> *-
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3500.
q2=.OI m2/sec2
TIME CHRS3
Figure
Contours of constant qd as a function of altitude
and time of day as predicted by the model using
the dynamic scale equation (compare with fig. A4)
44
-------
h-
I
CD
ti
HI
I
3500..
3000...
2500..
2000..
1500 ..
1000 ..
500..
12 15
TIME CHRSD
21
Figure 16:
Contours of constant length scale A as a function
of altitude and time of day as predicted by the
model using the dynamic scale equation (compare
with fig. A8).
-------
^ _ ,.~ ^ ,, _ ^ - -,---.".« 4-" -. - , j t - -, ,> ^. ", --, - .^i
L \- J. i A^L w I, 'O ^1 11 --- ( .^ £TJ IJ -1. v.> - V*. I i. 1 «^' 11. &. ~- \) * c. ) *s U- »
O'il4 uaiC Vila C LOr;3 Ui-f£a.a ft 1 Oil /., 1 lAt:Q y.\*
ana run until the ^ari^cles appear V; reach a stead;, st3te
(re-'ienibering that wa are mostly eorceaned with the response
picture below -A bout 10 n.) hitl t-nesc steady pr-ofzlc-s, thex^.
we change trie xo to Its ctr^-.i' value and track the response
of the boundar-y"'lay.;r tc th . ,, abrupt chatige I.c surface rough-
r:ess, The results fo.-' i.ota caseb are snown i]> fig, I7,
Cornparisor1, ^ith & great;- c.e'al of B-r-adl^y's do.ta shovjs gcod
agreement, i.are v>,e plat tat shearing stress :r>.cr-:r.alized oy
Its previcad titv..-.Jy jiia,v,e value) vci'S'u) ths x oibtance aown-
ritreaT. denoie:i ao ''fetvV'},, -'hi-ri L, ;tr,. r'uii3 ,1 r-e alloived to
r-oagh 13 -t/i ' - l,b;: «/:ille roaga-co-^n'oot'^
Our results -it-e \/-- ry h l '* : I a r t.-c tiie aiiner-loa
by Shir 19,
,Ir, -r of "-hi- '"* < ;^/ ' ~ -" ^ 3' 3^' >^; t ' .-' th, r- ; dJ *~ I '" ;.'!£ .^r she3,1
st re^t; rer';;. Li,r ,IO-,^T. ---,/H, \.j,; l"; Lho ; iu j^;cnce cT t ne S'>ep is
i
feltj w;iers the ecig1:' of toe i '.,\r-c boioauary lay*;: growing I
f'T'OTi y -.- i"1 ". 'f .-:>'