NASA TECHNICAL MEMORANDUM

NASA TM X-64554

STANDARD DEVIATION OF VERTICAL WIND SHEAR IN THE LOWER ATMOSPHERE

By George H. Fiehtl Aero- A s trodynamies Laboratory

December 17, 1970

--.,-s

NASA

George C. MarJbdZZ Space FZebt Center MathdZ Space FZigbt Center, AZabdmd

MSFC - Form 3190 <8eptembei 1968)

https://ntrs.nasa.gov/search.jsp?R=19710005748 2018-07-14T04:00:55+00:00Z

1. Report No.

TMX - 64 554 2. Government Accession No. 3. Recipient’s Cotolog No.

9. Performing Organizotion Name and Address

4. Ti t le and Subtitle

STANDARD DEVIATION OF VERTICAL W I N D SHEAR I N rm LOWER ATMOSPHERE

110. Work Unit No.

5. Report Dote

6. Performing Organization Code

. December 17* I g 7 O

George C. Marshall Space F l i g h t Center 11. Contract or Grant No.

Marshall Space F l i g h t Center, Alabama 35812 13. Type of Report ond Period Covered I

TECHNICAL MEMORANDUM 2. Sponsoring Agency Name and Address

7. Author(s1 George H. F i c h t l

NASA Washington, D. C. 20546

8. Performing Organization Report No.

> 14. Sponsoring Agency Code

IS. Supplementary Notes

16. Abstroct

The standard dev ia t ion CJ of v e r t i c a l two-point longi tudina l ve loc i ty f l u c t u a t i o n d i f f e rences i s analyied experimentally wi th eleven s e t s of turbulence measurements obtained a t the NASA 150-meter ground winds tower s i t e a t Cape Kennedy, F lo ida. It is concluded t h a t t he O/U*, i s propor t iona l t o ( f ~ ~ / u * ~ ) ~ ~ ~ , where the c o e f f i c i e n t of p ropor t iona l i t y is a func t ion of fZ/QO and u*o/fLo. f and Lo denote the Cor io l i s parameter and the s u r f a c e Monin-Obukhov s t a b i l i t y length , r e spec t ive ly ; Q~ is t h e s u r f a c e f r i c t i o n ve loc i ty ; & is the v e r t i c a l d i s t a n c e between the two poin ts over which the ve loc i ty d i f f e rence i s ca l cu la t ed ; and E is the mean he ight of the mid-point of t he i n t e r v a l & above na tu ra l grade. The r e s u l t s of t he ana lys i s a r e valid f o r 20 < -u,,/fL0 < 2000.

The quan t i t i e

17. Key Words I 18. Distribution Statement PUBLIC RELEASE

. D. Geissler I Direc tor , Aero-As trodynamics Lab.

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STANDARD DEVIATION OF VERTICAL W I N D SHEAR I N THE LOWER ATMOSPHERE

SUMMARY

The s tandard devia t ion cr of v e r t i c a l two-point longi tudina l veloc- i t y f l u c t u a t i o n d i f f e rences is analyzed experimentally wi th eleven s e t s of turbulence measurements obtained a t the NASA 150-meter ground winds tower s i t e a t Cape Kennedy Flor ida . It is concluded t h a t t he a/u*, is proport ional t o ( f a ~ / u , , > ~ ~ ~ , where t h e c o e f f i c i e n t of p ropor t iona l i t y is a func t ion of fZ/lqio and u*o/fLo. The q u a n t i t i e s f and Lo denote the Cor io l i s parameter and the s u r f a c e Monin-Obukhov s t a b i l i t y length, respec t ive ly ; %o is the s u r f a c e f r i c t i o n ve loc i ty ; & is the v e r t i c a l d i s tance between the two poin ts over which t h e v e l o c i t y d i f f e rence i s ca lcu la ted ; and f is the mean he igh t of the mid-point of t h e i n t e r v a l Clz above n a t u r a l grade. The r e s u l t s of t he ana lys i s a r e valid f o r 20 < -l&o/fL, < 2000.

I. INTRODUCTION

S t a t i s t i c a l es t imates of wind shear i n the p lane tary boundary l aye r a r e important i n t h e design of V/STOL a i r c r a f t ( v e r t i c a l / s h o r t take-off and landing a i r c r a f t ) . The STOL a i r c r a f t a r e s e n s i t i v e t o v a r i a t i o n s i n l i f t fo rces r e s u l t i n g from v e r t i c a l v a r i a t i o n s of the ho r i zon ta l wind during take-off and p a r t i c u l a r l y during landing operat ions VTOL a i r c r a f t con t ro l systems must be designed t o counterac t extreme t ipp ing moments r e s u l t i n g from v e r t i c a l v a r i a t i o n s of t he ho r i zon ta l wind f o r spec i f i ed acceptab le values of r i s k of exceeding the design wind shear condi t ions. S imi la r ly , wind shear information w i l l a l s o be necessary i n the design of t h e forthcoming NASA Space S h u t t l e f o r bo th the launch and landing phases. During t h e launch phase t h e S h u t t l e w i l l experience t ipping moments ; during the landing phase, i t w i l l experience f l u c t u a t i n g l i f t forces . The v e r t i c a l two-point longi tudina l v e l o c i t y d i f f e rences t h a t w i l l be experienced by these vehic les can be divided i n t o two pa r t s : the wind shear r e s u l t i n g from the mean flow, and the f l u c t u a t i n g or random part r e s u l t i n g from turbulence. The mean shea r can be ca l cu la t ed wi th cu r ren t ly a v a i l a b l e models of t h e mean flow wind p r o f i l e i n the atmos- pheric boundary l aye r [1,2, and 31. However, t he f l u c t u a t i n g p a r t must be spec i f i ed s t a t i s t i c a l l y f o r engineering problems, and t h i s r equ i r e s knowledge about t he s t a t i s t i c a l moments o r t he d i s t r i b u t i o n func t ion of the shear . second moment of wind shear . I n add i t ion , t he square r o o t of t he second

A g r e a t dea l of information can be gained by s tudying t h e

moment, t he s tandard devia t ion , of a random v a r i a b l e is used as a s tandard iz ing parameter t o render the random v a r i a b l e nondimensional. Accordingly, the ana lys i s i n t h i s r epor t is r e s t r i c t e d t o an examina- t i o n of t he second moment of v e r t i c a l two-point longi tudina l ve loc i ty d i f f e rences , o r , i n o ther words, wind shear .

It i s hypothesized t h a t the s tandard devia t ion of t he wind shear scaled wi th the su r face f r i c t i o n v e l o c i t y u * ~ i s a universa l func t ion of t he nondimensional parameters fDz/u*o, fz /%o, and uJrO/fLO, where f is the Cor io l i s parameter, Lo is the s u r f a c e Monin-Obukhov length, & is the v e r t i c a l d i s t ance between the two points over which the shear is ca l - cu la ted , and 'i is the mean he ight of t he mid-point of t he shear i n t e r v a l Llz above na tu ra l grade.

From the eleven cases of turbulence i n uns tab le a i r se l ec t ed f o r a n a l y s i s , i t is concluded t h a t cr/uJr0 i s indeed a func t ion of the non- dimensional parameters l i s t e d above. Empirical mathematical expressions a r e obtained t o represe t t h i s func t ion , It appears t h a t G / U J C ~ is pro- po r t iona l t o ( f & / ~ * ~ ) l ~ ~ , the c o e f f i c i e n t 05 propor t iona l i t y being a func t ion of fE/u,ko and u.zro/fLo. f o r 20 < -qco/fL0 < 2000.

The r e s u l t s of the ana lys i s a r e va l id

The author expresses h i s thanks t o M r . Archie Jackson and M r s . E l l a Mae McAllister of t he Marshall Space F l i g h t Center ' s Computation Labora- tory. M r . Jackson developed the program f o r the c a l c u l a t i o n of the two- poin t s tandard devia t ions . Mrs. McAllister developed the remaining programs which were used t o c a l c u l a t e the non-dimens iona l q u a n t i t i e s . I n add i t ion , thanks go t o M r . J u l i a n Nelson and M r . Douglas Mackiernan of the Aero-Astrodynamics Laboratory f o r t h e i r help i n the prepara t ion of t h i s r epor t and the ana lys i s of the data.

11. THE DATA SOURCE

The d a t a analyzed i n t h i s r epor t c o n s i s t of eleven s e t s of longi- t u d i n a l tu rbulen t v e l o c i t y f l u c t u a t i o n time h i s t o r i e s d i g i t i z e d a t 0.2-second i n t e r v a l s wi th 18,000 d a t a points p e r time h i s to ry . The longi tudina l v e l o c i t y f luc tua t ions were ca lcu la ted wi th hor izonta l wind speed and d i r e c t i o n data measured a t t h e 18-, 30-, 60-, 90-, 120-, and 150-meter l eve l s a t the NASA meteorological tower s i t e a t t h e Kennedy Space Center wi th Climet (model C1-14) wind sensors . The measurements were taken during the daytime i n uns tab le air . The computation procedures used t o c a l c u l a t e t h e long i tud ina l tu rbulen t v e l o c i t y f luc tua t ions a r e discussed by F i c h t l and McVehil i n re ference 4. Supporting temperature measurements were made a t the 18- and 30-meter l eve l s with Climet (model C1-016) a sp i r a t ed thermocouples. (Detai ls concerning the instrumentat ion a t the NASA tower s i t e can be found i n a r e p o r t by Kaufman and Keene [5]; the su r face roughness lengths assoc ia ted wi th t h i s s i t e a r e discussed by F i c h t l and McVehil i n re ference 4 . )

2

111. THEORETICAL CONSIDERATIONS

The instantaneous longi tudina l v e l o c i t y u(x,y, z , t) i n a ho r i zon ta l ly homogeneous atmospheric boundary l aye r can be represented as

where ; ( z ) is the temporal mean wind a t he igh t z , and u' (x ,y , z , t ) is a longi tudina l t u rbu len t v e l o c i t y f luc tua t ion . Upon evaluat ing equation (1) a t he ights z1 and z2 ( z 2 > zl) f o r f ixed values of x ,y , and t, we may c a l c u l a t e t he two-point v e l o c i t y d i f f e rence

where

The dependence of OU and aut on x and y is understood. For engineering purposes, t he quan t i ty AS(zl,z2) can be t r e a t e d as a known quan t i ty t h a t can be ca l cu la t ed wi th a power l a w wind p r o f i l e l ike

where a is a posit ive cons tan t , o r w i th the more e legant wind p r o f i l e formulations f o r t h e ba ro t rop ic and ba roc l in i c Ekman p lane tary boundary layers by Blackadar [l]. For values of z2 <, 30 my t h e mean wind p r o f i l e l a w appropr ia te f o r a Monin

-0 U(z) = -

k l

where %o and Lo denote the s t a b i l i t y length, zo is the

l a y e r can be used:

(In 5 0 - t(Z/L0)). (7)

s u r f a c e f r i c t i o n v e l o c i t y and Monin-Obukhov s u r f a c e roughness length , k, is von Karman's

3

cons tan t wi th numerical value approximately equal t o 0.4 and $(z/L0) is a universa l func t ion of z/Lo [ 3 ] . &'(z , z2 , t ) is a r ap id ly varying random funct ion of time, and thus mus t be t r ea t ed s t a t i s t i c a l l y .

On the o ther hand, t he quan t i ty

The longi tudina l time h i s t o r i e s were d i f fe renced i n the v e r t i c a l t o y i e ld f i f t e e n longi tudina l v e l o c i t y d i f f e rence time h i s t o r i e s f o r each case. Unbiased est imates of t he second moment were ca lcu la ted wi th the formula

where 0 is the standard dev ia t ion of t he random longi tudina l shear and n is the number of d a t a po in ts used i n the ca l cu la t ion .

IV. DIMENSIONAL ANALYSIS

The Monin-Obukhov s i m i l a r i t y hypothesis [3] pred ic t s t h a t the standard dev ia t ion of two-point longi tudina l v e l o c i t y d i f f e rences , scaled wi th the su r face f r i c t i o n v e l o c i t y u * ~ , a r e universa l funct ions of zl/Lo and z2/L0 o r ;/Lo and &/Lo, where

Thus , t he Monin-Obukhov s i m i l a r i t y hypothesis leads t o the conclusion t h a t

where F i s a universa l func t ion of ;/Lo and Az/L0. ceeds upward out of t he Monin l aye r i n t o the Ekman l aye r , i t is reason- a b l e t o hypothesize the a/%o is a func t ion of ;/Lo, &/Lo and other parameters t h a t c h a r a c t e r i z e t h e a c t i o n of Cor io l i s forces and ba roc l in i c e f f e c t s . Blackadar and Tennekes [SI have shown that, i n t h e ho r i zon ta l ly

However, as one pro-

4

homogeneous, n e u t r a l , baro t ropic Ekman l aye r , u*o/f is the appropr ia te length s c a l e , where f is the Cor io l i s parameter, This r e s u l t was pro- duced by analyzing the mean flow momentum conservat ion equations along with t h e tu rbu len t energy equation. Thus, as a working hypothesis , one might assume t h a t a/u,, is a func t ion of the independent va r i ab le s

az , z , Lo, U*,/f.

According t o Buckingham's theorem [7], the number of independent non- dimensional q u a n t i t i e s that can be constructed from t h i s l i s t of var iab les is three . Now the re a r e e ighteen poss ib l e ways t o s e l e c t t h i s group of t h ree independent nondimensional q u a n t i t i e s such t h a t and & appear w i th in each group. However, any one of t he poss ib l e groups can be derived from the remaining seventeen groups, Accordingly, we a r e f r e e t o s e l e c t any one group w e l i k e t o perform a data ana lys i s ; thus , we hypothesize t h a t

- = F ( & , e a u*O u*O , fL

This hypothesis includes the dominant e f f e c t s i n the Ekman layer : atmospheric s t a b i l i t y , and the a c t i o n of Cor io l i s forces i n the hor i - zontal mean flow momentum conservat ion equations. It does not include ba roc l in i c e f f e c t s r e s u l t i n g from a height-dependent hor izonta l pressure- g rad ien t force. Nevertheless , t he hypothesis given by equat ion (12) seems t o be appropr ia te i n view of the f a c t t h a t t he outs tanding f e a t u r e of t he Ekman l aye r i s the a c t i o n of Cor io l i s fo rces and t h a t inc lus ion of ba roc l in i c e f f e c t s w i l l only lead to refinements. This hypothesis w i l l be used t o analyze the standard dev ia t ion of vertical two-point lopgi tudina l v e l o c i t y f l u c t u a t i o n d i f f e rences . The procedures used i n the c a l c u l a t i o n of uJro and Lo a r e given i n appendix A. these c a l c u l a t i o n s and the r e s u l t s of the c a l c u l a t i o n s a r e given i n appendix B.

The d a t a used i n

V. DATA ANALYSIS

To es t imate the universa l func t ion F on the right-hand s i d e of equation (12) , the nondimensional s tandard dev ia t ion was p lo t t ed as a func t ion of f&/Q0 f o r var ious ca t egor i e s of f f / Q 0 and u*o/fLo. Each case is charac te r ized by a s i n g l e value of UJcO/fLO; however, w i th in each case, t he quan t i ty fZ/UkO can vary.

5

Figure 1 i s a p l o t of o / ~ ~ as a func t ion of fAz/u*, f o r the var ious ca tegor ies of fZ/uko and uJco/fLo. The d a t a a p p e a r t o show that, f o r each of the se l ec t ed ca t egor i e s , o/uJco is proport ional t o ( f & / q O ) l / 3 . I n add i t ion , wi th in each category of f f / w O , the values of o/ulc0 tend t o s o r t out according t o t h e parameter uJco/fLo; i .e . , f o r f ixed values fE/u*o and fAz/+o, the quan t i ty G / Q ~ tends t o be a decreasing func t ion of -Qo/fL0. There a l s o appears t o be a dependence of o/uJc0 on fZ/Qo; i . e . , f o r f ixed values of f&/Q0 and uJ,o/fLo, t he quan t i ty “/uJc0 is a decreas- ing func t ion fE/uk0.

.

To s e e these e f f e c t s b e t t e r , i t w a s assumed t h a t D/uJzo is proportional t o (f&/u,,)l/3 f o r a l l ca tegor ies of the remaining independent var iab les , so t h a t

1/ 3 - = G ( L y Z ) < ‘ o ) , u*O u*O 0.001

where G i s a universa l func t ion of fz/u*o and u;ro/fLo. w a s determined by f i t t i n g equation (13) t o the data i n f i g u r e 1 fo r each category of fz /Qo and wo/fLo. shown i n f i g u r e 2. c l e a r l y decreasing funct ions of fz/u*o and increasing funct ions of %,/fL0.

The funct ion G

The r e s u l t s of these ca l cu la t ions a r e We s e e from t h i s f i g u r e t h a t G and thus o/u,, a r e

To g ive the experimental r e s u l t s mathematical expression, we assume t h a t the curves i n f i g u r e 2 a r e shape-invariant wi th r e spec t t o v a r i a t i o n i n uJro/fLo and t h a t t r a n s l a t i o n only with r e spec t t o the G-axis is required t o produce a co l l apse of these curves. This means t h a t t he re e x i s t s a funct ion H(%o/fLo) such t h a t

I n view of t he experimental r e s u l t s i n f i g u r e 2 , these assumptions appear t o be reasonable. To c a l c u l a t e the func t ion H(u*,/fL0), we s u b t r a c t the experimental es t imate of G(fZ/u*o, -150) from each curve i n f i g u r e 2 and then average each of t he r e s u l t i n g d i f f e rence funct ions over t he experimental range of v a r i a t i o n fZ/u*o. mates of H(u*,/fL,) f o r the var ious values of uJ;o/fLo a r e given i n f i g u r e 3 . The func t ion

The experimental e s t i -

U,o/fLo H(u,,/fL0) = -0.072 I n (- ,,,)

6

appears t o summarize these r e s u l t s reasonably w e l l . To improve our e s t i - mate of G(fg/upco, -150), we s u b t r a c t H ( s o / f L ) as given by equation (15) from the experimental values of G(fz/%,, U.ko7fLo) given i n f i g u r e 2 and average the r e s u l t i n g d i f f e rence funct ions over t he experimental range of v a r i a t i o n of QO/fLO f o r each category of fE/uk0. mate of t he funct ion G ( f E / s 0 , -150) is depicted i n f i g u r e 4 , and the a n a l y t i c a l express ion

The revised e s t i -

summarizes these da t a . -150) 3 1 as f E / Q o + CQ.

height - invar ian t a s f E / S O + C Q .

asymptofic behavior

The r e s u l t s i n f i g u r e 4 show that G(fE/u*o, This means t h a t a / s o becomes asymptot ical ly

The func t ion (16) contains t h i s

I

The combination of equations (13) through (15) y i e lds

Z/U*O u*o/ f L b / U * O -=I 0 + 2 * 7 7 ,0.001 j5j2 - 0.072 In (- 150 o ) ] ~ 0.001 yi3 (17) u*o L

According t o equation (17) c r / ~ ~ becomes l a rge a s f z /Qo fAz/Q0 + w.

and the i n t e r v a l & increases . increases . However, equation (17) p r e d i c t s t h a t O/WO < 0 a t s u f f i c i e n t l y la rge values of - q o / f L 0 . This i s nonsense and is an apparent shortcoming of the empir ical formula (17). Negative values of o/wo a r e predicted with (17) when

0 and This means t h a t cr becomes l a r g e a s the he ight E decreases

I n add i t ion , a/+, decreases as - U * ~ / ~ L ,

2.77

0.072 In (- u*?fl") - 1 f U Q O > 0.001

\ 1'" ,

provided -u*,/fL0 > 4 x lo8. very r a r e i f we exclude t h e region of the e a r t h on and about t he equator. A phys ica l ly r e a l i z a b l e upper bound value on -wo/fLo i n unstable a i r is probably on the order of 10,000 i f we exclude the equator ia l region of

HOwever, values of -QO/fLO > 4 x lo8 a r e

7

t h e ear th . w i t h equation (17) i s only apparent and would n o t a r i se i n practical s i t u a t i o n s . s t a n t va lue as -Mo/fLo becomes l a r g e and i f th is occurs i n the i n t e r v a l 1,000 < -Mo/fLo < 10,000, then the e r r o r i n the es t imate of o / m 0 r e s u l t i n g from (17) i s approximately 5 percent. p o s i t i v e values of -u*,,/fL0, + w . The func t ion H(uko/fL0) should approach a f i n i t e va lue as -Uzto/fL0 + O because of the f i n i t e na tu re of cr. Accordingly, H(wo/fLo) does no t have logarithmic behavior as - s o / f L 0 + 0 as predic ted by (15). The p r e c i s e va lue of -14ro/fLo where H(Qo/fLo) depa r t s from t h e behavior s ta ted by (15) can only be d e t e r - mined wi th measurements of c r / ~ , \ ~ assoc ia t ed w i t h va lues of -Qo/fL0 < 20. I n view of t hese comments, i t is recommended t h a t (17) be used t o ob ta in estimates of cr only f o r those s i t u a t i o n s i n which u*o/fLo is contained wi th in the experimental range of v a r i a t i o n i n t h i s r e p o r t , i . e . , 20 5 - u * ~ / ~ L ~ S 2,000. t i o n (17) f o r 2,000 < -14ro/fL0 < 10,000 which probably have e r r o r s equal t o 5 percent.

Thus, t h e shortcoming of p red ic t ing nega t ive values of U / Q ~

The func t ion H(u,,/fL0) probably tends t o approach a con-

As -u*o/fLo 4 0 from

Estimates of a/uJc0 can be obtained wi th equa-

The c o r r e l a t i o n c o e f f i c i e n t of the long i tud ina l wind f l u c t u a t i o n s a t z1 and z2 i s

where cru(z) is t h e s tandard dev ia t ion of t he long i tud ina l wind f luc tua - t i ons a t he igh t z. F i c h t l and McVehil [ 4 ] f i n d t h a t f o r t h e uns tab le boundary l aye r

1 . 9 ( ~ / 1 8 ) - ' * ~ ~ . OLl

u*O

- =

This r e s u l t w a s obtained by e f f e c t i v e l y averaging toge ther measurements of a u / k o obtained during var ious degrees of i n s t a b i l i t y i n the f i r s t 150 m of the atmospheric boundary l aye r a t Cape Kennedy, F lo r ida . How- ever, t h e r e c e n t r e s u l t s of Panofsky and Mirabel la [9] a p p e a r t o show t h a t i s r e l a t i v e l y i n s e n s i t i v e t o changes i n s t a b i l i t y and thus u*o/fLO. so t h a t we w i l l assume au(zl) = cru(z2) and s e t a u / ~ * o = 1.5. of cr/u,, occurs a t z 2 80 m.

Now, (20) shows that ou/uJc0 is a slowly varying func t ion of z

Thus, we may w r i t e (19) as This va lue

( d % c O > 2

4.5 R(u ' ( z ~ ) , u ' ( ~ 2 ) ) 1 -

8

We may conclude from (17) and (21) t h a t i n the uns tab le boundary layer the i n t e r l e v e l c o r r e l a t i o n c o e f f i c i e n t of t h e long i tud ina l v e l o c i t y f luc tua t ions increases as the i n s t a b i l i t y of the boundary l aye r and fZ/Q0 inc rease and as f&/u*o decreases .

1 V I . CONCLUDING COMMENTS

1. We have experimental ly analyzed t h e s tandard dev ia t ion of v e r t i c a l two-point l ong i tud ina l v e l o c i t y f l u c t u a t i o n d i f f e rences , and

u*o/fLo. I n p a r t i c u l a r , is propor t iona l t o ( fm/Q0) . Although t h e ana lys i s d id not inc lude b a r o c l i n i c e f f e c t s , we believe them t o be of second order . (&)2/3 i s , i n a l l l i ke l ihood , r e l a t e d t o t h e f a c t t h a t t he s t r u c t u r e func t ion of longi tudina l v e l o c i t y f l u c t u a t i o n s a t a given he ight is proport ional t o ( L ? x ) ~ / ~ f o r s u f f i c i e n t l y s m a l l values of t h e ho r i zon ta l space l ag Ax when an i n e r t i a l subrange is present . of t h i s r e l a t i o n s h i p is not obvious; however, i t w i l l be the s u b j e c t of f u t u r e i n v e s t i g a t i o n by t h e author .

have found t h a t a/%o is indeed a func t ion of f E / S o , f&/yp; and

The f a c t t h a t a2 is propor t iona l t o

The precise na tu re

2. This s tudy is t h e f i r s t p a r t of a fou r -pa r t stwdy t o de f ine the s t a t i s t i ca l p rope r t i e s of f l u c t u a t i n g wind shears i n the atmospheric boundary layer . The second p a r t of t he s tudy w i l l be concerned wi th determining the dependence of the t h i r d and f o u r t h moments, p3 and p4, on f & / G o , fz/t+o, and u*O/fLo i n uns tab le a i r . be concerned wi th determining a class of d i s t r i b u t i o n func t ions which con- t a i n four parameters t h a t w i l l adequately desc r ibe the observed wind shear d i s t r i b u t i o n func t ions i n uns tab le a i r . Prel iminary r e s u l t s appear t o show t h a t t h e Pearson system of d i s t r i b u t i o n func t ions can be used t o r ep resen t t he da ta . The four parameters which w i l l appear i n the d i s t r i b u t i o n func t ion w i l l be completely s p e c i f i e d upon spec i fy- ing the, f i r s t four moments of t he wind shear . w i l l be used i n parts two and t h r e e of t h e t o t a l s tudy. The f o u r t h par t w i l l be concerned wi th extending the r e s u l t s of parts one, two, and th ree t o (1) the n e u t r a l boundary l aye r w i th turbulence d a t a f o r r e l a t i v e l y high wind s p e e d condi t ions (G(18m) (2) s t a b l e boundary l aye r s a s soc ia t ed wi th night t ime condi t ions. Accordingly, t h e o v e r a l l goa l i s t o develop models of a, p3 and p4 such t h a t t hese parameters can be ca l cu la t ed f o r given values of t he he ight parameters f and &z once the ex te rna l parameters Lo, so, and f are spec i f i ed . Subs t i t u t ion of 0, p3, p4 and t h e mean wind shear (see sec t ion 111) i n t o t h e d i s t r i b u t i o n func t ion of wind shear w i l l f a c i l i - t a te the c a l c u l a t i o n of wind shea r f o r any p e r c e n t i l e level of occur- rence f o r a g iven s ta te of t he boundary l aye r . Furthermore, t h e model w i l l permit c l ima to log ica l s t u d i e s of wind shear w i th r a t h e r s t r a i g h t - forward s t a t i s t i ca l procedures f o r s p e c i f i e d loca t ions upon spec i fy ing

The t h i r d p a r t w i l l

The da ta i n t h i s r e p o r t

2 15 m sec’l, s ay ) and

9

the c l imato logica l s ta t is t ics of Lo and Q~ ( j o i n t d i s t r i b u t i o n funct ion) . The d i s t r t b u t i o n func t ion of these ex te rna l parameters could be ext rac ted from standard meteorological observat ions.

10

I I I I , I l l , I I I I I I , l [ - I I I I 1 1 1 , 1

20 4n

u.u , , , , , , , , I I

I I

I I I I . * ,

0.0001 0.001 0.01 0.1 f A z / u , ,

Figure 1. Experimental values of a/u,, for various categories

of f Z / S O and -u*o/fLo

11

I I

I 0 m = I 2-

* I 0

I

i \ I

\ I

i ) I

\

I I

I I

I I I

t r

9

w 0

12

c

cu h. P 0 /

9 / / I1

d

t 2

13

0 , * O " 0 \ IN d 'c

I I

I I

I I

I I

.c 7 0

I I

? 9 "! t r

h 0 m Fl I

.L

7

-9 0

0 * a \ IN rc

0 -k

5

u

w

Q) u Y M 1-i Erc

14

APPENDIX A

Calculat ions of t h e Scaling Parameters u.ko and Lo

I n t h e uns tab le Monin l aye r , t h e dimensionless mean flow shear is a universal func t ion of z/Lo, so t h a t

where { ( z ) is the mean wind speed a t he ight z , k, is von Karman's cons t an t with numerical value approximately equal 0.4, uk0 is t h e su r face f r i c t i o n ve loc i ty , and pu(z/L0) is a universa l func t ion of z/Lo. is the su r face Monin-Obukhov s t a b i l i t y length , namely,

The quan t i ty Lo

uzo c 'j 5 k1g Ho

- Lo - - . I n t h i s equation Ho is the s u r f a c e hea t f lux , 'j and 3 denote the mean flow densi ty and Kelvin temperature, g is t h e a c c e l e r a t i o n of g rav i ty , and Cp is t h e s p e c i f i c h e a t a t cons tan t pressure. The dimensionless shear plcl is r e l a t ed t o the f l u x Richardson number through t h e experimentally derived r e l a t i o n s h i p

Pu = (1 - 18Ri) l l4 , (A-3)

which is given i n re ference 3. The f l u x Richardson number is defined as

(A-4)

where 'e is the mean p o t e n t i a l temperature a t he ight z . number i s a func t ion of z/Lo. to r e l a t e R i t o z/Lo, so t h a t

The f l u x Richardson We shall invoke t h e Businger hypothesis [8]

R i = z/Lo. (A-5)

15

Upon combining equations (A-l), (A-3), and (A-5) and in t eg ra t ing the r e s u l t i n g r e l a t ionsh ip , we f ind t h a t

G ( z ) = - u*O f i n Z - q(c , 3)) L 0 kl 0

where

- z/Lo

We have used the condi t ion t h a t C ( z o ) = 0 i n the de r iva t ion of (A-6), where zo is the su r face roughness length. Equation (A-7) can be evalu- a ted numerically f o r any value of z/Lo. may be s e t equal t o zero because the con t r ibu t ion t o q from the region 0 < -z/Lo < -zo/Lo is neg l ig ib ly small,

The lower bound of t h i s i n t e g r a l

Equations (A-4) through (A-7) can be used t o c a l c u l a t e the s c a l i n g v e l o c i t y Q~ and Lo. The procedure f o r ca l cu la t ing these q u a n t i t i e s is as follows: (1) c a l c u l a t e the g rad ien t Richardson number, equation (A-4) , with the mean flow wind and p o t e n t i a l temperature p r o f i l e d a t a i n the Monin l aye r , (2) c a l c u l a t e Lo wi th equation (A-5), and f i n a l l y (3) ca l - c u l a t e uk0 with (A-6) and (A-7).

The 18- and 30-meter temperature and wind d a t a were used t o es t imate the Richardson number. 18- and 30-meter l eve l s were assumed t o be logari thmic p r o f i l e s . The expressions f o r G ( z ) and ; ( z ) were d i f f e r e n t i a t e d wi th r e spec t t o z and evaluated a t the geometric he ight z = 23 meters t o y i e ld est imates of &/az and a"e/az f o r t he c a l c u l a t i o n of R i . used t o c a l c u l a t e Lo i n s t e p (2) above. and $(18m/Lo) were used t o c a l c u l a t e uJco i n s t e p (3).

The d i s t r i b u t i o n s of l ( z ) and i ( z ) between the

This Richardson number was The 18-meter l e v e l wind speed

The su r face roughness lengths zo t h a t were assoc ia ted wi th the NASA 150-meter meteorological tower s i t e and t h a t were used i n the c a l c u l a t i o n of u*o a r e given i n re ference 4.

16

APPENDIX A

Calculat ions of t h e Scaling Parameters u.ko and Lo

I n the uns tab le Monin layer , t he dimensionless mean flow shear is a universal func t ion of z/Lo, so that

where E(z) I s the mean wind speed a t he ight z, k, is von Karman's cons t an t with numerical value approximately equal 0.4, Q, is the su r face f r i c t i o n ve loc i ty , and flu(z/L0) is a universa l func t ion of z/Lo. is the su r face Monin-Obukhov stability length , namely,

The quan t i ty Lo

I n t h i s equation Ho is t h e su r face hea t f l ux , 2 and ? denote the mean flow densi ty and Kelvin temperature, g is t h e acce le ra t ion of g rav i ty , and Cp is t h e s p e c i f i c h e a t a t cons tan t pressure. is r e l a t ed t o the f l u x Richardson number through t h e experimentally derived r e l a t i o n s h i p

The dimensionless shear flu

gu = (1 - 18Ri) l l4 , (A-3)

which is given i n re ference 3. The f l u x Richardson number is defined as

(A-4)

where 5 is the mean p o t e n t i a l temperature a t he ight z. number i s a func t ion of z/Lo. t o r e l a t e R i t o z/Lo, so t h a t

The f l u x Richardson W e shall invoke t h e Businger hypothesis [8]

R i = z/Lo. (A- 5 1

15

Upon combining equations (A-l), (A-3), and (A-5) and in t eg ra t ing the r e s u l t i n g r e l a t ionsh ip , we f ind t h a t

Z ( z ) = - u*o f i n Z - q(T , k)} L 0 kl 0

where

- z/Lo

We have used the condi t ion t h a t ii(zo) = 0 i n the de r iva t ion of (A-6), where zo is t h e su r face roughness length. Equation (A-7) can be evalu- a ted numerically f o r any value of z/Lo. may be s e t equal t o zero because the con t r ibu t ion t o q from the reg ion 0 < -z/Lo < -zo/Lo is neg l ig ib ly small.

The lower bound of t h i s i n t e g r a l

Equations (A-4) through (A-7) can be used t o c a l c u l a t e the s c a l i n g The procedure f o r ca l cu la t ing these q u a n t i t i e s is v e l o c i t y %o and Lo.

as follows: (1) c a l c u l a t e the g rad ien t Richardson number, equation ( A - 4 ) , with the mean flow wind and po ten t i a l temperature p r o f i l e d a t a i n the Monin l a y e r , (2) c a l c u l a t e Lo with equation (A-5), and f i n a l l y (3) ca l - c u l a t e u.ko wi th (A-6) and (A-7).

The 18- and 30-meter temperature and wind d a t a were used t o es t imate the Richardson number. 18- and 30-meter l eve l s were assumed t o be logari thmic p r o f i l e s . The expressions f o r G ( z ) and ; ( z ) were d i f f e r e n t i a t e d wi th r e spec t t o z and evaluated a t t h e geometric he ight z = 23 meters t o y i e ld est imates of &/az and used t o c a l c u l a t e Lo i n s t e p (2) above. and $(18m/Lo) were used t o c a l c u l a t e uJco i n s t e p (3).

The d i s t r i b u t i o n s of G ( z ) and G ( z ) between the

f o r the c a l c u l a t i o n of R i . This Richardson number was The 18-meter l e v e l wind speed

The su r face roughness lengths zo t h a t were assoc ia ted wi th the NASA 150-meter meteorological tower s i t e and t h a t were used i n the ca l cu la t ion of u.ko a r e given i n re ference 4 .

16

APPENDIX B

Wind Speed and Temperature P r o f i l e Data and Other Parameters

The wind p r o f i l e and temperature d a t a t h a t were used i n t h i s repart a r e given i n Tables B-1 and B-2. The 18-meter l e v e l wind d i r e c t i o n is tabulated here because the s u r f a c e roughness length a t the NASA 150-meter tower f a c i l i t y is a func t ion of wind d i r e c t i o n . The values of Ri(23 m), Lo, and udco t h a t were ca l cu la t ed wi th these d a t a a r e given i n Tabfe B-3.

TABLE B-1

Table of Temperature P r o f i l e Data*

(Temperature i n OF)

Case N o .

299

305

310

319

35 5

365

366

406

445

551

554

1/23/68

1/26/68

2/8/68

2/26/68

3120168

31 28/68

31 281 68

41 71 68

51 7/68

6/29/68

61301 68

1315- 1400

1130-1230

915-1015

1030-1130

1330-1430

905-932

1145- 1245

1507-1537

1400-1415

947-1027

933-952

* (AT), = T(18m) - T(3m).

= T(30) - T(3m).

76.0

52.4

40.5

59.0

75.0

73.5

74.5

79.0

78.2

79.8

83.1

-1.41

-2.19

- .32

-1.28

-1.82

-1.44

-2.63

-2.02

-2.29

- .93

-1.40

-1.73

-2.66

-1.82

-2.63

- 2 . 2 1

-2.42

-3.13

-2.57

-2.54

-1.90

-1.91

1 7

TABLE B-2

Table of Wind Speed and Direction Data

Case No.

299

305

310

319

355

365

366

406

445

55 1

554

18 m Wind Direct ion

219"

338 O

298"

320"

83 " 95 " 87"

71 " 93 " 34 " 86"

G(18 m) (m sec- l )

5.95

8.50

9.98

4.81

3.87

4.98

6.12

4.36

9.62

2.92

4.01

G(30 m) (m see-l)

~

7.55

9.31

11.34

5.13

4.13

5.25

6.48

4.73

10.34

3.05

4.12

TABLE B-3

Table of Boundary Layer Parameters

Zase No.

299

305

310

319

355

365

366

406

445

55 1

554

Ri(23 m)

-0.065

-0.090

-0.032

-2.448

-0.557

-2.234

-0.504

-0.546

-0.234

-8.977

-5.100

-357

-259

-716

- 9

- 42

- 10

- 46

- 43

- 99

- 3

- 5

uk0(m sec- l )

0.52

0.76

1.20

0.48

0.36

0.67

0.56

0.40

0.98

0.41

0.49

18

REFERENCES

1. Blackadar, A. K. e t al. , "Flux of Heat and Momentum i n the Planetary Boundary Layer of the Atmosphere," The Pennsylvania State&Universi ty Mineral Indus t r ies Experiment S ta t ion , Dept. of Meteorology, Report AFCRL Contract No. AF99604)-6641.

2. Pandolfo, J. P., "Wind and Temperature P ro f i l e s f o r Constant-Flux Boundary Layers i n Lapse Conditions w i t h a Variable Eddy Conductivity t o Eddy Viscosi ty Ratio," J. Atmos. Sci. , 23, 495-502, 1966.

3. Lumley, J. L. and H. A. Panofsky, The St ruc ture of Atmospheric Turbulence, In te rsc ience Publ ishers , John Wiley & Sons, New York, 239 pp . , 1964.

4. F i ch t l , G. H. and G. E. McVehil, "Longitudinal and Latera l Spectra of Turbulence i n the Atmospheric Boundary Layer a t the Kennedy Space Center," J. Appl . Meteor., - 9, 51-63, 1970.

5. Kaufman, J. W. and L. F. Keene, "NASA's 150-meter Meteorological Tower Located a t the Kennedy Space Center, Flor ida," NASA TM X-53699, 1968.

6. Blackadar, A. K. and H. Tennekes, "Asymptotic S imi l a r i t y i n Neutral Barotropic Planetary Boundary Layers," J. Atmos. Sci., 25, 1015-1020, 1968.

7. Langhaar, H. L., "Dimensional Analysis and Theory of Models ," John Wiley and Sons, New York, 166 pp., 1951.

8. Panofsky, H. A., "Spectra of t he Horizontal Wind Components," NASA Contractor Report NASA CR-1410, 69-83, 1969.

9. Panofsky, H. A. and V. Mirabella, "Estimation of Variances," t o be published as a NASA CR, 1970, 56-70.

19

APPROVAL

STANDARD DEVIATION OF VERTICAL W I N D SHEAR I N THE LOWER ATMOSPHERE

by George H. F i c h t l

The information i n t h i s r e p o r t has been reviewed f o r s e c u r i t y clas- s i f i c a t i o n . Review of any information concerning Department of Defense or Atomic Energy Commission programs has been made by the MSFC Securi ty C l a s s i f i c a t i o n Off icer . This r epor t , i n its e n t i r e t y , has been de ter - mined t o be unc la s s i f i ed .

This document has a l s o been reviewed and approved f o r technica l accuracy.

Atmospheric Dynamics Branch

A ~ , W ! R v . E. D. Ge i s s l e r Di rec tor , Aero-As trodynamics Laboratory

20

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D r . Shih

Dep t. of Geosciences Purdue Univ. LaFayette, Ind. 47907 Attn: D r , P h i l Smith

American Meteor ol og ical Socie ty 45 Beacon S t . Boston, Mass. 02108

Federal Aviat ion Agency (3) Washington, D. C. 20546

National Center f o r Atmospheric Res. Laboratory of Atmospheric Sc i , Boulder, Colorado 80302 Attn: Library (2)

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MSFC-RSA A18