Indian Journal of Radio & Space PhysicsVol. 12. February 1983. pp. 23-27

Height Distribution of Refractivity StructureConstant from Radiosonde Observations

P K PASRICHA, MAHENDRA MOHAN, D K TIWARI & B M REDDY

Radio Science Division, National Physical Laboratory, New Delhi 110012

Received 22 April 1982

Radio refractivity structure constant C; and its height distribution under varied meteorological conditions are evaluatedfrom routine radiosonde observations. The mixing length theory in conjunction with Tatarskii's theoretical formulationpermits C; evaluation from mean meteorological fields. The C; values so deduced are normalized to the existing turbulenceconditions. obtained through an empirical relationship between thermal atmospheric stability and a parameter thatcharacterizes turbulence.

1 IntroductionThe scattered power in a tropospheric-scatter

system depends on the spatial distribution of radiorefractive index fluctuations in the turbulentatmosphere. The spectral distribution of refractivityfluctuations <I>(K) is characterized by two parameters,viz. the spectral intensity in terms of Tatarskii'sstructure constant C; and the spectral slope m in thehigh frequency region of the spectrum. The latterdetermines the size of the turbulent eddies in thescattering medium. The tropospheric-scatter experi-ments usually make use of spectral wave number ofeddies in the inertial subrange of the turbulencespectrum. The effective size of the eddies responsiblefor the scattering is given by the first Fresnel zone at thereceiver, i.e. (AL)!/2, A being the observing wavelengthand L the radio path length. Tropospheric radio wavepropagation in the Svband and at higher frequencieswith path length L '" 100 krn, and jln outer scale ofturbulence characteristic of the inertial subrangeLo ~ 100m, yields (AL)!/2 < Lo. At UHF and lowerfrequencies", however, (AL )1/2 ~ Lo. A typical estimateof C; lies in the range 10- !4 to 10 -15 corresponding toa height interval of 1 to 2 km (Refs. 2-5). In theliterature, C; values have not been quoted in relationto the m values of the corresponding turbulencespectrum under varied meteorological conditions.

Under certain assumptions, model height profiles ofC; have been computed using the routine radiosonde(mean) profiles of temperature, humidity and wind.Gossard" makes use of the basic definition of structureconstant in conjunction with the mixing lengthconcept 7 to compute (up to 4 km) C; for different airmass types. Such (large) C; values appropriate to ascale greater than L ° are normalized to turbulence'sscale by making use of a model of the height variation

01 optical refractivity structure put forth by Hufnagel".The height distribution of C; so computed is comparedwith the ones deduced from the data obtained with theMillstone Hill radar observations. VanZandt et al.9

have attempted model computations of C; (in therange 4 to 16 km) from a theoretical formulation validunder conditions of homogeneous and isotropicturbulence introduced by Tatarskii ' o. Lo occurs in theformulation of C; explicitly and is taken to be ~ 10 m.With other parameters (to be described in Section 2)having values appropriate to turbulence conditions,the refractivity gradient at the different radiosondelevels is appropriate to mean meteorologicalparameters. The efforts in this model computationsha ve been toward estimation of a fraction of thescattering volume that is turbulent. Model C; valuesare found to be in reasonable agreement with the VHFDoppler radar measurements that are averaged overthe scattering volume and over 1 hr in time.

This paper presents model calculations of C; and itsheight distribution from mean meteorologicalparameters by combining the earlier two approaches.The concept of mixing length enables various eddydiffusion coefficients to be computed from mean profilegradients at the radiosonde levels. A scale of turbulenceof the order of mixing length (and regarded equal toLo) along with the computed coefficients permits C;evaluation from the theoretical formulation ofTatarskii. The empirical relationship of Gjessing etal.'! between the atmospheric stability (in terms ofVaisala-Brunt frequency) and spectral slope (aparameter that characterizes turbulence) is invoked toobtain distribution of C; for turbulence conditions atthree 'significant' radiosonde levels. The values socomputed are compared with the model calculations ofGossard".

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INDIAN j RADIO & SPACE PHYS. VOL. 12. FEBRUARY 1983

2 Methodology of Structure Constant EvaluationThe turbulence structure constant for the radio

refractivity under conditions of homogeneous andisotropic turbulence is given by9.IO

C~ = aiex' L~/3 M2 ... (1)

ai == Fa2, a2 being the universal constant, ex' a ratio ofcoefficients of eddy conductivity and eddy viscosity, Loouter scale of turbulence, M the vertical gradient ofrefractivity, and F signifies fraction of the region that isturbulent. M is given by

,\1= _77.6XIO-Op(1 +).?500CJ)T2 T

[dT ( 7800 dq)]

x dz + ra - 1 + (15500q/T) dz ... (2)

fJ being the atmospheric pressure, T the absolutetemperature, q the specific humidity (::::::mixing ratio e)and ra the adiabatic lapse rate. In the present approach,Lo is regarded to be of the order of mixing length.

The mixing length is obtained explicitly from thevertical turbulent diffusion of water vapour,represented in the form

rz ( 1 I)p,.(Z) = pv(Zo)exp Jzo - c, - H dZ

Pv(Zo) being the water vapour density at the referencelevel, Pv(Z) the density at the subsequent radiosondelevels, H the scale height of the dry air, and (v themixing length in the case of eddy diffusion of watervapour. This mixing length is regarded to beapproximately equal to (strictly greater than) themixing length in the case of eddy diffusion ofmomentum. With the coefficient of eddy diffusivitydefined by K; = W ('" the vertical eddy flux of watervapour is given by

dqF,,= -pKvdz ... (4)

... (3)

p being the density of air and W the mean wind speed.Water vapour distribution significantly determines thevertical eddy flux of sensible heat in air. The effect isbrought about through the dependence of the specificheat of air on specific humidity, and through watervapour adding to air density and buoyancy'{. In anearly dry atmosphere (conditions normally prevailingabove ~ 1 krn), the moisture gradient determines thesensible heat flux \3, and the coefficient of eddyconductivity KH may be described in terms of the eddyflux of the water vapour as

KH = r~.?T\JL"F"PLp\d~ )... (5)

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L" being the latent heat of evaporation and Cp thespecific heat at constant pressure. The coefficient ofeddy viscosity (momentum) KM is defined in terms ofouter scale of turbulence as,

zldwlKM=Lo dz(J.' in Eq. (I) is ten given by the ratio KH/KM•

For turbulence conditions, the parameters aZ and ex'may be taken to be 2.8 (Ref. 14) and I respectively. Theconstant af, as introduced in Eq. (I), to be inferred frommean meteorological parameters, is expected to be lessthan I (Ref. 10). The constant ai in the presentapproach is the normalization factor to convert C~values, obtained from Eq. (I) with mean meteorologi-cal parameters, to turbulence conditions. Theestimation of C; for turbulence conditions associatedwith a given spatial averaging (and therefore staticstability) must invoke additional assumptions. Astraightforward approach is to look for an empiricalrelationship between static stability and a parameterthat characterises turbulence. Gjessing et at. II havedeveloped an empirical correlaiton between Vaisala-Brunt frequency v1 (a measure of thermal atmosphericstability) and the slope of the refractivity spectrum. Thebeam-swinging scatter experiment measures thea verage value of the slope of the refractivity spectrum<1>(K) in the height interval 2 to II km. The atmosphericstability parameter y2 is obtained from thesimultaneous radiosonde sounding observations givenby

y2 = .f..(dT + 0.055 T/ + 1_)T dz Cp

dT [d: being the temperature gradient per k ilornetre, gthe acceleration due to gravity; T and d'F[dz are theaverage values over I to 2 km height interval for thesignificant level' I' ,2 to 3 km for level '2' and 3 to 4.5 kmfor level '3'; T/ is the initial temperature of each of thethree significant levels. Actually, y2 should have beenaveraged over the entire height range. The parameterv2 was computed for three significant levels to ascertainthe validity of Eq. (7) for height interval less than the'prescribed one in the original formulaiton. Anapproximately equal value of y2 obtained at threelevels approves the sub-division of the height intervalat the significant levels. Therefore, the characteristics ofturbulence at the three significant levels shall remainthe same in the present approach. The spectral slope mcorresponding to a v2 value is given by the empiricalrelation

m= -40.6+7.08v2 ... (8)

From the set of data points presented by Gjessing etal.11 m varies from - 5/3 to - 22/3 corresponding todTid: variation from -8 to -5'Ckm-l. The lower

... (6)

... (7)

PASRICHA et al.: HEIGHT DISTRIBUTION OF REFRACTIVITY STRUCTURE CONSTANT

value of lapse rate may well set the limit for turbulenceoccurrence. The high spectrum slope values are likelyto be due to the presence of thin stratified layers in themedium. The parameter C~ associated with layeredstructures is expected to be of greater magnitudecompared to that of turbulent eddies. This is due to thecoherent nature of scattering (partial reflection) fromstratified layers compared to incoherent scattering inthe other situation. For turbulence conditions, fromthe little experimental data on C~, no one-to-onecorrespondence of C~ and m is found. In the absence ofsucha relationship, a C~ valueof5 x 10-15/1/-2,3 at 1.5km is taken to compute the normalizing constant ai,irrespective of the spectral slope value. It may be notedthat C~ so deduced for a Iml value greater than 11/3 (theKolmogorov's spectral slope) may reflect the presenceof stratified structures such that the computed C; valueis more than the actual value.

3 Model Calculation of Structure ConstantModel structure constant calculations have been

carried out, using the monthly mean fields oftemperature, humitity and wind at 0530 and 1730 hrsLT, for various sea,sons, for the coastal stationTrivandrum.

For the computations presented in this paper, in thelower troposphere (upto - 4.5 km), humidity and(indirectly) wind speed predominantly contribute tostructure constant. The smoothed height profiles ofmixing ratio of water vapour for the seasons, viz.premonsoon, monsoon, post-monsoon and winter, arepresented in Fig. 1. The actually observed resultantmean wind speed profiles for the correspondingseasons are also depicted in Fig. I. The mixing ratio forthe post-monsoon is higher compared to pre monson,

es

6

MIXING RATIOPOlloi

WIND SPEEDo P Ilol

5

4

2 1730hrs LTTRIVANORUM

p- PREMOSOONIl0l- MONSOONo- POST·MONSOONw- WINTER

OL- ~ ~ __~~ __~~~ __~5 10 15 20

MIXING RATIolg kg")

Fig. I-Smoothed vertical profiles of mixing ratio of water vapour at1730 hrs LT

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but reverse is the case for mean wind speed. Thus. theeddy water vapour flux is higher for the premonsooncompared to post-monsoon. The apparently in-significant differences in slopes of the mixing ratiodistributions, in fact, lead to substantial differences inmixing length distributions obtained from relation (3).In Eq. (3), the scale height of air is taken to be 6.5 km.The reference level IS the radiosonde levelcorresponding to a barometric pressure of 1000 mbar.The vertical profiles of mixing length so deduced arepresented in Fig. 2. The profiles of coefficient of eddyconductivity computed using the profiles of mixinglength and wind speed are also shown in Fig. 2. Inwinter, a decrease in the mixing length, overwhelmedby a decrease in the temperature gradient, results in asharp increase in the coefficient of eddy conductivity.At a given height, the mixing length is the largest for themonsoon season and the least for the winter season; thepost-monsoon mixing length being greater than thepremonsoon one. The mixing length is a measure of theefficiency of the turbulent diffusion; it increases withincreased mixing length. The enhanced turbulentdiffusion for the monsoon, aided by large windgradient, results in profound mixing of the atmosphericmedium. In general, turbulent dilTusion decreases withincreasing distance from the surface of the earth.Though eddy diffusion of water vapour is slower thanthe eddy diffusion of momentum, the (large) mixinglength for water vapour, obtained from Eq. (3), isregarded to represent eddy diffusion of momentum.The coefficient of eddy viscosity depends on the mixinglength and the vertical shear of mean wind speed[relation (6)]. The coefficient of eddy conductivity, onthe other hand, depends on the mean wind speed[relation (5)]. Thus the coelTicient of eddy conductivityis higher in the premonsoon than in the post-monsoonseason. This is to be compared with the fact that thetemperature gradient at different radiosonde levels ishigher in the prernonsoon than in the post-monsoonseason. The apparent increase of coefficient ofconductivity with height, for the winter season, in Fig.2, is due to a sharp fall in eddy diffusion with height. Itmay be pointed out that the temperature gradient is the

~-----------------------,.

17)0 ~,~ ,TTRlIJANDRUM

,1!

C~..L' .LI . --'--_--L_~~I --'-'-~I~III~j .~....LJ.........!

'0

Fig. 2-· Vertical profiles of mixing length corresponding 10 themixing ratio profiles of Fig. I.

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INDIAN J RADIO & SPACE PHYS, VOL. 12, FEBRUARY 1983

Table I-Model Structure Constant Calculations with ai = 10-4 for Trivandrum

Season Level 0530 hrs LT 1730hrsLT A model C;------,-,,--- variation

:x C; (m -23) m :x C; (m -13) m of Gossard"(xIO-") (x 3) (xlO-15) ( x3)

Pre monsoon I 1.2 7 -16 0.9 4 -IS Level I:

2 0.7 2 -IS 2.3 4 -- \3 5x10-15

3 0.6 0.4 -IS 2.6 2 -IS

Monsoon I 0.8 6 -15 1.5 5 -13 Level 2:

2 1.5 5 -19 2.5 6 -17 3X\O-1~

3 0.3 2 -\7 0.7 3 -IS

Post-monsoon I 0.6 3 -14 1.0 3 -14 Level 3:

2 U U U 0.7 2 -15 I x 10-"

3 U U U 0.2 0.2 --18

Winter 1 0.7 3 -16 0.3 I -9

2 U U U U U U3 U U U U U U

Levell: 1-1.5 km2: 2.5-3 km3: 4-4.5 km

U: Unresolved

highest in the winter. Due to enhanced turbulentdiffusion in the monsoon, the conductivity is thegreatest. This is contrary to the fact that thetemperature gradient is higher in the pre monsoon thanin the monsoon. The coefficient ~: in Eq. (I) is the ratioKH/KM. The values of a', at the two local times, and indifferent seasons are given in Table I. The c~computations carried out, using Eq. (I), with vi = 10-4

are presented in Table I. The reasons for choosing avalue of vi as 10-4 are described below. The spectralslope values at significant levels obtained fromrelations (7) and (8) are also presented in Table I. Asvisualized from Table I, the spectral slope at varioussignificant levels does not show marked variation. Thissuggests that turbulence associated with differentsignificant levels retains roughly the same characteris-tics. The spectral slope in the nighttime is greater thanin the daytime, since nighttime is marked by stratifiedlayered structures compared to turbulent eddies in thedaytime. The spectral slope for the nighttime-firstsignificant level is not large enough, since the averagingheight interval in Eqs (7) and (8) extends from r km andnot from ground. Further, the spectral slope is higherin the monsoon and post-monsoon than in thepremonsoon season or winter. The reason is that thetemperature gradient in the former seasons are lessthan in the latter ones. Thus increase in spectral slopemay be correlated with extended stability of theatmospheric medium, as presented empirically byGjessing t't al. In the absence of a systematicrelationship between C;, and spectral slope, thenormalization of the computed C~ is performed with atypical turbulence - C;, value of S x 10 - 15 m - 2 3 at aheight of I.S k m leading to ai:::;, 10-4.

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4 DiscussionThe decrease of C;, with height is evident in Table I.

The decrease is, however, more pronounced at the levelof 4-4.S krn. The model calculations of Gossard" showlarge variation of C;, about its mean value in a given airmass as well as variation from one air mass type toanother. A rather systematic decrease of C; taken fromthese calculations is included in Table I. A large scatterof C; from tropospheric scatter observations at aheight of I.S km lies in the range of 1.4 x 10- 13 to 4.4x 10- 16 m - 213 (Ref. IS). Vertical profile of C; with thePoker Flat phased array radar in Alaska shows adecrease of C; by an order of magnitude in the heightinterval of 2.5 to 4.5 km. In Table I, C; is higher, at thefirst significant level, in the nighttime than in thedaytime. At higher levels, however, C; is higher in thedaytime compared to nighttime value. This is to beexpected, since nighttime turbulence is marked bylayered structures and C; associated with such strata ishigher that that associated with turbulent eddies!".A higher C~ value at the first significant level in thenighttime compared to daytime is consistent with thestatistics of C;, obtained over a period of I yr inColorado at a height of 805 m (Ref. 4). From the limiteddata analyzed for a single coastal station, the heightvariation of C~ is more pronounced In theprernonsoon, post-monsoon and winter, in that order.The decrease is particularly enhanced at nighttime.This aspect brings out the limitations of themethodology adopted in this paper. The mixing ratioprofiles cannot be resolved for the scale heightcomputations (marked U-Unresolved in Table I) athigher levels corresponding to C; values of:::: 10-16m -2,3, a typical value at a height of 5 km. The

PASRICHA et al.: HEIGHT DISTRIBUTION OF REFRACTIVITY STRUCTURE CONSTANT

decrease of C; in the monsoon is not rapid, sinceturbulent diffusion is maintained at a steadymagnitude throughout the height interval.

In the methodology adopted in this paper, the choiceof af is rather oversimplified. The dependence of af onturbulence conditions (which vary from station tostation) may be visualized through its dependence onthe size of eddies characteristic of turbulence. Thewavelength of observations determines the effectivesize of eddies responsible for scattering. Thus af varieswith the different probing wavelengths. A relationshipbetween spectral slope and C~ shall enable appropriateC; be given for different turbulence conditions.

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