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Statistical analysis of ionospheric winds—II
JournalofAtmosphericand TerremtrialPhysics,1968,~rol.SO.pp. 907-917. PergamonPress. Printedin NorthernIreland : ~.:
Statistical analysis of ionospheric winds--H NORMAN W. ROSENBERG Air Force Cambridge Research Laboratories, Bedford, Massachusetts (Received 11 November 1967) Abstract--A statistical analysis of seventy midlatitude wind profiles between 90 and 150 km
altitude shows an rms wind speed ranging from 45 to 70 m/see and an rms shear ranging from 100 to 160 m/sec/scale ]at, (equivalent to 0.004-0.02 see-l). I f a scale-height normalized altitude is used, any vertical zonal or meridional profile can be approximated by an average, trend, and a dominant sine wave with a mean residual variance of 15 per cent. The mean wavelength of the dominant wave is 3.4 scale heights, and the mean energy content of the profile is 2J/g. Semidiurnal replacement of viscosity-dissipated energy is compatible with molecular viscosity between 100 and 120 km, but requires a 20-fold higher turbulent viscosity at lower altitudes. Both the mean Richardson and Reynolds numbers predict transition from turbulent to laminar flow between 110 and 115 kin.
INTRODUCTIOI~ ALMOST 200 night-time and twilight ionospheric wind profileshave been reported from chemical trailsby various investigatorssince 1960. About 70 of these profiles obtained at midlatitudesin the altituderegion 90 to 150 kin, have been used in the p r e s e n t analysis. Most of these were o b t a i n e d d u r i n g t h e y e a r s 1962-1966, r e a s o n a b l y well d i s t r i b u t e d in t i m e a n d season. Statistically-significant wind circulation p a t t e r n s h a v e been f o u n d in this g r o u p o f profiles a n d r e c e n t l y r e p o r t e d elsewhere (ROSENBERG, 1968). T h e p r e s e n t r e p o r t examines t w o wind p a r a m e t e r s , velocity, a n d shear, which are r e l a t i v e l y c o n s t a n t w h e n a p p r o p r i a t e l y averaged. More specifically, r m s shear p e r scale h e i g h t a n d rms t o t a l v e l o c i t y are f o u n d to be a l m o s t i n d e p e n d e n t of altitude. I n this paper, we discuss t h e statistical t r e a t m e n t of the original values, a n d resulting m e a n values of t h e t w o p a r a m e t e r s , which t h e n lead to m e a n values of c o n t r i b u t i n g waves, t u r b u l e n c e criteria, a n d e n e r g y balances.
EXPERIMENTAL DATA T h e original wind profiles were r e p o r t e d (BEDINGER e~ a~., 1966) a t vertical i n t e r vals o f a b o u t 1 k m a t altitudes below 110 l~m, a n d a t 2 t o 5 l~m intervals a t h i g h e r altitudes, w i t h t y p i c a l e x p e r i m e n t a l errors of 3 m/see in b o t h N S a n d E W v e l o c i t y c o m p o n e n t s . T o p r o v i d e i n t e r p o l a t e d values a t a n y desired a l t i t u d e spacing w i t h o u t smoothing, t h a t is, while m a i n t a i n i n g original wind values a t r e p o r t e d altitudes, we h a v e fitted e v e r y f o u r successive d a t a points in each o f t h e profiles to a cubic e q u a t i o n . S i x t y m i d ] a t i t u d e v e l o c i t y profiles h a v e b e e n s u p e r i m p o s e d b y a c o m p u t e r controlled m e c h a n i c a l p l o t t e r in Figs. 1-4. F i g u r e 1 presents N S a n d E W c o m p o n e n t profiles, f r o m which it is seen t h a t a l t h o u g h each profile has a n oscillating p a t t e r n , t h e w a v e l e n g t h s increase w i t h h e i g h t a n d t h e phases are r a n d o m l y d i s t r i b u t e d . F i g u r e 2 shows t h e a m p l i t u d e of t h e t o t a l v e l o c i t y v e c t o r , on a logarithmic scale 9o7
908
N.W.
ROS~-NBERO
180 160 140 .._1
,,::I:
120 I00
8_2000_
0
200 -200 WIND SPEED, M/S
0
200
Fig. 1. Wind component velocity vs. altitude.
vs, altitude, and it is seen that this parameter is restricted in its range of values, with a slight increase in average value with increasing altitude. The total shear per km (SKM) is defined as the vector connecting total velocity vectors at adjacent altitudes separated b y 1 kin. Figure 3 shows the amplitude of the total shear on a logarithmic scale vs. altitude; it is seen t h a t for the 60 trails graphed, this parameter also has a limited range of values, at any given altitude, although the mean shear value decreases markedly with increasing altitude. I t has previously been reported (I~osENBERG et al., 1966) t h a t the dominant oscillations in the NS and E W component profiles have wavelengths proportional to the scale height of the local pressure H~. Therefore, shears were also computed with respect to a scale-height normalized altitude, ZSH = S~ dZ/H~. The shear per scale height, Ss~, was computed b y multiplying shear per kilometer b y the local scale height in km: SSH ---- SK~ " H~. The scale-height values were taken from the 1962 U.S. Standard Atmosphere with no seasonal or time con'ections. The values of SSH are shown in Fig. 4, where it is seen that its mean value is practically heightindependent. AYERAGII~G PROCESS
The root mean square values of velocity and shear for 70 midlatitude profiles were obtained b y the following successive steps:
Statistical analysis of ionospheric winds--II
909
160
140 v UJ C~
D
I-I-d
120
100
80
I
I0 VELOCITY,
IO0 M/S
I000
Fig. 2. Total wind speed vs. altitude. (]) For each altitude Z of each profile, the magnitudes of velocity and shear were computed from equations (1) and (2). V 2f,z Siz
= [(V~,z+az -
=
° 2 VIv.z 4- V~,z
VNz) 2 + ( / ~ . z + a z -
(I)
V~.z)2]/d Z~
(2)
where V r . z and ST.Z are total velocity and total shear at height Z, and the subscripts N and E indicates NS and E W component velocities, also at height Z. The shear was taken for a height difference dZ in km equal to 0.2 scale heights. (2) Within each profile, r.m.s. VT. z and S t . z at each altitude were computed b y 2 2 z averaging the values of VT, z and ST, over an altitude-averaging interval in ]~m equal to 1.3 scale heights and taking the roots of the mean squares. (3) The logarithms of the r.m.s. VT, z and ST. z at each altitude were averaged over all profiles. N o t all profiles extended to the same lowest or highest altitude, so
910
N. ~V. ROSENBERG
160 1"
140 C~ D
120
5 100
80 I
I0
I00
SHEAR,CM/SVKM
Fig. 3. W i n d shear, m/sec/km vs. altitude.
the number of samples for which data was available increased from 20 profiles at 88 l~m to 70 profiles between 100 and 115 kin, and then decreased to 20 profiles at 150 kin. Logarithmic averaging was used to give a statistically normal distribution of values. This led to logarithmic standard deviations SD of 0.2 log units (45 per cent) both for velocity and for shear, practically independent of altitude. The standard error of the mean (SEM -- SD/(N -- 1)1/~) ranged from 5 per cent between 100 and 115 kin, where N = 70 independent samples, to 10 per cent at the altitudes (88 km and 150 km) where the sample count N decreased to 20. Therefore, differences between mean values which m a y be considered significant (at a 95 per cent confidence level) range from 10 per cent in the central region to 20 per cent at the top and bottom. The differencing interval mentioned in Step 1 and the averaging interval mentioned in Step 2 were selected and tested to assure t h a t the reported mean values were insensitive to these two details of the averaging process. The mean values of velocity, shear per kin, and shear per scale height are shown in Fig. 5 vs. altitude.
Statistical analysis of ionospheric winds--II
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160 140 120 a kin k._J
II0
I00
90 I
I0 I00 S H EAR,(M/S)/SH
I000
Fig. 4. Wind shear m/sec/SH vs. altitude. Seasonal segregation disclosed no significant differences at any altitude. I t is emphasized that calculations were carried out in meters/second and kilometers, with no assumptions regarding other atmospheric parameters, so that season-time uncertainties of these other parameters were not introduced. WAVELENGTH DETERMINATION I f the NS and E W components were pure sine waves with the same wavelength L and amplitude and with random phase, a number of independent samples at a given height would have an r.m.s, total velocity of V and an r.m.s, total shear S = 2~rV/L. I f the values of V and S were relatively constant over a wavelength, this equation could provide an estimate of the dominant wavelength from averaged estimates of V and S. For example, over the 100-150 km altitude region, the relatively height-independent r.m.s, velocity of 70 m/sec and shear per scale height of 120 m/sec/SH give L = 2~rF/S= 2~r(70/120) = 3-5 scale heights. Figure 5 shows the wavelength in scale heights L s s computed from the rms values of total velocity and shear at each altitude inserted into this equation. I t is emphasized that this method
912
N . W . ROS~.~ERG
I
I
I
140
120
LSH
VT~ SSH
SEM SD I--
I 0 0 tI 2
~SKM
I
I 5
I I0
/I 20
,4 50
/1 I I00 200
Fig. 5. Mean wind speed (Vr), mean wind shear per km (S~M), and mean wind shear per scale height (Ssa). The relative standard deviations of all throe parameters are about equal at 4-45 per cent and are shown by the error bar SD. The relative standard errors of the means are also equal, varying from 5 per cent in the center to 10 per cent at the top and bottom. The error bar SEM shows a 10 per cent standard deviation. The dominant wavelengths Lsm estimated as 2~rVr/SsH is also shown. d e p e n d s on a v e r a g i n g o v e r r a n d o m phase, a p p r o p r i a t e in this case b e c a u s e of t h e large n u m b e r of i n d e p e n d e n t samples. I t also d e p e n d s on m o s t of t h e e n e r g y b e i n g p r e s e n t in a single w a v e l e n g t h . I f this e s t i m a t e of w a v e l e n g t h is correct, a profile e x t e n d i n g f r o m 90 to 150 k m ( a b o u t 5.5 scale heights) c o n t a i n s o n l y 1-5 cycles of t h e d o m i n a n t w a v e l e n g t h . On such a s h o r t s a m p l e , s p e c t r a l analysis or F o u r i e r a n a l y s i s is n o t reliable for finding a d o m i n a n t w a v e w i t h precision. A n i n d e p e n d e n t e s t i m a t e of t h e " d o m i n a n t " w a v e l e n g t h w a s carried o u t for each c o m p o n e n t of e a c h profile b y finding t h e " b e s t f i t " single sine w a v e a f t e r r e m o v a l of a v e r a g e a n d t r e n d . This w a s done b y successive a p p r o x i m a t i o n s , v a r y i n g w a v e l e n g t h , phase, a n d a m p l i t u d e of a trial sine w a v e to m i n i m i z e t h e residual v a r i a n c e b e t w e e n it a n d t h e original d a t a . T h i s a n a l y s i s s h o w e d no significant differences b e t w e e n a v e r a g e N S a n d E W v a l u e s or t h e i r s t a n d a r d d e v i a t i o n s , so d a t a f r o m b o t h c o m p o n e n t s were p o o l e d to decrease t h e s t a n d a r d error of t h e m e a n . W i t h 140 samples, t h e s t a n d a r d e r r o r of t h e m e a n SEM, was o n l y 1/12 of t h e s t a n d a r d d e v i a t i o n SD.
Statistical analysis of ionospheric w i n d s - - I I
913
The mean best-fit wavelength was 3-3 scale heights with SD ---- 40 per cent, SEM ~- 3 per cent. Note that this estimate of wavelength, b y an independent method, agrees well with the 3-5 scale heights estimated from average velocity and shear. The mean r.m.s, fluctuation of the original profile was 47 m/see (SD -~ 34 per cent, SEM -~ 3 per cent), reduced b y the removal of trend and best-fit single wave to 18 m/sec. This is equivalent to a reduction in variance to (18/47) 2 --~ 15 per cent. In other words, 85 per cent of the mean variance of a profile can be accounted for b y a trend line plus a single dominant sine wave (with a scale-height proportional wavelength) TURBULENCE
The existence of turbulence in the wind structure below 110 km has been extensively studied and recently reviewed b y BLAI~IONT et al., 1967. I t seemed possible that the present study might allow a choice between two sources of turbulent instability. Two dimensionless parameters have been widely used in predicting turbulence. The first, the Reynolds number, is defined as LcV/.KV where Lc is a critical length, V the velocity, a n d / ~ V the kinematic viscosity. A high velocity V between two low-velocity boundaries separated b y a distance 2Lc (walls in the laboratory, and possibly velocity minima in the upper atmosphere) can cause an isotropic medium to develop a turbulent flow pattern. This cannot occur unless the Reynolds number exceeds some critical value, about 2000 in laboratory studies, although turbulence need not set in at that point. The mean Reynolds number is presented vs altitude in Fig. 6, based on the average velocity and a characteristic length Lc equal to a quarter of the mean dominant wavelength Ls~. Blamont reports a value of 600 as the criterion for appearance of turbulence, which he found to occur near 105 km in selected profiles (BLAMO~T et al., 1967). He used a shorter Lc, however, based on small scale fluctuations, so the discrepancy between the two estimates of Reynolds number is really an uncertainty in the correct choice of a characteristic length Lc. An alternative source of turbulence is that induced b y high shears in a nonisotropic medium. In the absence of shears, a gas whose temperature increases with height is generally stable. Excessive shear forces can, however, overcome the stabilizing forces of the temperature gradient, and lead to a nonisotropic turbulence in the medium. This stability ratio is described b y the Richardson number, which has been successfully used in predicting turbulence in the lower atmosphere. The Richardson number (Nm) is defined b y [(G/T)(dT/dZ ÷ TG]H)]/Ss 2 where g is the gravitational constant, T the temperature, dT/dZ the temperature gradient, G the specific heat ratio, H the local scale height, and Ss the shear in see -1 ( = SK~/1000). (Because Nm involves the square of the shears, the standard deviation and standard error of the mean are doubled, as shown in Fig. 6.) I f the Richardson number exceeds a critical value near unity, turbulence cannot exist. The bracketed term (Cm) m a y be regarded as describing the stabilizing forces; countered b y shear, which describes the destabilizing forces. The bracketed term (Cm), as shown in Fig. 6, is relatively constant to 120 km where it starts to decrease. The mean Richardson number has a value below 110 km near unity; that is, the atmosphere below 110 km is unstable. Above 110 km, shear per km decreases rapidly and at an altitude of 17
914
N.W.
I
ROSENBERG
I
I
I
,4o l-
SEM ReXlO -5
SD i ,o
-
NRi
m
I-._1
CRiXlO 4
I00 0.2
II
I
"K
I
1
I
0.5
I
2
5
I0
20
Fig. 6. Mean Reynolds number (NRe), moan Richardsons coefficien~ (Cm), and mean Riehardsons number (~JRI) VS. altitude. 112 km N m reaches 2, and at 114 km it reaches 5. Hence, the Richardson criterion would predict transition from unstable to stable flow in this height region. The mean Reynolds number, however, goes through 2000 (the accepted laboratory critical number) at 112 kin, and thus also predicts the transition from unstable to stable medium near this height. Some photographs show turbulence in regions of high velocity, b u t others show turbulence in regions of low velocity and high shear. I t is not possible to choose between the two criteria on the basis of this data. /~VERAGE ENERGY CONTENT AND DISSIPATION
We next consider the energy content and power dissipation represented b y the wind. Since energy content depends on V~ and power dissipation on S 2, the un"certainties--and hence standard errors of the m e a n - - a r e doubled because the squares of the experimental observables are used. This means that larger differences (20 per cent at the center, 40 per cent at the top and bottom) must occur before t h e y can be considered meaningful. The energy content per unit mass in J / k g is given b y V~/2. Since there is only a slight variation in V with height, the average value over all altitudes is 2000 J / k g :h 25 :per cent (Fig. 7). The 25 per cent deviation results largely from the 40 per cent increase in mean velocity between 90 and 100 km altitude. The mean viscous energy dissipation due to wind shear Ss expressed in sec -1 is
Statistical analysis of ionospheric winds--II
~
140
1
I
I
I/
915
I
I
EC J/KGxlO -3 ~.. 120
5
,,::[
I00
W/KGx I02
fl O.
0.2
I 0.5
1,," I
I 2
I -"1" 5 I0
I 20
Fig. 7. Mean energy content (EC), mean viscous dissipation rate (V/)), and mean time constant (TDx) vs. altitude. given in W/kg b y K V • Ss ~, where K V is the kinematic viscosity. I f the molecular viscosity is used for K V , the mean power dissipation ranges from 10 -a W]kg at 90 km to 200 × 10 -a W/kg at 150 km (Fig. 7). At altitudes above 110 kin, this viscosity is probably appropriate, and in this region our mean dissipation is within a factor of two of earlier estimates made b y H~r~.s (1967) from more limited data of Koc~s~ (1964). However, at altitudes below 110 lrm, the turbulent viscosity is more appropriate, and the actual dissipation is greatly above that shown in Fig. 7. Br.AMO~T (1967) estimates a turbulent diffusion coefficient 50 times higher than molecular, from which we might infer a similar ratio of viscosities. The ratio of energy content to energy loss rate represents a time over which the energy must be replenished ff both velocity and shear values are to be maintained constant. With molecular viscosities, the values of the time constant (shown in Fig. 7) range from 10 days at 90 km to 1 day at 103 k m to 4 hr at 150 kin. Between 105 and 125 km, the average value of ½ d a y within the statistical uncertainty is in agreement with a largely semidiurnal source. At lower altitudes, the loss rate is seriously underestimated b y using molecular viscosity. We might turn the problem around to estimate t h a t for a time constant of about a haft day, the "effective" viscosity must be about 20 times the molecular viscosity which is in reasonable agreement with Blamont's estimate (50 fold). Above 125 km, a higher rate of replacement of energy (than semidiurnal) is apparently demanded. On the other, hand, the variation in temperature and density necessary to bring the viscosity down
916
N . W . ROSENBERG
b y a factor of 3 (which would make the whole non-turbulent profile semidiurnal) is within the estimated variability of the U.S. Standard Atmosphere in this region. Between 120 and 150 km, there have been far fewer measurements of temperature and density than of wind, and their statistical reliability is lower than those of the winds and shears. NON-DEPENDENCE OF VELOCITY AND SHEAR ON ALTITUDE Energy injected from below in a transverse wave at a given wavelength will tend to increase the wave amplitude with increasing altitude to keep the energy per unit volume constant in a region of decreasing density. However, as the wave amplitude grows, so does its m a x i m u m shear. The increasing amplitude and/or shear creates a turbulent instability, causing most of the energy that would further increase the amplitude in the 90 to 105 k m region to drain off into turbulent viscous dissipation. Without turbulence, the amplitude would grow severalfold between 90 and 105 k m to offset the density decay. In contrast to the mean stability, the point-to-point atmospheric stability,can fluctuate from stable to unstable over a half wavelength of 1.7 scale heights = 8 km, so that layers of turbulent and laminar regions can alternate as reported by Blamont (1964). Above 110-120 kin, the atmospheric stability will not permit existence of turbulence. As the energy propagates to higher altitudes, further viscous loss in the wave can occur in either of two ways: by decreasing in amplitude or by increasing in wavelength. Consider an upward flow in a wave consistingof several components of differentwavelength. The higher-frequency components have a higher rate of shear loss,hence willhave dissipated to a greater extent at a given height altitude, and the apparent effective wavelength will increase. Our experimental observations show that there is no significantloss in mean amplitude but a continued increase in mean wavelength. The time constant for energy replacement ranges from a day to a few hours, but a moderate change in the viscosity estimate (which was estimated from a standard atmosphere which itselfhas a m a x i m u m uncertainty in the height region 120 to 150 kin) could bring the time constant to a half day throughout this region, implying a semidiurnal dominant mode. CONCLUSION A statistical analysis of 70 midiatitude and wind profiles at altitudes between 90 and 150 km leads to the following conclusions: (1) The rms total velocity increases from 45 m/sec ( ± 1 0 % ) at 90 km to an altitude-independent value of 70 m/sec (~-5-10%) above 100 km. (2) The rms total shear (in seconds -1) decreases from an altitude-independent value of 0.02 sec -1 (~:5-10 per cent) below 110 km to 0.004 sec -1 ± 10 per cent at 150 kin. (3) The rms total shear (in m/sec/scale height) is nearly independent of altitude. I t ranges from 100 m/sec/SH ± 10 per cent at 90 km to 160 m/sec/SH ± 5 per cent at 105 km to 120 m/sec/SH ~: 10 per cent above 115 km. (4) There is no significant seasonal dependence of these parameters. (5) The vertical profile of NS and E W components vs. a scale-height normalized altitude can be approximated b y average, trend, and best-fit sine waves. When this
Statistical analysis of ionospheric winds--II
917
is done, t h e m e a n f l u c t u a t i o n of t h e original d a t a , 47 m/sec, is r e d u c e d to a residual f l u c t u a t i o n of 18 m/sec. This implies t h a t 85 per cent of t h e profile v a r i a n c e can be described b y a n average, a t r e n d , a n d a single sine wave. T h e m e a n w a v e l e n g t h o f this best-fit sine w a v e is 3.3 scale heights. (6) An i n d e p e n d e n t e s t i m a t e of t h e m e a n w a v e l e n g t h is given b y 2~ (velocity/ shear) = 3.5 scale heights (7) T h e e n e r g y c o n t e n t per kg is n e a r - c o n s t a n t a t 2000 J / k g a t a l t i t u d e s a b o v e 100 km. (8) T h e e n e r g y dissipation rate, if one uses a molecular viscosity, requires semidiurnal r e p l a c e m e n t of e n e r g y b e t w e e n 110 a n d 120 km. A t lower altitudes, a t u r b u l e n t viscosity a b o u t 20-fold the molecular value is r e q u i r e d t o dissipate t h e e n e r g y c o n t e n t semidiurnally. (9) N o choice can be m a d e b e t w e e n R i c h a r d s o n a n d R e y n o l d s criteria for t r a n sition f r o m t u r b u l e n t to l a m i n a r flow. B o t h p r e d i c t transitions a t a b o u t 110 km, a n d b o t h would p e r m i t layers of t u r b u l e n c e a n d l a m i n a r flow b e t w e e n 100 a n d 115 kin. (10) T h e viscous dissipation of v e r t i c a l l y - p r o p a g a t i n g e n e r g y f r o m below could result in e i t h e r a decreasing a m p l i t u d e or in a w a v e l e n g t h increasing with height. I t is f o u n d to result almost e n t i r e l y in a w a v e l e n g t h increasing directly p r o p o r t i o n a l l y to local scale h e i g h t with no significant change in a m p l i t u d e .
ROSENBERG N. ~r. ROSE~rBERG 1~. W. and JUSTUS C.G. U.S. Standard Atmosphere. BLAMONT J'. E. and BARAT J'.
HI~ES C.O. KOCEA~SK~ A.
REFERENCES 1968 Ionospheric Winds--A Statistical Analy. sis, Space Science VIII, Interscience Publishers (in press). 1966 Radio Sci. 1, 149. 1962 U.S. Government Printing Office, Washington, D.C. (1962). 1967 Altitude Ann. Geophys. 23, 173. 1965 J. Geophys. Res. 70, 177. 1964 J. Geophys. Res. 69, 3651.
Reference is also made to the following unpublished material
BEDINGER J . F .
1966
BLAMOh~TJ. E. and HENRY C. FULLER R. F. and ED~,VARDSH. D.
1964 1966
JUSTUS C. G. and EDWARDS H. D.
1965
GCA Technical Report No. 66-7-N to NASA, GCA Corp., Bedford, Mass. Thesis, Univ. of Paris. Contract Report No. 4 under BRL Contract 169, Space Instruments Research Inc., Atlanta, Georgia. Tables of Observed Winds, Report under Georgia Tech Project A-434, Georgia Inst. of Technology, Atlanta, Georgia.
Statistical analysis of ionospheric winds--H NORMAN W. ROSENBERG Air Force Cambridge Research Laboratories, Bedford, Massachusetts (Received 11 November 1967) Abstract--A statistical analysis of seventy midlatitude wind profiles between 90 and 150 km
altitude shows an rms wind speed ranging from 45 to 70 m/see and an rms shear ranging from 100 to 160 m/sec/scale ]at, (equivalent to 0.004-0.02 see-l). I f a scale-height normalized altitude is used, any vertical zonal or meridional profile can be approximated by an average, trend, and a dominant sine wave with a mean residual variance of 15 per cent. The mean wavelength of the dominant wave is 3.4 scale heights, and the mean energy content of the profile is 2J/g. Semidiurnal replacement of viscosity-dissipated energy is compatible with molecular viscosity between 100 and 120 km, but requires a 20-fold higher turbulent viscosity at lower altitudes. Both the mean Richardson and Reynolds numbers predict transition from turbulent to laminar flow between 110 and 115 kin.
INTRODUCTIOI~ ALMOST 200 night-time and twilight ionospheric wind profileshave been reported from chemical trailsby various investigatorssince 1960. About 70 of these profiles obtained at midlatitudesin the altituderegion 90 to 150 kin, have been used in the p r e s e n t analysis. Most of these were o b t a i n e d d u r i n g t h e y e a r s 1962-1966, r e a s o n a b l y well d i s t r i b u t e d in t i m e a n d season. Statistically-significant wind circulation p a t t e r n s h a v e been f o u n d in this g r o u p o f profiles a n d r e c e n t l y r e p o r t e d elsewhere (ROSENBERG, 1968). T h e p r e s e n t r e p o r t examines t w o wind p a r a m e t e r s , velocity, a n d shear, which are r e l a t i v e l y c o n s t a n t w h e n a p p r o p r i a t e l y averaged. More specifically, r m s shear p e r scale h e i g h t a n d rms t o t a l v e l o c i t y are f o u n d to be a l m o s t i n d e p e n d e n t of altitude. I n this paper, we discuss t h e statistical t r e a t m e n t of the original values, a n d resulting m e a n values of t h e t w o p a r a m e t e r s , which t h e n lead to m e a n values of c o n t r i b u t i n g waves, t u r b u l e n c e criteria, a n d e n e r g y balances.
EXPERIMENTAL DATA T h e original wind profiles were r e p o r t e d (BEDINGER e~ a~., 1966) a t vertical i n t e r vals o f a b o u t 1 k m a t altitudes below 110 l~m, a n d a t 2 t o 5 l~m intervals a t h i g h e r altitudes, w i t h t y p i c a l e x p e r i m e n t a l errors of 3 m/see in b o t h N S a n d E W v e l o c i t y c o m p o n e n t s . T o p r o v i d e i n t e r p o l a t e d values a t a n y desired a l t i t u d e spacing w i t h o u t smoothing, t h a t is, while m a i n t a i n i n g original wind values a t r e p o r t e d altitudes, we h a v e fitted e v e r y f o u r successive d a t a points in each o f t h e profiles to a cubic e q u a t i o n . S i x t y m i d ] a t i t u d e v e l o c i t y profiles h a v e b e e n s u p e r i m p o s e d b y a c o m p u t e r controlled m e c h a n i c a l p l o t t e r in Figs. 1-4. F i g u r e 1 presents N S a n d E W c o m p o n e n t profiles, f r o m which it is seen t h a t a l t h o u g h each profile has a n oscillating p a t t e r n , t h e w a v e l e n g t h s increase w i t h h e i g h t a n d t h e phases are r a n d o m l y d i s t r i b u t e d . F i g u r e 2 shows t h e a m p l i t u d e of t h e t o t a l v e l o c i t y v e c t o r , on a logarithmic scale 9o7
908
N.W.
ROS~-NBERO
180 160 140 .._1
,,::I:
120 I00
8_2000_
0
200 -200 WIND SPEED, M/S
0
200
Fig. 1. Wind component velocity vs. altitude.
vs, altitude, and it is seen that this parameter is restricted in its range of values, with a slight increase in average value with increasing altitude. The total shear per km (SKM) is defined as the vector connecting total velocity vectors at adjacent altitudes separated b y 1 kin. Figure 3 shows the amplitude of the total shear on a logarithmic scale vs. altitude; it is seen t h a t for the 60 trails graphed, this parameter also has a limited range of values, at any given altitude, although the mean shear value decreases markedly with increasing altitude. I t has previously been reported (I~osENBERG et al., 1966) t h a t the dominant oscillations in the NS and E W component profiles have wavelengths proportional to the scale height of the local pressure H~. Therefore, shears were also computed with respect to a scale-height normalized altitude, ZSH = S~ dZ/H~. The shear per scale height, Ss~, was computed b y multiplying shear per kilometer b y the local scale height in km: SSH ---- SK~ " H~. The scale-height values were taken from the 1962 U.S. Standard Atmosphere with no seasonal or time con'ections. The values of SSH are shown in Fig. 4, where it is seen that its mean value is practically heightindependent. AYERAGII~G PROCESS
The root mean square values of velocity and shear for 70 midlatitude profiles were obtained b y the following successive steps:
Statistical analysis of ionospheric winds--II
909
160
140 v UJ C~
D
I-I-d
120
100
80
I
I0 VELOCITY,
IO0 M/S
I000
Fig. 2. Total wind speed vs. altitude. (]) For each altitude Z of each profile, the magnitudes of velocity and shear were computed from equations (1) and (2). V 2f,z Siz
= [(V~,z+az -
=
° 2 VIv.z 4- V~,z
VNz) 2 + ( / ~ . z + a z -
(I)
V~.z)2]/d Z~
(2)
where V r . z and ST.Z are total velocity and total shear at height Z, and the subscripts N and E indicates NS and E W component velocities, also at height Z. The shear was taken for a height difference dZ in km equal to 0.2 scale heights. (2) Within each profile, r.m.s. VT. z and S t . z at each altitude were computed b y 2 2 z averaging the values of VT, z and ST, over an altitude-averaging interval in ]~m equal to 1.3 scale heights and taking the roots of the mean squares. (3) The logarithms of the r.m.s. VT, z and ST. z at each altitude were averaged over all profiles. N o t all profiles extended to the same lowest or highest altitude, so
910
N. ~V. ROSENBERG
160 1"
140 C~ D
120
5 100
80 I
I0
I00
SHEAR,CM/SVKM
Fig. 3. W i n d shear, m/sec/km vs. altitude.
the number of samples for which data was available increased from 20 profiles at 88 l~m to 70 profiles between 100 and 115 kin, and then decreased to 20 profiles at 150 kin. Logarithmic averaging was used to give a statistically normal distribution of values. This led to logarithmic standard deviations SD of 0.2 log units (45 per cent) both for velocity and for shear, practically independent of altitude. The standard error of the mean (SEM -- SD/(N -- 1)1/~) ranged from 5 per cent between 100 and 115 kin, where N = 70 independent samples, to 10 per cent at the altitudes (88 km and 150 km) where the sample count N decreased to 20. Therefore, differences between mean values which m a y be considered significant (at a 95 per cent confidence level) range from 10 per cent in the central region to 20 per cent at the top and bottom. The differencing interval mentioned in Step 1 and the averaging interval mentioned in Step 2 were selected and tested to assure t h a t the reported mean values were insensitive to these two details of the averaging process. The mean values of velocity, shear per kin, and shear per scale height are shown in Fig. 5 vs. altitude.
Statistical analysis of ionospheric winds--II
911
160 140 120 a kin k._J
II0
I00
90 I
I0 I00 S H EAR,(M/S)/SH
I000
Fig. 4. Wind shear m/sec/SH vs. altitude. Seasonal segregation disclosed no significant differences at any altitude. I t is emphasized that calculations were carried out in meters/second and kilometers, with no assumptions regarding other atmospheric parameters, so that season-time uncertainties of these other parameters were not introduced. WAVELENGTH DETERMINATION I f the NS and E W components were pure sine waves with the same wavelength L and amplitude and with random phase, a number of independent samples at a given height would have an r.m.s, total velocity of V and an r.m.s, total shear S = 2~rV/L. I f the values of V and S were relatively constant over a wavelength, this equation could provide an estimate of the dominant wavelength from averaged estimates of V and S. For example, over the 100-150 km altitude region, the relatively height-independent r.m.s, velocity of 70 m/sec and shear per scale height of 120 m/sec/SH give L = 2~rF/S= 2~r(70/120) = 3-5 scale heights. Figure 5 shows the wavelength in scale heights L s s computed from the rms values of total velocity and shear at each altitude inserted into this equation. I t is emphasized that this method
912
N . W . ROS~.~ERG
I
I
I
140
120
LSH
VT~ SSH
SEM SD I--
I 0 0 tI 2
~SKM
I
I 5
I I0
/I 20
,4 50
/1 I I00 200
Fig. 5. Mean wind speed (Vr), mean wind shear per km (S~M), and mean wind shear per scale height (Ssa). The relative standard deviations of all throe parameters are about equal at 4-45 per cent and are shown by the error bar SD. The relative standard errors of the means are also equal, varying from 5 per cent in the center to 10 per cent at the top and bottom. The error bar SEM shows a 10 per cent standard deviation. The dominant wavelengths Lsm estimated as 2~rVr/SsH is also shown. d e p e n d s on a v e r a g i n g o v e r r a n d o m phase, a p p r o p r i a t e in this case b e c a u s e of t h e large n u m b e r of i n d e p e n d e n t samples. I t also d e p e n d s on m o s t of t h e e n e r g y b e i n g p r e s e n t in a single w a v e l e n g t h . I f this e s t i m a t e of w a v e l e n g t h is correct, a profile e x t e n d i n g f r o m 90 to 150 k m ( a b o u t 5.5 scale heights) c o n t a i n s o n l y 1-5 cycles of t h e d o m i n a n t w a v e l e n g t h . On such a s h o r t s a m p l e , s p e c t r a l analysis or F o u r i e r a n a l y s i s is n o t reliable for finding a d o m i n a n t w a v e w i t h precision. A n i n d e p e n d e n t e s t i m a t e of t h e " d o m i n a n t " w a v e l e n g t h w a s carried o u t for each c o m p o n e n t of e a c h profile b y finding t h e " b e s t f i t " single sine w a v e a f t e r r e m o v a l of a v e r a g e a n d t r e n d . This w a s done b y successive a p p r o x i m a t i o n s , v a r y i n g w a v e l e n g t h , phase, a n d a m p l i t u d e of a trial sine w a v e to m i n i m i z e t h e residual v a r i a n c e b e t w e e n it a n d t h e original d a t a . T h i s a n a l y s i s s h o w e d no significant differences b e t w e e n a v e r a g e N S a n d E W v a l u e s or t h e i r s t a n d a r d d e v i a t i o n s , so d a t a f r o m b o t h c o m p o n e n t s were p o o l e d to decrease t h e s t a n d a r d error of t h e m e a n . W i t h 140 samples, t h e s t a n d a r d e r r o r of t h e m e a n SEM, was o n l y 1/12 of t h e s t a n d a r d d e v i a t i o n SD.
Statistical analysis of ionospheric w i n d s - - I I
913
The mean best-fit wavelength was 3-3 scale heights with SD ---- 40 per cent, SEM ~- 3 per cent. Note that this estimate of wavelength, b y an independent method, agrees well with the 3-5 scale heights estimated from average velocity and shear. The mean r.m.s, fluctuation of the original profile was 47 m/see (SD -~ 34 per cent, SEM -~ 3 per cent), reduced b y the removal of trend and best-fit single wave to 18 m/sec. This is equivalent to a reduction in variance to (18/47) 2 --~ 15 per cent. In other words, 85 per cent of the mean variance of a profile can be accounted for b y a trend line plus a single dominant sine wave (with a scale-height proportional wavelength) TURBULENCE
The existence of turbulence in the wind structure below 110 km has been extensively studied and recently reviewed b y BLAI~IONT et al., 1967. I t seemed possible that the present study might allow a choice between two sources of turbulent instability. Two dimensionless parameters have been widely used in predicting turbulence. The first, the Reynolds number, is defined as LcV/.KV where Lc is a critical length, V the velocity, a n d / ~ V the kinematic viscosity. A high velocity V between two low-velocity boundaries separated b y a distance 2Lc (walls in the laboratory, and possibly velocity minima in the upper atmosphere) can cause an isotropic medium to develop a turbulent flow pattern. This cannot occur unless the Reynolds number exceeds some critical value, about 2000 in laboratory studies, although turbulence need not set in at that point. The mean Reynolds number is presented vs altitude in Fig. 6, based on the average velocity and a characteristic length Lc equal to a quarter of the mean dominant wavelength Ls~. Blamont reports a value of 600 as the criterion for appearance of turbulence, which he found to occur near 105 km in selected profiles (BLAMO~T et al., 1967). He used a shorter Lc, however, based on small scale fluctuations, so the discrepancy between the two estimates of Reynolds number is really an uncertainty in the correct choice of a characteristic length Lc. An alternative source of turbulence is that induced b y high shears in a nonisotropic medium. In the absence of shears, a gas whose temperature increases with height is generally stable. Excessive shear forces can, however, overcome the stabilizing forces of the temperature gradient, and lead to a nonisotropic turbulence in the medium. This stability ratio is described b y the Richardson number, which has been successfully used in predicting turbulence in the lower atmosphere. The Richardson number (Nm) is defined b y [(G/T)(dT/dZ ÷ TG]H)]/Ss 2 where g is the gravitational constant, T the temperature, dT/dZ the temperature gradient, G the specific heat ratio, H the local scale height, and Ss the shear in see -1 ( = SK~/1000). (Because Nm involves the square of the shears, the standard deviation and standard error of the mean are doubled, as shown in Fig. 6.) I f the Richardson number exceeds a critical value near unity, turbulence cannot exist. The bracketed term (Cm) m a y be regarded as describing the stabilizing forces; countered b y shear, which describes the destabilizing forces. The bracketed term (Cm), as shown in Fig. 6, is relatively constant to 120 km where it starts to decrease. The mean Richardson number has a value below 110 km near unity; that is, the atmosphere below 110 km is unstable. Above 110 km, shear per km decreases rapidly and at an altitude of 17
914
N.W.
I
ROSENBERG
I
I
I
,4o l-
SEM ReXlO -5
SD i ,o
-
NRi
m
I-._1
CRiXlO 4
I00 0.2
II
I
"K
I
1
I
0.5
I
2
5
I0
20
Fig. 6. Mean Reynolds number (NRe), moan Richardsons coefficien~ (Cm), and mean Riehardsons number (~JRI) VS. altitude. 112 km N m reaches 2, and at 114 km it reaches 5. Hence, the Richardson criterion would predict transition from unstable to stable flow in this height region. The mean Reynolds number, however, goes through 2000 (the accepted laboratory critical number) at 112 kin, and thus also predicts the transition from unstable to stable medium near this height. Some photographs show turbulence in regions of high velocity, b u t others show turbulence in regions of low velocity and high shear. I t is not possible to choose between the two criteria on the basis of this data. /~VERAGE ENERGY CONTENT AND DISSIPATION
We next consider the energy content and power dissipation represented b y the wind. Since energy content depends on V~ and power dissipation on S 2, the un"certainties--and hence standard errors of the m e a n - - a r e doubled because the squares of the experimental observables are used. This means that larger differences (20 per cent at the center, 40 per cent at the top and bottom) must occur before t h e y can be considered meaningful. The energy content per unit mass in J / k g is given b y V~/2. Since there is only a slight variation in V with height, the average value over all altitudes is 2000 J / k g :h 25 :per cent (Fig. 7). The 25 per cent deviation results largely from the 40 per cent increase in mean velocity between 90 and 100 km altitude. The mean viscous energy dissipation due to wind shear Ss expressed in sec -1 is
Statistical analysis of ionospheric winds--II
~
140
1
I
I
I/
915
I
I
EC J/KGxlO -3 ~.. 120
5
,,::[
I00
W/KGx I02
fl O.
0.2
I 0.5
1,," I
I 2
I -"1" 5 I0
I 20
Fig. 7. Mean energy content (EC), mean viscous dissipation rate (V/)), and mean time constant (TDx) vs. altitude. given in W/kg b y K V • Ss ~, where K V is the kinematic viscosity. I f the molecular viscosity is used for K V , the mean power dissipation ranges from 10 -a W]kg at 90 km to 200 × 10 -a W/kg at 150 km (Fig. 7). At altitudes above 110 kin, this viscosity is probably appropriate, and in this region our mean dissipation is within a factor of two of earlier estimates made b y H~r~.s (1967) from more limited data of Koc~s~ (1964). However, at altitudes below 110 lrm, the turbulent viscosity is more appropriate, and the actual dissipation is greatly above that shown in Fig. 7. Br.AMO~T (1967) estimates a turbulent diffusion coefficient 50 times higher than molecular, from which we might infer a similar ratio of viscosities. The ratio of energy content to energy loss rate represents a time over which the energy must be replenished ff both velocity and shear values are to be maintained constant. With molecular viscosities, the values of the time constant (shown in Fig. 7) range from 10 days at 90 km to 1 day at 103 k m to 4 hr at 150 kin. Between 105 and 125 km, the average value of ½ d a y within the statistical uncertainty is in agreement with a largely semidiurnal source. At lower altitudes, the loss rate is seriously underestimated b y using molecular viscosity. We might turn the problem around to estimate t h a t for a time constant of about a haft day, the "effective" viscosity must be about 20 times the molecular viscosity which is in reasonable agreement with Blamont's estimate (50 fold). Above 125 km, a higher rate of replacement of energy (than semidiurnal) is apparently demanded. On the other, hand, the variation in temperature and density necessary to bring the viscosity down
916
N . W . ROSENBERG
b y a factor of 3 (which would make the whole non-turbulent profile semidiurnal) is within the estimated variability of the U.S. Standard Atmosphere in this region. Between 120 and 150 km, there have been far fewer measurements of temperature and density than of wind, and their statistical reliability is lower than those of the winds and shears. NON-DEPENDENCE OF VELOCITY AND SHEAR ON ALTITUDE Energy injected from below in a transverse wave at a given wavelength will tend to increase the wave amplitude with increasing altitude to keep the energy per unit volume constant in a region of decreasing density. However, as the wave amplitude grows, so does its m a x i m u m shear. The increasing amplitude and/or shear creates a turbulent instability, causing most of the energy that would further increase the amplitude in the 90 to 105 k m region to drain off into turbulent viscous dissipation. Without turbulence, the amplitude would grow severalfold between 90 and 105 k m to offset the density decay. In contrast to the mean stability, the point-to-point atmospheric stability,can fluctuate from stable to unstable over a half wavelength of 1.7 scale heights = 8 km, so that layers of turbulent and laminar regions can alternate as reported by Blamont (1964). Above 110-120 kin, the atmospheric stability will not permit existence of turbulence. As the energy propagates to higher altitudes, further viscous loss in the wave can occur in either of two ways: by decreasing in amplitude or by increasing in wavelength. Consider an upward flow in a wave consistingof several components of differentwavelength. The higher-frequency components have a higher rate of shear loss,hence willhave dissipated to a greater extent at a given height altitude, and the apparent effective wavelength will increase. Our experimental observations show that there is no significantloss in mean amplitude but a continued increase in mean wavelength. The time constant for energy replacement ranges from a day to a few hours, but a moderate change in the viscosity estimate (which was estimated from a standard atmosphere which itselfhas a m a x i m u m uncertainty in the height region 120 to 150 kin) could bring the time constant to a half day throughout this region, implying a semidiurnal dominant mode. CONCLUSION A statistical analysis of 70 midiatitude and wind profiles at altitudes between 90 and 150 km leads to the following conclusions: (1) The rms total velocity increases from 45 m/sec ( ± 1 0 % ) at 90 km to an altitude-independent value of 70 m/sec (~-5-10%) above 100 km. (2) The rms total shear (in seconds -1) decreases from an altitude-independent value of 0.02 sec -1 (~:5-10 per cent) below 110 km to 0.004 sec -1 ± 10 per cent at 150 kin. (3) The rms total shear (in m/sec/scale height) is nearly independent of altitude. I t ranges from 100 m/sec/SH ± 10 per cent at 90 km to 160 m/sec/SH ± 5 per cent at 105 km to 120 m/sec/SH ~: 10 per cent above 115 km. (4) There is no significant seasonal dependence of these parameters. (5) The vertical profile of NS and E W components vs. a scale-height normalized altitude can be approximated b y average, trend, and best-fit sine waves. When this
Statistical analysis of ionospheric winds--II
917
is done, t h e m e a n f l u c t u a t i o n of t h e original d a t a , 47 m/sec, is r e d u c e d to a residual f l u c t u a t i o n of 18 m/sec. This implies t h a t 85 per cent of t h e profile v a r i a n c e can be described b y a n average, a t r e n d , a n d a single sine wave. T h e m e a n w a v e l e n g t h o f this best-fit sine w a v e is 3.3 scale heights. (6) An i n d e p e n d e n t e s t i m a t e of t h e m e a n w a v e l e n g t h is given b y 2~ (velocity/ shear) = 3.5 scale heights (7) T h e e n e r g y c o n t e n t per kg is n e a r - c o n s t a n t a t 2000 J / k g a t a l t i t u d e s a b o v e 100 km. (8) T h e e n e r g y dissipation rate, if one uses a molecular viscosity, requires semidiurnal r e p l a c e m e n t of e n e r g y b e t w e e n 110 a n d 120 km. A t lower altitudes, a t u r b u l e n t viscosity a b o u t 20-fold the molecular value is r e q u i r e d t o dissipate t h e e n e r g y c o n t e n t semidiurnally. (9) N o choice can be m a d e b e t w e e n R i c h a r d s o n a n d R e y n o l d s criteria for t r a n sition f r o m t u r b u l e n t to l a m i n a r flow. B o t h p r e d i c t transitions a t a b o u t 110 km, a n d b o t h would p e r m i t layers of t u r b u l e n c e a n d l a m i n a r flow b e t w e e n 100 a n d 115 kin. (10) T h e viscous dissipation of v e r t i c a l l y - p r o p a g a t i n g e n e r g y f r o m below could result in e i t h e r a decreasing a m p l i t u d e or in a w a v e l e n g t h increasing with height. I t is f o u n d to result almost e n t i r e l y in a w a v e l e n g t h increasing directly p r o p o r t i o n a l l y to local scale h e i g h t with no significant change in a m p l i t u d e .
ROSENBERG N. ~r. ROSE~rBERG 1~. W. and JUSTUS C.G. U.S. Standard Atmosphere. BLAMONT J'. E. and BARAT J'.
HI~ES C.O. KOCEA~SK~ A.
REFERENCES 1968 Ionospheric Winds--A Statistical Analy. sis, Space Science VIII, Interscience Publishers (in press). 1966 Radio Sci. 1, 149. 1962 U.S. Government Printing Office, Washington, D.C. (1962). 1967 Altitude Ann. Geophys. 23, 173. 1965 J. Geophys. Res. 70, 177. 1964 J. Geophys. Res. 69, 3651.
Reference is also made to the following unpublished material
BEDINGER J . F .
1966
BLAMOh~TJ. E. and HENRY C. FULLER R. F. and ED~,VARDSH. D.
1964 1966
JUSTUS C. G. and EDWARDS H. D.
1965
GCA Technical Report No. 66-7-N to NASA, GCA Corp., Bedford, Mass. Thesis, Univ. of Paris. Contract Report No. 4 under BRL Contract 169, Space Instruments Research Inc., Atlanta, Georgia. Tables of Observed Winds, Report under Georgia Tech Project A-434, Georgia Inst. of Technology, Atlanta, Georgia.