Estimation of potential-temperature gradient in turbulent stable layers using acoustic sounder measurements

Q. J. R. Meteoml. Soc. (ZOOO), 126, pp. 31-61

Estimation of potential-temperature gradient in turbulent stable layers using acoustic sounder measurements

By C. G. HELMIS*, J. A. KALOGIROS, D. N. ASIMAKOPOULOS and A. T. SOILEMES University of Athens, Greece

(Received 17 March 1998; revised I 1 December 1998)

SUMMARY The intensity of acoustic backscatter from a monostatic acoustic sounder is used to compute the profile of the

temperature structure parameter C?. The average value of C$ in low-height turbulent stable layers (temperature inversions) is used to infer the local potential-temperature gradient. The estimates of temperature jump in the first height inversion at the top of the unstable atmospheric boundary layer (ABL) are. compared with the direct measurements of the temperature profile by a tethered balloon, and the indirect estimation from the wave period of the waves present on the inversion. The agreement of the estimations with the direct measurements is satisfactory (at least a 0.4 degC accuracy). Also, the -4/3 similarity law of the C; profile in the unstable ABL is used to obtain estimates of the surface heat flux, Qo. The comparison of these estimates of Qo with eddy correlation measurements shows the high accuracy of this method. This direct connection of the C? profile with Qo can also be used to calibrate the backscatter intensity.

KEY WORDS: Acoustic sounder Boundary layer Temperature inversions

1. INTRODUCTION

The temperature increase (jump), A@, of the temperature inversion capping the convective boundary layer (CBL), and the surface heat flux, Qo, are essential parameters for its evolution and dispersion characteristics. These parameters can be obtained from measurements of the temperature structure parameter C: (proportional to the backscatter intensity) using a monostatic acoustic sounder. The values of these parameters obtained from run-average C: values, applying the methods described here to acoustic sounder data, correspond to a large volume of the atmosphere. These are, usually, more appropriate than one-point measurements (and especially ‘instant’ measurements of a temperature profile, for example with a tethered balloon) in atmospheric boundary layer (ABL) studies.

The height of the local maximum of the profile of C: is an accurate estimate of the lower part (near-base) of the temperature inversion capping the CBL (Kaimal et al. 1976; Caughey and Palmer 1979; Kaimal et al. 1982). The use of the average value of C: in the inversion and its theoretical dependence on A@ may give accurate estimates when a careful calibration and correction of the echo intensity measurements, mainly for the effect of sound absorption and excess attenuation, are applied. The methods described here for the estimation of A@ may be used, generally, for the estimation of the local potential-temperature gradient in stable layers, and are originally applied to acoustic sounder data. The results of these methods are compared with the results from a case- study over complex terrain using direct measurements of a tethered balloon, and the results from many experimental days over flat terrain using the relationship of Brunt- Vaisala frequency with the local potential-temperature gradient in stable layers.

The capability of the -4/3 similarity law of the C: profile in the CBL to give estimates of Qo has been examined by previous studies but with limited datasets (Coulter and Weseley 1980; Dubosclard 1982; Sorbjan et al. 1991). This work presents, also, a comparison of Qo estimates using the C; profile with the direct (eddy correlation) ones for an extended dataset acquired over a flat terrain, and proves the capability of the * Corresponding author: University of Athens, Department of Applied Physics, Laboratory of Meteorology, Panepistimioupolis, Building PHYS-5. 157 84, Athens, Greece.

31

32 C. G . HELMIS et al.

0 10 20 30 40 50 60 70 80 km

Figure 1. Map showing the greater Athens area with the experimental sites at the National Observatory of Athens (NOA) and Spata indicated by bullets (height contours of 200 m).

method to give reasonably accurate estimates. Thus, in addition, the direct link of C; with Qo in the CBL can be used, easily and accurately, to calibrate the backscatter intensity of a monostatic acoustic sounder when an independent estimate of Qo is available, in contrast to more difficult acoustic methods (Asimakopoulos et al. 1976, 1983). The independent estimates of Qo may come from direct in situ measurements or the analysis of the wind structure parameter C: which can be computed by vertical- velocity measurements (see section 3(b)).

2. INSTRUMENTATION AND EXPERIMENTAL SET-UP

The data used in this work were collected during the summer of 1995 at the flat, rural area (50 m above sea level (a.s.1.) with sparse trees) of Messogia Plain (Spata) in the Attiki peninsula, Greece, and on one experimental day (23 June 1996) in the centre of the urban area of Athens on top of the hill at the National Observatory of Athens (NOA, 110 m a d . ; see Fig. 1). The vertically oriented, monostatic acoustic sounders used at both sites were similar high-range systems with a 1.6 kHz operating frequency. Table 1 shows the operating parameters of the acoustic sounders. The amplitude of the backscatter signal is obtained from moving averages of absolute signal values (rectification). These amplitude measurements are interrelated due to the actual resolution (about 17 m) imposed by the length of the acoustic pulse. The profile of the backscatter intensity (square amplitude) was stored as the average of one every 80 s (16 time samples) and in 96 successive gates of 8.3 m each. Thus, the resulting profile of the backscatter intensity was actually smoothed (moving average of 17 + 8.3 m) relative to

POTENTIAL-TEMPERATURE GRADIENT FROM ACOUSTIC BACKSCATTER 33

TABLE 1. OPERATING PARAMETERS OF THE ACOUSTIC SOUNDERS

~ _ _ _ _

Operating frequency 1.6 kHz Pulse length 108 ms

Higher range 800 m Acoustic antenna

Pulse repetition rate 5.00 s

1.2 m parabolic disc

the real one. For vertical-velocity measurements (Doppler shift obtained from spectrum analysis using fast Fourier transform) the gates corresponded to 34 m.

At the Messogia Plain site, a 12 m high meteorological mast equipped with a UVW propeller anemometer (NEZII, Alcyon S. A.) and a fast responding temperature plat- inum wire (12.5 pm diameter) sensor was established close to the acoustic sounder. The sampling rate of these sensors was 1 s and they provided a direct method (eddy correlation) for estimating the surface heat and momentum fluxes under unstable atmospheric conditions. The sampling rate of 1 s is low if someone needs the spectral analysis of turbulent fluxes, but is sufficient if someone is interested only in the value of fluxes with the limitation of sufficiently fast response sensors (Wyngaard 1986). The slow sensor that we used is the UVW propeller anemometer. For unstable conditions, a cospectrum analysis (described in detail by Kalogiros et al. (1999)) shows that the loss in fluxes at the height of the meteorological mast (12 m) is about 10% for the heat flux and 5% for the momentum flux, which is a sufficient accuracy for our purposes. The tethered balloon used at the NOA site was equipped with a system measuring wind, temperature and humidity with a sampling rate of 1 s, that was developed in the Laboratory of Meteorology of the University of Athens (Soilemes et al. 1993). The time duration of the analysis runs was 30 minutes for the data collected at Messogia Plain and 15 minute half-overlapping runs (that is 7.5 minutes apart) for the data collected on 23 June 1996 at the NOA (because of the fast evolution of the temperature inversion observed on that experimental day).

3. PROCESSING OF THE BACKSCATTER INTENSITY

(a) Noise reduction andjixed echoes removal The acoustic environmental noise can be estimated by the received signal corre-

sponding to large distances from the receiver, where the noise is assumed to dominate over the atmospheric backscatter (Melling and List 1978). Also, the noise is assumed to have Gaussian real and imaginary components (Tatarskii 1971; Melling and List 1978). A noise level is estimated for a given level of significance using the Rayleigh distribution of the amplitude of noise (Papageorgas et al. 1993). In addition, the 80 s average profiles of the noise-reduced square amplitude were examined with a continuity check for possible temporary fixed echoes, for example by a moving tethered balloon (see the facsimile record in Fig. 7). These fixed echoes (sound reflections) were detected when comparing the measurements of successive 80 s average profiles at the corresponding heights. A strong increase of the square amplitude (for example, by a factor of 2) between the small (80 s) successive averaging time periods at the same height was assumed to be a fixed echo, and it was excluded from the calculation of the run-average profile. Because of the moving up-and-down target (tethered balloon) during the averaging time it was not possible to apply other, more elaborate, fixed-echo detection methods like the wavelet transform described by Kalogiros and Helmis (1999).

34 C. G . HELMIS eral.

(b) Calculation of C; and correction for propagation effects The received electrical power Pr (noise and sound reflection corrected) from the

backscatter of the acoustic pulse at a height z in a vertically oriented monostatic acoustic sounder, assuming a small contribution of humidity fluctuations as is the case over land, is (Brown and Hall 1978):

(1) where Er is the efficiency of acoustic-to-electrical conversion, Et is the efficiency of electrical-to-acoustic conversion, Pt is the transmitted electrical power, G is the antenna directivity gain factor, A, is the surface of the acoustic antenna, c is the speed of sound, t is the time length of the acoustic pulse, k is the wave number of the transmitted acoustic wave, T is the average air temperature, Q is the effective attenuation-absorption coefficient (assumed to be constant with height or a height average value) and L e is the excess attenuation factor. The case of sound refraction by strong winds or very stable atmospheric conditions is not considered here. The factor z -2 is compensated for by applying a linear ramp to the received signal, while the parameters in the outer brackets are the fixed operating ones and make up a proportionality factor. The parameters a, Le and C: are connected to the state of the atmosphere.

The proportionality of P, and C: is determined with the calibration of the intensity of the received signal against the direct estimates of C; under unstable atmospheric conditions, instead of more difficult acoustic methods (Asimakopoulos et af . 1976, 1983). The direct estimates of C; are computed using three frequencies in the inertial subrange of the power spectrum of the fast response measurements of temperature by the nearby 12 m high meteorological mast assuming local isotropy. These values of C; were 'extrapolated' to the first measurements height of the acoustic sounder (about 35 m) using the C? similarity law (see section 4(a)(i)) and an average calibration coefficient of 3.8 x

The sound-absorption coefficient is computed from the corresponding theoretical relations (Brown and Hall 1978), for an average state of the atmosphere. Another method to estimate the absorption coefficient is to consider it as an additional unknown parameter of the fit of the theoretical C; profile to the measured one (section 4(a)(i)). This method can be applied to data from the CBL, since the -4/3 similarity law for C: is a highly anticipated signature of convective conditions in the mixed layer.

Also, in the CBL case the excess-attenuation coefficient may be computed from the corresponding theoretical relation (Brown and Clifford 1976; Neff 1978), using the similarity profiles for C; and C: and operating parameters of the acoustic sounder. Since the turbulence near the acoustic antenna is contributing most of the excess attenuation, the exact profile of C; and C; is not significant. These theoretical profiles are computed from similarity relations and the estimates of Qo and the friction velocity, u*. An estimate of the latter parameter is obtained using measurements of the vertical- velocity variance 0: and the corresponding similarity relation (Kalogiros et af . 1999). An estimate of Qo is obtained by the direct measurements (eddy correlation) or the estimate (0.03 K m s-' accuracy and 0.90 correlation coefficient when compared with direct measurements) from the analysis of the measured C: profile (Kalogiros et af. 1999). According to the latter method, C: is estimated from the structure function of the vertical-velocity measurements which is calculated in the vertical direction among the measurement gates (assuming local isotropy). Note that this C: profile is corrected for the noise and sampling volume averaging effect. Then, under unstable

2 2 Pr = { ErEt Pt( G A,) ( c t / 2 ) (0.00389k ' 13/ T ') } exp( -2~7,) ( Le / Z )C, ,

K2m-2/3V-2 was found (section 4(a)(ii)).


atmospheric conditions, Q, may be estimated from the average value of the measured C: in the mixed layer using the relations C: = 2&2/3 (Tatarskii 197 1 ; Kaimal 1973) and E = 0.65(g/0)Q0 (Kaimal et al. 1976; Gaynor 1977; Weill et al. 1978; Caughey and Palmer 1979; Lenschow et al. 1980; Melling and List 1980), where E is the dissipation rate of the turbulent kinetic energy and g is the acceleration of gravity. Using this method, the excess attenuation was usually found to be 10-15% for the 1.6 kHz acoustic sounder.

The estimation of Qo by the C: profile can be used for an inter-calibration of the backscatter intensity using acoustic sounder measurements exclusively, since the C; profile in the mixed layer depends directly on Qo (section 4(a)). Actually, the estimates of Qo by both C; and C: profiles show very close agreement (especially in the middle of the day) when compared against the direct estimates (eddy correlation). The accuracy of the corresponding C? profile method has been examined by Kalogiros et al. (1999) and it was found to be quite satisfactory as described above. This method of calibration showed a similar factor of proportionality (5 x estimated using the direct C; estimates.

close to the factor 3.8 x

4. ANALYSIS OF c; MEASUREMENTS

In this section the similarity theory for the C'; profile in the CBL is presented, and it is applied to the profiles measured with the acoustic sounder. The estimation of Qo using the C; and the C: profiles measured with the acoustic sounder, or the estimation of Qo by direct measurements, can be used for the calibration of the backscatter intensity, as discussed in the previous section. Also, the calibrated profile of backscatter intensity proves to be a reasonably accurate method for an independent estimation of Q,.

(a) Application of the similarity relation for C:

(i) Similarity relation and profile fit. The similarity relation for C; in the unstable boundary layer and below the middle of the CBL that was used in this work is (Wyngaard et al. 1971; Kaimal et al. 1976; Fairall 1987):

where z is the height above ground, L = - U : / ( K g & / @ ) is the Monin-Obukhov length, K = 0.35 is the von K h 6 n constant, and 0 is the average potential temperature. Above the surface layer and in the mixed layer (1L1 < z < 0.55, where Zi is the height of the inversion), Eq. (2) becomes the well-known free convection -4/3 law (a significant indication of possibly unstable+onvective conditions):

However, the high-range acoustic sounder used in this work does not cover adequately the surface layer where Eq. (2) differs from Eq. (3). Instead, Eq. (3) describes accurately the C: profile in the mixed layer measured by the acoustic sounder. Above the middle of the CBL, the increase of C; relative to the z-4/3 law, because of the entrainment effect, is significant. There are various parametrizations that, however, have limited agreement with experimental data (Kaimal et al. 1976; Weill et al. 1980; Fairall 1987; Sorbjan 1988). In the inversion layer at the top of the CBL, the C: profile presents a characteristic

36 C. G . HELMIS eta/.

local maximum (because of the entrainment) that determines Zj. Its value is connected to the potential-temperature gradient in the inversion as it is shown in section 4(b). The similarity relation for the C; profile in the stable surface layer (Wyngaard et al. 1971) cannot be used in the case of the high-range acoustic sounder measurements used in this work since they do not cover the surface layer.

When the contribution of humidity fluctuations to the buoyancy flux is significant (which is the case over water surfaces), the temperature that is included in the heat flux or in the temperature structure parameter in Eqs. (2) and (3) should be replaced by the virtual temperature. In this case, the sound backscatter senses fluctuations of air density (or virtual temperature) to a reasonably close approximation (Coulter and Wesely 1980). Our data were collected over a land site that is usually characterized by a large Bowen ratio, and the humidity contribution is presumably insignificant.

The fit of the theoretical Eq. (2) or (3) to the experimental profile of C; is achieved by minimizing the absolute error (robust estimation). The significance of the fit is examined with an F-test on the function

n - n p r L F = - - n p - 1 1 -9’ (4)

where n is the number of points of the profile, np is the number of the parameters of the fit, and r is the correlation coefficient. The parameters of the fit are Qo (and u* if Eq. (2) is used) and the sound absorption coefficient a! according to Eq. (1) when it is not estimated by an average state of the atmosphere (see section 3(b)). Upper and lower limits of acceptable values of a! can be defined from the expected range of temperature and humidity values during the experimental run. The accurate estimation of a! is essential for the correct calculation of the C; value in the inversion which will be used to infer the local potential-temperature gradient according to section 4(b). The estimation of a! with the above method includes any possible refraction of the acoustic beam which results in a reception of the scattered sound off the main axis of the antenna.

(ii) Results of the similarity method. Figure 2 shows the calibration data collected in the time period 0630-1800 LST (LST is GMT plus 2 hours) of each day in the first week of the experiment at the Messogia Plain (flat terrain). The direct estimate of C; was ‘extrapolated’ from the 12 m height of the meteorological mast to the 35 m measurement height of backscatter intensity A2 using similarity theory. If all these data were used for the calibration of backscatter intensity in order to correspond to C;, a nonlinear calibration function with a slope that is dependent on the C; value is obtained. However, a careful inspection of the time evolution of A2 and the direct estimate of C; in Fig. 2 reveals that before 1200 LST the directly estimated C; values (computed as described in section 3(b)) show significant scatter (thus, they should be avoided for use in the calibration process) but evolve similarly with A2, while they decrease faster than A2 after 1400 LST.

In order to find out if the different decrease rate for the afternoon hours between directly estimated C; and A2 is due to the response of the acoustic sounder system or a deficiency of the method of direct estimation of C;, the time evolution of these parameters for several days was inspected. Figures 3(a) and 3(b) show two distinct cases for 11 and 26 June 1995, respectively. In the first case both A2 and the directly estimated C; follow each other very well until 1500 LST, but C? decreases more rapidly than A2 after that time. In the second case the values of both parameters are low during the early morning, reach a maximum around midday following each other well (except around


100.0 4

3- 1

8 10 12 14 16 18 6 Time (LST)

Figure 2. The time evolution of the backscatter intensity A2 at a height of 35 m, and the direct estimate of the temperature structure parameter C; ('extrapolated' from the 12 m height of the meteorological mast to 35 m using similarity theory) in the time period 0630-1800 LST of each day of the week 10-16 June 1995 at Messogia Plain. The thin solid lines are polynomial fits, while the thick ones show the average values of the corresponding

parameter in the time period between the two dashed lines.

1100 LST when the local sea breeze reached the experimental site), and decrease to low values again in the afternoon. However, the close agreement of A2 and directly estimated C; in the early morning (the values below the dashed line) does not hold for the same range of values in the afternoon. The reason for this difference was found to be a deficiency of the method of direct estimation of C;. More specifically, the increase of wind speed and, to a less degree, the decrease of the stability factor - z / L shown in Figs. 3(c) and 3(d), respectively, shift the frequency spectrum of temperature at the height of the meteorological mast towards higher frequencies (Kaimal et al. 1972). Thus, the frequencies that we assumed to be within the inertial subrange (our Nyquist frequency was just 0.5 Hz) and used for the direct estimation of C; usually lay outside it and into the 'energy-containing' (low wave numbers) subrange for the afternoon hours, giving estimates of C; lower than the real values. From the above analysis, it is concluded that the time period for a reliable calibration of backscatter intensity using the specific method and data for the direct estimation of C; is 1200- 1330 LST as shown in Fig. 2. The average calibration coefficient in that time period was 3.8 x K2m-2/3V-2.

Figure 4 shows an example of the fit of the similarity function to the measured C; profile (which is corrected using a that was found to be 0.00233 m-l or 1.01 dB/100 m) with the corresponding values of the parameters of the fit. The inversion capping the CBL was at about 570 m, as deduced from the local maximum of the profile. The well- defined smaller peak of C; at 350 m was the result of a temporary thermal turbulence structure (an increase of C;) between 300 and 500 m for only about 5 minutes just after 1035 LST, and corresponds to a first indication of the arrival of the front of the local sea breeze. The surface heat flux estimated from the eddy correlation method was

38 C. G. HELMIS et al.

3 q 0 1

b

o + 0 + 0

:- i 6 8 10 12 14 16 18

T h e (LST)

T i , I / / , I , , ,

T h e (LST) 8 10 12 14 16

1 O 0 I

t ' l ' l ' l ' l ' l ' i 6 8 10 12 14 16 18

Time (LST)

1.0,

3 - 0.1 y

6 8 10 12 14 16 T h e (LST)

8

Figure 3. (a) and (b) As in Fig. 2, but for (a) 11 June 1995 and (b) 26 June 1995. The dashed lines separate low values of C; from high ones. (c) and (d) The time evolution of (c) wind speed and (d) stability factor - z / L (see

text) at a height of 12 m (meteorological mast) above ground on 26 June 1995.

0.1 1 K m s-'. The fit did not give an estimate of u* due to incomplete coverage of the surface layer (below 50 m), even though its influence (a reduction relatively to the z-4/3 law) can be seen in the first three measurement heights.

Figure 5 shows the correlation of acoustic sounder estimates of Qo using the C; profile with the direct ones (eddy correlation) and their ratio with the time of day. This comparison also reveals the quality of the calibration of the backscatter intensity. The logarithmic scale damps out large excursions of the ratio (which can occur for small values of Qo and small absolute differences). Weak instability cases (-Zi/L < 4.5) are separated from strong instability ones (-Zi/L > 4.5). This figure shows that the C; profile method overestimates Qo during the early morning or the late afternoon hours (small values of Qo and weak instability), and slightly underestimates it at midday (high values of Qo). The rapidly changing atmospheric conditions during the early morning or the late afternoon hours limit the application of the similarity theory (Coulter and Wesely 1980). Also, at that time of day, the horizontal heat flux may be significant relatively to the small vertical one and, thus, increase C; according to the budget of temperature variance, while Qo might be less affected. The scatter of data even during midday (almost steady atmospheric conditions) is expected by the limits of validity of


h

E E Y

f 100;

1 E - 3 15-2 C T ~ ( K2W2I3)

Figure 4. The fit of the measured temperature structure parameter, C;, profile by the similarity relation for the run 1017-1047 LST on 4 September 1995 at the Messogia site using the 1.6 kHz acoustic sounder. The parameters

of the fit are also shown (see text for explanation).

the similarly theory and the fact that the acoustic sounder estimates (profile method) represent a larger volume of the atmosphere than the direct, one-point ones.

Table 2 presents the statistical information for the correlation of Qo estimates by the C; profile and the direct measurements. The correlation parameters presented are the number, N, of data points, the bias B , the random error root-mean-square error (R.M.S.E.), the accuracy P , (Chintawongvanich et al. 1989), the slope a and the intercept b of the regression line, and the correlation coefficient R . The difference of the values of the parameters a and R among the linear regression with zero forced and non-zero intercept, results from the bias of the least-squares criterion by the correlation at the lower values of Qo. This information shows the high accuracy and applicability of the C; profile method (many points of comparison) which is expected from the high occurrence of the -4/3 law in the unstable ABL.

(b ) The Dependence of C; on the potential-temperature gradient in turbulent stable layers

In this section various methods for the estimation of the potential-temperature gradient, a@/az, in turbulent stable layers using the estimation of C;, C:, o;, and the entrainment velocity in the case of a temperature inversion at the top of the CBL by a vertically oriented, monostatic acoustic sounder are presented. These physical parameters are measures of thermal turbulence generated locally in a stable layer by the stratification (temperature gradient) and the wind shear. The estimation of these parameters uses the backscatter echo intensity and the vertical-velocity measurements (section 3(b)). The proposed methods are based on time and space averaging by the acoustic sounder. Thus, assuming that they are accurate enough, ‘unrepresentative’ values introduced in the estimation of a@/az from direct, instant measurements of the temperature profile are avoided. This effect may be significant, for example, when using

40 C. G. HELMIS etal.

0.1 5

0.10

0.05

0.00 I I I I I I

0.00 0 05 0.13 2.15 0.20 direct Q, (Kms'l)

1 E - ? -? 4 6 8 10 12 14 16 18 20

Time (LST)

Figure 5. (a) The correlation and (b) the ratio (as a function of time of day) of the surface heat flux, Qo, estimates by the temperature structure parameter, C& profile measured by the 1.6 kHz acoustic sounder and the direct estimates for the whole experimental period at the Messogia site. The solid lines correspond to equality of the

estimates. For explanation of Z i / L see text.

a tethered meteorological balloon to probe instantly a fast evolving and intermittent elevated stable layer like the one at the top of the CBL in the first few morning hours (Hall ef al. 1975).


TABLE 2. PARAMETERS OF THE CORRELATION OF Q,, ESTIMATES OBTAINED APPLYING THE c; PRO- FILE METHOD TO THE DATA FROM THE 1.6 KHZ ACOUSTIC SOUNDER WITH THE DIRECT ESTIMATES FOR THE WHOLE EXPERIMENTAL PE-

RIOD AT MESSOGIA PLAIN

N 1654 B 0.002 R.M.S .E . 0.02 P 0.02 U 0.67 (0.94) b 0.02 R 0.79 (0.95)

See text for explanation of symbols. The values inside brackets refer to the regression line with zero intercept.

(i) The general case of turbulent stable layers. In a stable turbulent layer, under horizontally homogeneous and steady atmospheric conditions, the analysis of the temperature variance and turbulent kinetic energy budgets leads to the next equation for C; (Neff and Coulter 1986):

- where a@/az is the local potential-temperature gradient, K H = -w’O’/(a@/az) is the eddy diffusion coefficient for temperature and Rf = (g/@)w’e’/(u’w’aU/az) is the flux Richardson number. The overbar indicates run average and the primes indicate deviation from the corresponding run-average quantity.

Rf is connected with the gradient Richardson number R i = ( g / @ ) ( a @ / a z ) ( a U / az)-2 and the turbulent Prandtl number Pr = K , / K H = Ri/Rf, where K , = -u’w’/ ( a U / a z ) is the eddy diffusion coefficient for momentum. Assuming equal turbulent outer length-scales of temperature and momentum in a stable layer under steady-state conditions, it comes out from the budgets of temperature variance and turbulent kinetic energy and the form of the structure functions of temperature and wind speed in the inertial subrange that Pr - Ri = 1.6 and Rf = Ri / (Ri + 1.6) (Gossard et al. 1982).

In a stable turbulent layer, the coefficient K H is approximated by the next local scaling experimental relation (Nieuwstadt 1984; Hunt et al. 1985):

--

-

where Ni = (g/@)a@/az is the Brunt-Vaisala frequency and the average value of the constant CH varies over the range 0.17 f 0.08, and values below 0.10 were observed at heights above 150 m. Equation (6) implies that under stable atmospheric conditions the diffusion of heat takes place more by local mixing of air elements which are constrained in their vertical motions to a distance of aW/NB. Thus, using Eqs. (5) and (6):

In stable turbulent layers Ri usually varies between 0.1 and 0.3 so that the turbulence is maintained, which applies especially to elevated thin layers (Hall et al. 1975; Mahrt


and Lenschow 1976; Nieuwstadt 1984). A feedback mechanism balances the increase of layer thickness Ah due to turbulence, with a decrease of temperature change A@ or, usually, an increase of wind speed change AU, so that Ri % (g/O)AhAO/(AU)2 stays around its critical value of 0.25. This value represents a balance between the positive shear production and the negative (in stable layers) buoyant production in the turbulent kinetic energy budget. However, regions with R i up to 1.0 may correspond to sound backscatter regions, since turbulence once generated can persist up to values of Ri = 1.0 (Neff 1988). The factor containing Rf = R i ( R i + 1.6) in Eq. (7) does not vary significantly because of the weak 1/3 power dependence and, therefore, an average value of 0.25 may be used for R i . Thus, the above equation and the C; and 0; measurements with an acoustic sounder can give an estimate of the local potential- temperature gradient.

In order to avoid the uncertainty of the Ri value when wind speed measurements are not available, the equation (Tatarskii 1971; Kaimal 1973):

C; = 3.2Ng~-‘ /~

can be used, where E and No are the dissipation rates of turbulent kinetic energy and temperature variance, respectively. No can be approximated using the temperature variance budget under stable, horizontally homogeneous and steady-state conditions (Stull 1988) as:

-ao N~ = -wief-,

az

C; = 2213.

while E can be estimated by the velocity structure parameter:

Thus, using Eqs. (8a), (8b), (8c) and (6) the temperature structure parameter is given by:

where C?. C: and 0; can be measured by the acoustic sounder and a@/az is the unknown local potential-temperature gradient. However, the computation of C:, for example by the structure function of the vertical-velocity measurements (section 3(b)), is based on the hypothesis of local isotropy which may not hold under stable conditions (for instance, under the presence of gravity waves).

The gradient Ri can be estimated by the following relation and the already estimated gradient of the potential temperature:

which is similar to Eq. (16) of Gossard and Frisch (1987). Equation (10) comes out by Eqs. (7) and (9), eliminating 0; and using the relationship between Ri and Rf under stable conditions which was presented above. Also, Eq. (10) can come out from the budgets of temperature variance and turbulent kinetic energy assuming equal turbulent outer length-scales of temperature and momentum under steady-state stable conditions (Gossard et al. 1982). This equation does not include 0; and could be used inversely


to estimate the gradient of the potential temperature assuming an Ri value around the critical value of 0.25. However, unlike Eq. (7) this estimation would be more sensitive to the actual value of the unknown Ri .

Furthermore, it is possible to estimate other significant parameters of the stable layer. For example, the local heat flux can be computed by wfOf = -(3.2)-'(C,2/2)1/2{C;/ ( a @ / a z ) ) which comes out from Eqs. (8a), (8b) and (8c), while the wind speed gradient can be estimated by aU/az = ( ( g / 0 ) ( a 0 / a z ) / R i ) 1 / 2 using the assumption of an almost steady value of Ri at about 0.25.

In order to incorporate the contribution of humidity fluctuations in the intensity of backscattered sound (see section 4(a)(i)) and the turbulent kinetic energy budget, or when the humidity changes significantly with height, the temperature 0 in the above Eqs. (7), (9) and (10) should be replaced by the virtual temperature 0, = O( 1 + 0.61q), where q is the mixing ratio of water vapour. Also, C? should be replaced by the structure parameter of the virtual temperature C;, which is, in general, proportional to the intensity of backscattered sound to high accuracy (Coulter and Wesely 1980).

(ii) The case of the temperature inversion at the top of the CBL. For the temperature inversion at the top of the CBL, an additional method to estimate the average potential- temperature gradient (or else the potential-temperature jump) is based on the Wyngaard and LeMone (1980) model for the structure of the inversion under quasi-steady, horizontally homogenous and cloud-free conditions. Using their Eq. (42), which comes out from an analysis of the budget of temperature variance, the results of their appendix A and the assumption that the potential-temperature gradient at the base h, of the inversion (at the top of the mixed layer) is ro M 0, it comes out to a first-degree approximation for a = Ah/h, (Ah is the thickness of the inversion) that:

(1 1) The symbol () indicates an average value in the inversion layer, Wei is the entrainment velocity at the middle of the inversion Zi (approximately the height of the local maximum of C;, see sections 1 and 4(a)), A 0 is the potential-temperature jump across the inversion and r2 is the potential-temperature gradient at the top of the inversion h2. The factor AO/Zi that is not included by other researchers (Fairall 1984) is significant for low values of Zi, like the cases examined in our work. The quasi-steady conditions require that A 0 and a are constant (or change very slowly) with time. Equation (1 1) is changed a little when the contribution of the total derivative is not neglected in the budget of temperature variance. Thus, in general, the term AO(r2 + A0/2i)/6 is reduced by (0.05 + 0.01/a)(A@)2/Zi according to the results of Wyngaard and LeMone (1980).

C: can be computed from the vertical-velocity measurements of the acoustic sounder with the requirement of local isotropy, as already mentioned. The gradient r2 may be estimated under quasi-steady conditions by the parameters a, Ah and A 0 of the inversion using Eqs. (56) and (A.7) of Wyngaard and LeMone (1980):

-

(C?) = 1.6(C,2/2)-'/2W,iA0(r2 + AO/Zi)/6.

r2 = (A@/Ah){ 1 + (a/2))/{ 1 + (a1 1/12)). (12) The near-middle height Zi and the thickness Ah of the inversion layer-entrainment

zone are estimated from the height and the width of the local maximum of C;, respectively, as described below. First, the backscatter intensity is corrected for propagation effects (see section 3(b)). If such a correction (especially for the sound absorption) is not applied, it is very difficult to detect the local maximum of C; of the entrainment zone

44 C. G . HELMIS er af.

when that is at high altitudes. Next, the 80 s average profiles of C; (see section 2) are processed with an algorithm that searches for significant changes of C; with height in order to detect the first (lowest) significant local minimum and the following significant local maximum of C; which are expected in the CBL according to section 4(a)(i). The 80 s average profiles are smooth enough in order to apply our detection algorithm which is similar to the one described in the appendix of Kalogiros et al. (1995). The height of the local maximum of C; is assumed to be the ‘instantaneous’ Zi height, while its width includes a local turbulence effect and a small trend effect (broadening) due to the 80 s averaging time. The ‘instantaneous’ Zi are used to estimate the trend during the analysis run which is 15 or 30 minutes (section 2). Finally, the heights and the C: values around the first significant local maximum of the run average C; profile are used as an independent variable and weighting factor, respectively, in order to estimate the run average Zi (the first-order moment) and Ah (square root of the second-order central moment). The linear trend effect for the run estimated by the ‘instantaneous’ Zi is removed in the Ah estimation. A somewhat different approach to estimate run average Zi and Ah has been used by Beyrich and Gryning (1998). However, they did not apply any sound absorption correction to C; profiles (missing, thus, high-level inversions), they did not include the local turbulence effect (they estimated the run average Ah by the variance of ‘instantaneous’ Zi) and their ‘instantaneous’ Zi corresponded to a long (especially for the first morning hours of rapid evolution of the CBL and, thus, a significant trend of Zi) time averaging of 5 minutes and 3-points running mean (that was 30 m height averaging) profiles.

The entrainment velocity Wei = aZi/at - W (Stull 1988) can be estimated from the change of ‘instantaneous’ Zi with time during the analysis run, assuming that large-scale subsidence, W, is weak (typical values of 0.005-0.01 m s-l or 18-36 m per hour at a height of 1000 m and smaller values closer to the ground) relative to aZi/at. However, this assumption may be a significant limitation for the application of this method, and it may be necessary to estimate the large-scale subsidence from the horizontal divergence of the wind speed or the warming rate above the ABL. In the case that the change of Zi with time is not clear or negative, a theoretical expression for Wei should be used. This can be obtained from the conservation equation of mean temperature at the base ho of the inversion and its integral form over the inversion (Wyngaard and LeMone 1980). The above process leads to the next equation for We0 at height ho:

Qo .- U

l + a - a Y A@’ We0 =

where Y * 1/2 - G2/12, G2 = r2Ah/AO and Wei (1 + ~/2)Weo. Qo can be estimated from the similarity analysis of the C; profile in the mixed layer (section 4(a)). Equation (13) incorporates the effect of wind shear (Fairall 1984) in the measured non- dimensional thickness a of the inversion.

Once more, if the contribution of humidity fluctuations with time or height is significant in the intensity of backscattered sound (like when the humidity jump across the inversion is significant) then the temperature in the above equations (including the terms of the structure parameter of temperature and the heat flux) should be replaced by the virtual temperature.

5 . EXPERIMENTAL RESULTS

In the next two subsections, results of the application of the theoretical relations of section 4(b) to acoustic sounder measurements from the first temperature inversion


at the top of the CBL in order to estimate the potential temperature jump A @ are presented. This inversion is easily detected by the local maximum of the C$ profile, and corresponds to a characteristic jump of the potential temperature which varies with time. Thus, the corresponding cases can provide a good test for the application of the method. In addition, Eq. (1 1) is applied only to the case of an inversion at the top of the CBL, which is of interest for the dispersion of air pollutants in the urban environment. Also, the parallel evolution of the parameters that control the evolution of the CBL, like Qo, u* , Zi, Ah, A @ , AU and r2, during the presented experimental days is examined. Especially, the time change of A @ obeys the next equation (Deardorff 1979):

aA@ Qo

at h0 -- - we2r2 - -,

where We= is the entrainment velocity at the top of the inversion (estimated as described in section 4(b)(ii)) and ho = Zi - Ah/2. This equation represents a balance between the tendency of A@ to increase due to the growth of the CBL and to decrease due to the surface heating of the mixed layer. Also, it agrees with Eqs. (12) and (1 3) assuming quasi-steady conditions ( a A @ / a t = aa/at = 0).

(a) Comparison with direct measurements In this subsection, the estimates of the jump of virtual potential temperature, A@",

applying Eqs. (7), (9) and (1 1) to acoustic sounder measurements are compared with direct measurements obtained with a tethered meteorological balloon on an experimental day at the NOA site. The comparison refers to virtual potential-temperature gradients according to section 4(b), because the effect of the humidity jump across the inversion (see Fig. 6(d)) is significant. During this experimental day (23 June 1996), the sky was clear and a moderate north-westerly synoptic wind was observed. A relatively strong low-height (120-150 m) inversion was established in the first morning hours and later it was eroded. After 1000 LST the local sea breeze dominated the experimental site. The acoustic sounder data were processed in 15 minute runs with a 7.5 minute overlap in order to track the rapid evolution of the inversion.

Figure 6 shows the profiles of the atmospheric parameters of interest measured with the tethered balloon (the height is the distance from the ground which is at 110 m a.s.1.). The higher part (above 350 m) of the last profile (0809-0839 LST) corresponds to the characteristics of the overlying free atmosphere (a 4 m s-' west wind, a 5 g kg-' water vapour mixing ratio and a low stability of 1 WlOO m). Below this layer there are two turbulent height layers at 150 m and 300 m, which are also shown in the facsimile record of the acoustic sounder in Fig. 7. According to the profiles shown in Fig. 6, the lower and thinner turbulent layer is generated by a strong temperature inversion (thermal turbulence) with a 1.5 K increase of the potential temperature in about 35 m and a significant change of the wind direction. The overlying, thicker and less stable turbulent layer is characterized by a strong change of the wind speed (mechanical turbulence). The surface layer up to 150 m is slightly stable in the early morning and later it becomes isothermal (mixed layer)-with a 1 K increase of temperature at all heights due to redistribution of the heat by solar radiation on the ground-above the first 50 m (unstable layer). This layer is quite moist (10 g kg-') and the wind direction is from the sea (south to south-west), except for the first profile when the orientation of the tethered balloon was restricted by the very light wind. On the other hand, the atmosphere above the strong inversion at 150 m is dry (6 g kg-' and south-west to west-north-west wind). The characteristic step change of humidity or wind direction in this inversion layer is a good tracer of its bounds.

46 C. G . HELMIS etal.

400 P

(a) -

300-

E - 5

100-

- 8:098:39LST

I I 0 2 4 6

wind speed (Ins-')

t 400

(4 300-

E E 200- Y

.P I

. - * . . 7:47-8:01 LST

d- 8:088:39LST

293 296 299 302 3 Potential temperature (K )

15

400

(b)

300

h

Y E

t *0°

100

0

- - - . 6:216:55 LST

?'

.-

. I I I - , I t

90 180 2jo 3 wind direction (deg )

3001

E 200 7 F

l o o 1

7:478:01 LST - 8:098:39LSl - - * - . 300

T - E 200

100

0 4 6 8 10

Water vapour mixing ratio (gkg-l )

o+ 4

7:478:01 LST - 8:098:39LSl

Water vapour mixing ratio (gkg-l )

2

Figure 6. The profiles of (a) wind speed, (b) wind direction, (c) potential temperature, and (d) water vapour mixing ratio measured with the meteorological balloon on 23 June 1996 at the National Observatory of Athens

site.

The low height inversion at 150 m is not destroyed until 0800 LST because of the slow increase of Qo and the significant stability (2 WlOO m, as it is shown later in Table 3) of the overlying layer. These characteristics result in a small rise and delayed erosion of the inversion according to Eq. (14). Qo does not increase rapidly because of a local southerly wind (shown in the last two profiles of wind speed as a local maximum jet). The estimate of Qo using the C: profiles measured by the acoustic sounder (section 3(b)) is shown in Fig. 8(a). The estimation of u* by the u i profile is described by Kalogiros et al. (1999).

The mechanism of the destruction, finally, of the low height inversion can be understood using the facsimile record of the acoustic sounder in Fig. 7. According to this, in the time period 0730-0750 LST there is an intrusion of dry air from the free atmosphere down to the low height inversion. This can be seen, also, in the second and


Figure 7. The facsimile record of the acoustic sounder on 23 June 1996 at the National Observatory of Athens site.


0.06 7 0.50

0.05

-1

y 0.03

0.02

0.40

0.30 I 0.20

(by %9 c 0.10

0.0 -1 6 7 8 9

Tim (Lsr)

""F o 06290644LST 07lBO734LsT

1 E-4 1 E-3 1 E-2

C( (K2m-",

160

120

P 5i

80 - 40

0

Figure 8. The evolution of (a) the parameters of the convective boundary layer: Q, by the C; profile, and u* by the u: profile measured with the acoustic sounder, (b) Zi, Ah of the inversion, (c) the various A& estimates, and (d) the C:" profiles on 23 June 1996 at the National Observatory of Athens site. See text for explanation of

symbols.

third tethered balloon profiles where there is a small change of wind direction from west-north-west to south-west and drier air above the inversion relative to the first and the last profiles. Because of the low stability of the free atmosphere, the low height inversion is rapidly destroyed (see the time period 0747-0801 LST of descent of the second balloon flight that is visible in the first part of the facsimile record in Fig. 7) and the mixed layer quickly extends up to the higher stable turbulent layer at 300 m (with subsequent variations down to 200 m). The last potential-temperature profile (0809- 0839 LST) shows that there is a true mixing (a constant with height wind speed, wind direction, potential temperature and water vapour mixing ratio) from 30 m above ground up to the base of the remaining lower inversion at 203 m f 3 3 m which was the location of the inversion at the time of the balloon flight according to the second part (0753- 0925 LST) of the facsimile record in Fig. 7. This figure shows, also, that this inversion was at 300 m in the time period 075348 16 LST. The small thermal turbulence structures at 200 m that are seen at the same time period are the remnant of the intrusion of dry air from the free atmosphere down to the initial low height (150 m) inversion as described above.


TABLE 3. THE ESTIMATES OF VARIOUS PARAMETERS OF THE INVERSION AT THE TOP OF THE CONVECTIVE BOUNDARY LAYER FROM (a) ACOUSTIC SOUNDER MEASUREMENTS AND (b) THE CORRESPONDING DIRECT MEASUREMENTS BY THE TETHERED METEOROLOGICAL BALLOON DURING THE MORNING OF 23 J U N E 1996 AT THE NATIONAL OBSERVATORY OF

ATHENS SITE ~

Time (LST) Zi (m) Ah (m) AOv1.2.3 (K) AU(m s-I) Rb r2 (W100 m)

6:21-6:55 (b) 148 34 1.2 1.4 0.68 2.2 6:29-6:44 (a) 148 52 1.6, 1.2, - 2.3 0.53 2.5 7121-7~45 (b) 161 33 0.9 2.6 0.14 2.0 7:19-7:34 (a) 128 54 1.1.0.8, 1.0 2.0 0.45 1.7 7:47-8:01 (b) 163 32 0.4 2.4 0.07 1 . 1 7:45-8:00 (a) 133 52 0.4, 0.4.0.6 2.3 0.19 1 .o 8109-8139 (b) 203 67 1.4 3.4 0.27 2.1 8120-8:35 (a) 195 60 1.3, 1.0,0.8 2.6 0.37 1.9

See text for explanation of column headings and bold type.

With the destruction of the low height inversion the ground heat is, also, redistributed by turbulence up to 300 m. Thus, the value of Qo estimated by the C; profile, which is actually representative of the state of the atmosphere in the layer up to the temperature inversion, temporarily decreases (Fig. 8(a)). This may be the result of an increase of the vertical-turbulent-transport term in the budget of heat flux due to an increase of wind shear near the surface (see the balloon data in the time period 0747-0801 LST of the rapid destruction of the initial low height inversion). Fast tower measurements were not available on that day in order to estimate surface fluxes using the eddy correlation method. Further increase of Qo (intense thermal activity) results in the destruction of the higher turbulent stable layer at 0930 LST. After 1000 LST, the local sea breeze reaches the experimental site and the south-south-westerly wind increases significantly (not shown here).

Figures 8(b) and 8(c) show the time series of various parameters of the first height inversion above the mixed layer. The estimates A@,,l, A@,2 and A@,3 were obtained by Eqs. (7), (9) and (1 l), respectively. The height Zi of the middle and the thickness Ah of the inversion are estimated from the average height and the width of the local maximum of the C; (proportional to echo intensity) profiles (see section 4(b)(ii)). These profiles, that $ere measured by the acoustic sounder and correspond to the time periods of the balloon profiles, are shown in Fig. 8(d). The sound absorption coefficient is 1.8 x m-l in the moist surface layer below the low height inversion and 2.2 x m-' in the overlying dry layer according to the direct measurements of the profiles of temperature and humidity. The actual value used for the correction of the C;, profile that was estimated by the acoustic sounder measurements was 2.3 x m-' which corresponds to an average temperature of 25 "C and a relative humidity of 50%, but the difference in the estimated C; profile was not significant. The evolution of the inversion parameters in these figures agrees with the previous description. Especially, the abrupt decrease of A@,, during the destruction of the low height inversion and the following increase (where now the parameters correspond to the higher stable turbulent layer) are obvious. The differences among the A@,,], A O V 2 and AOV3 estimates are explained in section 5(b).

Table 3 compares the estimates of A@,,, Ri, AU and r2 by the acoustic sounder data indicated by (a), with the direct measurements by the tethered balloon indicated by (b) for about the same time periods. The actual upper and lower limits of the temperature


inversion layer were estimated by examining the profiles of potential temperature, water vapour mixing ratio (which is a less ambiguous indication of the inversion layer than the profile of potential temperature) and wind direction in order to detect a layer of abrupt change of one of these parameters (see Fig. 6). Then, the.height Zi was estimated as the middle of the inversion layer and Ah as its thickness. The virtual potential-temperature jump AOv, the change of the wind speed AU = (Au2 + Au2)'/*, where u and u are the horizontal components of the wind speed, and the average (bulk) Richardson number Rb = (g/C3v)(A@v/Ah)/(AU/Ah)2 were computed by the change of the values of the parameters a,, u and u across the inversion layer. Also, the stability I'2 of the overlying atmosphere was estimated as the average gradient of potential temperature of the layer just above this inversion.

Using the acoustic sounder measurements, the parameters A@, Ah and Zi were estimated as described above (see the discussion of Fig. 8), the ambient potential- temperature gradient was estimated by the relation r2 = (A@,,/Ah)(l + 0.5a)/(l + l la /12) according to Eq. (12), where a = Ah/(Zi - OSAh), the average Richardson number was estimated by Rb = (C+,/C,2){(g/@,)/(AOV/Ah)} according to Eq. (lo), and the wind speed change was estimated by AU = {(g/Ov)AhAOv/Rb}'/2. The estimates of AO, shown bold in Table 3 are the ones (the larger) used for the estimation of r2, Rb and AU by the acoustic sounder data. In the first run, there is no estimate of A0,3 because the atmospheric conditions did not correspond to an unstable state (the -4/3 law, which is expected to hold in the mixed layer, was not observed in the profile of C;,; see Fig. 8(d)). In the last run, the significant error of AOv3 is due to the rapid intrusion of the dry air of the free atmosphere and, thus, the assumption of quasi- steady conditions does not hold even approximately. The low values of A@,, during the destruction of the low-height inversion (at 150 m) have been recorded by the direct measurements with the meteorological balloon, too.

The results in Table 3 show that the estimation of the middle of the inversion Zi from the local maximum of the C+v profile, generally, underestimates it. This implies that it corresponds to a height closer to the base of the inversion as already mentioned in section 1. Also, the estimation of Ah by the width of the C+v profile overestimates the actual value, probably because the local maximum of C:, extends to the mixed layer due to entrainment (section 4(a)(i)). An additional factor for this overestimation is the actual spatial resolution ( I 7 + 8.3 m) of the C:, profile due to the finite length of the acoustic pulse and the applied moving-average filter (see section 2). The accuracy of the estimation of AOv especially by Eqs. (7) and (9) is high (0.4 K maximum error) and the following of its evolution with time is quite satisfactory. This comparison indicates that the AeV1 estimate may be closer to the actual value of AOv. The quality of the estimation of the ambient potential-temperature gradient is also satisfactory. The estimates of Rb by the acoustic sounder data follow the evolution of the values computed from the direct measurements, but usually overestimate them because of the errors in A@,, and Ah. The relatively high value (about 0.7) of Rb in the first run is probably due to the use of average gradients of temperature and wind speed for its estimation (Stull 1988). The estimates of AU by the acoustic sounder data are significantly influenced by the errors in A@", Ah and Rb. It should be noted that the tethered balloon measurements have a low height resolution (30-50 m) and represent 'instant' estimates. Thus, differences with the estimates by the acoustic sounder are expected (for example, the corresponding height and thickness of the inversion may differ).


(b) Correlation of the various A @ estimates In this section the correlation of the various A@ estimates (AO1, A 0 2 and A 0 3 ) is

examined using acoustic sounder data collected at the Messogia Plain in order to show their consistency and general behaviour. These data come from eight experimental days during the summer of 1995 when a low-height temperature inversion was observed in the facsimile record of the acoustic sounder for at least two to three morning hours. Since no humidity profiles where measured, in this section we refer to potential-temperature changes A 0 instead of A@".

For two of these experimental days (one with a strong inversion and another with a weak inversion) the evolution of Qo, u*, Zi, Ah, AU and r2 is examined and it is shown to be consistent with the development of the Ah0 estimates according to Eq. (14). These two case-studies were selected because wave motions on the inversion were clear and, thus, an estimation of the Brunt-Viiisala frequency NB and the potential-temperature gradient was possible since ZVi % ( g / @ ) A @ / A h . NB is an upper limit of the frequency of non-evanescent gravity waves developed on the inversion (Stull 1976; Carruthers and Moeng 1987). The wave period can be estimated from the echo intensity (or the facsimile record) and the vertical-velocity time series. Even though this comparison is a qualitative one, it is a good indication for the application of the acoustic sounder method.

In the first case-study (16 June 1995), the facsimile record of the acoustic sounder presented in Fig. 9 shows wave motions on the low height (200-300 m) inversion with a time period of about 3 minutes and small amplitude before and after the onset of thermal plumes at the surface around 0700 LST. This small time period corresponds to a relatively strong local gradient of the potential temperature (3.6 WlOO m) according to the estimation by the Brunt-Vaisala frequency. Thus, high values of A@ are expected, in agreement with the corresponding estimates in Fig. 10. The parameters Zi, Ah, AO1, A02 and A03 were estimated by the acoustic sounder data similarly with section 5(a). The r2 estimates were based on A03 and Eq. (12), while the AU estimates were based on A03 and a constant Ri = 0.25. The estimates of Qo and the friction velocity in this experimental site (Fig. 10(a)) were obtained by the direct measurements (eddy correlation). The inability of the inversion layer to rise quickly indicates a strong ambient temperature gradient r2 which agrees with the corresponding estimates in the same figure. The interruption of this small trend to rise in the time period 0800 LST to 0900 LST agrees (see Eq. (14)) with the approach to a layer of increased stability (3.5 W100 m) at about 300 m, which can be seen in the facsimile record to exist even before 0700 LST, and the temporary fall of Qo. After 0800 LST the intensity A 0 of the inversion decreases rapidly. This is due to the increase of Qo and the destruction of ambient stability by the increased north-easterly wind speed (and, thus, an increase of mechanical turbulence and mixing) which was observed in the direct measurements with the meteorological mast (not shown here). The inversion at the top of the CBL is visible in the facsimile record until 1000 LST when it has become too weak.

The various estimates of A 0 agree very well with each other (the A01 estimate is intermediate of the other two), except the overestimation by A03 at the two runs when the inversion almost stopped to rise. In these runs a theoretical estimate of Wei according to Eq. (1 3) was used in Eq. (1 1) instead of the ascending rate of Zi, which was practically zero or negative. The estimation of entrainment velocity by Eq. (13) assumes quasi-steady conditions and, probably, gives low estimates in our case, leading to an overestimation of A 0 by Eq. (1 1). From the time series shown in Fig. 10, it is, also, characteristic of the parallel evolution of Ah and Zi, leading to an almost constant in the


Figure 9. The facsimile record of the acoustic sounder on 16 June 1995 at the Messogia site.


0.08

0.06

-f !! y 0.04

0.02

0.00

1

0.0 -1 2.0 0.0 6 7 8 9 10 6 7 8 9 10

The (Lsr) (Lsr)

Figure 10. The evolution of (a) the Q, and u* parameters of the convective boundary layer, (b) Zi, Ah of the inversion, (c) the various A@ estimates, and (d) the ACJ, rz estimates on 16 June 1995 at the Messogia site. See

text for explanation of symbols.

time ratio a = Ah/ ho. Figure 10 shows that both the ambient temperature gradient (1- 3.5 WlOO m, as already mentioned) and the wind shear in the inversion (2.54.5 m s-l wind step) were strong.

On the latter experimental day (25 August 1996), the facsimile record of the acoustic sounder in Fig. 11 shows that before 0700 LST there is no thermal turbulence above 150 m. This observation implies a weak environmental potential-temperature gradient, in agreement with the rapid ascent of the layer of the low-height inversion which reaches 450 m at 0900 LST. The facsimile record shows, also, another stable layer above 400 m at 0800 LST which has a descending trend and is connected with the intensification of the lower inversion ( A 0 increase, see Fig. 13(c)) and the generation of significant wave motion on it. These wave motions are evident, also, in the vertical- velocity measurements shown in Fig. 12. The time period of these wave motions is about 6-10 minutes which implies a weak potential-temperature gradient of 0.3-0.9 W100 m. After 0900 LST an increase of the north-east wind speed to 4 m s-', according to the meteorological mast data, resulted in a weakening of the inversion, but increased noise in the acoustic sounder system did not permit the observation of the evolution of the inversion. The appearance of an inversion at 300400 m after 1030 LST is due to the

54 C . G . HELMIS er al.

Figure 1 I . The facsimile record of $e acoustic sounder on 25 August 1995 at the Messogia site.


Figure 12. Time series of the vertical velocity between 200 and 450 m in the time period 08274913 LST on 25 August 1995 at the Messogia site.

arrival of the local sea breeze at the experimental site. The sea breeze arrival is, also, connected with the fall of the surface heat and momentum flux at 1000 LST (Fig. 13(a)).

The comparison of the various A 0 estimates in Fig. 13(c) shows a similar behaviour, with the A 0 1 estimate intermediate of the other two, and a small overestimation by A 0 3 as in the first case-study. This will be also shown in the correlation between the three estimates for all the runs examined during the experimental period that follows. The close agreement of A 0 3 and A 0 1 is also due to the good estimation of the entrainment velocity, since the inversion (the first local maximum of C:) presents a clearly detected and steady ascending trend (0.035 m s-I). In the first two runs the A 0 estimates present a decrease with time, something that can be understood from Eq. (14) since r2 is almost zero as already mentioned, while Qo increases to positive values. After 0800 LST the A 0 estimates increase due to the encounter with the overlying stable layer mentioned above. This layer should be characterized by sufficient potential-temperature gradient r 2 (about 1 WlOO m) in order to overcome the heating of the mixed layer and to give the observed positive change of A 0 (0.5 W30 minutes) according to Eq. (14). The peak of the estimate of r2 around 0830 LST in Fig. 13(d) agrees with this expectation. The stabilization of A 0 around 0900 LST is due to equilibrium between the two terms in Eq. (14). Further increase of Qo and the low inversion height at 1100 LST lead to smaller values of A@. It is worth mentioning that the evolution of the Ah estimates (Fig. 13(b)) is similar to the A 0 estimates which is probably connected with the almost steady value, near the critical value of 0.25, of Ri.

Figure 14 shows the correlation of A 0 2 and A 0 3 with the intermediate A 0 1 estimate for the whole runs from the eight experimental days examined. The last estimate is considered as more reliable (see section 5(a), too) and the reference for the comparison. This can be attributed to the fact that Eq. (7) contains only the measurement of the average vertical-velocity variance in the inversion in contrast to the inclusion of C: (estimated using the local isotropy assumption) in Eq. (9), and Wei and the assumption of quasi-steady conditions in Eq. (1 1) which may be sources of significant error. Also, Eq. (1 1) applies only to the inversion at the top of the CBL case. On the other hand, the vertical-velocity variance, which appears in Eq. (7), is more easily measured and with sufficient accuracy by the acoustic sounder in the inversion turbulent layer where the signal-to-noise ratio is relatively high.

56 C. G. HELMIS er al.

0.12 0.70 (a)

0.08

I r 8 0.04 - 8

0.00

0.60

0.50 - I 0.40 3 -

a, 10.30

7 8 9 10 11 12 T i (rn

0.8 -

0.6 -

' 0.4-

8 -

0.2 -

7 8 9 10 11 Tkne (LST)

100 30 7 8 9 10 11 12

mM (Lsr)

4.0

3.0 A

Ti! - z

2.0

1 .o

Figure 13. As Fig. 10 but for 25 August 1995.

The overestimation by A 0 3 in Fig. 14(b) is probably due to the underestimation of the entrainment velocity estimated from the trend of the maximum of the C; profile during the analysis run ignoring the effect of subsidence. The dispersion of these estimates is connected to the assumption of quasi-steady conditions that may not hold in the early morning hours examined here. The small underestimation by A 0 2 in Fig. 14(a) may be caused by a small underestimation of C:. This is quite possible in stable layers where the outer scale of turbulence is relatively small and the hypothesis of local isotropy may not hold. The assumption of an almost steady value at about 0.25 for Ri in Eq. (7) is justified by the high correlation of A 0 2 and A 0 1 , since the first estimate does not include this assumption. The actual value of CH that was used in Eqs. (7) and (9) was 0.06 instead of the average observed value of 0.17 f 0.08. This is in agreement with the trend to values below 0.10 at heights above 150 m found by Hunt et al. (1985). The accuracy of the value of CH used in Eqs. (7) and (9) can be concluded by the agreement between the A 0 1 and A 0 2 estimates (and not with any direct estimate of A@) that come out from the application of these two equations. Actually, this agreement was the basic criterion (taking into account the range of values found in the literature) for the choice of the value of CH. This choice is, also, supported by the agreement of A 0 1 and A 0 2 with the independent estimate A 0 3 by Eq. (1 1) as well as the results of the

POTENTIAL-TEMPERATURE GRADIENT FROM ACOUSTIC BACKSCATTER

1.5-

E a $4 1.0-

0.5 -

.

0.0

57

. * . 9 .

..*# *. * 8 *b - - . .*.;

I , I ' / '

I $

1 .o -

-

1.6

- 1.2 Y

* . . . .. 9

: ' .;* I . .. .:

. . .- . . * . .:

0.8

0.4

5 .. ** .* 0 . 0 , I I I , 1 1 I , I

0 40 80 120 160 200 Ah (m)

Figure 14. The correlation of (a) A 0 2 and A@, (b) A@3 and A@1, (c) A 0 1 and Ah, and (d) a = A h / h , and 2, for all the runs examined during the whole experimental period at the Messogia site for the summer of 1995.

See text for explanation of symbols.

comparison with direct measurements described in the previous section. Figure 14(c) shows that for the specific experimental days the potential-temperature gradient in the inversion at the top of the CBL varies in the range 0.5-1.5 WlOO m. Also, the ratio a = A h / h o shown in Fig. 14(d) does not vary significantly, as it is assumed under quasi- steady conditions.

6. CONCLUDING REMARKS

This work describes the estimation of the potential-temperature gradient in turbulent stable layers using the backscatter intensity and the vertical-velocity measurements of a vertically oriented, monostatic acoustic sounder. The backscatter intensity is used to compute the structure parameter of temperature C: after a calibration of the system using direct C; measurements in the CBL. These direct measurements are 'extrapolated' to the first height of measurements of the acoustic sounder using the corresponding similarity relation. Another, equally accurate and simple, method to calibrate the system is based on the estimation of Qo by both C; and C: profiles in the CBL. This latter method uses exclusively acoustic sounder data providing, thus, an inter-calibration of

2.0 30r 0.0 1

0.0 1 .o 2.0 3 .0 (0

l . O T 1

d I

200 300 400 500 600 4 (m)


the system. Ct and cr; are computed by the vertical-velocity measurements of the acoustic sounder. Specifically, the structure function of the vertical velocity in the vertical direction is used to estimate C:.

The calculation of C;, C; and 0; by the acoustic sounder enables the estimation of the potential-temperature gradient a@/az (generally, the gradient of the virtual potential temperature) in turbulent stable layers using theoretical relations which are based on the budgets of the temperature variance and the turbulent kinetic energy. Two general equations ((7) and (9)) relating these parameters with the local potential-temperature gradient in a stable layer were presented. The uncertainty due to the usually valid assumption of a constant Richardson number, near the critical value of 0.25, in the first equation is absent in the second one. Instead of assuming a constant Ri, the wind speed gradient may be estimated by Doppler measurements of a three-axial acoustic sounder and, thus, the only unknown in Eq. (7) would be the potential-temperature gradient. An empirical constant CH which relates the local heat flux to the gradient a@/az and 0; is present in both equations. Both uncertainties can be avoided if Eq. (10) is used and the wind speed gradient is estimated by Doppler measurements of a three-axial acoustic sounder.

In the case of the temperature inversion at the top of the CBL, an additional method (Eq. (1 1)) for the connection of the space averaged C; with the temperature jump A@ based on Wyngaard and LeMone’s (1980) model of the inversion was presented. This equation needs an estimate of the entrainment velocity in addition to an estimate of C:. The entrainment velocity is usually computed from the rate of change of the height of the inversion Zj that corresponds to the local maximum of the C; profile, ignoring any possible large-scale subsidence. In this case, an estimation of the potential-temperature gradient of the free atmosphere above the inversion is also possible.

The various relations for estimating a@/az using the measurements by a monostatic acoustic sounder were tested on cases of an inversion at the top of the CBL. The comparison of the A@ estimates with direct measurements of the temperature profile acquired with a tethered meteorological balloon during an experimental day proved their satisfactory accuracy (0.4 K at least), especially for the estimates obtained by Eq. (7) which involves the easily measured and with sufficient accuracy C; and cr; parameters. This comparison, also, showed the ability of the method to provide satisfactory estimates of the local Richardson number and the potential-temperature gradient of the overlying free atmosphere. Two other case-studies were examined with the approximate value of a@/az inferred by Ni. This was estimated by the wave motions on the inversion. The time evolution of various parameters that control the evolution of the CBL was shown to be consistent with the development of the A@ estimates using the theoretical equation for the time change of A@ and the turbulent structure observed in the facsimile record of the acoustic sounder. The high correlation of the estimates of A@ by Eqs. (7), (9) and (1 1) for the whole runs from eight experimental days confirmed the validity of an almost constant Ri in turbulent stable layers and the accuracy of the assumed value of the CH constant. Also, an underestimation of entrainment velocity (due to the neglect of large-scale subsidence or the use of the theoretical relation (13)) that resulted in an overestimation of A@ by Eq. (1 1) was concluded.

Other remote sensing techniques, like the Radio Acoustic Sounding System (RASS), can give the profile of (virtual) temperature, but with limited resolution (down to 50 m), relatively high first height of measurements (down to 100 m) and temperature precision of 1 K (May et al. 1990). Typical values of the potential-temperature jump A@ and the thickness of the inversion at the top of the CBL over land in the first few morning


hours are 0.5-2 K and 50-100 m, respectively, (Hall et al. 1975; Dubosclard 1982), which are at the limits of the RASS precision and resolution. The acoustic sounder methods presented here can be an alternative and complementary, low-cost solution for the estimation of the temperature gradient in the case of low height and shallow thickness inversions.

Asimakopoulos, D. N., Cole, R. S., Caughey, S. J. and Crease, B. A.

Asimakopoulos, D. N., Moulsley, T. J., Helmis, C. G., Lalas, D. P. and Gaynor, J. E.

Beyrich, F. and Gryning, S.

Brown, E. H. and Clifford, S. F.

Brown, E. H. and Hall, F. F.

Carmthers, D. J. and

Caughey, S. J. and Palmer, S. G. Moeng, Chin-Hoh

Chintawongvanich, P., Olsen, R. and Biltoft, C. A.

Coulter, R. L. and Wesely, M. L.

DeardorfT, J. W.

Dubosclard. G.

Fairall. C. W.

Gaynor, J. E.

Gossard, E. E. and Frisch, A. S.

Gossard, E. E., Chadwick, R. B., Neff, W. D. and Moran, K. P.

Hall Jr, F. F., Edinger, J. G. and Neff, W. D.

Hunt, J. C. R., Kaimal, J. C. and

Kaimal, J. C.

Kaimal, J. C., Wyngaard, J. C., Izumi, Y. and Cote, 0. R.

Kaimal, J. C., Wyngaard, J. C., Haugen, D. A., Cote, 0. R., Izumi, Y., Caughey, S. J. and Readings, C. J.

Gaynor, J. E.

1976

1983

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1976

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1987

1979

1989

1980

1979

1982

1984

1987

1977

1987

1982

1975

1985

1973

1972

1976

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