[1] The diurnal variation of the rate of dissipation of turbulent energy over Arecibo (18°N, 67°W) for the 1989 Arecibo Initiative on Dynamics of the Atmosphere (AIDA '89) campaigns from 1–10 April and 2–9 May 1989 have been estimated using the random velocity field deduced from imaging Doppler interferometry data. A method for extraction of the vertical component of the gravity wave spectrum has been applied to the random velocity. There is a tendency for the turbulence to maximize at dawn and dusk. Estimates of the RMS vertical gravity wave component range from 2 to 5 m s^{−1}, and estimates of the RMS turbulent energy range from 5 to 100 mW kg^{−1}. Estimates of the vertical eddy diffusivity have also been calculated, ranging from <50 to 300 m^{2} s^{−1}. The buoyancy scale ranges from <100 m to 1 km. A cautionary note is added regarding the applicability of the Kolmogorov theory above 100 km.

[2] The capabilities of the Middle Atmosphere Structure Associated Radiance (MAPSTAR) imaging Doppler interferometry (IDI) radar have been described in detail by Brosnahan and Adams [1993]. In a more recent paper, Roper and Brosnahan [1997] have detailed the application of the structure function technique to the measurement of ɛ, the rate of dissipation of turbulent energy, using data collected by the MAPSTAR radar over Arecibo on 10 April 1989 during the second of the 1989 Arecibo Initiative on the Dynamics of the Atmosphere (AIDA '89) campaigns. Turbulent dissipation rates of from <10 mW kg^{−1} to 60 mW kg^{−1} were measured over the interval from 1200–1416 hours Atlantic Standard Time (AST), with the lower values being measured around 75 km altitude, increasing to a maximum at 90 km. Because signal reflection occurred at a virtual height of 96 km, values above 93 km were considered unreliable. In a more recent paper, Roper [2000] has developed an alternative to the structure function approach with a formulation based on the vertical random velocity component as measured by IDI. In this paper, this revised formulation is used to extend the measurements to successive two hourly averages (local mean solar time) calculated hour by hour for the AIDA '89 intervals 1–10 April and 2–9 May 1989 in order to investigate the diurnal variation of the turbulence.

[3] The background wind has been calculated as the two hourly mean using the MAPSTAR imaging Doppler interferometry data as described by Adams et al. [1985]. The daily mean plus diurnal and semidiurnal zonal and meridional components have also been calculated using the analysis of Groves [1959] as in work by Roper et al. [1993] and Turek et al. [1995]. Note that the scatterer echo rate is high enough for the Groves analysis to be applied on a day to day basis [see Roper et al., 1993]. As yet, no attempt has been made to extract gravity wave spectra from this data, although an attempt is made to subtract the gravity wave velocity from the measured random velocities (see section 2). For details of turbulence production at these altitudes, as caused primarily by breaking gravity waves, see Roper [1996] and Roper and Brosnahan [1997, and references therein].

2. Turbulence Measurements

[4]Roper [1998] reviewed the established methods for medium frequency (MF) radar measurements of ɛ, the rate of dissipation of turbulent energy, with particular emphasis being placed on the structure function formulation used by Roper [1966] in the analysis of radio meteor data. In this formulation, ɛ estimates are based on the measurement of line of sight velocities separated in space. This technique has its limitations when applied to imaging Doppler interferometry, in that the positions of identifiable scattering entities are determined only in the horizontal and are thus less than the true vector separations. Thus the turbulent dissipation rates estimated will be even further biased as upper limits. More recently, Roper [2000] has derived an equation

where ɛ is the rate of dissipation of turbulent energy, σ is the RMS turbulent velocity deduced from the velocity of the buoyancy scale as measured by IDI [see Roper, 2000] and ω is the Brunt-Väisälä frequency. One of the major factors inhibiting a full definition of turbulence parameters, and particularly the stability of the background atmosphere, is the lack of ambient temperature information. To calculate the Brunt-Väisälä for the intervals in question, the CIRA (1986) temperature profiles for April and May at 20°N (see Figure 1) are used. Note that the constant of proportionality in equation (1) is 1/2 to 1/3 of the generally accepted values [see, e.g., Weinstock, 1978; Hocking, 1985].

[5] The plots of the rate of dissipation of turbulent energy ɛ presented in this paper are computed from the RMS turbulent velocity σ = 0.77u_{β} [Roper, 2000], using the random velocities u_{β} of the IDI scatterers (measured line of sight minus the two hourly background velocity component) with zenith angles less than 3° recorded in 3 km altitude intervals from 72 to 114 km using (1). At least six days must contribute to each height/time entry in the superposed diurnal epoch, and the reflection point height must be above that entry for it to be counted.

[6] Note that a correction is applied to u_{β} to minimize the influence of gravity wave contamination as proposed by Roper [1998]. This correction relies on the universally observed gravity wave energy spectrum

which, when transformed to length space, yields

where v is the wave amplitude, or

[7] To apply this assumption to the data at any given height, the data is divided into successive concentric annuli of 3 km width (seven all together to cover the range from zero to 21 km from the zenith), and the line of sight RMS velocity deviation from that of the two hourly mean wind for all the scatterers within each annulus is calculated. A straight line is then fitted to these seven points, with the constant of the linear fit being determined as the gravity wave amplitude at zero zenith angle. The goodness of fit of the straight line is also calculated, and used to determine whether or not to subtract the gravity wave amplitude (weighted by the significance of fit) from the measured velocity determined from all scatterers within 3° of zenith. An example of this fit appears as Figure 12 of Roper [1998]. Figure 2 presents histograms of the distribution of R^{2}, the square of the correlation coefficient between the data and the straight line fit. The universal nature of the saturated gravity wave spectrum is again validated, with 80% of the April and 65% of the May data having correlation coefficients R > 0.7. Optical imaging during AIDA confirmed that there was greater gravity wave activity in April than in May. If the source of the turbulent energy is part of the gravity wave spectrum, it would seem logical that the turbulent energy would be greater in April rather than in May. While this is true of the measured values, it is not true for the corrected values, as is shown later. Thus the gravity wave component “extraction” technique may offset to some extent the fact that the values of the measured turbulence parameters in general represent upper limits, in that the theory underlying the turbulence calculations (as with all current radar estimates) considers the turbulence to be stationary and homogeneous.

3. Interval From 1–10 April 1989

[8] Data collected by the MAPSTAR IDI radar over Arecibo (18°N, 67°W) from 0000 hours local time 1 April to 2400 hours 10 April are used in this first study. The mean plus diurnal and semidiurnal components calculated using the method of Groves [1959] appear in Figure 3 and Table 1. Note that, while E layer reflection can occasionally be responsible for a lack of wind data above 90 km around noon (transmitted frequency was 3.175 MHz), the long-term means computed here, together with the interpolation inherent in the analysis, allow some assessment of lower thermospheric data for not only this interval, but also intervals lacking data at dawn and dusk. The Groves analysis, as presented by Turek et al. [1995] when applied to the 5–10 April interval, shows in their Figure 12 the primacy of the mean plus tidal components in the description of the background wind field for both the IDI and Arecibo incoherent scatter data. Maximum background wind gradients are of the order of 15 m s^{−1} km^{−1} in the morning and 10 m s^{−1} km^{−1} in the evening. As pointed out by Roper and Brosnahan [1997], values of wind shear that can initiate turbulence in the upper mesosphere by driving the Richardson number (Ri) below 1/4 are of order 30 m s^{−1} km^{−1} and are seldom achieved by the background wind. Thus the background wind is rarely a source of turbulence at upper mesospheric/lower thermospheric altitudes. However, while the shears present are not sufficient to produce turbulence, they do help in the destabilization of the region. Of course, a significant diurnal variation in the temperature profile could push Ri values lower (or higher, for that matter). Also, since the pulse width of the MAPSTAR radar is 4.5 km, higher shears at smaller scales, such as those associated with short-wavelength gravity waves, are not resolved.

Table 1. Mean, Diurnal, and Semidiurnal Profiles for the Interval 1–10 April 1989 at Arecibo (18°N, 67°W)

Height, km

Mean, m s^{−1}

Error, m s^{−1}

24 Hour

12 Hour

Amplitude, m s^{−1}

Error, m s^{−1}

Phase, hours

Error, hours

Amplitude, m s^{−1}

Error, m s^{−1}

Phase, hours

Error, hours

East-West (Zonal) Components

114

−6

4

36

6

23

0.9

36

6

6

0.4

110

−3

3

30

4

20

0.4

33

4

6

0.4

106

−7

2

26

2

21

0.4

31

2

7

0.4

102

−8

1

13

2

4

0.4

29

2

7

0.3

98

−9

1

32

1

8

0.3

25

1

7

0.3

94

−11

1

36

1

10

0.3

18

1

6

0.2

90

−16

1

30

1

13

0.3

20

1

5

0.2

86

−23

1

22

1

17

0.3

21

1

4

0.2

82

−28

1

31

2

23

0.4

9

2

4

0.4

78

−26

2

42

2

1

0.4

9

2

10

0.7

74

−16

3

14

3

2

0.8

15

3

10

0.4

70

−9

4

25

6

15

1.0

13

6

10

1.1

North-South (Meridional) Components

114

−3

4

23

5

18

0.9

11

4

4

0.7

110

−2

2

25

3

19

0.4

31

3

3

0.4

106

−8

2

35

2

17

0.4

26

2

4

0.4

102

−14

1

29

2

20

0.4

21

1

4

0.4

98

−15

1

34

1

24

0.3

3

1

4

0.6

94

−11

1

31

1

2

0.3

4

1

10

1.0

90

−4

1

21

1

6

0.3

1

1

12

1.6

86

1

1

21

1

10

0.4

6

1

3

0.4

82

4

1

11

2

14

0.6

6

2

2

0.4

78

4

2

19

2

20

0.4

5

2

0

1.3

74

3

2

11

3

21

0.8

4

3

9

1.8

70

1

3

16

4

22

1.3

6

5

6

1.4

Vertical Components, cm/s

114

−44

46

97

66

11

2.8

64

106

1

1.0

110

−79

22

125

36

7

1.4

30

120

2

0.4

106

−52

16

124

28

6

1.3

23

143

2

0.4

102

−14

10

89

19

6

1.2

17

141

2

0.3

98

11

8

56

16

5

0.9

15

115

2

0.2

94

22

6

47

13

3

0.6

12

73

2

0.3

90

28

5

61

11

3

0.6

12

30

3

0.6

86

43

7

85

14

2

0.4

13

18

5

0.9

82

75

10

104

17

2

0.9

16

33

5

0.9

78

121

15

113

24

2

1.1

22

59

4

1.2

74

154

22

112

32

2

1.4

30

109

4

1.3

70

119

39

122

54

3

1.8

50

174

4

1.4

[9]Figure 4 presents the height-time variation of v_{gw}, the RMS vertical component of the gravity wave velocity, deduced as outlined in section 2. The blank portions of the plot indicate lack of data or inability to satisfy the gravity wave component determination criteria.

[10]Figure 5 is a plot of the height-time variation of σ, the RMS vertical component of the turbulent velocity (0.77u_{β} − v_{gw}). This plot shows all of the heights and times that σ was measured. Calculation of the correlation between v_{gw} and σ yields R = 0.41 for the 210 height-time samples, indicating that the vertical component of the gravity wave velocity is a possible source of the turbulence.

[11]Figure 6 presents contour plots of the diurnal variation in ɛ, the rate of dissipation of turbulent energy for the same height-time periods as the σ plots. Maxima occur at dawn and dusk. Dissipation rates vary from <5 mW kg^{−1} to just over 100 mW kg^{−1}, almost twice the 60 mW kg^{−1} maximum at 90 km of Roper and Brosnahan [1997] for their midday interval on 10 April. Note that the Roper and Brosnahan [1997] maximum of 60 mW kg^{−1} has been revised by Roper [2000] to 40 mW kg^{−1}. The corrected daily average dissipation (Table 2) over the 72 to 114 km height range for the 10 day interval is 25 mW kg^{−1}.

Table 2. Measured Values for 1–10 April 1989 at Arecibo (18°N, 67°W)

Z, km

Measured Values

Values With Gravity Wave Contamination Minimized

ω, s^{−1}

σ, m s^{−1}

ɛ, m^{2} s^{−3}

K_{z}, m^{2} s^{−1}

L_{β}, m

σ, m s^{−1}

v_{gw}, m s^{−1}

ɛ, m^{2} s^{−3}

K_{z}, m^{2} s^{−1}

L_{β}, m

114

.025

3.8

71

120

650

3.1

0.9

43

110

510

111

.025

3.3

65

100

580

2.8

4.0

39

90

430

108

.025

3.4

77

120

600

2.4

4.1

27

60

360

105

.026

4.1

88

150

690

2.7

4.0

36

80

420

102

.024

4.1

77

160

700

2.5

3.8

33

90

420

99

.024

4.1

83

180

750

2.3

3.7

29

80

390

96

.023

3.9

72

170

730

2.1

3.9

25

70

360

93

.022

3.8

59

180

710

2.1

4.1

23

70

380

90

.020

3.8

50

180

780

2.1

4.0

21

90

440

87

.019

3.7

50

170

790

2.5

3.8

29

120

530

84

.019

3.9

59

190

840

2.3

3.5

25

110

480

81

.020

3.5

55

130

730

1.6

2.7

13

50

320

78

.020

3.2

59

110

710

1.6

2.1

14

50

310

75

.020

3.5

57

120

720

1.8

2.6

17

70

370

72

.010

3.8

62

150

850

2.0

2.4

20

90

430

[12]Lindzen [1968] has predicted that the equatorial diurnal tide should be unstable in the neighborhood of the mesopause. Zimmerman and Murphy [1977] used this prediction to explain observed differences in observations of turbulence at equatorial latitudes and midlatitudes. While the shears associated with the diurnal tide are substantial (Figure 3), they do not correlate well with ɛ (Figure 6). Roper [1966] found a seasonal correlation of ɛ with the diurnal tidal amplitude at 92 km over Adelaide (35°S), but had insufficient data to look at either height profiles or diurnal variations.

[13]Roper [2000] has developed a set of self-consistent equations for the estimation of turbulence parameters. There is, however, an error therein in the determination of the turbulent diffusivity. Roper's equation (15) should read

which then leads to

which has been used here to estimate the diurnal variation of the turbulent momentum diffusivity. These results are presented in Figure 7. Maxima do occur at dawn and dusk, as they must since they are based on the same velocity fields as the dissipation rates.

The diurnal variation of L_{β} is plotted in Figure 8.

4. Interval From 2–9 May 1989

[15]Figures 9, 10, 11, 12, 13, and 14 and Table 3 present results similar to those already discussed, but for the interval 2–9 May. In contrast to the April interval, where the diurnal tide dominated the whole height range, here the meridional semidiurnal is larger above 90 km. Correlation between v_{gw} and σ yields R = 0.27 for the 274 height-time samples (cf. April above), indicating that for this interval the estimation of the vertical component of the gravity wave spectrum may be questionable. Again, the turbulent dissipation rate appears to maximize at dawn and dusk. Note that the higher values of turbulent dissipation rate (>70 mW kg^{−1}) observed in a few isolated areas are probably spurious, for reasons discussed in section 5. The corrected daily average dissipation (Table 4) over the 72 to 114 km height range for the 8 day interval is, again, some 25 mW kg^{−1}. No correlation with mean wind shears is apparent in the May data.

Table 3. Mean, Diurnal, and Semidiurnal Profiles for the Interval 2–9 May 1989 at Arecibo (18°N, 67°W)

Height, km

Mean, m s^{−1}

Error, m s^{−1}

24 Hour

12 Hour

Amplitude, m s^{−1}

Error, m s^{−1}

Phase, hours

Error, hours

Amplitude, m s^{−1}

Error, m s^{−1}

Phase, hours

Error, hours

East-West (Zonal) Components

114

2

7

20

9

16

2.0

40

9

3

0.4

110

5

4

29

5

‘6

1.4

27

5

3

0.4

106

9

2

31

3

‘6

0.4

22

3

4

0.4

102

13

2

28

3

‘6

0.4

21

3

4

0.4

98

14

2

24

2

16

0.4

24

2

5

0.3

94

10

1

23

2

14

0.3

25

2

6

0.3

90

1

2

21

2

14

0.4

24

2

5

0.3

86

−11

2

12

2

16

0.8

22

2

5

0.4

82

−22

2

16

4

21

0.9

20

3

5

0.4

78

−27

3

27

5

24

1.2

13

5

3

0.6

74

−23

5

22

7

2

1.4

18

7

1

0.8

70

−12

8

18

12

7

1.9

32

12

0

1.2

North-South (Meridional) Components

114

5

6

12

9

12

2.8

19

8

1

1.2

110

2

4

18

5

13

1.4

27

5

1

0.4

106

−1

2

18

3

14

1.2

34

3

1

0.4

102

−7

2

8

3

17

0.9

34

2

2

0.4

98

−13

1

8

2

0

0.7

28

2

3

0.3

94

−16

1

11

2

3

0.7

20

2

3

0.3

90

−13

1

8

2

5

0.7

12

2

4

0.3

86

−5

1

6

2

11

0.8

8

2

3

0.4

82

2

2

8

2

16

0.9

10

3

3

0.4

78

5

3

11

4

20

1.2

9

4

4

1.3

74

0

4

16

5

23

1.4

6

5

7

1.8

70

−5

6

12

8

1

2.1

13

8

8

1.4

Vertical Components, cm/s

114

−119

80

359

112

18

1.3

176

105

9

1.3

110

−150

36

223

52

17

1.4

133

49

8

1.3

106

−134

26

129

39

17

1.5

91

37

8

1.2

102

−97

18

70

28

18

1.7

57

27

8

1.1

98

−54

16

52

24

20

1.8

37

25

8

0.8

94

−14

14

58

22

21

1.5

46

22

9

1.0

90

16

14

77

23

18

1.4

83

23

9

1.0

86

37

17

121

27

18

1.4

132

27

9

0.7

82

50

20

197

31

18

1.3

182

31

9

0.8

78

55

31

302

44

18

1.4

217

45

8

0.9

74

52

44

429

61

18

1.4

219

62

8

1.1

70

40

74

575

103

18

1.5

192

105

7

1.4

Table 4. Measured Values for 2–9 May 1989 at Arecibo (18°N, 67°W)

Z, km

Measured Values

Values With Gravity Wave Contamination Minimized

ω, s^{−1}

σ, m s^{−1}

ɛ, m^{2} s^{−3}

K_{z} m^{2} s^{−1}

L_{β}, m

σ, m s^{−1}

v_{gw}, m s^{−1}

ɛ, m^{2} s^{−3}

K_{z}, m^{2} s^{−1}

L_{β}, m

114

0.24

3.3

51

130

570

3.0

0.6

40

110

510

111

0.25

3.1

48

110

530

2.4

3.9

29

70

400

108

0.26

3.3

52

120

530

2.8

3.9

40

90

440

105

0.26

3.1

48

100

500

2.0

3.2

22

50

320

102

0.25

3.2

50

110

550

2.1

3.6

23

60

350

99

0.24

3.3

50

130

570

2.3

3.6

28

80

390

96

0.23

3.5

52

150

640

2.3

4.0

28

90

420

93

0.22

3.6

54

170

670

2.3

4.1

27

80

420

90

0.21

3.6

48

150

680

2.5

4.0

29

100

490

87

0.20

3.7

46

190

750

2.3

4.0

26

110

490

84

0.20

3.5

43

160

720

2.0

3.8

19

80

420

81

0.20

3.3

42

150

680

2.5

3.9

27

100

500

78

0.21

3.2

42

140

630

2.3

2.9

22

80

450

75

0.20

3.1

40

140

640

2.3

2.9

21

80

470

72

0.19

2.9

42

150

690

2.3

2.7

20

80

480

5. Errors Associated With the Estimated Turbulent Dissipation and Diffusivity

[16] There is considerable smoothing involved in the production of the contour plots of v_{gw}, ɛ, σ, K_{z} and L_{β}. The plots are constructed from hour-by-hour two hourly means of the scattering point parameter data plotted hourly in 3 km height bins. In the plots, only those times with values on six or more days are plotted. There are also gaps in the data, particularly at night when the electron density falls.

[17] One of the indicators of the significance of the turbulent dissipation rates is the average number of scatterers within 3° of zenith in two hourly intervals contributing to each calculation, as indicated for the April interval in Figure 15a, for example. Those regions with less than 100 scatterers per interval produce ɛ values that are less reliable than those intervals with more than 150 scatterers. On the basis of this criterion, the high values in April (90 mW kg^{−1} and greater) of Figure 6 estimated around 0400 hours and 2100 hours at 97 km have more weight than similar values estimated around 0400 hours at 78 km and around 2200 hours at 75 km. However, reference to the vertical gravity wave velocity of Figure 4 shows that no gravity wave correction is applied at 0400 hours either at 78 or 97 km or at 2100 hours at 97 km, whereas the correction is applied at 2200 hours at 75 km.

[18] Similarly, for the 2–9 May results, the scatterer plots of Figure 15b would indicate that the ɛ values below 80 km throughout the day are the least reliable. In particular, the high values around 72 to 75 km at various times during the morning and evening, and particularly the >70 mW kg^{−1} around 1900 hours, are probably spurious. However, note also that the otherwise significant values >70 mW kg^{−1} around 2200 hours at 87 km have no gravity wave correction applied.

6. Turbopause Height

[19] An attempt has been made to determine the turbopause height from the contour plots of the rate of dissipation of turbulent energy. Because of the rapid rise in the kinematic viscosity above 95 km altitude, there is a need to consider ɛ_{min}, the minimum rate of turbulent dissipation that would be inferred from values of the kinematic viscosity. The kinematic viscosities used here, as with the Brunt-Väisälä frequency ω, are from the COSPAR '86 model. ɛ_{min} has been considered by several authors [see, e.g, CIRA, 1986; Hall et al., 1998; Roper, 1998] to be the dimensionally correct

where υ is the kinematic viscosity and ω is the Brunt-Väisälä frequency. A formulation consistent with the work of Roper [2000] and the equations used here can be derived from Roper's estimate of the turbulent diffusivity as amended in section 3, viz.,

with kinematic viscosity υ substituted for K, yielding

A turbulence Reynolds number

where ɛ is the measured rate of dissipation of turbulent energy, should have a value of 1 at the turbopause.

[20]Figures 16a and 16b present the variation of the diurnal mean turbulent dissipation rate with height for the April and May intervals, respectively, together with the variance associated with each measurement and the variation with height of ɛ_{min}. While this variance contains a measure of the error associated with each measurement, it is principally a measure of the variation with time of ɛ itself. The error in the measurement of ɛ, which is dependent not only on instrument parameters but also the scatterer rate, is estimated to be of order 2 mW kg^{−1} for two hourly mean scatterer rates greater than 150 in each 3 km height gate (see Figure 15). The variation of ɛ_{min} with height is dependent principally on the kinematic viscosity, which undergoes more than a thousandfold increase in the height range from 70 to 110 km.

[21] The turbopause is, by definition, the height at which ɛ equals ɛ_{min}. From the contour plots of Figures 6 and 12 the turbopause is ill defined. The plots of Figure 16 show measured values of ɛ < ɛ_{min} above 108 km. This anomaly is discussed in section 9.

[22] In terms of the eddy diffusion coefficient K_{z}, the turbulent Reynolds number is

When K_{z} = υ, Re = 1 and turbulence ceases. The K_{z} means for each campaign interval as presented in Figure 17 show, as they must, a behavior similar to the dissipation rate profiles.

[23] One of the factors determining the applicability of the Kolmogorov regime is the relationship between the large-scale and small-scale ends of the spectrum, here defined as L_{β} and the Kolmogorov microscale η. For the theory to be viable,

As can be seen from Figure 18, this relationship is >1000 at 72 km, 40 at 100 km, but drops to 6 at 114 km. Thus the Kolmogorov theory may not be applicable at the higher altitudes.

7. Intermittency

[24] All radar measurements of turbulent dissipation rate to date use formulae which assume that the turbulence being measured is stationary and homogeneous. If a characteristic velocity is being measured, as with pulse Doppler broadening or the residual velocity characterizing IDI measurements, an average of the local velocities characterizing the turbulence, which is known to be intermittent in space and time, is recorded. Thus only an average of the localized turbulent dissipation rates or diffusivities from areas where the scattering (and hence the turbulent intensity) is greater than their surroundings is sampled, producing more than the actual average rate of dissipation of turbulent energy or diffusivity. Measurements of C_{n}^{2}, which are volume averages, come closer, but even these are over estimates if the scattering is coming from only a turbulent layer or patch within the pulse volume. If the turbulence was stationary and homogeneous within the sampled pulse volume, then IDI returns would come from spurious scatterers aligned along the wind direction as predicted by the volume scattering theory as extended by Vandepeer and Reid [1995]. As shown by Turek et al. [1998], and as confirmed by further analysis, an alignment of scatterers is rarely observed in the AIDA data (in less than 5% of the over 400 hours of radar frames examined). Because of their inability to track scatterers from frame to frame, Roper and Brosnahan [1997] estimated the lifetime of individual scatterers to be some 100 s, about half the Brunt-Väisälä period. The intermittency in scatterer appearance is well illustrated in Figure 11 of Roper [1998], in which the number of echoes per 102.4 s radar frame within 2° of zenith at 84 km are plotted from 1300 to 1416 hours on 10 April. In some of those frames, there are no identifiable scatterers; in others, as many as eight. Again, in dozens of frames considered from both the April and May data sets, no individual scatterers could be tracked from one frame to the next. Methods for deriving an intermittency factor (based on the probability of occurrence of identifiable scatterers and their spatial distribution) for application to the data to provide better estimates of σ, ɛ, K_{z} and L_{β} are being investigated.

8. Summary of Results

[25] Mean profiles of the turbulence parameters estimated for the April and May intervals are presented in Tables 2 and 4. To enable comparison with the measurements of others, the measured as well as the corrected turbulence parameters are presented. Note that the work of others could produce values of ɛ from 2 to 3 times those presented here, depending on the constant used in equation (1).

[26] Recently, Holdsworth et al. [2001] have presented results of turbulent velocity estimates for the period March 1999 to June 2000, using the Buckland Park (35°S, 139°E) MF radar. They found RMS turbulent velocities from 2 to 6 m s^{−1} over the height range from 80 to 95 km for March 1999, the period of maximum diurnal tide during their total measurement interval. Their results can be compared to the profile of the mean turbulent velocity σ for the Arecibo April interval (Table 2), a time of significant diurnal tidal amplitude, as seen from Table 1. The uncorrected values of σ range from 2.9 at 78 km to 4.1 at 102 km.

9. Conclusions

[27] Comparison of the magnitudes of these results with those of others is difficult in that the constants of proportionality in (1) and (2) are those of Roper [2000] (as amended here for turbulent diffusivity) and are different from the constants usually used. There is also very little data on turbulent intensities at these altitudes in equatorial latitudes. However, the diurnal variations observed (maxima at dawn and dusk) are considered significant but not absolute.

[28] A diurnal variation in the rate of dissipation of turbulent energy is observed in both the April and May data, with maxima occurring at dawn and dusk. The dissipation rates have diurnal variations over the height range from 72 to 114 km of <10 to 90 mW kg^{−1} in April and <5 to 70 mW kg^{−1} in May, with maxima of 100 mW kg^{−1} possibly reached occasionally, although data at such maxima are sparse. There does not appear to be any correlation between the turbulent intensity and the mean shear with altitude in either data set. There are indications of a turbopause around 105 km in both data sets during daylight hours, with a possible nighttime maxima above 110 km in April and somewhat lower in May. Measurements made up to 114 km indicate that the model Brunt-Väisälä and viscosity profiles may not describe the actual profiles well, in that they predict a mean turbopause near 110 km which is not observed. The turbulence does not decrease, let alone cease, above the predicted turbopause altitude. There is the possibility that ion drag destabilization of the neutral atmosphere is being observed at the higher altitudes [see, e.g., Hall, 2000], but if that is the case, would not one expect at least some discontinuity in the measurements?

[29] Since the eddy diffusion coefficient K_{z} calculated using (2) uses the same velocity field as (1), and the equations are self-consistent, contour plots for the same data intervals are similar to the dissipation rate plots but with different contour line values. As expected, the diurnal variations in diffusivity averaged over each of the April and May intervals (Figures 7 and 13) and the buoyancy scales L_{β} (Figures 8 and 14) maximize at dawn and dusk. Note that the diffusivity values presented here, which range from <10 to 300 m^{2} s^{−1}, are larger (by a factor of up to 2) than those that appear in the recent literature, since they are based on the set of self-consistent equations developed by Roper [2000] as corrected in section 5. The buoyancy scales are consistent with those measured by others. As pointed out in section 6, consideration of the relationship between the buoyancy length scale L_{β} and the Kolmogorov microscale η casts some doubt as to the applicability of the Kolmogorov theory above 100 km.

Acknowledgments

[30] One of us (R.G.R.) thanks an anonymous reviewer for pointing out that the K_{z} values reported here are thermal, not momentum, diffusivity. This research has been supported by the Atmospheric Sciences (Aeronomy) Division of the National Science Foundation under grants ATM-9728629 and ATM-0228219, by the Air Force Office of Scientific Research under contract F49620-89C-0022, by the National Aeronautics and Space Administration under grant NGT-50864, and by the National Astronomy and Ionosphere Center (NAIC). NAIC is operated by Cornell University under a cooperative agreement with and funding from the National Science Foundation.