a. Basic principle

The method is a classical one (Suomi and Businger, 1959) and has already been successfully used for ground-based instruments (Mitsuta 1966). Each sonic source emits a short pulse of compression–dilatation waves that propagate with a phase velocity c relative to the air. The characteristics of these waves can be derived from the general sound propagation theory, where sound “rays” in an elastic medium are defined as lines drawn normally to the propagating wave front (appendix Aa).

This concept allows a fairly good description of the sound wave field, that may be considered as spherical at a distance from the source greater than 10 wavelengths.

After that, the transducer setting in the measurement volume gives a very easy set of equations (appendix Ab) to determine the following:

• the sound velocity c (m s⁻¹),

  

  

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• the relative wind speed V (m s⁻¹),

  

  

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• and the temperature of the dry air T (K),

  

  

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In these above equations, L is the distance between E1–R1 and E2–R2; γ_a = c_pa/c_υa, the ratio of specific heats (J kg⁻¹ K⁻¹) at constant pressure c_pa and at constant volume c_υa of dry air; and R_a = R*/M_a, is the specific constant of dry air with R* (J K⁻¹) the universal gas constant and M_a (kg), the molecular weight of dry air.

b. Rear convergent and speed correction

Wind tunnel tests have shown that the faster the airspeed is, the greater is the signal attenuation. Due to the sensor’s cylindrical shape, a conical convergent base is set at the rear exhaust part for slowing down the flow in the measurement volume.

If the upstream velocity out of the cylinder is noted V_o and if the velocity in one point in the measurement volume is noted V_i, then a coefficient μ equals to the ratio V_i/V_o may define the airstream slowing down. Thus, this AUSAT measurement of the velocity can afford an interesting other way to obtain the aircraft’s true air speed V_o.

Wind tunnel measurements and computational fluid dynamics gave μ equal to 0.36 and indicated that it only depends on the sensor’s geometry. This value was found independent of the incident flow angle; the AUSAT was tested with an angle of attack up to 4°. Therefore,

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c. Dynamic heating

But the decrease of velocity leads to an increase of upstream temperature T_o to the inner temperature T_i. The overheating ΔT = T_i − T_o, and thus the correction to apply to the measured temperature may be calculated by using the thermodynamic relation for compressible flows during an adiabatic process of an ideal gas:

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The above relation, true for incompressible and compressible flows does not require further compressibility correction. Thus we obtain

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Since V_i is measured, this correction only requires to know the value of μ. This correction is included in the software and computed in real time.

d. Humidity correction

Therefore, the AUSAT is self-containing for the measurement of the aircraft true airspeed V_o but provides an air temperature T_o that must be corrected a posteriori by taking into account the air humidity. Note that the aircraft true airspeed as measured by AUSAT does not need this humidity correction.

The temperature T_o obtained by combining relations (3) and (5) is not the true air temperature. Indeed, the so-calculated temperature is correct for strictly dry air and not for the humid air in the atmosphere because, in this case, γ and R are both a function of the specific humidity of the air.

This correction is a linear function of two independent variables, even if q (kg kg⁻¹) and T_o (K) are not really independant in the atmosphere. Thus, the magnitude of ΔT_o may be calculated for different values of q and T_o. For example, if the temperature ranges between 250 and 290 K and the humidy varies between 3 and 7 g kg⁻¹, the humidity correction is then between 0.39° and 1.05°C. For the same humidity range, at T_o = 270 K, ΔT_o varies from 0.42° to 0.98°C. It also can be seen that an uncertainty of 0.1 g kg⁻¹ yields to an error of 0.015°C.

Note that in formula (5) for the dynamic heating, using c_pa for dry air in place of c_p for humid air, with c_p = c_pa(1 + 0.84q), yields to an error on T smaller than 0.04°C when q is less than 10 g kg⁻¹ and smaller than 0.07°C for q = 20 g kg⁻¹.

a. Influence of the turbulence

An acoustic wave in a fluid is extremely sensitive to air turbulences, which deform wave fronts in a very chaotic way. So, in air turbulence, received signals show strong amplitude instabilities. Turbulence may also be responsible for phase instabilities of the received signals. Even weak, these perturbations may induce a spurious noise in the high-frequency domain.

These difficulties have been overcome, first by an optimization of the sensor’s shape to reduce air turbulence in the vicinity of the transducers, and secondly by using an electronic detection device with low sensitivity to the small amplitude instabilities: transducers are settled into a cavity in the sensor wall so they do not create a barrier to the airflow and then a local turbulence in their vicinity; further, the measurement frequency is at 1024 Hz, near the upper limit imposed by the sensor dimensions. The transit time of the sonic wave along the upwind path is in the order of 1 ms when V_o = 100 m s⁻¹, so the time interval between two measurements cannot be less.

The AUSAT sensor measures the mean velocity of an acoustic pulse in the measuring volume. It may be shown (appendix B) that it constitutes a natural frequency filter and calculations show that for a true airspeed of about 100 m s⁻¹ (inner speed V_i about 40 m s⁻¹) frequencies larger than 300 Hz are strongly damped. So, with a frequency measurement of 1024 Hz the Shannon’s law is complied and moreover, as successive pulses have a comparable amplitude, it is easy to realize a correct detection with a delay line (section 5).

b. Refraction of the sound rays

Interface between the airstream and each of the four cavities is provided by a thin grid. In the cavity the air is quiet. Then, when acoustic rays pass through the grid, they are submitted to a sharp wind shear. It can be shown that in such conditions, the sound rays are refracted as light electromagnetic waves through a dioptre (Snell’s law). An approximate theory (Morse and Ingard 1986) gives a method to calculate the deviation of a sound ray as a function of the wind shear. This phenomenon induces an additional attenuation of the received signal as it has been observed during wind-tunnel tests of AUSAT. For large values of the shear (airspeed near 120 m s⁻¹), the refracted ray may travel along the boundary layer and no energy is transmitted to the receiver.

At this point, the presence of the rear convergent is completely justified. Indeed, turbulence in the pipe and signal attenuation increase quickly with the airspeed. Moreover, symmetry deficiencies in the sensor’s geometry due to construction faults involve measurement errors that increase also with the airspeed. Finally, due to the refraction phenomenon, the sensor could not work at the cruise speed of a classical research aircraft without its rear convergent.

a. General comparison of the time series of three different thermometers

A simple method to compare response time of different thermometers is to obtain their response to a very sharp step of temperature. Such steps are not easy to find in clear air but are usual at the border of cumulus clouds. So, some results obtained during a constant level flight with penetrations in two cumulus turrets during the experience TRAC 98 are presented hereafter.

Figure 9 shows temperature measurements respectively obtained by the AUSAT and by a 102 E4 AL non-deiced Rosemount probe, during a constant-level flight with penetrations into two cumulus cloud turrets. The liquid water content measured by a PMV-100 Gerber probe is also in this figure. This last probe is situated 6.50 m behind the thermometers and data have to be corrected for the time lag approximately equals to 50 ms or 10 data at a rate of 200 Hz taking into account the outer aircraft velocity (100 m s⁻¹) and the inner slow down velocity in the AUSAT.

AUSAT temperature presented in this chapter is not calibrated and not linearly corrected to fit the Rosemount temperature in clear air (see section 8c). Thus, data cannot be used for a quantitative comparison but they give a first insight on respective response times.

During penetrations, both thermometers show that the in-cloud air temperature is lower than the surrounding air temperature by about 1.5°C, suggesting an overshooting effect at these levels (in this case, the air temperature in cloud is cooler than out of cloud).

Note the very fast response time of the AUSAT compared to the Rosemount, at the entrance as well as at the exit of the cloud.

On the static pressure record (Fig. 10), we can observe that the aircraft slightly climbed during the first penetration but kept a fairly constant altitude during the second penetration. Consequently, the AUSAT temperature slowly decreases in the first turret and remains fairly constant in the second turret.

During the first cloud turret penetration (lasting 4 s and corresponding to a distance of 400 m), the Gerber probe indicates a more continuous liquid water content than during the second one. The Rosemount record shows a more pronounced temperature decrease than the AUSAT, perhaps due to wetting of the Rosemount housing (Lenschow et al. 1974; Heymsfield et al. 1979;Harmer et al. 1990).

On an enlargement of one of these cloud penetrations (Fig. 11), temperature measurements obtained by the AUSAT is compared to those from a prototype of a very fast thermometer built by Météo France (Villain et al. 1996). This last thermometer, also experimented during these flights, uses a very thin tungsten wire (5-μm diameter) placed in a small cylindrical housing and protected from water droplets by an helicoidal device.

The thin-wire thermometer was mounted on the front part of the fuselage, but unfortunately, not on the same side as the AUSAT. Both signals display fluctuations of the same amplitude, which appear to have physical signification. But, some discrepancies can be seen between the two signals. They are due to the high resolution of the two thermometers, which do not catch the same phenomena at their two different locations.

As may be seen in Fig. 11, the thin-wire thermometer has a double time constant, probably due to its housing as for the Rosemount (Rosemount Engineering Company 1965; Lawson 1988; Harmer et al. 1990).

Furthermore, at the end of the penetration, the thin-wire thermometer record shows a spike, which could be interpreted as an evaporative cooling one.

b. Estimate of the AUSAT response time from time series

Figure 12 is an enlargement of another cloud penetration performed during spring 1997. These data concern a time period of 1.5 s (each graduation corresponds to 0.05 s). From these data, the measured AUSAT response time appears smaller than 50 ms. The Rosemount and the Météo France’s thin wire present a comparable double response time, in the order of 1 s, due to their housing.

c. Soundings

AUSAT and Rosemount temperature during a sounding are shown from 1020 to 700 hPa (Fig. 13). Once again, AUSAT data are not linearly corrected to fit Rosemount data in clear air.

Figure 14 compares both temperatures after adjustment. Before it, AUSAT data overestimated temperatures under 18°C and underestimated them over 18°C.

d. Estimate of the AUSAT response time and band pass from power spectra

From homogeneous turbulence theory, power spectra of an atmospheric variable should follow the expected −5/3 power ratio. Power spectra of different sensors are shown in Fig. 15 together with (−5/3) slope straight lines. Spectra have been smoothed (on six iterations) using a 1024-point Hamming window on the same time series.

Figures 15a and 15b show the Rosemount 102 and the Lyman-α spectra respectively with a bandwith of 20 and 40 Hz. Data from this hygrometer are used for AUSAT humidity correction. One should note the exceedingly steep slope of the Rosemount spectrum, which implies that temperature given by this probe should not be used at frequencies higher than a few hertz.

The third spectrum (Fig. 15c) concerns the AUSAT data before the humidity correction, filtered at 50 Hz by the output filter. The fourth spectrum (Fig. 15d) is for the same AUSAT data but after humidity correction by the Lyman-α data. The similarity of the two last spectra shows that a very little noise is added by the humidity correction. For this sample, the bandwith is better than 30 Hz. Practical response time may be deduced from these spectra by taking the inverse of the bandwith equal to 33 ms.

e. AUSAT in-cloud temperature measurement

It has been said that sonic anemometers–thermometers should not work at all in cloudy air: large signal lessening due to considerable absorption and diffusion by cloud droplets, reduced sound velocity inducing temperature underestimation by several degrees. The data presented in Figs. 9, 11, and 12 show that the AUSAT works and its temperature is very comparable to those of Rosemount and thin-wire sensors.

However, comparing in-cloud and out-cloud AUSAT temperature to the Rosemount and to the Météo France’s thin wire, it can be observed that amplitude variation of AUSAT temperature seems greater than the Rosemount one but absolutely comparable to the thin-wire one. Thus, taking into account that, before entering the clouds, the AUSAT temperature is greater than the Rosemount by 0.55°C, the in-cloud AUSAT temperature appears lower than Rosemount by about 0.50°C. Of course, it is well known that Rosemount cannot really be considered as a reference for in-cloud temperature measurements: it is supposed to be sometimes lesser by several tenths of degrees (up to 1.9°C) than the real temperature, due to evaporative cooling of droplets trapped by the sensing element (Harmer et al. 1990; Lenschow and Pennel 1974).

However, one may question whether a possible evaporative cooling of the cloudy air in the AUSAT measurement channel may be due to the slowdown. An estimation of the possible effect of cloud droplet evaporation on temperature in the measurement channel can be made by taking into account the dynamic heating correction for AUSAT probe and a diagram giving the theoretical lifetime of an evaporating drop as a function of its radius and of the relative humidity of the environment (Lawson 1988). This diagram is based on a very classic hypothesis and theory of microphysics (Pruppacher and Klett 1980), in particular on the fact that the growth of the evaporation of a cloud droplet is a rather slow process due to the low diffusion velocity of water molecules through oxygen and nitrogen molecules.

Let the in-cloud temperature be equal to 6.5°C, the mean static pressure P = 844.3 hPa, and q = 6.3 g kg⁻¹ (as measured by the Lyman-α hygrometer). We obtain an humidity of 88.4%, indicating an initial in-cloud subsaturation of 11.6% (at saturation e_w = 9.67 hPa and r_w = 7.12 g kg⁻¹). This subsaturation may perhaps be explained by the fact that we are near the cloud top where entrainment and mixing are important. The 4.33°C dynamic increase of temperature inside the cylinder yields to an hygrometry of 65.8%, thus to a subsaturation of 34.2%.

Figure 16 shows that for droplets whose radius is 5 μm the evaporation time lays between 1 and 8 s for relative humidity ranging between 20% and 90%. For droplets of 20 μm this time varies from 12 to more than 80 s for the same range of humidity. As the flow within the AUSAT is about 40 m s⁻¹, assuming evaporation on a 40-cm path (crossing time of 10 ms), it may be infered that there is not enough time for droplets to totally evaporate. By writing the time constant evaporation τ_r,

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We can calculate that for time t, a droplet of initial radius r_o and mass m_o falls to a radius r and a mass m such as

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From the diagram, for an initial radius r_o = 5 μm and a relative humidity ≈ 70%, the time constant is τ = 3 s. As the crossing time t = 10 ms, we obtain m/m_o = 0.990 that is a relative variation of mass of 1% and r/r_o = 0.997. With a measured liquid vapor content of 0.25 g kg⁻¹, the evaporated water is in the order of 2.5 × 10⁻³ g kg⁻¹, resulting in an evaporative cooling of about 6 × 10⁻³°C. Thus, if the classical theory from which this diagram is deduced may be applied to AUSAT, although some hypothesis have not been complied with (droplets at rest, stationarity), this effect cannot be invoked in order to explain the observed difference between in-cloud AUSAT and Rosemount temperatures.

Even with greater liquid water content, (up to 1 g kg⁻¹) this variation of drop mass does not seem to have an effect on the measurement (cooling up to 2.5 × 10⁻²°C). In an evaporative process, only the droplet is essentially cooling down, due to the low thermal diffusitivity of the air. Thus, it is highly probable that, even for larger partial evaporation, the very short time taken by the droplet to cross the cylinder does not allow a sensible cooling of the cylinder air bulk.

But now, it is interesting to consider the humidity correction applied to the AUSAT data. Out of cloud, humidity q is on the order of 3 g kg⁻¹, yielding to ΔT_o = 0.43°C. The in-cloud value of q, as measured by the Lyman-α hygrometer, is equal to 6.3 g kg⁻¹, yielding to ΔT_o = 0.91°C (by taking q_w = 7.12 g kg⁻¹ we would find ΔT_o = 1.03°C). Before humidity correction, AUSAT data always remain above Rosemount. So, we can see that the observed difference of 0.5°C between AUSAT and Rosemount in-cloud temperatures may be explained by the strong change in humidity observed when the aircraft penetrates the cloud.

Moreover, according to some authors (Lenschow 1986; Hartman et al. 1990), in some cases of high hygrometry, Rosemount temperature data may also be higher than the correct temperature when salt crystals are trapped on the sensing element. This salt becomes deliquescent before saturation is reached and increases the probability of the occurrence of sensor wetting: condensation on the sensor and corresponding latent heat deposition may result in temperature higher than the correct of several tenths of degrees. These possibilities of contamination errors are pointed out by Rosemount Inc. (Stickney et al. 1981).

So, if the incertitude on in-cloud Rosemount temperature is assumed to be in the order of ±1°C, then the observed 0.5°C in-cloud deviation between AUSAT and Rosemount data remains in the uncertainty domain of Rosemount in-cloud measurement. However, we are aware that some more complete theoretical and practical investigations are to be conducted before we know all the possible errors in our in-cloud temperature measurement.

f. AUSAT true airspeed measurement

On board the Merlin IV, the true airspeed is calculated from data measured by dynamical and thermodynamical sensors such as a five-hole radome and an impact thermometer (Brown et al. 1983). Complete calculation of true airspeed needs data provided by several other instruments generally with a very different response time and bandpass, so that even if they are, at last, delivered at 200 Hz, the classic true airspeed data do not constitute a fast measurement of this parameter, nor an absolute reference (Nacass 1992). Approximation of measurements of static temperature, static and dynamic pressures, position error, angles of attack and sideslip, and by neglecting the air humidity (Khelif et al. 1999), induce an estimation of accuracy on the calculation of the true airspeed that may reach up to 1 m s⁻¹ (Morera 1995).

The AUSAT airspeed, compared to the classical measurement (Fig. 17), shows that the two determinations are very close. For example, for an airspeed varying between 75 and 95 m s⁻¹, the difference remains under 0.40 m s⁻¹ (Fig. 18). There is no direct correlation between the AUSAT true airspeed and the angles of attack and sideslip (Figs. 19, 20) except the correlation from the dynamics of flight (Etkin 1972). The same conclusion can be drawn about the correlation between these two angles and the difference between the AUSAT true airspeed and the radome one (Fig. 21) where the variations are under the limit of a real influence.

So, the AUSAT corroborates the rather complicated algorithm used in atmospheric research for the true airspeed determination and provides a different and very simple way to obtain, at high frequency, a precise value of the true airspeed.

g. Characterization of adiabatic ascents in a convective planetary boundary layer

In order to test thermometer performances, a calibration flight is used. It was performed during a pre-ACE-2 field experiment calibration flight, in June 1997, above the Beauce Plain in France. The Merlin IV flew through updrafts of a slightly convective boundary layer with cumulus clouds. The adiabatic updrafts are identified thanks to the total water mixing ratio r_t (kg kg⁻¹).

With the water evaporation/condensation latent heat L (J kg⁻¹) and the liquid water mixing ratio r₁ (kg kg⁻¹), the liquid water potential temperature θ₁ (K) is calculated with the following formula:

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Average conditions near ground level at 1006 hPa were a potential temperature θ_g = 295 K and a mixing ratio r_g = 6.3 g kg⁻¹. The vertical air velocity computed by the Merlin IV is noted as “w” (m s⁻¹).

Five legs were performed consecutively at altitudes of 1550, 1700, 2000, 2300, and 2450 m and at the aircraft speed nearby 100 m s⁻¹. The first leg was run in clear air under the cloud bases, the second near condensation level, the third and fourth across clouds, and the last one at cloud top. Figures 22–25 display, versus the time in seconds, for each of these legs: w, r_t and r₁, θ_1A, θ_1R and θ_1A − θ_1R. Notice that θ_1A and θ_1R are displayed on the same graph, the latter being offset by 0.5 K.

On the two lower level plots (Figs. 22, 23) the upper part of the boundary layer appears highly heterogeneous: broad areas of nearly uniform r_t (≈6 g kg⁻¹) are separated by chaotic areas with r_t values ranging from 4.5 to 5.5 g kg⁻¹. In high r_t areas, liquid water potential temperature θ₁ is also nearly uniform (≈293.5 K) while vertical air velocity is positive and in the order of 1–2 m s⁻¹. These areas may be interpreted as well-formed homogeneous updrafts expected to raise adiabatically up to some level above which entrainment will yield mixtures with lower r_t and higher θ₁. Two such drafts are visible in the first leg plot and four in the second one, among which three display a small amount of liquid water (r₁ ≈ 0.05 g kg⁻¹). In adiabatic ascents, total water mixing ratio and liquid water potential tremperature have to be constant; the first two leg results are consistent with an adiabatic ascent mixing ratio (r_a ≈ 6 ± 0.1 g kg⁻¹) and both thermometers agree with an adiabatic ascent liquid water potential temperature (θ_a ≈ 293.5 ± 0.1 K).

On the third leg (Fig. 24), the aircraft penetrated a cumulus cloud (r₁ ≈ 0.4 g kg⁻¹) 300 m above its basis. Again, r_t reaches r_a, which is compatible with an adiabatic utpdraft. Here, θ_1R and θ_1A are both within 0.1 K of θ_a.

In the next higher-level leg (Fig. 24), r_t reaches only 5.5 g kg⁻¹ during cumulus cloud penetration suggesting some entrainment and mixing at this level. For both thermometers, θ₁ hardly passes below 294 K. A clear updraft is also crossed in which r_t reaches 4 g kg⁻¹ and θ₁ 294.8 K.

In the last leg (Fig. 25), the aircraft flew through the cumulus cloud top. The updraft with the strongest mixing ratio (4 ⩽ r_t ⩽ 5 g kg⁻¹) and the lowest liquid water potential temperature (295 ⩽ θ₁ ⩽ 295.8 K) is surrounded by two well-defined downdrafts with minimal r_t (1 and 1.5 g kg⁻¹) and maximal θ₁ (≈297 K).

In all this flight, the same general features are observed as in the next in 1998:

• Globally, the AUSAT performs correctly: the temperature difference with the Rosemount stays below 0.2 K, except in a few points.

• The AUSAT variability is higher than the Rosemount one; high-frequency fluctuations account for ±0.1 K/arp temperature variations (1 or 2 K in less than 0.5 s) induce temperature differences reaching 0.5 K (e.g., legs 3 and 4 at times t = 407 s and t = 583 s); temperature fluctuations with a characteristic time of 1 or 2 s induce temperature differences ±0.2 K (leg 3, 405 s < t < 425 s).

• In-draft temperatures given by the AUSAT are lower than those given by the Rosemount.

In addition, in-draft temperatures from both thermometers are compatible with the adiabaticity given by r_t measurements.

Thus, this test would look quite satisfactory, were it not for the cold bias of the AUSAT relative to the Rosemount. This bias is manifest in two instances: in clouds, and in clear but humid updrafts.

The first point, which was already analyzed in section 8e, can be examined more closely. It turns out that the temperature differences are larger when r₁ is low (r₁ < 0.2 g kg⁻¹). See, for instance, cloud crossing of leg 4, where r₁ decreases with time, while the absolute value of the difference θ_1A − θ_1R increases). Especially during leg 2, where liquid water amounts involved in updrafts are very low (<0.05 g kg⁻¹), the temperature differences lie between 0.3 and 0.4 K. The second point appears clearly, for instance, at times 595–598 s (leg 4) when the aircraft flies through a clear updraft: the AUSAT temperature drops by 1.5 K while the Rosemount temperature drops only by 1.1 K.

It seems very difficult to disentangle the contributions of the two thermometers to these biases. The present observations would be compatible with the Rosemount being too slow to follow fast temperature changes and being polluted by salt and thus warmed up by condensation when humidity is close to saturation. Also, neither evaporative cooling of the AUSAT measurement channel, nor sound velocity change by cloud drops, would make the AUSAT temperature colder when little liquid water is present compared to situations when r₁ is large. On the other hand, since the cold bias manifests itself even in clear humid drafts, one will have to investigate the effect of different gas laws on the behavior of sound velocity when humidity varies, even though such deviations from usual perfect gas law are not mentionned in studies of ground-based sonic thermometers.

To sum up, during this 1997 test flight the AUSAT provided fast and accurate temperature. Improving still its accuracy will demand understanding the discrepancies between AUSAT and Rosemount probe, which will require new field experiments with increased care, especially preventing pollution of the Rosemount probe by salt. Behavior of sound velocity in mixtures of air and water vapor also need to be investigated.