An acoustic investigation of the near-surface turbulence on Mars

Turbulence within the lower atmosphere of Mars, which is in direct contact with the surface (namely, the atmospheric surface layer), is a critical process as it drives the transport of heat, dust, and trace gases, all of which have a global impact on the overall dynamics of the atmosphere (Petrosyan , 2011; Read , 2017). Detailed knowledge of the turbulent processes on all spatial and time scales is, therefore, key to understanding the whole climate system of Mars. On a diurnal time scale, the atmospheric surface layer of Mars is prone to large thermal gradients that generate a strong convection-driven turbulence during the daytime, leading to large temperature fluctuations compared to its counterpart on Earth (Chide , 2022; de la Torre Juárez , 2023; Munguira , 2023). At night, the thermal inversion inhibits the convection, but it may be replaced by shear-induced turbulence at some specific locations and seasons (Chatain , 2021; Pla-García , 2023). The current understanding of small scale processes, especially the turbulence energy cascade, relies on terrestrial-based models that have been adapted for Mars. Therefore, in situ observations are crucial to validate such models. These small scale processes have been described by the Insight mission in terms of pressure fluctuations up to 2 Hz (Banfield , 2020; Temel , 2022), but no data at higher frequency exist due to the lack of suitable sensors. Moreover, such small scales cannot be resolved by atmospheric models, e.g., large eddy simulations, and are instead parameterized in the latter example. Indeed, in situ observations of the atmospheric boundary layer processes are crucial for testing large eddy simulations predictions, which sometimes include different small scale diffusion and turbulent scheme, yielding very different predictions (Bertrand , 2016).

The random inhomogeneities in a turbulent atmosphere are known to greatly affect sound propagation by scattering the phase and amplitude of acoustic waves. The statistics of these scattering properties has been extensively described by theoretical (Ishimaru, 1997; Tatarskii, 1967) and experimental works (Blanc-Benon and Juvé, 1993). The resulting mathematical formalism has been used to describe the distribution of the turbulent scales in the Earth's atmospheric layer (Wilson , 1999). Here, the objective of this paper is to test if this formalism can be applied to data obtained on Mars and, for the first time, to infer small scale properties of its atmospheric surface layer turbulence.

The NASA Perseverance rover, which landed on Mars on February 18, 2021 (Farley , 2020), offers a unique combination of a microphone and controlled sound source at various distances (Maurice , 2022). The microphone of the SuperCam instrument (Maurice , 2021; Mimoun , 2023) records the Mars soundscape for the first time between 20 Hz and 50 kHz, as well as artificial sounds produced by the rover and its equipment. Since landing, several results have been published (Chide , 2022; Lorenz , 2023; Murdoch , 2022; Stott , 2023), highlighting the great potential of acoustic investigations for Mars. The main artificial sound source recorded by the microphone is created by the SuperCam pulsed laser, which vaporizes a few nanograms of matter at the surface, and subsequently generates a plasma that extends to several millimeter in size (Vogt , 2022). The optical spectrum from this plasma is used to measure the elemental composition of the targeted rocks and soils with an analytical method laser-induced breakdown spectroscopy (Maurice , 2021; Wiens , 2020). As the plasma expands in Mars' atmosphere, it generates a shockwave (Seel , 2023) that results in an N-shaped acoustic wave (2–25 kHz), which can be recorded by the microphone. The absolute amplitude of this pressure signal has been used to characterize the unique sound propagation properties of the low-pressure CO₂-dominated Mars atmosphere, i.e., large sound absorption and seasonal variation of the acoustic impedance (Chide , 2023). However, these laser sparks are produced and recorded at a fast cadence of 3 Hz over 10 s, which is reminiscent of laboratory experiments designed to study the propagation of spherical waves through turbulent medium (Averiyanov , 2011; Yuldashev , 2017). Therefore, the analysis presented here focuses on the variance of the amplitude and travel time of these acoustic waves induced by thermal fluctuations to statistically characterize the near-surface turbulence using the acoustic formalism (Blanc-Benon and Juvé, 1993).

This is the first acoustic experiment ever performed on Mars and for which, unfortunately, experimental parameters are not as controlled as they would be in laboratory experiments. In this context, we have considered the following approximations: we place ourselves in the framework of geometric acoustics, which means that diffraction effects are neglected. Second, we only consider temperature fluctuations, and wind velocity fluctuations are ignored. Furthermore, we assume that temperature fluctuations do not depend on the height over the first 2 m above the ground, where the experiment is performed. All these hypotheses are discussed in the text and could be explored and refined in future research.

The geometry of the experiment, the dataset considered, and the quantities used [travel time and scintillation index (SI)] are presented in Sec. II, along with a discussion on the validity of this experiment with regard to the various lengths and time scales. Then, Sec. III is devoted to the study of travel times and testing various turbulence models that can account for the nonlinear behavior of travel times with distance. Section IV deals with acoustic scintillation to discriminate which of the turbulence models tested represents the Mars data the best. In short, a power spectrum for the eddy size distribution, their characteristic scales, and the associated temperature fluctuations are proposed to describe the turbulence around noon local time.

The laser and microphone are mounted at about 2.1 m above the ground (Fig. 1). The longitudinal axis, z axis, connects the target to the microphone. The plasma is at z = 0 and the microphone is at z = x, which represents the propagation distance. Because the laser is focused onto targets at various distances from the rover for sample analysis, the propagation distance varies from target to target. Note that for each target, there is a different z axis as targets are located at different distances from the rover.

The dataset consists of data acquired by SuperCam at the top of Perseverance rover mast over the first 800 Sols (Martian solar day; 1 Sol = 88 775 s; Sol 0 is the landing day, February 18, 2021) of the Mars 2020 mission (Farley , 2022). Acoustic data closer than 2 m and farther than 8 m were removed: data closer than 2 m corresponds to the on board calibration targets at the rear of the rover and result in a different propagation path above the artificially heated rover deck. Targets farther than 8 m were also discarded because their signal-to-noise ratio of the associated audio signal is lower than two.

Acoustic recordings are performed at 100 ksamples/s (Mimoun , 2023). Although the microphone bandwidth starts at 20 Hz, laser-induced electromagnetic interference associated with subsystem heating dominates signals below 1 kHz (Maurice , 2021). However, 95% of the acoustic energy produced by a laser-induced spark is between 2 and 15 kHz (Maurice , 2022). Therefore, the instrument limitation below 1 kHz is of no consequence for the current investigation.

SuperCam operates the laser at a repetition rate of 3 Hz by bursts of 30 shots, 29 of which are recorded by the microphone (the 30th shot is not recorded). For a surface target, such a sequence is usually repeated 5–10 times on fresh sampling points separated by a few millimeters. Per burst, the first three shots are discarded because they are potentially affected by dust on the surface (Lasue , 2018). Therefore, a series of 26 shots per burst is used and recorded in less than 9 s. Occasionally, there are bursts of 125 shots (or 150 shots) to penetrate deeper into the rocks. These bursts are separated into data packets of 29 shots before applying the process above.

To support chemistry data, SuperCam acquires high resolution images of each target sampled by the laser. These images provide geological context for the geochemical analysis and interpretation. Manual classification of those images for this study results in three categories: rocks, the vast majority of the targets that are hard natural surfaces on the ground; soils, for unconsolidated targets, including regolith and drill tailings; drill holes, a few attempts to shoot inside rover's drilled holes. However, because of a very particular geometry that induces a lot of echoes, shots inside drill holes are discarded.

Finally, to consolidate the dataset, some recordings have been discarded when the focus of the laser is known to be bad or the determination of the N-wave arrival time is ambiguous. On the contrary, windy recordings are not rejected even when wind noise is detected in the time series. Note that distances are not evenly distributed: targets at close distances are investigated more often than those at long distance. Hence, 50% of the rock targets are between 2 and 2.8 m. Targets are sampled during the daytime from 07:06 to 19:07 local true solar time (LTST), and 70% of the recordings are acquired between 10:00 and 14:00 LTST because of operational constraints. This time period also corresponds to the maximum of turbulence intensity during a Sol. Therefore, unless mentioned otherwise, most of this study is based on data collected between 10:00 and 14:00 LTST to maintain a similar turbulence level. In total, this dataset relies on 70 000+ recordings, assembled as 2762 consecutive sets of 26 recordings (13% are soils and 77% are rocks), corresponding to 262 different targets sampled between 2 and 8 m.

When the laser pulse hits the target with an irradiance greater than 1 GW cm⁻², it vaporizes a few nanograms of matter, which subsequently becomes a plasma. The plasma expands at supersonic speed into the ambient atmosphere, creating a shock wave. Its speed quickly decays to the local speed of sound and propagates as an N-wave (Chen , 2000).

The acoustic recordings of the N-waves cover 60 ms around each laser pulse. The start of the recording window is precisely triggered on the laser ignition. The time for the laser beam to hit the target is negligible. First, the plasma expansion is supersonic: after about 23 μs, the pressure wave reaches the speed of sound; 15 mm have been covered at an average speed of 714 ms⁻¹ (Seel , 2023), i.e., about three times the sound speed, which is about 255 ms⁻¹. The travel time is given by the rise of the pressure wave since the laser was fired minus 23 μs as we only consider the sonic propagation. Because of the uncertainty on the travel time detection, we estimate the global error on the propagation time to be 10 μs. Over bursts of 26 recordings, the quantities that are measured in this study are the mean travel time, $⟨ δ t ⟩$⁠, and the variance of the travel time, $σ t 2 = ⟨ ( δ t − ⟨ δ t ⟩ ) 2 ⟩$⁠. For $x =$ 2, 4, and 6 m, $⟨ δ t ⟩ ≃$ 7.8, 15.6, and 23.5 ms, respectively. For the same distances, $σ t 2 ≃ 2 × 10 − 9 s 2 , 6 × 10 − 9 s 2 , and 13 × 10 − 9 s 2$⁠, respectively.

Figure (2a) shows SI values for the different populations of our dataset at all local times. For rocks, SI values in the range 0.002–3. Soils show a wider range of SI values, particularly high, up to ten. Indeed, the laser-induced acoustic signal in soils decreases drastically over 26 shots because of the formation of a several millimeter deep pits in such a loose material (Alvarez-Llamas , 2023). This adds another source of variability in the shot-to-shot acoustic signal and explains the higher values. For this reason, the soil population is excluded for the remainder of the study. SI values on rocks also depend on distance and local time as depicted in Fig. 2(b). This behavior is consistent with the idea developed in Sec. IV. SI is driven by turbulence; SI values are low at dusk and dawn and large at midday. It follows the diurnal evolution of the turbulence on Mars: a strong turbulence during the daytime and a more stratified atmosphere during the nighttime (Spiga, 2019). Moreover, SI increases with distance: the farther away the targets are, the more turbulence cells the acoustic wave goes through, and the more variability is created.

To describe the propagation of acoustic waves through a turbulent medium, the log of the acoustic amplitude, χ, is also used and, particularly, its variance: $σ χ 2$ (Tatarskii, 1961). Like SI, $σ χ 2$ is independent of the absolute amplitude of the pressure, making it an easy-to-handle quantity.

To be sure of the validity of this acoustic experiment on Mars over 9 s (i.e., the duration of 26 shots at 3 Hz), three conditions need to be verified over this time scale: (i) the laser energy is stable, (ii) the laser beam itself is not disturbed by atmospheric turbulence, and (iii) the turbulence scales are small compared to the characteristic scale of the experiment and large compared to the acoustic wavelength.

• Shot-to-shot variations of the laser energy over a burst induce variations of the irradiance on target, which create an acoustic scintillation that does not depend on turbulence. Laboratory experiments over 30 shots have demonstrated that the laser energy is stable at 3% root mean square (RMS; Maurice , 2021), which gives $SI ∼ 10 − 3$⁠. This can be observed as a noise floor for the significance of SI. Measured values of SI are above this threshold (Fig. 2);

• atmospheric turbulence will affect the propagation of the laser beam between the microphone and targets on the ground: it causes local variations in the refractive index, which, in turn, induces scintillation of the laser beam itself if the turbulence is strong enough. For the sake of brevity, this discussion is deferred to  Appendix A. There, it is verified that within the limits of this experiment, the most extreme atmospheric turbulence produces a scintillation $SI < 10 − 3$ of the laser for targets between 2 and 8 m. This is below what is measured on Mars for acoustic scintillation (see Fig. 2). The Mars' atmosphere is not dense enough; and

• the theory of wave propagation in turbulent media, to which we will refer, is founded on the statistical representation of random media (Iooss , 2000). To make the statistical inference possible, waves must traverse several heterogeneities along their propagation path. It will be revealed later that the turbulence scale is $L ≃$ 11 cm. Propagation lengths between 2 and 8 m are 20–70 times the characteristic scale of the turbulence. The same theory uses the parabolic approximation to connect fluctuations of a spherical wave to the variations of the medium of propagation. This approximation is valid if the dominant wavelength, λ, is shorter than the size of the heterogeneities (Rytov , 1987). Indeed, at 8 kHz, λ is about 3 cm and, therefore, is smaller than L.

In conclusion, the different length scales, aided by the low density of the Martian atmosphere and the stability of the setup, make the value of SI meaningful. The only limitation is given by the inclined geometry (Fig. 1), which would not be used in a laboratory experiment: the laser shoots downward, and it can be imagined that the distribution of turbulent structures along the propagation path is not homogeneous (Martins , 2021).

In this section, the dataset is restricted to recordings performed between 10:00 and 14:00 LTST such that it consists of waves that have propagated in similar turbulent conditions (still 70% of the data).

In this study, different representations of the perturbation field, i.e., the distribution of the turbulent feature sizes, are tested. They are represented by their power density spectrum, $ϕ ( k )$⁠, as a function of the wavenumber k (see Fig. 3):

• Gaussian field: $ϕ ( k ) ∝ exp ( − k 2 L 2 )$⁠;

• pure Kolmogorov field: $ϕ ( k ) ∝ ( k 2 L 2 ) − 11 / 6$⁠; and

• generalized von Karman field: $ϕ ( k ) ∝ ( 1 + k 2 L 2 ) − p − 3 / 2$⁠, where p is a parameter. Note that when p = 1/3, it is commonly referred as the von Karman spectrum.

For large wavenumbers, i.e., for very small eddies, viscous effects become important and the energy is dissipated. This viscous dissipation range begins for eddies smaller than the inner scale of turbulence, l₀. To take into account the dissipation of energy at large wavenumbers, the generalized von Karman field is modified by an $exp ( − k 2 l 0 2 )$ term that marks the end of the inertial regime. For Mars, it is assumed that $l 0 =$ 0.02 m (Maurice , 2022; Petrosyan , 2011), but this value may vary with the time of the day and season. In this study, this modified von Karman spectrum is not considered.

The Gaussian field drops very fast with the wavenumber, k. The pure Kolmogorov spectrum decays less rapidly with the wavenumber than the Gaussian field. However, it has no finite bound for low k values. Then, a more sophisticated shape is the generalized von Karman spectrum, which is defined beyond the outer scale of turbulence, L, with a shallower slope than that for the Gaussian field. This is more realistic as it converges to a pure Kolmogorov shape for large values of k and reaches a plateau at low values of k. This spectrum is the first to come to mind to describe the planetary surface layer (Kaimal , 1972).

To start this study, let us assume the simplest perturbation field and that ϵ is Gaussian in all directions, i.e., fluctuations are very smoothed. Such a distribution is given by only one scale of inhomogeneity: the outer scale of turbulence, $L Gauss$⁠. Then, the spectral density is $ϕ ( k ) ∝ exp ( − k 2 L Gauss 2 )$ and the Gaussian covariance is $N ( u ) = exp ( − u 2 )$⁠. It is found that $A 1 = 1 2 π$ and $A 2 = − π$ (see the details in  Appendix B).

First, the variance of travel time is examined. Figure 4 shows the application of this Gaussian model to this Mars dataset. At first order, the variance of the travel time follows the quadratic trend expected from Eq. (5). Considering $c 0 = 254.9 ± 0.1$ ms⁻¹ (see later in Fig. 5) in the case of the Gaussian model, the fit of Eq. (5) to the data leads to a first characterization of the fluctuation field, ϵ: the typical size of the turbulent eddies is $L Gauss = 14 ± 1$  cm (⁠ $± 1 σ$⁠) and $σ ϵ , Gauss 2 = 7.4 10 − 4 ± 0.7 10 − 4$ (⁠ $± 1 σ$⁠). Still, a significant scattering is observed in Fig. 4, which might be attributed to non-negligible diffraction effects. Indeed, Eq. (5) is valid in the geometrical acoustics approximation (Iooss , 2000), which is valid when the wave parameter $D = x / ( k L Gauss 2 )$ (Ostashev and Wilson, 2017) is much lower than one. Here, D is lower than one for x closer than 4 m (the majority of our dataset) and is no higher than four for 8 m. Therefore, this experiment might be at the limit of the weak diffraction regime (⁠ $D ≪ 1$⁠), especially for longer propagation distances. It is also consistent with the fully saturated scattering regime of the acoustic wave at such distances (see Fig. 6). Moreover, data from targets beyond 7 m seem to deviate from the fitted solution. Actually, Eq. (5) is valid if the second term is relatively small compared to the first term (Iooss , 2000). Or, for distances longer than 7 m, the quadratic term is more than twice the value of the Chernov model such that it supports that Eq. (5) might be no longer valid at long distances. On the other hand, the scattering of the data points might also be attributed to turbulence characteristics that are not necessarily the same from one recording to the other due to seasonal variations and wind fluctuations, which are not taken into account in this study.

The distribution of mean of travel times with distance given by Eq. (4) is now examined. A precise fit of the second order deviation is not expected as this quantity is close to the 10 μs precision. The first-order fit of the mean travel time as a function of distance (see inset in Fig. 5) gives $c 0 = 254.9 ±$ 0.1 m·s⁻¹ (⁠ $± 1 σ$⁠), which is the value used above as an input for the calculation of $σ ϵ , Gauss 2$ and $L Gauss$⁠. Considering an ideal gas, it corresponds to a temperature $T 0 =$ 242 K, which is a value representative of those measured at Jezero crater (Munguira , 2023; Rodriguez-Manfredi , 2023). As expected, the determination of the time shift, $⟨ δ t ⟩ − x / c 0$⁠, is poorly constrained (see Fig. 5). However, the data show a negative time shift (i.e., a negative slope in the fit in Fig. 5), which is in agreement with the model predicted by Eq. (4). In fact, acoustic waves favor the shortest path and, therefore, statistically, the effective velocity is higher than the mean velocity of the medium. This is similar to the propagation of light in a medium with varying indices and the choice of the shortest path (Fermat's law). This effect is often called the fast path effect in the literature (Blanc-Benon and Juvé, 1993; Codona , 1985): a shorter travel time is expected for unsaturated and partially saturated scattering regime while a pulse delay remains for the fully saturated regime (see Sec. III D). The latter might be observed for distances farther than 7 m, which are likely to fall in the fully saturated regime. However, the dispersion of this dataset does not allow to draw further quantitative conclusions on the fast path effect.

Hence, with $σ ϵ 2 = σ ϵ , Gauss 2 = 7.4 × 10 − 4$⁠, it gives $σ T =$ 3 K, which is $± 1.4 %$ of T₀. This value is consistent with the fluctuations reported in Jezero (de la Torre Juárez , 2023).

The results above were derived for a Gaussian spectrum. In this section, the other turbulence distributions mentioned above (see Fig. 3) are being tested, with the exception of the pure Kolmogorov spectrum for which the approach of Iooss (2000) for the travel times does not work because such a spectrum has no finite bound for low k values (the coefficients A₁ and A₂ cannot be calculated).

For the generalized von Karman spectrum, coefficients A₁ and A₂ are more difficult to constrain as the spectrum depends on an additional parameter, p. A₁ and A₂ for the generalized von Karman spectrum have been calculated in  Appendix B. The parameter p = 1/3 leads to the classical $− 11 / 6$ exponent for a Kolmogorov cascade (Tatarskii, 1961). For such a model of turbulence, $L vKarman , p = 1 / 3 = 1.63 L Gauss$ and $σ ϵ , vKarman , p = 1 / 3 2 = 1.28 σ ϵ , Gauss 2$⁠. For a shallower spectrum with $p = − 1 / 6$⁠, we find that $L vKarman , p = − 1 / 6 = 0.85 L Gauss$ and $σ ϵ , vKarman , p = − 1 / 6 2 = 2.7 σ ϵ , Gauss 2$⁠. The choice for $p = − 1 / 6$ is justified later.

Finally, it has to be noticed that Eq. (5) only gives the intensity of the fluctuation field and its characteristic scale for a given distribution. It does not help to constrain which distribution is the most appropriate to describe the Mars atmosphere. Rather, this will be the focus of Sec. IV.

Λ and $Φ$⁠, computed for the Gaussian field characteristics and generalized von Karman field characteristics, are projected on a Flatté plot in Fig. 6. As the distance increases, the likelihood of saturation increases. For both models, the acoustic signal is not saturated before 3 m. It is fully saturated beyond 6 m with the Gaussian model hypothesis and beyond 7 m with the generalized von Karman model hypothesis. This is consistent with the deviation of the travel time from the Iooss model in Fig. 4 for distances longer than 7 m. In between, most of the points lie in the partially saturated. Of course, this representation is not definitive as the parameters of the models can vary. However, it allows us to verify that our acoustic experiment on Mars is well sized, coincidentally, to probe the near-surface turbulence of Mars. More can be performed from Flatté theory, in particular, looking back at time series and the shape of N-waves when the signal is said to be saturated (Yuldashev , 2017); this will be the subject of a future study.

The amplitude of the sound pressure wave is now investigated. Figure 7 displays sound amplitude and sound speeds for four 29 shot recordings. The greater the dispersion of the amplitude, the larger the SI value is by definition of SI. As discussed in Chide (2023), this amplitude scattering is attributed to turbulence: the morning recordings [Figs. 7(a) and 7(b)] are more stable, and those around noon [Figs. 7(c) and 7(d)] are much more scattered. For the same local time (i.e., assuming a similar level of turbulence), a longer distance contributes to an increase in SI. Moreover, the travel time evolution follows the same trend but with lower variance, which supports the hypothesis that the amplitude scattering is mostly controlled by turbulence. As was performed in Sec. III, the statistics on SI are compared here with different turbulence spectra.

The comparison of SI and $σ χ 2$ (the variance of the lognormal sound pressure) with theoretical models of propagation through thermal turbulence allows the turbulence regime applicable to Mars to be constrained (Blanc-Benon and Juvé, 1993). The relationship between SI and $σ χ 2$ can be described following three solutions that depend on the strength of the perturbations.

Figure 8 shows SI values as a function of $σ χ 2$ for rocks at all distances acquired between 10:00 and 14:00 LTST, and they are compared with the three aforementioned models. The match with the small perturbation model is very good over nearly 2 orders of magnitude in SI and $σ χ 2$⁠. Deviation from the low-turbulence model occurs when SI > 0.3, which is in line with the literature (Blanc-Benon and Juvé, 1993). Most of the points resulting from a longer propagation distance (⁠ $x >$ 7 m, yellow points) better follow the I-K distribution and tend to fall in the intermediate regime. This confirms the observations in the Flatté plot (Fig. 6), where these long distance points are partly saturated. This also explains why these points deviate from the Iooss model for the variance of the travel time (see Fig. 4). Last, the saturation when SI values are leveling off is not reached.

As previously highlighted in Chide (2023) and Figs. 7 and 8, SI varies with the propagation distance and its evolution law as a function of the distance allows to better constrain the turbulence spectrum at the origin of these variations.

Equation (14), applied to the different fields introduced in Sec. III, leads to

• $σ χ 2 ∝ x 2.9$ for a Gaussian field (analytical solution);

• $σ χ 2 ∝ x 1.83$ for a pure Kolmogorov field (analytical solution);

• $σ χ 2 ∝ x 1.65$ for a generalized von Karman field with p = 1/3 (numerical solution); and

• $σ χ 2 ∝ x 1.25$ for a generalized von Karman field with $p = − 1 / 6$ (numerical solution).

Figure 9 shows $σ χ 2$ values as a function of the propagation distance for midday recordings (between 10:00 and 14:00 LTST). For points between 2 and 6 m (corresponding to the weak perturbation regime as per Fig. 2), data can be best fitted with $σ χ 2 ∝ x 1.25$⁠. This corresponds to a generalized von Karman field with $p = − 1 / 6$⁠.

In Sec. III, the turbulence characteristics for different temperature fluctuation fields were computed by comparing the variance of the travel time with Eq. (5). A summary of these parameters (outer scale of turbulence, L; turbulence intensity, $σ ϵ 2$⁠; and temperature fluctuations, σ_T) for each fluctuation field tested is provided in Table I. However, this does not constrain the validity of one model over the others. Here, in Sec. IV, the evolution of the variance of the lognormal sound amplitude with distance tends to validate a generalized von Karman distribution with a parameter $p = − 1 / 6$⁠. Therefore, the acoustic dataset presented in this study, restricted to the midday period (10:00–14:00 LMST), is best represented by a turbulence field that has a power spectrum of $ϕ ( k ) ∝ ( 1 + k 2 L 2 ) − 4 / 3$⁠, an outer scale of turbulence of $L =$ 11 cm and an intensity of $σ ϵ 2 = 2 × 10 − 3$⁠, corresponding to a temperature fluctuation of 6 K over 9 s.

First, the sizes of the largest eddies found here are less than an order of magnitude larger than the theoretical end of the inertial regime, i.e., the inner scale of turbulence, which could be estimated for Mars at 2 cm (Maurice , 2022; Petrosyan , 2011). Note that the value for the inner scale of turbulence is not constant and may vary as a function of the local time and season. However, as no direct measurements are currently available, this value it taken as an order of magnitude. It tends to confirm that the inertial subrange on Mars is very small compared to its extent on Earth, as shown by previous in situ observations (Davy , 2010; Tillman , 1994). Then, the spectral slope of the power spectra found for the daytime data, $− 4 / 3$⁠, is shallower than the slope of $− 11 / 6$ currently predicted for temperature fluctuations by Kolmogorov theory for the inertial subrange [Eq. (6.26) in Ostashev and Wilson (2015)], reinforcing what is found on Mars for the temperature (de la Torre Juárez , 2023) and wind velocity spectra (Viúdez-Moreiras , 2022). In fact, Kolmogorov's theory of turbulence is based on the assumption of isotropic turbulence, which is not true for Mars or the Earth's atmospheric boundary layer. It supports Mars observations of the InSight pressure data (Banfield , 2020; Temel , 2022), which show a pressure spectrum that is flatter than expected. This may be explained by a strong stratification over the first few meters during the day, which introduces anisotropy, and a low thermal inertia of the ground, which induces strong thermal swings of the atmosphere and, thus, intermittency. It is also supported by the power spectra of temperature fluctuations recorded by the Perseverance MEDA weather station (de la Torre Juárez , 2023), which show a different slope as a function of the time of day due to intermittent and sudden events (dust devils, convective cells, etc.). Another source of anisotropy comes from the inclined geometry of the experiment itself (Fig. 1), which means that the distribution of scales is not perfectly homogeneous along the propagation path. Indeed, this ground blocking effect damps the vertical fluctuations of the largest scales (Ostashev , 2023). Therefore, the outer scale of turbulence, L, measured here is likely underestimated compared with a measurement in a free atmosphere. Moreover, in this study, temperature fluctuations have been considered to model the perturbation field, ϵ. However, shear- and buoyancy-produced velocity fluctuations may also contribute to the perturbation field and, therefore, have a contribution to the effective power spectra of fluctuations [Eq. (7.30) in Ostashev and Wilson (2015)]. However, analyses of the Mars pressure and wind speed spectra (Murdoch , 2023) have shown that these two quantities do not follow the Kolmogorov theory either. They tend to show shallower slopes, which are likely explained by anisotropy near the ground. The results presented in this study with temperature fluctuation spectrum confirm the observations presented by Murdoch (2023). On a final note, the results presented are averaged over 800 Sols of data (more than one Martian year) with no distinction on the season or any meteorological event that might have happened during this period. However, Martian turbulence parameters computed here are likely to vary with the time of year or during specific weather, e.g., dust storm or gravity waves (Temel , 2022). This will be the focus of an upcoming study.

In conclusion, the Perseverance rover is performing an original active acoustic experiment to infer, for the first time, the characteristics of the near-surface acoustic turbulence on Mars during the maximum of the convective period, which is between 10:00 and 14:00 LTST. Statistics on the travel time (mean value and variance) and amplitude (SI) of the laser-induced acoustic waves are compared with an extensive theoretical work on the propagation of spherical waves in thermal turbulence (Blanc-Benon and Juvé, 1993; Iooss , 2000; Tatarskii, 1961).

The study of the travel time as a function of the propagation distance helps to establish the key characteristics of the thermal turbulence (outer scale, L, and temperature standard deviation, σ_T), comparing three different fluctuation fields (Gaussian, pure Kolmogorov, and generalized von Karman). All in all, as derived from this analysis, the daytime outer scale of turbulence of the Mars atmosphere ranges between 11 cm (smooth generalized von Karman) and 23 cm (for a standard generalized von Karman) with a temperature fluctuation that ranges between 3 K (Gaussian field) and 6 K (smooth generalized von Karman) over 9 s. Moreover, in the case of a Gaussian field, the variance of the travel times diverges from the linear Chernov model exactly where the density of occurrence of the first caustic reached its maximum. The study of the mean travel time itself shows that the acoustic waves travel, on average, faster than the mean speed of sound (fast path effect; –0.5% at 6 m). Finally, using dimensionless strength and diffraction parameters (Flatté, 1983), it is revealed that the scattering of the acoustic signal begins to saturate beyond 7 m. A study of the SI helps to distinguish which of the tested fields best represents the data. Data match very well the theoretical model (Andrews , 1988; Prokhorov , 1975; Tatarskii, 1961): most of the recordings lie within a weak- to moderate-perturbation field of turbulence, also called the Rytov's solution. Considering the simplifying assumptions assumed in this study, a model of SI variation with distance (Ishimaru, 1978) favors a generalized von Karman spectrum with a shallower slope than the Kolmogorov cascade: $ϕ ( k ) ∝ ( 1 + k 2 L 2 ) − 4 / 3$⁠, where $L =$ 11 cm and $σ T =$ 6 K. These values are within the ranges of temperature fluctuation probability density functions reported in de la Torre Juárez (2023). This result is averaged over the first 800 Sols of the mission. Overall, by only considering thermal fluctuations, the agreement between the two independent datasets, travel time and scintillation, with respect to sound propagation models is quite good, and shows that the theory of acoustic wave propagation in random media is a promising tool to probe the turbulence near the surface of Mars. It also presents a new and interesting framework to adapt theoretical terrestrial models to Martian conditions.

To go further, the simplifying hypotheses used to perform this study should be investigated, especially to better determine whether the contribution of the wind velocity fluctuations is important. Now that the shape of the turbulence spectrum has been estimated, this methodology should be applied at different times of the Martian year to highlight any change of the turbulence properties with season. In particular, acoustic data recorded during dust storms (Lemmon , 2022) could help to validate if such conditions have an impact on the energy cascade (Temel , 2022). This requires a good sampling on the propagation distance over a short period of time (typically 10 Sols) to fit the evolution of the SI. However, propagation distance is not evenly sampled during the mission as targets are usually selected for the geological rational. This is even more the case when it comes to study the nocturnal conditions for which only a couple of recordings are available. It would be necessary to collect more nighttime data to further characterize the shear-induced turbulence observed at Jezero crater (Pla-García , 2023) and detect the intermittent turbulence highlighted on InSight pressure data (Banfield , 2020). For such purposes, a dedicated sound propagation campaign could be required. To complement this study, microphone recordings in passive mode (without any source other than atmospheric motions) give direct access to pressure fluctuations at the boundary between the inertial and dissipative regimes (Maurice , 2022). Numerous 167 s long recordings, at all local times and even during the night, are currently acquired at 25 kHz. Their study will allow a better understanding of the inner scale of turbulence (i.e., the dissipation length) and its variations as a function of wind intensity, local time, and season. Last, the quantities that have just been calculated (spectral slope, intensity, and outer scale of turbulence) are related to fluctuations in refractivity index, that is, density fluctuations. Therefore, they can be used to parameterize these turbulence processes in numerical large eddy simulations, which cannot be resolved at such a small time scale.

The authors thank the Mars2020 Science, operation, and engineering teams for their work to support the mission that has enabled the scientific research presented in this manuscript. This project was supported in the United States by the National Aeronautics and Space Administration's Mars Exploration Program and in France is conducted under the authority of Centre National d'Etudes Spatiales. B.C. is supported by the Director's Postdoctoral Fellowship from the Los Alamos National Laboratory, Grant No. 20210960PRD3. F.S. and S.S. are supported by German Aerospace Center. A.E.S. is supported by a Centre National d'Etudes Spatiales postdoctoral fellowship. The authors thank the three reviewers for their thorough reviews, which greatly improved the manuscript.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The data that support the findings of this study are openly available in the Planetary Data System under the SuperCam Archive Bundle in the SuperCam calibrated audio data collection.

Finally, assuming a maximum value of 30 K for the potential temperature gradient near the surface (Munguira , 2023), $L 0 =$ 30 cm, a pressure of 610 Pa, and a temperature 240 K, the laser scintillation ranges from $10 − 7$ to $6 × 10 − 4$ for distances between 1 and 10 m. This is a negligible value compared to the SI values considered in this sound propagation experiment.

In the body of this study, different turbulence models are tested: a Gaussian distribution and variants of the Kolmogorov distribution. The objective of this appendix is to calculate for each of these spectra: (i) the coefficients A₁ and A₂ that are necessary for the analysis of the mean and variance of travel times [Eqs. (4) and (5)] and (ii) the variations of the variance of the lognormal of the sound pressure, $σ χ 2$⁠, as a function the propagation distance, x, to support the variations of SI.

For a Gaussian spectrum, Eqs. (B2) and (B3) yield $N ( u ) = exp ( − u 2 )$⁠. Then, Eq. (B4) gives $A 1 = 1 2 π$ and $A 2 = − π$⁠.

The Kolmogorov spectrum is not realistic as it increases continuously at low wavenumbers. Thus, it is not possible to derive a covariance function that can be normalized at zero, nor coefficients A₁ and A₂.