Solar Wind Turbulence Around Mars: Relation Between The Energy Cascade Rate And The Proton Cyclotron Waves Activity
Nahuel Andrés, Norberto Romanelli, Lina Z. Hadid, Fouad Sahraoui, Gina DiBraccio, Jasper Halekas
mmanuscript submitted to
Geophysical Research Letters
Solar Wind Turbulence Around Mars: Relation Between TheEnergy Cascade Rate And The Proton Cyclotron WavesActivity
N. Andr´es , , N. Romanelli , , L. Z. Hadid , , F. Sahraoui , G. DiBraccio , J. Halekas Instituto de Astronom´ıa y F´ısica del Espacio, CONICET-UBA, Ciudad Universitaria, 1428, Buenos Aires, Argentina Departamento de F´ısica, UBA, Ciudad Universitaria, 1428, Buenos Aires, Argentina Solar System Exploration Division, NASA Goddard Space Flight Center, Greenbelt, MD, USA CRESST II, University of Maryland, Baltimore County, Baltimore, MD, USA. LPP, CNRS, ´Ecole polytechnique, Institut Polytechnique de Paris, Sorbonne Universit´e, F-91128 Palaiseau, France European Space Agency, ESTEC, Noordwijk, Netherlands Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, USA
Key Points: • We estimate, for the first time, the incompressible energy cascade rate obtained in the solarwind near Mars. • We find that the nonlinear cascade of energy is slightly amplified when proton cyclotron wavesare present in the plasma. • These statistical results show the presence of Alfv´enic and non Alfv´enic turbulent fluctuationsin a magnetic dominant regime.
Abstract
The first estimation of the incompressible energy cascade rate at magnetohydrodynamic (MHD)scales is obtained in the plasma upstream of the Martian bow shock, using MAVEN observationsand an exact relation derived for MHD turbulence. The energy cascade rate is computed for eventswith and without proton cyclotron wave (PCW) activity, for time intervals when MAVEN was inthe solar wind with no magnetic connection to the bow shock. It is shown that the nonlinear cas-cade of energy at the MHD scales is slightly amplified when PCWs are present in the plasma. Theanalysis of the normalized cross helicity and residual energy for the turbulent fluctuations shows thepresence of Alfv´enic and non-Alfv´enic fluctuations in a magnetic dominant regime for the majorityof the cases.
Plain Language Summary
Throughout its radial expansion from the Sun, the solar wind develops a strongly turbulentregime, which can be characterized by in situ observations of proton density, velocity and magneticfield fluctuations. Turbulence serves as a reservoir of energy that cascades through the inertialrange down to the smallest scales, where it is dissipated by kinetic effects. For the first time, wecompute the energy cascade rate which is transferred though different scales in the inertial range.This energy rate is computed for cases with and without proton cyclotron waves activity, whenMAVEN was in the solar wind. Our results show that the energy cascade rate is emphasized whenwaves are present in the plasma.
Corresponding author: Nahuel Andr´es, [email protected], [email protected]://orcid.org/0000-0002-1272-2778 –1– a r X i v : . [ phy s i c s . s p ace - ph ] J u l anuscript submitted to Geophysical Research Letters
Turbulence is a unique phenomenon present in several space environments, like the solarcorona (Hendrix & Van Hoven, 1996; Dmitruk et al., 2002), planetary environments (Sahraoui etal., 2020) or the solar wind (Bruno & Carbone, 2005; W. Matthaeus & Velli, 2011). In particular,solar wind turbulence is partially characterized by an inertial range, where energy is transferredwithout dissipation through different spatial and temporal scales (e.g., Frisch, 1995). Typically,in the largest magnetohydrodynamic (MHD) scales, the solar wind magnetic spectrum presents a − / ∼ –2–anuscript submitted to Geophysical Research Letters frames) and the solar wind is also responsible for a Doppler shift that defines the observed wavefrequency near the local proton cyclotron frequency in the spacecraft reference frame (e.g., Russellet al., 1990; Brain, 2002; Mazelle et al., 2004; Romanelli et al., 2013, 2016; Ruhunusiri et al., 2015,2016; Liu et al., 2020). Variability in the proton cyclotron waves (PCWs) occurrence rate has beenobserved based on Mars Global Surveyor magnetic field data (Romanelli et al., 2013; Bertucci et al.,2013) and more recently with MAVEN Magnetometer (MAG) observations (Romanelli et al., 2016;Jakosky et al., 2015a; Connerney et al., 2015). In particular, Romanelli et al. (2016) have analyzedMAG observations between October 2014 and March 2016. The authors reported that the PCWsoccurrence rate upstream of the Martian bow shock varies with time and takes higher values near theMartian perihelion. Such long term trend was associated with higher hydrogen exospheric densitiesaround that orbital position (derived from numerical simulations) and was also in agreement withthe long term trend observed in the irradiances in the 121-122 nm range by MAVEN extreme ultra-violet monitor (EUVM) measurements (Eparvier et al., 2015), which provide a proxy to study thetemporal variability of the photoionization frequency of the neutral H exosphere.Ruhunusiri et al. (2017) have characterized magnetic energy spectra in the Mars plasma en-vironment using the MAVEN MAG observations, in the frequency range 0.005 Hz to 16 Hz. Bycomputing the spectral indices for the magnetic energy, the authors showed a wide range of values inthe upstream solar wind and the magnetosheath plasma. Also, they observed a seasonal variabilityof the spectral indices, indicative of a clear connection with the seasonal variability of the PCWs.Nevertheless, to the best of our knowledge, no estimation of the energy cascade rate has been re-ported yet in the Martian plasma environment. In the present Letter, we aim to extend the currentstate of knowledge of the solar wind turbulence upstream the Martian shock by computing for thefirst time the energy transfer rate using an exact relation for fully development turbulence. Usingboth magnetic field and plasma moments observations at ∼ . The three-dimensional (3D) incompressible MHD equations can be written as, ∂ u ∂t = − u · ∇ u + u A · ∇ u A − ρ ∇ ( P + P M ) + f k + d k , (1) ∂ u A ∂t = − u · ∇ u A + u A · ∇ u + f m + d m , (2) ∇ · u = 0 , (3) ∇ · u A = 0 (4)where we have defined the incompressible Alfv´en velocity u A ≡ B / √ πρ (where ρ the meanmass density) and P M ≡ ρ u / u and u A , are expressed in speed units. Finally, f k,m are respectively a mechanical and the curl of theelectromotive large-scale forcings, and d k,m are respectively the small-scale kinetic and magneticdissipation terms (Andr´es, Mininni, et al., 2016; Banerjee & Kritsuk, 2018).Using Eq. (1)-(4) and following the usual assumptions for fully developed homogeneous tur-bulence (i.e., infinite kinetic and magnetic Reynolds numbers and a steady state with a balancebetween forcing and dissipation (see, e.g. Andr´es & Sahraoui, 2017), an exact relation for incom-pressible MHD turbulence can be obtained as (Politano & Pouquet, 1998a,b), − ε = ρ ∇ (cid:96) · (cid:104) ( δ u · δ u + δ u A · δ u A ) δ u − ( δ u · δ u A + δ u A · δ u ) δ u A (cid:105) , (5)where ε is the total energy cascade rate per unit volume. Fields are evaluated at position x or x (cid:48) = x + (cid:96) ; in the latter case a prime is added to the field. The angular bracket (cid:104)·(cid:105) denotes anensemble average (Batchelor, 1953), which is taken here as time average assuming ergodicity. Finally,we have introduced the usual increments definition, i.e., δα ≡ α (cid:48) − α . Here is we are interestedin estimating ε from Eq. (5), which is fully defined by velocity and magnetic field increments (orfluctuations) that we can estimate from MAVEN observations. –3–anuscript submitted to Geophysical Research Letters
The MAVEN Magnetometer (MAG) provides vector magnetic field measurements with a 32Hz maximum sampling frequency and absolute vector accuracy of 0.05% (Connerney et al., 2015).MAVEN’s Solar Wind Ion Analyzer (SWIA) is an energy and angular ion spectrometer coveringan energy range between 25 eV/q and 25 keV/q with a field of view of 360 ◦ × ◦ (Halekas et al.,2015). In this study, we have analyzed the MAVEN MAG and SWIA data sets as follows. Magneticfield observations with 32 Hz cadence are analyzed to discriminate events in the pristine solar windwith PCWs and without wave activity. To estimate the energy cascade rate at MHD scales (i.e.,frequencies below ∼ . ∼ ∼ For sets A and B ( ∼
330 orbits per set), during time periods when MAVEN was traveling inthe solar wind with no connection to the shock (Gruesbeck et al., 2018), we looked for intervals inwhich the number density fluctuation level was lower than 20% (to be as close as possible to theincompressibilty condition). Moreover, in order to have reliable estimate of the energy cascade rate ε (both its sign and its absolute value (Halekas et al., 2017)) we only consider the events in whichthe θ uB (the angle between the magnetic and velocity field) was relatively stationary (Andr´es et al.,2019). The long time intervals that fulfil these criteria were divided into a series of sample eventswith a duration of 30 minutes. This duration ensures having at least one correlation time of theturbulent fluctuations (Hadid et al., 2017; Marquette et al., 2018). Finally, for set A (set B) weconsidered only cases when PCWs activity was present (absent). By doing this, we can assess theeffects that the PCWs may have on the solar wind turbulence. This selection eventually resulted in184 and 208 events for sets A and B, respectively.Figure 1 shows two examples of the typical events analyzed in the present Letter (panels (a)-(h)show an example from set A, and panels (i)-(p) from set B). Figure 1 (a)-(f) show the time series forthe proton and Alfv´en velocity field components in Mars-centered Solar Orbital (MSO) coordinatesystem (where the x -axis points from Mars to the Sun, z -axis is perpendicular to Mars’ orbital planeand is positive toward the ecliptic north; the y -axis completes the right-handed system). Figure1 (g)-(h) show the angle between the magnetic and velocity field θ uB and the density fluctuationlevel (i.e., ∆ n/ (cid:104) n (cid:105) ), respectively. Both examples show approximately the same level of densityfluctuations and the same θ uB angle. Finally, the Supporting Information shows that both sets Aand B have similar distributions for the density, velocity and magnetic fluctuation values. To determine if a given time interval presents PCWs activity or not, we used a criterion similarto the one in Romanelli et al. (2016). An event is considered to present PCWs activity when themagnetic energy power spectral density (PSD) displays an increase in a frequency interval centered –4–anuscript submitted to
Geophysical Research Letters around the local proton cyclotron frequency f ci when compared to two contiguous windows of width0.2 f ci . More precisely, max { PSD[ B ( f )] | . f ci . f ci } > max { PSD[ B ( f )] | . f ci . f ci } , max { PSD[ B ( f )] | . f ci . f ci } (6)where max corresponds to the maximum value in the PSD in the corresponding window.Figure 2 (a) and (b) show the PSD for all the events in sets A and B, respectively. Forreference, we plot a straight line with Kolmogorov-like slope (i.e., -5/3) in both cases. As weexpected, all events near the Martian perihelion (i.e., set A) show a clear peak in their PSD nearthe proton cyclotron frequency f ci . Moreover, all the cases analyzed in the present Letter show aKolmogorov-like slope in the MHD scales (see, Ruhunusiri et al., 2017). The inset in Figure 2 (a)and (b) show the MAVEN location (where R MSO = (cid:112) y + z ) for each event for sets A andB, respectively. Finally, the gray dashed line corresponds to best fit of the bow shock extract fromGruesbeck et al. (2018). To compute the right hand side of Eq. (5), we constructed temporal correlation functions ofthe different turbulent fields at different time lags τ in the interval [4,1800] s, which allows coveringthe MHD inertial range (Ruhunusiri et al., 2017; Hadid et al., 2017). More precisely, assuming theTaylor hypothesis (i.e., τ ≡ (cid:96)/V , where V is the mean plasma flow speed), Eq. (5) can be expressedas a function of time lags τ . Therefore, for each event in both sets, we compute ε ( τ ).Figure 3 (a) and (b) show the absolute value of the energy cascade rate as a function of thetime lag ( τ ) for both sets. Figure 3 (c) shows the histogram for the (log) mean values log (cid:104)| ε |(cid:105) MHD inthe MHD scales ( τ = 5 × − . × s). It is worth emphasizing that if ε is changing significantlyin amplitude and/or sign, then the resulting mean values would not be reliable (see, e.g., Hadid etal., 2018; Andr´es et al., 2019). Therefore, as we mentioned before, we kept only the intervals forwhich the cascade rate shows a constant (negative or positive) sign for all the time lags in the MHDrange. By doing so, the mean value of ε for each event is robust and so is its absolute value (Coburnet al., 2015; Hadid et al., 2018). The only limitation of analyzing the non-signed ε is related to thedirect vs. inverse nature of the energy cascade rate. This is because the convergence of the signof ε is more stringent than its absolute value (see, Coburn et al., 2015; Hadid et al., 2018), thusdemanding a much larger statistical sample than the one considered in the present work. For bothdata set A and B, the cascade rate values are lower than the averaged value observed at 1 AU, ε ∼ − − − J m − s − (Hadid et al., 2018). Also, it is worth mentioning that the energycascade rate slightly increases when PCWs are present in the solar wind, based on our statisticalanalysis. The cross helicity H c = (cid:104) u · u A (cid:105) and the total energy E T ≡ ( (cid:104)| u | (cid:105) + (cid:104)| u A | (cid:105) ) / u and u A are the proton and Alf´en velocities fluctuations) are the two rugged invariant of the idealincompressible MHD model (see Eqs. 1-4). The dimensionless measure of the normalized cross-helicity corresponds to σ c ≡ H C /E T , with − ≤ σ c ≤
1. Usually, fluctuations with | σ c | ∼ σ r ≡ ( (cid:104)| u | (cid:105) − (cid:104)| u A | (cid:105) ) /E T .This parameter also range between -1 and 1.Figure 4 shows the scatter plot of σ r as a function of σ c , for both sets A and B, respectively.The colorbar corresponds to the mean value of the energy cascade rate in the MHD scales (cid:104)| ε |(cid:105) MHD .The statistical results show a wide variety of possible values of σ r and σ c , independently of thepresence of PCWs. However, for set B, the events gather around | σ c | ∼ .
75 and σ r ∼ − . –5–anuscript submitted to Geophysical Research Letters
In the present work, we analyzed two data sets by considering separately the cases with (setA) and without PCWs (set B). In agreement with previous studies, our findings are consistent withthe seasonal variability of PCWs (Romanelli et al., 2013; Bertucci et al., 2013; Romanelli et al.,2016). We confirmed that such variability is not the result of biases associated with the spatialcoverage of MAVEN or with changes in the background velocity or magnetic fields.Our statistical results show slopes compatible with a Kolmogorov scaling in the largest MHDscales in both sets. Ruhunusiri et al. (2017) determined spectra of magnetic field fluctuations in orderto characterize turbulence in the Mars plasma environment. Using 512 s sliding windows, the authorsfound that magnetic spectrum slopes present different values. In particular, they found that theslope is typically ∼ − . θ uB are approximately constant; ii)the sliding window size used in Ruhunusiri et al. (2017) may not include enough correlation timesto yield reliable PSD slopes; and iii) we are separating between PCWs and no waves events, whileRuhunusiri et al. (2017) included all the available data. It is worth mentioning that Gurnett etal. (2010) showed that the magnetic field fluctuations have a Kolmogorov scaling using magneticfield values derived from electron cyclotron echoes from Mars Express observations. Also, the f − / spectrum for the magnetic energy is theoretically compatible with our constant energy cascade rateassumption (Andr´es, Mininni, et al., 2016; Andr´es, Galtier, & Sahraoui, 2016).We found that the energy cascade rate at Mars ( ∼ | ε | of at least 1 order ofmagnitude with respect to the value at 1 AU (i.e., 10 − − − J m − s − ) (Hadid et al., 2018).However, for the data set A, we observe a slight increase in the transfer of energy when waves arepresent in the plasma. Our results suggest that PCWs at the sub ion scales may affect the turbulenceproperties at the MHD scales. In other words, while Eq. (5) is valid only in the MHD inertial range,our results suggest that the instabilities and consequent nonlinear waves at frequencies ∼ f ci mayaffect the largest MHD scales (Osman et al., 2013; Hadid et al., 2018). However, although severaltheoretical papers have shown that newborn planetary ions are capable of providing the free energyfor the presence of PCWs (e.g., Brinca, 1991), the PCWs observed upstream from the Martianbow shock are nonlinear and likely not saturated (Cowee et al., 2012). While a increase in | ε | in correlation with PCWs activity has not been observed before, an analysis of the local velocitydistribution functions is still needed to better characterize the growing stage of the observed PCWsand its connection with the reported results.While both sets show similar values in the parameter space of number density, velocity andAlfv´en velocity fields values, our results show a wide variability in the possible values of σ c and σ r . In particular, the events in set B correspond to Alfv´enic and magnetic dominant fluctuations( | σ c | ∼ .
75 and σ r ∼ − . σ r values with amajority gathering around σ r ∼ − .
25 and σ r ∼ − .
4, respectively. This majority of events in themagnetic dominant regime is compatible with previous results between 1 and 8 AU (Roberts et al.,1990; Bruno et al., 2007; Halekas et al., 2017). In particular, Halekas et al. (2017) have investigatedthe spatial distributions of σ r and σ c using 30 minutes time intervals with a 45 s cadence. Separatingobservations into four subsets based on the B y sign and the time range (near perihelion or aphelion),the authors found that the temporal decrease in σ c appears to be equally present in all upstreamregions sampled by MAVEN. Our results using 4 s or 45 s (not shown here) cadence exhibit a similarstatistical trend. Therefore, the PCWs activity is not affecting significantly the mean value of thestatistical distributions of σ r and σ c . Slight differences with Halekas et al. (2017) are probably dueto the considered selection criteria.Finally, in this study we have not computed the compressible component of the energy cascaderate (Banerjee & Galtier, 2013; Andr´es & Sahraoui, 2017). In particular, we expect to obtain an –6–anuscript submitted to Geophysical Research Letters strong increases in the nonlinear cascade rate of energy in the Martian magnetosheath, wherecompressibility plays a major role, like in the Earth’s magnetosheath (Hadid et al., 2018; Andr´es etal., 2019). Furthermore, a possible seasonal variability of the incompressible and/or compressibleenergy cascade rate may be present in the Martian environment. These studies will be part of futureworks.
Acknowledgments
N.A., L.H.Z. and F.S. acknowledge financial support from CNRS/CONICET Laboratoire Interna-tional Associ´e (LIA) MAGNETO. We thank the entire MAVEN team and instrument leads for dataaccess and support. N.A. acknowledge financial support from the Agencia de Promoci´on Cient´ıfica yTecnol´ogica (Argentina) through grants PICT 2018 1095 3123. MAVEN data are publicly availablethrough the Planetary Data System ( https://pds-ppi.igpp.ucla.edu/index.jsp ). References
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Figure 1.
Time series for two examples from sets A and B. In particular, the proton and Alfv´en velocityfield components (in MSO coordinate system), the angle between magnetic and velocity fields and the densityfluctuation level, respectively. –13–anuscript submitted to
Geophysical Research Letters
Figure 2.
Magnetic power spectra density for both sets A and B, respectively. Inset: MAVEN location ofeach event in MSO reference frame and the bow shock best fit.
Figure 3.
Energy cascade rate (absolute value) as a function of the time lag for sets (a) A and (b) B, re-spectively. (c) Histogram of log (cid:104)| ε |(cid:105) MHD for both sets.–14–anuscript submitted to
Geophysical Research Letters
Figure 4.
Scatter plot of σ r as a function of σ rr