Debye-scale Solitary Structures in the Martian Magnetosheath

On 2015 February 9, the MAVEN spacecraft (orbital time 4.5 hr) made approximately five passes around Mars. During these passes, LPW observed several bipolar electric field pulses in the medium frequency range. We could identify a total of 450 bipolar electric field pulses. One example of such a series of bipolar electric field pulses is shown in Figure 1(a) and its spectrogram is shown in Figure 1(b). Time on the x-axis is in milliseconds starting after 9.9651 UT hr (09:57:54:360 hh:mm:ss:mss). The plotted electric field is the y component of electric field (i.e., E_y ) recorded in spacecraft coordinate system. In panel (b), the ion plasma frequency is shown by the black dashed horizontal line. Another example of a series of bipolar pulses is shown in Figure 2. Here, the time on the x-axis is in milliseconds starting after 10.0517 UT hr (10:03:06:120 hh:mm:ss:mss). The important features of the observed events are that the electric field pulses have durations (peak-to-peak time difference) of ≤ 1 ms and their frequencies are close to the ambient ion plasma frequency. We examined the ambient plasma conditions during the occurrence of the electric field pulses, as shown in Figures 1 and 2. Figure 3 shows the electron (black) and ion (red) densities in panel (a), the angle between magnetic field ( B ) and plasma bulk flow velocity ( V _i ) in the Mars Solar Orbital (MSO) coordinate system in panel (b), the magnetic field components [B_x , B_y , B_z ] and the total magnetic field strength B in spacecraft coordinate system in panel (c), the angle between magnetic field ( B ) and E_y in the spacecraft coordinate system in panel (d), and the plasma bulk flow velocity components [V_ix , V_iy , V_iz ] along with the total bulk flow speed (V_i ) in the MSO coordinate system in panel (e) as a function of time in UT hours. The start time of example 1 and example 2 (shown in Figures 1 and 2) are marked by vertical dashed lines in panel (a). One can see that the electron and ion plasma densities vary in the close range between 10 and 20/cc. From Figure 3(b) we can see that the angle between the solar wind velocity and the ambient magnetic field vector is around 110°, which suggests that the solar wind is quasi-perpendicular to the ambient magnetic field. As can be seen in Figure 3(d), the angle, θ, between E_y and B is approximately 30°, which indicates that the observed electric field structures are quasi-parallel to the ambient magnetic field. This confirms the electrostatic nature of the observed electric field pulses. Therefore, we can treat them as ESWs. As can be seen in panel (e), the solar wind bulk speed was around 150 km s⁻¹ during this time. These electric field pulses are quasi-parallel to the ambient magnetic field. In such a situation, there could be some weak magnetic field fluctuations associated with this wave mode but it cannot be checked as MAVEN magnetic field measurement has a sampling capacity of 32 Hz (Connerney et al. 2015), whereas ion-acoustic electric field pulses have frequency close to a few kilohertz.

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In the previous section, we looked at two examples wherein a series of electric field pulses were observed in Mars' atmosphere. We identified such distinct electric field pulses and noted their characteristics, like the maximum electric field (E_max), the minimum electric field (E_min), the duration δt (peak-to-peak time difference), and their corresponding local time, altitude, ambient electron (n_e ), and ion (n_i ) density. The ambient plasma density is used to calculate the ion and electron plasma frequencies (i.e., ${\omega }_{ps}=\sqrt{{n}_{s}{e}^{2}/{m}_{s}{\epsilon }_{0}}$; here s represents species). In Figure 4(a), the occurrences of bipolar electric field pulses are denoted by red "+" signs on the three-dimensional trajectory of MAVEN (green color) around Mars on 2015 February 9. Overall, we identified 450 bipolar electric field pulses. The minimum $| {E}_{\min }|$ and maximum E_max electric fields associated with these pulses are shown in Figure 4(b). The minimum and maximum amplitudes of the electric field pulses are in close range, which indicates that these electric field structures are nearly symmetric in nature. The strength of the electric field of these pulses varies between 1 and 25 mV m⁻¹. Next, we examined the altitude and local time of these electric field pulses. The normalized probability distribution function of local time encountered by the MAVEN spacecraft during its trajectory around Mars (black) and the normalized probability distribution function of local time corresponding to the occurrence of electric field pulses (red color) are shown in Figure 4(c). Here, the sum of normalized distribution is 1, and it represents the total number of occurrences available to estimate respective distributions. For example, we had a total of 11,655 local time occurrences of the MAVEN spacecraft during its trajectory around Mars and a total of 450 simultaneous occurrences of local time corresponding to the time of the electric field pulses. In a similar way, we obtained the normalized probability distribution function for the altitude encountered by the MAVEN spacecraft during its trajectory around Mars (black) and the normalized probability distribution function of altitudes corresponding to the occurrence of electric field pulses (red color) as plotted in Figure 4(d). Figures 4(c)–(d) reveal that these electric field pulses are predominantly seen in the dawn, 5–6 LT and dusk sector 15–18 LT at an altitude of 1000–3500 km.

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In a further investigation, we considered the bipolar electric field pulses that are observed at an altitude of 1000–3500 km (total of 371 events). The distribution of the pulse duration, δt of electric field pulses, and their associated frequencies, f_ef = 1/δt, are plotted in Figures 5(a) and 5(b), respectively. The distribution of the ion plasma frequency (f_pi = ω_pi/2π) and the electron plasma frequency (f_pe = ω_pe/2π) calculated from the ambient ion and electron densities are plotted in Figures 5(c) and (d), respectively. It may be noted that simultaneous, electron density information was only available for 117 electric field pulses. We found that the duration and frequency of electric field pulses are dominantly seen in the range of 0.2–1 ms i.e., 1–5 kHz, respectively, with an average duration of electric field pulses of 0.60 ms and average frequency of 2.1 kHz. The ambient plasma conditions suggest that the ion plasma frequency varies between 0.2 and 1 kHz with an average close to 0.65 kHz, whereas the electron plasma frequency varies between 20 and 50 kHz with an average close to 35 kHz. It clearly indicates that the electric field pulses observed at an altitude of 1000–3500 km have a frequency close to the ion plasma frequency, which is much less than the electron plasma frequency, i.e., f_pi ≤ f_ef ≪ f_pe. This hints that the generation of these electric field pulses might have been associated with ion dynamics. Also, one can see frequency broadening in the spectrograms shown in Figures 1(b) and 2(b). Here, the peak average PSD is localized close to ion plasma frequency; however, there is a broadening (Δf) in the spectra, which is estimated to be 300–400 Hz. Here, Δf is the width in the frequency domain, where the peak average PSD decreases to 50%. Generally, ion-acoustic waves have frequencies close to or less than f_pi. However, as suggested by Akimoto & Winske (1985), when these waves are observed the wave frequency may get Doppler shifted by a few kilohertz due to the motion of the solar wind. This could be a possible reason for getting f_pi ≤ f_ef < <f_pe for observed electric field pulses. These ion-acoustic type solitary waves can have different generation mechanisms, e.g., they can be ion-acoustic solitary waves (Kakad et al. 2013) or ion holes (Akimoto & Winske 1985; Aravindakshan et al. 2020) driven by an ion-beam instability. In order to look into their possible generation mechanisms, we performed a fluid simulation, which is discussed in the next subsections.

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In order to examine the ambient plasma conditions, we looked into ion and electron flux observations. The MAVEN spacecraft carry Solar Wind Ion Analyzer (SWIA) and Solar Wind electron Analyzer (SWEA) instruments that record the ion and electron flux. On 2015 February 9, SWIA observed ion energy flux as a function of time and energy as depicted in Figure 6(a). Similarly, SWEA observed electron energy flux as a function of time and energy, as shown in Figure 6(c). As detailed in Table 1, bipolar pulses are observed in 10 different time slots marked by vertical dotted lines in Figure 6(e), which depicts the variation in the altitude of the MAVEN spacecraft as a function of time on 2015 February 9. For the respective time slots, we averaged the ion and electron fluxes to get their energy spectra. These estimated ion and electron flux are plotted as a function of energy in Figures 6 (b) and (d), respectively, with different colored dashed–dotted lines. The black curves in Figures 6(b) and (d) represent the average flux. It can be seen that the ion flux and electron flux peaks are close to 300 and 60 eV, indicating the difference in the temperatures of the ambient plasma populations. We considered these two plasma populations while modeling these solitary pulses. Mars does not have an intrinsic global magnetic field like Earth does, but it has an induced magnetosphere resulting from the solar wind interactions with the upper atmosphere of Mars (Nagy et al. 2004; Ma et al. 2004). We used the model proposed by Trotignon et al. (2006) to get the location of Mars' bow shock and magnetopause, which is shown in Figure 7(a) superimposed with a red "+" sign, representing the occurrences of 371 electric field pulses. In addition, the direction of the ambient magnetic field in the dusk sector, during the occurrence of electric field pulses shown in Figures 1 and 2, is depicted by the black arrow. Figure 7(a) reveals that the solitary pulses are observed dominantly in the magnetosheath region of Mars. A sketch of the ambient plasma and magnetic field conditions along with electric field pulses is illustrated in Figure 7(b). We considered the simulation model based on this schematic. From Figure 3(b), one can see that the solar wind is quasi-perpendicular (at an angle of ∼110° in the +B direction); therefore, the solar wind ions and electrons have a velocity component antiparallel to the magnetic field such that speed is approximately ${V}_{i}\cos (\theta )$, where θ = 180°−110°. Along similar lines, as shown in Figure 3(d), the angle between B and E_y is around 30°, which affirms the quasi-parallel propagation of the bipolar pulses illustrated in the schematic shown in Figure 7(b).

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We considered a homogeneous, collisionless two species plasma consisting of electrons and ions (H⁺ ions) in the simulation model. The ambient plasma parameters for ions and electrons are given in Table 2. The ions and electrons are considered to be fluid and their dynamic is incorporated into the simulation model using the following model equations viz., the continuity, momentum, and pressure equations of each species, and the Poisson equation (Kakad et al. 2014) given by

$$\begin{eqnarray}&&\displaystyle \frac{\partial {n}_{j}}{\partial t}+\displaystyle \frac{\partial ({n}_{j}{v}_{j})}{\partial x}=0,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {v}_{j}}{\partial t}+{v}_{j}\displaystyle \frac{\partial {v}_{j}}{\partial x}+\displaystyle \frac{1}{{m}_{j}{n}_{j}}\displaystyle \frac{\partial {P}_{j}}{\partial x}-\displaystyle \frac{{q}_{j}}{{m}_{j}}E=0,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {P}_{j}}{\partial t}+{v}_{j}\displaystyle \frac{\partial {P}_{j}}{\partial x}+{\gamma }_{j}{P}_{j}\displaystyle \frac{\partial {v}_{j}}{\partial x}=0,\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial E}{\partial x}=\sum _{j}{q}_{j}{n}_{j}/{\epsilon }_{0}.\end{eqnarray} \tag{ 4 }$$

Here, the electric field E = −∂ϕ/∂x and the variables n_j , P_j , and v_j are the plasma density, thermal pressure, and velocity of species j, respectively. The subscripts j = e and j = i are, respectively, used for electrons and ions. m_j and q_j represent the mass and the charge of species j, respectively. For electrons q_e = −e and ions q_i = e., ₀ is the electric permittivity. In Equation (3), electrons and ions are treated as adiabatic with the same adiabatic index γ_e = γ_i = 3. The above set of equations is solved numerically. The spatial derivatives in these equations are solved using the fourth order central finite difference method and time derivatives are integrated using the leap-frog method to achieve second order accuracy. The details of the development of the simulation model are given in Kakad et al. (2013). We used a compensated filter to eliminate the small wavelength modes linked with such numerical noise (Lotekar et al. 2016; Kakad et al. 2016b). These numerical schemes are highly stable, and in the past, several electrostatic solitary wave structures have been modeled using such fluid simulations in multispecies plasmas (Kakad et al. 2014, 2016a). Δx and Δt are respectively considered as the grid size in spatial and time domain, and their values are taken in such a way that it fulfills the Courant–Friedrichs–Lewy condition, i.e., $c\tfrac{{\rm{\Delta }}t}{{\rm{\Delta }}x}\leqslant 1$, which is necessary for the convergence of the explicit finite difference method. Here, c is the speed of light. We performed two simulation runs in a one-dimensional system with the periodic boundary conditions by considering the observed ambient parameters as (i) T_e = 60 eV, T_i = 300 eV, n_i0 = 15/cc, n_e0 = 15/cc, V_di = V_de = −50 km s⁻¹ (2) T_e = 30 eV, T_i = 300 eV, n_i0 = 15/cc, n_e0 = 15/cc, V_di = V_de = −50 km s⁻¹ (see Table 2 for more details). In the simulations, we considered the real mass ratio, i.e., m_i /m_e = 1836. The background electron and ion densities satisfy the quasi-neutrality, i.e., n_e0 = n_i0 = n₀. For electron temperature, two values are considered based on the observations, i.e., 60 and 30 eV (see Figure 6(d)). The values of ω_pi, ω_pe, λ_i , and λ_e for the considered parameters are 5.1 × 10³ rad s⁻¹, 2.18 × 10⁵ rad s⁻¹, 33.3 m, and 14.9 m, respectively. To initiate the simulations, we used a localized Gaussian-type initial density perturbation in the equilibrium electron and ion densities given by

$$\begin{eqnarray}&&\delta n={\rm{\Delta }}n\,\exp \left[-{\left(\displaystyle \frac{x-{x}_{c}}{{l}_{0}}\right)}^{2}\right].\end{eqnarray} \tag{ 5 }$$

Here, Δn and l₀ are the amplitude and width of the initial density perturbations, respectively. Thus, the perturbed density at t = 0 is n_j (x) = n_j0 + δn. We consider the simulation system length as L_x , and x_c = L_x /2 is the center of the simulation system. We performed two simulation runs for the parameters given in Table 2. For these simulation runs, we consider the time interval dt = 1 × 10⁻³, the grid spacing dx = 0.2, system length L_x = 7000, l₀ = 1, and Δn = 0.1. Here, time is expressed in units of ω_pi ⁻¹, space is in λ_di, and density is in units of n_i0.

The ambient plasma temperatures of ions and electrons are such that T_i > T_e , but V_thi < V_the. The ratio of V_the/V_thi ∼ 19.2 for run-1, and V_the/V_thi ∼ 13.5 for run-2. For such plasma conditions, we expect the presence of Langmuir and ion-acoustic modes. For simulation run-1, the space-time evolution of the electric field generated in the simulation system is shown in Figure 8(a). The ω − k dispersion obtained from the fast Fourier transform of the electric field in space and time during t = 0−50ω_pi is shown in Figure 8(b). The ion and electron plasma frequency values are marked with black horizontal dashed lines. Also, the ion-acoustic speed obtained from their linear dispersion relation ${V}_{{IA}}=\sqrt{({\gamma }_{e}{K}_{B}{T}_{e}+{\gamma }_{i}{K}_{B}{T}_{i})/{m}_{i}}=1.9{V}_{{thi}}$ is shown with white dashed−dotted lines. The generation and evolution of the electric field in the simulation system are shown in a video, which is provided as supplementary material and can be directly accessed online at (https://doi.org/10.5281/zenodo.6590897). From Figure 8, one can see that Langmuir and ion-acoustic modes are generated in the system. The Langmuir mode is the fastest and propagates ahead of the ion-acoustic mode. We observed two nearly identical ion-acoustic solitary waves propagating parallel and antiparallel to the magnetic field, and their speeds are V_ph = 1.64V_thi and V_ph = −2.12V_thi, respectively. To examine the characteristics of these ion-acoustic solitary waves, we tracked their maximum and minimum electric field amplitude, peak-to-peak distance (termed as width), and speed. In Figure 9, the spatial variation in electric field amplitude for the ion-acoustic solitary pulses propagating in (a) parallel to B (c) antiparallel to B, at ω_pi t = 30 (blue color) and ω_pi t = 35 (red color) are shown. Next, the maximum E_max and the absolute value of the minimum electric field $| {E}_{\min }|$ associated with the ion-acoustic solitary pulse propagating parallel to B are plotted as a function of time in panel (b), whereas their widths are plotted in panel (d). In Figure 9(d), the δx^R and δx^L , respectively, indicate the width of ion-acoustic solitary pulses propagating parallel and antiparallel to B. One can see that the maximum electric field amplitude evolves with time, and it approaches to nearly constant amplitude in a stable region around ω_pi t = 30–40, which is shown by the horizontal dashed lines in panels (c) and (d). In the stable region, the speed, width, duration, maximum electric field, and maximum potential of ion-acoustic pulses propagating in parallel and antiparallel to the direction of the magnetic field are (i) V_ph = 1.64 V_thi, δx = 5.4λ_di, δ = 0.64 ms, E_max = 8.33 mV m⁻¹, and ϕ = 1065 V, and (ii) V_ph = 2.12V_thi, δx = 5.2λ_di, δ = 0.48 ms, E_max = 8.33 mV m⁻¹, and ϕ = 1065 mV. In a similar way, we analyzed the simulation run-2, and the speed, width, duration, maximum electric field, and maximum potential of ion-acoustic pulses propagating in parallel and antiparallel directions are (i) V_ph = 1.56V_thi, δx = 5.6λ_di, δt = 0.70 ms, E_max = 4.13 mV m⁻¹ and ϕ = 500 mV, and (ii) V_ph = 2.04V_thi, δx = 5.5λ_di, δt = 0.52 ms, E_max = 4.13 mV m⁻¹, and ϕ = 500 mV. In general, statistical analysis of 371 solitary waves indicates that the duration (δt) of these pulses lies in the range of 0.2–1 ms. In order to convert these durations into spatial scales, one needs information about the phase speed of these bipolar pulses. As the phase speed of these structures is not available from observations, one can use the phase speed estimated from the simulation. For run-1, the phase speed is [1.64V_thi, 2.12V_thi], and for run-2, the phase speed is [1.56V_thi, 2.04V_thi]. Here, the two values in the square bracket represent the phase speed of ion-acoustic mode propagating parallel and antiparallel to the magnetic field. By considering these phase speeds, the 0.2–1 ms durations translate to 55–340 m, which is ∼1.65–10λ_di. This exercise clearly indicates that the observed electric field pulses have spatial scales in the order of a few ion Debye-scale lengths.

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In an earlier study, it was shown that the results from fluid and particle-in-cell simulations are similar for initial density perturbations of ≤10% (Kakad et al. 2014). In our present simulations, we used an initial density perturbation of ∼10%; therefore, the results obtained from fluid and particle-in-cell simulations are less likely to deviate. In our simulation, we used a density perturbation as a free energy source to perturb the equilibrium conditions, but one can use a velocity perturbation as well (Lotekar et al. 2019). It may be noted that most of the electric field pulses are seen in the magnetosheath region of Mars in the altitude range of 1000–3500 km. This is a region between the magnetopause boundary and bow shock boundary. In these boundary regions, the ambient plasma conditions like density and velocity change drastically and they can act as a source of perturbation for the generation of ion-acoustic waves. We cannot deny the possibility of the presence of some kinetic instability because unlike on Earth the induced mini-magnetosphere of Mars is highly dynamic. In our simulations, we could see Langmuir waves moving ahead of the ion-acoustic solitary waves. Whereas in the observations, we did not observed Langmuir-type wave packets in the vicinity of the ion-acoustic waves. As we know, the Langmuir mode is the fastest electrostatic mode, which is generated in plasma, where V_the > V_thi. Its group speed is $\sqrt{3}{V}_{\mathrm{the}}$ (i.e., 5630 km s⁻¹), whereas the phase speed of the ion-acoustic mode is 250–350 km s⁻¹ in the Martian plasma of interest in this study. This indicates that the Langmuir mode travels much faster and being a dispersive mode it will dissipate faster. In such circumstances, it is difficult to detect them in the vicinity of an ion-acoustic wave. From simulations, the estimated spatial scales of ion-acoustic waves come out to be few λ_di. We saw that the ion-acoustic mode drifts with a phase speed of [2.12V_thi, 1.64V_thi]. In the absence of ambient plasma flow, one expects the phase speed to be 1.9V_thi, which is estimated from the linear dispersion relation. This change in phase speed due to the ambient plasma can give rise to a change in wave frequency of the order of 200–300 Hz, which is in general agreement with the frequency broadening observed in Figures 1(b) and 2(b) around the peak average spectral power.

In general, for the given plasma parameters, the system can support solitary wave solutions with different characteristics. The allowed range of these solutions can be obtained from nonlinear theory. For this purpose, the Sagdeev pseudo-potential technique (Sagdeev 1966) is applied, and the analytical solutions of arbitrary amplitude solitary waves in a stationary reference frame are obtained. Over the past several decades this technique has been extensively used to model plasma waves in space and astrophysical plasmas (Maharaj et al. 2013; Verheest et al. 2013; Lakhina et al. 2014). We developed a nonlinear fluid-theoretical framework by taking the ambient plasma parameters given in Table 2 and applying the Sagdeev pseudo-potential technique as detailed in Kakad et al. (2013). The Sagdeev pseudo-potential analysis provides all possible solutions of stable solitary wave structures supported by the given plasma system. The Sagdeev pseudo-potential analysis suggests that the system supports the generation of two wave modes such that one is propagating parallel and another antiparallel to the magnetic field. The Mach number of these two wave modes is represented by M⁺ and M⁻, respectively. We estimated the allowed range of Mach numbers of solitary waves propagating parallel and antiparallel to the magnetic field. For the parameters of run-1 and run-2, the theoretically allowed ranges of Mach numbers are M = 1.6−2.25 and M = 1.52−2.14, respectively. To demonstrate this technique, as an example, in Figures 10(a)–(c) we plotted (i) the Sagdeev pseudo-potential S(ϕ, M) as a function of the electrostatic potential ϕ and (ii) the associated electric field as a function of space, and the wave potential as a function of space, respectively for the Mach numbers M⁺ =1.62 and 1.63 by considering the parameters of run-1. As the Mach numbers of these solitary pulses are available, one can deduce the duration of the solitary pulses. The allowed range of Mach numbers, electric field amplitudes, maximum potential, widths, and time durations of ion-acoustic solitary wave pulses are given in Table 3 for the plasma parameters of run-1 and run-2. From Table 3, one can see that the maximum electric field and time durations allowed by theory for the plasma parameters of run-1 and run-2 are in excellent agreement with nonlinear theory. In Figure 10(d), we plotted E_max as a function of δt observed by MAVEN in the magnetosheath region of Mars. We included the E_max and δt of the modeled ion-acoustic solitary wave pulses obtained through simulations (red: run-1, and blue: run-2) and theory (green). One can see that the characteristics of the modeled solitary pulses are in good agreement with the observations. This confirms that most of the ion Debye-scale electric field pulses observed in the magnetosheath region of Mars are ion-acoustic solitary waves.

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