Transmission of foreshock waves through Earth's bow shock

Abstract

The Earth's magnetosphere and its bow shock, which is formed by the interaction of the supersonic solar wind with the terrestrial magnetic field, constitute a rich natural laboratory enabling in situ investigations of universal plasma processes. Under suitable interplanetary magnetic field conditions, a foreshock with intense wave activity forms upstream of the bow shock. So-called 30 s waves, named after their typical period at Earth, are the dominant wave mode in the foreshock and play an important role in modulating the shape of the shock front and affect particle reflection at the shock. These waves are also observed inside the magnetosphere and down to the Earth's surface, but how they are transmitted through the bow shock remains unknown. By combining state-of-the-art global numerical simulations and spacecraft observations, we demonstrate that the interaction of foreshock waves with the shock generates earthward-propagating, fast-mode waves, which reach the magnetosphere. These findings give crucial insight into the interaction of waves with collisionless shocks in general and their impact on the downstream medium.

Keywords: Magnetospheric physics.

Fig. 1. Overview of the simulation and wave activity in the foreshock and magnetosheath.

a, Colour map of the magnetic field strength fluctuations in the simulation plane at time t = 500 s from the beginning of the simulation. We subtract <B>_50s, which is a 50 s average of the field magnitude, from B to reveal the fluctuations of the magnetic field magnitude. The black curve shows the approximate magnetopause location. The black arrows show the IMF direction, and the purple arrows depict the shock normal direction n_shock at two positions along the bow shock. b, PSD of the total magnetic field fluctuations at the three locations marked by coloured circles in a. c, PSD of the magnetic field fluctuations parallel and perpendicular to the mean magnetic field at the virtual spacecraft location in the magnetosheath. The perpendicular directions are defined such that B_⊥1 lies in the simulation (x–y) plane while B_⊥2 completes the right-handed set.

Fig. 2. Virtual spacecraft observations in the foreshock and magnetosheath.

a,b,f,g, Time series of the magnetic field strength and ion density (a and f), and of the magnetic field components (b and g). c–e,h–j, Wavelet power spectrum of the magnetic field strength where P is the wave power (c) and (h), wavelet cross-correlation (CC) of the magnetic field strength and density fluctuations (d and i), and compressibility of the magnetic field fluctuations, defined as the wave power parallel to the mean magnetic field P_// divided by the total wave power P_//+ P_⊥1+ P_⊥2 (e and j). The data were extracted at (x = 12R_E, y = 0R_E) (left) and (x = 8R_E, y = 0R_E) (right). The dashed pink line in c–e and h–j shows the foreshock wave period predicted using the Takahashi et al. formula. Note that the time series used for the magnetosheath wavelet power spectra have been high-pass filtered to remove low-frequency variations due to boundary motion (with a cut-off at 40 s), to better highlight the wave power in the relevant period range. The hatched area in c–e and h–j shows the cone of influence, where edge effects are dominant, while the solid black line marks the 95% significance level. Source data

Fig. 3. Wave activity along the Sun–Earth line.

a–c, Maps of the magnetic field strength (a) and its B_y (b) and B_z (c) components along the Sun–Earth line, as a function of x and time. The data have been high-pass filtered, with a cut-off at 40 s, to highlight the relevant frequencies, as indicated by the subscript “filt” in the variable names. The negative magnetic field strength values are due to this filtering. The cyan contour in a marks where the ion density reaches twice its solar wind value, a proxy for the shock position. The black contour in b and c marks where the magnetosonic Mach number M_ms = 1. The coloured lines in the magnetosheath indicate streamlines originating from two locations associated with typical plasma velocities: bulk speed (green), Alfvén speed (blue) and fast magnetosonic speed (pink). The dashed lines correspond to an earthwards propagation in the plasma rest frame, and the dotted lines to a sunwards propagation. The outward motion of the bow shock is due to the two-dimensional (2D) setup of our simulation, as interplanetary magnetic field lines pile up in front of the magnetosphere. d, Time–position map of the magnetosonic Mach number. The white contour marks where M_ms = 1. e,f, Dispersion plots obtained from the 2D Fourier transform of the magnetic field strength (e) and B_z component (f) between x = 6.5R_E and x = 9R_E, using unfiltered data to which a Hann window has been applied along both dimensions. On the horizontal axis, the frequencies are normalized to the ion cyclotron frequency Ω_ci and the wavenumber, on the vertical axis, to the proton inertial length d_p. The solid yellow lines show the Courant–Friedrichs–Lewy (CFL) condition, which is the maximum speed at which information can travel in the simulation. The median bulk speed in the magnetosheath, at the locations used to calculate the 2D Fourier transform ($v_{{\mathrm{bulk}}_x}=48$ km s⁻¹) is indicated by the dashed green line. The dash-dotted blue lines and the dotted pink lines indicate sunwards and earthwards propagation at the median Alfvén speed (v_A = 145 km s⁻¹) and median fast magnetosonic speed (v_ms = 360 km s⁻¹), respectively, in the plasma rest frame.

Fig. 4. MMS observations in the foreshock and the magnetosheath on 14 February 2020.

a,b, The magnetic field strength (black) and electron density (red) (a) and the magnetic field components (b) as functions of time. c–h, Wave properties during two sub-intervals marked with vertical solid and dashed lines in a and b in the foreshock (c–e) and the magnetosheath (f–h): the cross-correlation between the magnetic field strength and electron density fluctuations (c and f), the wavelet trace power (P_tr) spectrum (d and g) and the magnetic field compressibility, defined as the power of the magnetic field fluctuations along the mean magnetic field direction P_∥ divided by the total magnetic field wave power (e and h). The dot-dashed lines denote the expected foreshock wave frequency.

Fig. 5. Experimental wave properties obtained from MMS observations.

The data set includes all events listed in Extended Data Table 1. a–d Orientation of the wavevectors, with negative k_x corresponding to earthwards propagation, as a function of the angle θ_kB between the wavevector and magnetic field. The data are divided between four ranges of θ_kB values to distinguish between nearly parallel (θ_kB ≈ 0° or θ_kB ≈ 180°), nearly perpendicular (θ_kB ≈ 90°) and intermediate propagation direction. The percentage in each panel indicates the fraction of data points within this θ_kB range. The data points marked in red (blue) correspond to those points found within the red (blue) areas in e–h and are thus consistent with the fast wave (Alfvén wave) solution from linear Vlasov theory. The points outside both areas are left in black. e–h, Recovered plasma frame wave frequencies (normalized to the ion cyclotron frequency Ω_ci) as a function of the wavevectors (normalized to the proton inertial length d_p). These are separated by the orientation of the wavevector with respect to the mean-field direction. The red areas denote the fast wave solutions from linear Vlasov theory, and the blue areas denote the solutions expected for the Alfvén wave solutions. The percentages in red (blue) indicate the fraction of data points found within the red (blue) area. The solutions are calculated using the extreme θ_kB values for each angle range and isotropic ion and electron temperatures. The extremes of proton and electron plasma β are β_p = [5, 20], and β_e = [1, 3], and the ratio of Alfvén speed to the speed of light is 2 × 10⁻⁴. The error bars on ω are derived from the s.d. of the velocity component in the direction of the obtained wavevector.

Fig. 6. Schematic of the interaction of foreshock waves with the shock.

Summary of our findings and the scenario we propose for the interaction of foreshock waves with the shock and the resulting waves and structures in the magnetosheath. The wave propagation is shown from left to right, from the foreshock away from the shock (light purple) to the magnetosheath (green). The relevant properties of the 30 s waves are indicated in the foreshock (left) and just upstream of the shock (second from the left). The processes occurring upon their interaction with the shock are marked in the third box (dark purple). The resulting waves and structures in the magnetosheath are listed in the rightmost box (green).

Extended Data Fig. 1. Virtual spacecraft observations near the bow shock.

From top to bottom: (a) magnetic field strength (black) and ion density (red), (b) magnetic field components, (c) wavelet power spectrum of the magnetic field strength, and (d) cross-correlation of the magnetic field strength and density fluctuations. The pink dashed line in panel d shows the predicted foreshock wave period using the Takahashi et al. formula. The black contours in panels c and d delineate the 95% confidence interval of the wavelet power spectrum. The virtual spacecraft is positioned in (x = 9.6 R_E; y = 0 R_E) and is initially located in the foreshock, then crosses the shock shortly after t = 400 s and remains in the magnetosheath afterwards, due to the outward bow shock motion. Between t = 400 − 450 s, fast-mode oscillations at the foreshock wave period are observed just downstream of the shock, consistent with the observations reported by Liu et al.. Source data

Extended Data Fig. 2. Total pressure variations in the magnetosheath caused by the foreshock waves.

Time-position maps of the magnetosonic Mach number (a) and the total pressure (b) along the Sun-Earth line. The total pressure is calculated as the sum of the thermal pressure and the magnetic pressure. The white contour marks where the magnetosonic Mach number Mms = 1.

Extended Data Fig. 3. Magnetosheath wave activity away from the Sun-Earth line.

Left-hand side: time series of the magnetic field and plasma parameters extracted at a virtual spacecraft positioned in (x = 8 R_E; y = -5 R_E). The panels show, from top to bottom: (a) the magnetic field strength (in black) and the ion density (red), (b) the magnetic field components, (c) the wavelet power spectrum of the magnetic field strength, (d) the cross-correlation of the magnetic field strength and density fluctuations, and (e) the compressibility of the magnetic field fluctuations, defined as the wave power parallel to the mean magnetic field divided by the total wave power. The pink dashed line in panel d shows the predicted foreshock wave period using the Takahashi et al. formula. The black contours in panels c-e delineate the 95% confidence interval of the wavelet power spectrum and the hatched area marks the cone of influence. The time series used for panels c-e have been high-pass filtered to remove low frequency variations due to boundary motion (with a cutoff at 40 s), to better highlight the wave power in the relevant period range. Right-hand side: Dispersion plot obtained from the 2D Fourier transform of the magnetic field strength in the magnetosheath between x = 5 R_E and x = 8 R_E at y = - 5 R_E using unfiltered data to which a Hann window has been applied along both dimensions. On the horizontal axis, the frequencies are normalised to the ion cyclotron frequency Ω_ci and the wave number, on the vertical axis, to the proton inertial length d_p. The solid yellow lines show the Courant-Friedrichs-Lewy condition, which is the maximum speed at which information can travel in the simulation. The median bulk speed in the magnetosheath, at the locations used to calculate the 2D Fourier transform is indicated by the green dashed line. The blue dash-dotted lines and the pink dotted lines indicate sunward and earthward propagation at the median Alfvén speed and median fast-magnetosonic speed, respectively, in the plasma rest frame.

Extended Data Fig. 4. Power spectral density of the foreshock and magnetosheath fluctuations observed by MMS on 14 February 2020.

Power spectral density of the total magnetic field fluctuations calculated for the intervals marked by the vertical lines in Fig. 4 in the foreshock (black) and in the magnetosheath (red). The vertical dashed lines indicate the predicted foreshock wave frequency for each interval from the Takahashi et al. formula.

Extended Data Fig. 5. Wave phase speeds from MMS observations.

Total magnetic power as a function of the wave phase speeds v_ph = ω/k. The phase speeds correspond to the recovered points obtained from the Bellan method and are organised as a function of the orientation of the wavevector, with a negative speed indicating earthward propagation. The data points in black are the same as shown in Fig. 5, for which the current density was obtained from the particle measurements. Those in red are based on the current density calculated using the curlometer method. The results from both approaches are in excellent agreement. The vertical blue line denotes the Alfvén speed. The dashed lines denote the phase speeds for largest and smallest phase speeds $v_{\mathrm{A}}\cos \left({\theta }_{\mathrm{kB}}\right)$ and the shaded cyan area denotes the region where the speeds are consistent with Alfvén waves or advected structures.

Extended Data Fig. 6. Wave activity in a 1D local shock simulation with Alfvén Mach number M_A = 6.9.

Proton density (a), magnetic field strength (b) and magnetic field components (c) along the x axis at t = 500 s from the beginning of the simulation. (d) and (e) Dispersion plots obtained from the 2D Fourier transform of the magnetic field strength (d) and By component (e) between x = -10 R_E and x = -5 R_E (marked by the black bar in panel (b)) and t = 250 - 500 s, using unfiltered data to which a Hann window has been applied along both dimensions. On the horizontal axis, the frequencies are normalised to the ion cyclotron frequency Ω_ci and the wave number, on the vertical axis, to the proton inertial length d_p. The solid yellow lines show the Courant-Friedrichs-Lewy condition, which is the maximum speed at which information can travel in the simulation, the blue dash-dotted lines the median Alfvén speed, and the pink line the median fast-magnetosonic speed in the magnetosheath at the locations used to calculate the 2D Fourier transform. The median bulk speed is indicated by the green dashed line.

Extended Data Fig. 7. Wave activity in a 1D local shock simulation with Alfvén Mach number M_A = 4.

Proton density (a), magnetic field strength (b) and magnetic field components (c) along the x axis at t = 500 s from the beginning of the simulation. (d) and (e) Dispersion plots obtained from the 2D Fourier transform of the magnetic field strength (d) and By component (e) between x = -10 R_E and x = -5 R_E (marked by the black bar in panel (b)) and t = 250 - 500 s, using unfiltered data to which a Hann window has been applied along both dimensions. On the horizontal axis, the frequencies are normalised to the ion cyclotron frequency Ω_ci and the wave number, on the vertical axis, to the proton inertial length d_p. The solid yellow lines show the Courant-Friedrichs-Lewy condition, which is the maximum speed at which information can travel in the simulation, the blue dash-dotted lines the median Alfvén speed, and the pink line the median fast-magnetosonic speed in the magnetosheath at the locations used to calculate the 2D Fourier transform. The median bulk speed is indicated by the green dashed line.

Extended Data Table 1. List of all events used in the spacecraft data analysis, including important parameters for each interval.

Intervals during which the MMS satellites were located in the quasi-parallel subsolar magnetosheath. The upstream interplanetary magnetic field (IMF) vector and cone angle are obtained either from the OMNI data set or directly from measurements from the ACE or Wind spacecraft propagated to the bow shock, depending on the data availability for each event. The IMF cone angle provides a good estimate of the shock θ_Bn angle upstream of MMS, because the spacecraft are located in the subsolar region. The magnetosheath wave period is obtained as the peak of the power spectral density of the magnetic field strength during the interval that is closest to the predicted foreshock wave period, given between parentheses. The last column provides the observed (predicted) foreshock wave period when MMS probed the foreshock shortly before or after the magnetosheath interval. Those events for which no data were available in the foreshock are marked with ‘-’. Note that the solar wind and IMF conditions somewhat differ between the magnetosheath and foreshock intervals, hence the slightly different wave periods, but which show good agreement with the predicted values.