In this section we focus our analysis on high-resolution burst data measured during time 14:31:36–14:32:22  UTC, which contains shock no. 4 of Fig. a. All shocks have similar wave content and heating/acceleration capacity, which can be seen in Fig. . However, the magnitude of compression increases slightly in the earthward direction, which can be seen in Fig. c. On the other hand, the shock speed decreases in the earthward direction.

Figure a–c show reduced one-dimensional distribution functions measured by the FPI instrument. The ion measurements are transformed to a Cartesian coordinate system in which the x^ axis is along the E×B direction, the z^ axis is along the magnetic field and the y^=z^×x^ axis is along the electric field, forming the E×B reference system in velocity space (vE×B,vE,vB). We have used the convection electric field E=-Vp×B to construct these coordinates, where Vp is the velocity of the maximum of the distribution function. Colour spectrograms show phase space density F(vE×B,t), F(vE,t), F(vB,t) integrated over two other velocities with time resolution of 0.15 s corresponding to the sampling time of the instrument. The vertical lines labelled with A–E mark positions of ion distribution functions shown in Fig. . The shock ramp, identified with the B and Te profiles in Fig. d and e, is within blue vertical lines.

Electrons are mostly in quasi-adiabatic regime, |χe| < 1, which means that the temperature shown in Fig. d follows the quasi-adiabatic relation (Eq. ), i.e. Te∝B1-α at shocks . On the other hand, ions are in strongly stochastic regime with |χp|∼ 50, computed with waves f < 64 Hz. The temperatures Ti⟂ and Ti∥ measured by FPI show the perpendicular ion temperature elevated to 200 eV in the foot and ramp of the shock from the isotropic temperature 20 eV measured in the solar wind. The high perpendicular ion temperatures in the foot of shocks are artefacts of the presence of multiple beams in the perpendicular plane; see Fig. b. These beams produce a large velocity spread from the mean velocity, making high temperature from moment computations. Individual beams have lower temperatures than the magnetosheath plasma, whereas Fig. d shows the opposite. Secondary beams are produced by the Ẽ×B acceleration, as we will show further in the text.

In Fig. f we show the time–frequency spectrogram of the field measured by the electric field double-probe instruments with sampling rate of 8192 s-1. Overlaid is the lower hybrid frequency flh and the proton cyclotron frequency fcp. Lower hybrid drift waves have been observed in the dayside magnetosphere by many authors . They can be identified in the frequency range fcp–flh, as discussed extensively in previous papers . This frequency range also contains ion whistler waves, which could originate from mode conversion of lower hybrid waves on density striations .

The vertical striations seen in the spectrogram (Fig. f) represent cascades of instabilities: LHD → MTS → ECD extending form fcp up to nfce in the range of a few kilohertz. The presence of ECD instability at shocks has been reported in several papers . Lower hybrid waves generated in the shock propagate upstream in Fig. f and appear to be associated with particles in Fig. b.

The velocity of the shock in Fig.  determined from inter-spacecraft timing is (-12.0, -9.7, 2.1) or 15.6 kms-1 (GSE) in spacecraft frame. Time lags between two signals were determined with the least squares method for the ramp interval. Strong wave activity in the magnetic signal sampled at 64 Hz introduces some uncertainty into the results. We have used multiresolution wavelet decomposition to remove high frequencies which produce jitter. Wavelet decomposition was chosen instead of low-pass filtering to avoid introducing phase distortions. The least squares values were minimised for signals at frequencies f = 0–2 Hz, which were used to determine the shock velocity. The neighbouring frequency level f < 4 Hz gave a velocity difference of ∼2 kms-1, which we assumed corresponds to the error of the analysis. This value also corresponds to the speed of the spacecraft.

The upstream magnetic field was steady, (1, -5, -3) nT, making angle ∠BN = 71∘ with the shock-normal direction. The proton gyroradius is 100 km in the solar wind, going up to 200 km in the shock. The ion inertial length is 80 km in the solar wind, going down to 40 km in the shock. The time duration of the shock ramp within the blue vertical lines is 9 s. With the derived shock speed of 15 kms-1, this implies a ramp thickness of 135 km or 1 proton gyroradius (rp). The shock comprising the ramp and foot would have thickness 2rp embracing the whole proton orbit, which can be inferred from data presented in Fig.  and in particular from the ion temperature in Fig. d. These values agree with many other estimates of shocks thickness and motion published by other authors. However, shock thickness scalings based on ion inertial length or the hybrid gyroradius (rpre)1/2 are not supported by measurements.

The FPI instrument cannot resolve accurately the small thermal spread of the solar wind beam. Furthermore, the double-beam structure seen in Fig. b artificially increases the ion temperature in the ramp and foot of the shock (Fig. d), so the values of the ion gyroradius are likely overestimated.

In Fig.  we show decomposition of ṼE×B, which corresponds to the energisation capacity of waves in the frequency range 1–256 Hz. The decomposition can be compared with the measured distribution functions shown in Fig. a–c. It indicates that the observed ions can be accelerated by the Ẽ×B mechanism. Acceleration capacity of waves increases with frequency and goes well over 1000 kms-1 for f > 256 Hz.

We shall now inspect the measured ion distribution functions shown in Fig.  in columns A–E, which correspond to events marked in Fig. . Each picture shows a two-dimensional reduced distribution function in the reference system (vE×B,vE,vB). The distributions are averages of three measurements with sampling time 0.15 s each. Magenta circles mark positions of the primary beam in the measured distributions.

Event A shows partly thermalised ions in the magnetosheath with some remaining non-gyrotropic features. The crescent-like structure in distribution A1 is characteristic for Ẽ×B acceleration, which will be explained further. Event B shows ion distribution downstream of the shock peak, and event C shows the distribution on the upstream side of the peak. Event D is in the middle of the shock ramp, and E is in the foot of the shock.

All distributions are strongly non-gyrotropic and some are with separated beams. Similar distributions with double-beam structures have been reported by many authors and interpreted usually in terms of “specularly reflected ions, non-specularly reflected ions, gyrating ions, gyro-phase-bunched ions” or simply “shock reflected ions”.

Particularly puzzling are multiple peaks in the perpendicular plane (first row). Ions reflected from magnetic barriers could acquire a different parallel velocity component V∥. But they are in the same electric field, so they should have the same V⟂≈VE×B velocity component as the original solar wind beam, with possible modifications by temperature-dependent gradient drifts. However, we observe secondary beams in all directions in the perpendicular plane, with similar parallel velocities, which appears to be at odds with standard plasma physics. Ion distribution functions shown in Fig. a–c are inconsistent with the concept of reflection, which should produce a reflected ion beam in Fig. c (parallel direction) and possibly in Fig. a (E×B direction). Instead, the secondary beam is observed in Fig. b (E direction), which can be explained by the stochastic resonant acceleration (SRA) mechanism presented in the next section.

In this section we shall argue that the presented observations are consistent with the Ẽ×B acceleration . First, we should distinguish between the convection E×B drift of ∼ 500 kms-1 used to establish the coordinate system and the wave electric drift ṼE×B=Ẽ×B/B2 at higher frequencies. For wave amplitudes of ∼ 50 mVm-1 in a magnetic field of 7 nT, the latter is ṼE×B∼ 7000 kms-1, corresponding to the gyration speed of a 250 keV proton, which can explain acceleration of ions in quasi-parallel shocks .

Usually, particles do not obey the electric drift in waves with frequencies higher than the gyrofrequency or wavelengths shorter than the gyroradius. In such situations, the effects average to zero over the wave period or wavelength. However, when the electric field gradient ∂xEx=kxẼx, with electrostatic waves with wave vector kx, exceeds the stochastic condition in Eq. (), the particle can be accelerated to the value in Eq. () within a fraction of the gyroperiod.

The mechanism of Ẽ×B acceleration by electrostatic waves can be studied with the Lorentz equation . The previous model is generalised here for waves propagating in arbitrary direction in the perpendicular plane to the magnetic field B0=(0,0,B0). The position r and velocity v of an ion with mass m and charge q are determined by the equation mdv/dt=q(E+v×B0) together with dr/dt=v. We assume that convection electric field Ey convects plasma into electrostatic wave Ewsin⁡(ωDt-k⋅r) propagating in the (x,y) plane at angle α to the x direction, with the Doppler-shifted frequency ωD in the observer's frame. By using dimensionless variables with time normalised by ωc-1, space by k-1 and velocity by ωc/k (with ωc=qB0/mp being the angular ion cyclotron frequency), the normalised equations of motion for a test ion in a stationary (shock related) frame are 5duxdt=(χwcos⁡α)sin⁡Φ+uy,6duydt=(χwsin⁡α)sin⁡Φ-(ux-χd),7dxdt=ux;dydt=uy.

Here, Φ=ΩDt-xcos⁡α-ysin⁡α is the wave phase with the Doppler-shifted angular frequency ΩD=ωD/ωc=Ω+χdcos⁡α with respect to Ω=ω/ωc in the plasma frame. The normalised amplitudes of the Ẽ×B drift and the convection drift are, respectively, 8χw=EwB0kωc;χd=EyB0kωc.

Please note that χw=χp represents here the stochastic wave parameter given by Eq. (). By setting χd=0, we obtain equations in the plasma frame of reference.

The most efficient energisation occurs on the acceleration lane , which corresponds to u0=Ω that matches particle velocity with phase speed of waves. The initial conditions for the here-presented solutions are chosen in such a way that the gyration velocity v0 at t=0 is aligned with the (kx,ky,0) vector or alternatively with the phase velocity of waves so that in the plasma frame we have 9ux0=u0cos⁡α,uy0=u0sin⁡α; where u0=v0k/ωc=krc, and rc=v0/ωc is the gyroradius.

Generally the equations have chaotic solutions, because for χw>1 the solutions are very sensitive for initial conditions and have positive Lyapunov exponent . They are representative of deterministic chaos. The parameters of these equations are the following: Ω, χw, χd, u0 and α, which can be varied to fit particular physical conditions.

Figures  and  show examples of solutions applicable to the foot/upstream region of the shock in Fig. e, where the assumption B≈const. is valid, and we see ions accelerated in the y direction, presumably by lower hybrid waves in Fig. f. The frequency Ω=25 is below the lower hybrid frequency Ωlh≈43, and the ratio χw/χd=Ew/Ey=8 is realistic.

At time t=-1 the proton in Fig.  has initial gyration energy K0=u02 and is drifting earthward with the convection speed ux=χd=5. During time t=0-0.4 it experiences a burst of waves with amplitude χw=40. We see that the gyration energy K=ux2+uy2 has increased more than 4 times after a couple of wave periods. The acceleration is in the uy direction, while ux is constant, which could correspond to Fig. , column E, row 1.

The convection electric field Ey is an essential element in the shock surfing (or surfatron) acceleration (SSA) by waves in front of shocks and in shock drift acceleration (SDA) models based on the magnetic gradient drift. The situation observed in Fig. b, where particles are accelerated along the convection electric field Ey, suggests that we may have the surfatron case here.

To illuminate the significance of Ey for acceleration of particles, we show in Fig.  similar solutions as in Fig.  but with convection switched off by setting χd=0. It can be seen that energisation of particles does not depend on the value of Ey, so the positioning of secondary ion beams along vE in measurements is circumstantial.

The position of accelerated particles in the perpendicular plane (vx,vy)≡(vE×B,vE) is controlled by the wave propagation direction. For waves propagating upstream, α = 180∘ (-x direction) and it is in the positive vy direction, while for α=0 it is in the negative vy direction. At time 14:31:50  UTC in Fig. b we see ions accelerated in the negative vE direction, which is most likely due to downstream-propagating waves as in case of Fig. . For waves propagating at α =  ± 90∘, the acceleration is in the vx or equivalently vE×B direction, which is also observed in measurements. By changing the wave propagation angle α and the amplitude of waves χw, we can reproduce any secondary ion peak which can be found in Fig. , row 1. A free gyration after acceleration would produce crescent-like structures seen in most distributions.

It can also be seen that the duration of wave activity is not an essential factor. In Fig.  we show the electric field Ex seen by the particle along the trajectory made in Fig. . The energisation occurs only during short coherence time after t=0 by means of vxEx>0. The work done by the electric field on the vx component is transferred to the vy component by the Lorentz force vxBz. The mechanism is inherently bursty and works only during short coherence times of a few wave periods. After decoherence, the waves do not affect particles anymore, as can be seen in Figs. –. We can conclude that stochastic particle energisation by waves is performed in a sequence of coherent resonant interactions. This leads to the concept of stochastic resonant acceleration (SRA) as a complementary description of the Ẽ×B wave mechanism. The coherence/resonance is between the wave phase speed ω/k⟂ and the particle initial gyration velocity v⟂ (not drift velocity) in the plasma reference frame. This resonance should not be confused with a better-known parallel resonance (ω-nωc)/k∥=v∥.

The stochastic condition in Eq. () is necessary for energisation of particles. When χw<1, no acceleration can be produced by Eqs. ()–(), irrespective of the values of other parameters. The convection electric field Ey plays no role in the Ẽ×B energisation, which could be anticipated. Indeed, transformation between the plasma frame of reference (where Ey=0, χd=0) and the shock fixed frame with the convection electric field cannot involve Ey in particle energisation as both are equivalent inertial systems.

Secondary beams in the perpendicular plane, such as those seen in Figs. b and  are commonly observed in front of quasi-perpendicular shocks and have been usually described as shock-reflected ions. In shock reflection scenarios it has been usually assumed that the Ex field which makes the cross-shock potential is also responsible for the reflection. Contrary to this popular belief, a strong Ex field does not reflect ions upstream but accelerates them in the y direction through the Ẽ×B mechanism in a stochastic resonant way.

The heating maps published by show that stochastic heating is most efficient for electrostatic waves in the frequency range (0.1–10) fcp with the maximum efficiency depending on the value of χ. Kinetic simulations which can resolve frequencies around ∼ fcp, e.g. , , and , exhibit signatures of ions accelerated by the SRA mechanism. However, these accelerated ions have been described as “shock reflected” by authors being unaware of the SRA mechanism.