Particle Acceleration and Alfv'en Wave Generation at ... - Springer Link

ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2018, Vol. 126, No. 5, pp. 636–644. © Pleiades Publishing, Inc., 2018. Original Russian Text © S.N. Taneev, S.A. Starodubtsev, E.G. Berezhko, 2018, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2018, Vol. 153, No. 5, pp. 765–775.

NUCLEI, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS

Particle Acceleration and Alfv’en Wave Generation at the April 4, 2001 Interplanetary Shock S. N. Taneev, S. A. Starodubtsev*, and E. G. Berezhko Shafer Institute of Cosmophysical Research and Aeronomy, Siberian Branch, Russian Academy of Sciences, pr. Lenina 31, Yakutsk, 677980 Russia *e-mail: [email protected] Received November 23, 2017

Abstract—Based on the theory of diffusive shock acceleration of charged particles, we investigate the formation of proton spectra and the generation of an Alfv’en turbulence spectrum by accelerated (storm) particles at the interplanetary shock front in the event occurred at 14:22 UT on April 4, 2001. We formulate a scenario whereby a satisfactory theoretical description of the observed proton and Alfv’en wave spectra obtained from measurements becomes possible: the propagation of solar cosmic rays in interplanetary space before the interplanetary shock arrival at the Earth’s orbit creates an enhanced level of Alfv’en turbulence in the solar wind, which reduces the diffusion coefficient of particles and increases the efficiency of their acceleration. DOI: 10.1134/S106377611804009X

1. INTRODUCTION The development of a diffusive shock acceleration theory (see, e.g., [1, 2] and references therein) as applied to phenomena in interplanetary space is necessary for a detailed understanding of the formation of high-energy ion spectra at shock fronts. In this paper we continue to investigate the diffusive shock acceleration of particles at interplanetary shock fronts and the generation of Alfv’en waves by accelerated ions (for our previous studies, see [3–5]). The rationale for the subject of our research is detailed in the Introduction to our previous paper [5], while here we comprehensively study the acceleration of particles observed in the quasi-parallel interplanetary shock event on April 4, 2001 at 14:22 UT onboard the ACE spacecraft located at the libration point L1. The goal of this paper is to compare the theory with the experimentally established spectra of accelerated particles and Alfv’en waves generated by them in this event. 2. MEASUREMENTS AND THE METHOD OF CALCULATING THE SOLAR WIND TURBULENCE We used information about the various experiments performed onboard the ACE spacecraft located at the libration point L1 as the initial data characterizing the state of the interplanetary medium in the period of interest to us. For the verified data obtained through the measurements of various instruments, see [6].

Turbulence (perturbations of mainly the interplanetary magnetic field (IMF) with a scale length less than 1 AU (AU stands for astronomical unit)) in the solar wind largely determines the behavior of various kinds of physical processes in the interplanetary medium. Therefore, we used 1-s measurements of the IMF components obtained in the MAG experiment to calculate the spectral IMF turbulence characteristics and to determine the Alfv’en wave spectra. We calculated the spectra of magnetic field fluctuations in several steps. (1) First, the IMF components (Bx, By, and Bz) were transformed from the initial geocentric solar ecliptic (GSE) coordinate system to a new coordinate system where the x' axis is directed along the mean magnetic field vector B. Thus, we determined the new IMF components Bx' , By' , and Bz' . This operation is necessitated by the fact that the Alfv’en waves have a maximum amplitude in a plane perpendicular to the direction of the mean magnetic field [7], i.e., in the Y 'Z' plane. (2) Then, for 20-min time intervals of measurements we performed the standard operation of data reduction to zero mean and their filtering with a digital recursive filter in the frequency range δν = 10–3– 0.5 Hz. (3) Next, using the standard Blackman–Tukey method with the application of a Tukey correlation

636

B, nT

PARTICLE ACCELERATION AND ALFV’EN WAVE GENERATION

25 20 15 10

J, particle cm−2 s−1 sr−1 MeV−1 106

(a)

105 104

5 0

103

n, cm−3

U, km s−1

800 700

637

102 (b)

47–66 keV 66–114 114–190 190–310 310–580 580–1050 1050–1890 1890–4750

101

600

100

500 400 16 14 (c) 12 10 8 6 4 2 0 04:00 16:00 04:00 16:00 04:00 16:00 UT

Fig. 1. IMF magnitude (a), solar wind velocity (b) and density (c) as a function of time. The dashed vertical line marks the detection time of the interplanetary shock observed on April 4, 2001, by the ACE spacecraft in the MAG and SWEPAM experiments.

window [8], we calculated the fluctuation spectra of the quantity

B⊥' = By' 2 + Bz' 2, which we took as the fluctuation spectra of Alfv’en waves in the frequency range under study. To obtain information about the state of the solar wind (its density, velocity, and temperature), we invoked the measurements with various time resolutions from the SWEPAM experiment. To construct the particle energy spectra averaged over 20-min time intervals, we used 5-min proton measurements in eight different differential channels in the EPAM/LEMS120 experiment spanning the energy range from 47 to 4750 keV. To obtain information about the spacecraft coordinates, the detection time of the interplanetary shock, and its characteristics (the angle between the mean IMF direction and the normal to the interplanetary shock front, the components of the normal, the shock velocity, the shock compression ratio, etc.), we used the well-known database of interplanetary shocks observed by the ACE spacecraft and presented at the Harvard-Smithsonian Center for Astrophysics site [9].

10−1 10−2 10−3

00:00 08:00 16:00 00:00 08:00 16:00 00:00 08:00 16:00 UT Fig. 2. Time dependence of the proton flux (April 2–4, 2001) recorded onboard the ACE spacecraft in eight differential energy channels in the EPAM/LEMS120 experiment. The solid and dashed vertical lines mark the solar flare time and the interplanetary shock detection time, respectively.

3. THE APRIL 4, 2001 INTERPLANETARY SHOCK EVENT A powerful X20 solar flare accompanied by the generation of solar cosmic rays (SCRs) and a coronal mass ejection occurred on April 2, 2001, in the active region NOAA AR9393 with coordinates N14W82 at 21:51 UT [10]. The interplanetary shock arrived at the Earth’s orbit already at the beginning of the second half of April 4 (see Fig. 1). Figure 1 presents the IMF magnitude, the solar wind velocity and density recorded on April 2–4, 2001, as a function of time. It follows from this figure that the interplanetary shock was recorded on April 4, 2001, at 14:22 UT. According to the Harvard-Smithsonian Center for Astrophysics information, this shock belongs to the class of quasi-parallel ones—the mean angle between the mean IMF direction and the normal to the shock front is 26.2°.

It can be seen from Fig. 2 that a particle flux increase up to energies of the order of MeV is observed in the same period against the background of SCR fluxes in the upstream region of the shock. Such particles are commonly called storm ones and they are often (but by no means always!) recorded at the fronts of quasi-parallel interplanetary shocks. Generally, these can be the ions accelerated and/or raked up at the shock front. The theory can answer the question about their nature.

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4. THE MODEL The problem under consideration was first formulated in detail by Lee [11]. Subsequently, it was corrected by Gordon et al. [12] and by us [3–5]. A detailed description of the model used here is given in our previous paper [5]. Here we will dwell only on its main elements. The particle acceleration is most efficient at the nose of the shock with the highest velocity VS at which the magnetic field lines B make a small angle ψ with the normal to the shock front n (ψ ≲ 45°). Therefore, we will represent the traveling shock front along the normal n as a segment of a spherical surface with radius RS(t) increasing in time t with velocity VS = dRS/dt and corresponding to a constant solid angle ΩS. Within this solid angle ΩS we will assume the angle ψ to be constant in the range of heliocentric distances 0.2 AU ≤ r ≤ re = 1 AU under consideration, just as the velocity of the flow w directed radially away from the Sun. Here and below, the subscript “e” points to the value at the Earth’s orbit. Below all quantities and variables are taken along the normal n to the shock front. If a physical quantity (or variable) has a different direction, then its projection onto the normal n is marked with a prime. Since the characteristic transverse size L⊥ of the nose (i.e., the acceleration region) is large enough (L⊥ ~ RS) and since the fast particles are strongly magnetized (κ|| ≫ k⊥, here κ||(k⊥) is the parallel (perpendicular) particle diffusion coefficient with respect to the IMF), the approximation of spherical symmetry in our case implies that all physical quantities are functions of only one spatial variable, the heliocentric distance r. In this case, the transport equation for the particle distribution function f(r, v , t) in the region r > RS is

(

)

∂f ∂f ∂f f = 1 ∂ r '2 κ ' − w' − ∂t r '2 ∂r ' ∂r ' ∂r ' τ⊥ 2 ⎡ ∂(r ' w ')⎤ ∂f +⎢ 12 v , ∂r ' ⎥⎦ ∂v ⎣3r ' where v is the ion velocity;

(1)

κ' = κ|| cos 2 ψ + κ⊥ sin 2 ψ ≈ κ || cos 2 ψ is the particle diffusion coefficient; w' is the velocity component of the particle-scattering centers in the solar wind along n whose role is performed by Alfv’en waves; τ⊥ is described in more detail below. The diffusion coefficients in Eq. (1) are defined by the following relations [11–13] (see also [5, 14]):

ρ v v 2B 2 (2) , κ||κ⊥ = B , −1 2 3 8π ωB E w (k = ρB ) where ρB = v /ωB is the gyroradius, ωB = ZeB/Amc is the gyrofrequency, e is the elementary charge, Z is the κ|| =

2

2

charge number, m is the proton mass, A is the mass number c is the speed of light,

E w (k ) = d (δB 2 /8π)/d ln k

(3)

is the energy density of Alfv’en waves. Particles are scattered due to their interaction only with the waves whose wave number k is equal to the inverse particle gyroradius ρB. The spectrum of Alfv’en waves in the undisturbed solar wind, E w0(r, k ) = E w0(RS , k ),

(4)

is a mixture of waves E w+ ( E w− ) propagating away from the Sun (toward the Sun) relative to the solar wind frame;

E w = E w+ + E w−. Since the waves are generated by particles in a narrow region (r – RS ≪ RS) upstream of the shock (r ≥ RS), the assumption made in (4) that the background spectrum of Alfv’en waves is taken at the position of the shock front RS is quite justified. The particle-scattering centers (Alfv’en waves) in the solar wind frame have the following velocity along (opposite to) the IMF in the region r ≥ RS: cc = ca (E w+ − E w− )/E w ,

(5)

where ca = B/ 4πρ is the Alfv’en velocity and ρ is the density of the medium. The velocity of the scattering centers in Eq. (1) is then

w' = w cos ϕ + cc cos ψ,

(6)

where ϕ is the angle between the solar wind velocity w and the normal n to the shock front. The last term on the right-hand side of Eq. (1) describes the adiabatic particle deceleration in an expanding flow, which is one of the factors limiting the accelerated particle spectrum at high energies. The next-to-last term in Eq. (1) describes the particle escape from the acceleration region due to perpendicular diffusion with a time scale τ⊥ = L2⊥ / κ⊥' . The actual values of the diffusion coefficient κ⊥' = κ⊥sin2ψ are such that the term f/τ⊥ has little effect on the particle acceleration. As in our previous papers [3–5], here we adopted L⊥ = 0.6RS, corresponding to ΩS = 1.26 sr. Note that the angle ΩS affects only the total number of shock-produced accelerated particles (which is directly proportional to ΩS) and has no effect whatsoever on their distribution within the cone with the apex angle ΩS. We neglect the shock modification by the inverse effect of accelerated particles, because their pressure, as will be shown below, is much lower than the dynamic pressure of the medium on the shock front:


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Pm = ρu12.

(7)

Thus, the shock front is treated as a discontinuity at which the velocity of the medium relative to the shock front u = VS – wcosϕ undergoes a jump from u1 at x = –0 (r = RS + 0) to u2 = u1/σ at x = +0 (r = RS – 0), where

σ = 4/(1 + 3/M 2f )

(8)

is the shock compression ratio, VS is the shock velocity along the normal n to the shock front, Mf = u1/cs is the Mach number, cs = γ g kBT f /m is the sound speed, Tf is the proton temperature in the narrow transition layer of the shock front in the range of distances –0 ≤ x ≤ +0 (RS + 0 ≥ RS ≥ RS – 0), and kB is the Boltzmann constant. The adiabatic index of the plasma was taken to be γg = 5/3. The x coordinate system is associated with the shock front (x = 0) at the center of the segment of a spherical surface with radius RS and is directed oppositely to the normal n to the shock front (toward the Sun). The subscript 1 (2) marks the quantities that correspond to a point immediately upstream (downstream) of the shock. The distribution function at the shock front (x = 0) satisfies the condition

(

) (

u1' − u2' ∂f ∂f ∂f v = κ||cos 2ψ − κ||cos 2ψ 3 ∂v ∂x 1 ∂x where u ' = u − cc cos ψ

) + Q , (9) 0

⎛ ⎞ u1' u' = σ⎜ ⎟ ⎝ u − σcc cos ψ ⎠1 u2'

N s,inj 4πvinj 2

δ(v − vinj )H (r − r0 )

The choice of the injected particle velocity vinj that, by its definition, separates the slow (thermal) and fast (accelerated) particles in the unified spectrum is to some extent arbitrary. Basically, it is limited only by the validity of the diffusion approximation based on Eq. (1) for the entire range v ≥ vinj under consideration. Therefore, we assume, as usual, that vinj = λ cs 2,

(13)

where λ > 1 (see, e.g., [18]) and cs2 = u1 γ g (σ − 1) + σ/M 2f /σ is the sound speed downstream of the shock. We used λ = 4 in our calculations. The injection energy of particles of species “s” is defined by the expression εs, inj = Amvinj /2. Since the shock front is the only source where particles are injected into acceleration, the problem should be solved under the following initial and boundary conditions: f (r, v, t0 ) = 0,

(10)

(11)

was obtained by taking into account the fact that ca2 = σ ca1 and the angle ψ2 = arccos(cosψ1/σ) between the normal n to the shock front and the interplanetary magnetic field B in the region x > 0 (r < RS) was determined from the condition B2cosψ2 = B1cosψ1 for a perpendicular IMF transition through the discontinuity boundary (shock front). The source

Q0 = u1

Since there is no well-developed theory of the injection mechanism (or, to be more precise, theory of a strong collisionless shock), the parameter η is free in our model. Based on experimental data analysis [15] and numerical simulations of quasi-parallel collisionless shocks [16, 17], we can only specify the possible range of values for this parameter: η = 10–3–10–2.

2

is the velocity of the scattering centers relative to the shock front. The “effective” shock compression ratio

σeff ≡

639

(12)

concentrated at the shock front provides the injection into acceleration of some fraction ηs = Ns, inj of the particles of species “s” with a number density Ns1 = Ns(r = RS + 0). The Heaviside step function H(r) in this expression shows that the source “switched on” at r = r0.

f (r = ∞, v, t ) = 0.

(14)

These imply the absence of background particles with energies in the range under consideration in the solar wind. As in our previous studies, we assume that the medium in the downstream region (r < RS) is perturbed much more strongly than in the upstream region (r > RS), which ensures that κ||2 ≪ κ||1. This assumption allows the second term on the right-hand side of Eq. (9) to be neglected. As a result, the solution of the problem ceases to depend on any peculiarities of the region r < RS. In this case, the particle distribution function in this region can be easily found from Eq. (1) if we neglect the diffusion term, which is justified by the adopted condition, for any given velocity field w (for more details, see, e.g., [19]). The background wave spectrum Ew0 is modified due to the generation of Alfv’en waves by accelerated particles and their damping on thermal protons. Therefore, the transport equation for Alfv’en turbulence is ∂E w± ∂E ± + u1± w = ±ΓE w± − L, ∂t ∂x

where


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Γ(k ) = ξ

8π ca 2 2 kc v

∞

3

∑ s

ω (Ze)2 ⎛ κ|| ⎜ v = B ⎞⎟ Am ⎝ k ⎠

m2

(16)

2 ⎛ ω ⎞ ∂f d vv 3 ⎜1 − 2 B 2 ⎟ ⎝ k v ⎠ ∂x vmin is the growth (damping) rate of the waves excited by accelerated particles [12];

×

∫

vmin = max(vinj, ωB /k ); “s” is the ion species (to simplify the formulas, the ion species index on the corresponding quantities is omitted); L is the wave damping rate on thermal protons, which we take into account when comparing our calculations with the experimental data; (17) u1± = VS − w cos ϕ ∓ ca cos ψ. In comparison with that derived by Lee in [11, 13], the diffusion coefficient κ|| in Eq. (16) was factored outside the integral sign. The latter changes significantly the dynamics of the Alfv’en waves produced by accelerated ions [4, 5]. As Gordon et al. [12] showed, their procedure of deriving Eq. (16) for the growth rate is more consistent than that used previously by Lee [11, 13]. Equation (16) is also consistent with the results from McKenzie and Völk [20] and seems to us more correct in form (factoring the particle diffusion coefficient κ|| outside the integral sign). As experimental data show [21], the background spectrum of Alfv’en waves at the Earth’s orbit deviates in shape from a purely power-law dependence Ew0(ν) ∝ ν–λ: it exhibits a break at νb ~ 0.3 Hz following which the spectrum Ew0(ν) becomes steeper—for a power-law approximation Ew0(ν) ∝ ν–λ the spectral index, on average, is λ = 3, which we adopted in our calculations. The frequency

νb = [kb cos ψ(w cos ϕ + cc cos ψ)]1 /2π, at which the break is observed is determined from the condition [21]

kb = ωB /(ca + vT ),

(18)

where vT = kBTw /m is the thermal proton velocity and Tw is the solar wind proton temperature in the region x < 0 (r > RS). The high-frequency part of the Alfv’en wave spectrum is subjected to damping on thermal protons. To take it into account, we added the following term to the wave transport equation (15):

L = Γ p (E w± − E w±0 ).

(19)

As the wave dissipation rate we use an approximation of the numerical solution to the Vlasov equation [22] for parallel (antiparallel) wave propagation relative to B:

2 ⎡⎛ kc ⎞2 ⎤ ⎡ 2 ⎛ ωp ⎞ ⎤ Γ p (k ) = m1ωB ⎢⎜ ⎟ ⎥ exp ⎢−4m3 ⎜ ⎟ ⎥ , ⎝ kc ⎠ ⎥⎦ ⎢⎣⎝ ω p ⎠ ⎥⎦ ⎢⎣

(20)

where ωp = 4πN pe 2 /m is the proton plasma frequency, m1 = 0.66β0.43, m2 = 1.17 + 0.4β0.4, m3 = 0.31/β0.26; β = 8πkBNpTw/B2 is the plasma parameter.

In order that the spectra E w±0 (k) chosen from particular considerations be the solution of Eq. (15) in the absence of accelerated particles (Γ = 0), the term ‒Γp E w±0 was added to the source L. In this case, far from the shock front, where there are virtually no accelerated particles, L = 0 and, hence, the required solution of Eq. (15) is ensured: E w± = E w±0. Waves are excited by accelerated particles on a relatively small scale within which the source L acts only on the waves excited by accelerated particles, which is what is required. The coefficient ξ before the right-hand side of Eq. (16) defines the degree of Alfv’en wave generation. At ξ = cosψ Eq. (16) coincides with what was obtained by Gordon et al. in [12] to within a numerical factor: we used 8 instead of 32 due to the change of the numerical factor in the denominator in Eq. (2) for the diffusion coefficient κ||. The problem (1)–(20) formulated above is solved numerically. The numerical solution algorithm and the applied numerical methods are briefly described in [14].

5. RESULTS AND DISCUSSION Let us apply the proposed model to investigate the event under consideration. The shock-undisturbed solar wind velocity in this event is w = 457.6 km s–1. From the measured IMF components in the GSE coordinate system, (–5.0, 1.9, 3.1) × 10–5 G, we find the magnetic field strength Be = 6.2 × 10–5 G and the cosine of the angle cos χ between Bx = –5.0 × 10–5 G and the direction to the Sun x: cosχ = –5.0/6.2 = –0.8. Since the IMF in the period under consideration is nearly radial (cosχ = ‒0.8), we use the simplified relation B = Be(re/r)2 to describe its strength in space. The particle number density in the undisturbed solar wind is described by a similar dependence: N = Ne(re/r)2. The proton number density at the Earth’s orbit is N = 5 cm–3. For simplicity, we restrict our analysis to the protons and α-particles whose fraction relative to the proton number density from measurements is 0.02; ρ = 1.08mN. We adopt a dependence of the proton temperature in space, according to the measurements by Eyni and


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Steinitz [23], in the form T = Te(re/r), while the mean proton temperature in the solar wind undisturbed by the interplanetary shock at the Earth’s orbit is Tw = 1.1 × 105 K. The shock compression ratio in the event was determined to be σ = 1.72 ± 0.44. However, σ = 2.1 is best suited for describing the proton spectrum at the shock front at the time of its detection. This requires assuming that the proton temperature in the narrow transition layer of the shock front at the Earth’s orbit is Tf = 4 × 106 K. In our calculations σ changes with distance (0.2–1)re in the range 1.97–2.1, while the “effective” shock compression ratio lies within the range σeff = 2.26–2.5. The angle ψ between the normal n to the shock front and the IMF lines B determined by eight different methods, (13.7 ± 7.4, 17.7 ± 9.5, 15.2 ± 12.5, 18.0 ± 12.8, 19.0 ± 12.8, 39.5 ± 4.4, 43.2 ± 4.0, 43.6 ± 4.0)°, has a wide range of variations with a mean of 26.2°. The angle ψ1 = ψ = 39.5° is best suited to the experiment for our calculations. Downstream of the shock ψ2 = arccos(cosψ1/σ) = 53.5°. We determine the angle ϕ between the normal n to the shock front and the solar wind velocity w (the radial direction r away from the Sun) from the mean cosine of the angle, cos(180° – ϕ) = –0.885, between the normal component Nx and r: ϕ = 28°. According to the measurements [24], the background spectrum of Alfv’en waves

the event recorded onboard the ISEE-3 spacecraft near the Earth’s orbit at 01:21 UT on April 5, 1979. An analysis of the Alfv’en turbulence based on experimental data shows that for the equatorial slow solar wind there are slightly more waves propagating along IMF lines away from the Sun E w+0 than toward the Sun E w−0 [26]. In our calculations, just as in [5, 25], we adopted E w+0 = 0.7Ew0 and E w−0 = 0.3Ew0. The shock velocity VS in the radial direction r away from the Sun calculated by eight methods, (884.4 ± 403.2, 1000.6 ± 440.4, 959.0 ± 429.4, 967.3 ± 431.7, 973.6 ± 432.9, 950.6 ± 399.7, 903.5 ± 376.7, 893.9 ± 372.3) km s–1, has a wide range of variations. We chose its mean to be VS(re) = 941.6 km s–1. Following Watari and Detman [27], we specify a dependence of the shock velocity on heliocentric distance in the form

E w±0(ν) = E w±0[kcosψ = 2πν/(wcosϕ ± cacosψ)]/ν, (21) where

For the calculation to correspond to the experiment at the point of measurements r = re, the injection rate η of particles into acceleration at their injection energy εinj was taken to be η = 10–3. According to [5], we determine the coefficient ξ before Eq. (16):

ν = k cos ψ(w cos ϕ ± ca cos ψ)/2π is the frequency perceived by a stationary observer in the frequency range ν > 10–3 Hz of interest to us, in the undisturbed solar wind is

Ew0 ∝ ν −5/3. Accordingly, in our calculations we used a background turbulence spectrum in the form E w±0(k, r ) = E0± (k /kinj )−λ (r /re )−δ,

(22)

where kinj = ωBe/vinj is the wave number corresponding to the waves that resonantly interact with protons with velocity vinj at r = re. Given that the IMF turbulence in the interplanetary medium after the SCR passage before the interplanetary shock arrival is perturbed, we use the most suitable values to describe the measurements in the event under consideration:

E0 = E0+ + E0− = 3.2 × 10−13 erg cm −3,

λ = 0.

As in our previous paper [25], we take δ = 5. These parameters coincide with those used in [5] to describe

(23) VS (r ) = V0(r0 /r )0.28, –1 where we adopt V0 = 1300 km s at the initial point r0 = 0.2re. Since all calculations in our model are performed along the normal n to the shock front, VS (r ) = VSn(r ), where VSn is the shock velocity along n, at the Earth’s orbit VSn = VS (re ) cos ϕ = 831.4 km s −1.

(24) ξ = (cos 2 ψ)γ−3 cos ψ, where γ = 3σ/(σ − 1). In our case, for σ = 2.1 and ψ = 39.5° we have

ξ = ξ0 cos ψ,

ξ0 = 0.24.

(25)

Figure 3 presents the differential (in kinetic energy ε) intensity J of accelerated protons (26) J (ε) = v 2 f (v )/m at the shock front at the time of its detection at 14:22 UT on April 4, 2001, determined by us from the measurements of particle fluxes. The horizontal segment in the lower right corner indicates the range of the eighth differential channel of measurements of particles with energies from 1890 to 4750 keV at the level of the proton flux amplitude at this time. It can be seen from Figs. 2 and 3 that in addition to accelerated particles, low-energy SCRs are present in this channel. As can be seen from Fig. 3, the proton spectrum calculated at the shock front describes well the mea-


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J, particle cm−2 s−1 keV−1

Ew × 1010, G2 Hz−1

104

101

14:20 UT 04:00 UT

100

102

10−1 100 10−2 10−2

101

102

103

10−3

10−1

100 ν, Hz

ε, keV

Fig. 3. Differential (in kinetic energy ε) intensity J of accelerated protons at the front of the interplanetary shock recorded at 14:22 UT on April 4, 2001. The horizontal segment in the lower right corner indicates the range of the eighth differential channel of measurements of particles with energies from 1890 to 4750 keV at the level of the proton flux amplitude at this time. The solid curve corresponds to the calculation at the shock front (d = 0), while the dashed curve is for d = 0.01 AU along the normal n from it.

surements. Our calculation at d = 0.01 AU along the normal n from the shock front gives an idea of the behavior of the calculated spectra with increasing distance from the front. Figure 4 presents the spectrum of Alfv’en waves Ew calculated from the results of IMF measurements as a function of frequency ν,

E w (ν) = E w[kcosψ = 2πν/(wcosϕ + cc cosψ)]/ν (27) for two times, 04:00 and 14:20 UT on April 4, 2001, before the interplanetary shock arrival. The solid curve corresponds to the calculation at the shock front (d = 0). The dotted line marks the background level of waves we chose. For the sum of the waves Ew = E w+ + E w− traveling away from the Sun, E w+ , and toward the Sun, E w− , the frequency perceived by a stationary observer is

ν = k cos ψ(w cos ϕ + cc cos ψ)/2π.

10−2

(28)

At the Earth’s orbit the protons with injection energy εinj = 8.5 keV resonantly interact with Alfv’en waves with frequency νinj = 2.6 × 10–2 Hz. It can be seen from Fig. 4 that the calculated wave spectrum reproduces well the measured one for 14:20 UT immediately before the shock arrival at 14:22 UT on April 4, 2001.

Fig. 4. Alfv’en wave spectrum Ew for two times, 04:00 and 14:20 UT on April 4, 2001, before the interplanetary shock arrival. The solid curve corresponds to the calculation at the shock front (d = 0). The dashed line marks the chosen background level of waves.

The low-frequency part of the wave spectra (ν < νinj) reflects the shape of the accelerated particle spectra, while their high-frequency part corresponding to frequencies ν > νinj reflects the damping of waves on thermal protons. The distinctive feature of the dynamics of Alfv’en waves is a monotonic increase in their energy at all frequencies, which radically distinguishes it from the wave dynamics calculated using the wave growth rate deduced by Lee [11, 13]. In the latter case, the growth of the wave amplitude is significantly nonmonotonic, so that the wave amplitude at intermediate times exceeds considerably the subsequently established values [3, 28, 29]. The source L describing the damping on thermal ions causes the spectrum of the waves excited by accelerated particles Ew at frequencies ν ≳ 0.05 Hz to deviate toward the background spectrum. The analysis performed in [30] shows that although the damping of waves changes radically their spectrum in the high-frequency region ν ≳ 0.05 Hz, the reduction in the total energy content of the waves due to this process is only 6%. In this regard a significantly more important factor is the presence of waves propagating toward the shock front in the background spectrum that are absorbed by accelerated particles [30]. The sum of the accelerated particle pressures ∞

∑ Am ∫ d vv

Ps = 4 π 3

s

4

f s (RS , v ),

(29)

vinj

where “s” is the ion species, at the shock front (RS = 0) has a maximum at the Earth’s orbit:


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I, particle cm−2 s−1 keV−1 103

47–66 keV

102

190–310 310–580

101

580–1050

643

4. This is explained by the introduction of the coefficient ξ0 = 0.24 into Eq. (16). As the Earth’s orbit is approached, the particle pressure P and the Alfv’en wave energy density E tend to a constant value, confirming the conclusion that the particle acceleration by interplanetary shocks at 1 AU is close to a quasi-stationary state [31]. As an example, the dotted lines in Fig. 5 indicate the proton measurements for five different energy channels. The solid lines correspond to them from the calculation using the formula ε1

1050–1890

∫

I = J (d, ε)d ε/(ε2 − ε1),

100

(32)

ε1

10

12

14

Time, h

Fig. 5. Comparison of the particle flux gradients before the shock arrival measured onboard the ACE spacecraft near the Earth’s orbit on April 4, 2001, with those we calculated. The dotted lines indicate the proton measurements for five differential channels. The solid lines correspond to the calculation. The vertical dashed line marks the shock detection time.

P = Ps /Pm = 3.8 × 10−2, where Pm is the dynamic pressure of the medium (7). The smallness of Ps compared to Pm justifies the neglect of the shock modification by the accelerated particle pressure. The energy density of the Alfv’en waves self-consistent with the accelerated particles ∞

∫

E s = [E w (RS , ν) − E w0(RS , ν)]d ν

(30)

0

at the shock front (RS = 0) also has a maximum at the Earth’s orbit:

E = E s /E B = 4.4 × 10−2, where EB = B2/8π is the energy density of the regular magnetic field. According to [12, 20], the energy density E of the Alfv’en waves generated by accelerated particles at the shock front (RS = 0) can be estimated: E ' = (u1' cos ψ /ca )(Ps /ρ1u1' 2 ).

(31)

Under the condition E w+ ≫ E w−0, it corresponds to a quasi-stationary state and, as follows from Eq. (16), is determined by the accelerated particle pressure Ps [20]. The difference between the E ' approximation and the E calculation decreases with increasing distance r, and the estimate E ' = 0.19 at the Earth’s orbit exceeds the E calculation approximately by a factor of

where ε1 and ε2 are the lower and upper energies of the channel boundaries, respectively; d is the distance from the shock front along the normal n before the particle flux measurement. The vertical dashed line marks the shock detection time. Satisfactory agreement between the measurements and calculations can be seen here. It follows from a comparison of the results of our calculations and the theory presented in this figure that an efficient particle acceleration at the interplanetary shock front in the event under consideration occurred up to an energy of ~1 MeV. Note also that in the low-energy channels, for example, 47–66 keV (see Fig. 5), the observed prolonged (about four hours) proton flux increase toward the shock front can be the flux of particles not accelerated but raked up by the shock. It should be noted that we considered 16 quasi-parallel interplanetary shock events recorded by the ACE spacecraft in 1998–2010. As a result, we concluded that an efficient particle acceleration at them occurred only in the presence of “seed” Alfv’en turbulence. The latter could be produced during the propagation in the solar wind by SCRs generated by the same source on the Sun as the interplanetary shock. The generation of Alfv’en waves in the interplanetary medium by SCR fluxes was considered in detail in a number of theoretical works (see, e.g., [32–34]). 6. CONCLUSIONS The following conclusions can be drawn from our analysis. (1) A comparison of the calculated spectra of accelerated protons and the spectra of Alfv’en waves generated by ions with the measurements performed near the front of the interplanetary shock recorded by the ACE spacecraft on April 4, 2001, at 14:22 UT shows that the theory of diffusive shock acceleration being developed by us is capable of satisfactorily reproducing the experimental data. (2) The results obtained here completely confirm the conclusions reached previously by Berezhko and Taneev [5] when applying this theory to the event


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recorded by the ISEE-3 spacecraft on April 5, 1979, at 01:21 UT, whose detailed experimental results are presented in the papers of various authors [31, 35, 36]. (3) An analysis of the results of various studies of charged (storm) particle acceleration events at quasiparallel shock fronts shows that they occur according to the following scenario: the SCR passage in interplanetary space before the shock arrival at the Earth’s orbit perturbs the IMF, creating an enhanced (“seed”) level of Alfv’en turbulence. This causes the particle diffusion coefficient to decrease and, in turn, according to the diffusive shock acceleration theory, increases the particle acceleration efficiency at interplanetary shock fronts. We can satisfactorily describe the proton and Alfv’en turbulence spectra established in various experiments using the theory only based on this assumption. Thus, a comparison of the diffusive shock acceleration theory with the spectra of accelerated particles and Alfv’en waves generated by them that we established experimentally in the event under consideration shows that the theory we develop reproduces satisfactorily the experimental data. Our results can be useful for analyzing the energy exchange in inhomogeneous media that are of interest in plasma physics and astrophysics. ACKNOWLEDGMENTS We thank the design teams of the EPAM, MAG, and SWEPAM instruments installed on the ACE spacecraft and the ACE Science Center for providing the data and the Harvard-Smithsonian Center for Astrophysics for providing open access to the database of interplanetary shocks. This work was supported in part by Program no. 23 of the Presidium of the Russian Academy of Sciences. REFERENCES 1. E. G. Berezhko, V. K. Elshin, G. F. Krymskii, and S. I. Petukhov, Generation of Cosmic Rays by Shock Waves (Nauka, Novosibirsk, 1988) [in Russian]. 2. E. G. Berezhko and G. F. Krymskii, Sov. Phys. Usp. 31, 27 (1988). 3. E. G. Berezhko, S. I. Petukhov, and S. N. Taneev, Astron. Lett. 24, 122 (1998). 4. E. G. Berezhko and S. N. Taneev, in Proceedings of the 31st International Conference on Cosmic Rays, Łódź, 2009, icrc0201. 5. E. G. Berezhko and S. N. Taneev, Astron. Lett. 42, 126 (2016). 6. http://www.srl.caltech.edu/ACE/ASC/level2/index.html. 7. I. N. Toptygin, Cosmic Rays in Interplanet Magnetic Fields (Nauka, Moscow, 1983) [in Russian].

8. R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra from the Point of View of Communications Engineering (Dover, New York, 1958). 9. https://www.cfa.harvard.edu/shocks. 10. https://umbra.nascom.nasa.gov/SEP/seps.html. 11. M. A. Lee, J. Geophys. Res. 88, 6109 (1983). 12. B. E. Gordon, M. A. Lee, E. Möbius, et al., J. Geophys. Res. 104, 28263 (1999). 13. M. A. Lee, J. Geophys. Res. 87, 5063 (1982). 14. E. G. Berezhko, S. N. Taneev, and K. J. Trattner, J. Geophys. Res. 116, A07102 (2011). 15. K. J. Trattner, E. Möbius, M. Scholer, et al., J. Geophys. Res. 99, 13389 (1994). 16. M. Scholer, K. J. Trattner, and H. Kucharek, Astrophys. J. 395, 675 (1992). 17. K. J. Trattner and M. Scholer, Ann. Geophys. 11, 774 (1993). 18. E. G. Berezhko, V. K. Elshin, and L. T. Ksenofontov, J. Exp. Theor. Phys. 82, 1 (1996). 19. E. G. Berezhko and S. N. Taneev, Astron. Lett. 29, 530 (2003). 20. J. F. McKenzie and H. J. Völk, Astron. Astrophys. 116, 191 (1982). 21. R. J. Leamon, C. W. Smith, N. F. Ness, et al., J. Geophys. Res. 103, 4775 (1998). 22. S. P. Gary and J. E. Borovsky, J. Geophys. Res. 109, A06105 (2004). 23. M. Eyni and R. Steinitz, J. Geophys. Res. 83, 4387 (1978). 24. C.-Y. Tu and E. Marsh, Space Sci. Rev. 73, 1 (1995). 25. E. G. Berezhko and S. N. Taneev, Astron. Lett. 39, 393 (2013). 26. T. S. Horbury, Plasma Turbulence and Energetic Particles in Astrophysics, Ed. by M. Ostrowski and R. Schlickeiser (Obserw. Astron., Krakow, Uniw. Jagiellonski, 1999). 27. S. Watari and T. Detman, Ann. Geophys. 16, 370 (1998). 28. E. G. Berezhko and S. N. Taneev, Kosm. Issled. 29, 582 (1991). 29. E. G. Berezhko, S. I. Petukhov, and S. N. Taneev, Astron. Lett. 28, 632 (2002). 30. E. G. Berezhko and S. N. Taneev, Astron. Lett. 33, 346 (2007). 31. L. C. Tan, G. M. Mason, G. Gloeckler, and F. M. Ipavich, J. Geophys. Res. 93, 7225 (1988). 32. D. V. Reames, Astrophys. J. Lett. 342, L51 (1989). 33. E. G. Berezhko, Sov. Astron. Lett. 16, 483 (1990). 34. R. Vainio, Astron. Astrophys. 406, 735 (2003). 35. K.-P. Wenzel, R. Reinhard, T. R. Sanderson, and E. T. Sarris, J. Geophys. Res. 90, 12 (1985). 36. T. R. Sanderson, R. Reinhard, P. van Nes, et al., J. Geophys. Res. 90, 3973 (1985).


Translated by V. Astakhov

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