Laminar shock stage

The shock structure in the initial stages of my simulations (tΩci= 7.5) is shown in Fig. 6.5. The shock front is located at x/λsi∼ 11. The density increase in the downstream is averaging around ∼ 3, consistent with the value of R= 2.83 derived from Eq. 3.23 in Sec. 3.2.1. The last two panels of Fig. 6.5 show details of the upstream region of the shock.

The features in Bzare fully compatible with those in Ey, hence Eymap is not showed here. The shock front is located at x ' 11λsi. Upstream of the shock, at x & 12λsi, one can see plane-wave fluctuations in Bzmagnetic field (see 2D map ofδBz= Bz−B0in Fig. 6.5c and transversely-averaged profile in Fig. 6.5e), that move with the speed of light away from the shock (i.e., their wave vector is along positive x, kBz= kxˆx). They are polarised in the z-direction, parallel to the large-scale magnetic field. The associated fluctuations in Ey are transverse both to the mean magnetic field and to the propagation wave vector. These waves are then electromagnetic waves of the X-mode type. In the shock upstream one can also see longer-wavelength longitudinal fluctuations in Ex electric field (Figs. 6.5b and 6.5d) with k_Ex= kxˆx and moving also with the speed of light. The normalised amplitude of these electrostatic waves averaged over the three oscillations observed, Ey/B0c ' 1.8 · 10⁻², is an order of amplitude smaller than the X-mode waves amplitude. Note that already in this very early stage of the simulation the shock surface

Figure 6.5: Distribution of the normalised electron number density N_e/N0(a) and the structure of Ex electric field (b) and turbulent Bz magnetic field (c) components at time tΩci = 7.5.

Logarithmic scaling is applied to the density map. The scaling for electromagnetic fields is also logarithmic, but sign-preserving (e.g., sgn(Bz) · {2+ log[max(|Bz|/B0,10⁻²)]}), so that field amplitudes below 10⁻²B₀are not resolved. Field amplitudes are normalised to the initial magnetic field strength. Panel (d) shows transversely averaged profile of the electric field, hExi, upstream of the shock, and panel (e) the profile of δBz taken in the middle of the box along y/λsi= 6.

is perturbed by the developing shock-front ripples.

The emission of X-mode polarised electromagnetic waves from the shock towards the upstream region is a signature of SMI [e.g., Hoshino, 2008, Iwamoto et al., 2017, 2018]. Linear dispersion relation for this waves for parameters used in this study is

presented in Sec. 4.1. Direct comparison with the linear theory will be discussed in the following sections, since the limited extension of the shock precursor at this early stage restricts the dynamic range of wave-vectors. However, here I demonstrate that the waves observed in the precursor of my mildly relativistic shock have been generated through the SMI mechanism by analysing the wave power spectrum, shown for the oscillations in Bz in Fig. 6.6a. I choose the region upstream of the shock at x/λsi= 13 − 18 (see Fig. 6.5). The waves emitted immediately after the beam reflection off the conducting wall and now localised in the x/λsi ∼ 20 − 23 region are heavily affected by the initial conditions and hence are not considered in this analysis.

One can see in the spectra that the waves are mostly parallel with k_Bz,x/λsi' 10 − 20, though a weak oblique component is also present. The wave spectrum has a cutoff at lower wave-numbers and the wave power is mostly to the right of the white solid line. This line represents an estimate of the theoretical cutoff wavenumber. This cutoff represents the minimum wave number above which the waves can propagate upstream ahead of the shock. It expresses the requirement that the upstream-directed group velocity of the precursor waves is larger than the shock velocity. To estimate this wavenumber, one can start from the the X-mode waves dispersion relation in the upstream, Eq. 4.32:

η⁰²= yy→ c²k⁰²

ω⁰² = 1 − ω²_pe

ω⁰²−Ω²_ce. (6.2)

Here and in the following, upstream rest frame quantities are denoted with prime. In the upstream plasma rest frame, the wavenumber is k⁰= kx. One observes that ifω⁰ωpe

we retrieve the dispersion relation in a weekly magnetized plasma,ω⁰²= k⁰²c²+ω²_pe. In case ofω⁰→ ∞ the dispersion relation tends to the one for a wave in vacuumω⁰²= k⁰²c². Considering thatΩ²_ce= ω²_peσ²and by performing a Lorentz transformation one obtains the dispersion relation in the simulation rest frame:

ω²

ω²_pe = σ

γ²(1+ β η cosθ)+ 1

1 −η² . (6.3)

As we consider precursor waves propagating towards the upstream, −π/2 < θ < π/2 is the angle between the x−axis and the wave propagation direction. One observes that, in the first term on the right hand side (1 + βηcosθ) is always equal or greater than

Figure 6.6: Fourier power spectra for B_zfield fluctuations and Exin the early stage of the shock evolution, t= 7.5Ω⁻¹_ci , calculated upstream of the shock in the region x/λsi= (13−18) (compare Fig. 6.5). The solid white line represents the precursor wave theoretical cutoff wavenumber.

unity. I then estimate the maximum contribution to the dispersion relation by this term asσ/γ², and the dispersion relation in its approximated form can be written:

ω² ω²_pe ≈ σ

γ²+ 1

1 −η². (6.4)

This gives

ω ≈ r

(1+ σ

γ²)ω²pe+ c²k², (6.5) where the terms in the fourth power ofωpewere discarded. The waves group velocity is then:

vgr= ∂ω

∂k = c²k

q(1+_γ^σ2)ω²_pe+ c²k²

. (6.6)

From the group velocity one can estimate the cutoff wavenumber for the precursor waves propagating in the upstream. Equating the x−component of the wave group velocity v_gr,x= vgrcosθ with the shock velocity, cβshone obtains:

kx= βshΓsh

s (1+ σ

γ²)ω²pe

c² + k²_y. (6.7)

To calculate the theoretical cutoff wavenumber I used the shock velocities measured from the simulations.

In the magnetized electron-proton plasma the interaction of the precursor waves

with the upstream medium should lead to the generation of electrostatic fluctuations, so-called wake-field. Fig. 6.6b shows the Fourier power spectrum of Ex in the same upstream region that was selected to obtain Fig. 6.6a. Parallel component (with ky≈ 0) is clearly present at k_Ex,xλsi' 2.5 − 3.0. Higher-intensity oblique spectral component at kx& 4 also exists for Ex. As discussed in the next section, these waves are electromagnetic SMI-induced modes generated at oblique angles due to the shock ripples already emerging at this stage. The signal at kxλsi∼ 1 − 4 and kyλsi∼ 3 − 20 is due to upstream filamentation and is described in detail in Sec. 6.2.2(c).

The wavelength of the waves in the Ex component propagating in the x−direction can be inferred from Fig. 6.6 to be λEx ∼ 3.2λsi. Under the assumption that wake-field waves are generated via Raman scattering, it is expected that if in the upstream frame the frequency of the electromagnetic pump wave is larger thanωpe, then for the Langmuir wave ω⁰_L'ωpe, and k⁰_L' 1/λse [Kruer, 1988, Hoshino, 2008], This condition is met in our simulation, since the frequencies of the precursor wavesω⁰/Ωce& 1 and Ωce/ωpe' 2.25 (see Sec.4.1). After performing the Lorentz transformation we obtain the wavenumber, kL, of the electrostatic waves in the downstream (simulation) frame:

kL =γ0k⁰_L (1 −β ω⁰_L

c k⁰_L) 'γ0 1

λse(1 −β ωpeλse

c )

=γ0

λsi

r m_i

me (1 −β).

(6.8)

For our parameters, this gives the theoretical wavenumber kExλsi' 1.86, corresponding to the wavelength ofλEx' 3.4λsi. A slight discrepancy with the observed signal is due to a coarse sampling for the wavenumbers allowed by a limited extension of the precursor wave region at this stage (compare Fig. 6.5). A better agreement is observed in the later stage (see Fig. 6.11b in Sec. 6.2.2(b), in which the Fourier spectra are calculated in a region of double width.