Rankine–Hugoniot Relations and Magnetic Field Enhancement in Turbulent Shocks

In fast collisionless shocks, the density and magnetic field increase and the plasma is heated. The compression and heating are ultimately determined by the Rankine–Hugoniot relations connecting the upstream and downstream parameters. The standard Rankine–Hugoniot relations take into account only mean upstream and downstream parameters. Observations at the Earth's bow shock show that the downstream magnetic field does not always relax to a uniform state, but large amplitude magnetic oscillations persist. Here, these Rankine–Hugoniot relations are extended to such turbulent shocks where the mean downstream magnetic field is accompanied by magnetic fluctuations. It is shown that the turbulent magnetic field pressure may substantially exceed the pressure of the mean field, while the density compression and heating may be only weakly affected. Thus, strong amplification of the rms magnetic field can be achieved at the expense of a modest reduction of plasma heating.


Introduction
Collisionless shocks are one of the most ubiquitous phenomena in space plasmas.They are encountered in all plasma environments, from a lab (Niemann et al. 2014) to a galaxy cluster (Markevitch & Vikhlinin 2007), and at all spatial scales, from millimeters to kiloparsecs.Supernova remnant (SNR) shocks are one of the most efficient accelerators of charged particles in the Universe (Miceli 2023).SNR shocks can be studied only with remote measurements of the electromagnetic radiation coming from the accelerated and heated particles at the shock.The heliosphere is the only place where collisionless shocks can be studied with in situ measurements, although these shock Mach numbers are well below those for SNR shocks.The results of studies of heliospheric shocks are often extrapolated to SNR shocks with the assumption the physical processes are similar.
In fast collisionless shocks, the density, the plasma temperature, and the magnetic field increase upon crossing the shock.The standard Rankine-Hugoniot (RH) relations, which connect the upstream and downstream parameters, place the absolute upper limit on the possible density and magnetic compressions ρ d /ρ u 4, B d /B u 4. Hereafter, u denotes upstream and d denotes downstream.The standard RH relations assume that the plasma parameters and the magnetic field arrive at uniform upstream and downstream values and do not take into account fluctuations.Observations of the electromagnetic emission from SNR shocks imply magnetic amplification larger than that predicted by RH relations (Vink & Laming 2003;Völk et al. 2005;Helder et al. 2012).Mechanisms of the magnetic field amplification due to interaction with the preexisting turbulence excited in the upstream region have been proposed (Zank et al. 2002;Völk et al. 2005;Terasawa et al. 2007;Niemiec et al. 2008;Adhikari et al. 2016;Guo et al. 2021;Nakanotani et al. 2022;Wang et al. 2022).
Figure 1 shows three magnetic field components measured by the Magnetospheric Multiscale (MMS) mission, probe 1, at the 2015 November 4/04:38:42 quasi-perpendicular shock crossing.The angle between the upstream magnetic field and the model shock normal (Farris & Russell 1994) is θ Bn ≈ 66°.The Alfvénic Mach number is the ratio of the normal component of the upstream flow velocity in the shock frame V u to the upstream Alfvén speed It can be easily seen that the level of fluctuations in the upstream region −20 < t < −10 is much lower.Indeed, in the upstream region, B turb /B u ≈ 0.1.
Figure 2 shows three magnetic field components measured by the MMS mission, probe 1, at the 2015 October 21/ 07:05:05 quasi-parallel shock crossing, θ Bn ≈ 33°, M ≈ 9.For the choice of the upstream region 50 < t < 100 and the downstream region −500 < t < −100, we get B d /B u ≈ 2.5 and B turb /B u ≈ 4. Thus, the turbulent magnetic pressure exceeds the mean magnetic pressure.In this case, B turb /B u ≈ 0.5, although the upstream region is small and the precision is lower.
In both cases, the relative level of turbulence in the downstream region is much larger than it is in the upstream region, while the absolute level is 1 order of magnitude larger.Other observations show a typically low or moderate level of solar wind turbulence at the scales of interest (Sahraoui et al. 2020;Fraternale et al. 2022).The level of turbulence behind the shock transition (in the downstream region) is typically 1 order of magnitude higher (Pitňa et al. 2017;Zank et al. 2021;Zhao et al. 2021).Note that the term turbulence may be somewhat misleading, since the downstream magnetic fluctuations may be simply persisting coherent magnetic oscillation, due to the downstream ion gyration and slow gyrophase mixing (Balikhin et al. 2008;Gedalin 2015;Gedalin et al. 2015).In this case, the spatial scale of these fluctuations is of the order of the downstream gyroradius of the ions.For RH relations, the averaging is over a much larger distance.On the other hand, the spatial scale of the fluctuation is much larger than the gyroradius of the shock-heated electrons, so they emit electromagnetic radiation in a local field that may significantly exceed the mean field.It has been found observationally on the Earth's bow shock that very long waves with a period of tens of seconds pass from the foreshock and propagate in the magnetosheath toward the magnetopause (Turc et al. 2023).Recent simulations show enhancement of the turbulence on transmission through the shock (Trotta et al. 2021(Trotta et al. , 2022(Trotta et al. , 2023)).In these simulations, there is a preexisting turbulence with amplitudes higher than those observed and wavelengths much smaller than those observed.The simulation box was not large enough to catch the far downstream state of the plasma and magnetic field, and the simulations focused on the modifications of the shock front itself.The transmission of the turbulence through the shock was studied theoretically (Zank et al. 2021), but the turbulence was decoupled from the mean fields.A step toward theoretical incorporation of the turbulence in the shock conditions was done within the Burgers equation for an incompressible fluid without a magnetic field (Zank et al. 2002).It was shown that the magnetic field in shocks may be amplified due to the large scale (wavelength of 10 1 -10 2 shock widths) upstream density fluctuations (Giacalone & Jokipii 2007).RH relations with the turbulence included were introduced ad hoc by Terasawa et al. (2007), without addressing the magnetic field.
In the present paper, we study the implications of the inclusion of fluctuating variables in RH relations.We focus on strong shocks, M ? 1, where the contribution of the upstream turbulence can be neglected relative to the pressure and energy of the directed flow.The downstream region also contains a regular mean magnetic field and large amplitude magnetic fluctuations, which may be random or coherent.Irrespective of the nature of the fluctuations, we shall call them turbulence.We investigate the possibility that the downstream turbulent magnetic pressure is a sizable fraction of the total pressure.An upper limit is placed on the turbulent magnetic compression in a strong parallel shock.

RH
The magnetohydrodynamical (MHD) RH relations are nothing but the mass, momentum, and energy conservation laws, together with the requirements of the constancy of the normal component of the magnetic field and the tangential component of the electric field.In an ideal MHD, the requirement that the electric field E vanishes in the flow frame is added.The corresponding equations read: Here, i = x, y, z, the velocity V is the hydrodynamic velocity, p is the kinetic pressure (assumed isotropic for simplicity), ò is the internal energy, and δ ij is the Kronecker tensor.The shock normal is in the x-direction and In the equation of state (7), the polytropic index γ is constant.Ley us denote by á¼ñ proper averaging over x, y, z, t.Then, where we have taken into account that á ñ = E 0 x .Let us split each variable into the mean part and the fluctuating part: , . 1 3 In what follows, we omit the symbol of averaging, that is, á ñ  B B, etc.Then, the Equations ( 8)-( 12) take the form: and we assume that means of triple products vanish, e.g., á ñ = rv 0 2 .The modified RH relations, in principle, depend on all pairwise averages.A detailed study of the implications in the general case will be done elsewhere.Here, we focus on the simplest case of a parallel shock where the mean magnetic field remains constant across the shock.

Parallel Shock with Incompressible Turbulence
In what follows, we restrict ourselves to a parallel shock, V ⊥ = 0, B ⊥ = 0, E ⊥ = 0, and incompressible turbulence In this case, Equations (14)-( 18) reduce to the following system: x Equation ( 23) is characteristic for Alfvén turbulence, in which case 2 2 so that eventually the energy conservation reads the turbulent magnetic amplification is ≈45.The mean field is not amplified at all in a parallel shock.

Conclusions
Observations at the Earth's bow shock show that there exist shocks in which the upstream region is uniform, but the downstream region is filled with large amplitude magnetic oscillations superimposed on the mean magnetic field.The effective pressure of these oscillations may greatly exceed the pressure of the mean magnetic field.At present, it is not clear why such oscillations persist well into the downstream region.Magnetic amplification derived from the standard RH relations, incorporating only the mean field, is incorrect for such shocks, so modifications are necessary.We have shown that the effective magnetic amplification may be much larger.The spatial scales of the magnetic fluctuations are substantially larger than the gyroradii of thermal electrons, but may be substantially smaller than the gyroradius of a high-energy accelerated electron and certainly smaller than the subparsec scales resolved by X-ray polarimetry of SNR shocks (Ferrazzoli 2023).Therefore, thermal electrons emit electromagnetic radiation in a local magnetic field that may be much stronger than the present estimates the rms magnetic field for the subparsec-scale magnetic turbulence.On the other hand, high-energy accelerated electrons emit in strong but rapidly varying magnetic fields, and it is possible that the jitter emission for these electrons is a more appropriate approach than the synchrotron emission (Medvedev 2000;Medvedev et al. 2007).If the physics of collisionless shocks at M ∼ 10 2 -10 3 does not differ drastically from the physics at M ∼ 10, the above conclusions about the rms magnetic field amplification may be relevant to observations of the electromagnetic emission from SNR shocks and their interpretation.

Figure 1 .
Figure 1.Three components of the magnetic field measured by MMS1 at the 2015 November 4/04:38:42 shock crossing.The Alfvénic Mach number is M ≈ 14 and the shock angle is θ Bn ≈ 66°.

Figure 3 .
Figure 3. Left: dependence of the density compression N = ρ d /ρ u on the turbulent magnetic pressure pr = á ñ X b V 8 u u 2 2 .Right: dependence of the normalized downstream ion temperature ( ) T m V 2 p u 2 on the turbulent magnetic pressure pr = á ñ X b V 8 u u 2