Cosmic Ray Streaming Research Articles

Cosmic ray feedback from supernovae in dwarf galaxies

The regulation of the baryonic content in dwarf galaxies is a long-standing problem. Supernovae (SNe) are supposed to play a key role in forming large-scale galactic winds by removing important amounts of gas from galaxies. SNe are efficient accelerators of non-thermal particles, so-called cosmic rays (CRs), which can substantially modify the dynamics of the gas and conditions to form large-scale galactic winds. We investigate how CR injection by SNe impacts the star formation and the formation of large-scale winds in dwarf galaxies, and whether it can produce galaxy star-formation rates (SFR) and wind properties closer to observations. We ran CR magneto-hydrodynamical simulations of dwarf galaxies at high resolution (9 pc) with the adaptive mesh refinement codeRAMSES. Those disc galaxies are embedded in isolated halos of mass of 1010and 1011 M⊙, and CRs are injected by SNe. We included CR isotropic and anisotropic diffusion with various diffusion coefficients, CR radiative losses, and CR streaming. The injection of CR energy into the interstellar medium smooths out the highest gas densities, which reduces the SFR by a factor of 2–3. Mass outflow rates are significantly greater with CR diffusion, by 2 orders of magnitudes for the higher diffusion coefficients. Without diffusion and streaming, CRs are inefficient at generating winds. CR streaming alone allows for the formation of winds but which are too weak to match observations. The formation of galactic winds strongly depends on the diffusion coefficient: for low coefficients, CR energy stays confined in high density regions where CR energy losses are highest, and higher coefficients, which allow for a more efficient leaking of CRs out of dense gas, produce stronger winds. CR diffusion leads to colder and denser winds than without CRs, and brings outflow rates and mass loading factors much closer to observations.

The Role of the Parker Instability in Structuring the Interstellar Medium

Abstract The Parker instability, a Rayleigh–Taylor-like instability of thermal gas supported against gravity by magnetic fields and cosmic rays, is thought to be dynamically important for galaxy evolution, possibly promoting molecular cloud formation and the galactic dynamo. In previous work, we examined the effect of three different cosmic-ray transport models on the Parker instability: decoupled (γ c = 0), locked to the thermal gas (γ c = 4/3), and coupled to the gas with streaming by self-confinement. We expand on that work here by considering radiative cooling, a smooth gravitational potential, and simulations into the nonlinear regime. We determine that cosmic-ray transport away from compression points, whether by diffusion or streaming, is the largest driver of the instability. Heating due to cosmic-ray streaming is also destabilizing and especially affects the nonlinear regime. While cooling depressurizes the dense gas, streaming cosmic rays heat and inflate the diffuse extraplanar gas, greatly modifying the phase structure of the medium. In 3D, we find that the fastest growth favors short-wavelength modes in the horizontal direction perpendicular to the background magnetic field; this is imprinted on Faraday rotation measure maps that may be used to detect the Parker instability. The modifications to the Parker instability that we observe in this work have large implications for the structure and evolution of galaxies, and they highlight the major role that cosmic rays play in shaping their environments.

Chaotic cold accretion in giant elliptical galaxies heated by AGN cosmic rays

ABSTRACT Black hole feedback plays a central role in shaping the circumgalactic medium (CGM) of elliptical galaxies. We systematically study the impact of plasma physics on the evolution of ellipticals by performing three-dimensional non-ideal magnetohydrodynamic simulations of the interactions of active galactic nucleus (AGN) jets with the CGM including magnetic fields, and cosmic rays (CRs) and their transport processes. We find that the physics of feedback operating on large galactic scales depends very sensitively on plasma physics operating on small scales. Specifically, we demonstrate that (i) in the purely hydrodynamical case, the AGN jets initially maintain the atmospheres in global thermal balance. However, local thermal instability generically leads to the formation of massive cold discs in the vicinity of the central black hole in disagreement with observations; (ii) including weak magnetic fields prevents the formation of the discs because local B-field amplification in the precipitating cold gas leads to strong magnetic breaking, which quickly extracts angular momentum from the accreting clouds. The magnetic fields transform the cold clouds into narrow filaments that do not fall ballistically; (iii) when plasma composition in the AGN jets is dominated by CRs, and CR transport is neglected, the atmospheres exhibit cooling catastrophes due to inefficient heat transfer from the AGN to CGM despite Coulomb/hadronic CR losses being present; (iv) including CR streaming and heating restores agreement with the observations, i.e. cooling catastrophes are prevented and massive cold central discs do not form. The AGN power is reduced as its energy is utilized efficiently.

Probing Cosmic-Ray Transport with Radio Synchrotron Harps in the Galactic Center

Abstract Recent observations with the MeerKAT radio telescope reveal a unique population of faint nonthermal filaments pervading the central molecular zone, a region rich in molecular gas near the Galactic center. Some of those filaments are organized into groups of almost parallel filaments, seemingly sorted by their length, so that their morphology resembles a harp with radio-emitting “strings.” We argue that the synchrotron-emitting GeV electrons of these radio harps have been consecutively injected by the same source (a massive star or pulsar) into spatially intermittent magnetic fiber bundles within a magnetic flux tube or via time-dependent injection events. After escaping from this source, the propagation of cosmic-ray (CR) electrons inside a flux tube is governed by the theory of CR transport. We propose to use observations of radio harp filaments to gain insight into the specifics of CR propagation along magnetic fields of which there are two principle modes: CRs could either stream with self-excited magnetohydrodynamic waves or diffuse along the magnetic field. To disentangle these possibilities, we conduct hydrodynamical simulations of either purely diffusing or streaming CR electrons and compare the resulting brightness distributions to the observed synchrotron profiles of the radio harps. We find compelling evidence that CR streaming is the dominant propagation mode for GeV CRs in one of the radio harps. Observations at higher angular resolution should detect more radio harps and may help to disentangle projection effects of the possibly three-dimensional flux-tube structure of the other radio harps.

Thermal instability of halo gas heated by streaming cosmic rays

ABSTRACT Heating of virialized gas by streaming cosmic rays (CRs) may be energetically important in galaxy haloes, groups, and clusters. We present a linear thermal stability analysis of plasmas heated by streaming CRs. We separately treat equilibria with and without background gradients, and with and without gravity. We include both CR streaming and diffusion along the magnetic-field direction. Thermal stability depends strongly on the ratio of CR pressure to gas pressure, which determines whether modes are isobaric or isochoric. Modes with $\boldsymbol {k \cdot B }\ne 0$ are strongly affected by CR diffusion. When the streaming time is shorter than the CR diffusion time, thermally unstable modes (with $\boldsymbol {k \cdot B }\ne 0$) are waves propagating at a speed ∝ the Alfvén speed. Halo gas in photoionization equilibrium is thermally stable independent of CR pressure, while gas in collisional ionization equilibrium is unstable for physically realistic parameters. In gravitationally stratified plasmas, the oscillation frequency of thermally overstable modes can be higher in the presence of CR streaming than the buoyancy/free-fall frequency. This may modify the critical tcool/tff at which multiphase gas is present. The criterion for convective instability of a stratified, CR-heated medium can be written in the familiar Schwarzschild form dseff/dz < 0, where seff is an effective entropy involving the gas and CR pressures. We discuss the implications of our results for the thermal evolution and multiphase structure of galaxy haloes, groups, and clusters.

Shock-accelerated cosmic rays and streaming instability in the adaptive mesh refinement code Ramses

Cosmic rays (CRs) are thought to play a dynamically important role in several key aspects of galaxy evolution, including the structure of the interstellar medium, the formation of galactic winds, and the non-thermal pressure support of halos. We introduce a numerical model solving for the CR streaming instability and acceleration of CRs at shocks with a fluid approach in the adaptive mesh refinement code RAMSES. CR streaming is solved with a diffusion approach and its anisotropic nature is naturally captured. We introduce a shock finder for the RAMSES code that automatically detects shock discontinuities in the flow. Shocks are the loci for CR injection, and their efficiency of CR acceleration is made dependent on the upstream magnetic obliquity according to the diffuse shock acceleration mechanism. We show that the shock finder accurately captures shock locations and estimates the shock Mach number for several problems. The obliquity-dependent injection of CRs in the Sedov solution leads to situations where the supernova bubble exhibits large polar caps (homogeneous background magnetic field), or a patchy structure of the CR distribution (inhomogeneous background magnetic field). Finally, we combine both accelerated CRs with streaming in a simple turbulent interstellar medium box, and show that the presence of CRs significantly modifies the structure of the gas.

Cosmic ray small-scale anisotropies in quasi-linear theory

The distribution of arrival directions of cosmic rays is remarkably isotropic, which is a consequence of their repeated scattering in magnetic fields. Yet, high-statistics observatories like IceCube and HAWC have revealed the presence of small-scale structures at levels of 1 part in 10,000 at hundreds of TeV, which are not expected in typical diffusion models of cosmic rays. We follow up on the suggestion that these small-scale anisotropies are a result of cosmic ray streaming in a particular realisation of the turbulent magnetic field within a few scattering lengths in our local Galactic neighbourhood. So far, this hypothesis has been investigated mostly numerically, by tracking test particles through turbulent magnetic fields. For the first time, we present an analytical computation that through a perturbative approach allows predicting the angular power spectrum of cosmic ray arrival directions for a given model of turbulence. We illustrate this method for a simple, isotropic turbulence model and we find remarkable agreement with the results of numerical studies.

Role of cosmic-ray streaming and turbulent damping in driving galactic winds

ABSTRACT Large-scale galactic winds driven by stellar feedback are one phenomenon that influences the dynamical and chemical evolution of a galaxy, redistributing material throughout the circumgalatic medium. Non-thermal feedback from galactic cosmic rays (CRs) – high-energy charged particles accelerated in supernovae and young stars – can impact the efficiency of wind driving. The streaming instability limits the speed at which they can escape. However, in the presence of turbulence, the streaming instability is subject to suppression that depends on the magnetization of turbulence given by its Alfvén Mach number. While previous simulations that relied on a simplified model of CR transport have shown that super-Alfvénic streaming of CRs enhances galactic winds, in this paper we take into account a realistic model of streaming suppression. We perform three-dimensional magnetohydrodynamic simulations of a section of a galactic disc and find that turbulent damping dependent on local magnetization of turbulent interstellar medium (ISM) leads to more spatially extended gas and CR distributions compared to the earlier streaming calculations, and that scale heights of these distributions increase for stronger turbulence. Our results indicate that the star formation rate increases with the level of turbulence in the ISM. We also find that the instantaneous wind mass loading is sensitive to local streaming physics with the mass loading dropping significantly as the strength of turbulence increases.

Magnetohydrodynamic Particle-in-cell Simulations of the Cosmic-Ray Streaming Instability: Linear Growth and Quasi-linear Evolution

Abstract The gyro-resonant cosmic-ray (CR) streaming instability is believed to play a crucial role in CR transport, leading to the growth of Alfvén waves at small scales that scatter CRs, and impacts the interaction of CRs with the interstellar medium (ISM) on large scales. However, extreme scale separation (λ ≪ pc), low CR number density (n CR/n ISM ∼ 10−9), and weak CR anisotropy (∼v A/c) pose strong challenges for proper numerical studies of this instability on a microphysical level. Employing the recently developed magnetohydrodynamic particle-in-cell method, which has unique advantages to alleviate these issues, we conduct 1D simulations that quantitatively demonstrate the growth and saturation of the instability in the parameter regime consistent with realistic CR streaming in the large-scale ISM. Our implementation of the δf method dramatically reduces Poisson noise and enables us to accurately capture wave growth over a broad spectrum equally shared between left- and right-handed Alfvén modes. We are also able to accurately follow the quasi-linear diffusion of CRs subsequent to wave growth, which is achieved by employing phase randomization across periodic boundaries. Full isotropization of the CRs in the wave frame requires the pitch angles of most CRs to efficiently cross 90° and can be captured in simulations with relatively high wave amplitude and/or spatial resolution. We attribute this crossing to nonlinear wave–particle interaction (rather than mirror reflection) by investigating individual CR trajectories. We anticipate that our methodology will open up opportunities for future investigations that incorporate additional physics.

Bell-instability-mediated Spectral Modulation of Hadronic Gamma-Rays from a Supernova Remnant Interacting with a Molecular Cloud

Abstract Supernova remnants (SNRs) are believed to be the site of galactic cosmic-ray acceleration. However, the details of the cosmic-ray acceleration are still not well understood. Gamma-ray observation is a promising method to study cosmic-ray acceleration in the SNRs, because a hadronic gamma-ray can trace high-energy cosmic-rays above ∼GeV energy. Conventional theory predicts that the hadronic gamma-ray shows a flat νF ν spectrum from the pion-creation threshold energy to the maximum energy of diffusive shock acceleration. In this paper, by employing numerical simulations that solve a hybrid system of the magnetohydrodynamics of a molecular cloud and diffusive propagation of cosmic-rays, we demonstrate that the hadronic gamma-ray spectrum can be harder than the conventional one and that the modulated spectrum becomes consistent with observations. The modification mechanism is explained as follows: The cosmic-rays accelerated at the supernova blast wave shock propagate into a clump of a molecular cloud. The cosmic-ray streaming in the cloud induces the so-called Bell instability that induces Alfvén waves in the cloud. The induced magnetic field fluctuations prevent further cosmic-ray incursion by diminishing the diffusion coefficient for the cosmic-rays below ∼1 TeV energy. This mechanism reinforces recent claims of a similar spectral modification by magnetic field amplification around a molecular cloud by Inoue et al. and Celli et al.

Cosmic-ray hydrodynamics: Alfvén-wave regulated transport of cosmic rays

Star formation in galaxies appears to be self-regulated by energetic feedback processes. Among the most promising agents of feedback are cosmic rays (CRs), the relativistic ion population of interstellar and intergalactic plasmas. In these environments, energetic CRs are virtually collisionless and interact via collective phenomena mediated by kinetic-scale plasma waves and large-scale magnetic fields. The enormous separation of kinetic and global astrophysical scales requires a hydrodynamic description. Here, we develop a new macroscopic theory for CR transport in the self-confinement picture, which includes CR diffusion and streaming. The interaction between CRs and electromagnetic fields of Alfv\'enic turbulence provides the main source of CR scattering, and causes CRs to stream along the magnetic field with the Alfv\'en velocity if resonant waves are sufficiently energetic. However, numerical simulations struggle to capture this effect with current transport formalisms and adopt regularization schemes to ensure numerical stability. We extent the theory by deriving an equation for the CR momentum density along the mean magnetic field and include a transport equation for the Alfv\'en-wave energy. We account for energy exchange of CRs and Alfv\'en waves via the gyroresonant instability and include other wave damping mechanisms. Using numerical simulations we demonstrate that our new theory enables stable, self-regulated CR transport. The theory is coupled to magneto-hydrodynamics, conserves the total energy and momentum, and correctly recovers previous macroscopic CR transport formalisms in the steady-state flux limit. Because it is free of tunable parameters, it holds the promise to provide predictable simulations of CR feedback in galaxy formation.

Non-linear diffusion of cosmic rays escaping from supernova remnants – II. Hot ionized media

We study the problem of the escape and transport of Cosmic-Rays (CR) from a source embedded in a fully ionised, hot phase of the interstellar medium (HIM). In particular, we model the CR escape and their propagation in the source vicinity taking into account excitation of Alfv\'enic turbulence by CR streaming and mechanisms damping the self-excited turbulence itself. Our estimates of escape radii and times result in large values (100 pc, $2\times10^5$ yr) for particle energies $\lesssim20$ GeV and smaller values for particles with increasing energies (35 pc and 14 kyr at 1 TeV). These escape times and radii, when used as initial conditions for the CR propagation outside the source, result in relevant suppression of the diffusion coefficient (by a factor 5-10) on time-scales comparable with their (energy dependent) escape time-scale. The damping mechanisms are fast enough that even on shorter time scales the Alfv\'enic turbulence is efficiently damped, and the ratio between random and ordered component of the magnetic field is $\delta B/B_0\ll 1$, justifying the use of quasi-linear theory. In spite of the suppressed diffusion coefficient, and then the increased residence time in the vicinity (<200 pc) of their source, the grammage accumulated by CRs after their escape is found to be negligible (at all energies) as compared to the one accumulated while diffusing in the whole Galaxy, due to the low density of the HIM.

On the maximum energy of protons in the hotspots of AGN jets

We study particle acceleration and magnetic field amplification in the termination shocks (hotspots) of radiogalaxy jets. The cut-off of the synchrotron spectrum in the hotspots of powerful radiogalaxies is typically observed between infrared and optical frequencies, indicating that the maximum energy of non-thermal electrons accelerated at the jet termination shock is about 1 TeV for a canonical magnetic field of 100 μG. Based on theoretical considerations and observational data we show that the maximum energy of electrons cannot be constrained by synchrotron losses as usually assumed, unless the jet density is unreasonable large and most of the jet kinetic energy goes to non-thermal electrons. The maximum energy is ultimately determined by the ability to scatter particles back and forth the shock, and this limit applies to both electrons and protons. Therefore, the maximum energy of protons is also about 1 TeV when radiative cooling is not efficient. We show that non-resonant hybrid (Bell) instabilities generated by the streaming of cosmic rays can grow fast enough to amplify the jet magnetic field up to 100 μG and accelerate particles up to the maximum energies observed in the hotspots of radiogalaxies.

The Role of Cosmic-ray Transport in Shaping the Simulated Circumgalactic Medium

Abstract The majority of galactic baryons resides outside of the galactic disk in the diffuse gas known as the circumgalactic medium (CGM). While state-of-the art simulations excel at reproducing galactic disk properties, many of them struggle to drive strong galactic winds or to match the observed ionization structure of the CGM using only thermal supernova feedback. To remedy this, recent studies have invoked nonthermal cosmic ray (CR) stellar feedback prescriptions. However, numerical schemes of CR transport are still poorly constrained. We explore how the choice of CR transport affects the multiphase structure of the simulated CGM. We implement anisotropic CR physics in the astrophysical simulation code Enzo and simulate a suite of isolated disk galaxies with varying prescriptions for CR transport: isotropic diffusion, anisotropic diffusion, and streaming. We find that all three transport mechanisms result in strong, metal-rich outflows but differ in the temperature and ionization structure of their CGM. Isotropic diffusion results in a spatially uniform, warm CGM that underpredicts the column densities of low ions. Anisotropic diffusion develops a reservoir of cool gas that extends farther from the galactic center, but disperses rapidly with distance. CR streaming projects cool gas out to radii of 200 kpc, supporting a truly multiphase medium. In addition, we find that streaming is less sensitive to changes in constant parameter values like the CR injection fraction, transport velocity, and resolution than diffusion. We conclude that CR streaming is a more robust implementation of CR transport and motivates the need for detailed parameter studies of CR transport.

The requirement for mixing-heating to utilize bubble cosmic rays to heat the intracluster medium

I conduct simple analytical estimates and conclude that mixing by vortices is a more efficient process to transfer the cosmic ray energy of jet-inflated bubbles to the intracluster medium (ICM) than streaming of cosmic rays along magnetic field lines. Jets and the bubbles they inflate transfer heat to the ambient gas in cooling flows in cluster of galaxies and in galaxies. The internal energy of the jet-inflated bubbles is dominate by very hot thermal gas and/or cosmic rays. Cosmic rays that stream along magnetic field lines that connect the bubbles with the ICM heat the ICM as their energy is dissipated there. I find that about half of the cosmic ray energy is dissipated in the bubbles themselves. I also find that the ICM volume that the cosmic ray streaming process heats is only about five times as large as the volume of the bubbles. The outcome of heating by streaming only is that the cosmic rays form a larger bubble filled with very hot thermal gas. Therefore, there is a requirement for a more efficient process to transfer the internal energy of the bubbles to the ICM. I suggest that this process is heating by mixing that operates very well for both cosmic rays and the very hot thermal gas inside the bubbles. This leaves mixing-heating to be the dominant heating process of cooling flows.

The Parker Instability with Cosmic-Ray Streaming

Abstract Recent studies have found that cosmic-ray transport plays an important role in feedback processes such as star formation and the launching of galactic winds. Although cosmic-ray buoyancy is widely held to be a destabilizing force in galactic disks, the effect of cosmic-ray transport on the stability of stratified systems has yet to be analyzed. We perform a stability analysis of a stratified layer for three different cosmic-ray transport models: decoupled (Classic Parker), coupled with γ c = 4/3 but not streaming (Modified Parker), and finally coupled with streaming at the Alfvén speed. When the compressibility of the cosmic rays is decreased the system becomes much more stable, but the addition of cosmic-ray streaming to the Parker instability severely destabilizes it. Through comparison of these three cases and analysis of the work contributions for the perturbed quantities of each system, we demonstrate that cosmic-ray heating of the gas is responsible for the destabilization of the system. We find that a 3D system is unstable over a larger range of wavelengths than the 2D system. Therefore, the Parker instability with cosmic-ray streaming may play an important role in cosmic-ray feedback.

Powering of Hα Filaments by Cosmic Rays

Abstract Cluster cool cores possess networks of line-emitting filaments. These filaments are thought to originate via uplift of cold gas from cluster centers by buoyant active galactic nuclei (AGNs) bubbles, or via local thermal instability in the hot intracluster medium (ICM). Therefore, the filaments are either the signatures of AGN feedback or feeding of supermassive black holes. Despite being characterized by very short cooling times, the filaments are significant Hα emitters, which suggests that some process continuously powers these structures. Many cool cores host diffuse radio mini halos and AGN injecting radio plasma, suggesting that cosmic rays (CRs) and magnetic fields are present in the ICM. We argue that the excitation of Alfvén waves by CR streaming, and the replenishment of CR energy via accretion onto the filaments of high-plasma-β ICM characterized by low CR pressure support, can provide the adequate amount of heating to power and sustain the emission from these filaments. This mechanism does not require the CRs to penetrate the filaments, even if the filaments are magnetically isolated from the ambient ICM, and it may operate irrespectively of whether the filaments are dredged up from the center or form in situ in the ICM. This picture is qualitatively consistent with non-thermal line ratios seen in the cold filaments. Future X-ray observations of the iron line complex with XARM, Lynx, or Athena could help to test this model by providing constraints on the amount of CRs in the hot plasma that is cooling and accreting onto the filaments.

Penetration of Cosmic Rays into Dense Molecular Clouds: Role of Diffuse Envelopes∗

Abstract A flux of cosmic rays (CRs) propagating through a diffuse ionized gas can excite MHD waves, thus generating magnetic disturbances. We propose a generic model of CR penetration into molecular clouds through their diffuse envelopes, and identify the leading physical processes controlling their transport on the way from a highly ionized interstellar medium to the dense interior of the cloud. The model allows us to describe a transition between a free streaming of CRs and their diffusive propagation, determined by the scattering on the self-generated disturbances. A self-consistent set of equations, governing the diffusive transport regime in an envelope and the MHD turbulence generated by the modulated CR flux, is characterized by two dimensionless numbers. We demonstrate a remarkable mutual complementarity of different mechanisms leading to the onset of the diffusive regime, which results in a universal energy spectrum of the modulated CRs. In conclusion, we briefly discuss implications of our results for several fundamental astrophysical problems, such as the spatial distribution of CRs in the Galaxy as well as the ionization, heating, and chemistry in dense molecular clouds.

A New Numerical Scheme for Cosmic-Ray Transport

Abstract Numerical solutions of the cosmic-ray (CR) magnetohydrodynamic equations are dogged by a powerful numerical instability, which arises from the constraint that CRs can only stream down their gradient. The standard cure is to regularize by adding artificial diffusion. Besides introducing ad hoc smoothing, this has a significant negative impact on either computational cost or complexity and parallel scalings. We describe a new numerical algorithm for CR transport, with close parallels to two-moment methods for radiative transfer under the reduced speed of light approximation. It stably and robustly handles CR streaming without any artificial diffusion. It allows for both isotropic and field-aligned CR streaming and diffusion, with arbitrary streaming and diffusion coefficients. CR transport is handled explicitly, while source terms are handled implicitly. The overall time step scales linearly with resolution (even when computing CR diffusion) and has a perfect parallel scaling. It is given by the standard Courant condition with respect to a constant maximum velocity over the entire simulation domain. The computational cost is comparable to that of solving the ideal MHD equation. We demonstrate the accuracy and stability of this new scheme with a wide variety of tests, including anisotropic streaming and diffusion tests, CR-modified shocks, CR-driven blast waves, and CR transport in multiphase media. The new algorithm opens doors to much more ambitious and hitherto intractable calculations of CR physics in galaxies and galaxy clusters. It can also be applied to other physical processes with similar mathematical structure, such as saturated, anisotropic heat conduction.

On the maximum energy of non-thermal particles in the primary hotspot of Cygnus A

We study particle acceleration and magnetic field amplification in the primary hotspot in the northwest jet of radiogalaxy Cygnus A. By using the observed flux density at 43 GHz in a well resolved region of this hotspot, we determine the minimum value of the jet density and constrain the magnitude of the magnetic field. We find that a jet with density greater than $5\times 10^{-5}$ cm$^{-3}$ and hotspot magnetic field in the range 50-400 $\mu$G are required to explain the synchrotron emission at 43 GHz. The upper-energy cut-off in the hotspot synchrotron spectrum is at a frequency < $5\times 10^{14}$ Hz, indicating that the maximum energy of non-thermal electrons accelerated at the jet reverse shock is $E_{e, \rm max} \sim 0.8$ TeV in a magnetic field of 100 $\mu$G. Based on the condition that the magnetic-turbulence scale length has to be larger than the plasma skin depth, and that the energy density in non-thermal particles cannot violate the limit imposed by the jet kinetic luminosity, we show that $E_{e,\rm max}$ cannot be constrained by synchrotron losses as traditionally assumed. In addition to that, and assuming that the shock is quasi-perpendicular, we show that non-resonant hybrid instabilities generated by the streaming of cosmic rays with energy $E_{e, \rm max}$ can grow fast enough to amplify the jet magnetic field up to 50-400 $\mu$G and accelerate particles up to the maximum energy $E_{e, \rm max}$ observed in the Cygnus A primary hotspot.