 between the magnetic field growth and the maximum energy of the accelerated cosmic rays. In Section 3 we quantitatively show the various stages of the magnetic field amplification as it results from the NRH instability in MHD simulations. In Section 4 we describe how the maximum cosmic ray energy can be self-consistently calculated as a function of time, requiring enough escaping cosmic rays to let the NRH instability grow fast enough to confine the cosmic rays. We assess the results in light of the various types of SNe and environments. In Section 5 we evaluate what this means in terms of which sources can accelerate PeV cosmic rays, and how the various sources add up to make up the Galactic CR population. In Section 6 we illustrate the result of this analysis with some simulations, where we integrate the previously calcuated cosmic-ray current with simulations of the evolution of young SNRs. Finally, we reiterate the implications of our results in Section 7.

COSMIC RAY CURRENT SOURCE
If cosmic rays are efficiently accelerated at the shock, it can be assumed that a certain fraction χ of the shock's kinetic energy is transferred to the relativistic particle population and that a fixed fraction of this energy is carried upstream by escaping CR. For a given CR energy flux the electric current carried by the CR is inversely proportional to the energy of each escaping CR. The return current that is consequently carried by the thermal plasma to maintain neutrality can give rise to a number of current driven instabilities (for an overview see Schure et al. 2012).
We are investigating the effect of this current on the magnetic field growth and stability of the upstream plasma in the regime of the non-resonant hybrid instability (Bell 2004). To our knowledge, this is the instability that allows for the fastest magnetic field growth in plasmas representative of young SNRs. Its maximum growth rate γ and corresponding wave number k are (in cgs units): with j the return current resulting from balancing the escape current of cosmic rays, B the magnetic field parallel to j, and ρ the density of the plasma. If we, for now, assume that the instability grows at the maximum rate, the number of e-foldings for magnetic field amplification at a fixed radius R upstream of (or at) the variable shock radius Rs is given by: with us the shock velocity. A surface area dilution applies in the entire upstream region, resulting in a (Rs/R) 2 dependency for j at a radius R when the CR are accelerated by the shock at radius Rs. Fixing the cosmic ray energy density Ucr as a fraction of the shock kinetic energy, by fixing χ and defining Ucr = χρu 2 s gives: for the current from escaping cosmic rays with charge q upstream of the shock, given the number of cosmic rays as a function of energy follows a powerlaw with a slope dN/dE ∝ E −2 (or N ∝ E −1 ) (see also Bell et al. 2013).
There are various indications that the source spectrum in fact deviates from the canonical powerlaw slope as derived from diffusive shock acceleration. A flatter spectrum is expected if nonlinear shock modification is taken into account (e.g. Vladimirov et al. 2006), whereas oblique high-velocity shocks, or other reasons, may generate a steeper spectrum (e.g. Bell et al. 2011). Clearly, the number of escaping cosmic rays and thus the current becomes a different function of Emax. In a related analysis applied to planar shocks propagating at constant velocity,  discuss how χ (their ηesc) varies in response to non-linear CR pressure feedback onto the shock structure and the resulting effect on Emax. The crucial factor determining Emax is the fraction of the available energy flux ρu 3 s given to CR at the highest energy.
Depending on whether ρ now has a dependency on R as in a stellar wind environment (circumstellar medium, CSM), or not, as in a standard Type Ia environment (interstellar medium, ISM), solving the integral equation 2 for fixed us constrains Emax to behave as: This is the energy at which the particle spectrum starts to roll over. The difference between this description of Emax and that constrained by either the Hillas criterion that restricts the energy based on the size of the system (Hillas 1984), and the Lagage-Cesarsky requirement (Lagage & Cesarsky 1983) that the rate at which energy is gained determines the maximum energy, is the inclusion and outcome of magnetic field amplification. Given the magnetic field is amplified to levels that are reasonably expected in the presence of the NRH instability, both Hillas and Lagage-Cesarsky limits are less restrictive on Emax than the fact that an escape current is needed to amplify the field to levels that allow for confinement of the cosmic rays. In other words, even though Emax is lower using the Hillas and Lagage-Cesarsky limits with the background magnetic field strength, allowing for magnetic field growth alleviates this restriction and, if enough amplification is required by using the appropriate value for γτ as we will discuss below, Emax subsequently is restricted by allowing for enough escape current rather than by time dependence or size of the accelerator. By substituting Eq. 4 into Eq. 3, and evaluating for R = Rs, we can write the current in terms of γτ as: Which values of χ and γτ should be used are up for discussion and should ultimately be determined by comparison with observations. From self-similar dimensional arguments we expect χ and γτ to be independent of the shock velocity provided the Mach number is large and Emax is much greater than the proton rest mass energy, but this assumption can be questioned. However, to first approximation some bounds for these parameters can be estimated, where we assume they are independent of the absolute value for the shock velocity. The parameter χ is a measure of how efficient energy is transferred to the cosmic ray population, as a fixed fraction of the downstream thermal energy density. From the Rankine-Hugoniot jump conditions, the downstream (postshock) thermal energy density is U th = 9 8 ρ0u 2 s. If 30% of that is transferred to cosmic rays, this translates to a value for χ = 0.34 (to be compared to η ln(Emax/Erest) in Bell et al. (2013) and others).
A rough lower and upper bound on γτ can be estimated in various ways. A lower limit can be found by requiring B ⊥ ≡ (B 2 2 + B 2 3 )/2 ≈ B0 (where B0 is the background field, aligned with the first dimension, and B ⊥ the magnitude of the magnetic field in the directions perpendicular to that) for confinement (necessary for Bohm diffusion), such that γτ > ln(B0/B ⊥,0 ). If the initial fluctuation is around 5% of that of the background magnetic field, this translates to a value of γτ > 3 as a very minimal value (and upper limit to Emax). Any value higher than that, will require a lower energy cut-off in the spectrum to allow for a higher escape current. A tentative upper bound can be estimated by looking at the value for the saturated magnetic field, and approximating that growth does not slow down, which in reality it will. As such, this yields a lower limit on the upper bound, since, as we will see in the next section, γ will decrease when B ⊥ /B0 ≈ 2 − 3, where B ⊥ is the average of the magnitude B ⊥ over the 2nd dimension (in 2D simulations). Very early on, while the cosmic ray energy is not yet built up, the dE/dt restriction as posed in Lagage & Cesarsky (1983) causes Emax to be lower than described by Eq. 4. This results initially in a large escape current and thus a temporary very high value for γτ, allowing for more rapid growth of Emax. We find that within the first year, this allows for the cosmic ray energy to reach levels that from thereon are restricted by the magnetic field amplification requirement. At no point at these early stages does the magnetic field as required for sufficient amplification exceed the saturation value of the magnetic field, which we will discuss next.
The value for the saturated magnetic field can be estimated by looking at when c/4π(∇ × B) becomes comparable to the cosmic ray current, which is when the size of the fluctuations and the gyroradius of the cosmic rays are approximately equal (Bell 2004), such that rg ≈ λ (= 2π/k), meaning E/qB ≈ Bc/2j, with j ≈ ckB/(4π) the cur-rent at the smallest growing scale, being at k = 2kmax. This gives an estimate of the saturation magnetic field 4πχρu 3 s /c ln(Emax/(mpc 2 )) on the scale of the Larmor radius of a particular CR. This spatial component of the magnetic energy density is independent of the CR energy. If we suppose that the magnetic energy density per logarithmic interval in scalelength is the same on all scales ranging from the Larmor radius of a GeV proton to the Larmor radius of a PeV proton, the estimated characteristic saturation magnetic field integrated over all scales is For Bsat B0, and B 2 nl ≈ B 2 0 (B ⊥,0 /B0) 2 exp(2γτ ), we then find the following 'upper limit' to γτ of Because of the logarithmic dependence on the initial fluctuations, γτ is not very sensitive to which value we assume for that. Putting in numbers provides an upper bound of γτ ≈ 9, a factor 3 larger than our absolute lower limit. We will get back to this with our simulations and numerical solutions in Sections 4 and 6.