The paper discusses the amplification of magnetic fields and the diffusive shock acceleration of cosmic rays (CRs) in supernova remnants (SNRs). The author, A. R. Bell, explores how cosmic rays can generate magnetic fluctuations upstream of the shock, which in turn amplify the magnetic field. This amplification can significantly increase the maximum attainable energy of CRs in SNRs. Key points include: 1. **Linear Theory**: The linear theory is developed to understand how cosmic rays excite MHD turbulence, leading to the amplification of magnetic fields. The waves excited are not Alfvén waves but nearly purely growing modes, which grow preferentially at wavelengths that are not resonant with the CR Larmor radius. 2. **Non-linear Simulation**: Non-linear simulations show that the magnetic field can be amplified by orders of magnitude from its seed value. The simulations also reveal that the growth rate of the magnetic field is proportional to the square root of the CR current for fixed wavenumber. 3. **Maximum CR Energy**: The maximum energy attainable by CRs is limited by the condition that the acceleration time should be less than the age of the SNR. The paper provides a new condition for wave growth based on the efficiency of CR production and the shock velocity. 4. **Non-linear Saturation**: The paper discusses the non-linear saturation of wave growth due to the tension in magnetic field lines. This saturation occurs when the tension in the magnetic field lines becomes strong enough to oppose the driving force from the CR current. 5. **General Application**: The process of magnetic field amplification described in the paper can be applied to other sites of diffusive shock acceleration, such as in the early universe or in other astrophysical environments. The paper concludes that the amplification of magnetic fields by cosmic rays is a crucial process that can significantly enhance the maximum energy of CRs in SNRs, potentially explaining the observed energy spectrum of CRs in our galaxy.The paper discusses the amplification of magnetic fields and the diffusive shock acceleration of cosmic rays (CRs) in supernova remnants (SNRs). The author, A. R. Bell, explores how cosmic rays can generate magnetic fluctuations upstream of the shock, which in turn amplify the magnetic field. This amplification can significantly increase the maximum attainable energy of CRs in SNRs. Key points include: 1. **Linear Theory**: The linear theory is developed to understand how cosmic rays excite MHD turbulence, leading to the amplification of magnetic fields. The waves excited are not Alfvén waves but nearly purely growing modes, which grow preferentially at wavelengths that are not resonant with the CR Larmor radius. 2. **Non-linear Simulation**: Non-linear simulations show that the magnetic field can be amplified by orders of magnitude from its seed value. The simulations also reveal that the growth rate of the magnetic field is proportional to the square root of the CR current for fixed wavenumber. 3. **Maximum CR Energy**: The maximum energy attainable by CRs is limited by the condition that the acceleration time should be less than the age of the SNR. The paper provides a new condition for wave growth based on the efficiency of CR production and the shock velocity. 4. **Non-linear Saturation**: The paper discusses the non-linear saturation of wave growth due to the tension in magnetic field lines. This saturation occurs when the tension in the magnetic field lines becomes strong enough to oppose the driving force from the CR current. 5. **General Application**: The process of magnetic field amplification described in the paper can be applied to other sites of diffusive shock acceleration, such as in the early universe or in other astrophysical environments. The paper concludes that the amplification of magnetic fields by cosmic rays is a crucial process that can significantly enhance the maximum energy of CRs in SNRs, potentially explaining the observed energy spectrum of CRs in our galaxy.