{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"183528746f18c7367b8a2cfaa6de4d5af096e2040efe5f5ed88732eeedc318ce","created":"2026-10-03T07:18:08Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"258432c942ad91aef0a6adc457287f1bf071e66b0610240016f2778a49f7e383","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"plasma.collisionless-shocks.field-amplification-saturation","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Cosmic rays streaming ahead of a shock can amplify the magnetic field, which helps trap particles and accelerate them to higher energy. How strong the amplified field becomes, which decides whether supernova remnants can accelerate protons to about 1e15 electron volts, is unsettled.","posed_since":"","precise":"Cosmic-ray-current-driven instabilities (notably the nonresonant Bell instability) amplify the upstream field of fast shocks. Determine the saturated amplitude deltaB/B_0 and its coherence length as functions of shock speed $v_{\\mathrm{sh}}$ and cosmic-ray acceleration efficiency, including the saturation mechanism (expansion of current-driven cavities, back-reaction on the cosmic rays, nonlinear damping). An answer is deltaB/B_0(v_sh, efficiency) from kinetic or hybrid simulations converged over many growth times, with an analytic saturation model from which the maximum proton energy at young supernova remnants follows.","problem_ref":null,"references":"","settled_by":"Kinetic simulations resolving saturation over many growth times, consistent with field strengths inferred from the thin X-ray rims of young supernova remnants.","status_note":"","title":"Saturation of cosmic-ray-driven field amplification at fast 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