1. Ion Heating Induced by Waves Russell M. Kulsrud Princeton Plasma Physics Laboratory

2. First Anomalous Resistivity Induced by Waves Unstable Waves Wave Growth Force on electrons Opposite to k if e >0.

3. Cont. This Force on electrons balances force due to electric field E

4. Cont. Nonlinear s steady state If e NL =0 e = e L If i NL =0 e = e L + e NL =- i >0

5. Ion and Electron Heating Quasilinear Theory Kinetic Equation Rate of change of electron momentum: Rate of change of electron energy: Rate of change of electron heat:

6. Cont. Now so Now from electron force balance so

7. Ions Steady State

8. Interpretation Train braking on tracks. Tracks are rigid so in rest frame no energy is exchanged Train loses directed energy which appears as heat. For waves, they are also rigid at V Heat of electrons R¢ V e Heat of ions R¢ V i Total heat w j 2 For collision V i << V e, no ion heating For wave V i ¼ V e, ion heating

9. Conclusion Simple formula for w Total heating of ions plus electrons is w j 2, split according to,