Basic Plasma Physics PrinciplesGordon EmslieOklahoma State University - PDF document

1 Basic Plasma Physics PrinciplesGordon EmslieOklahoma State University Single particle orbits; driftsMagnetic mirroringMHD EquationsForce-free fieldsResistive DiffusionThe Vlasovequation; plasma waves Single particle orbits Motion in a Uniform Magnetic Field Is this an electron or an ion DriftsNow let 3 Resistive Diffusiont = ’u(vuB) + D The ratio of the two terms on the RHS:/ D vL/D ~ 4vL/is known as the magnetic Reynolds numberS. Fouur S 1, the plasma is essentially diffusion-free, for S he dynamics are driven by resistive diffusion.For a flare loop, V ~ V~ 10cm s, L ~ 10cm and-3/2-17. This S ~ 10, and the plasma should be almost perfectly frozen in.The timescale for energy release should be of order L/D ~ (this is of order the timescale for resistive decay of current in an inductor of inductance L/cresistance R = L/L/L). For solar values, this is s ~ 10years! Summary to Date•Solar loops are big (they have a high •Solar loops are good conductors•Solar loops have a low ratio of gas to magnetic pressure The plasma in solar loops is tied to the magnetic field, and the motion of this field determines the motion of the plasma trapped on it Also:It is very difficult to release energy from such a high-conductivity, high-inductance system! ??? The VlasovEquationNote that we have still . A proper solution of the plasma equations be obtained from the particle densities and currents. The equation that accounts for this is called the Vlasovequation Phase-space Distribution FunctionThis is defined as the number of particles per unit volume of space per unit volume of velocity space:At time t, number of particles in elementary volume of space, with velocities in range ,t) d(r,v,has units cm(cm s The BoltzmannEquationThis equation expresses the fact that the net gain or loss of particles in phase space is due to collisionaldepletion:Df/Dtt+ f+ f= (The Boltzmannequation takes into account the self-consistent evolution of the and fields through the appearance of the acceleration term The VlasovEquationThis is a special case of the Boltzmannequation, with no collisionaldepletion term:t+ f+ f= 0,i.e.,t+ f+ (q/m) (f= 0. The Electrostatic VlasovEquationSetting B = 0, we obtain, in one dimension for simplicity, with q = -e (electrons)t+ v x-(eE/m) v= 0.Perturb this around a uniform density, equilibrium (E = 0) state ft + v x -(eE/m) Also consider Poisson’s equation (= -4ne The Electrostatic VlasovEquationNow consider modes of the formg ~ exp(i[kx-t])Then the Vlasovequation becomes+ ivkg–(eE/m) dg/dv= 0–kv)g= (ieE/m) dg/dvand Poisson’s equation isikE= -4Combining,ikE= -i(4/m) E/dvdv/(–kv)Simplifying, and defining the plasma frequencythrough /m,1-(/dvdv/(v–/k) = 0.This is the dispersion relationfor electrostatic plasma waves. 4 The Electrostatic VlasovEquationIntegrating by parts, we obtain an alternative form1-(dv/(1 -kv/= 0.For a plasma, g(v), so that we obtain1-() = 0, i.e., The Electrostatic VlasovEquationFor a warm plasma, we expand the denominator to get1-(dv[1 + 2kv/+ 3ki.e. 1-() [1 + 3k&#xv-5.;耀wher&#xv000;e T/mis the average thermal speed. This gives the dispersion relationT/m) k(cf.for EM waves) Dispersion relationsElectrostatic waves in a warm plasma:/m) kludes motion of ions):=[k)/m1/2(note electrons effectively provide quasi-neutrality)Upper hybrid waves (includes B):= eB/m Dispersion relationsAlfvénwaves:/[1 + (VMagnetoacousticwaves:) + c= angle of propagation to magnetic field)etc., etc. Two-Stream Instability1-(dv/(1 -kv/= 0.For two streams,s,į(v-U) + (v+U)],so thatatȦ-kU]) + ((Ȧ+kU]) = 1.This is a quadratic in –2(–2 = 0,with solution+ 4k1/2There are solutions with negative and so imaginary (exponentially growing) solutions. v Two-Stream InstabilityDistribution with two maxima (one at zero, one at the velocity of the “beam”) is susceptible to the two-stream instability.This generates a large amplitude of plasma waves and affects the energeticsof the Two-Stream InstabilityThis can also happen due to an “overtaking”instability –fast particles arrive at a location earlier than slower ones and so create a local maximum in f. SummaryHigh energy solar physics is concerned with the physics of plasma, which is a highly interacting system of“Plasma physics is complicated”(J.C. Brown & D.F. Smith, 1980)