Transcript Plasma
Chapter 6: Equilibrium and stability 6.1 Introduction 6.2 Hydromagnetic Equilibrium 6.3 The concept of 6.4 Diffusion of Magnetic field into a plasma 6.5 Classification of Instabilities 6.6 The Gravitational Instability 6.1 Introduction 6.2 Hydromagnetic Equilibrium The general problem of equilibrium is complicated. Several physical concepts are easily gleaned from the MHD equations. Steady state and g=0, p j B B 0 j V j B p g t The single fluid equation of motion. From first equation, there is a balance of forces between the pressure-gradient force and the Lorentz force. p B ( j B ) B j B p / B 2 B n ( KTi KTe ) B2 The Diamagnetic current is generated by the pressure-gradient across B. • J and B are each perpendicular to p The hydromagnetic equilibrium the density is constant along a line of force . • p 0 s 6.3 The concept of 1 p j B 0 ( B ) B 1 0 [( B ) B 12 B 2 ] B2 1 ( p ) 0 ( B ) B 20 B2 p const . 20 nkT 2 B / 20 In Fusion reactors 1%. is about If 1 , there are two regions: a region of plasma without field, and a region of field without plasma 6.4 Diffusion of Magnetic field into a plasma The diffusion of a magnetic field into a palsma. The regions will stay separated if the plasma has no resistivity. If the resistivity is finite, the plasma can move through the field and vice versa. This diffusion need time. The diffusion time can be calculated E B E V B j For simplicity, let us assume that the plasma is at rest. B j t B 0 j B 2 ( B) B t 0 0 B 2 0 L t / B B0e 0 L2 The time can also be interpreted as the time for annihilation of the magnetic field. As the field lines move through the plasma, the induced currents cause ohmic heating of the plasma. This energy comes from the energy of the field. B 2 j 0 L 2 0 L2 B2 B2 2( ) 0 2 0 Thus is essentially the time it takes for field energy to be dissipated into Joule heat. 6.5 Classification of Instabilities In non perfect thermodynamic equilibrium in which all forces are balance and a time-independent solution is possible, the free energy can cause waves to be selfexcited; the equilibrium is then unstable. An instability is always a motion which decreases the free energy and bring the plasma closer to true thermodynamic equilibrium. There are four main categories instabilities: Streaming instabilities, Universal instabilities Rayleigh-Taylor Instabilities and Kinetic instabilities. Streaming instabilities Either a beam of energetic particles travels through the plasma, or a current is driven through the plasma so that the different species have drifts relative to one another. The drift energy is used to excite waves, and oscillation energy is gained at the expense of the drift energy in the unperturbed state. Rayleigh-Taylor Instabilities: The plasma has a density gradient or a sharp boundary, so that it is not uniform Universal instabilities The plasma pressure tends to make the plasma expand, and the expansion energy can drive an instability. This type of free energy is always present in any finite plasma, and the resulting wave are called universal instabilities. Kinetic instabilities If the velocity distributions are not Maxwellian, there is a deviation from thermodynamic equilibrium; and instabilities can be driven by the anisotropy of the velocity distribution. For example, if T// and T are different, an instability called the modified Harris instability can arise. 6.7 The Gravitational Instability In a plasma, a Rayleigh-Taylor instability can occur because the magnetic field acts as a light fluid supporting a heavy fluid (plasma). In curved magnetic fields, the centrifugal force on the plasma due to particle motion along the curved lines of force acts as an equivalent “gravitational” force. In the equilibrium state, the ions obey Mn0 (v0 )v0 en0v0 B0 Mn0 g If g is a constant, v0 will be also, then M g B0 g v0 yˆ 2 e B0 c If a ripple develop in the interface as the result of random thermal fluctuations, the drift v0 will cause the ripple to grow. The drift of ions cause a charge to build up on the sides of the ripple, and an electric field develops which changes sign as one goes from crest to though in the perturbation. To find the growth rate, we can perform the usual linerized wave analysis. M (n0 n1 ) (v0 v1 ) (v0 v1 ) (v0 v1 ) e(n0 n1 )[ E1 (v0 v1 ) B0 ] M (n0 n1 ) g t Mn0 (v0 )v0 en0v0 B0 Mn0 g M (n0 n1 )(v0 )v0 e(n0 n1 )v0 B0 M (n0 n1 ) g v1 Mn0 (v0 )v1 en0 ( E1 v1 B0 ) t For perturbation of the form exp[ i(ky t )] , we have M ( kv0 )v1 ie( E1 v1 B0 ) For Ex 0 and c2 ( kv0 ) 2 , the solution is vix Ey vex B0 Ey B0 viy i vey 0 kv0 E y c B0 The perturbed equation of continuity for ions is n1 (n0v0 ) (v0 )n1 n1 v0 t (v1 )n0 n0 v1 (n1v1 ) 0 first order equation: in1 ikv0 n1 vix n0 ikn0 viy 0 because ve 0 0 and vey 0 in1 vex n0 0 note that we have used the plasma approximation. For low frequency, it is ok. From the equation of motion and the equation of continuity, we obtain: ( kv0 )n1 i n1 i Ey B0 Ey B0 n0 ikn0 kv0 E y c B0 n0 0 ( kv0 )n1 (n0 kn0 kv0 n1 c ) n0 kv0 [ k v g (n0 / n0 )] 1 4 2 2 0 0 M g B0 g v0 yˆ 2 e B0 c ( kv0 ) v0c n0 / n0 1 2 0 1 2 there is instability if the frequency is complex, that is, if 1 4 k 2 v02 g (n0 / n0 ) 0 From this we see that instability requires g and n0 / n0 to have opposite sign. This is just the statement that the light fluid is supporting the heavy fluid. Since h can be used to model the effects of magnetic field curvature, we see that stability depends on the sign of the curvature. The growth rate is about Im() [ g (n0 / n0 )] 1 2 This instability is sometimes called a flute instability for following reason. In a cylinder, the waves travel in the direction if the forces are in the r direction.