No equilibrium

1.    No equilibrium Neutrally stable Metastable equilibrium • • Equilibrium and Stability Stable equilibrium Unstable equilibrium • • Equilibrium with linear stability and nonlinear instability Equilibrium with linear instability and nonlinear stability

2. Hydromagnetic Equilibrium From the MHD equations (single fluid MHD equations) (Lecture 2). For a steady state and g = 0, p = jxB (4.2-1) and from Maxwell’s equations xB = 0j (4.2-2) Equation (4.2-1) tells us that the p force is balanced everywhere by the Lorentz force (jxB), and that j and B are perpendicular to p. This latter condition is not trivial considering that the surfaces of constant p may be nested toroidal surfaces. Using ((4.2-2) to eliminate j from 4.2-1) (B2/(20)) + (1/0)(B.)B = p (4.2-3) The first term on the left hand side is a “magnetic pressure” while the second term is related to forces arising from the inhomogeneity in the field (essentially the curvature of the field). If we neglect the curvature of the field then we find (B2/(20)) = p (for a B field without curvature) (4.2-4)

3. Linear Z Pinch Consider a cylindrical axisymmetric system (coordinates r, , z with unit vectors in these directions er, e, ez) in which a total longitudinal current (I(a) in the z direction) is established between an anode and a cathode (radius a). This current has an associated azimuthal field B. Using j = jz(r)ez. then B = B(r)e, = 0I(r)/(2r)e, and p = dp/drer where jz(r) = 1/(2r)dI(r)/dr. Now from (4.2-1) and using the ideal gas law (p(r) = 2n(r)kBT where T is uniform) we have 0I(a)2 = -82kBT∫0adrr2dn/dr (4.2-5) where using integration by parts we have 0I(a)2/8 = NkBT (4.2-6) where N is the total number of particles in the plasma ∫0adr2rn = N (4.2-7) Equation (4.2-6) is the so-called Bennett Pinch relation and the linear z pinch was one of the original configurations in magnetic fusion research h because it seemed to offer a simple way of forming, heating (Ohmically) and confining a fusioning plasma. Stability and particle and energy confinement issues have prevented it from achieving this but it is still useful for producing energetic radiation (neutrons and X-rays).

4. Linear Theta Pinch Consider a cylindrical axisymmetric system (coordinates r, , z with unit vectors in these directions er, e, ez) in which an azimuthal current in the  direction) is induced in the plasma. This current has an associated longitudinal field Bz. This means that j = j,e, B = Bz(r)ez, p = dp/drer, where 0 j = -dBz/dr From (4.2-1) we have d(p + B2/(20))/dr = 0 (4.2-8) This means that sum of the magnetic pressure and plasma pressure (p + B2/(20)) is constant. across the radius. The decrease of magnetic pressure in the plasma is caused, of course, by the diamagnetic current (j = Bx(jxB)/B2 = Bxp/B2), and the size of this effect is usually expressed in terms of the plasma beta  which is the ratio of plasma pressure to magnetic pressure, p/(B2/(20)). For low  plasmas the diamagnetic effect is small and so it doesn’t matter whether in the definition of  one uses the vacuum field or the plasma-modified field. This parameter is a measure of the confinement efficiency of the magnetic field and for economic fusion reactors one wants it to be as high as possible. Once again, fast theta pinches seemed originally to be a simple way of forming, heating and maintaining a plasma, but stability and particle and energy confinement issues have prevented it from achieving this.

6. Major axis Poloidal direction Equatorial or mid-plane Toroidal direction

9. JET MAST NCSX H1-MNRF Variations of this type of toroidal magnetic confinement

10. Diffusion of a Magnetic Field into a Plasma From the single fluid MHD equations of lecture 2, xE = - B/t (4.3-1) and E + vxB = j (4.3-2) For simplicity let us assume that the plasma is at rest and the field lines are moving into it. Then v = 0 and we Have B/t = -x(j) = - (/0)x(xB) = - ((/0)((.B) - 2B)(4.3-3) as .B = 0, we are left with a diffusion equation B/t = (/0) 2B (4.3-4) This can be solved by separation of variables. To have a rough estimate of the solution we take L as the scale length of the spatial variation of B. Then we have B/t = (/(0L2))B (4.3-5) so that B = B0exp(t/) (4.3-6) where  = (0L2 )/ which is the characteristic time for the penetration of a field into a plasma.

11. Plasma Instabilities Streaming instabilities. In this case, either a beam of energetic particles travels through the plasma, or a current is driven through the plasma so that different species have drifts relative to one another. The drift energy is excites waves and oscillation energy is gained at the expense of drift energy in the unperturbed state. 2) Rayleigh-Taylor instabilities. The plasma has a density gradient or sharp boundary so that it is not uniform. In addition, an external, non-electromagnetic force is applied to the plasma. It is this force which drives the instability. Universal instabilities. Even when there are no obvious driving forces such as electric or gravitational field, a plasma is not in perfect thermodynamic equilibrium as long as it is confined. The plasma pressure tends too expand the plasma and this expansion energy can drive an instability. This type of free energy is always present in any finite plasma, hence the resulting waves are referred to as universal instabilities. Kinetic instabilities. If the velocity distribution is not Maxwellian there is a deviation from thermodynamic equilibrium and instabilities can be driven by this anisotropy.