Outline Lecture IV: Magnetized plasma stability Wojciech Fundamenski UKAEA Fusion, Culham Science Centre, Abingdon and Plasma Physics Group, Imperial College, London 22nd May 2009
Outline Part I: 1 Hydrodynamic waves and instabilities 2 Ideal MHD waves in a uniform plasma in a stratified plasma Ideal in a confined plasma Ideal in a general screw pinch Flute-reduced MHD 3 Non-homogenous shear Alfvén waves Current driven ideal MHD instabilities: kink modes Pressure driven ideal MHD instabilities: ballooning modes Resistive MHD instabilities: tearing modes
Outline Part II: Drift-fluid and kinetic waves and instabilities 4 Drift waves and instabilities 5 Kinetic waves and instabilities
Part I
Something to consider... Let the nature of a fluid be assumed to be such that of its parts, which lie evenly and are continous, that which is under lesser pressure is driven along by that under greater pressure. Archimedes (c. 500 BC) Dynamical equilibrium does not guarantee dynamical stability. In general, an equilibrium is said to be stable if the system remains bounded, i.e., confined to its neighbourhood, after being subjected to a small perturbation. There are many different classifications of stability, e.g., linear stability implies small perturbations, while non-linear stability allows for perturbations of arbitrary size.
Aspects of dynamical stability The former is equivalent to spectral stability, which occurs when all eigenvalues of the linearized dynamical operator have real parts which are positive or zero. The most general formulation of linear stability is the energy principle, which states that an equilibrium point is stable when it represents a minimum of the potential energy of the system. Most dynamical systems have both stable and unstable equilibria, e.g., the lowest and highest position of a pendulum. Having identified an equilibrium point, one should next investigate its stability properties and, if the point proves unstable, the physical mechanism responsible for the instability.
Hydrodynamic stability of a stratified fluid By way of introduction to plasma instabilities, let us consider the stability properties of a stratified neutral fluid in the presence of a gravitational field g = gê g. Note that MHD differs from hydrodynamics only by the appearance of the Lorentz force term J B in the momentum conservation equation and by an additional evolution equation for B, including the resistivity η. Moreover, the presence of B provides a preferred direction. When B 0, or β 1, isotropy is restored and MHD reduces to hydrodynamics.
Hydrodynamic (neutral fluid) equations The fluid is assumed to conserve mass, momentum and energy, d t ρ + ρ V = 0, (1) ρd t V + p + π = ρg, (2) d t p + γp V + (γ 1)[π : V + q] = 0, (3) where d t = t + V is the convective time derivative, γ = c p /c v is the ratio of heat capacities at constant pressure and volume, π is the viscous stress and q is the heat flow. The momentum equation, albeit with a specific formulation of the viscosity tensor, is known as the Navier-Stokes equation.
Collisional closure of hydrodynamic equations In Lecture V we will see that π and q can be related to the velocity and temperature gradients, respectively, as π = µw, q = κ T, T = p/n = pm/ρ, (4) W = 2{ V} 2 3 ( V)I, W ij = xj V i + xi V j 2 3 δ ij V,(5) W = 2 V + 1 3 ( V), { V} = 1 2 ( V + V ), (6) W : V = W : { V} = 2{ V} : { V} 2 3 ( V)2, (7) where µ is the dynamic viscosity, κ is the heat conductivity and W is the (traceless) rate-of-strain tensor ; it is also customary to introduce the kinematic viscosity, ν = µ/ρ, and heat diffusivity, χ = κ/n = κm/ρ. Note that since W = W, W : V = W : V and { V} = { V}, then W : V = 1 2 (W + W ) : V = 1 2 (W : V + W : V ) = W : { V}.
Hydrostatic equilibrium and the thin-layer approximation The equilibrium conditions of hydrostatics are obtained by setting d t = 0 and V = 0 in (1)-(3), which yields p = ρg, κ T = const. (8) If the flow is non-dissipative (ν = 0 = χ), then (1)-(3) reduce to d t ρ + ρ V = 0 = d t T + γt V, (9) d t V + ρ 1 p = g, (10) where (9) yields the polytropic relation, p ρ γ, or T ρ γ 1. Combined with (8), it leads to the so called byrotropic equilibrium ρ(z) = ρ(0)(1 (1 γ 1 )z/l g ) 1/(γ 1), where z is the vertical distance and L g = p(0)/ρ(0)g is a characteristic length scale. When γ = 1 one finds ρ(z) = ρ(0) exp( z/l g ). A version of (1)-(3) for a layer of thickness L L g is known as the thin layer, thin atmosphere, or Boussinesq, approximation.
Acoustic (sound) waves Together with (10) this relation allows the propagation of compressible (acoustic, sound) waves with a phase velocity equal to the sound speed, V S, ω 2 = (kv S ) 2 v ph ω k = V S = γp γt ρ = m. (11) Acoustic waves are longitudinal disturbances of the form ρ 1, p 1, V 1 k exp(ik x iωt), where k is the wave vector and ω is the wave frequency. Since their group velocity, v gr k ω = V S is independent of k, acoustic waves are non-dispersive a fact which can be ultimately traced to the absence of an intrinsic length scale in hydrodynamics. Acoustic waves, which are the only waves possible in a uniform neutral fluid, are the primary means of establishing and restoring the hydrostatic equilibrium (8).
Incompressible flows ( V = 0) and the vorticity eq n If the flow is incompressible ( V = 0), which pertains when the characteristic flow velocity is small compared to the sound speed, i.e., when the Mach number, M S = V /V S 1, then (1) and (3) reduce to d t ρ = 0, d t T = (γ 1)[2νm{ V} : { V} + χ 2 T ]. (12) Moreover, the Navier-Stokes equation (2) can then be replaced by, d t Ω = Ω V + ν 2 Ω, Ω V, (13) where Ω is the fluid vorticity. The vorticity equation (13) is obtained by taking the curl of (2), which eliminates the pressure gradient. Here we used the fact that V V = 1 2 (V V) V V = 1 2 V 2 V Ω and that (V Ω) = V Ω + Ω V for incompressible flows.
Kelvin s vorticity theorem It is worth noting that V may be computed a posteriori from 2 V = Ω. Since Ω is just the vorticity, (13) is known as the vorticity equation in the fluid frame of reference; it may also be written in the laboratory frame as t Ω = (V Ω) + ν 2 Ω. (14) The above equations imply that, for incompressible flows, vorticity can only be created/destroyed by vortex stretching Ω V and viscous dissipation ν 2 Ω. In the absence of viscosity, the total circulation of the flow must remain constant and the vorticity is effectively frozen into the flow, d t Ω ds = d t V ds = d t V dl = 0, (15) S a result known as Kelvin s vorticity theorem. S C
Incompressible and non-dissipative flows As a consequence of (15), an initially irrotational flow (Ω = 0) must remain so at all times. The velocity can then be written as the gradient of some potential (V = φ) and the incompressibility condition, V = 0, may be recast in the form of Laplace s equation, 2 φ = 0. Finally, if the flow is both incompressible and non-dissipative, then d t ρ = 0 = d t T, d t V + ρ 1 p = g, d t Ω = Ω V. (16)
MHD version of the vorticity eq n There is a clear resemblance between the vorticity, Ω = V, the current density, µ 0 J = B and the magnetic field, B = A. This similarity is based on a general version of Kelvin s theorem, which states that d t S Q ds = 0 for any vector field satisfying t Q = (V Q). Hence, the conservation of magnetic flux in ideal MHD, and of circulation in ideal HD, stems from same vector relation. For future reference we also write down the MHD version of (13), d t Ω = Ω V + (B J J B)/ρ + ν 2 Ω, (17) which shows that the Lorentz force, J B, provides a source of vorticity.
Rayleigh-Taylor (R-T) instability Consider two fluids having different densities and zero relative velocity. This system has two possible hydrostatic equilibria: the heavy fluid is 1 above (ρ a > ρ b ), 2 below (ρ a < ρ b ), the lighter fluid. Of these, only (2) is stable, since it clearly represents the lowest energy state of the system. In contrast, the gravitational potential energy stored in (1) can be converted into kinetic energy, driving the heavier fluid down, and the lighter up, in some complicated 3D motion. The physical mechanism leading to interchange of the lighter and heavier fluids is the bouyancy force, g ρ, experienced by elements of the lighter fluid, and the process is known as an interchange, or Rayleigh-Taylor (R-T), instability.
Rayleigh-Taylor (R-T) instability Source: MIT
Rayleigh-Taylor (R-T) instability Source: DOE SCIDAC
Brunt-Vaisala frequency It also applies to a stratified fluid in which the density increases gradually with height (ê g ρ < 0), e.g., in a fluid heated from below, provided the fluid expands with increasing temperature, which is the case for nearly all fluids, e.g., for an ideal gas, in which p ρt, one finds ln ρ/ ln T = 1 and α ρ = 1/T > 0. The instability occurs when the ê g T is super-adiabatic, i.e., when the specific entropy, s = (c p /γ) ln(pρ γ ), decreases with height, N 2 g ( ln T (1 1/γ) ln p) = g ln(pρ γ )/γ > 0, (18) where N is the Brunt-Vaisala frequency of internal surface waves, see below.
Rayleigh-Benard (R-B) convection Combined with equilibrium (8) and ideal gas law, this gives the Schwarzschild instability condition, ê g T > (1 1/γ)mg. (19) The above variant of the R-T instability, in which ρ is caused by T, is known after Rayleigh and Benard (R-B). In the presence of constant heating, (3) requires constant energy flow from the bottom to top of the system. When (19) is satisfied, the system becomes R-B unstable, leading to upward flow of warm, light fluid, and downward flow of cold, heavy fluid. The resulting thermally driven flow is known as R-B, or thermal, convection.
Rayleigh-Benard (R-B) convection Sources: U.Goettingen (left), Van Dyke (right)
Kelvin-Helmholtz (K-H) instability If the light and heavy fluids have a finite relative velocity along their horizontal interface, ê g (V a V b ) 0, then the kinetic energy of this flow can amplify vertical perturbations. In the non-linear phase of this process, the surface ripples evolve into complicated vortices, destroy the smooth interface and effectively mix the two fluids. This mechanism, known as the Kelvin-Helmholtz (K-H) instability, applies equally well to a sheared flow, e.g., when the horizontal velocity changes with height, ê g V 0.
Kelvin-Helmholtz (K-H) instability Sources: UCAR
R-T and K-H stability criteria The linear stability criteria for both R-T and K-H instabilities in ideal, incompressible fluids, can be derived by linearising (16) around the hydrostatic equilibrium (8), and assuming small, vertical perturbations of the two-fluid interface, ξ exp(ikx iωt), where x is the distance and k the wave number along the surface. Since the initial flows are irrotational (Ω = 0) everywhere, except at the interface, which is thus a thin vorticity sheet, the perturbed velocity may be written as V 1 = φ 1. Denoting quantities above and below the interface by subscripts a and b, respectively, and the vertical distance away from the surface as z, we choose φ 1a and φ 1b so that V 1a and V 1b decrease exponentially with z : φ 1a exp(ikx + kz iωt) and φ 1b exp(ikx kz iωt).
Surface gravity waves Inserting these quantities into (16), introducing the effect of surface tension, T S by adding the term (T S /ρ) 2 x ξ on the left hand side of (10), and matching the conditions at the interface, yields the phase velocity of surface gravity waves (Choudhuri, 1998), ω k = ρ [ ] bv b + ρ a V a g ρb ρ a ± + k2 T S ρ aρ b (V b V a ) 2 ρ b + ρ a k ρ b + ρ a g(ρ b + ρ a ) (ρ b + ρ a ) 2. (20) Since all perturbed quantities are assumed to vary with time as exp( iωt), the real part of ω represents the frequency of sinusoidal oscillations, while the imaginary part is the rate of exponential growth or decay, i.e., when ω = Re(ω) 0, the wave propagates along the surface with velocity (20), when Im(ω) < 0, it is quickly damped, and when Im(ω) > 0 it quickly grows without bound. The last case is just the instability we seek.
R-T instability criterion The condition for R-T instability follows from (20) with V a = V b, ρ b + k 2 T S /g < ρ a, (21) which agrees with the intuitive result (ρ b < ρ a ) for T S = 0 and shows that surface tension has a strong restoring effect on small scale (large k) perturbations. To appreciate this effect, imagine that an R-T unstable system is contained in a vertical cylinder. As the diameter d of the cylinder is reduced, k min = 2π/d increases until, for a sufficiently small d, k 2 min > g(ρ a ρ b )/T S is achieved and the R-T instability is suppressed. Indeed, since the volume to surface ratio scales linearly with d, one would expect surface tension to dominate at sufficiently small scales.
K-H instability criterion Similarly, we obtain the condition for the K-H instability, [ ρb ρ a g k + kt S g ] < ρ aρ b (V b V a ) 2, (22) (ρ b + ρ a ) which indicates that an R-T stable system (ρ b > ρ a ) becomes K-H unstable for sufficiently large velocity discontinuity V a V b. Although short scales (large k) are more unstable, they are also more effectively stabilized by surface tension, so that the K-H instability is most likely to occur at some intermediate k, determined from (22).
Internal gravity waves In a stratified fluid the surface gravity waves are replaced by internal gravity waves, and the stabilising effect of surface tension by that of viscosity and heat diffusivity. The dispersion relation for internal gravity waves may be derived by linearising (1)-(3) about the equilibrium point (8). In the absence of velocity shear, this yields (k g N/k) 2 = ω 2 + iωk 2 (ν + χ) νχk 4, (23) where k g = k ê g and N is the Brunt-Vaisala frequency (18), which in the incompressible limit (γ ) reduces to N 2 = g ln ρ = α ρ g T = α ρ g z T. The system becomes (R-T) unstable when ω 2 < 0 or (k g N/k) 2 < k 4 (ν 2 + χ 2 2νχ). (24)
in incompressible, ideal fluids Since the right hand side is always positive, the diffusive terms have a stabilising effect, predominantly at small scales (large k), allowing an incompressible fluid to support a finite temperature gradient, z T, and a compressible fluid to support a super-adiabatic gradient, in excess of (19). In an incompressible ( V = 0, γ ), non-dissipative (ν = 0 = χ), fluid, (23) and (24) become (ωk/k g N) 2 = 0 and ê g T > 0, or z T < 0, respectively. The relationship between waves and instabilities, as outlined above, is common to both fluid and plasma dynamics. Magnetized plasma equilibria are thus subject to a range of instabilities, most of which can be associated with corresponding plasma waves.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Three types of MHD instabilities The underlying reason for fluid, and specifically magnetohydrodynamic, instabilities has been succinctly summarized by Biskamp (2003), A fluid, in particular a plasma, becomes unstable when the gradient of velocity, pressure, or magnetic field exceeds a certain threshold, which occurs, roughly speaking, when the convective transport of momentum, heat, or magnetic flux is more efficient than the corresponding diffusive transport by viscosity, thermal conduction, or resistivity. There are hence three types of instabilities, which play a fundamental role in macroscopic plasma dynamics: the Kelvin-Helmholtz instability driven by a velocity shear; the Rayleigh-Taylor instability caused by the buoyancy force in a stratified system; and current-driven MHD instabilities in a magnetized plasma.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Linearized MHD equations Before investigating MHD instabilities, all of which are related to plasma non-uniformity, it is worthwhile to review the basic MHD waves in a uniform magnetized plasma. For this purpose we subject an MHD equilibrium to small perturbations of the form exp(ik x iωt) (denoting equilibrium and perturbed quantities by subscripts 0 and 1, respectively), which is equivalent to Fourier transforming the MHD equations in space and time and neglecting products of perturbed quantities, such as V 1 B 1, ωρ 1 = ρ 0 k V 1, E 1 + V 1 B 0 = 0, (25) ωρ 0 V 1 = kp 1 + B 0 J 1 /i ik B 1 = µ 0 J 1, (26) ωp 1 = γp 0 k V 1, k E 1 ωb 1 = 0. (27)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Perturbed velocity eq n Note: Fourier transform reduces the differential operators to algebraic ones: t iω and ik. Since equilibrium is assumed to be stationary, V 0 = 0, then d t = t + V iω + iv 1 k. In the above, we also neglected the source terms and replaced the ideal gas ratio of specific heats 5 3 by some general value γ. Note that B = 0 implies k B 1 = 0, which also follows from (27). After simplification, (25)-(27) reduce to a single equation for the perturbed velocity, V 1, [ ] ω 2 B0 ρ 0 V 1 = (k B 0 ) + γp 0 k k V 1 k B0 (k V 1 ) B 0. µ 0 µ 0 (28)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD dispersion relation in a uniform plasma This velocity can have both longitudinal (k V 1 ) and transverse (k V 1 ) components, thus allowing both compressional and shear deformations. We next choose a Cartesian coordinate system (x, y, z) in which the the z-axis is aligned with the magnetic field (ê z = b 0 = B 0 /B). Without loss of generality, we can assume the wave to propagate in the in the y z plane, so that k = k ê z + k ê y. The vector equation (28) then yields three algebraic equations for the velocity components V 1x, V 1y, V 1z. The eigenmodes of this system may be found by requiring the determinant of the matrix to vanish, which yields the following dispersion relation, [ ω 2 (k V A ) 2] [ ω 4 (ωkv max ) 2 + (kk V S V A ) 2] = 0, (29)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Roots of the MHD dispersion relation where V S is the sound speed (11), V A is the Alfvén speed, Vmax 2 = VS 2 + V A 2 and k2 = k 2 + k2. The three roots of this equation, namely, ωa 2 = k2 V A 2, ω2 ± = 1 2 k2 Vmax (1 2 ± ) 1 α 2, (30) α 2 = 4(k /k) 2 (V A V S /V 2 max) 2, (31) represent the three possible MHD waves in a uniform plasma. Since the discriminant in (30) is positive definite, all three waves are stable. This follows because k /k 1 and (a 2 + b 2 ) 2 /a 2 b 2 = 2 + a 2 /b 2 + b 2 /a 2 4, so that α 2 1.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Shear Alfvén wave The first root (ω A ) corresponds to the shear Alfvén wave which travels along the field lines (k b) at a phase velocity equal to the Alfvén speed, ω A /k = ±V A The Alfvén waves are non-dispersive, with same group velocity v gr = k ω A = ±V A for all k. It consists of purely transverse, and hence incompressible (V 1 k = 0), plasma motions and magnetic field displacements (B 1 V 1 ). The origin of the shear Alfvén wave is the combination of ion inertia and field line tension, T B, as discussed in Lecture III.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Magnetosonic waves The second (ω + ) and third (ω ) roots of (30) represent the fast and slow magnetosonic waves, which as the name suggests are a combination of Alfvén and sound waves. They can propagate both along and across B 0, k b 0 ω 2 + = k 2 V 2 max, ω 2 = 0, (32) k b 0 1 2 < ω2 +/k 2 V 2 max < 1, 0 < ω 2 /k 2 V 2 max < 1 2.(33) The fast wave includes in-phase oscillations of plasma pressure and magnetic stress, p + T B, with T B being the Maxwell stress tensor, while the slow wave includes out-of-phase oscillations; the perturbations to p and T B are compressional, while those to T B are tensional, i.e., involve transverse shearing.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Low and high beta MHD waves Recall that T B > 0 is indicative of magnetic pressure, which counteracts compressional deformations, while T B < 0 represents magnetic tension, which counteracts bending and/or twisting deformations. Hence, the MHD momentum equation can be written as ρd t V + (p + T B ) = 0, see Lecture III. In the limits of small and large plasma pressure, compared to the magnetic pressure, i.e., when β (V S /V A ) 2 1 or 1, the fast wave reduces to the Alfvén and acoustic waves, respectively, while the slow wave vanishes entirely, β 1 ω 2 + k 2 V 2 A, ω2 0, (34) β 1 ω 2 + k 2 V 2 S, ω2 0. (35) The low β perpendicular wave (32), involves purely longitudinal deformations and is known as the compressional Alfvén wave.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation R-T instability in a magnetized fluid We next consider the effect of finite B 0 on the stability of a plasma in a gravitiation field, i.e., on the MHD version of the R-T and K-H instabilities. In ideal MHD (ν = χ = η = 0), (23) becomes (k g N/k) 2 + k 2 V 2 A = ω2, k g ê g (k ê g ) (36) which shows that magnetic field has a stabilising effect on the R-T instability provided that k = k b does not vanish. The effect is largest for k = k, when vertical motions (V 1 ê g ) involve bending deformations of the magnetic field lines.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Flute-like perturbations most unstable Such deformations excite shear Alfvén waves, which transport the free energy released by the bouyancy force away from the point of disturbance. For k g = k = k, the phase velocity becomes ω 2 /k 2 = (N/k) 2 + VA 2 and the waves are R-T unstable when N 2 > (kv A ) 2. As a result, the R-T instability in a magnetized plasma proceeds by internal gravity waves perpendicular to both g and b, i.e., by B-field aligned (k g b, k 0, k = k ), or flute-like, perturbations for which the stabilising effect of magnetic tension T B, which plays a role analogous to the surface tension T S, is minimized. The name refers to the flutes in a grecian column which are resembled by field aligned perturbations in a cylindrical plasma column.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation K-H instability in a magnetized fluid Similarly, neglecting surface tension and assuming ρ a = ρ b for simplicity, one finds the dispersion relation for the K-H instability across a vorticity sheet in a non-dissipative magnetized plasma, ω k = 1 2 (V b + V a ) ± which is the ideal MHD version of (20). V 2 A 1 4 (V b V a ) 2, (37) Once again magnetic field has a stabilising effect, with the K-H instability suppressed entirely when V A > V a V b /2, i.e., when the kinetic energy of the flow discontinuity can be transported away from the disturbance by the shear Alfvén wave.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Linearized MHD force balance We are now ready to consider the stability properties of a magnetically confined plasma equilibrium, i.e., a non-uniform, toroidal plasma supporting a radial pressure gradient. We begin with the MHD momentum equation, recall Lecture II, replacing p by the divergence of the pressure tensor, p = p + π, f ρd t V + π = J B p, d t = t + V. (38) Here we denoted the left hand side, which represents the plasma inertia and/or the net force acting on a fluid element, by f; note that f = 0 in equilibrium, i.e., in the absence of flow, V = 0 = π. Let us consider f as an operator acting on a small displacement away from equilibrium ξ(x), f(ξ) = J 1 B 0 + J 0 B 1 p 1. (39)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Linearized MHD force operator, f Here we linearized f, neglecting products of small quantities; the perturbed pressure, p 1, current, J 1, and field, B 1, are found by linearising the MHD fluid equations and integrating over time (for p 1 and B 1 only), µ 0 J 1 = B 1, B 1 = (ξ B 0 ), p 1 = γp 0 ξ ξ p 0. (40) Inserting (39) into (38) gives the explicit form of the linear force operator, f(ξ) = µ 1 0 ( B 0) B 1 +µ 1 0 ( B 1) B 0 + (γp 0 ξ+ξ p 0 ). (41)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Properties of f This operator has several important properties: 1 it is self-adjoint, η f(ξ)dx = for any two displacements ξ and η; ξ f(η)dx, (42) 2 the squares of its eigenvalues ω 2 are purely real for all discrete normal modes ξ(x, t) = ξ ω (x) exp( iωt), such that its eigenvalues ω are either real or purely imaginary, 3 as a result, all discrete normal modes are orthogonal, (ωn 2 ωm) 2 ρξm ξ n dx = 0; (43)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Properties of f 4 f(ξ) permits both discrete and continuum eigenvalues, although the latter are only allowed in the stable domain, i.e., for ω 2 0, and represent wave continua; 5 consequently, the transition from unstable to stable domains must occur via the marginally stable state, defined by ω = 0 = f(ξ ω ). The above properties suggest that normal mode analysis is well suited for investigating MHD spectral stability of confined plasmas, and indeed, it is the most common technique used for this purpose.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD energy principle I An alternative approach relies on variational techniques and the energy principle, which must be satisfied for MHD linear stability, namely δ 2 W [ξ(x)] > 0, (44) where δ 2 W is the second variation of the potential (free) energy; the first variation δw vanishes in equilibrium. This variation is equal to the work done in displacing the plasma element against the force f, given by (39), δ 2 W [ξ] = 1 ξ f(ξ)dx. (45) 2 The relation between the two approaches is precisely the same as that between the Schrödinger and Heisenberg pictures of quantum mechanics
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD energy principle II Inserting (41) into (45) and simplifying yields the following expression for δ 2 W [ξ], 1 2 p γp 0 ξ 2 + (ξ p 0 )( ξ ) J 0 (B 1 ξ ) + B2 1 µ 0 dx + 1 2 s (p 1 + B 0 B1 µ 0 )ξ ds + 1 B1 2 2 v µ 0 dx, (46) where the subscripts p, v and s denote integrals over the plasma volume, δ 2 W p, the vacuum envelope, which is assumed to surround the plasma, δ 2 W v, and the plasma-vacuum surface, δ 2 W s, respectively. The surface integral, which follows from application of Gauss s theorem, vanishes for purely tangential displacements, e.g., for a perfectly conducting boundary.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Internal and external MHD instabilities This suggests a natural division of MHD instabilities into internal, or fixed boundary, modes, which do not deform the plasma-vacuum surface, and external, or free boundary, modes, which include such deformations. The fixed boundary form of the MHD energy principle may be obtained by neglecting the surface and vacuum terms, separating ξ, B 1 and J 1 into parallel and perpendicular components, and effecting several cancelations involving ξ,
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD energy principle III δ 2 W p [ξ] = 1 2 p γp 0 ξ 2 2(ξ p 0 )(ξ κ) J 0 (B 1 ξ ) + B2 1 µ 0 + B2 0 µ 0 ξ + 2ξ κ 2 dx > 0. (47) Let us consider the energy associated with each of the terms above: 1 fluid compression acoustic waves, 2 flux tube interchange R-T instability, 3 flux tube twisting kink instability; the name originates in the kinking (buckling or twisting into a spiral) of a current filament, 4 magnetic tension, or flux tube bending shear Alfvén waves, 5 magnetic compression compressional Alfvén waves.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD energy principle IV Only the kink and interchange terms above can lower the plasma free energy, thus destabilizing the equilibrium. These two terms are known collectively as the Newcomb terms. The interchange term typically changes sign as one travels poloidally around the flux surface, i.e., it is positive (destabilizing) on the outer side of the torus and negative (stabilizing) on the inner side. However, global stability is determined by the volume, i.e., flux surface, average of all the terms in (47). In contrast, the compressional and tensile terms, which perturb the total stress tensor, p + T B, always increase this energy thus providing a stabilising effect.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation General plasma vorticity eq n As a result, the most stringent stability criteria are imposed by those perturbations which satisfy (p 1 + T B1 ) = 0. In other words, an equilibrium is MHD stable, if an only if, it is stable against incompressible displacements. Let us now return to (38) to derive the primary dynamical relation for plasma fluid instabilities. Taking a parallel projection of the curl of (38) yields the general plasma vorticity equation, b ( f 2κ f) = b [B 2 (J /B) + 2κ p ], (48) where f now denotes the fluid inertia, i.e., the left side of (38).
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Shear Alfvén law This relation, also known as the shear Alfvén law, is of fundamental importance in the theory of confined plasma stability. It is worth noting that (J /B) and p in (48) originate in the gradients of parallel and perpendicular currents. To illustrate this fact, (48) may be recast term by term into a charge conservation equation, J = 0, with J = J + J + J p, or J p = J J, (49) expressing the total current continuity (quasi-neutrality) as the sum of polarisation, parallel return and diamagnetic currents, J p = (ρ/b)b ( t + V E )V E, J = b p/b. (50)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Implications of quasi-neutrality Here it is useful to recall the discussion of J = 0 in Lecture III: In steady state (d t = 0), we recover the equilibrium relation, (J + J ) = 0, whereas transiently the net compression of J + J gives rise to the polarisation current, and hence to perpendicular plasma motion. Depending on the ordering of v E, this motion can either be fast (v E /v ti 1), as in the MHD ordering assumed here, or slow (v E /v ti δ i ), as in the drift ordering. While global equilibrium can be maintained by a parallel return current, polarisation currents are always generated on small scales. They are the origin of so-called micro-instabilities (e.g., drift waves, interchange modes, etc.) which are responsible for plasma turbulence.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Bending/twisting and interchange perturbations Due to the ever-present interchange forcing, micro-instabilities exhibit a ballooning character, i.e., are most active in regions of unfavourable magnetic curvature. In short, the two terms on the right hand side of (48), b ( f 2κ f) = b [B 2 (J /B) + 2κ p ], whose energies are given by the third and fourth and second terms in (47), represent vorticity sources due to gradients of parallel current and scalar pressure (or diamagnetic current) and give rise to instabilities involving flux tube bending/twisting and interchange, respectively.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Pressure gradient driven instabilities I Let us first consider the pressure gradient term in (48). This term, which can also be written as 2b κ p, vanishes when κ p = 0, i.e., when κ p, and is largest when (b κ) p. It vanishes for the equilibrium pressure gradient p 0 at the inboard and outboard mid-planes of the torus, since there κ p 0 R, but generates a vorticity source of 2κ n k p 1 due to diamagnetic (poloidal) pressure variation. Here we assume an eikonal of S = k x, and decompose k into diamagnetic and radial components, ξ exp(ik x), k = k b + k ê = k b + k ê + k ê. (51)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Pressure gradient driven instabilities II Substituting for p 1 with p 1 ξ p 0, (40), we see that such poloidal variation is caused by flute-like radial displacements, generating a vorticity source of 2k ξ κ n p 0. We thus expect the condition for interchange instability to be related to the sign of the product κ n p 0 or, more generally, of κ p 0. This is confirmed by the second term in (47), which indicates that ξ lowers the potential energy of the plasma, δ 2 W p [ξ], and hence drives the interchange instability, when (ξ κ)(ξ p 0 ) > 0.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Pressure gradient driven instabilities III This corresponds to the outboard side of the torus, where κ p 0 > 0, which defines the region of unfavourable, or bad, magnetic curvature. Similarly, κ is said to be favourable, or good, when κ p 0 < 0, which occurs on the inboard side of the torus. Based on this preliminary analysis, we may conclude that the magnetic curvature κ plays the role of negative gravity, g, and subjects the plasma fluid to a buoyancy force proportional to κ p 0.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Pressure gradient driven instabilities IV By expressing the grad-b and curvature GC drifts in gravitational form, recall Lecture II, this negative gravity is seen to represent the centripetal acceleration of a charged particle (fluid element) moving along a curved magnetic field line (flux tube). The buoyancy force is easily understood in the particle picture: following a flute-like perturbation ξ exp(ik θ) of the flux surface r = const, the curvature GC drift, v κ e s κ b e s ê, which has opposite sign for ions and electrons, leads to a poloidal electric field, and thus to a radial (outward) E B drift.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Pressure gradient driven instabilities V It is worth stressing that the plasma remains quasi-neutral at all times. In effect, the radial plasma motion is required to prevent the separation (polarisation) of charge that would otherwise ensue due to an unchecked curvature drift i.e., v κ (J ) leads to V E (J p ) necessary to ensure that (50) is satisfied. Hence, the frequently cited explanation of the radial plasma motion as being due to charge polarisation is not, strictly speaking, correct.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Current driven instabilities Let us next examine the current gradient term (48), which, when linearized, can be decomposed into two parts, B 2 b (J /B) B 0 B 1 (J 0 /B 0 ) + B 0 B 0 (J 1 /B 0 ), (52) corresponding to the Newcomb terms in (47). Here, the first part, which contains gradients of the equilibrium parallel current, corresponds to term (iii) in (47), associated with twisting deformations and the kink instability. Similarly, the second part, which contains gradients of the perturbed parallel current, corresponds to term (iv) in (47), associated with bending deformations (note that µ 0 J 1 = B 1 is largest when b 1 b 0 ) and the familiar stabilising effect of magnetic tension.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Magnetic differential eq n Hence, the displacement ξ(x) (or more accurately, the normal mode amplitude ξ ω (x), as discussed earlier) is most likely to grow when it exhibits a flute-like character, b 0 = 0 0, b 0 = B 0 /B 0, (53) so that field line bending is minimized. The inhomogenous form of (53), namely f = A/B, is known as the magnetic differential equation and has a solution if, and only if, it satisfies the Newcomb solubility conditions, f = A/B (A/B)dl = 0 = (A/B θ ) mn. (54)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Spectral MHD stability: Fourier expansion To analyse the spectral MHD stability of the plasma, we express ξ(x) in flux co-ordinates (r f, θ f, ζ f ), and Fourier expand ξ(r f, θ f, ζ f ) in terms of the poloidal and toroidal harmonics with mode numbers m and n, and Fourier components ξ mn (r f ), which we assume to be flux surface labels, ξ(r f, θ f, ζ f ) = m,n ξ mn (r f ) exp(imθ f inζ f ). (55) Combining (55) and (53) yields a criterion for flute-like perturbations, B θ ( θf +q ζf )ξ 0, ξ mn (r f ) 0, m/n q, (56) which states that ξ mn (r f ) must vanish on all ergodic flux surfaces,
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Resonant flux surfaces most unstable so that ξ becomes localized to resonant flux surfaces on which ξ mn (r f ) are field line labels. Since k = (m qn) θ f, (56) may also be written as k (m qn)ψ p/ gb k k, k 0 (57) which is clearly satisfied near resonant surfaces, q m/n. Note that k k implies that the frequency of the magnetosonic waves, ω + (32), greatly exceeds that of the shear Alfvén waves, ω A (30), such that the former equilibrate on the Alfvén time scales considered, and can thus be neglected in MHD stability analysis. This equilibration establishes a quasi-static perpendicular force balance which leads to a purely electrostatic perpendicular electric field, E = ϕ; this relation is employed in both flute reduced-mhd and in DHD turbulence, see Lecture VI.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD instabilities in a general screw pinch I* Although the above equations contain all the essential elements of MHD stability, they are too complicated for analytic treatment in all but a few simple systems. Below we examine two such simple systems, the general screw pinch and the cylindrical tokamak, both of which neglect the effect of toroidal curvature, so that κ = κ r (r)ê = b 2 Pê /r is purely poloidal, and hence a flux surface label. Likewise, both systems exhibit θ and z (or ζ) symmetry, so that the displacement ξ is easily Fourier transformed in θ and z, and requires only a single radial function, ξ(x) = ξ mn (r) exp(imθ inζ) = ξ mn (r) exp(ik P rθ ik T z).
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD instabilities in a general screw pinch II* As a result of these symmetries, the Fourier components ξ mn (r) in (55) become decoupled, i.e., evolve separately, allowing the energy variation (46) and the full eigenmode equation for ξ mn (r) to be evaluated explicitly In short, the model retains the essential elements of both kink and interchange modes while dispensing with most of the algebraic complexities related to toroidal geometry. It is thus very useful for introducing both instabilities.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD instabilities in a general screw pinch III* Writing the displacement ξ(x) as ξ b + ξ ê + ξ ê, where b = b P ê θ + b T ê z, ê = b T ê θ b P ê z, (58) ξ = ξ P b P + ξ T b T, ξ = ξ P b T ξ T b P, (59) the plasma energy variation may be calculated as (Freidberg, 1991), δ 2 [ W p [ξ] a k 2π 2 = f ξ 2 R 0 /µ + gξ2 dr + k ] 0 k 2 Ba 2 ξ a 2, = d r, (60) 0 where k and k are the parallel and diamagnetic components of k, a
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD instabilities in a general screw pinch IV* k = k T b T + k P b P, k = k T b P k P b T, (61) k = k T b T k P b P, k 2 = k 2 T + k2 P, (62) k T and k P are its axial (toroidal) and poloidal components, k T = n/r 0, k P = m/r, (63) and f (r) and g(r) are functions of radial position given by ( ) k B 2 ( f (r) = r = rk2 T B2 T k k 2 1 m ) 2 = B2 P qn rk 2 (qn m)2,(64) ( ) g(r) = 2 k T k µ 0p 0 + k 2 r 2 1 + 2 k2 T k f (r) k 2 k r 2. (65)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD instabilities in a general screw pinch V* The vacuum energy variation is found as [ δ 2 W v [ξ] Λw a 2 k 2 ] 2π 2 = Ba 2 ξ a 2 (66) R 0 /µ 0 m where Λ w is a geometrical function of the minor radius, a, and the conducting wall radius, b, involving modified Bessel functions taking the argument k T a and k T b. For k T b 1, this function may be estimated as Λ w 1 + (a/b)2 m 1 (a/b) 2 m, k T b 1, a < b, (67) which for a/b 1 reduces to 1 + 2(a/b) 2 m 1, while for a/b 1 becomes singular as [(1 a/b) m ] 1 ; in this limit, a finite ξ a leads to an infinite increase of δ 2 W v and is thus prohibited, i.e., ξ (a) 0. a
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD instabilities in a general screw pinch VI* Imposing the condition that ξ (r) minimise δ 2 W p (60), where we neglect the vacuum energy δ 2 W v (66) for simplicity, allows us to derive the corresponding eigenmode equation for ξ (r). Setting the variation of (60) with respect to ξ (r) to zero, yields an Euler-Lagrange equation, [f (r)ξ ] g(r)ξ = 0, (68) with f (r) and g(r) defined above. This MHD dispersion relation is the cylindrical version of (29), and describes all MHD waves and instabilities in a marginally stable screw pinch equilibrium.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD instabilities in a general screw pinch VII* We next write down the corresponding full eigenmode problem for ξ(x), f[ξ(x)] + ρω 2 ξ(x) = 0 (69) where f[ξ(x)] is given by (41). For a general screw pinch this problem takes the form of a second order ODE for the radial displacement ξ, [F(r)(rξ ) ] G(r)rξ = 0, (70) where F(r) and G(r) are functions of frequency, wave number and radius, F(r) = ρv max 2 (ω 2 ωa 2 )(ω2 ωh 2 ) r (ω 2 ω+ 2 )(ω2 ω 2 ) (71)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD instabilities in a general screw pinch VIII* G(r) = ρ(ω2 ωa 2 ) ( 4k 2 + T V 2 ) A B2 P (ω 2 ωs 2) r µ 0 r 3 (ω 2 ω+ 2 )(ω2 ω 2 [ ) B 2 ( ) + P µ 0 r 2 2kT GB P V 2 max (ω 2 ωh 2 ) ] µ 0 r 2 (ω 2 ω+ 2 )(ω2 ω 2 ). (72) Here, the Alfvénic and magneto-sonic frequencies, ω A and ω ±, defined in (30), are supplemented by the sonic and hybrid frequencies, ω S and ω H, ω 2 S = k2 V 2 S, ω2 H = k2 V 2 H, V 2 H = V 2 A V 2 S /V 2 max, (73) where V A and V S are the Alfven and sound speeds, see Lecture III, and V 2 max = V 2 A + V 2 S ; finally, G = k PB T k T B P.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation MHD instabilities in a general screw pinch IX* The eigenmode equation (70) is a more general version of the marginally stable dispersion relation (68), and describes all possible in a general screw pinch. Indeed, it reduces to the latter result when the marginal stability condition (ω = 0) is imposed on (71)-(72). The agreement between the two results confirms our earlier assertion that a marginally stable eigenmode of the MHD force operator f[ξ(x)] (41) must minimise the free energy of the plasma; this is a direct consequence of the self-adjointness of f, see (42).
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Flute-reduced MHD I* The above results indicate that toroidal plasmas are most unstable to flute-like perturbations ξ exp(ik x) which satisfy (53). These are in general radially localized to resonant flux surfaces, although in large aspect ratio (small ɛ) tokamaks, where all displacements satisfy (53), b 0 k /k r/r 0 q = ɛ/q O(ɛ) 1. (74) they also apply on ergodic flux surfaces. To show this, we first note that in cylindarical geometry, the poloidal and axial components of k are k P = m/r and k T = n/r 0, while parallel and diamagnetic components become k R 0 = m/q n, k r m, (75) so that the ratio k /k O(ɛ/q) is small everywhere when ɛ 1 and q O(1).
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Flute-reduced MHD II* Here we assume that the perturbation is localized to the flux surface r = const and thus exclude the radial component of k. Hence k = b (k b) reduces to the projection in the diamagnetic direction k = k ê. More generally, k should be replaced by k = k. Of course, k 1 is not true for all perturbations when ɛ 1, although it is always satisfied close to resonant flux surfaces, q m/n. Hence, it is useful to derive a version of the MHD (and more generally of DHD) model in which flute-like displacements are assumed from the outset.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Flute-reduced MHD III* Such a model, known a flute-reduced MHD may be derived by imposing the flute-ordering (74) on the MHD equations. This is accomplished by applying a multi-scale expansion in the small parameter ɛ qk /k 1, to the MHD, or more generally to DHD, equations and retaining only leading order terms. The resulting equations are a useful reference point for the study of stability and dynamics of toroidal plasmas. Below we outline the derivation, broadly following the original account of Strauss (1976), yet making explicit the multiple-scale expansion.
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Flute-reduced MHD IV* Multi-scale expansion involves separation of slow, O(1), and fast, O(ɛ 1 ), spatial variations into (independent) co-ordinates X and x, and slow, O(ɛ 2 ), and fast, O(1), temporal variations into co-ordinates T and t, x ɛ 1 x + X, t t + ɛ 2 T. (76) Hence, the net gradient can be expressed as the sum of slow, 0 = X, and fast, 1 = x = + O(ɛ), gradients, and the net time derivative as the sum of slow, T, and fast, t, time derivatives, ɛ 1 + 0, t t + ɛ 2 T. (77) Similarly, any quantity A is expanded in powers of ɛ into an equilibrium part A 0 and the flute-perturbed parts A 1, A 2,...,
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Flute-reduced MHD V* satisfying (74), as follows A = A 0 (X, T ) + ɛ n A n (x, X, t, T ). (78) n=1 Hence the fast variables x and t measure the variation of the flute-perturbed quantities, and the slow variables X and T that of equilibrium quantities. First, we assume the existence of an MHD equilibrium for B 0, J 0 and p 0, J 0 B 0 = 0 p 0, 0 B 0 = 0, µ 0 J 0 = 0 B 0. (79) We next note that in a tokamak, parallel projection of Ohm s law expresses the balance between inductive EMF and collisional resistivity, so that the equilibrium electric field is purely parallel,
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Flute-reduced MHD VI* t A 0 = E A 0 = E 0 = η J 0, E 0 = ϕ 1 t A 1 = 0, (80) and the first order potentials must vanish. Flute-reduction then allows us to express the electromagnetic (vector ) field in terms of two scalar fields ψ 1 and ϕ 2. Due to the flute-ordering, (74), the perturbed electric field is mainly transverse and electrostatic E 1 E 1 ϕ 2. (81) The perturbed magnetic field, which follows from B = 0 and (74), implies B 1 0 to leading order. Hence, B 1 may be written as B 1 = A 2 = (A 2 +b 0 A 2 ) = B 1 b 0 + ψ 1 b 0, (82)
in a uniform plasma in a stratified plasma in a magnetically confined plasma in a general screw pinch in the flute-reduced approximation Flute-reduced MHD VII* where ψ 1 = A 2 denotes the parallel vector potential. As a result, the parallel gradient, (B/B 0 ), can be simplified as = B 0 0 + B 1 = b 0 0 B 1 0 {ψ 1, } b 0 0 { ψ 1, }, (83) where breve denotes division by B 0 and {A, } is the Poisson bracket, Ă = A/B 0, {A, } b 0 A = b 0 A. (84) The first term in the parallel gradient represents the variation along the equilibrium field, while the second term represents variation due to field line bending and perpendicular gradients. By nature of (74), only transverse deflection B 1 = ψ 1 b 0 contributes to the latter.