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Pressure-Driven Instabilities

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Pressure-Driven Instabilities Pressure-driven instabilities Ben Dudson Department of Physics, University of York, Heslington, York YO10 5DD, UK 20th February 2014 Ben Dudson Magnetic Confinement Fusion (1 of 23) Previously In the last lecture we looked at the basics of plasma instabilities Analysed in the same way as waves: Expand equations into equilibrium and perturbed quantities Linearise by keeping only terms linear in a perturbed quantity @y Linear equations of the form F y have solutions @t · exp ( i!t) For/ instabilities,− ! has an imaginary component usually called γ which is the growth rate. Examples are the Sausage and Kink instabilities in z-pinches Driven unstable by plasma pressure and current Can be stabilised by introducing additional magnetic fields as bending field-lines act to restore the equilibrium An application is the Kruskal-Shafranov condition and q > 1 for stability in tokamaks. Ben Dudson Magnetic Confinement Fusion (2 of 23) Analysis of plasma stability There are usually several ways to study a plasma instability: Linearising fluid equations Sometimes just considering particle orbits and drifts is enough Calculate the energy available and the energy needed to overcome stabilising effects Ben Dudson Magnetic Confinement Fusion (3 of 23) Single-particle picture of sausage instability In a Z-pinch the magnetic field is curved around the axis: B κ B J × R κ = (b ) b · ∇ In curved magnetic fields particles drift: v 2 v 2 jj RC B jj v R = 2× = κ b Ω RC B − Ω × Since Ω = qB=m, the sign is different for electrons and ions. For a z-pinch the drift is along the axis. Ben Dudson Magnetic Confinement Fusion (4 of 23) Single-particle picture of sausage instability The curvature drift is in opposite directions for electrons and ions along the axis of the plasma (z or φ direction) If there is a perturbation to the plasma, then this leads to charge separation and electric field in the z direction E B drift is then in the radial direction × E B × κ B × B κ - + E - + E + - E B × For a z-pinch, this E B drift is such that it enhances the original perturbation instability.× ) Ben Dudson Magnetic Confinement Fusion (5 of 23) Plasma stability depends on the relative directions of the magnetic field curvature and the pressure gradient. These are known as good curvature and bad curvature. Good and bad curvature If we reverse the direction of the curvature, so put the plasma on the outside of the z-pinch (reverse pressure gradient), then we get: E + - E B × + - - + E B × E κ B × B κ Here the E B acts to reduce the perturbation stable. × ) Ben Dudson Magnetic Confinement Fusion (6 of 23) Good and bad curvature If we reverse the direction of the curvature, so put the plasma on the outside of the z-pinch (reverse pressure gradient), then we get: E + - E B × + - - + E B × E κ B × B κ Here the E B acts to reduce the perturbation stable. × ) Plasma stability depends on the relative directions of the magnetic field curvature and the pressure gradient. These are known as good curvature and bad curvature. Ben Dudson Magnetic Confinement Fusion (6 of 23) Good and bad curvature Low pressure p The relative directions of κ r curvature κ and pressure High pressure gradient has an important effect on plasma stability Figure : Bad curvature: κ p > 0 · r Instabilities driven by bad High pressure p curvature are analogous to r Rayleigh-Taylor fluid instability and are called κ Interchange instabilities Low pressure Figure : Good curvature: κ p < 0 · r Ben Dudson Magnetic Confinement Fusion (7 of 23) Good and bad curvature in tokamaks In a tokamak, the toroidal field is curved and the pressure is highest in the core on the outboard side (large R) the curvature is bad, whilst) on the inboard side (small R) the curvature is good Many tokamak instabilities have maximum amplitudes on the outboard side, called ballooning type modes Low pressure Bad curvature High pressure p r Low pressure κ Bφ Ben Dudson Magnetic Confinement Fusion (8 of 23) Analogous to a ball on a hill Unstable Stable Energy and plasma stability To calculate whether a plasma is unstable we need to consider both sources of instability such as pressure gradients, but also stabilising effects Last lecture we saw that bending field-lines produced a force F which opposed the motion i.e F v < 0 · This means that the instability is having to do work to bend the field-lines To be unstable, the energy available has to be greater than the energy needed to overcome this force Ben Dudson Magnetic Confinement Fusion (9 of 23) Energy and plasma stability To calculate whether a plasma is unstable we need to consider both sources of instability such as pressure gradients, but also stabilising effects Last lecture we saw that bending field-lines produced a force F which opposed the motion i.e F v < 0 · This means that the instability is having to do work to bend the field-lines To be unstable, the energy available has to be greater than the energy needed to overcome this force Analogous to a ball on a hill Unstable Stable Ben Dudson Magnetic Confinement Fusion (9 of 23) Energy and plasma stability The difference between stable and unstable situations is the change in potential energy δW due to a small perturbation δW < 0 Unstable: potential energy converted to kinetic energy δW > 0 Stable: kinetic to potential, then oscillates δW = 0 Marginal: Like a ball on a flat surface A plasma is stable if δW > 0 for all possible perturbations, and unstable if any perturbation results in δW < 0 To calculate δW we'll use the ideal MHD equations... Ben Dudson Magnetic Confinement Fusion (10 of 23) Ideal MHD linearisation To calculate the change in energy from a small perturbation we need to first linearise the equations: n = n0 + n1 v = v 0 + v 1 ::: which after substituting into the ideal MHD equations, and assuming a stationary equilibrium v 0 = 0 gives: @ n1 = n0 v v n0 @t − r · 1 − 1 · r @ 1 1 1 v 1 = p1 + ( B1) B0 + ( B0) B1 @t mi n0 −∇ µ0 r × × µ0 r × × @ p1 = γp0 v v p0 @t − r · 1 − 1 · r @ B = (v B ) @t 1 r × 1 × 0 Ben Dudson Magnetic Confinement Fusion (11 of 23) Ideal MHD dispacement In ideal MHD all perturbed quantities n1, v 1, p1, and B1 can be written in terms of a single displacement ξ (x) which is the distance the fluid has moved from equilibrium. @ξ The velocity is v = 1 @t Substitute this into the other equations: @ @ξ @ξ n1 = n0 n0 @t − r · @t − @t · r Since the equilibrium quantities do not depend on time, this can be integrated trivially: n1 = n0 ξ ξ n0 ) − r · − · r Similarly: p1 = p0 ξ ξ p0 − r · − · r B = ξ B 1 r × 1 × 0 Note: This only works for ideal MHD: resistivity breaks this Ben Dudson Magnetic Confinement Fusion (12 of 23) Ideal MHD normal modes Substituting into the equation for velocity gives: @2ξ 1 1 m n = ξ p + γp ξ + ( B ) B + ( B ) B i 0 t2 0 0 1 0 0 1 @ r | · r {z r · } µ0 r × × µ0 r × × −p1 This is a linear operator 2 @ ξ mi n0 = F ξ Fancy form of ma = F @t2 Solutions are linear ξ (x; t) = ξ (x) e−i!t and include the shear Alfv´enand magnetosonic waves. Hence we can write an eigenvalue equation with eigenfunction ξ 2 mi n0! ξ = F ξ − Ben Dudson Magnetic Confinement Fusion (13 of 23) Ideal MHD energy principle To calculate the work done on the plasma, we can start with the power: force velocity × d Z @ξ δW = d3xF ξ dt − · @t dV @ξ Integrate this by parts: u = F ξ, = dt @t Z t Z Z @F ξ δW = d3xF ξ ξ + dt d3xξ ) − · t=0 · @t @F ξ @F ξ @ξ By writing = , this becomes: @t @ξ · @t ! Z t Z Z @F ξ @ξ δW = d3xF ξ ξ + dt d3xξ − · t=0 · @ξ · @t Ben Dudson Magnetic Confinement Fusion (14 of 23) Ideal MHD energy principle Because ideal MHD is Hermitian, ! ! @F ξ @ξ @F ξ @ξ @ξ ξ = ξ = F ξ · @ξ · @t · @ξ · @t · @t and so this becomes: Z t Z Z @ξ δW = d3xF ξ ξ + dt d3xF ξ − · t=0 · @t | {z } =−δW Hence 1 Z δW = d3xF ξ ξ −2 · Ben Dudson Magnetic Confinement Fusion (15 of 23) Ideal MHD energy equation (standard form) Substituting the expression for F ξ from earlier: 1 1 F ξ = ξ p0 + γp0 ξ + ( B1) B0+ ( B0) B1 r · r r · µ0 r × × µ0 r × × then integrating over plasma and vacuum regions gives the following standard form of the ideal MHD energy equation: Z " 2 1 3 B1 2 ∗ δWp = d x j j + γp ξ ξ (J0 B1) 2 µ0 r · − · × # + ξ p ξ∗ Plasma terms ? · r r · ? Z 2 1 3 B1 δWV = d x j j Vacuum (always stabilising) 2 µ0 I 1 B0 B1 ∗ δWS = γp ξ · ξ dS Surface terms −2 r · − µ0 · Ben Dudson Magnetic Confinement Fusion (16 of 23) Ideal MHD energy equation (intuitive form) The plasma contribution δWp can be rearranged: δWp = " 1 Z B 2 d3x j 1j Field-line bending 0 2 µ0 ≥ B2 2 + + 2 Magnetic compression 0 ξ? ξ? κ µ0 r · · ≥ 2 +γp0 ξ Plasma compression 0 r · ≥ 2 ξ p κ ξ∗ Pressure/curvature drive, + or − ? · r · ? − # B ξ b j Parallel current drive, + or − 1 · ? × jj − This is a very useful form of the energy equation because it makes clear the balance between destabilising and stabilising effects Ben Dudson Magnetic Confinement Fusion (17 of 23) Instability drives The first three terms in this equation are always 0 and so are stabilising, but the last two can be positive (stabilising)≥ or negative (destabilising): 2 ξ p κ ξ∗ depends on p and κ: if p κ > 0 − ? · r · ? r r · then this is destabilising.
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