22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 14: Formulation of the Stability Problem

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1 .65, MHD Theory of Fusion Systems Prof. Freidberg Lecture 4: Formulation of the Stability Problem Hierarchy of Formulations of the MHD Stability Problem for Arbitrary 3-D Systems. Linearized equations of motion. Normal mode eigenvalue approach 3. Variational approach 4. Energy principle 5. Extended Energy Principle Linearized Equations of Motion Assume we have an equilibrium satisfying static J B = p V = E = B = u J ρ arb. B = Q = Q x 3D in general Linearize the Equation Q( x,t) = Q ( x) + Q ( x,t) ρ + ρ v = ρ Mass: + ρ v = Energy: dp rp v dt + = p + v p + rp v = amp L: μ J = B μ J = B B : B = B = Faraday: Momentum: B = E = v B dv ρ = J B p dt B = ( v B) v ρ = J B + J B p.65, MHD Theory of Fusion Systems Lecture 4 Prof. Freidberg Page of 8

2 Simplify the PDE s by introducing the displacement sector ξ and appropriate initial conditions a. v ξ = t b. ξ is the plasma displacement away from equilibrium Initial Conditions: Assume the plasma is in its equilibrium position moving away with a small velocity ξ x, = B x, = ρ x, = p x, = ξ v ( x,) = ( x,) Simplify the equations ( B = not needed, redundant) Express all quantities in terms of ξ Mass: Energy: ρ + ρ v = ρ + ρξ = ( p +ξ p + rp ξ ) = ( B ξ B ) = Faraday: Ampere μ J = B Momentum: ρ = ( ρξ ) p p rp = ξ ξ B = ( ξ B) μ J = ξ B v ρ = J B + J B ρ a. ξ ξ ρ = F( ξ ) Cξ ( x,) =, ( x,) = given, + B.C. b. ( B ) F ξ = ξ B B + ξ B + ξ p + rp ξ μ a μ.65, MHD Theory of Fusion Systems Lecture 4 Prof. Freidberg Page of 8

3 Initial Value Approach Solve the linear equations of motion Advantages: a. directly gives time evolution of the system b. fastest growing mode automatically appears c. good stand for nonlinear calculations Disadvantages: a. more information contained than required to determine stability Normal Mode Approach b. extra work is required analytically and numerically to determine this information c. tough to find marginal stability A more efficient procedure that treats one mode at a time. The initial ξ ( x, ) can be decomposed into normal modes. Each mode is then analyzed separately 3. To do this we fourier analyze in time i t = ω ξ = ξ Q x,t Q x e x,t x e ω i t 4. Why is this legitimate? 5. Note: The equations for ρ,b,j,p no explicit ω's appear. We find (drop subscript) do not have any time durations. Hence, ρ = ρξ p = ξ p rp ξ B ( B) = ξ ( B ) μ J = ξ.65, MHD Theory of Fusion Systems Lecture 4 Prof. Freidberg Page 3 of 8

4 The momentum equation becomes (F has no time derivation) ω ρξ = F ( ξ ) ( ξ ) = + F J B J B p We only need B.C. ω eigenvalue This is normal mode approach Advantages: a. more amenable to analysis b. directly addresses stability question (examine from ω ) c. more convenient numerically Disadvantages: a. cannot be generalized for nonlinear calculation Properties of F b. still relatively complicated. To proceed further (variational approach, energy principle) we need to understand the properties of the force operator F ( ξ ). We show that a. F ( ξ ) is self adjoint b. ω is purely real c. the normal modes are orthogonal 3. Self adjointness ( procedures) a. subtle but elegant b. direct but complicated The basic self adjoint property is associated with the conservation of energy; there is no dissipation in the system Self Adjoint Property a. η F( ξ ) dr = ξ F( η) dr where ξ and n are any two arbitrary, independent sectors satisfying the boundary conditions.65, MHD Theory of Fusion Systems Lecture 4 Prof. Freidberg Page 4 of 8

5 b. Simple self adjoint equations: ξ η ξdr, η dr = ξ η dr x x ξ η c. A non self adjoint equation: η dr = ξ dr x x Direct Demonstration: Very Tedious Calculation. Assume n ξ = n η= as plasma boundary (can be generalized) μ. η F( ξ ) dr = dr ( B ξ )( B η ) + rp( ξ)( η) ( )( ) B + ξ + ξ κ η + η κ μ F 4B B ( ξ κ)( η κ ) + ( η ξ : ) p + μ μ is self adjoint by inspection: switch ξ and η, get the same result. Show that ω is real ω ρ ξ = ξ > d r ξ. F *. ω ρ ξ = ξ ( ξ) * dr F dr * 3. Similarly * F( * ) ω ρ ξ = ξ > drξ real operator * * * 4. ω ρ ξ = ξ ( ξ ) dr F dr 5. Subtract the equations * * * dr dr F F ω ω ρξ = ξ ξ ξ ξ = because of the self adjoint property.65, MHD Theory of Fusion Systems Lecture 4 Prof. Freidberg Page 5 of 8

6 6. Therefore ω = ω * ω is real This has important consequences. At marginal stability we note that by definition ω i =. In ideal MHD ω r = also!! This is a big help. There is no need to find 3. ω ρξ = F ( ξ ) ωr real real ξ real r, t r e ω i t 4. That ξ is real is not initially obvious ξ = ξ 5. We continue to allow complex ξ to simplify fourier analysis in space, later on. i ( r) ( r) e im θ τ kz ξ = ξ for example.65, MHD Theory of Fusion Systems Lecture 4 Prof. Freidberg Page 6 of 8

7 Show that the Normal Modes are Orthogonal. Consider two normal modes (assume real ξ now) m m F m n ω ρ ξ = ξ > ξ m n F n m ω ρ ξ = ξ > ξ. Subtract dr dr ( ωn ω ) m ρξm ξ n = ξn ( ξm) ξm ( ξ n) dr F F dr = by the self adjoint property 3. For n m n m, ω ω ρξ n ξ m dr = orthogonal property 4. For n=m choose ρξ m dr = orthonormal Spectrum of E In general F exhibits both discrete eigenvalues and continua Spectrum: F F ω ρ ξ = ξ ξ = ω + initial conditions ρ The points where ω + do not exist define the spectrum of F ρ F.65, MHD Theory of Fusion Systems Lecture 4 Prof. Freidberg Page 7 of 8

8 Examples Continua significantly complicate MHD analysis for general initial value problems. They require more than just picking up the pole contributions from the displace transform. However the continua lie on stable side of the spectrum and thus do not affect stability. Accumulation points: these provide a simple necessary condition for stability..65, MHD Theory of Fusion Systems Lecture 4 Prof. Freidberg Page 8 of 8