Formal Theory of MHD Stability:Energy Principle

Andrei N. Simakov

Los Alamos National Laboratory

CMPD & CMSO Winter SchoolJanuary 2008

Outline

• Definition of stability

• Linear stability formulation as

– initial value problem– normal mode eigenvalue problem– variational principle– Energy Principle

• Classification of ideal MHD instabilities

• Example: stability of θ pinch

• Good textbook: J. P. Freidberg, Ideal Magnetohydrodynamics (Plenum Press,New York, 1987), chapters 8 and 9.

1

Preliminary Remarks

• Ideal MHD Equations (d/dt ≡ ∂/∂t + V ·∇):

dρ/dt + ρ ∇ · V = 0, dp/dt + γ p ∇ · V = 0

ρ dV /dt = c−1 J ×B −∇p, E + c−1 V ×B = 0

∂B/∂t + c∇×E = 0, ∇×B = (4π/c) J , ∇ ·B = 0

• Assume there exists a static equilibrium:

V 0 = E0 = 0, c−1 J0 ×B0 = ∇p0, ∇×B0 = (4π/c) J0, ∇ ·B0 = 0

• Is it stable or unstable? Linear stability analysis is usually sufficient, no needfor non-linear studies

• Reason: unstable ideal MHD modes usually destroy the plasma ⇒ have toavoid them, not study the details of plasma self-destruction

2

Definition of Stability

• Linearize all quantities of interest about their equilibrium values

q(r, t) = q0(r) + q1(r, t), q1(r, t) = q1(r) exp(−iωt), |q1|/|q0| � 1

• By definition,

Im(ω) > 0 ⇒ (exponential) instability

Im(ω) ≤ 0 ⇒ (exponential) stability

• Word of caution: this definition implicitly assumes that the modes arediscrete.

• They are not necessarily in the stable part of the ideal MHD spectrum(continuum of modes may exist), but they are in the unstable part of thespectrum for all systems studied so far

3

Energy Principle: Plan of Attack

• Energy Principle is a powerful method for testing ideal MHD stability inarbitrary 3D magnetic confinement configurations

• Will derive it by going through the following steps

– Formulate the stability problem as an initial value problem for plasmadisplacement from an equilibrium position

– Cast the initial value problem in the form of a normal-mode eigenvalueproblem

– Transform the eigenmode formulation into a variational form

– Reduce the variational formulation to the Energy Principle

4

Initial Value Formulation

• Assume a static ideal MHD equilibrium and linearize all quantities about it:

q(r, t) = q0(r) + q1(r, t)

• Introduce plasma displacement vector ξ defined as V 1 ≡ ∂ξ/∂t

• Choose initial conditions:

ξ(r, 0) = B1(r, 0) = ρ1(r, 0) = p1(r, t) = 0

∂ξ(r, t)/∂t ≡ V 1(r, 0) 6= 0

• These conditions correspond to the situation when at t = 0 the plasma is in itsexact equilibrium but is moving away with a small velocity V 1(r, 0)

5

Initial Value Formulation: Linearized Equations

Mass: ∂ρ∂t + ∇ · ρV = 0 ⇒ ∂ρ1

∂t + ∇ · ρ0V 1 = 0 ⇒ ρ1 = −∇ · ρ0ξ

Energy: dpdt + γp∇ · V = 0 ⇒ ∂p1

∂t + V 1 ·∇p0

+ γp0∇ · V 1 = 0⇒ p1 = −ξ · ∇p0

− γp0∇ · ξ

Faraday’slaw:

∂B∂t =∇× (V ×B)

⇒ ∂B1∂t =

∇× (V 1 ×B0)⇒ B1 =

∇× (ξ ×B0)

Ampere’slaw:

(4π/c)J = ∇×B ⇒ (4π/c)J1 = ∇× B1 ⇒ (4π/c)J1 =∇×∇×(ξ×B0)

∇ ·B: ∇ ·B = 0 ⇒ ∇ · B1 = 0 ⇒ redundant

Mom-m: ρdVdt =

c−1J ×B −∇p⇒ ρ0

∂V 1∂t = c−1J1×B0

+c−1J0×B1−∇p1

⇒ ρ0∂2ξ∂t2

= F (ξ)

6

Initial Value Formulation: Summary I

ρ0∂2ξ

∂t2= F (ξ),

F (ξ) =14π

(∇×Q)×B0+14π

B0×(∇×Q)+∇(ξ·∇p0+γp0∇·ξ), Q ≡ ∇×(ξ×B0) = B1

+

ξ(r, 0) = 0,∂ξ(r, 0)

∂t= given

+

boundary conditions

7

Initial Value Formulation: Summary II

• Advantages

– directly gives time evolution of the system– fastest growing mode automatically appears– good start for a nonlinear calculation

• Disadvantages

– more information is contained than is required for determining stability– extra analytical and numerical work is required to obtain this information

8

Boundary Conditions I

• Plasma is surrounded by a perfectly conducting wall (n is outward-pointingnormal vector)

– (n×E)|rwall= 0

– (n ·B)|rwall= 0

– n× (E + c−1V ×B)|rwall= 0 ⇒ (n · V )|rwall

= 0 ⇒ (n · ξ)|rwall= 0

9

Boundary Conditions II

• Plasma is surrounded by a vacuum region (which is described by∇× B = ∇ · B = 0, with B the vacuum magnetic field)

– (n · B)|rwall= 0

– plasma surface is free to move ⇒ (n · ξ)|rplasma= arbitrary

– [[n ·B]]|rplasma= 0 (with [[· · · ]] denoting a jump across the plasma

surface)

– [[n×B]]|rplasma= (4π/c)K , with K the surface current density

– [[p + B2/8π]]|rplasma= 0

• See Sec. 3.2 of the Ideal Magnetohydrodynamics for further discussion

10

Normal Mode Formulation I

• Assume for perturbed quantities q1(r, t) = q1(r) exp(−iωt)

• Obtain−ω2ρ0ξ = F (ξ),

F (ξ) =14π

(∇×Q)×B0+14π

B0×(∇×Q)+∇(ξ·∇p0+γp0∇·ξ), Q ≡ ∇×(ξ×B0) = B1

+

boundary conditions

• ∂ξ(r, 0)/∂t can be decomposed into normal modes, and each normal mode canbe analyzed separately

• From now on will drop for simplicity all subscripts “0” denoting equilibriumquantities

11

Normal Mode Formulation II

• Advantages

– the method is more amenable to analysis– directly addresses stability question (sign of Im(ω))– more computationally efficient

• Disadvantages

– cannot be generalized for nonlinear calculations– still relatively complicated

• The usefulness of this approach relies upon the assumption that for theproblems of interest the eigenvalues are discreet and distinguishable so thatthe concept of exponential stability is valid

12

Properties of the Force Operator F (ξ)

• To proceed further (variational approach, Energy Principle) we need tounderstand the properties of the force operator F (ξ)

• We will next see that

– F (ξ) is self-adjoint– ω2 is purely real (so that ω is either purely real or purely imaginary)– the normal modes are orthogonal

13

Self-adjointness of F (ξ) I

• It is associated with the conservation of energy (there is no dissipation in thesystem)

• F is self-adjoint if for any arbitrary, independent vectors ξ and η satisfyingappropriate boundary conditions∫

dr η · F (ξ) =∫

dr ξ · F (η)

• Examples of self-adjoint operators:∫

dr η · ξ,∫

dr η · ∂2ξ∂x2 =

∫dr ξ · ∂2η

∂x2

• Example of a non-self-adjoint operator:∫

dr η · ∂ξ∂x = −

∫dr ξ · ∂η

∂x

14

Self-adjointness of F (ξ) II

• Can demonstrate self-adjointness directly via a tedious calculation. E.g.assuming n · ξ = n · η = 0 on plasma boundary can show∫

dr η · F (ξ) = −∫

dr

[(B ·∇ξ⊥)(B ·∇η⊥)

4π+ γp(∇ · ξ)(∇ · η)

+B2

4π(∇ · ξ⊥ + 2κ · ξ⊥)(∇ · η⊥ + 2κ · η⊥)

−B2

π(κ · ξ⊥)(κ · η⊥) + (ξ⊥η⊥ : ∇∇)

(p +

B2

8π

) ]• In obtaining this expression wrote ξ = ξ⊥ + ξ‖b and η = η⊥ + η‖b, where ‖

and ⊥ refer to the equilibrium B

• κ ≡ b ·∇b is the magnetic field curvature

• Can generalize the proof for the boundary conditions allowing a vacuum region

15

Real ω2

• Dot the equation −ω2ρ ξ = F (ξ) with ξ∗ and integrate over the plasma volume:

ω2

∫dr ρ|ξ|2 = −

∫dr ξ∗ · F (ξ)

• Repeat for the complex conjugate equation −(ω∗)2ρ ξ∗ = F (ξ∗) dotted with ξ:

(ω∗)2∫

dr ρ|ξ|2 = −∫

dr ξ · F (ξ∗)

• Because of self-adjointness of F ,

[ω2 − (ω∗)2]∫

dr ρ|ξ|2 = 0 ⇒ ω2 = (ω∗)2

• So, both ω2 and ξ are purely real

• Consequence: ω2 > 0 ⇒ stable oscillations, ω2 < 0 ⇒ exponential instability

16

Orthogonality of the Normal Modes

• Consider two discreet normal modes (ξm, ω2m) and (ξn, ω2

n) for which

−ω2mρ ξm = F (ξm), −ω2

nρ ξn = F (ξn)

• Dot the ξm-equation with ξn and vice versa, integrate over the plasma volume,subtract the results, and use self-adjointness of F to get

(ω2m − ω2

n)∫

dr ρ ξm · ξn = 0

• For non-degenerate discreet normal modes ω2m 6= ω2

n so that∫dr ρ ξm · ξn = 0

• Therefore, such modes are orthogonal with weight function ρ

17

Elements of Variational Calculus I• Consider the classic Sturm-Liouville problem [f(x) and g(x) are known]

ddx

(f(x)

dy

dx

)+ (λ− g(x))y = 0, y(0) = y(1) = 0

• Need to accurately evaluate the eigenvalue λ

• Cast the differential equation in the form of an equivalent integral relation

• Substitute analytic “guesses” (trial functions) in place of the trueeigenfunction to estimate the eigenvalue

• The integral relation is variational if the estimated λ exhibits an extremum forthe actual eigenfunction. This extremum is equal to the actual eigenvalue

• If the integral relation is variational the estimated eigenvalue is more accuratethan the trial function

18

Elements of Variational Calculus II

• Multiply the Sturm-Liouville differential equation by y and integrate over0 ≤ x ≤ 1

• Employing integration by parts and boundary conditions for y obtain

λ =∫

dx (fy′2 + gy2)∫dx y2

• Prove that this form is variational. Assume that trial function y0(x) gives λ0

and trial function y(x) = y0(x) + δy(x), with δy a small arbitrary perturbationsuch that δy(0) = δy(1) = 0, gives λ = λ0 + δλ. Then,

δλ =∫

dx [f(y0 + δy)′2 + g(y0 + δy)2]∫dx (y0 + δy)2

−∫

dx (fy′20 + gy20)∫

dx y20

19

Elements of Variational Calculus III

• Taylor expanding for small δy obtain

δλ ≈2

∫dx [δy′fy′0 + δy y0(g − λ0)]∫

dx y20

• Integrating the first term by parts yields

δλ = −2

∫dx δy[(fy′0)

′ + (g − λ0)y0]∫dx y2

0

• If y0 is a solution to the original differential equation, δλ = 0 for an arbitrary δy

20

Elements of Variational Calculus IV

• Can come up with an infinite number of integral relations for λ, which arenot variational

• E.g. multiplying the original differential equation by h(x)y(x) instead of y(x)and integrating over x gives

λ =∫

dx {hfy′2 + [hg − 0.5(fh′)′]y2}∫dxhy2

• This integral relation is not variational since the function that makes δλ = 0does not satisfy the original differential equation unless h(x) = 1

21

Elements of Variational Calculus V

• Since δλ = 0 for the true eigenfunction y0(x), a variational estimate for λobtained with some trial function is more accurate than the trial function itself

• To prove this write y = y0 + δy, with |δy|/|y0| � 1

• Substituting y into the variational expression for λ gives

λ = λ0 +∫

dx [f(δy′)2 + g(δy)2]∫dx y2

0

+ O[(δy)3]

• The estimate λ differs from the true eigenvalue λ0 by a correction of orderO[(δy)2], which is one order more accurate than the trial function

22

Variational Formulation of Linearized Ideal MHDStability Problem I

• Because of self-adjointness of F (ξ) the normal mode formulation of thelinearized ideal MHD stability problem can be easily cast in the variational form(Bernstein et al., 1958)

• Dotting −ω2ρ ξ = F (ξ) with ξ∗ and integrating over the plasma volume yields

ω2 =δW (ξ∗, ξ)K(ξ∗, ξ)

, δW (ξ∗, ξ) ≡ −12

∫dr ξ∗·F (ξ), K(ξ∗, ξ) ≡ 1

2

∫dr ρ|ξ|2

• δW represents the change in potential energy associated with the perturbationand is equal to work done against the force F (ξ) in displacing the plasmaby ξ. K is proportional to the kinetic energy

• The above integral representation is variational

• To prove this let ξ → ξ + δξ, evaluate δω2, and set it to zero

23

Variational Formulation of Linearized Ideal MHDStability Problem II

ω2 + δω2 =δW (ξ∗, ξ) + δW (δξ∗, ξ) + δW (ξ∗, δξ) + δW (δξ∗, δξ)

K(ξ∗, ξ) + K(δξ∗, ξ) + K(ξ∗, δξ) + K(δξ∗, δξ)

• For small δξ,

δω2 =δW (δξ∗, ξ) + δW (ξ∗, δξ)− ω2[K(δξ∗, ξ) + K(ξ∗, δξ)]

K(ξ∗, ξ)

• Using self-adjointness of F and requiring δω2 = 0 gives∫dr {δξ∗ · [F (ξ) + ω2ρ ξ] + δξ · [F (ξ∗) + ω2ρ ξ∗]} = 0

• Since δξ is arbitrary this implies −ω2ρ ξ = F (ξ) ⇒ the formulation isvariational

24

Variational Formulation of Linearized Ideal MHDStability Problem III

• The variational formulation and the normal mode eigenvalue formulation of thelinearized ideal MHD stability problem are equivalent

• Advantages of the variational formulation

– allows use of trial functions to estimate ω2

– can be applied to multidimensional systems efficiently

• Disadvantages of the variational formulation

– still somewhat complicated– gives more information than minimum required

25

Energy Principle: Preliminary Comments

• Growth times of ideal MHD instabilities are short, τinst ∼ r/Vthermal,i . 50 µs

• Experimental times are much longer, τexp > 1 ms

• Ideal MHD instabilities normally terminate the plasma

• Therefore, it is usually much more important to know whether the systemis ideal MHD stable or not and to determine the conditions for avoidinginstabilities, rather than to know precise growth rates (which can easily beestimated)

• In these cases, the variational formulation can be simplified further, giving theEnergy Principle

• The Energy Principle exactly determines stability boundaries but onlyestimates growth rates

26

Energy Principle: Formulation

• Variational formulation:

ω2 =δW

K, δW = −1

2

∫dr ξ∗ · F (ξ), K =

12

∫dr ρ|ξ|2

If all ω2 ≥ 0, then the system is stableIf any ω2 < 0, then the system is unstable

• Energy Principle:

If δW (ξ∗, ξ) ≥ 0 for all allowable ξ, then the system is stableIf δW (ξ∗, ξ) < 0 for any ξ, then the system is unstable

• The underlying physics: a system is unstable if a perturbation can reduce itspotential energy

27

Energy Principle: Proof• Assume that F allows only discreet normal modes, ξn, which form a complete

set of basis functions (see the textbook for a more general proof):

−ω2nρ ξn = F (ξn),

∫dr ρ ξ∗n · ξm = δnm

• Can expand an arbitrary trial function as ξ(r) =∑

n anξn(r)

• Evaluate δW :

δW = −12

∫dr ξ∗ · F (ξ) = −1

2

∑n,m

a∗nam

∫dr ξ∗n · F (ξm)

= −12

∑n,m

a∗nam

∫dr ξ∗n · (−ω2

mρ ξm) =12

∑m

ω2m|am|2

• It a ξ exists that makes δW < 0 ⇒ at least one ω2m < 0 ⇒ instability

• If all ξ make δW > 0 ⇒ all ω2m > 0 ⇒ stability

28

Extended Energy Principle I

• The Energy Principle calculated with

F (ξ) =14π

(∇×Q)×B0+14π

B0×(∇×Q)+∇(ξ·∇p0+γp0∇·ξ), Q ≡ ∇×(ξ×B0) = B1

is valid both when the plasma is surrounded by a conducting wall and when it isisolated from this wall by a vacuum region

• In the first case application of the Energy Principle is straightforward since theonly boundary condition for ξ is (n · ξ⊥)|rwall

= 0

• When the vacuum region is present, direct application of this form of theEnergy Principle is cumbersome because

– boundary conditions for ξ on the plasma surface are complicated– this form of δW is not very intuitive since the vacuum contribution does not

appear explicitly

29

Extended Energy Principle II• After a lengthy calculation can rewrite δW in the following intuitive form

δW = δWF︸︷︷︸fluid energy

+ δWS︸︷︷︸surface energy

+ δWV︸ ︷︷ ︸vacuum energy

• The surface energy, δWS =12

∫S

dS |n · ξ⊥|2n ·[[

∇(

p +B2

8π

)]]

• [[q]] denotes a jump in q from vacuum to plasma; δWS = 0 unless plasmasurface currents are present

• The vacuum energy, δWV =12

∫V

dr|B1|2

4π

• Vacuum magnetic field is obtained from ∇× B1 = ∇ · B1 = 0, subject to theboundary conditions

(n·B1)|rwall= 0, (n·B1)|rplasma

= {B1·∇(n·ξ⊥)−(n·ξ⊥)(n·∇B1·n)}|rplasma

30

Extended Energy Principle III

• The fluid energy,

δWF =12

∫P

dr

[|Q⊥|2

4π+

B2

4π|∇ · ξ⊥ + 2κ · ξ⊥|2 + γp|∇ · ξ|2

−2(∇p · ξ⊥)(κ · ξ∗⊥)− J‖(ξ∗⊥ × b ·Q⊥)

]• Individual terms have the following physical interpretation:

– |Q⊥|2 represents magnetic field bending energy ⇒ stabilizing– |∇ · ξ⊥ + 2κ · ξ⊥|2 represents magnetic field compression energy ⇒

stabilizing– γp|∇ · ξ|2 represents plasma compression energy ⇒ stabilizing– ∇p represents pressure gradient instability drive ⇒ can be destabilizing– J‖ represents parallel current instability drive ⇒ can be destabilizing

31

Incompressibility I

• ξ‖ only contributes to δW through the term δW‖ ≡∫

Pdr γp|∇ · ξ|2

• Let’s minimize δW‖ with respect to ξ‖ by allowing ξ‖ → ξ‖ + δξ‖ and thensetting the variation of δW‖ to zero:

δ(δW‖) =∫

P

dr γp(∇ · ξ)∇ ·(

δξ‖B

B

)= −

∫P

drδξ‖

BB ·∇[γp(∇ · ξ)]

= −∫

P

drδξ‖

Bγ p B ·∇(∇ · ξ)

• The boundary term vanishes since n ·B = 0 on the plasma surface. We alsoused the fact that B ·∇p = 0

• The minimizing condition is B ·∇(∇ · ξ) = 0

32

Incompressibility II

• For most configurations the operator B ·∇ is nonsingular and the minimizing

condition becomes ∇ · ξ = 0

• The most unstable modes are incompressible ⇒ δW‖ =∫

Pdr γp|∇ · ξ|2 = 0

• There are two cases when ∇ · ξ cannot be set to zero

– system possesses a special symmetry (B ·∇ vanishes identically for certainspecific modes)

– closed field line systems (B ·∇ is not singular but a periodicity constraintmust be taken into account)

33

Special Symmetry: m = 0 mode in Z-pinch

• B = Bθ(r)θ, ξ = ξ(r) exp[i(mθ + kz)]

∇ · ξ = ∇ · ξ⊥ + B ·∇(

ξ‖

B

)= ∇ · ξ⊥ +

Bθ

r

∂

∂θ

(ξ‖

Bθ

)= ∇ · ξ⊥ +

imξ‖

r

• For m = 0 ξ‖ does not appear, so that ∇ · ξ = ∇ · ξ⊥ and

δW‖ =∫

P

dr γ p |∇·ξ⊥|2

34

Close Field Line Systems

• Have to satisfy a periodicity constraint on each field line, ξ‖(l) = ξ‖(l + L),where l is arc length and L is the total length of a given line

• Need to add a homogeneous solution when solve B ·∇(∇ · ξ) = 0:

∇ · ξ = F (p) ⇒ξ‖

B= −

∫ l

0

dl

B∇ · ξ⊥ + F (p)

∫ l

0

dl

B

• Then, periodicity requires

F (p) = 〈∇ · ξ⊥〉 ≡∮

dlB∇ · ξ⊥∮

dlB

• Consequently, δW‖ =12

∫P

dr γ p | 〈∇ · ξ⊥〉 |2

35

Classification of MHD Instabilities• Internal or fixed boundary modes. Plasma surface is not perturbed,

(n · ξ⊥)|rplasma= 0. Equivalent to moving conducting wall onto the plasma

surface

• External or free boundary modes. Plasma surface is allowed to move,(n · ξ⊥)|rplasma

6= 0. Often the most severe stability limitations

• Current driven (or kink) modes. J‖ is the dominant destabilizing term.Important in tokamaks, RFPs (Kruskal-Shafranov instability, sawtoothoscillations, disruptions). In general, the modes have long wavelengths, low m,n. Cure: tight aspect ratio, low current, peaked current profiles, conductingwall.

• Pressure driven modes. ∇p is the dominant destabilizing term. Special cases:interchange or flute modes, ballooning modes, sausage instability. Important intokamaks, stellarators, RFPs, mirrors. Normally have long parallel and shortperpendicular wavelengths. Cure: low β, magnetic field shear, average favorablecurvature.

36

Summary I

• We reviewed basics of ideal MHD stability theory

• A plasma is said to be exponentially stable if Im(ω) ≤ 0 for every discreetnormal mode of the system

• General 3D stability problem is first formulated as an initial value problem andthen transformed into a normal mode eigenvalue problem: −ω2ρ ξ = F (ξ)

• The force operator F is self-adjoint, so that ω2 is purely real. ω2 > 0 indicatesstability, ω2 < 0 – instability

• Using self-adjointness of F can recast the normal mode eigenvalue problem intothe form of a variational principle

• If only stability boundaries are required, can obtain from the variational principlea powerful minimizing principle, known as the Energy Principle. The system isstable if and only if δW ≥ 0 for all allowable plasma displacements

37

Summary II

• Can minimize δW once and for all with respect to ξ‖ for arbitrary geometries.For most systems this leads to ∇ · ξ = 0, i.e. the most unstable perturbationsare incompressible

• Based on this analysis, ideal MHD instabilities can be conveniently separatedinto internal (fixed boundary) or external (free boundary), and current driven(kinks) of pressure driven

38

Example: Stability of θ Pinch I

• B = Bz(r)z, p = p(r)

• Equilibrium

– ∇ ·B = 0 is satisfied identically– Ampere’s law gives J = Jθ(r)θ = −(c/4π)dBz(r)/dr– pressure balance gives d(p + B2

z/8π)/dr = 0

p(r) + Bz(r)2

8π = B20

8π

– B0 is the externally applied field

• Equilibrium is independent of θ and z ⇒ can write plasma displacement as

ξ(r) = ξ(r) exp[i(mθ + kz)] , with m and k poloidal and axial wave numbers

• ∇ · ξ = 0 ⇒ ξ‖ ≡ ξz = (i/kr)[(rξr)′ + imξθ] ⇒ as long as k 6= 0 the most

unstable perturbations are incompressible

39

Example: Stability of θ Pinch II

• Next, evaluate δWF

• It is easy to show that

– Q⊥ = ikBzξ⊥ = ikBz(ξrr + ξθθ) ⇒ |Q⊥|2 = k2B2z(|ξr|2 + |ξθ|2)

– ∇ · ξ⊥ = 1r(rξr)′ + im

r ξθ ⇒|∇ · ξ⊥|2 = 1

r2|(rξr)′|2 + m2

r2 |ξθ|2 + imr2 [(rξ∗r)′ξθ − (rξr)′ξ∗θ ]

– κ = 0– J‖ = 0

• Then (Z is the length of the plasma column, a is its radius),

δWF =Z

4

∫ a

0

dr W (r)r,

W (r) = B2z

{k2(|ξr|2 + |ξθ|2) +

|(rξr)′|2 + m2|ξθ|2 + im[(rξ∗r)′ξθ − (rξr)′ξ∗θ ]r2

}40

Example: Stability of θ Pinch III

• ξθ appears only algebraically

• use identity |a ξθ + b|2 = a2|ξθ|2 + a(b∗ξθ + b ξ∗θ) + |b|2, valid for arbitrary real aand complex b and ξθ, to combine all ξθ terms by completing the squares asfollows (k2

0 ≡ k2 + m2/r2):

W (r) = B2z

{∣∣∣∣k0ξθ −im

k0r2(rξr)′

∣∣∣∣2 +k2

k20r

2

[|(rξr)′|2 + k2

0r2|ξr|2

]}

• ξθ appears only in the first term. Since this term is positive, its minimum valueis zero. So, δWF is minimized by choosing

ξθ =im

k2r2 + m2(rξr)′

41

Example: Stability of θ Pinch IV

• Then,

δWF =Z

4

∫ a

0

dr rk2B2

z

k2r2 + m2

[|(rξr)′|2 + (k2r2 + m2)|ξr|2

]

• δWF > 0 for any k2 > 0 and δWF → 0 as k2 → 0

• So, for any β, θ pinch is positively stable for perturbations with finite axialwavelengths and marginally stable for perturbations with very long axialwavelengths

• Inclusion of the vacuum term removes the restriction that ξr(a) = 0 butdoes not alter the stability: δWF remains positive, δWV ≥ 0 by definition,and δWS = 0 for profiles that decay smoothly to zero at r = a

42

Example: Stability of θ Pinch V

• Physical explanation for the stability:

– J‖ = 0 ⇒ no current driven modes– κ = 0 ⇒ no pressure driven modes– any perturbation bends or compresses B ⇒ both are stabilizing influences

43

Homework: Stability of Z Pinch

• Use Ampere’s law and pressure balance equation to obtain a Z pinch equilibriumcondition

• Use energy principle to analyze stability of this equilibrium

– write the displacement as ξ(r) = ξ(r) exp[i(mθ + kz)]– consider cases of m = 0 and m 6= 0 modes separately– show that m 6= 0 are incompressible, while compressibility must be retained

for m = 0 modes– in each case, white δWF and show that ξz enters only algebraically– complete squares with respect to ξz and obtain expressions for ξz that

minimize δWF

– observe, that for the m 6= 0 case the most unstable modes have k →∞, andtake this limit

– obtain necessary and sufficient stability conditions for m 6= 0 modes[8πrp′ + m2B2

θ > 0] and m = 0 modes [−rp′/p < 2γB2θ/(4γp + B2

θ)]– come up with a physical picture for the instability

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