Lecture 38 Self-Adjointness of the Ideal-MHD Force Operator

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Lecture 38
Self-Adjointness of the Ideal-MHD Force Operator

Overview

This appendix explains why the static ideal-MHD force operator is self-adjoint and why that fact matters. The point is not merely formal. Self-adjointness is the operator statement behind the energy principle of Lecture 15: it is what guarantees that the quadratic form \(\delta W\) is real, that \(\omega ^2\) is real, and that distinct normal modes are orthogonal in the natural density-weighted inner product.
There is also an important caution. A self-adjoint ideal-MHD operator can still possess a continuum spectrum. The Alfvén continuum is not a failure of Hermiticity; it is a consequence of singular coefficients in the reduced wave equation.

Historical Perspective

The self-adjoint structure of static ideal MHD was clarified in the classic work of Bernstein, Frieman, Kruskal, and Kulsrud, together with Kruskal and Oberman, and then sharpened geometrically by Newcomb’s field-line formulation Bernstein et al. (1958); Kruskal and Oberman (1958); Newcomb (1960). One of the deepest later insights was that this clean variational structure belongs specifically to static ideal MHD. Frieman and Rotenberg showed how the problem changes in the presence of equilibrium flow, where gyroscopic terms spoil the simple Hermitian form Frieman and Rotenberg (1960). This appendix is therefore the mathematical companion to the energy-principle lecture and to the later appendix on flow and moving boundaries.

Caution

It is very easy to start from \(\int \vect {\eta }^*\!\cdot \vect {F}(\vect {\xi })\,dV\) and get lost in apparently asymmetric intermediate steps. The right object to study is not the unsymmetrized force term by itself, but the bilinear form obtained by polarizing the quadratic functional \(\delta W[\vect {\xi }]\). Once that is done, the symmetry becomes transparent.

38.1 The operator statement and the natural inner products

In Lecture 15 the linearized equation of motion for a static ideal-MHD equilibrium was written as

\vect {F}(\vect {\xi }) = -\omega ^2\rho \vect {\xi }, \tag{D.1}

which is the normal-mode form of Eq. (15.13). The force operator \(\vect {F}\) is linear in the displacement \(\vect {\xi }\).

There are really two inner products in the story.

The ordinary \(L^2\) inner product. The operator itself is symmetric in the ordinary volume inner product

\langle \vect {\eta },\vect {\xi }\rangle \equiv \int _P \vect {\eta }^{*}\cdot \vect {\xi }\,dV, \tag{D.2}

provided the usual ideal-MHD boundary conditions remove the surface terms.

The density-weighted inner product. Because the eigenvalue problem is generalized, with the positive weight \(\rho \) on the right-hand side of (D.1), the mode orthogonality is naturally expressed in

\langle \vect {\eta },\vect {\xi }\rangle _\rho \equiv \int _P \rho \,\vect {\eta }^{*}\cdot \vect {\xi }\,dV. \tag{D.3}

Tutorial

Why the problem is a generalized Hermitian eigenvalue problem. The operator statement \[ \vect {F}(\vect {\xi })=-\omega ^2\rho \vect {\xi } \] looks slightly different from the textbook form \(\mathcal L\xi =\lambda \xi \) because the inertia is carried by \(\rho \). The clean way to think about it is:
1.
\(\vect {F}\) is symmetric in the ordinary \(L^2\) product (D.2);
2.
the eigenvalues are \(-\omega ^2\);
3.
orthogonality of distinct eigenfunctions is with respect to the positive weight \(\rho \), namely (D.3).

So the static ideal-MHD normal-mode problem is the plasma analogue of a generalized Hermitian matrix problem \(Kx=\lambda Mx\), with stiffness matrix \(K\sim -\vect {F}\) and mass matrix \(M\sim \rho \).

38.2 The clean proof: self-adjointness from the energy principle

The most economical proof starts from the energy principle itself. In Lecture 15, the bilinear form was defined by

\delta W(\vect {\eta },\vect {\xi }) \equiv -\frac 12\int _P \vect {\eta }^{*}\cdot \vect {F}(\vect {\xi })\,dV. \tag{D.4}

The actual potential energy is the quadratic functional \(\delta W[\vect {\xi }]\equiv \delta W(\vect {\xi },\vect {\xi })\), as in Eq. (15.15).

Step 1: polarize the quadratic form. To keep the algebra transparent, suppress complex conjugation for one moment and work with real trial displacements. Let

\mathcal W(\alpha ,\beta ) \equiv \delta W[\alpha \vect {\xi }+\beta \vect {\eta }], \tag{D.5}

where \(\alpha \) and \(\beta \) are independent real parameters. Because \(\delta W\) is quadratic,

\mathcal W(\alpha ,\beta ) = \alpha ^2\delta W[\vect {\xi }] + 2\alpha \beta \,\delta W(\vect {\eta },\vect {\xi }) + \beta ^2\delta W[\vect {\eta }]. \tag{D.6}

Therefore

\delta W(\vect {\eta },\vect {\xi }) = \frac 12\left .\frac {\partial ^2 \mathcal W}{\partial \alpha \,\partial \beta }\right |_{\alpha =\beta =0}. \tag{D.7}

Since mixed partial derivatives commute,

\delta W(\vect {\eta },\vect {\xi }) = \delta W(\vect {\xi },\vect {\eta }). \tag{D.8}

For Fourier amplitudes, one restores the Hermitian form simply by replacing one factor with its complex conjugate:

\delta W(\vect {\eta },\vect {\xi }) = \delta W(\vect {\xi },\vect {\eta })^{*}. \tag{D.9}

Step 2: translate symmetry of \(\delta W\) into symmetry of \(\vect {F}\). Using (D.4), Eq. (D.9) becomes

-\frac 12\int _P \vect {\eta }^{*}\cdot \vect {F}(\vect {\xi })\,dV = -\frac 12\int _P \vect {F}(\vect {\eta })^{*}\cdot \vect {\xi }\,dV. \tag{D.10}

Hence

\boxed { \int _P \vect {\eta }^{*}\cdot \vect {F}(\vect {\xi })\,dV = \int _P \vect {F}(\vect {\eta })^{*}\cdot \vect {\xi }\,dV. } \tag{D.11}

This is the self-adjointness statement already quoted in Eq. (15.51); here we have seen exactly where it comes from. The key point is that the force operator is the Fréchet derivative of a scalar potential-energy functional.

Why this proof is the right one. One can certainly manipulate the linearized force term by term. But that route is messy because individual intermediate contributions do not look symmetric until they are combined correctly. The variational proof avoids that clutter and makes the physics clearer: static ideal MHD is self-adjoint because it is a small-oscillation problem around an equilibrium of a potential-energy functional.

38.3 A direct fixed-boundary bilinear form

The variational proof is the clean one, but it is also helpful to see the bilinear form explicitly. For fixed-boundary perturbations with \(\xi _n=0\) at the plasma surface and with gravity temporarily set to zero, Eq. (15.25) reduces to the plasma volume integral

2\delta W[\vect {\xi }] = \int _P dV\left [ \frac {|\vect {Q}|^2}{\muo } - \vect {\xi }\cdot (\J \times \vect {Q}) + \gamma p\,(\divergence \vect {\xi })^2 + (\vect {\xi }\cdot \grad p)\,\divergence \vect {\xi } \right ], \tag{D.12}

where

\vect {Q}\equiv \curl (\vect {\xi }\times \B ). \tag{D.13}

Polarizing (D.12) gives

\begin{aligned}2\delta W(\vect {\eta },\vect {\xi }) = \int _P dV\Bigg [ &\frac {\vect {Q}_{\eta }^{*}\cdot \vect {Q}_{\xi }}{\muo } - \frac 12\,\vect {\eta }^{*}\cdot (\J \times \vect {Q}_{\xi }) - \frac 12\,\vect {\xi }\cdot (\J \times \vect {Q}_{\eta }^{*}) \nonumber \\ &+ \gamma p\,(\divergence \vect {\eta }^{*})(\divergence \vect {\xi }) + \frac 12(\vect {\eta }^{*}\cdot \grad p)\,\divergence \vect {\xi } + \frac 12(\vect {\xi }\cdot \grad p)\,\divergence \vect {\eta }^{*} \Bigg ].\end{aligned} \tag{D.14}

Now the symmetry is manifest: under the interchange \(\vect {\eta }\leftrightarrow \vect {\xi }\) followed by complex conjugation, every term in (D.14) is unchanged.

Magnetic term in one line. The positive-definite line-bending term comes from a single integration by parts. Starting from the first part of the magnetic force,

\begin{aligned}-\int _P \vect {\eta }^{*}\cdot \frac {1}{\muo }(\curl \vect {Q}_{\xi })\times \B \,dV &= -\frac {1}{\muo }\int _P (\curl \vect {Q}_{\xi })\cdot (\B \times \vect {\eta }^{*})\,dV \nonumber \\ &= \frac {1}{\muo }\int _P \vect {Q}_{\xi }\cdot \curl (\vect {\eta }^{*}\times \B )\,dV \nonumber \\ &= \frac {1}{\muo }\int _P \vect {Q}_{\eta }^{*}\cdot \vect {Q}_{\xi }\,dV,\end{aligned} \tag{D.15}

where the boundary term vanishes under the fixed-wall ideal boundary condition. This is the clean source of the \(|\vect {Q}|^2\) term in the quadratic form.

If gravity is retained. In the interchange lectures one keeps the term involving \(\grad \Phi _g\). Its polarization is also symmetric:

\begin{aligned}2\delta W_g(\vect {\eta },\vect {\xi }) = -\int _P dV\Bigg [ &\frac 12(\vect {\eta }^{*}\cdot \grad \Phi _g)\,\divergence (\rho \vect {\xi }) \nonumber \\ &+ \frac 12(\vect {\xi }\cdot \grad \Phi _g)\,\divergence (\rho \vect {\eta }^{*}) \Bigg ].\end{aligned} \tag{D.16}

So gravity changes the stability physics, but it does not destroy self-adjointness.

38.4 Consequences: real \(\omega ^2\), orthogonality, and continua

Reality of \(\omega ^2\). Take the ordinary inner product of (D.1) with \(\vect {\xi }\):

\int _P \vect {\xi }^{*}\cdot \vect {F}(\vect {\xi })\,dV = -\omega ^2\int _P \rho |\vect {\xi }|^2\,dV. \tag{D.17}

Complex conjugation gives

\int _P \vect {F}(\vect {\xi })^{*}\cdot \vect {\xi }\,dV = -\omega ^{*2}\int _P \rho |\vect {\xi }|^2\,dV. \tag{D.18}

Using (D.11), the left-hand sides are equal, so

\omega ^{*2}=\omega ^2. \tag{D.19}

Thus \(\omega ^2\) is real. The sign of \(\omega ^2\) then decides whether the response is oscillatory, marginal, or exponentially growing, exactly as discussed in Lecture 15.

Orthogonality of distinct eigenmodes. Let \(\vect {\xi }_m\) and \(\vect {\xi }_n\) satisfy

\vect {F}(\vect {\xi }_n)=-\omega _n^2\rho \vect {\xi }_n, \qquad \vect {F}(\vect {\xi }_m)=-\omega _m^2\rho \vect {\xi }_m. \tag{D.20}

Multiply the first equation by \(\vect {\xi }_m^{*}\), integrate over volume, and then use self-adjointness to move \(\vect {F}\) onto \(\vect {\xi }_m\):

\begin{aligned}-\omega _n^2\int _P \rho \,\vect {\xi }_m^{*}\cdot \vect {\xi }_n\,dV &= \int _P \vect {\xi }_m^{*}\cdot \vect {F}(\vect {\xi }_n)\,dV \nonumber \\ &= \int _P \vect {F}(\vect {\xi }_m)^{*}\cdot \vect {\xi }_n\,dV \nonumber \\ &= -\omega _m^2\int _P \rho \,\vect {\xi }_m^{*}\cdot \vect {\xi }_n\,dV.\end{aligned} \tag{D.21}

Therefore

(\omega _n^2-\omega _m^2) \int _P \rho \,\vect {\xi }_m^{*}\cdot \vect {\xi }_n\,dV = 0. \tag{D.22}

If \(\omega _n^2\neq \omega _m^2\), then

\boxed { \int _P \rho \,\vect {\xi }_m^{*}\cdot \vect {\xi }_n\,dV = 0. } \tag{D.23}

So distinct ideal-MHD eigenmodes are orthogonal in the density-weighted inner product (D.3).

Why a continuum does not contradict self-adjointness. Self-adjointness does not mean the spectrum must be purely discrete. A Hermitian differential operator may still have a continuous spectrum when the effective radial equation becomes singular or when the domain is unbounded. In ideal MHD the classic example is the Alfvén continuum: at a resonant surface the coefficient of the highest radial derivative vanishes, and the local solutions become singular even though the underlying operator remains Hermitian. The toroidal Alfvén-eigenmode lecture is therefore perfectly consistent with the present appendix: the continuum comes from singular structure, not from any failure of self-adjointness.

Where the symmetry is lost. The clean Hermitian structure belongs to static ideal MHD. With equilibrium flow, the Frieman–Rotenberg problem acquires gyroscopic terms that are antisymmetric rather than symmetric Frieman and Rotenberg (1960). With resistivity, the induction equation becomes diffusive and the eigenvalue problem becomes non-Hermitian. With kinetic closures, resonant denominators and frequency-dependent operators replace the simple potential-energy structure. That is why the energy principle is so powerful for ideal problems and so limited once dissipation, flow, or resonances become important.

Takeaways

There are three equivalent ways to remember this appendix.
1.
Static ideal MHD is a small-oscillation problem about an equilibrium of a scalar potential-energy functional.
2.
Because \(\delta W\) is a symmetric quadratic form, the linearized force operator is self-adjoint.
3.
Self-adjointness implies real \(\omega ^2\), \(\rho \)-weighted orthogonality of distinct modes, and a clean variational stability theory — but it does not exclude the existence of continua.

That is the mathematical backbone of the ideal-MHD stability lectures.

Bibliography

I. B. Bernstein, E. A. Frieman, M. D. Kruskal, and R. M. Kulsrud. An energy principle for hydromagnetic stability problems. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 244(1236):17–40, 1958. doi:10.1098/rspa.1958.0023.

M. D. Kruskal and C. R. Oberman. On the stability of plasma in static equilibrium. Physics of Fluids, 1(4):275–280, 1958. doi:10.1063/1.1705885.

William A Newcomb. Hydromagnetic stability of a diffuse linear pinch. Annals of Physics, 10 (2):232–267, 1960. doi:10.1016/0003-4916(60)90023-3.

E. Frieman and Manuel Rotenberg. On hydromagnetic stability of stationary equilibria. Reviews of Modern Physics, 32(4):898–902, 1960. doi:10.1103/RevModPhys.32.898.