Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 64 / 4

The equilibrium quantities $\rho ,p,\mathbf{B},\Phi$ are real, and the gravitational Hessian ${\Phi }_{,ij}={\partial }_i{\partial }_j\Phi$ is symmetric. The crucial tensor property isThis follows directly from the displayed expression for $V$: the products ${\delta }_{ij}{\delta }_{kl}$ and ${\delta }_{il}{\delta }_{jk}$ are unchanged by interchanging the pairs $(i,j)$ and $(k,l)$; $B_j{B}_l{\delta }_{ik}$ is also unchanged; and the two terms involving $B_i{B}_j{\delta }_{kl}$ and $B_k{B}_l{\delta }_{ij}$ exchange with each other. All coefficients are real.

Multiply $F\mathbf{\xi }$ by $\rho {\mathbf{\eta }}^{\ast }$ and integrate by parts over all space. For admissible displacement fields, the boundary contribution at infinity vanishes: the fluid pressure and density are confined to the body, and the magnetic stress decays in the surrounding conducting medium. The conducting exterior is included in the integration. For a piecewise description, the usual displacement and traction matching conditions cancel the two opposite interface contributions; alternatively, take a smooth transition and then its limit. ThusThe first term is a Hermitian pairing because ${\Phi }_{,ij}={\Phi }_{,ji}$, and the second is Hermitian because $V_{ijkl}=V_{klij}$. Taking the complex conjugate of the same expression with $\mathbf{\eta }$ and $\mathbf{\xi }$ interchanged and relabelling $(i,j)\leftrightarrow (k,l)$ reproduces it. HenceThis establishes the required self-adjoint operator identity in the mass-weighted inner product, on the physical displacement domain with the stated decay and interface conditions. The negligible-density exterior can be treated as a small-positive-density conducting limit when defining this weighted pairing; its magnetic contribution is retained even as its inertia becomes negligible.

Self-adjointness gives $⟨\mathbf{\xi },F\mathbf{\xi }⟩=⟨F\mathbf{\xi },\mathbf{\xi }⟩=⟨\mathbf{\xi },F\mathbf{\xi }{⟩}^{\ast }$, soThe integrated expression from part (a) also writes it explicitly asThis is the quadratic change in potential energy associated with the perturbation, including pressure, magnetic and imposed-gravity restoring effects. Its sign need not be positive term by term.

The perturbation kinetic energy is $T=\frac{1}{2}⟨\dot{\mathbf{\xi }},\dot{\mathbf{\xi }}⟩$, and the total quadratic perturbation energy isSince the equilibrium and $F$ are time independent, differentiating and using the self-adjoint operator identity givesThus $\boxed{dE/dt=0}$ along the linearized motion. This conserved kinetic-plus-restoring energy is the basis of the magnetohydrodynamic energy principle.

First suppose $W[\mathbf{\xi }]>0$ for every nonzero admissible displacement. The conserved energy from part (b) then bounds both $T$ and $W$ by $E$. In particular,where $\Vert \mathbf{\xi }{\Vert }_{\rho }^2=⟨\mathbf{\xi },\mathbf{\xi }⟩$. The positive quadratic form $\Vert \dot{\mathbf{\xi }}{\Vert }_{\rho }^2+2W[\mathbf{\xi }]$ defines the perturbation energy norm and is conserved. Small perturbations in this norm remain small, proving energetic linear stability. Also $\Vert \mathbf{\xi }(t){\Vert }_{\rho }\le \Vert \mathbf{\xi }(0){\Vert }_{\rho }+\sqrt{2E}t$, excluding exponential displacement growth. For any admissible normal mode $\mathbf{\xi }=\mathbf{\zeta }{e}^{-i\omega t}$, multiplication of $-{\omega }^2\mathbf{\zeta }=F\mathbf{\zeta }$ by ${\mathbf{\zeta }}^{\ast }$ gives directlyso all such frequencies are real. No variational principle has been assumed.

To prove instability, suppose some admissible ${\mathbf{\xi }}_0$ has $W_0=W[{\mathbf{\xi }}_0]<0$, and put $N_0=\Vert {\mathbf{\xi }}_0{\Vert }_{\rho }^2>0$. Choose the initial velocityIts kinetic energy exactly cancels $W_0$, so $E=0$ for the entire subsequent motion. Write $N(t)=\Vert \mathbf{\xi }(t){\Vert }_{\rho }^2$. Since $F$ is self-adjoint,Zero energy gives $⟨\mathbf{\xi },F\mathbf{\xi }⟩=\Vert \dot{\mathbf{\xi }}{\Vert }_{\rho }^2$, and hence $N^{\prime \prime }=4\Vert \dot{\mathbf{\xi }}{\Vert }_{\rho }^2\ge 0$. Initially $N^{\prime }(0)=2\alpha N_0>0$, so $N$ stays positive and its logarithm is defined. The Cauchy-Schwarz inequality now givesTherefore $(\ln N{)}^{\prime }(t)\ge (\ln N{)}^{\prime }(0)=2\alpha$, and integration provesThis is an exponentially growing perturbation. Scaling the initial displacement and velocity down leaves $\alpha$ unchanged, so arbitrarily small initial perturbations display the same linear growth mechanism. It proves that a negative-energy direction gives exponential growth in a self-adjoint wave equation, without assuming a discrete spectrum or quoting a trial-function variational theorem.

Positive $W$ on nonzero displacements gives energetic stability; a negative direction gives instability. For mathematical precision, a uniform bound on the mass-weighted displacement norm follows if $W\ge \frac{1}{2}{\omega }_{\min }^2\Vert \mathbf{\xi }{\Vert }_{\rho }^2$ for some ${\omega }_{\min }>0$, giving $\Vert \mathbf{\xi }{\Vert }_{\rho }^2\le 2E/{\omega }_{\min }^2$. Strict positivity alone in an infinite-dimensional space need not supply such a uniform spectral gap; the standard magnetohydrodynamic energy principle asserts energetic stability and absence of growing modes. Cases with zero directions are marginal and require separate consideration.