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Let $L$ be a core-scale length, $B_{\ast }$ a representative strength of the prescribed poloidal magnetic field, and $v_A=B_{\ast }/\sqrt{{\mu }_0\rho }$ the Alfvén speed. A consistent weakly damped, small-amplitude ordering isThe first ratio is the Lehnert number. It makes the wave slow relative to rotation and allows almost geostrophic flow, with the azimuthal velocity approximately constant along a rotation-axis cylinder. The small wave amplitude gives $|B_{\varphi }|\sim \sqrt{{\mu }_0\rho }|U|\ll B_{\ast }$. Poloidal advection then changes neither the background field nor the wave substantially over one wave period. If there are no-slip boundary layers, also require their damping rate $\sqrt{\nu \Omega }/L\ll \sigma$, or otherwise retain their torque; a small bulk viscosity alone does not remove that boundary torque.

In the rotating magnetohydrodynamic momentum equation, axisymmetry removes the azimuthal pressure derivative. The neglected nonlinear azimuthal advection has size $|{\mathbf{u}}_p|U/L$, and the neglected viscous term has size $\nu U/L^2$. The retained azimuthal Coriolis force is $2\Omega u_r$. A typical compatible local balance has $|u_r|\sim \sigma U/\Omega$, small compared with $U$ when the Lehnert number is small. The azimuthal Lorentz force density can be written using $\nabla \cdot {\mathbf{B}}_p=0$:The physical prefactor is $1/({\mu }_0\rho )$: the extra $i$ printed inside the first reduced momentum equation is a typographical error, also contradicted by the subsequent real-valued system.

In the resistive induction equation, neglecting poloidal advection of the perturbation and magnetic diffusion leaves the winding of the prescribed poloidal magnetic field by differential rotation. ThusThis is the small-amplitude torsional Alfvén wave approximation, with a time-independent background and no external azimuthal forcing.

We now average over the full height of a rotation-axis cylinder. The sphere is impenetrable. Integrating incompressible flow over the volume inside any such cylinder gives zero flux through its cylindrical side, since there is no flux through its spherical caps. ThereforeThis is why the Coriolis force disappears from the cylinder average; it need not vanish locally. An axisymmetric field in the insulating exterior is a potential field with $B_{\varphi }=0$, so continuity gives $B_{\varphi }=0$ on the sphere. Consequently both the end flux terms and the moving-endpoint terms in the magnetic-stress integral vanish. Expanding the divergence givesSince $U$ is independent of $z$, its integrated time derivative is $2z_b{U}_t$. Defining $T={\int }_{-z_b}^{z_b}{B}_r{B}_{\varphi }dz$ givesThe stationary background and the fact that $U/r$ depends only on $r,t$ also givesoTogether these form the cylinder-averaged torsional Alfvén wave system.

Assume $H>0$ on the cylinders carrying the wave, with regular endpoint limits. If $H$ vanishes on an interior cylinder, then $B_r$ and $T$ vanish there and that cylinder has no radial magnetic coupling; the displayed $H^{-1}$ energy term must be interpreted on the coupled region rather than literally dividing by zero. Since $H$ is time independent, differentiation of the proposed conserved energy of a cylinder-averaged torsional wave givesRegularity at the rotation axis makes the lower endpoint vanish, while vanishing magnetic torque at the collapsed outer cylinder makes the upper endpoint vanish. This provesThe magnetic term has a direct interpretation. Decompose $B_{\varphi }=(T/H)B_r+B_{\varphi }^{\perp }$, with $\int B_r{B}_{\varphi }^{\perp }dz=0$. Then $\int B_{\varphi }^{2}dz=T^2/H+\int (B_{\varphi }^{\perp }{)}^{2}dz$, and the induction equation gives $(B_{\varphi }^{\perp }{)}_t=0$. Thus, apart from a common dimensional factor and a stationary orthogonal magnetic component, the conserved quantity is the physical kinetic plus magnetic energy.

For the frequency estimate, put $V=U/r$ and eliminate $T$:Locally this is a wave equation with speedIt is the cylinder-averaged radial field component that matters, not simply the dipole measured at the Earth's surface. Taking a cylinder height of order $6\times 10^6\mathrm{m}$, $\rho \sim 10^4\mathrm{kg}{\mathrm{m}}^{-3}$, and an assumed internal cylindrical radial field $B_{r,\mathrm{rms}}\sim 0.2$--$2\mathrm{mT}$ givesFor a standing wave of radial length $L_c\sim 2$--$3.5\times 10^6\mathrm{m}$, the cyclic frequency is of order $c_T/(2L_c)$ and the angular frequency is of order $\pi c_T/L_c$. HenceGeometry and the assumed internal field can shift the estimate substantially. As a concrete calibration, the 2010 torsional-wave study inferred a cylindrical radial rms field near $2\mathrm{mT}$ and a roughly four-year crossing time, with an observed six-year recurrence. A much weaker assumed interior field instead gives the older multi-decadal estimate. The paper supplies no numerical interior field strength, so a unique numerical frequency cannot be deduced without such an assumption.