Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 67 (Stellar and Planetary Magnetic Fields)

Use a planar flux function $F=B_{0}z+a$ and writeThis automatically satisfies the solenoidal magnetic-field constraint. There is a sign convention to settle: with the usual definition $\mathbf{B}=\nabla \times \mathbf{A}$, the magnetic vector potential is $\mathbf{A}=-F\hat{\mathbf{y}}$. The positive sign printed for $A_y$ instead gives the opposite field, or corresponds to defining $\mathbf{B}=-\nabla \times \mathbf{A}$. We use the standard curl convention and the flux $F$ oriented along the prescribed positive $B_0$; the flux evolution and all transport identities below are unchanged by reversing the overall potential convention.

For planar incompressible flow, substitute this representation into the resistive induction equation. The two in-plane components implyAn additive time-dependent constant in $F$ leaves the field unchanged and removes $C(t)$. Since ${\nabla }^2(B_{0}z)=0$, this givesThis is the scalar advection-diffusion equation underlying the planar anti-dynamo theorem, with the imposed-field source retained.

Put $M(z)=\bar{a}(z)$, $a=M+a^{\prime }$ and $F_c(z)=\overline{w{a}^{\prime }}$. Stationarity of the horizontal averages, $\bar{w}=0$, and incompressibility giveHere the horizontal derivative of the advective flux has zero average. Integration gives $F_c=\eta M^{\prime }+C$. Averaging once more, with $⟨M^{\prime }⟩=0$, identifies $C=⟨w{a}^{\prime }⟩$. ThusDecay or periodic averaging removes the endpoint contribution from the mean derivative. The periodic assumption used later replaces decay at vertical infinity; a nonzero periodic function cannot also decay there.

Subtract the averaged equation from the scalar equation and multiply by $a^{\prime }$. The average of the time derivative is zero by stationarity; the advective term becomes a boundary flux and vanishes by incompressibility and the stated boundary averaging. Terms depending only on $z$ multiply $\overline{a^{\prime }}=0$. Integration by parts in the diffusion term therefore givesSubstituting $F_c=\eta M^{\prime }+C$ and using $⟨M^{\prime }⟩=0$ yields the Zeldovich planar flux balanceBoth terms on the right are dissipative. In a nontrivial stationary forced problem this requires transport opposing the imposed flux gradient.

For finite-energy localized fields, enforcing decay at both vertical infinities literally gives $C=0$ from the mean-flux boundary values; the corresponding integral balance then forces both dissipation terms to vanish. The later nonzero-transport, periodic case therefore replaces that localized boundary setting rather than satisfying both simultaneously.

Now assume $B_0\ne 0$ and a positive turbulent magnetic diffusivity ${\eta }_1$. DefineSince $F_c=-{\eta }_1{B}_{0}f$ and $⟨f⟩=1$, its spatial mean is $C=-{\eta }_1{B}_0$. HenceThe Zeldovich planar flux balance becomesBecause $C\ne 0$, the fluctuation cannot vanish, and a nontrivial fluctuation has $G>0$ under the stated boundary or zero-mean periodic conditions. ThereforeIf ${\eta }_1\gg \eta$, then $\Vert f-1{\Vert }_{L^2}^2<\eta /{\eta }_1\ll 1$: the transport profile must be nearly constant in mean square. This does not, without additional regularity, imply a uniform pointwise bound on narrow spikes.

For the final estimate use the supplied Poincaré inequality in the periodic setting and the Cauchy-Schwarz inequality:Insert the expression for $G$, cancel $B_0^2$, and multiply by $V$. Setting $y=r_{t}V\in [0,1)$ givesConsequently the variance bound for planar turbulent magnetic transport isThe factor $1/4$ comes from maximizing $y(1-y)$, so the weaker supplied Poincare constant already suffices. Under a zero vertical mean at each horizontal location, the usual period-$d$ Fourier bound is even stronger, with $4{\pi }^2/d^2$ in place of ${\pi }^2/d^2$.

Write $m_0={\mu }_0\rho$ and treat the imposed mean magnetic field as uniform on the forcing scale. In first-order smoothing, retain terms linear in the fluctuating fields in their equations, while keeping their quadratic correlation when calculating the mean-field electromotive force. The inertial fluctuation, nonlinear induction term and quadratic fluctuating Lorentz force density are neglected in the steady response. The linear magnetic force isAbsorb the gradient into the total pressure. The resulting equations arewith $\nabla \cdot \mathbf{u}=\nabla \cdot \mathbf{b}=0$. Taking the divergence of the first equation gives ${\nabla }^{2}P=0$ for solenoidal forcing. In a periodic or homogeneous Fourier calculation its nonconstant part therefore vanishes.

For each Fourier mode with wavevector $\mathbf{q}$, $|\mathbf{q}|=k$, define $\ell =\mathbf{B}\cdot \mathbf{q}$ andSolving the two coupled linear equations gives the monochromatic coupled magnetic and velocity responseThe magnetic coupling suppresses modes with large projection of their wavevector along the imposed field.

To calculate the mean-field electromotive force, group each pair of opposite wavevectors into a real mode and average its spatial phase. Put $\mathbf{n}=\mathbf{q}/k$, and define its forcing correlationsFor any solenoidal real mode $\mathbf{v}=\mathbf{a}\cos (\mathbf{q}\cdot \mathbf{x})+\mathbf{c}\sin (\mathbf{q}\cdot \mathbf{x})$, its two amplitude vectors are perpendicular to $\mathbf{q}$. Direct multiplication givesAlso, integration by parts gives $⟨(L\mathbf{g})\times (L\mathbf{f})⟩=-{\ell }^2⟨\mathbf{f}\times \mathbf{g}⟩$, where $L=\mathbf{B}\cdot \nabla$. Expanding the response products therefore yieldsSumming over independent real modes, or integrating over their angular spectrum, is the required general answer for $⟨\mathbf{u}\times \mathbf{b}⟩$.

At zero field, $\mathcal{E}(0)={\sum }_{\mathbf{q}}{\mathbf{C}}_{\mathbf{q}}/(\nu \eta k^4)$. Thus the stated zero-field condition removes the unweighted sum of the cross correlations. For anisotropic correlated forcing it does not necessarily remove their differently weighted sum at nonzero field. The displayed general formula retains that allowed contribution. It disappears if cross forcing is absent mode by mode, or under the jointly isotropic statistics assumed for the final part.

The helicity content can also be expressed directly using the actual fluctuating fields. Define their kinetic helicity $H_u(\mathbf{q})=⟨{\mathbf{u}}_{\mathbf{q}}\cdot \nabla \times {\mathbf{u}}_{\mathbf{q}}⟩$ and current helicity $H_b(\mathbf{q})=⟨{\mathbf{b}}_{\mathbf{q}}\cdot \nabla \times {\mathbf{b}}_{\mathbf{q}}⟩$. Substitute $\mathbf{f}=\nu k^2\mathbf{u}-L\mathbf{b}/m_0$ and $\mathbf{g}=\eta k^2\mathbf{b}-L\mathbf{u}$ into their cross correlation. This gives the exact identityIt shows the competing kinetic and magnetic contributions without introducing a spurious singularity when the factor on the left vanishes. For a monochromatic field, ${\mathbf{a}}_b=\nabla \times \mathbf{b}/k^2$ is a magnetic vector potential, so $H_b=k^2⟨{\mathbf{a}}_b\cdot \mathbf{b}⟩$: current helicity is $k^2$ times magnetic helicity. In the isotropic weak-field limit, with zero-field fluctuations ${\mathbf{u}}_0,{\mathbf{b}}_0$, the scalar alpha effect isThis identifies both signs and their different viscous and magnetic relaxation times.

For jointly isotropic forcing let $h_f,h_g$ now denote the total forcing helicities, and set $S_h=\nu h_g/m_0-\eta h_f$. The cross-forcing contribution is even in $\mathbf{B}$, whereas isotropy requires any vector depending only on $\mathbf{B}$ to be parallel to it and odd under reversal. Hence that contribution vanishes. A natural symmetric alpha tensor isOnly its longitudinal coefficient contributes to $\mathcal{E}={\alpha }_{\parallel }\mathbf{B}$. Introduce $h=|B|/[k\sqrt{m_0\nu \eta }]$ and $A_0=\nu \eta k^4$. Angular integration givesTherefore the strong-field quenching of an isotropic electromotive force isprovided $S_h\ne 0$. If the two helicity contributions cancel exactly, this electromotive force vanishes.

There is a tensor qualification to the printed scaling claim. The natural transverse coefficient instead contains $\frac{1}{2}{\int }_0^1(1-{\mu }^2)(1+h^2{\mu }^2{)}^{-2}d\mu \sim \pi /(8h)$ and scales as $|B{|}^{-1}$. The relation ${\mathcal{E}}_i={\alpha }_{ij}{B}_j$ by itself leaves transverse components undetermined. One may choose the effective representative ${\alpha }_{ij}={\alpha }_{\parallel }{\delta }_{ij}$, for which the requested $|B{|}^{-3}$ scaling holds. It is the observable longitudinal response that is fixed, rather than every component of a unique field-dependent tensor.

Let the layer depth be $d$, its imposed temperature drop $\Delta T$, and its dimensional vertical field $B_0\hat{\mathbf{z}}$. Use length $d$, thermal time $d^2/\kappa$, velocity $\kappa /d$, temperature $\Delta T$ and field $B_0$ as units. Write the total dimensionless field as $\hat{\mathbf{z}}+\mathbf{b}$, and temperature as the conductive profile plus $\theta$. The Boussinesq equations coupled to the resistive induction equation areThe dimensionless parameters in this normalization areHere ${\alpha }_T$ is the thermal expansion coefficient, $\nu$ is kinematic viscosity, $\kappa$ is thermal diffusivity, and $\eta$ is magnetic diffusivity. The first parameter is the Rayleigh number and $\sigma$ is the Prandtl number.

The displayed dispersion relation fixes the magnetic normalization: its $Q$ must be the thermal-diffusion magnetic-field parameter above. The conventional Chandrasekhar number is instead $Q_{\mathrm{Ch}}=B_0^2{d}^2/({\mu }_0\rho \nu \eta )$, so $Q=\zeta Q_{\mathrm{Ch}}$. If $Q$ is reserved for that conventional definition, the magnetic term in the printed polynomial requires an extra factor $\zeta$. We keep $Q$ for the printed polynomial throughout the following calculation.

Linearize about the conductive state. Absorbing linear magnetic pressure into $p$, the magnetic force becomes $\sigma Q{\partial }_z\mathbf{b}$. The pressure-eliminated vertical momentum equation isThe normal mode implicit in the given relation has $w,\theta \propto \sin \pi z$ and $b_z\propto \cos \pi z$, as for stress-free, fixed-temperature boundaries with the corresponding vertical-field magnetic conditions. Put $x=k^2$, $p_0={\pi }^2$ and $m=x+p_0={\beta }^2$. Taking growth proportional to $e^{st+i\mathbf{k}\cdot {\mathbf{x}}_h}$ gives the amplitude equationsTheir determinant isThis is the stated vertical-field magnetoconvection dispersion relation after accounting for its field-strength convention. The determinant derivation remains valid even at values where an intermediate division by $s+m$ or $s+\zeta m$ would not be allowed.

For steady marginality set $s=0$. One obtainsFor oscillatory marginality write $s=i\omega$ with $\omega \ne 0$. Dividing the polynomial by $m$ gives a real cubic $s^3+A{s}^2+Bs+C_0$, whereIts real and imaginary parts imply ${\omega }^2=B$ and $C_0=AB$. Solving for the oscillatory convection threshold givesandA physical oscillatory branch therefore requires $0<\zeta <1$ and positive frequency squared. In particular an algebraic continuation of $R_o$ into ${\omega }^2<0$ is not an oscillatory onset.

The steady-Hopf merger in vertical-field magnetoconvection occurs at zero frequency:Substitution verifies $R_s=R_o$. Equivalently the merged horizontal wavenumber satisfiesA finite-wavenumber intersection exists only when this expression is positive. At the intersection the growth polynomial has a double zero root: the nonzero-frequency neutral branch ends there.

For the requested merger at a magnetoconvection neutral-curve minimum, first note that a curve of the form $R=[A_{\ast }{m}^3+C_{\ast }m]/x$ has derivative zero exactly whenIts unique positive minimum has $x>p_0/2$, because the left side increases there from zero to infinity. At this minimum $R_c=2A_{\ast }{m}^3/p_0$.

Merger at the steady minimum. For $R_s$, the stationarity condition is $m_s^2(2x_s-p_0)=Q{p}_0^2/\zeta$. Combining it with the merger equation givesThe corresponding critical Rayleigh number is $R_{s,c}=2(x_s+p_0{)}^3/p_0$. For fixed positive $\sigma$ and large $Q$, set $\zeta =1-{\delta }_s$ and retain the leading powers. ThenMore generally, before taking $\zeta \to 1$, the steady minimum obeys $k_s^2\sim [Q{\pi }^4/(2\zeta ){]}^{1/3}$.

Merger at the oscillatory minimum. Define $A_o=(\sigma +\zeta )(1+\zeta )/\sigma$. The stationarity condition for $R_o$ isCombining it with the zero-frequency relation instead givesHere $R_{o,c}=2A_o(x_o+p_0{)}^3/p_0$. For fixed $\sigma >0$ and $Q\gg 1$, setting $\zeta =1-{\delta }_o$ givesThese two merger conditions are distinct parameter surfaces, not simultaneous assertions about the same minimum. Both have the strong-field wavenumber selection in magnetoconvection scaling $k=O(Q^{1/6})$.

If the first subleading Rayleigh corrections are desired, let $X_s=[Q{p}_0^2/(2\zeta ){]}^{1/3}$ and $X_o=[\sigma Q{p}_0^2/(2(1+\sigma )(1+\zeta )){]}^{1/3}$. With parameters bounded in the above regime,These retain the $O(Q^{2/3})$ corrections, including the effect of the $O(Q^{-1/3})$ diffusivity deficit at a merger.

A planetary magnetic field is evidence about electrical conductivity, fluid motion and thermal evolution, but its presence alone does not identify a present planetary dynamo. One must separate internal global fields, crustal remanent magnetization and currents driven by the solar wind. The following discussion uses the observational picture available by 2009, including the early MESSENGER flybys and the Voyager, Galileo and Cassini results.

Outside a planet's internal electric currents, the internal component is approximately a potential magnetic field. Its spherical harmonic of degree $\ell$ decreases as $r^{-(\ell +2)}$. Measurements at different positions and altitudes can therefore distinguish a deep, low-degree source from localized crustal anomalies and external magnetospheric currents. Repeated observations, radio emissions, aurorae and particle measurements supply complementary evidence for a magnetosphere, while slow secular variation and records of past reversals test the internal field's evolution.

For Earth, ground observatories and satellite measurements reveal a predominantly dipolar field of several tens of microtesla at the surface, tilted relative to the rotation axis and changing with time. Magnetic minerals in rocks record a much older field and repeated geomagnetic reversals. These observations, together with seismic evidence for a liquid outer core, support an operating planetary dynamo in conducting iron alloy. Core temperatures exceed the relevant Curie temperatures, so ordinary permanent remanent magnetization cannot explain the main field. Thermal convection, cooling and compositional convection associated with inner-core growth supply buoyancy; the Coriolis force organizes the motion. The observed nonaxisymmetric components and time dependence are consistent with a dynamic source, rather than a fixed bar magnet.

Mariner 10 and the first two MESSENGER flybys established a weak but coherent intrinsic field at Mercury, with roughly dipolar structure closely aligned with its rotation axis. It is much weaker than Earth's surface field, but its global character favors a deep source over an explanation solely by localized crustal patches. A planetary dynamo in a fluid part of Mercury's large iron-rich core is the natural interpretation, supported by evidence for a molten core component. Light elements can depress the alloy's freezing temperature, and thermal or compositional buoyancy can maintain motion. Slow rotation is not enough to rule out dynamo action. The limited pre-orbital measurements do not uniquely determine the detailed core structure or the mechanism responsible for the field's weakness.

For Venus, spacecraft including Pioneer Venus found no appreciable internally generated global field. Nevertheless its ionosphere interacts with the solar wind, producing magnetic draping, a bow shock and an induced magnetosphere. Those external signatures are not evidence for an internal planetary dynamo. Venus is comparable to Earth in size and may possess a substantial conducting core; the absence of a detectable dynamo therefore need not mean that all of its core is solid. Insufficient core heat loss or stable stratification can prevent the required convection. Its slow rotation can affect flow organization, but rotation rate by itself is not a sufficient explanation of the contrast with Earth or Mercury.

For Mars, Mars Global Surveyor found strong, spatially localized crustal anomalies, especially in ancient southern terrain, rather than an appreciable present global dipole. This is remanent magnetization, indicating that an ancient field magnetized the crust while it formed and cooled. The distribution of anomalies and weakly magnetized major impact basins constrain when that ancient planetary dynamo operated; crustal magnetization is a record of past activity, not a present global dynamo. Modern interaction with the solar wind also produces an induced magnetosphere. Cooling and changes in core heat extraction can explain the cessation of dynamo-driving convection; the observations alone do not prove complete solidification of the core.

For Jupiter, radio emissions, its intense radiation belts and measurements by Pioneer, Voyager and Galileo demonstrate a strong global intrinsic field and an enormous magnetosphere. The field is broadly dipolar but tilted and contains higher multipoles. It is generated in a deep conducting fluid rather than by a permanent solid magnet: the principal conducting region is metallic hydrogen in the high-pressure interior. Cooling and gravitational energy release maintain convection, while rapid rotation organizes the flow. Plasma supplied by Io and currents in the magnetosphere strongly modify the observed external field, but cannot account for the internally sourced planetary component. Jupiter's large magnetic moment also reflects its large size; a moment ratio should not be confused with the ratio of local surface field strengths.

For Saturn, Pioneer, Voyager and Cassini measurements likewise show an intrinsic field and a magnetosphere, with a dipole strikingly close to the rotation axis. A planetary dynamo in conducting metallic hydrogen is supported by the same broad interior argument as for Jupiter. Interior cooling and differentiation can provide power, including energy associated with helium separation. The near alignment does not invalidate the Cowling anti-dynamo theorem: that theorem excludes an exactly axisymmetric isolated dynamo field, not a predominantly axisymmetric exterior with undetected nonaxisymmetric structure in its source region. A stably stratified conducting layer above the active region is one possible way to attenuate the observable nonaxisymmetric components; it is a model, not a uniquely established consequence of the observations.

For Uranus, Voyager 2 in 1986 observed an intrinsic field with a dipole tilted by about $59^{\circ }$ to the rotation axis and significantly displaced from the centre. Higher multipoles are substantial, so a centred dipole is an inadequate description. The field and associated magnetosphere indicate an active planetary dynamo, despite the planet's small intrinsic heat flux. That low luminosity constrains its thermal evolution and may indicate inhibited deep mixing; it does not demonstrate an absence of all conducting-fluid motion. A conducting, volatile-rich fluid shell is a plausible source region. The term ice giant describes bulk composition, not a claim that the dynamo region is ordinary frozen ice.

For Neptune, Voyager 2 in 1989 also found a highly non-dipolar, displaced intrinsic field, with a dipole tilt near $47^{\circ }$. Its field, radio emissions and magnetosphere imply another active planetary dynamo. Its larger intrinsic heat flux than Uranus supplies evidence for substantial continuing interior energy transport. Electrically conducting water-ammonia-rich mixtures at high pressure, with ionic or otherwise mobile charges, can supply the conducting region without requiring the metallic hydrogen layer characteristic of gas giants. Numerical dynamos in convecting shells surrounding more stably stratified interiors can produce strongly nonaxisymmetric, multipolar fields resembling those of the two ice giants. Such models show that their unusual field geometry is compatible with dynamo action, but field measurements do not uniquely identify the shell's composition or thickness.

The maintenance mechanism is governed by the resistive induction equation,A conducting fluid needs a sufficiently large magnetic Reynolds number and a flow geometry that regenerates magnetic flux rather than merely amplifying one component temporarily. Three-dimensional stretching and twisting can couple poloidal and toroidal fields; strictly planar regeneration and a strictly axisymmetric magnetic field are restricted by the relevant anti-dynamo theorems. Rotation influences this regeneration through the Coriolis force, but does not provide an inexhaustible power source merely by being present.

Without regeneration, the characteristic magnetic diffusion time is ${\tau }_{\eta }\sim L^2/\eta$. For a terrestrial core scale of order $10^6$ metres and diffusivity of order one square metre per second, this is of order $10^4$–$10^5$ years, far shorter than geological ages. Long-lived global fields therefore require continual energy input. A schematic magnetic energy balance iswhen boundary energy fluxes vanish. Mechanical work against the Lorentz force density supplies the energy lost as Ohmic heating. Thermal cooling, latent heat release, compositional convection and differentiation supply that mechanical power. Magnetic feedback eventually changes the flow and saturates the field. Strong conductivity without an adequate energy source is not sufficient for a dynamo.

The 2009 observations support active global dynamos in Mercury, Earth, Jupiter, Saturn, Uranus and Neptune; Venus has no detected appreciable intrinsic dynamo field, and Mars preserves evidence of an extinct earlier dynamo in its crust. Induced fields and surviving remanence explain why planets without a current global dynamo can still display magnetic phenomena. The diversity of field geometries and strengths constrains, rather than uniquely solves, the diversity of planetary conducting regions and their convective histories.