Solution (source code)

= Solution

\b[Oceanic <magnetic anomalies>.] Newly formed oceanic <crust> contains magnetic minerals. As it cools through their <magnetic blocking temperatures>, it acquires <thermoremanent magnetization> recording the ambient field polarity. Continuing <seafloor spreading> carries that record away from the ridge. <Geomagnetic reversals> therefore generate approximately paired <marine magnetic stripes> parallel to the ridge, with older <crust> farther from its axis. A <magnetometer> measures the superposed field above this magnetized material; subtracting a reference field reveals <magnetic anomalies>. The anomaly sign and shape depend on <magnetization> direction, latitude, source depth and geometry, so a positive measured anomaly is not universally identical to normal rock polarity.<Developing the theory by United States Geological Survey>[Sections Magnetic striping and polar reversals, and Seafloor spreading and recycling of oceanic crust]{ref}

Matching a sequence of stripes to an independently dated reversal chronology gives plate ages and spreading histories. A stripe of width $w$ formed over interval $\Delta t$ implies <half-spreading rate> $v=w/\Delta t$ under steady symmetric spreading; full separation <speed> is $2v$. Unequal widths on opposite flanks can reflect asymmetric spreading or later deformation. Alteration, remagnetization and overlapping magnetic sources complicate inversion. \b[The sequence correlation is stronger evidence than an isolated stripe or a field-strength peak.]

\b[<Rayleigh-Bénard convection>.] Consider a horizontal fluid layer of thickness $d$, hotter below by $\Delta T>0$. The conductive state is motionless with a linear <temperature> <gradient>. A rising warm parcel is buoyant, but <viscosity> resists its motion and thermal <diffusion> erases its <temperature> anomaly. In the <Boussinesq approximation>, <density> variations are retained in <buoyancy> while the flow is otherwise incompressible. The competing effects are measured by the <Rayleigh number> and <Prandtl number>,
$$
\operatorname{Ra}=\frac{g\alpha\Delta T d^3}{\nu\kappa},\qquad
\operatorname{Pr}=\frac\nu\kappa.
$$
For stress-free impermeable, fixed-temperature boundaries, linear stationary disturbances with horizontal <wavenumber> $q/d$ give
$$
\operatorname{Ra}(q)=\frac{(q^2+\pi^2)^3}{q^2}.
$$
Minimizing gives $q_c=\pi/\sqrt2$ and the <critical Rayleigh number for an isothermal layer>, $\operatorname{Ra}_c=27\pi^4/4\simeq657.5$. Two rigid no-slip, fixed-temperature boundaries instead give approximately $1708$; the critical value is not independent of <boundary conditions>.<Boundary layer turbulence by Massachusetts Institute of Technology>[Chapter 20, §20.4, critical value for two rigid isothermal plates]{ref}

Above onset, <buoyancy> can sustain overturning rolls and <heat> transfer exceeds pure conduction, with <Nusselt number> $\operatorname{Nu}>1$. Nonlinear advection eventually limits the disturbance; larger driving can produce time-dependent or turbulent states. The ideal Boussinesq problem does not require hexagonal cells: planform selection depends on boundaries and non-Boussinesq effects. <Convection> driven by <surface tension> at a <free surface> is a distinct mechanism. \b[An adverse <temperature> <gradient> supplies driving, but overcoming <diffusion> and <viscosity> requires a finite threshold.]