For a stress-free, fixed-temperature layer with vertical magnetic perturbation at its boundaries, take vertical velocity and temperature proportional to $\sin \pi z$, vertical magnetic perturbation proportional to $\cos \pi z$, and horizontal magnetic perturbations proportional to $\sin \pi z$. With $s=k^2+{\pi }^2$, the vertical velocity, temperature and magnetic amplitudes form a three-variable linear system whose determinant is the stated polynomial. The Prandtl number is $\sigma$, magnetic-to-thermal diffusivity ratio is $\zeta$, and field strength is the Chandrasekhar number $Q$. The steady marginal curve is $R^{(e)}=[s^3+Q{\pi }^{2}s]/k^2$.

In thermal-diffusion units, let an imposed field point along $\mathbf{c}=(\sin \alpha ,0,\cos \alpha )$ and consider a Fourier mode of wavevector $(k,0,m)$. Put $q=k^2+m^2$ and $d=k\sin \alpha +m\cos \alpha$. The growth exponent $s$ of its buoyant branch satisfieswhere $\sigma$ is the Prandtl number, $\zeta$ the magnetic-to-thermal diffusivity ratio, $Q$ the Chandrasekhar number, and $R$ the Rayleigh number. The resistive induction equation gives $\hat{\mathbf{b}}=id\hat{\mathbf{u}}/(s+\zeta q)$, and thermal transport gives $\hat{\theta }=\hat{w}/(s+q)$. Projecting momentum perpendicular to the wavevector removes pressure; the vertical component of the projected buoyancy is $R(k^2/q)\hat{\theta }$. Substitution gives the displayed polynomial. The stationary neutral curve is $R=q^3/k^2+Qq{d}^2/k^2$. No vertical boundary conditions are imposed in this plane-wave calculation.

In nonlinear magnetoconvection, magnetic flux separation is coexistence of weak-field regions with vigorous broad convection and strong-field regions with inhibited or small-scale convection. Flux expulsion provides a mechanism for segregation. Conserved mean vertical magnetic flux implies $f{B}_c+(1-f)B_q=B_0$ for an idealized two-region area fraction $f$. Weak field in convecting regions therefore requires enhanced field elsewhere. The existence and stability of the segregated state depend on parameters and boundaries.

Subcritical magnetoconvection means finite-amplitude convection on the stable side of the linear conductive-state threshold. At large magnetic Reynolds number, flux expulsion can reduce interior magnetic inhibition enough to sustain a nonlinear convective cell even when infinitesimal disturbances decay in the unredistributed imposed field. Stable subcritical states can produce hysteresis and magnetic flux separation. A schematic saturated amplitude equation $\dot{A}=\mu A+c|A{|}^{2}A-d|A{|}^{4}A$, $c,d>0$, has an unstable lower branch and stable upper radial branch for $-c^2/(4d)<\mu <0$, coexisting with stable $A=0$. This illustrates bistability; it is not a derivation of coefficients for all magnetoconvection geometries.

The ratio of magnetic diffusivity $\eta$ to thermal diffusivity $\kappa$ controls their relative relaxation times. In nonrotating vertical-field magnetoconvection, oscillatory linear onset in the simple stress-free layer requires $0<\zeta <1$, so thermal perturbations diffuse faster than magnetic perturbations. This is distinct from the magnetic Prandtl number, $\nu /\eta$.

For a unit-depth Boussinesq layer with stress-free boundary conditions, fixed boundary temperatures, imposed field along $y$, and horizontal magnetic perturbations at the boundaries, put $a_h^2=k^2+m^2$ and $\Lambda =a_h^2+{\pi }^2$. Poloidal vertical velocity, temperature and vertical field proportional to $e^{st+ikx+imy}\sin \pi z$ have amplitudes satisfyingThe determinant gives the stated dispersion relation. Here $\sigma$ is the Prandtl number, $\zeta$ the magnetic-to-thermal diffusivity ratio, $Q$ the Chandrasekhar number and $R$ the Rayleigh number. The steady neutral curve is $R={\Lambda }^3/a_h^2+Q{m}^2\Lambda /a_h^2$.

// Target: fluid-mechanics.bigb

A Galerkin method approximation can retain one roll streamfunction mode, two temperature modes and two magnetic-flux perturbation modes in vertical-field magnetoconvection. Projection onto these five modes, rather than pointwise equality of the truncated fields, gives a closed system of five ordinary differential equations. Products generate higher spatial harmonics that are discarded by orthogonal projection. The system has symmetry $(a,b,c,d,e)\mapsto (-a,-b,c,-d,e)$ and hence admits paired steady roll branches.

When the magnetic field supplies instantaneous damping rather than an evolving magnetic perturbation, the roll amplitudes obey a two-variable linear system with trace $-[s+\sigma (s+Q{\pi }^2/s)]<0$. Its determinant vanishes at the displayed stationary threshold. Therefore its initial instability cannot be a Hopf bifurcation. This reduced model must not be confused with a three-variable magnetoconvection model carrying finite magnetic diffusivity and an additional oscillatory mode.

Use depth $d$, time $d^2/\kappa$, stream-function scale $\kappa$, temperature perturbation scale $\Delta T$, and magnetic flux-function scale $B_{0}d$. With $\mathbf{u}=(-{\psi }_z,0,{\psi }_x)$, $\mathbf{B}=(-A_z,0,A_x)$ and $A=x+\chi$, advection is $J(\psi ,f)={\psi }_x{f}_z-{\psi }_z{f}_x$. The temperature background contributes ${\psi }_x$ to the heat equation and the imposed vertical field contributes ${\psi }_z$ to the perturbation induction equation. Taking the curl of the magnetohydrodynamic momentum equation gives a magnetic term $\sigma \zeta QJ(A,{\nabla }^{2}A)$, where $\sigma =\nu /\kappa$, $\zeta =\eta /\kappa$, $R=g\alpha \Delta T{d}^3/(\nu \kappa )$ and $Q=B_0^2{d}^2/({\mu }_0\rho \nu \eta )$.

A Galerkin method using one convective streamfunction mode and two Cartesian magnetic flux function modes yields the displayed system. With $\alpha =\pi /\sqrt{3}$, use $\psi =4\sqrt{2}a\sin (\alpha x)\sin (\pi z)$ and $A=(3\sqrt{2}/\pi )b\sin (\alpha x)\cos (\pi z)+(\sqrt{3}/\pi )c\sin (2\alpha x)$, time $\tau =(4{\pi }^2/3)t$, $q=9Q/(16{\pi }^2)$ and $r=9R/(64{\pi }^4)$. The origin's planar trace is $\sigma (r-1)-\zeta$ and determinant $\sigma \zeta (1+q-r)$; the third eigenvalue is $-\zeta$. The stationary threshold is $r=1+q$, and the oscillatory threshold is $r=1+\zeta /\sigma$ when $q>\zeta /\sigma$. They meet in a symmetric double-zero point at $(q,r)=(\zeta /\sigma ,1+\zeta /\sigma )$. Higher spatial harmonics generated by the quadratic terms are discarded, so this is a projected approximation rather than an exact invariant subspace of the original equations.

For a smooth Anosov magnetic flow on a closed oriented surface, a smooth function plus a smooth one-form can be a smooth flow coboundary only when the function is zero and the one-form is exact. Equivalently their sum has zero integral along every closed orbit precisely in that case, using the smooth Livšic theorem. This is the nontrivial magnetic transform kernel theorem. Exact one-forms give the converse directly, since $F(w\circ \pi )=dw(v)$. The theorem does not require pointwise negative Gaussian curvature.

Two uniform horizontal magnetic fields on either side of a horizontal interface exert no bulk force, but their magnetic pressures differ. For a common-temperature isothermal atmosphere with sound-speed squared $c_T^2=\mathcal{R}{T}_0$ and gravity $g$, hydrostatic equilibrium gives $p_i=c_T^2{\rho }_i$ and $p_i(z)=p_i(0)e^{-z/H}$, $H=c_T^2/g$. Continuity of magnetohydrodynamic total pressure gives the displayed density jump. A stronger field below means denser gas above, supplying a Rayleigh-Taylor instability even though each half-space is hydrostatic.

// Target: continuum-mechanics.bigb

A compressible intensification mechanism for a thin, nearly vertical flux tube in a superadiabatic atmosphere. An unstable downflow drains material and lowers internal gas pressure; approximate lateral balance $p_i+B^2/(2{\mu }_0)=p_e$ then permits the tube to contract and its field to strengthen at nearly conserved magnetic flux. It relies on compressibility and thermal evolution, which are absent from a strict Boussinesq approximation. Dynamic photospheric structures need not satisfy thin-tube equilibrium, so this mechanism does not explain every intense field concentration.

Equating magnetic energy density $B^2/(2{\mu }_0)$ to kinetic energy density $\rho U^2/2$ defines this order-of-magnitude field. It marks when a weak-field kinematic treatment of a given flow can fail, not a universal upper bound on concentrated fields. Gas-pressure confinement, compression, partial evacuation and dynamic pressure can allow stronger local fields.

The PHYWE guide Magnetic moment in the magnetic field makes the magnetic dipole moment measurable. A circular test current loop sits between a pair of Helmholtz coils, with separate supplies and meters for the test electric current $I$ and field-producing electric current $I_H$. The test current loop is mounted on a torsion dynamometer. The guide varies $I_H$, the number of turns, the test electric current, diameter, and the angle between the magnetic dipole moment and magnetic field. Its angular settings are spaced by $15°$. The leads hang loosely and are twisted together to suppress extra torque; the instrument zero is checked frequently because moving leads can shift it. Its reported results include approximately linear dependence on each electric current and on winding count, and approximately quadratic dependence on diameter.^[ref]

The readout is a balance of torques. According to the manufacturer's Torsion dynamometer 0.01 N operating instructions, a metallic band supplies the restoring torque. First set the indication to zero with the applied force absent and center the lever using the zero knob. Apply the load, then turn the indicating knob until the lever returns to its original zero position. The scale then reads a force equivalent, not a torque directly. The quoted scale division is $0.1\mathrm{mN}$. For a force applied perpendicular to the calibration lever arm at distance $\ell$, the conversion is $\tau =F\ell$; a current loop applies a couple, so its torque must be compared with the calibrated restoring torque.^[ref]

For a quantitative repeat, independently check that conversion with known forces at a measured perpendicular lever arm, and check the imposed magnetic field at the test current loop. These are proposed calibration checks, not additional measurements reported here. The ideal Helmholtz coil model predictswhere $N_H$ and $R_H$ are the winding count and radius of each field-producing Helmholtz coil, distinct from $N$ and $R$ of the test current loop. Derive $B_0$ by adding the on-axis magnetic field of a circular current loop at distances $R_H/2$ from both coils. Independently calibrating $B_0$ and the restoring torque lets a fitted torque amplitude determine $m$ as amplitude divided by $B_0$, to compare with $NI\pi R^2$.

Useful additional controls are to reverse only $I$, reverse only $I_H$, and reverse both: the signed torque should reverse in either single reversal and be unchanged in the double reversal. Record baselines with each electric current separately zero, and subtract a nearby zero reading to reduce torsion balance zero drift. Repeat angular settings in both directions to reveal lead-dependent offsets. These are suggested controls. Limits include finite magnetic field uniformity, uncertainty in effective winding area and angular zero, meter calibration, Joule heating, and residual lead torque. An apparently sinusoidal trace alone does not establish the absolute magnetic dipole moment. No raw laboratory data or new measured values are claimed here.

A nonuniform angular velocity $f(r)$ winds an azimuthal magnetic mode into phase $m(\varphi -f(r)t)$. Its radial phase gradient grows as $-mt{f}^{\prime }(r)$, creating small scales even when its ideal amplitude does not decay. The corresponding leading resistive damping rate is $\eta m^2{t}^2{f}^{\prime }(r{)}^2$, so a local short-wavelength estimate gives damping factor $\exp [-\eta m^2{f}^{\prime }(r{)}^2{t}^3/3]$. Hence magnetic diffusion can become important at time $[\eta m^2{f}^{\prime 2}{]}^{-1/3}$ although it is initially small. This is a local phase-mixing estimate, not a globally exact solution in a finite radial domain: boundary forcing, field stretching and spatial variation of $f^{\prime }$ must also be retained there.

For disturbances constant along the horizontal magnetic field, the minimized energy density is $\rho g[-(\ln \rho {)}^{\prime }-\rho g/(\gamma p+C)]|{\xi }_z{|}^2$. If its coefficient is nonnegative everywhere, the full energy is nonnegative. If it is strictly negative at a point of a smooth equilibrium, a compactly supported vertical displacement in a neighbouring interval gives negative energy. These interchange motions compress gas and magnetic pressure together.

A horizontal flux tube displaced without bending preserves $B/\rho$ by magnetic flux freezing. If thermal exchange keeps its temperature equal to its new surroundings, lateral magnetic pressure balance gives $(c_T^2+v_A^2)({\rho }_i^{\prime }-{\rho }^{\prime })=\rho B(B/\rho {)}^{\prime }/{\mu }_0$. Here $c_T^2=p/\rho$, $v_A^2=B^2/({\mu }_0\rho )$, and primes denote the response to a small vertical displacement from an equilibrium. Its linear buoyancy acceleration is $-gB(B/\rho {)}^{\prime }\xi /[{\mu }_0(c_T^2+v_A^2)]$. Thus local interchange is unstable when $B(B/\rho {)}^{\prime }<0$, equivalently $(B^2/{\rho }^2{)}^{\prime }<0$. This fast-thermal-exchange criterion differs from the adiabatic localized interchange criterion for a magnetized atmosphere.

The supersymmetric M2-brane worldvolume theory includes embedding bosons and fermions and has kappa symmetry. In eleven dimensions its fermionic embedding coordinate is a thirty-two-component Majorana spinor.

The transverse harmonic function $H$ solves the field equations away from membrane sources. Positive centres contribute $Q_a/|y-y_a{|}^6$ in eight dimensions. Its Killing spinors are $H^{-1/6}{ϵ}_0$ with one oriented membrane projector, giving sixteen supersymmetries. The three-form's sign and membrane orientation must be chosen together.