Magnetoconvection

Magnetoconvection is thermal convection coupled to the evolution and forces of a magnetic field. Conserved magnetic flux can supply a slow field that must be retained along with the convective complex amplitude, rather than being eliminated into a single local amplitude equation.

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 three-mode porous magnetoconvection, set $\mu =r-1-q$ and assume $0<q<\zeta /\sigma$. The transverse eigenvalues are negative. Centre-manifold invariance givesSubstitution into the first equation gives the displayed positive linear and cubic coefficients. Thus the bifurcation is a subcritical pitchfork bifurcation: the origin is stable for $\mu <0$ and unstable for $\mu >0$, while the small nonzero branches at $a^2=-{\zeta }^2\mu /(3q)$ are unstable. Simply setting $\dot{b}=0$ misses the time-derivative contribution responsible for the common factor $(1-\sigma q/\zeta {)}^{-1}$.

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 $Q=B_0^2{d}^2/({\mu }_0\rho \nu \eta )$.

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.

The primary roll and temperature modes force only a horizontally uniform temperature mode proportional to $\sin (2\pi z)$ at quadratic order. Its amplitude is $c=-\alpha ab/(8\pi )$. Its feedback on the primary temperature mode saturates magnetoconvection, giving the displayed positive squared amplitude above threshold. Thus a simple roll onset is a supercritical pitchfork bifurcation for every $Q\ge 0$. The three-mode Fourier truncation and full perturbation calculation agree through the cubic solvability condition because all forced quadratic modes are retained; this is not an assertion that the truncation gives an exact finite-amplitude PDE solution.

// Destination: dynamical-systems.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.

The conduction linearization has two negative eigenvalues and a cubic factor $s^3+A{s}^2+Bs+C$, where $A=\sigma +1+\zeta$, $B=\sigma (1-r)+\sigma \zeta (1+q)+\zeta$, $C=\sigma \zeta (1+q-r)$. A double zero requires $B=C=0$, giving the displayed Bogdanov–Takens bifurcation location for $0<\zeta <1$. The remaining cubic eigenvalue is $-A$. The imaginary-pair condition $AB=C$ yields $r_H=(\sigma +\zeta )(1+\zeta )/\sigma +\zeta (\sigma +\zeta )q/(\sigma +1)$ and ${\omega }^2=\sigma \zeta (1-\zeta )q/(\sigma +1)-{\zeta }^2$. Thus the oscillatory onset exists precisely for $q>q_{\ast }$ when $0<\zeta <1$.

For positive diffusivity ratio $\zeta$ and $0<\phi <4$, define $\mu =(4-\phi ){\zeta }^2/\phi$. The nonzero steady branch has $b=a/(1+a^2)$, $c=a^2/(1+a^2)$, $d=\mu a/[\zeta (\mu +a^2)]$, and $e=a^2/(\mu +a^2)$. Substitution yields the displayed branch through $r=1+q$. Its initial slope in $a^2$ has the sign of $\phi q(2-\phi )+(4-\phi ){\zeta }^2(1+q)$. Positive slope is a supercritical pitchfork bifurcation direction, negative slope a subcritical pitchfork bifurcation direction. This geometric criterion alone does not assert stability against every perturbation.

Put $m=(4-\phi ){\zeta }^2/\phi$. The derivative of the stationary branch of five-mode magnetoconvection at $a^2=0$ is $r_2=1+q+q(2-\phi )/m$. For fixed positive $q$ and small $\zeta$, it changes from a forward to a backward pitchfork bifurcation direction at the displayed value. Expanding the exact numerator $q\phi (2-\phi )+(4-\phi ){\zeta }^2(1+q)$ gives this result. The expansion in amplitude requires $a^2\ll m$; branch direction alone does not determine stability against all other modes.

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

Restrict the horizontal-field magnetoconvection dispersion relation to $k=0$, $x=m^2>0$. The steady neutral curve becomes $R={\pi }^{2}Q+(Q+3{\pi }^2)x+x^2+3{\pi }^4+{\pi }^6/x$. Its unique minimum satisfies $x^2(Q+3{\pi }^2+2x)={\pi }^6$, since $R^{\prime \prime }(x)=2+2{\pi }^6/x^3>0$. Expanding at large Chandrasekhar number gives $x\sim {\pi }^3{Q}^{-1/2}$ and the stated asymptotic expansion. Unlike field-aligned rolls, these disturbances bend the field, and their stationary threshold grows with field strength.

// Target: fluid-mechanics.bigb

At a fixed horizontal wavenumber magnitude, the steady neutral curve in the horizontal-field magnetoconvection dispersion relation is minimized by $m=0$. The disturbance is then constant along the imposed magnetic field; it induces no field bending and incurs no magnetic tension penalty. Minimizing $(k^2+{\pi }^2{)}^3/k^2$ gives the displayed stationary threshold for every $Q\ge 0$. These convection rolls have axes parallel to the imposed field. This assertion concerns steady onset.

// Target: fluid-mechanics.bigb

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, $\nu /\eta$.

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.

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.

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.

At inclination $\alpha =\arctan \sqrt{2}$ and vertical wavenumber one, the nonmagnetic minimizing wavenumber $k=-1/\sqrt{2}$ also satisfies $\mathbf{c}\cdot \mathbf{k}=0$. The stationary threshold is the sum of $F(k)\ge 27/4$ and a nonnegative magnetic penalty that vanishes there. Therefore $k_c=-1/\sqrt{2}$ and $R_c=27/4$ for every $Q\ge 0$. The selected disturbance is constant along the imposed field, so the linear resistive induction equation produces no magnetic perturbation and there is no field-line tension penalty.

For fixed vertical wavenumber one, write the stationary threshold as $F(k)+QG(k)$, with $F=(1+k^2{)}^3/k^2$ and $G=(1+k^2)(\cos \alpha +k\sin \alpha {)}^2/k^2$. The nonmagnetic minima are $k=\pm 1/\sqrt{2}$ with $F=27/4$; a small positive field selects the negative one. Expanding the stationarity condition gives $k_c=a-Q{G}^{\prime }(a)/F^{\prime \prime }(a)+O(Q^2)$, $a=-1/\sqrt{2}$, andFor large $Q$, the magnetic penalty forces $k$ toward its unique zero $b=-\cot \alpha$. Since $G^{\prime \prime }(b)>0$, $k_c=b-F^{\prime }(b)/[Q{G}^{\prime \prime }(b)]+O(Q^{-2})$ and $R_c={\csc }^6\alpha /{\cot }^2\alpha -F^{\prime }(b{)}^2/[2Q{G}^{\prime \prime }(b)]+O(Q^{-2})$. Thus strongly magnetized rolls align their wavevector perpendicular to the imposed field, rather than continuing to raise their stationary threshold proportionally to field strength.

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 $R^{(e)}=[s^3+Q{\pi }^{2}s]/k^2$.

For one stress-free planar roll mode, thermal-time rescaling reduces the linear dynamics to $a^{\prime }=\sigma (-a+rb-\zeta qc)$, $b^{\prime }=a-b$, $c^{\prime }=a-\zeta c$. Its characteristic polynomial is $(s+\sigma )(s+1)(s+\zeta )-\sigma r(s+\zeta )+\sigma \zeta q(s+1)$. For $\sigma ,\zeta >0$ and $q\ge 0$, steady marginality is $r_s=1+q$. Oscillatory marginality is $r_H=(\sigma +\zeta )(1+\zeta )/\sigma +\zeta (\sigma +\zeta )q/(\sigma +1)$, with ${\omega }_H^2=\zeta [\sigma (1-\zeta )q/(\sigma +1)-\zeta ]$. A genuine Hopf bifurcation onset requires ${\omega }_H^2>0$; a formal imaginary-root formula with negative frequency squared is not an oscillation. The Routh-Hurwitz stability criterion selects the first crossing, while nonlinear coefficients determine criticality.

At the minimum of the steady vertical-field magnetoconvection dispersion relation, stationarity gives $R_c=2(k_c^2+{\pi }^2{)}^3/{\pi }^2$. With nonmagnetic value $R_0=27{\pi }^4/4$, eliminating the critical wavenumber gives the stated relation and $k_c^2={\pi }^2[\frac{3}{2}(R_c/R_0{)}^{1/3}-1]$. These hold for every nonnegative Chandrasekhar number, including its zero-field limit.

Both steady and oscillatory neutral curves have the form $R=C(k^2+p{)}^3/k^2+DQp(k^2+p)/k^2$, with $p={\pi }^2$. Their exact minimizing condition is $x^2(2x+3p)=p^3+(D/C)Q{p}^2$, where $x=k^2$. For fixed positive coefficients and large Chandrasekhar number, $x\sim [D{p}^{2}Q/(2C){]}^{1/3}$ and $R_{\min }=DpQ+3C^{1/3}(D{p}^{2}Q/2{)}^{2/3}+O(Q^{1/3})$. The optimum balances transverse diffusion against the magnetic penalty, leading to increasingly narrow rolls.

A nonzero imaginary root of the vertical-field magnetoconvection dispersion relation requires $0<\zeta <1$ and $Q>\zeta (1+\sigma )s^2/[\sigma (1-\zeta ){\pi }^2]$. Its marginal curve is $R^{(o)}=C{s}^3/k^2+DQ{\pi }^{2}s/k^2$, with $C=(\sigma +\zeta )(1+\zeta )/\sigma$ and $D=\zeta (\sigma +\zeta )/(1+\sigma )$. A formal minimum of that algebraic curve is a physical oscillatory threshold only if the frequency squared there is positive.

For thermal-time magnetoconvection, a physical oscillatory neutral mode has ${\omega }^2=\sigma (1-\zeta )Q_{\kappa }{\pi }^2/(1+\sigma )-{\zeta }^2(k^2+{\pi }^2{)}^2>0$. The steady and oscillatory neutral curves meet when this frequency tends to zero. Thus $0<\zeta <1$ is necessary and the displayed relation locates the merger. A formal oscillatory-curve minimum with negative frequency squared is not a physical onset.

At a steady-curve minimum, differentiation and the steady-Hopf merger in vertical-field magnetoconvection give the first displayed wavenumber. At an oscillatory-curve minimum they give the second. For fixed positive Prandtl number $\sigma$ and large $Q_{\kappa }$, the corresponding merger diffusivity deficits are $1-{\zeta }_s\sim [(1+\sigma )/\sigma ]({\pi }^2/(4Q_{\kappa }){)}^{1/3}$ and $1-{\zeta }_o\sim [(1+\sigma ){\pi }^2/(16\sigma Q_{\kappa }){]}^{1/3}$. The two parameter surfaces are distinct: coincidence of neutral curves at one wavenumber does not imply coincidence of their minimizing wavenumbers.

For an imposed field $B_0$ in a layer of depth $d$, the Chandrasekhar number compares magnetic tension with viscous and magnetic-diffusive effects. Here $\eta$ is magnetic diffusivity and $\nu$ is kinematic viscosity. In thermal-diffusion units the linear magnetic-force coefficient is $Q\zeta$, where $\zeta =\eta /\kappa$ and $\kappa$ is thermal diffusivity.

With thermal diffusivity $\kappa$ and magnetic diffusivity $\eta$, the standard Chandrasekhar number uses $\nu \eta$ in its denominator. A thermal-diffusion alternative uses $\nu \kappa$ and is $\zeta =\eta /\kappa$ times the standard parameter. In thermal-time magnetoconvection equations, the linear magnetic-force coefficient is $\sigma Q_{\kappa }=\sigma \zeta Q_{\mathrm{Ch}}$. Specifying the convention prevents a missing diffusivity factor in a dispersion relation.

A representative scaled amplitude model has complex $A$ and real $B$ withPeriodic boundary conditions conserve $\int BdX$, because the right-hand side is a total second derivative. The slow field can change long-wave convection roll stability and permit localized pulses. For zero mean $B$, the uniform convection rolls are $A=R{e}^{iqX}$, $R^2=1-q^2$, $B=0$.

For a real steady amplitude and zero-mean field, twice integrating the field equation yields $B=\kappa (⟨A^2⟩-A^2)$, $\kappa =\mu /\sigma$. The amplitude equation becomes $A^{\prime \prime }+(1-\kappa ⟨A^2⟩)A+(\kappa -1)A^3=0$. Using $(R\mathrm{sech}cX{)}^{\prime \prime }=c^{2}A-2c^2{A}^3/R^2$ gives $\kappa ⟨A^2⟩=1+c^2$ and $R^2=2c^2/(\kappa -1)$, requiring $\kappa >1$. Its finite-window mean is $2R^2\tanh (cL/2)/(cL)$. For $cL\gg 1$, consistency gives $c+c^{-1}=4\kappa /[L(\kappa -1)]$. The leading roots coalesce at $c=1$; for $L>2$ the leading upper parameter limit is $\kappa \le L/(L-2)$, while retaining the finite-window factor makes the necessary inequality strict. The isolated sech profile has an exponentially small derivative mismatch on a periodic interval, so exact periodic tails need correction. For $L\le 2$ the algebra supplies no finite upper bound of this form.

Linearize around $A=R{e}^{iqX}$ and write $r=R^2>0$. For sideband wavenumber $l$, the sum and difference of the two amplitude sidebands, together with the field amplitude $b$, evolve underExpanding $\det (\lambda I-M_l)$ givesAt $l=0$, the eigenvalues are $-2r,0,0$; the field's constant mode must be removed if its mean is prescribed. Nonzero long-wave field modes are still dynamically active.

Putting $\lambda =l^2\nu$ into the sideband cubic and retaining $l^4$ givesFor $r=1-q^2>0$, $\mu /\sigma >(1-3q^2)/(1-q^2)$ makes the constant coefficient negative. There is a positive real $\nu$, so sufficiently small nonzero $l$ grows. On a finite period $L$, however, only $l=2\pi j/L$ are allowed. For $q=0$, $\mu =2$, $\sigma =1$, $L=1$, every nonzero allowed mode has negative growth rates despite this long-wave inequality. Thus an arbitrarily-small-wave-number conclusion cannot be asserted without the domain-size qualification.