Magnetohydrodynamics (MHD)

Equating magnetic energy density

B^{2} / (2 μ_{0})

to kinetic energy density

ρ 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.

Convective collapse

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 μ_{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.

Magnetostrophic balance ( $2 ρ Ω \overset{z}{^} \times u = - \nabla p + f_{b} + J \times B$ )

The leading force balance in a rapidly rotating conducting fluid when inertial and viscous forces are small, while Coriolis force, pressure, buoyancy and Lorentz force density remain significant. The Rossby number and Ekman number measure the omitted inertia and bulk viscosity. Rotation does not balance an arbitrary magnetic torque: impenetrable axisymmetric boundaries impose the Taylor constraint. Magnetic diffusion can remain in the induction equation even when viscosity is neglected in momentum balance.

Taylor constraint ( $\int_{C_{s}} (J \times B)_{ϕ} d ϕ d z = 0$ )

In inviscid magnetostrophic balance within an impenetrable axisymmetric container, with no azimuthal buoyancy or external mechanical torque, the cylindrically integrated Lorentz force density must vanish on every cylinder coaxial with rotation. The azimuthal pressure derivative integrates to zero, and the cylindrical Coriolis integral is proportional to net radial flux, zero by incompressibility and impermeability. Multiplying by cylinder area and lever arm gives the equivalent zero-torque form. This rotating-fluid condition is distinct from magnetic-helicity relaxation to a force-free plasma state. Its original primary source is Taylor's 1963 rotating-fluid paper.

Viscous regularization of the Taylor constraint ( $T_{L} (s) + T_{ν} (s) = 0$ )

With small viscosity, the azimuthal cylindrical balance permits Lorentz force torque to be cancelled by viscous stress. For ordinary no-slip rotating boundaries an Ekman layer gives stress of order

ρ Ω LU E

, larger than the order-

E

bulk viscous correction at fixed velocity. Thus bounded small-

E

flows approach a magnetostrophic Taylor state but need not satisfy its constraint exactly at finite viscosity. An order-one uncompensated magnetic torque would require a divergent flow as

E \to 0

, invalidating the assumed small-inertia balance. Stress-free boundaries remove the leading no-slip friction and require their own torque/regularity conditions.

Geostrophic flow preserving the Taylor constraint ( $D T_{B} [\nabla \times (u \times B) + η \nabla^{2} B] = 0$ )

For an instantaneous magnetostrophic Taylor state, the momentum equation leaves a geostrophic null component

u_{g} = s ω (s) \hat{ϕ}

. Decompose the velocity into this component and a particular ageostrophic flow, substitute the induction equation into the displayed derivative constraint, and solve the resulting linear equation for

ω

with magnetic boundary conditions imposed. When magnetic surface torque vanishes, the geostrophic part is

μ_{0}^{- 1} \partial_{s} [s^{3} \int_{C_{s}} B_{s}^{2} d ϕ d z ω^{'}]

. Boundary terms generally cannot be discarded for an insulating exterior; the primary analysis is Hardy and collaborators' boundary-compatible geostrophic-flow construction. Regularity and an angular-momentum normalization fix remaining homogeneous freedoms when the magnetic coupling is nondegenerate.

Magnetostrophic Taylor state

A divergence-free magnetic field satisfying the Taylor constraint on all geostrophic cylinders. It is a compatibility condition for slow rotating-fluid balance, not a local force-free condition. To retain such a state as magnetic induction proceeds, the velocity must also preserve the time derivative of the cylindrical torque constraint.

Mean-field electrodynamics ( $E = ⟨ u \times b ⟩$ )

Mean-field electrodynamics applies averaging and a response closure to magnetic induction in a fluctuating conducting fluid. The mean equation contains the correlated mean-field electromotive force in the title. Scale separation permits a local expansion into alpha transport, turbulent diffusion and higher gradients; general anisotropic or finite-memory statistics require tensorial and nonlocal responses. A mean-field dynamo is a self-exciting application of this framework, not a consequence of averaging alone.

Kinematic magnetic dynamo ( $D_{t} B = (B \cdot \nabla) u$ )

A kinematic magnetic dynamo studies magnetic field amplification by a prescribed fluid flow, neglecting the field's feedback on the velocity through the Lorentz force. With incompressibility and zero magnetic diffusivity, a Lagrangian field vector evolves by its local velocity-gradient matrix. A growth law derived in this approximation cannot hold indefinitely once magnetic stresses significantly change the flow.

Gaussian white-noise magnetic stretching ( $\dot{B}_{i} = σ_{im} B_{m}$ )

For a trace-free isotropic Gaussian white noise velocity gradient with covariance

⟨ σ_{im} (t) σ_{jn} (t^{'})⟩ = κ_{2} δ (t - t^{'}) [δ_{ij} δ_{mn} - (δ_{im} δ_{jn} + δ_{in} δ_{jm}) /4]

, interpret the multiplicative equation as the Stratonovich integral limit of smooth velocity fluctuations. The causal Furutsu–Novikov formula has endpoint weight one half. It gives

P_{t} = (κ_{2} /2) \partial_{i} [(B^{2} δ_{ij} - B_{i} B_{j} /2) \partial_{j} P]

for the vector probability density function.

Lognormal magnetic-field amplification ( $ln (B / B_{0}) \sim N (3 D t, 2 D t)$ )

Set

x = ln (B / B_{0})

and

f = BF

in the radial magnetic-field Fokker-Planck equation. Then

f_{t} = D f_{xx} - 3 D f_{x}

, so

f

is a Gaussian distribution drifting at speed

3 D

. For initial magnitude

B_{0} > 0

F = [B 4 π D t]^{- 1} exp [- (ln (B / B_{0}) - 3 D t)^{2} / (4 D t)]

. This lognormal distribution has moments

B_{0}^{q} e^{D q (q + 3) t}

; the width of its logarithm grows as

2 D t

Magnetic stretching moment growth ( $\partial_{t} ⟨ B^{q} ⟩ = D q (q + 3) ⟨ B^{q} ⟩$ )

Twice integrating the radial magnetic-field Fokker-Planck equation by parts gives the displayed law when moments and endpoint terms are controlled. In particular

⟨ B^{2} ⟩

grows at rate

10 D = 2 γ

, with

γ = 5 κ_{2} /4

. The corresponding magnetic energy density is

⟨ B^{2} ⟩ / (8 π)

in Gaussian units.

Radial magnetic-field Fokker-Planck equation ( $F_{t} = D \partial_{B} (B^{2} F_{B} - 2 BF), D = κ_{2} /4$ )

For an isotropic initial ensemble in Gaussian white-noise magnetic stretching,

F (B) = 4 π B^{2} P (B)

is the magnitude probability density function. The radial vector equation

P_{t} = D (B^{2} P_{BB} + 4 B P_{B})

gives the displayed conservative Fokker-Planck equation. Its integral is conserved under zero endpoint probability flux. For an anisotropic initial vector distribution, use the angularly averaged magnitude density instead; isotropic noise alone does not make arbitrary initial data isotropic.

Theta pinch ( $B = B (r) \hat{z}$ )

A theta pinch is a cylindrical plasma configuration with axial magnetic field and azimuthal electric current. For a static equilibrium without gravity, the radial Lorentz force gives

d [p + B^{2} / (8 π)] / d r = 0

in Gaussian units. This straight-field configuration has no field-line curvature drive of the usual interchange instability.

Positive ideal energy of a theta pinch ( $δ W \geq 0$ )

For an admissible Fourier fluid Lagrangian displacement

ξ (r) e^{im θ + ik z}

, put

D_{⊥} = (r ξ_{r})^{'} / r + im ξ_{θ} / r

and

D = D_{⊥} + ik ξ_{z}

. Force balance cancels every magnetic-field-gradient cross term in the magnetohydrodynamic energy principle, leaving an integrand

γ p ∣ D ∣^{2} + B^{2} [∣ D_{⊥} ∣^{2} + k^{2} (∣ ξ_{r} ∣^{2} + ∣ ξ_{θ} ∣^{2})] / (4 π)

. For

p \geq 0

γ > 0

, this is nonnegative. Neutral displacements can exist, so stability here means absence of negative-energy exponentially growing ideal modes, not strictly positive energy for every displacement.

Hall magnetohydrodynamics (Hall MHD)

Hall magnetohydrodynamics retains bulk ion inertia but freezes magnetic flux into the Electron velocity

u_{e} = u - j / (e n)

. With constant density, negligible electrical resistivity and Electron inertia, Gaussian units give

j = c \nabla \times B / (4 π)

and

\partial_{t} B = \nabla \times [(u - c \nabla \times B / (4 π e n)) \times B]

. The ion velocity still satisfies the magnetohydrodynamic momentum equation. Neglecting that velocity gives electron magnetohydrodynamics, a further approximation rather than a synonym for the complete Hall system.

Fast-time electron-MHD limit of Hall magnetohydrodynamics ( $l ≪ d_{i}, τ ≪ l / v_{A}$ )

For field amplitudes of order the guide field, the ion Lorentz-force response over time

τ

Δ u / v_{A} = O (v_{A} τ / l) ≪ 1

. The electron-current velocity is of order

v_{A} d_{i} / l

, larger than a bulk velocity of order

v_{A}

. Thus the leading Hall magnetohydrodynamics induction equation becomes the closed electron magnetohydrodynamics equation, while ions remain fixed to leading order on this fast scale. The first nonzero ion response need not have zero acceleration. This limit retains the fast branch of incompressible Hall-MHD wave dispersion, not the slow branch; very oblique modes must separately satisfy the fast-time ordering.

Incompressible Hall-MHD wave dispersion ( $(ω^{2} - k_{∥}^{2} v_{A}^{2})^{2} = ω^{2} k_{∥}^{2} v_{A}^{2} k^{2} d_{i}^{2}$ )

For a uniform guide field, write the transverse perturbation in velocity units as

b

. Linear momentum gives

ω u = - k_{∥} v_{A} b

. Linear induction gives

ω b = - k_{∥} v_{A} u + i k_{∥} v_{A} d_{i} k \times b

. The helicity decomposition of a transverse Fourier mode then yields the displayed dispersion relation. Its two positive frequency magnitudes are

∣ k_{∥} ∣ v_{A} [1 + (k d_{i})^{2} /4 \pm k d_{i} /2]

, approaching the Alfvén wave frequency for

k d_{i} ≪ 1

. For

k d_{i} ≫ 1

, the fast branch is a whistler wave and the slow branch tends to

∣ k_{∥} ∣ v_{A} / (k d_{i})

Electron magnetohydrodynamics ( $\partial_{t} B = \nabla \times (u_{e} \times B)$ )

Electron magnetohydrodynamics neglects ion motion and Electron inertia while retaining transport of the magnetic field by the Electron current. At constant Electron number density,

u_{e} = - J / (e n) = - c \nabla \times B / (4 π e n)

. Flux freezing in this velocity gives

\partial_{t} B = - \frac{c}{4 π e n} \nabla \times [(\nabla \times B) \times B] .

For a uniform guide field, the solenoidal linear equation implies

ω^{2} = (c B_{0} /4 π e n)^{2} k_{∥}^{2} k^{2}

. With

v_{A} = B_{0} / 4 πn m_{i}

and

d_{i} = c m_{i} / (4 π e^{2} n)

, the coefficient is

c B_{0} / (4 π e n) = v_{A} d_{i}

Whistler wave ( $ω_{\pm} = \pm v_{A} d_{i} k_{∥} k$ )

In the inertia-free, constant-density electron magnetohydrodynamics model with uniform guide field

B_{0} \hat{z}

, a transverse magnetic plane wave obeys

ω b = i v_{A} d_{i} k_{∥} k \times b

. The helicity decomposition of a transverse Fourier mode gives the displayed dispersive branches with opposite circular polarizations. This electron-current mode is called a whistler wave; see Lyutikov's analysis of electron magnetohydrodynamics. Its pressure-free induction equation should not be identified with a general kinetic Alfvén model.

Reduced electron magnetohydrodynamics ( $\partial_{t} Ψ = v_{A}^{2} d_{i} \nabla_{∥} b, \partial_{t} b = - d_{i} \nabla_{∥} \nabla_{⊥}^{2} Ψ$ )

With

k_{∥} / k_{⊥} \sim δ B / B_{0} ≪ 1

, define

b = δ B_{∥} / B_{0}

and

δ B_{⊥} / B_{0} = \hat{z} \times \nabla_{⊥} Ψ/ v_{A}

to leading order. Let

\nabla_{∥} = \partial_{z} + {Ψ, \cdot} / v_{A}

, where

{f, g} = f_{x} g_{y} - f_{y} g_{x}

. The leading Electron velocities are

u_{e ⊥} = v_{A} d_{i} \hat{z} \times \nabla_{⊥} b

and

u_{e ∥} = - d_{i} \nabla_{⊥}^{2} Ψ

. Substitution into frozen-in induction yields

\partial_{t} Ψ = v_{A}^{2} d_{i} \nabla_{∥} b, \partial_{t} b = - d_{i} \nabla_{∥} \nabla_{⊥}^{2} Ψ.

Linearization gives

ω^{2} = v_{A}^{2} d_{i}^{2} k_{∥}^{2} k_{⊥}^{2}

, agreeing with the full EMHD relation to leading anisotropic order. If

b

varies along the guide field, an order-smaller perpendicular gradient correction restores exact solenoidality; the two-field representation is not an exact decomposition at arbitrary anisotropy.

Stratified magneto-Coriolis wave ( $K^{2} (ω^{2} - a^{2})^{2} - 4 Ω^{2} k_{z}^{2} ω^{2} + H h^{2} (ω^{2} - a^{2}) = 0$ )

An incompressible Boussinesq fluid with uniform magnetic field, uniform vertical rotation and uniform temperature gradient supports waves combining magnetic tension, Coriolis force and buoyancy. Put

K = ∣ k ∣

h^{2} = K^{2} - k_{z}^{2}

a^{2} = (B \cdot k)^{2} / (μ_{0} ρ)

and

H = gα β

, with

\partial_{z} T_{0} = - β

. Eliminating pressure and the temperature and magnetic perturbations gives the displayed dispersion relation for nonzero frequencies. Stable thermal stratification has

H < 0

. Clearing frequency denominators requires checking any resulting zero roots in the original equations; a stationary magnetostatic-buoyancy balance can exist independently of the propagating branches.

Rapid-rotation slow magneto-Coriolis branch ( $ω_{s}^{2} \sim a^{2} (K^{2} a^{2} - H h^{2}) / (4 Ω^{2} k_{z}^{2})$ )

For fixed wavevector with

k_{z} \neq = 0

, fixed nonzero Alfvén frequency

a

, and

∣Ω∣ \to \infty

, the two roots in

X = ω^{2}

of the stratified magneto-Coriolis wave polynomial have

X_{f} \sim 4 Ω^{2} k_{z}^{2} / K^{2}

and the displayed slow root. This follows from the sum of roots being

4 Ω^{2} k_{z}^{2} / K^{2} + O (1)

and their product being

a^{2} (K^{2} a^{2} - H h^{2}) / K^{2}

. The slow branch is oscillatory for

K^{2} a^{2} > H h^{2}

and growing or decaying for the reversed inequality. At equality its frequency vanishes and the original stationary equations must be used.

Lehnert number ( $Le = v_{A} / (Ω L)$ )

The Lehnert number compares an Alfvén speed to a rotation speed on length scale

L

. With

Le ≪ 1

, magnetic-wave frequencies are small compared with the rotation frequency, allowing nearly geostrophic flow. Ideal weakly damped magnetic waves also require

η / L^{2}

and

ν / L^{2}

to be small compared with their wave frequency; a small Lehnert number alone does not guarantee weak diffusion.

Magnetohydrodynamic momentum equation ( $ρ D_{t} u = - \nabla p + J \times B + ρ ν \nabla^{2} u$ )

For constant mass density

ρ

, kinematic viscosity

ν

and magnetic permeability

μ_{0}

, the incompressible momentum equation is

\partial_{t} u + (u \cdot \nabla) u = - ρ^{- 1} \nabla p + (μ_{0} ρ)^{- 1} (\nabla \times B) \times B + ν \nabla^{2} u .

It is the Navier-Stokes equation with Lorentz force density. The magnetic force decomposes into magnetic tension and a magnetic pressure gradient. A uniform streamwise pressure gradient need not make ordinary pressure independent of transverse coordinates.

Resistive magnetohydrodynamics

Resistive magnetohydrodynamics retains finite magnetic diffusivity in the coupled evolution of a conducting fluid and its magnetic field. The resistive induction equation allows field diffusion relative to the fluid, while the magnetohydrodynamic momentum equation includes the Lorentz force density. It contrasts with the zero-diffusivity limit of ideal magnetohydrodynamics.

Magnetic reconnection

Magnetic reconnection changes connectivity between magnetic field lines. It requires departure from smooth magnetic flux freezing, for example through finite resistivity or other nonideal terms concentrated in a current sheet. Ideal viscous magnetic relaxation dissipates mechanical energy while retaining frozen connectivity, so viscosity alone does not provide reconnection.

Magnetic skin layer ( $δ = 2 η / ω$ )

A magnetic skin layer is the thin region penetrated by an oscillating magnetic field in a conductor. For locally tangential boundary amplitude

C (x)

, negligible material motion and variations along the boundary much slower than those across it, the resistive induction equation gives

B_{x} = C (x) e^{- (1 + i) y / δ}

with time convention

e^{iω t}

. The solenoidal magnetic-field constraint requires the smaller component

B_{y} = C^{'} (x) e^{- (1 + i) y / δ} δ / (1 + i)

to leading order. The tangential-field approximation therefore does not mean that the normal component is exactly zero at finite magnetic diffusivity.

Magnetic skin-layer streaming slip ( $U = δ^{2} (C^{2})^{'} / (16 μ_{0} ρ ν)$ )

In a thin magnetic skin layer next to a rigid no-slip boundary condition, vertical pressure balance gives

p = P - C^{2} e^{- 2 y / δ} / (4 μ_{0})

. With negligible inertia in the layer, the tangential Stokes flow balance is

ρ ν u_{yy} = p_{x}

. Matching zero leading shear outside the layer gives

u = U (1 - e^{- 2 y / δ})

and the displayed effective tangential slip. The physical wall still has zero velocity; the slip is a boundary condition for the outer flow extrapolated through the unresolved layer. It points toward larger squared boundary-field amplitude.

Mean Lorentz force in a magnetic skin layer ( $\overline{F}_{y} = C^{2} e^{- 2 y / δ} / (2 μ_{0} δ)$ )

For real boundary amplitude

C

and

q = (1 + i) / δ

, the leading electric current density is

j_{z} = qC e^{- q y} / μ_{0}

. Averaging products of real harmonic fields gives

\overline{F} = Re [(\nabla \times B) \times B^{*}] / (2 μ_{0})

. Its leading normal component is the displayed positive force. The first candidate tangential contribution is proportional to

Re (q / q^{*}) C C^{'} = 0

, so retaining the small normal field does not introduce a tangential force of that order.

Flux expulsion

Flux expulsion is redistribution of magnetic flux out of the interiors of persistent circulating flows into their edges and surrounding regions. In two-dimensional incompressible flow, a magnetic flux function obeys

\partial_{t} A + u \cdot \nabla A = η \nabla^{2} A

. At large but finite magnetic Reynolds number, circulation stretches gradients of

A

, enabling resistive smoothing and weak interior magnetic field with strong field near the circulation boundary. Finite magnetic diffusivity is essential to the eventual rearrangement; exact ideal magnetic flux freezing alone does not justify a flux-free steady cell. Magnetic back reaction can limit the process.

Phase mixing in magnetic flux expulsion ( $t_{exp} \sim (L / U) Rm^{1/3}$ )

Differential circulation of a weak magnetic field winds its flux function into progressively smaller radial scales. In a smooth eddy with angular-velocity gradient of order

U / L^{2}

, its radial wavenumber grows as

k_{r} \sim U t / L^{2}

. Resistive damping becomes substantial when

η \int_{0}^{t} k_{r}^{2} d t \sim 1

, giving

t \sim (L^{4} / (η U^{2}))^{1/3} = (L / U) Rm^{1/3}

. This estimate presumes differential rotation and negligible magnetic feedback; a rigidly rotating eddy does not create this phase mixing. Finite magnetic diffusivity then permits weak interior field and boundary concentration, unlike exact ideal flux freezing.

Cubic-time resistive damping in a differentially rotating cell ( $a \sim a_{0} exp [- η m^{2} (Ω^{'})^{2} t^{3} /3]$ )

In a smooth interior region with locally constant shear

Ω^{'}

, a wound azimuthal Fourier mode has radial wavenumber

k_{s} ≃ - m Ω^{'} t

. Integrating its resistive damping rate

η k_{s}^{2}

gives the exponential decay in the title. The characteristic time is

τ_{c} = [3/ (η m^{2} Ω^{'2})]^{1/3}

. For

m = 1

and

Ω^{'} = - Ω_{0} / d

Ω_{0} τ_{c} = (3 Rm)^{1/3}

, where

Rm = Ω_{0} d^{2} / η

. This is a high-magnetic Reynolds number interior transient on a time long compared with the rotation time but short compared with

d^{2} / η

. Physical magnetic field components may have additional algebraic factors from taking derivatives of the wound magnetic vector potential. A maintained exterior field, an axial core and a circulation-edge boundary layer prevent interpreting this estimate as an exact, uniform, infinite-time solution.

Dynamo action

Dynamo action is growth or sustained maintenance of a magnetic field by conducting-fluid motion despite magnetic diffusion. In the kinematic problem the velocity is prescribed and the resistive induction equation is linear in the magnetic field. Magnetic feedback through the Lorentz force density matters once the field becomes dynamically important.

Radial-flow dynamo energy bound ( $q^{2} \geq η^{2} D / M$ )

Let a solenoidal flow be confined to a sphere

V

, with the insulating boundary condition for the radial magnetic scalar. Define

N = \int_{V} P^{2}

D = \int_{R^{3}} ∣\nabla P ∣^{2}

M = \int_{R^{3}} ∣ B ∣^{2}

and

q = max_{V} ∣ u \cdot x ∣

. Integration by parts inside and outside the sphere gives

\frac{1}{2} N^{'} = - \int_{V} Q B \cdot \nabla P - ηD \leq q M D - ηD .

The Cauchy-Schwarz inequality therefore makes

q^{2} \geq η^{2} D / M

necessary whenever

N^{'} \geq 0

and

D > 0

. A uniform strict violation with a positive gap forces decay, using a Sobolev inequality to bound

N

by a constant times

D

. A bounded statistically steady field instead requires

q_{*}^{2} \geq η^{2} ⟨ D ⟩ / ⟨ M ⟩

, with

q_{*} = sup_{t} q (t)

. This is a necessary condition for dynamo action, not a sufficient criterion or a pointwise assertion at every instant of an arbitrary time-dependent solution.

// Target: analysis.bigb

Dynamo quenching ( $α (B) = α_{0} / (1 + B^{2} / B_{*}^{2})$ )

Dynamo quenching is the reduction or modification of magnetic-field generation by magnetic feedback on the generating flow and its correlations. The Lorentz force can change shear or turbulence, while magnetic helicity affects mean-field regeneration. A phenomenological scalar example is

α (B) = α_{0} / (1 + B^{2} / B_{*}^{2})

. In the illustrative amplitude equation

\dot{B} = [c α (B) - d] B

with

c, d > 0

, a growing field saturates at

B^{2} = B_{*}^{2} (c α_{0} / d - 1)

when

c α_{0} > d

. This demonstrates the negative feedback mechanism without claiming that algebraic quenching is a complete solar closure.

Algebraic alpha quenching ( $α (B) = α_{0} - p ∣ B ∣^{2}$ )

Algebraic alpha quenching prescribes a field-dependent alpha effect as a closure for nonlinear mean-field dynamo feedback. The polynomial law

α = α_{0} - p ∣ B ∣^{2}

with

p > 0

decreases alpha, so it can saturate positive alpha but makes negative alpha more negative. It therefore differs importantly from the sign-preserving rational law

α_{0} / (1 + ∣ B ∣^{2} / B_{*}^{2})

. Its solutions must satisfy nonnegative squared-amplitude conditions; an assertion based only on large

∣ α_{0} ∣

need not hold for both signs.

Alpha-quenching rotating wave ( $ω = \pm 1$ )

A rotating wave in a complex dynamo quenching model has constant amplitude and a common phase factor

e^{iω t}

. In the two-hemisphere model with toroidal coupling parameter

ϵ_{2} = 0

, eliminating the poloidal amplitudes gives

\frac{2 ω - i ( 1 - ω ^{2} )}{Ω} (B_{1} B_{2}) = (α_{0} - p ∣ B_{1} ∣^{2} ϵ_{1} ϵ_{1} α_{0} - p ∣ B_{2} ∣^{2}) (B_{1} B_{2}) .

The matrix is Hermitian, so its eigenvalue is real and a nonzero rotating wave with

Ω \neq = 0

has

ω = \pm 1

. Writing

a = α_{0} - 2 ω /Ω

, pure dipole and quadrupole waves have

∣ B_{j} ∣^{2} = (a - ϵ_{1}) / p

and

(a + ϵ_{1}) / p

, respectively, when those quantities are positive.

Unequal-hemisphere dynamo rotating wave ( $∣ B_{1, 2} ∣^{2} = (a \pm a^{2} - 4 ϵ_{1}^{2}) / (2 p)$ )

For nonzero coupling

ϵ_{1}

, an alpha-quenching rotating wave can be made real in its toroidal amplitudes by a common phase rotation. Set

a = α_{0} - 2 ω /Ω

. The amplitude equations are

(a - p B_{1}^{2}) B_{1} + ϵ_{1} B_{2} = 0

and

(a - p B_{2}^{2}) B_{2} + ϵ_{1} B_{1} = 0

. Subtracting after cross multiplication gives

(B_{1}^{2} - B_{2}^{2}) (p B_{1} B_{2} + ϵ_{1}) = 0

. Unequal magnitudes therefore require

B_{1} B_{2} = - ϵ_{1} / p

and

B_{1}^{2} + B_{2}^{2} = a / p

. They exist exactly when

a > 2∣ ϵ_{1} ∣

and have the displayed squared amplitudes. Exchanging hemispheres yields the companion solution. This breaks equatorial reflection symmetry and combines both dipole and quadrupole parity in a mean-field dynamo. Existence does not alone prove nonlinear stability.

Equivariant three-mode dynamo normal form ( $B \in R, A, V \in C$ )

An axisymmetric magnetic mode is unchanged by axial rotations, while an equatorial magnetic mode and a corresponding velocity mode transform as

A, V \mapsto e^{i θ} (A, V)

. Magnetic reversal sends

(B, A, V)

(- B, - A, V)

. For magnetic evolution linear in magnetic amplitudes, rotational equivariance requires

\dot{B} = B f (∣ V ∣^{2}) + 2 Re [A^{*} V g (∣ V ∣^{2})]

and

\dot{A} = A h (∣ V ∣^{2}) + B Vj (∣ V ∣^{2}) + A^{*} V^{2} ℓ (∣ V ∣^{2})

, with

f

real and the other functions complex. Velocity monomials

B^{p} A^{q} (A^{*})^{r} V^{s} (V^{*})^{t}

must have even

p + q + r

and weight

q - r + s - t = 1

. These conditions enumerate the cubic normal form without assuming reflection symmetry or nonlinear magnetic self-interactions.

Magnetic onset on a rotating velocity branch ( $(μ_{1} - e s^{2}) [μ_{2}^{2} + (ω_{2} - ω_{3})^{2}] = 2 c_{2} μ_{2} s^{2}, s^{2} = μ_{3} / d_{4}$ )

For real

c_{2}, d_{4}

and the simplified magnetic amplitude equations, the pure-velocity branch is

V = s e^{i ω_{3} t}

with

A = B = 0

. In its rotating frame the magnetic linearization on

(B, Re A, Im A)

has rows

(μ_{1} - e s^{2}, 2 s, 0)

(c_{2} s, μ_{2}, - Δ)

and

(0, Δ, μ_{2})

, where

Δ = ω_{2} - ω_{3}

. Its zero determinant gives the displayed onset relation. A nonzero magnetic branch needs a kernel involving both magnetic amplitudes, a transverse parameter crossing and nondegenerate nonlinear saturation; determinant zero alone is insufficient in degenerate cases.

Normalization of nonzero dynamo coupling coefficients ( $c_{1} = 1, ∣ c_{3} ∣ = 1$ )

For nonzero coefficients of

A^{*} V

\dot{B}

and

A B

\dot{V}

, a relative phase rotation makes

c_{1}

real positive. Positive rescalings of

A, V, B

then normalize both magnitudes to one. The phase of

c_{1} c_{3}

is invariant under these rescalings and remains as the phase of normalized

c_{3}

. Zero couplings cannot be normalized to one; they are degenerate parameter cases.

Anti-dynamo theorem

An anti-dynamo theorem gives a geometric or analytic restriction excluding self-sustained dynamo action under stated magnetic diffusivity, domain and boundary conditions. It excludes regeneration against resistive decay, not all transient amplification or field induced by an external source. Important restrictions include Cowling anti-dynamo theorem, the toroidal-velocity anti-dynamo theorem, and the planar anti-dynamo theorem. Backus' necessary condition for dynamo action instead bounds the stretching required for possible regeneration.

Periodic two-coordinate anti-dynamo energy bound ( $Δ ⟨ B_{z}^{2} ⟩ \leq Q^{2} ⟨ A_{0}^{2} ⟩ / (4 η^{2} k^{2})$ )

For a solenoidal three-component velocity depending on two periodic coordinates and a zero-mean magnetic field with the same dependence, the in-plane flux potential satisfies a homogeneous advection-diffusion equation. The Poincaré inequality gives

⟨ A^{2} ⟩ \leq ⟨ A_{0}^{2} ⟩ e^{- 2 η k^{2} t}

. Integrating the vertical stretching term by parts bounds it by

Q ⟨ A^{2} ⟩ ⟨ ∣\nabla B_{z} ∣^{2} ⟩

, where

Q = ∥\nabla u_{z} ∥_{\infty}

. Maximizing against resistive dissipation and integrating gives the displayed transient bound. Splitting off half the dissipation instead gives

E^{'} + η k^{2} E \leq (Q^{2} / η) ⟨ A_{0}^{2} ⟩ e^{- 2 η k^{2} t}

, proving vertical energy decays. This permits transient generation by a nonzero vertical velocity while excluding sustained fields in this zero-mean, z-independent periodic class.

Planar anti-dynamo theorem

In the standard two-dimensional setting, with planar velocity and magnetic field independent of the third coordinate and homogeneous energy-closed boundary conditions, the in-plane magnetic potential is an advected diffusing scalar. Its quadratic integral decreases, preventing regeneration of a dynamo field. A third field component independent of that coordinate also obeys a passive advection-diffusion equation. A flow depending on two coordinates but possessing a third velocity component is not covered by this strictly planar formulation.

Zeldovich planar flux balance ( $B_{0} ⟨ w a^{'} ⟩ = - η ⟨ ∣\nabla a^{'} ∣^{2} ⟩ - η ⟨ \overset{a}{ˉ}_{z}^{2} ⟩$ )

For a stationary planar advected magnetic flux potential, average its advection-diffusion equation horizontally and split it into mean and fluctuation. The mean transport is

\overline{w a^{'}} = η \overset{a}{ˉ}_{z} + C

, with

C = ⟨ w a^{'} ⟩

. Multiplying the fluctuation equation by

a^{'}

and applying integration by parts gives

⟨ \overline{w a^{'}} \overset{a}{ˉ}_{z} ⟩ + B_{0} C = - η ⟨ ∣\nabla a^{'} ∣^{2} ⟩

. A vanishing mean derivative removes the constant term in the first correlation, yielding the displayed dissipative identity. The boundary assumptions must eliminate surface terms; a periodic layer replaces, rather than accompanies, decay at vertical infinity.

Toroidal-velocity anti-dynamo theorem

For a bounded conducting sphere with uniform magnetic diffusivity and an insulating exterior, an incompressible flow tangent to concentric spheres cannot sustain dynamo action. Neither the velocity nor the magnetic field needs to be axisymmetric. The radial magnetic scalar has a homogeneous advection-diffusion equation, so the poloidal field decays. Once it is absent, the toroidal magnetic potential also has a homogeneous scalar advection-diffusion equation. This is a theorem about spherical flow geometry, not about a general cylindrical flow with an axial component.

Cowling anti-dynamo theorem

An isolated axisymmetric magnetic field cannot be sustained by ordinary resistive dynamo action with scalar positive magnetic diffusivity. Its poloidal magnetic flux function satisfies a homogeneous scalar advection-diffusion equation with no toroidal-to-poloidal source. The maximum principle excludes regeneration of that flux; after it decays the toroidal field has no lasting source either. This constrains the symmetry of the field. An axisymmetric flow may still generate a nonaxisymmetric field, and a turbulent mean-field electromotive force is not covered by the unmodified scalar-induction proof.

Magnetic concentration by an incompressible stagnation flow

The linear incompressible flow

u = (- ω x, - ω y, 2 ω z)

compresses the transverse plane and stretches the axial direction. An axial magnetic field obeys

B_{t} - ω r B_{r} = 2 ω B + η Δ_{⊥} B

. In ideal magnetohydrodynamics, its amplitude grows as

e^{2 ω t}

while its transverse width decreases as

e^{- ω t}

. Thus its magnetic energy per unit axial length grows as

e^{2 ω t}

, even when opposite-polarity angular sectors give zero signed magnetic flux. Finite magnetic diffusivity limits the amplification of localized finite-flux Gaussian fields.

Self-similar magnetic mode in a stagnation flow ( $B_{z} = f (t) b (r / g (t)) cos (m ϕ)$ )

For magnetic concentration by an incompressible stagnation flow, one possible normalization gives

(g^{2})^{'} = 2 η - 2 ω g^{2}

f^{'} / f = 2 ω - λ η / g^{2}

, and

b^{''} + (q^{- 1} + q) b^{'} + (λ - m^{2} / q^{2}) b = 0

. The mode parameter

λ

is not fixed by decay at infinity alone. The regular axisymmetric Gaussian has

λ = 2

b = e^{- q^{2} /2}

and conserved axial magnetic flux, with energy proportional to

g^{- 2}

. The radial equation also has regular algebraic-tail solutions.

Backus' necessary condition for dynamo action ( $S R^{2} / η \geq π^{2}$ )

For an isolated bounded conductor of uniform magnetic diffusivity

η > 0

, contained in a sphere of radius

R

and matched to a decaying potential field in an insulating exterior, dynamo action requires maximum stretching rate

S \geq η π^{2} / R^{2}

. Here

S

bounds the largest eigenvalue of the rate-of-strain tensor throughout the flow and time. Use fluid boundary conditions eliminating the stretching boundary term, for example a no-slip boundary condition, and no imposed energy input. The proof combines the magnetic free-decay spectral bound with the magnetic energy equation. This is a necessary condition, not a sufficiency criterion; changing magnetic boundary conditions changes the spectral constant.

Maximum-strain bound on dynamo growth ( $g_{B} \leq sup_{x, t} λ_{m a x} (e (x, t))$ )

Under the energy-closed boundary assumptions of Backus' necessary condition for dynamo action, the magnetic energy equation gives

\dot{E}_{B} \leq 2 s (t) E_{B}

after discarding nonnegative resistive dissipation. Integrating gives an upper bound by the time average of the largest spatial eigenvalue of the rate-of-strain tensor. Thus the field-amplitude exponent is at most its space-time supremum. The exponent of squared energy is twice the field-amplitude exponent.

Mean-field dynamo

A mean-field dynamo evolves a slowly varying spatially averaged magnetic field. Correlations of fluctuating velocity and magnetic field supply a mean-field electromotive force. Its local linear response includes the alpha effect; gradients of the mean field can contribute additional transport terms. The distinction between a uniform test field used to measure a response coefficient and a spatially varying mean field used to obtain growth is essential.

Mean-field shearing-wave transient amplification ( $t_{peak} ≃ ∣ α ∣ T / (η k_{0})$ )

For

k (t) = k_{0} 1 + (t / T)^{2}

, factor resistive damping out of the shearing-mode equations by

R = e^{- D} G

S = e^{- D} H

D = η \int k^{2} d t

. They imply

G^{''} = α^{2} k^{2} G

, so a growing WKB approximation has

G \propto k^{- 1/2} e^{∣ α ∣ \int k d t}

and

H ≃ i sgn (α) k G

. Its magnetic energy is proportional to

k exp [2∣ α ∣ \int k d t - 2 η \int k^{2} d t]

. At long times the positive quadratic exponent is overtaken by negative cubic resistive damping. A well-separated growing interval exists for sufficiently small positive magnetic diffusivity, with

η k_{0} /∣ α ∣ ≪ 1

at fixed nonzero

∣ α ∣ k_{0} T

, and the displayed leading peak time. This is transient amplification, not an asymptotically self-sustaining dynamo.

Dipole and quadrupole parity in a mean-field dynamo

In a reflection-symmetric mean-field dynamo model, dipole parity can be represented by an even poloidal-potential amplitude and an odd toroidal-field amplitude; quadrupole parity reverses these assignments. The names describe the reflection classes appropriate to the model, not arbitrary electric multipoles. A complex amplitude

P = A + i B

allows a phase rotation by

i

to interchange the two classes when the unperturbed equations have that phase symmetry.

Coupled-hemisphere dynamo parity threshold ( $F_{d} = Ω^{2} (α_{0} - ϵ_{1})^{2} /4 - ϵ_{2} (α_{0} - ϵ_{1})$ )

In a two-hemisphere mean-field dynamo, dipole and quadrupole subspaces can reduce to

\dot{A} = a B - A

\dot{B} = b A - B

, with

(a, b) = (α_{0} - ϵ_{1}, i Ω - ϵ_{2})

and

(α_{0} + ϵ_{1}, i Ω + ϵ_{2})

respectively. Their eigenvalues satisfy

(λ + 1)^{2} = ab

. Strict decay is equivalent to

Re (ab) + [Im (ab)]^{2} /4 < 1

. Thus dipole decay requires

F_{d} < 1

, while quadrupole decay requires

F_{q} = Ω^{2} (α_{0} + ϵ_{1})^{2} /4 + ϵ_{2} (α_{0} + ϵ_{1}) < 1

. Their difference is

F_{q} - F_{d} = α_{0} (Ω^{2} ϵ_{1} + 2 ϵ_{2})

, which selects the first unstable parity when starting from a state stable in both subspaces.

Parity splitting of a mean-field dynamo threshold

An antisymmetric perturbation can split the thresholds of two degenerate mean-field dynamo parity classes. At first order, project the perturbation equation onto the adjoint null mode using the Fredholm alternative. If multiplying the unperturbed complex mode by

i

leaves the parameter-derivative matrix element unchanged but reverses the perturbation matrix element, the two first-order threshold shifts are equal and opposite.

Alpha-Omega dynamo ( $α Ω$ )

An alpha-Omega mean-field dynamo couples toroidal-field generation by the Omega effect to poloidal-field regeneration by the alpha effect. A simple Fourier model is

\dot{A} = α_{0} f (t) B - η k^{2} A

\dot{B} = ik Ω A - η k^{2} B

, where

A

is a component of the vector potential for the poloidal field and

B

is a toroidal-field amplitude. For bounded

∣ f ∣ \leq 1

, the bounded-modulation alpha-Omega growth estimate controls the maximum amplification.

Plane-layer alpha-Omega dynamo threshold ( $∣ D ∣_{c} = 32/ (33), ∣ k_{c} ∣ = π / (3 d)$ )

For a mean-field dynamo with uniform alpha, shear

πV / d

, diffusivity

η > 0

and fixed vertical wavenumber

π / d

, a horizontal Fourier mode obeys

(s + η (k^{2} + π^{2} / d^{2}))^{2} = i α πVk / d

. With

D = α V d^{2} / (π^{2} η^{2})

and

h = ∣ k ∣ d / π

, the growing branch has positive real part exactly when

∣ D ∣ > 2 (1 + h^{2})^{2} / h

. Differentiation gives

3 h^{4} + 2 h^{2} - 1 = 0

, hence

h = 1/ 3

. Equality is a neutral oscillation, not exponential growth. The sign of

D

reverses wave propagation without changing the growth threshold.

Alpha-squared Omega dynamo threshold ( $Ω_{0}^{2} = 4 η^{2} ℓ^{4}$ )

For constant coefficients and a transverse wavenumber

ℓ \neq = 0

, a planar mean-field dynamo retaining both alpha feedback and shear has dispersion

(σ + η p)^{2} = α^{2} p + i α Ω k

p = k^{2} + ℓ^{2}

. Marginal stability gives

α_{m}^{2} = 4 η^{4} p^{4} / (4 η^{2} p^{3} + Ω^{2} k^{2})

. Differentiate with respect to

k^{2}

: an interior minimum solves

4 η^{2} p^{3} + Ω^{2} (3 k^{2} - ℓ^{2}) = 0

. Its derivative with respect to

k^{2}

is positive, so a unique positive minimizing

k^{2}

exists only above the displayed shear threshold. Below it the minimum is at

k = 0

. The sign of the marginal oscillation frequency is fixed by

2 η p ω = α Ω k

Bounded-modulation alpha-Omega growth estimate ( $g_{m a x} = Θ (∣ α_{0} Ω ∣^{2/3} η^{- 1/3})$ )

In the two-component alpha-Omega dynamo with

∣ f (t) ∣ \leq 1

, apply the weighted energy estimate for two coupled modes with

a = ∣ α_{0} ∣

c = ∣ k Ω∣

, and

d = η k^{2}

. The field-amplitude growth rate is at most

∣ α_{0} Ω k ∣ - η k^{2}

. Optimizing over

k

gives the upper constant

3/ 2^{8/3}

multiplying

(α_{0}^{2} Ω^{2} / η)^{1/3}

. The admissible choice

f = 1

gives a lower constant

3/ 2^{10/3}

, so the maximum possible growth has the displayed scaling. An arbitrary chosen modulation need not grow. The statement concerns the model with freely adjustable wavenumber; physical scale-separation restrictions can constrain that optimization.

Omega effect ( $\partial_{t} B_{tor} \sim Ω B_{pol}$ )

Differential rotation or shear stretches a poloidal magnetic field into a toroidal magnetic field. This is the Omega effect. By itself it need not regenerate the poloidal component; the alpha effect provides one possible feedback mechanism in an alpha-Omega dynamo. Without feedback, shear amplification can be transient or algebraic rather than sustained exponential dynamo action.

First-order smoothing approximation

First-order smoothing neglects the fluctuating nonlinear velocity-magnetic-field product in the fluctuation resistive induction equation, but retains

⟨ u \times b ⟩

in the mean-field electromotive force. For a constant test field it gives

\partial_{t} b - η \nabla^{2} b = (B_{0} \cdot \nabla) u

. Its validity requires the omitted terms to be small compared with the retained forcing and diffusion terms.

Isotropic alpha effect of three helical traveling waves ( $α = - \frac{η k ^{2}}{3 [( η k ^{2} ) ^{2} + Ω ^{2} ]} ⟨ u \cdot \nabla \times u ⟩$ )

Three equal-amplitude transverse waves along orthogonal axes, with the same circular polarization, each satisfy

\nabla \times u_{j} = σk u_{j}

σ = \pm 1

. Distinct spatial phases have zero cross averages. The periodic first-order smoothing response to a uniform test field satisfies

(\partial_{t} - η Δ) b_{j} = (\overline{B} \cdot \nabla) u_{j}

. Writing

a = η k^{2}

, the response is

b_{j} = k \overline{B}_{j} [a \partial_{ϕ} u_{j} - Ω u_{j}] / (a^{2} + Ω^{2})

. Since

u_{j} \times \partial_{ϕ} u_{j} = - σ U^{2} e_{j}

, summing yields the displayed isotropic alpha effect. It includes the resistive and temporal weighting absent from an unqualified instantaneous helicity rule.

Helicity formula for isotropic first-order smoothing ( $α = - \frac{1}{3} \int_{0}^{\infty} ⟨ u (t) \cdot \nabla \times e^{η τ Δ} u (t - τ)⟩ d τ$ )

For a locally uniform mean magnetic field, first-order smoothing solves

\partial_{t} b - η Δ b = (\overline{B} \cdot \nabla) u

. Inserting its causal diffusion propagator into the mean-field electromotive force and taking the trace of the isotropic response tensor gives the displayed relation to kinetic helicity. When diffusion during a correlation time is negligible, it reduces to

α = - τ_{H} ⟨ u \cdot \nabla \times u ⟩ /3

. At small magnetic Reynolds number, a quasistatic monochromatic velocity instead gives

α = - ⟨ u \cdot \nabla \times u ⟩ / (3 η k_{0}^{2})

. The effective correlation time and diffusion weighting should not be silently discarded.

Quasistatic magnetic response of a helical shearing wave ( $E ≃ - B_{0} H / (η ∣ k ∣^{2})$ )

For background shear

Ω y x

and uniform field

B_{0} x

, first-order smoothing gives the magnetic shearing-wave amplitude equation

\dot{c} = Ω c_{y} x + i k_{x} B_{0} v - η K^{2} c .

If magnetic relaxation is fast compared with shear,

c_{y}, c_{z} ≃ i k_{x} B_{0} (v_{y}, v_{z}) / (η K^{2})

after free transients decay. Thus the doubled Fourier-mode electromotive quantity

E = Re (v_{y} c_{z}^{*} - v_{z} c_{y}^{*})

- B_{0} H / (η K^{2})

, where

H = i k_{x} (v_{y} v_{z}^{*} - v_{z} v_{y}^{*})

. The spatially averaged mean-field electromotive force of real waves has an extra factor

1/2

. Using the helicity invariant of an inviscid shearing wave, and assuming the approximation holds uniformly in time with

Ω k_{x} \neq = 0

, gives

\int_{- \infty}^{\infty} E d t ≃ - \frac{π B _{0} H ( 0 )}{2 η ∣Ω k _{x} ∣ k _{x}^{2} + k _{z}^{2}} .

(2)

Indeed

K^{2} = A + Ω^{2} k_{x}^{2} t^{2}

A = k_{x}^{2} + k_{z}^{2}

, and substitution

z = ∣Ω k_{x} ∣ t / A

uses

\int_{- \infty}^{\infty} (1 + z^{2})^{- 2} d z = π /2

Monochromatic coupled magnetic and velocity response ( $D = ν η k^{4} + (B \cdot q)^{2} / (μ_{0} ρ)$ )

For a steady forced Fourier mode with

∣ q ∣ = k

, coupled viscous and magnetic response gives

u = [η k^{2} f + i (B \cdot q) g / (μ_{0} ρ)] / D

and

b = [ν k^{2} g + i (B \cdot q) f] / D

. Their mean-field electromotive force contains both forcing helicities and a possible weighted cross-forcing correlation. Jointly isotropic forcing removes the latter; its vanishing at zero mean field alone is insufficient for anisotropic forcing.

Oscillatory magnetic response in first-order smoothing ( $a = η k^{2}, d = a^{2} + Ω^{2}$ )

For a velocity Fourier mode with wavenumber

k

, define

a = η k^{2}

d = a^{2} + Ω^{2}

and

C = i (B_{0} \cdot k) \hat{u}

. A cosine-forced mode has periodic response coefficients

(p, q) = (a C, Ω C) / d

, while a sine-forced mode has

(- Ω C, a C) / d

. Homogeneous transients decay as

e^{- a t}

. The periodic response is a particular solution, not arbitrary initial data.

Parker dynamo wave

A Parker dynamo wave is a traveling or oscillatory mode of a mean-field dynamo coupling poloidal and toroidal magnetic-field components through an alpha effect and shear. A reduced complex-amplitude model can represent fluctuating alpha by a time-dependent coupling. Its periodically switched growth must be calculated from the monodromy matrix of a periodic linear system, rather than by averaging noncommuting generators.

Local alpha-Omega dynamo wave dispersion ( $(p + η_{T} k^{2})^{2} = i α S k$ )

Take a right-handed local Cartesian frame with

x

towards the equator,

y

azimuthal and

z

radially outwards in the northern hemisphere. A local radial shear

U_{y} = S z

and a poloidal magnetic field represented by

A y

give the alpha-Omega dynamo equations

A_{t} = α B + η_{T} A_{xx}

and

B_{t} = S A_{x} + η_{T} B_{xx}

. Substitution of a Fourier mode proportional to

e^{pt + ik x}

gives the displayed dispersion relation. The growing branch has

Re p = - η_{T} k^{2} + ∣ α S k ∣/2

and

Im p = sgn (α S k) ∣ α S k ∣/2

. Its phase velocity is

- Im p / k

: it propagates equatorwards when

α S < 0

, polewards when

α S > 0

. The result assumes local constant coefficients and no meridional circulation in a star; advection and spatially separated field regeneration can alter migration.

Periodically reversing Parker dynamo coupling ( $tr N = cosh (2 D T) + cos (2 D T)$ )

For

\dot{A} = D^{2} f (t) B

and

\dot{B} = i A

, with

f

alternating between

+ 1

and

- 1

for intervals of length

T

, the monodromy matrix of a periodic linear system has determinant one and trace

cosh (2 D T) + cos (2 D T) > 2

. Its reciprocal Floquet multipliers give positive dominant Floquet growth rate for

D, T > 0

, despite the zero mean of

f

. An exceptional initial state in the stable eigenspace decays.

Alpha effect ( $E = α \overline{B}$ )

The alpha effect is the part of a mean-field electromotive force linear in the mean magnetic field itself,

E = α \overline{B} + \dots

. The alpha tensor can be anisotropic; it need not be a scalar multiple of the identity. Its curl can couple transverse mean-field components and produce an alpha-squared dynamo.

Rapidly fluctuating alpha effect ( $\overline{α B} = - C \partial_{x} \overline{B}$ )

In a Parker dynamo wave model, an alpha effect of the form

α = ϵ^{- 1} α_{0} sin (ω t / ϵ)

has zero time average but can generate a finite averaged coupling. The method of multiple scales gives

A = - (α_{0} / ω) cos (ω τ) \overline{B}

and

B = - (Ω α_{0} / ω^{2}) sin (ω τ) \partial_{x} \overline{B}

, where

τ = t / ϵ

. Consequently

C = Ω α_{0}^{2} / (2 ω^{2})

. With diffusion operator

\partial_{x}^{2} - ℓ^{2}

, the averaged Fourier mode growth exponents are

- k^{2} - ℓ^{2} \pm ∣ k ∣ Ω C

. For freely adjustable real

k

, growth requires

Ω C > 4 ℓ^{2}

. The result is an asymptotic averaged model for fixed nonzero

ω

, not an exact replacement at finite

ϵ

Strong-field quenching of an isotropic electromotive force ( $α_{∥} \propto ∣ B ∣^{- 3}$ )

For isotropic monochromatic forcing, the longitudinal coefficient in a mean-field electromotive force is proportional to

\int_{0}^{1} μ^{2} (1 + h^{2} μ^{2})^{- 2} d μ = [arctan h - h / (1 + h^{2})] / (2 h^{3})

, where

h

is proportional to mean-field strength. This gives generic

∣ B ∣^{- 3}

quenching. The natural angular alpha tensor is anisotropic in a nonzero mean field: its transverse coefficient instead contains

\int_{0}^{1} (1 - μ^{2}) (1 + h^{2} μ^{2})^{- 2} d μ

and scales as

∣ B ∣^{- 1}

. Only the longitudinal contraction enters

E = α B

. An effective scalar representation may therefore have every diagonal entry equal to the longitudinal coefficient without being the unique natural response tensor.

Alpha-squared dynamo ( $α^{2}$ dynamo)

An alpha-squared dynamo generates a mean magnetic field using an alpha effect without requiring an additional large-scale shear coupling. For

α = α_{0} diag (1, 1, 0)

and a transverse mean field varying as

e^{i Kz}

, the growth rates are

- η K^{2} \pm α_{0} K

. A nonzero

α_{0}

therefore allows growth for sufficiently long wavelengths, provided the domain and scale separation permit them.

Homogeneous alpha-squared dynamo growth criterion ( $σ_{\pm} = \pm α k - (η + β) k^{2}$ )

For constant isotropic transport coefficients and zero mean flow, a divergence-free Fourier mode has two circular polarizations that diagonalize curl with eigenvalues

\pm k

. The mean induction equation therefore has the displayed growth rates. Growth requires an admissible

k

with

∣ α ∣ > (η + β) k

. Freely variable wavenumber gives

k_{*} = ∣ α ∣/ [2 (η + β)]

, provided it respects mean-field scale separation. A uniform field has zero curl and cannot grow by constant alpha alone; boundaries fix the longest available spatial mode.

Anisotropic alpha-squared dynamo ( $\partial_{t} B = \nabla \times (α B) + η \nabla^{2} B$ )

An alpha-squared dynamo may use a tensorial alpha effect rather than an isotropic coefficient. For constant alpha tensor

α

and magnetic diffusivity

η > 0

, a steady nonzero Fourier mode with wavevector

K

satisfies

η K^{2} B = i K \times (α B)

and

K \cdot B = 0

. Anisotropy can change both the critical alpha magnitude and the preferred wavevector direction.

Uniaxial alpha dynamo threshold ( $γ^{2} = (q^{2} + m^{2})^{2} / (m^{2} + δ q^{2})$ )

For

α = diag (α_{0}, α_{0}, α_{1})

, set

γ = α_{0} / η

δ = α_{1} / α_{0}

, and

q^{2} = k^{2} + l^{2}

. A steady nonzero Fourier mode obeys the displayed condition when the denominator is nonzero. At fixed

q > 0

and

0 < δ < 1

, minimizing over

m^{2} \geq 0

gives

m_{*}^{2} = (1 - 2 δ) q^{2}

and

γ_{m i n}^{2} = 4 (1 - δ) q^{2}

for

δ < 1/2

; for

δ \geq 1/2

it gives

m_{*}^{2} = 0

and

γ_{m i n}^{2} = q^{2} / δ

. This is a boundary-constrained threshold of an anisotropic alpha-squared dynamo.

Alpha tensor ( $α_{ij}$ )

The alpha tensor is the linear map from a uniform test magnetic field to the corresponding mean-field electromotive force. Its components satisfy

E_{i} = α_{ij} B_{0 j}

. Anisotropic helical flows can produce a real symmetric rank-two response, such as

α_{0} diag (1, 1, 0)

Mean-field electromotive force ( $E = ⟨ u \times b ⟩$ )

The mean-field electromotive force is the vector correlation

E = ⟨ u \times b ⟩

. Its curl enters the mean resistive induction equation. It is a local vector with units of velocity times magnetic field, not the scalar circuit voltage called electromotive force. A space-time average of periodic modes is often used to measure its transport coefficients.

Monochromatic small-Reynolds-number mean electromotive force ( $E = R_{m} E^{(1)} + R_{m}^{2} E^{(2)} + \dots$ )

Suppose a steady solenoidal velocity satisfies

Δ u = - u

, and spatial averaging permits integration by parts without boundary terms. For a uniform test magnetic field

B

, the small-magnetic Reynolds number expansion gives

b_{1} = (B \cdot \nabla) u

and

Δ b_{2} = - \nabla \times (u \times b_{1})

. Using the self-adjointness of the Laplacian inside the average,

E^{(1)} = ⟨ u \times (B \cdot \nabla) u ⟩, E^{(2)} = ⟨ u \times \nabla \times [u \times (B \cdot \nabla) u]⟩ .

The second equality concerns the averaged mean-field electromotive force; it does not assert that the quadratic forcing itself is monochromatic.

// Target: fluid-mechanics.bigb

Antisymmetry of the second-order monochromatic alpha tensor ( $α_{ij}^{(2)} = - α_{ji}^{(2)}$ )

For

C = u \times (B \cdot \nabla) u

, the cross product identity

(u \times \nabla \times C)_{i} = u_{j} \partial_{i} C_{j} - u_{j} \partial_{j} C_{i}

and integration by parts give

E_{i}^{(2)} = - ⟨ \partial_{i} u_{j} C_{j} ⟩

. Its alpha tensor is

α_{ij}^{(2)} = - ⟨ \partial_{i} u \cdot (u \times \partial_{j} u)⟩ = ⟨ u \cdot (\partial_{i} u \times \partial_{j} u)⟩ .

Interchanging

i, j

reverses the cross product, showing that the coefficient is an antisymmetric matrix. Such a coefficient represents a turbulent magnetic pumping contribution rather than the symmetric part of the alpha effect.

// Target: fluid-mechanics.bigb

Symmetry of the first-order monochromatic alpha tensor ( $α_{ij}^{(1)} = α_{ji}^{(1)}$ )

The first coefficient of the alpha tensor is

α_{ij}^{(1)} = ϵ_{ik l} ⟨ u_{k} \partial_{j} u_{l} ⟩

. Its contraction with the Levi-Civita symbol is

ϵ_{p ij} α_{ij}^{(1)} = ⟨ u_{j} \partial_{j} u_{p} - u_{p} \partial_{j} u_{j} ⟩ = 0,

by incompressibility and vanishing averages of total derivatives. Every antisymmetric second-rank tensor in three dimensions is detected by this contraction, so the coefficient is a symmetric matrix. The proof needs no Fourier analysis.

// Target: fluid-mechanics.bigb

Turbulent magnetic pumping ( $E_{pump} = γ \times \overline{B}$ )

Turbulent magnetic pumping is an effective transport of the mean magnetic field produced by correlations in inhomogeneous or anisotropic turbulence. Its contribution to the mean-field electromotive force is

E_{pump} = γ \times \overline{B}

. In the mean induction equation this enters as

\nabla \times (γ \times \overline{B})

, the same algebraic form as advection by a velocity

γ

. Thus it supplements the mean material velocity without requiring an equal mean mass flow. Downward pumping can help retain or couple fields in a solar dynamo.

Integrated electromotive response of a finite cyclonic event ( $E_{x} = π B_{0} a^{4} (x cos x - sin x) / x^{2}$ )

For the ideal magnetic response to a cylindrical cyclonic event, the plane-integrated

x

-component of

u \times B

at duration

t

E_{x} (t) = π B_{0} \int_{0}^{\infty} [t r^{2} (f g^{'} - g f^{'}) cos (f t) - 2 r g sin (f t)] d r .

This follows from

(u \times B)_{x} = (r f B_{z} + g A_{r}) cos ϕ - (g / r) A_{ϕ} sin ϕ

and azimuthal integration. If boundary terms vanish, integration by parts gives the equivalent form

π B_{0} \int_{0}^{\infty} r^{2} g^{'} [sin (f t) + t f cos (f t)] d r

. For

f = g = a^{2} - r^{2}

r < a

, zero outside, the response is

E_{x} (t) = π B_{0} a^{4} \frac{x c o s x - s i n x}{x ^{2}}, x = a^{2} t .

(2)

The value at zero is its continuous limit zero. It is initially

- π B_{0} a^{6} t /3

, but changes sign repeatedly at the nonzero roots of

tan x = x

, with oscillatory envelope of order

t^{- 1}

. The flow's kinetic helicity density is

2 f^{2} \geq 0

, showing that the sign of a finite-duration ideal response is not fixed by the instantaneous helicity sign.

Space-time average of periodic modes ( $⟨ \cdot ⟩_{x, t}$ )

A space-time average over periodic modes integrates over the spatial periodic cell and one temporal period. For real spatial Fourier modes sharing a wavevector,

⟨ Re (v e^{i k \cdot x}) \times Re (w e^{i k \cdot x}) ⟩_{x} = Re (v^{*} \times w) /2

. A matching sine or cosine temporal factor supplies another

1/2

. Opposite wavevectors can have nonzero spatial cross averages, so their temporal coefficients must be checked rather than discarded merely because the wavevectors are distinct.

Hartmann flow

Hartmann flow is a fully developed viscous conducting-fluid flow driven along a planar channel while a uniform magnetic field crosses the walls. Induction and the Lorentz force density couple its velocity and tangential induced field. For no-slip, normal-field walls separated by

2 L

, the velocity is a hyperbolic cosine profile that approaches plane Poiseuille flow at zero imposed field and develops a uniform core with Hartmann layers at large Hartmann number.

Hartmann flow rate with normal-field walls ( $Q = \frac{2 G L ^{3}}{ν} \frac{H c o t h H - 1}{H ^{2}}$ )

For a streamwise pressure gradient

- ρG

and walls at

z = \pm L

with zero tangential magnetic field, the volumetric flow rate per unit span is

Q (H) = \frac{2 G L ^{3}}{ν} \frac{H c o t h H - 1}{H ^{2}} .

Its zero-field value is

2 G L^{3} / (3 ν)

; it decreases strictly for

H > 0

and is asymptotic to

2 G L^{3} / (ν H)

. The partial-fraction expansion of the hyperbolic cotangent proves monotonicity directly.

Hartmann layer ( $δ \sim L / H$ )

A Hartmann layer is the thin viscous-magnetic boundary layer next to a wall crossed by the applied magnetic field. For channel half-width

L

and large Hartmann number

H

, its thickness is

δ \sim L / H = μ_{0} ρ ν η /∣ B_{0} ∣

. It restores the no-slip boundary condition and the specified magnetic wall condition outside the nearly uniform Hartmann flow core.

Hartmann number ( $H = ∣ B_{0} ∣ L / μ_{0} ρ ν η$ )

The Hartmann number measures the strength of magnetic coupling relative to viscous and resistive effects in a channel:

H = \frac{∣ B _{0} ∣ L}{μ _{0} ρ ν η} = ∣ B_{0} ∣ L \frac{σ _{c}}{ρ ν} .

Here

L

is the channel half-width,

ν

the kinematic viscosity,

η

the magnetic diffusivity and

σ_{c}

the electrical conductivity. In Hartmann flow, large

H

produces thin magnetic-viscous wall layers and a flatter, slower core for fixed pressure gradient.

Normal magnetic field boundary condition ( $B \times n = 0$ )

A normal-field wall imposes zero tangential magnetic field:

B \times n = 0

for the wall unit normal

n

. In a channel with imposed normal field

B_{0} \hat{z}

, an induced tangential component

b (z) \hat{x}

therefore satisfies

b = 0

on each wall. This condition is distinct from the no-slip boundary condition on fluid velocity.

Resistive induction equation ( $\partial_{t} B = \nabla \times (u \times B) + η \nabla^{2} B$ )

For a solenoidal vector field

B

and constant magnetic diffusivity

η

, the magnetic induction equation is

\partial_{t} B = \nabla \times (u \times B) + η \nabla^{2} B, \nabla \cdot B = 0.

It combines Faraday's law with the moving-conductor relation for electrical conductivity, neglecting displacement current. With incompressible flow, its advective form is

\partial_{t} B + (u \cdot \nabla) B = (B \cdot \nabla) u + η \nabla^{2} B

Shearing-coordinate magnetic flux equation ( $\partial_{t} a = η [(\partial_{s} - im t Ω^{'})^{2} + s^{- 1} (\partial_{s} - im t Ω^{'}) - m^{2} / s^{2}] a$ )

For a planar magnetic field written as

B = \nabla \times (A z)

and a circular velocity

u = s Ω (s) ϕ

, the resistive induction equation reduces, after choosing the additive gauge transformation, to

\partial_{t} A + Ω \partial_{ϕ} A = η \nabla^{2} A

. A Fourier mode

A = a (s, t) e^{im [ϕ - Ω (s) t]}

therefore obeys the equation in the title. Indeed, each radial derivative acting on the exponential replaces

\partial_{s}

\partial_{s} - im t Ω^{'}

, while the azimuthal Laplacian contributes

- m^{2} / s^{2}

. At zero magnetic diffusivity,

a

is constant in time, but the physical magnetic field can still be amplified algebraically by differential rotation.

Radial magnetic induction scalar ( $P = B \cdot x$ )

For solenoidal velocity and magnetic field, with constant magnetic diffusivity

η

, the resistive induction equation implies

\partial_{t} P + u \cdot \nabla P = B \cdot \nabla Q + η Δ P, Q = u \cdot x .

The product rule gives

D_{t} P = x \cdot D_{t} B + u \cdot B

and

Δ P = x \cdot Δ B + 2\nabla \cdot B

. Consequently the radial velocity, through

Q

, is the source term for the radial magnetic field.

// Target: fluid-mechanics.bigb

Insulating boundary condition for the radial magnetic scalar ( $[P] = [\partial_{r} P] = 0, Δ P_{ext} = 0$ )

At a spherical interface with an insulating exterior, equal magnetic permeability on both sides and no surface current, all components of the magnetic field are continuous. The solenoidal magnetic-field constraint then makes

P = B \cdot x

and

\partial_{r} P

continuous. A current-free exterior has

Δ P = 0

. For an isolated field decaying at infinity, its degree-

l

spherical harmonic satisfies

\partial_{r} P_{l m} = - (l + 1) P_{l m} / a

at a sphere of radius

a

. This boundary condition incorporates the exterior field without setting the radial field to zero.

// Target: fluid-mechanics.bigb

Magnetic diffusion ( $\partial_{t} B = η_{m} \nabla^{2} B$ )

Finite electrical conductivity permits resistive smoothing of a magnetic field. With uniform coefficients and no motion the field satisfies a diffusion equation; with motion the diffusion term supplements the curl of

u \times B

. The magnetic Reynolds number compares these effects.

Magnetic phase mixing under differential rotation ( $exp [- η m^{2} f^{'} (r)^{2} t^{3} /3]$ )

A nonuniform angular velocity

f (r)

winds an azimuthal magnetic mode into phase

m (ϕ - f (r) t)

. Its radial phase gradient grows as

- m t f^{'} (r)

, creating small scales even when its ideal amplitude does not decay. The corresponding leading resistive damping rate is

η m^{2} t^{2} f^{'} (r)^{2}

, so a local short-wavelength estimate gives damping factor

exp [- η m^{2} f^{'} (r)^{2} t^{3} /3]

. Hence magnetic diffusion can become important at time

[η m^{2} f^{'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^{'}

must also be retained there.

Exact quadratic-shear magnetic flux solution ( $μ^{2} = 4 i η mω$ )

For azimuthal flow

u_{ϕ} = s (Ω_{0} + ω s^{2})

, a planar magnetic flux mode

s^{m} g (t) e^{im ϕ - im Ω_{0} t - i s^{2} f (t)}

obeys the resistive induction equation if

\dot{f} = mω - 4 i η f^{2}

and

\overset{g}{˙} / g = - 4 i η (m + 1) f

. With

f (0) = 0

and

g (0) = 1

, their solution is

f = (mω / μ) tanh μ t

and

g = (cosh μ t)^{- (m + 1)}

. Choose

Re μ > 0

. For positive

η, ω

, the fixed-position field decays asymptotically at rate

(m + 1) 2 η mω

. Differential rotation creates small spatial scales, accelerating magnetic diffusion.

Magnetic free-decay spectral bound ( $\int_{D} ∣\nabla \times B ∣^{2} \geq \frac{π ^{2}}{R ^{2}} \int_{R^{3}} ∣ B ∣^{2}$ )