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The solenoidal constraint on the magnetic field reduces to ${\partial }_x{B}_x+{\partial }_z{B}_z=0$. Locally, or globally in a simply connected cross-section, it permits a Cartesian magnetic flux function satisfyingThe transverse magnetic field lines are contours of $\psi$, because $\mathbf{B}\cdot \nabla \psi =0$. The independent component $B_y$ supplies the twist of the flux tube; it is not restricted by the solenoidal constraint.

For the Lorentz force, take the curl explicitly:Its cross product with $\mathbf{B}$ has transverse components $-({\nabla }^2\psi )\nabla \psi -B_y\nabla B_y$ and $y$ component ${\psi }_x{B}_{y,z}-{\psi }_z{B}_{y,x}$. ConsequentlyThe term $-B_y\nabla B_y/{\mu }_0$ is the transverse gradient of the magnetic pressure associated with the axial field, while the other terms include magnetic tension.

Use $D={\partial }_t+w{\partial }_z$, $w=v_z$, and a prime for ${\partial }_z$. For the plane-parallel magnetohydrodynamic flow with imposed shear, $\nabla \cdot \mathbf{u}=w^{\prime }$, whileThe solenoidal magnetic field constraint gives $B_z^{\prime }=0$. The $z$ component of the MHD induction equation then gives ${\partial }_t{B}_z=0$. Thus $B_z$ is constant in both space and time.

Use the magnetic tension and magnetic pressure decompositionThe continuity equation and momentum equation now reduce toThe transverse components of the MHD induction equation areThe terms involving $a$ respectively accelerate the flow across the imposed shear flow and wind the transverse magnetic field. The closure is the isothermal equation of state $p=c_s^2\rho$, as printed in the PDF.

The logarithmic density term is the barotropic fluid energy for $p=c_s^2\rho$, rather than the thermodynamic heat content of an isolated gas. DefineFor any scalar $q$, the continuity equation implies ${\partial }_t(\rho q)+{\partial }_z(\rho wq)=\rho Dq$. Therefore $De=-c_s^2{w}^{\prime }$, $D\Phi =wg$, and dotting the momentum equations with $\rho \mathbf{v}$ givesThe MHD induction equation gives the complementary magnetic energy balanceAdding these two identities, and collecting the product derivatives, yields the barotropic magnetic energy equationHere $E=E_h+E_B$. The two stress terms in $S$ describe magnetohydrodynamic shear work: the maintained background shear flow can supply energy to, or remove energy from, the perturbation flow and magnetic field. The sign depends on the off-diagonal total stress; this is why the energy excluding the background shear flow is not generally conserved. Changing ${\rho }_0$ merely adds a constant multiple of the conserved density to $E$ and the corresponding mass flux to $F$, leaving $S$ unchanged.

For a steady flow, mass conservation gives $\rho w=\text{constant}$ and ${\rho }^{\prime }/\rho =-w^{\prime }/w$. Define the Alfvén velocity ${\mathbf{v}}_a=\mathbf{B}/\sqrt{{\mu }_0\rho }$ and $C=w^2-v_{az}^2$. Combining the horizontal momentum and MHD induction equations without dividing by $C$ givesDot these identities with $B_x/({\mu }_0\rho )$ and $B_y/({\mu }_0\rho )$, respectively:The vertical momentum equation and the isothermal equation of state giveMultiply by $C$ and eliminate the magnetic derivative to obtainNo division by $w^2-v_{az}^2$ was needed in deriving this necessary relation.

The magnetosonic critical speeds in the $z$ direction areThe plus sign gives the fast magnetosonic wave speed and the minus sign the slow magnetosonic wave speed. Thus the differential coefficient is $(w^2-c_{\mathrm{f}}^2)(w^2-c_{\mathrm{s}}^2)$. A smooth outflow proceeding from below both speeds to above both must normally pass through both magnetosonic critical speeds. At each crossing, the right-hand side must also vanish; this is the regularity at a magnetosonic point condition. The derivative coefficient changes sign at each nondegenerate crossing. For $w>0$ and $w^{\prime }>0$, the driving term must be positive below the slow point, negative between the points and positive above the fast point.

The Alfvén speed component satisfies $c_{\mathrm{s}}^2\le v_{az}^2\le c_{\mathrm{f}}^2$. With downward gravity $g>0$, the gravitational term is consequently nonnegative at the slow point and nonpositive at the fast point. The magnetohydrodynamic shear work contribution must balance it at each point, and can provide the upward driving needed to pass the fast point. For $a=0$ and strictly positive $g$, generic separated slow and fast points cannot satisfy the required zero numerator; special vanishing-gravity or coincident-speed cases need separate treatment.

There is also Alfvén-point compatibility in a plane-parallel sheared flow. At $w^2=v_{az}^2$, the original transverse equations requireThese restrictions are not generally visible as a zero of the scalar differential coefficient, which there equals $-v_{az}^2(v_{ax}^2+v_{ay}^2)$. In particular, a strictly accelerating regular solution must have $B_x=0$ at that point. The scalar relation is therefore a necessary wind equation, not a substitute for regularity of all the original ideal magnetohydrodynamic equations. Degenerate cases such as a purely longitudinal magnetic field can merge characteristic speeds and reduce the number of distinct critical conditions.

Use the central-field identity and substitute the prescribed relation between the toroidal magnetic field and axial magnetic field:The flux-tube axial field virial identity bounds the last average by $B_p^2$, while $p(0)\ge 0$. Therefore the uniform-twist central axial field bound isThe stronger exact expression is $B_z(0{)}^2=(1+{\gamma }^2)B_p^2-2{\mu }_{0}p(0)-(4{\mu }_0{\gamma }^2/s_0^2){\int }_0^{s_0}sp(s)ds$.

In SI units, magnetostatic equilibrium without gravity is $\nabla p={\mu }_0^{-1}(\nabla \times \mathbf{B})\times \mathbf{B}$. For a purely axial magnetic field, the magnetic tension term $(\mathbf{B}\cdot \nabla )\mathbf{B}$ vanishes. Thus the radial balance isMatching the magnetohydrodynamic total pressure to the field-free exterior gives the gas pressure and axial field bound:The inequality follows directly from nonnegative gas pressure. It is the magnetic pressure limit for an untwisted flux tube, without inward hoop tension.

The azimuthal derivative of a cylindrical vector Fourier mode must include basis rotation: the azimuthal dependence refers to the components in the rotating cylindrical basis. Since ${\partial }_{\varphi }\hat{\mathbf{s}}=\hat{\mathbf{\varphi }}$ and ${\partial }_{\varphi }\hat{\mathbf{\varphi }}=-\hat{\mathbf{s}}$, a vector amplitude $\mathbf{v}\propto e^{im\varphi }$ obeysAlso $\mathbf{B}=(J/2)(-y,x,0)$ in Cartesian coordinates, so $(\mathbf{v}\cdot \nabla )\mathbf{B}=(J/2)\hat{\mathbf{z}}\times \mathbf{v}$. Consequentlyand the basis-rotation terms cancel in the ideal magnetohydrodynamic induction equation:Here the printed $J$ is the coefficient in the specified magnetic field: $\nabla \times \mathbf{B}=J\hat{\mathbf{z}}$. If $J_{\mathrm{phys}}$ denotes SI electric current density, then $J={\mu }_0{J}_{\mathrm{phys}}$. The field and all subsequent printed coefficients are mutually consistent with this normalization.

For a normal mode with $\sigma \ne 0$, induction gives $\mathbf{b}=imJ\mathbf{u}/(2\sigma )$. Substitution into the momentum equation, including the Coriolis force, gives the reduced velocity equationThus the coefficients are the printed $\lambda$ and $\nu$. Let $C=J^2/({\mu }_0\rho )$. Inserting the allowed relation $\lambda =i\delta \nu$ and multiplying by $\sigma$ yieldsHence the uniform-current rotating magnetohydrodynamic wave dispersion isThe radicand is real. For $|m|\ge 2$, $2m\delta -m^2<0$ because $|\delta |<1$, so both roots are purely imaginary. For $m=0$ the roots of the quadratic are also imaginary; any zero-frequency case must be checked in the original equations, since the elimination divided by $\sigma$. A positive real part requiresFor integer $m$ this is possible only when $|m|=1$ and $m\delta >1/2$. Taking the conventional representative $m\ge 0$ gives the printed exception $m=1$, with $\delta >1/2$ and sufficiently large $J$. Literally allowing negative integers also gives $m=-1$, $\delta <-1/2$; complex conjugation maps $(m,\delta ,\sigma )$ to $(-m,-\delta ,\overline{\sigma })$, so these describe the conjugate real disturbance. Purely imaginary roots describe neutral oscillatory modes; a repeated root at threshold does not itself prove boundedness of arbitrary initial perturbations.

Use perturbations $\mathbf{v},\mathbf{b},p,\vartheta$ for velocity, magnetic field, magnetohydrodynamic total pressure and buoyancy displacement variable. SetThis is advection by the background Keplerian shearing sheet. Subtracting the equilibrium before retaining first-order terms gives the nine linearized equations:The azimuthal coefficient $\Omega /2$ combines the Coriolis acceleration with perturbation advection of the background shear. The $-3\Omega b_x/2$ term is the winding of radial magnetic field by that shear. The tidal force cancels when the equations at a fixed position are subtracted; it does not add a separate Eulerian perturbation force. Magnetic pressure is already included in $p$, leaving only linear magnetic tension. In this normalization $\vartheta$ has dimensions of length because $D_0\vartheta =v_x$.

When the imposed magnetic field vanishes, the dynamical dispersion relation becomesThe nonstationary hydrodynamic branch therefore has ${\sigma }^2=-(N^2+{\Omega }^2)$. The radial Solberg–Høiland instability criterion in this local model isThe radial buoyancy frequency supplies either restoration or a driving force, while the radial epicyclic frequency $\kappa =\Omega$ supplies rotational restoration. Zero sum is marginal, and positive sum gives stable oscillations. The neutral roots do not themselves signify exponential growth.

Because both equilibrium pressure and dimensionless specific entropy decrease outward, their radial gradients have the same sign. The leading minus sign in the expression for $N^2$ therefore makes $N^2<0$: the radial stratification is adverse. For smooth profiles varying over radius $r$, $|{\partial }_{r}P|\sim P/r$ and $|{\partial }_{r}S|\sim 1/r$, soFor a thin disk, the magnitude of this negative radial buoyancy frequency squared is much less than ${\Omega }^2$. Rotation stabilizes the hydrodynamic mode despite the adverse entropy gradient. This estimate assumes gradients on the global radial scale; a sharp thermal feature may be different. It addresses the ideal nondiffusive equations here. Processes such as convective overstability require additional thermal relaxation and are not ruled out by this particular criterion.

A cataclysmic variable is a close semidetached binary in which a white dwarf accretes from a cool, usually low-mass donor star filling its Roche lobe. In an ordinary hydrogen-rich system the donor star is often near the lower main sequence. In a nonmagnetic cataclysmic variable (a disk-fed CV), the white dwarf's magnetic field is too weak to control the flow over the disk. This leaves a characteristic disk-fed geometry, summarized in the original schematic below. NASA's introduction to cataclysmic variables describes the basic components.

Roche-lobe overflow passes through the inner Lagrange point $L_1$. A nearly ballistic stream bends in the rotating frame and strikes the outer accretion disk, producing an accretion-disk stream-impact hot spot. Its retained angular momentum prevents direct radial infall. Viscous evolution of an accretion disk transports angular momentum outward while gas moves inward through a nearly Keplerian accretion disk. Close to the white dwarf, gas slows from orbital rotation toward the stellar rotation in an accretion-disk boundary layer. The disk, accretion-disk stream-impact hot spot, accretion-disk boundary layer, white dwarf and donor star all contribute to the spectrum and, where the orbital inclination permits, to the eclipses of an eclipsing binary. In particular, “nonmagnetic” describes the accretor's control of the flow; it does not imply that the donor star cannot sustain a magnetic field.

The gravitational power available from accretion is approximately $L_{\mathrm{acc}}=G{M}_{\mathrm{WD}}{\dot{M}}_{\mathrm{acc}}/R_{\mathrm{WD}}$. For a slowly rotating white dwarf and a thin steady Keplerian accretion disk, the specific energy changes from approximately zero far out to $-G{M}_{\mathrm{WD}}/(2R_{\mathrm{WD}})$ at the inner disk. Roughly half the available power is radiated by the accretion disk; the rest is released in the accretion-disk boundary layer as the orbital kinetic energy is dissipated. Stellar rotation and departures from a steady thin disk change this partition.

A classical nova has a different energy source. Transferred hydrogen accumulates on the white dwarf; compression heats the base of its envelope until temperature-sensitive hydrogen burning accelerates. Electron degeneracy pressure initially weakens the expansion response to heating, helping a thermonuclear runaway develop. The envelope subsequently expands and ejects material, producing a large optical outburst followed by a decline as the ejecta expand and residual burning ends. The white dwarf usually survives, so continued accretion can build another fuel layer. A rough recurrence scale is the ignition-envelope mass divided by the mean accretion rate; both this mass and the rate vary strongly among systems. The event is an envelope eruption, and the retained fraction is not automatically unity. Starrfield, Iliadis and Hix's nova calculations explains this nuclear mechanism. A classical nova is powered by unstable nuclear burning on the white dwarf.

A dwarf nova undergoes recurrent, shorter brightenings powered principally by enhanced gravitational accretion. The hydrogen-ionization disk instability creates cold, mostly neutral and hot, ionized branches of the disk's accretion-disk thermal S-curve, separated by unstable equilibria. In quiescence the cool disk stores matter because inward transport is slow. Once a critical surface density is reached, a heating transition puts the disk into a hotter, more state with higher effective viscosity: the inward accretion rate and luminosity rise and the disk drains. A cooling transition returns it to quiescence, completing the cycle. A persistent increase in the donor star's transfer rate is not required. Sufficiently high transfer rates can keep the disk on its hot stable branch, giving a nova-like variable rather than ordinary disk cycles. Lasota's disk-instability analysis and Hameury's disk-instability review develop this picture. A dwarf-nova outburst is a disk instability, not a white-dwarf thermonuclear explosion. The names classify mechanisms and need not identify permanently distinct binaries: a nova-producing binary can also possess an unstable disk between nuclear eruptions.

The cataclysmic-variable orbital-period distribution is not smooth. For ordinary hydrogen-rich cataclysmic variables, prominent features are a cataclysmic-variable period gap around two to three hours, a cataclysmic-variable period minimum near eighty minutes, and an accumulation near that minimum. These are population features rather than absolute exclusions. Selection effects matter: luminous high-accretion rate systems are easier to find than faint evolved systems. Helium-transferring binaries have a different period range and are not described by the hydrogen-rich minimum. Gänsicke and collaborators' period-minimum study documents the observed accumulation.

The Roche-lobe-filling period-density relation makes the orbital period a measure of donor structure. Combining $R_d\simeq 0.46a(M_d/M{)}^{1/3}$ with Kepler's third law givesAs the donor star loses mass, its stellar radius response exponent $\zeta =d\log R_d/d\log M_d$ impliesA donor with $\zeta >1/3$ evolves toward shorter orbital periods. When its effective response falls below $1/3$, continued mass loss instead lengthens the orbital period: this is the cataclysmic-variable period bounce. A very low-mass donor may be substellar and increasingly affected by electron degeneracy pressure; the ideal degenerate scaling $R_d\propto M_d^{-1/3}$ illustrates the reversal. The precise cataclysmic-variable period minimum depends on thermal disequilibrium and the strength of orbital angular momentum loss. Near a turning point $|\dot{P}|$ is small; for an approximately steady evolutionary flow of systems, the number per period interval scales as $N(P)\propto 1/|\dot{P}|$, explaining the accumulation. Knigge, Baraffe and Patterson's donor-based evolutionary study relates the donor star's response to these period features.

Long-term transfer is driven mainly by losses of orbital angular momentum, rather than by disk outbursts. Magnetic braking of a binary star removes the cool donor star's spin through a magnetized stellar wind. Tidal synchronization makes the orbit replenish that spin, so the wind extracts orbital angular momentum. This is usually the dominant standard driving mechanism above the cataclysmic-variable period gap. Gravitational-wave emission from a binary system supplies a baseline loss, especially important below the gap. For a weak-field, slowly moving circular orbit, the circular gravitational-wave inspiral givesBecause ordinary transfer from the lighter donor star tends to expand its Roche lobe if orbital angular momentum is conserved, an external loss is needed to sustain contact. In the conservative contact approximation, the binary mass-transfer contact equation readswhere the stellar radius response exponent must match the evolutionary timescale and additional donor expansion has been neglected. The positive denominator on the stable branch makes ${\dot{J}}_{\mathrm{ext}}<0$ drive ${\dot{M}}_d<0$. Nova ejecta can introduce additional nonconservative binary mass transfer. Knigge's evolutionary discussion describes the standard loss mechanisms and their limitations.

In the disrupted magnetic braking model, relatively rapid mass loss above the cataclysmic-variable period gap keeps the donor star inflated relative to stellar thermal equilibrium. When the donor approaches the fully convective star transition, the model postulates a substantial reduction in magnetic braking of a binary star. The donor star can contract within its Roche lobe, suppressing Roche-lobe overflow near the upper edge of the gap. Gravitational-wave emission from a binary system continues to shrink the detached binary; near the lower edge the Roche lobe again reaches the donor radius and transfer resumes. No mass transfer is needed during the detached crossing. The density relation predicts a radius ratio $(3/2{)}^{2/3}\simeq 1.31$ between contact at three and two hours if the masses remain nearly fixed, illustrating the required inflation before detachment. The torque reduction is a model ingredient, not a claim that all fully convective stars lose their magnetic fields. Zorotovic and collaborators' detached-binary study tests the predicted detached population in the gap.

The standard formation channel starts with an initially wider binary star containing two main sequence stars. The initially more massive component evolves first and becomes a giant. Unstable Roche-lobe overflow can engulf the companion in a common envelope. Drag causes inward orbital motion, releasing two-body orbital energy and transferring angular momentum to the envelope. If the envelope is expelled before merger, a close detached binary survives, containing the exposed core, which becomes a white dwarf, and the lower-mass companion. This explains how a binary becomes much tighter than the giant progenitor's radius would have allowed. The outcome depends on envelope binding and the efficiency of energy deposition, summarized approximately by the common-envelope energy formalism; ejection is not guaranteed. Ivanova and collaborators' common-envelope analysis discusses the relevant physics and uncertainties.

Subsequent magnetic braking of a binary star and gravitational-wave emission from a binary system shrink the detached binary until its companion fills its Roche lobe. Stable Roche-lobe overflow, for a suitable binary mass ratio and stellar radius response exponent, then creates a cataclysmic variable. Its later nuclear eruptions, disk cycles and secular orbital period evolution occur on different timescales. The formation sequence is a wide binary, envelope ejection, a close detached white-dwarf binary, and angular-momentum-driven contact.

Use the Minkowski metric with signature $(-,+,+,+)$. The source point is the intersection of the worldline with the observation event's past light cone: $R^{\mu }{R}_{\mu }=0$ and $R^0>0$. In components this is $c(t-t_r)=|x-y(t_r)|$ with $t_r<t$. For a timelike worldline the retarded solution is unique in the usual complete-source setting.

Writing $R=|x-y(t_r)|$, $n=(x-y)/R$ and $u^{\mu }=\gamma (c,v)$ gives $R\cdot u=-\gamma cR(1-n\cdot v/c)$. Thus the Lienard-Wiechert potentials are $\varphi =q{\mu }_0{c}^2/[4\pi R(1-n\cdot v/c)]$ and $A=q{\mu }_{0}v/[4\pi R(1-n\cdot v/c)]$. At leading order in $1/r$ and first order in $v/c$, set $R\simeq r,n\simeq \hat{r}$ and expand the denominator:

Now use $t_{-}=t-r/c$ and let $a=\dot{v}(t_{-})$. Derivatives of $1/r$ and $\hat{r}$ give near-field terms of order $r^{-2}$; the radiation terms come from ${\partial }_t{t}_{-}=1$ and $\nabla t_{-}=-\hat{r}/c$. The electric field and magnetic field are thereforeThey are transverse to the observation direction. The Poynting vector isIntegrating over a sphere uses $\int {\sin }^2\theta d\Omega =8\pi /3$, yielding the nonrelativistic Larmor formula $P=q^2{\mu }_0|a{|}^2/(6\pi c)$. For simple harmonic motion, $|a{|}^2=A^2{\omega }^4{\cos }^2(\omega t_{-})$, whose period average is $A^2{\omega }^4/2$. Multiplication by the period $2\pi /\omega$ givesThis is the leading radiation energy in the nonrelativistic and far-field approximation.

For the magnetic part of the Lorentz force, $\mathbf{F}=q\mathbf{v}\times \mathbf{B}$. Since velocity has physical dimension $L{T}^{-1}$, the magnetic field has dimension

For the field generated by the infinitely long, axially symmetric shell currents, axisymmetry and the right-hand rule give $\mathbf{B}=B_{\varphi }(r){\mathbf{e}}_{\varphi }$. A circular loop has $\oint \mathbf{B}\cdot d\mathbf{\ell }=2\pi r{B}_{\varphi }(r)$. Its enclosed electric current is zero inside the inner shell, $I$ between the shells and $2I$ outside both. Ampère's law therefore givesThe positive ${\mathbf{e}}_{\varphi }$ direction is counterclockwise viewed from positive $z$. The ideal current sheets make the tangential magnetic field discontinuous at each shell; these formulas specify the three open regions, with no additional imposed background field.

For a moving closed circuit $C(t)$ bounding an oriented surface $S(t)$, Faraday's law isHere $\mathbf{E}$ is the electric field, $\mathbf{B}$ the magnetic field, $\mathbf{v}$ the local velocity of a circuit element, and ${\mathcal{E}}_{\mathrm{em}}$ the electromagnetic electromotive force per unit charge around the circuit. The line orientation and normal $\mathbf{n}$ obey the right-hand rule. The flux derivative is a total derivative and includes both changing field and changing surface. The term $\mathbf{v}\times \mathbf{B}$ is the motional electromotive force. A battery contributes an additional non-electromagnetic source electromotive force.

For motion to the right in $B_0{\mathbf{e}}_z$, $B_0>0$, the induced current is clockwise. Its own magnetic field points into the page inside the circuit, opposing the increase in external flux, as required by Lenz's law. As that current changes, its changing flux also produces an electromotive force opposing the change in current: the circuit has inductance.

Consequently the algebraic current law used in parts (i)-(iv) must be replaced by a dynamical circuit equation that includes the derivative of the self-flux. The induced current cannot in general adjust instantaneously to the bar velocity. Energy is temporarily stored in the self-field and can later return to the motion or be dissipated; the simple exponential braking law can change. Self-field forces and changes of inductance with circuit geometry may also matter. The magnetic self-field changes the transient dynamics but does not create energy: mechanical energy, magnetic-field energy and Joule heat must be included in the conservation balance. A quantitative answer would require the circuit's inductance and its dependence on $s$.

For the oscillatory magnetic field, Faraday's law gives $\mathbf{B}=(1/\omega )\mathbf{k}\times \mathbf{E}$. Indeed its time derivative equals $-\nabla \times \mathbf{E}$, and the dispersion relation makes it satisfy the remaining Maxwell equations. ThusFor $\omega =ck$, its amplitude is $E_0/c$. The magnetic field, electric field and direction of propagation are mutually perpendicular. A constant background magnetic field could be added; the expression above is the wave-associated field.

Use the fully developed flow branch of Hartmann flow. Translation invariance along the walls makes the velocity and induced magnetic field functions of $z$ alone. Incompressible flow and zero normal velocity at the walls give $u_z^{\prime }=0$ and hence $u_z=0$. The zero divergence of the magnetic field makes $B_z$ constant, equal to the imposed $B_0$. There is no forcing in $y$; the homogeneous $y$-components obey the same coupled viscous-resistive equations as the $x$-components, with zero boundary data. Multiplying them by $u_y$ and $b_y/({\mu }_0\rho )$, integrating by parts and adding givesThus $u_y=b_y=0$, establishing the asserted forms on this fully developed flow branch. This is a symmetry reduction of the steady channel model, rather than a claim that every possible flow in a channel is translation invariant.

The current density and Lorentz force density areThe advective acceleration vanishes because $\mathbf{u}\cdot \nabla =u(z){\partial }_x$ acts on fields independent of $x$. The $x$-component of the magnetohydrodynamic momentum equation therefore givesThe $z$-component must also balance: it requires ${\partial }_z(p+b^2/(2{\mu }_0))=0$. Hence a compatible pressure is $p(x,z)=p_{\ast }-\rho Gx-b(z{)}^2/(2{\mu }_0)$; the specified streamwise pressure gradient does not require the ordinary pressure to be uniform in $z$. This accounts for the magnetic pressure of the induced field.

The resistive induction equation with constant magnetic diffusivity gives ${\partial }_t\mathbf{B}=\nabla \times (\mathbf{u}\times \mathbf{B})+\eta {\nabla }^2\mathbf{B}$. Since $\mathbf{u}\times \mathbf{B}=(0,-B_{0}u,0)$, its curl has $x$-component $B_0{u}^{\prime }$. Steadiness consequently givesFinally the no-slip boundary condition gives $u(\pm L)=0$, and the normal magnetic field boundary condition gives $b(\pm L)=0$. The coupled equations and all wall conditions follow directly from momentum balance and magnetic induction.