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The curvature vector of a magnetic field line parametrized by arc length. For circular lines, it points towards their center and has magnitude $1/R$. The curvature contribution to magnetic tension is $B^2\mathbf{\kappa }/(4\pi )$ in Gaussian units.

For a resolved magnetic field, the unaveraged tensor $B_i{B}_j$ describes magnetic tension; the isotropic magnetic pressure is treated separately in the magnetohydrodynamic total pressure. Its spatial or ensemble average need not have matrix rank one. Statistical isotropy in three dimensions gives $⟨B_i{B}_j⟩=⟨B^2⟩{\delta }_{ij}/3$, by taking the trace. In Alfvén velocity units this tensor has dimensions of squared velocity.

For two incompressible streams of equal density with a common tangential magnetic field parallel to their velocities, decaying normal modes and interface displacement/total-pressure continuity give $\rho (\sigma +k_x{U}_1{)}^2+\rho (\sigma +k_x{U}_2{)}^2=2k_x^2{B}^2$ in units ${\mu }_0=1$. The displayed dispersion relation follows, with $v_A^2=B^2/\rho$. Thus $|\Delta U|<2v_A$ excludes exponential Kelvin-Helmholtz instability; equality is marginal. Above threshold the growth rate is $|k_x|\sqrt{(\Delta U{)}^2/4-v_A^2}$. The transverse horizontal wavenumber affects the decay length through $k=(k_x^2+k_y^2{)}^{1/2}$ but not this growth threshold. Magnetic tension supplies the stabilizing restoring force.

Magnetic tension is the restoring stress associated with bending magnetic field lines. In a divergence-free field, the Lorentz-force density splits into a magnetic-pressure gradient and the tension term $(\mathbf{B}\cdot \nabla )\mathbf{B}/{\mu }_0$ in SI units.

In ideal magnetohydrodynamics, a small displacement of a uniform magnetic field obeys $\delta \mathbf{B}=\nabla \times (\mathbf{\xi }\times {\mathbf{B}}_0)$. For ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$, a horizontal Fourier mode proportional to $e^{ikz}$ with $\nabla \cdot \mathbf{\xi }=0$ gives the displayed field and force perturbations. The linear magnetic pressure perturbation vanishes because $\delta B_z=0$. The remaining magnetic tension acts as a restoring spring, with Alfvén frequency $k{v}_A$.

A long-wavelength transverse disturbance of an initially isotropic magnetic second-moment tensor behaves as an elastic shear wave when small-scale rearrangements of the tangle are omitted. For $v_A^2=⟨B^2⟩/3$, its fluid displacement obeys ${\partial }_t^2\mathbf{\xi }=v_A^2{\nabla }^2\mathbf{\xi }$ and $\nabla \cdot \mathbf{\xi }=0$ on the oscillatory branch. The tensor perturbation is $\delta M_{ij}=v_A^2({\partial }_i{\xi }_j+{\partial }_j{\xi }_i)$. A zero mean magnetic field does not imply zero restoring magnetic tension.

For constant mass density $\rho$, kinematic viscosity $\nu$ and magnetic permeability ${\mu }_0$, the incompressible momentum equation isIt 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.

Magnetohydrodynamics couples an electrically conducting fluid to a magnetic field. The electric current density $\mathbf{J}$ produces a Lorentz force density $\mathbf{J}\times \mathbf{B}$, and fluid motion changes the magnetic field through induction.

For constant magnetic diffusivity $\eta$, a common nonrelativistic model includeswith $\nabla \cdot \mathbf{B}=0$. Ideal magnetohydrodynamics neglects the diffusion term; resistive magnetohydrodynamics retains it. The ratio of induction $UB/L$ to diffusion $\eta B/L^2$ is the magnetic Reynolds number $\mathrm{Rm}=UL/\eta$.

Magnetic tension supplies a restoring force for an Alfvén wave, with speed $v_A=B/\sqrt{{\mu }_0\rho }$ in a homogeneous medium. Magnetic pressure and ordinary pressure can both oppose compression. These effects explain why conducting-fluid waves and equilibria differ from those of an unmagnetized gas. ^[ref]

The magnetohydrodynamic total pressure is $\Pi =p+B^2/(2{\mu }_0)$, the sum of gas pressure and magnetic pressure. The remaining Lorentz force is magnetic tension, so this scalar does not include the entire magnetic stress.

The magnetorotational instability allows a weak magnetic field to destabilize some differentially rotating flows that are stable to axisymmetric hydrodynamic displacements. A useful picture connects two fluid elements by magnetic tension: the inner element loses angular momentum and moves inward, while the outer element gains angular momentum and moves outward. Decreasing angular velocity amplifies their separation. This makes the instability relevant to angular momentum transport in an accretion disk.

For a local, unstratified, incompressible ideal-MHD model with vertical magnetic field $B_z$, take an axisymmetric perturbation proportional to $e^{st+ikz}$. Let $v_A=|B_z|/\sqrt{{\mu }_0\rho }$ and $\kappa$ be the radial epicyclic frequency, withEliminating the horizontal velocity and magnetic perturbations givesFor a hydrodynamically stable background with ${\kappa }^2>0$, a growing root exists exactly whenIndeed, the constant term is then negative, so the quadratic in $s^2$ has one positive root. If the constant term is nonnegative, both roots are nonpositive: their sum is negative and their discriminant is ${\kappa }^4+16{\Omega }^2{k}^2{v}_A^2$. The vertical-field local limit follows from the original linear analysis.^[ref]

In a Keplerian disk, the unstable interval is $0<k^2{v}_A^2<3{\Omega }^2$. Maximizing the positive root gives $k^2{v}_A^2=15{\Omega }^2/16$ and growth rate $s_{\max }=3\Omega /4$. Strong tension stabilizes shorter wavelengths. In a finite disk the unstable wavelengths must also fit the vertical domain; magnetic diffusivity, viscosity and nonideal magnetic coupling can modify or suppress them. Thus a negative angular-velocity gradient alone is not a universal sufficient condition for every magnetized flow. Linear growth also does not by itself specify the eventual turbulent stress.

Parker instability is an undular magnetic buoyancy instability of a gravitationally stratified atmosphere supported partly by magnetic pressure. A bent horizontal magnetic field can permit gas to drain along its magnetic field lines, leaving buoyant rising regions. Magnetic tension resists bending, so wavelengths and boundary conditions affect the instability threshold. For a magnetized isothermal atmosphere, the unrestricted-wavevector threshold is given by the magnetic buoyancy energy criterion.

For $k_{\parallel }\ne 0$ the Alfvén wave becomes a firehose instability when its squared angular frequency is negative:Equality is marginal, rather than exponential growth. The pressure anisotropy must have negative sign; its absolute magnitude alone is insufficient.

For the intergalactic plasma, a negative relative pressure anisotropy of $-10^{-2}$ reaches this fluid threshold only for $\beta >200$. In particular, the literal pair $\beta =100$ and relative pressure anisotropy $-0.01$ is stable because the threshold is $-0.02$. Since the quoted numbers are approximate and allow larger plasma beta, this weakly magnetized plasma is susceptible, rather than necessarily unstable at every quoted value. For example, $\beta =300$ and relative pressure anisotropy $-0.01$ are unstable. Positive pressure anisotropy does not excite this branch.

For the fusion-device plasma, $\beta \sim 10^{-2}$ requires a negative relative pressure anisotropy below roughly $-200$, whereas its magnitude is only about $0.1$. The stabilizing magnetic tension dominates, so it is not susceptible to this fluid firehose instability under the stated conditions. Indeed, nonnegative perpendicular pressure requires the relative pressure anisotropy to be no smaller than $-1$; at such a small plasma beta even this extreme cannot satisfy the threshold.

Magnetic flux freezing ties a transverse fluid displacement to a bend in the magnetic field lines. For the uniform background, the transverse displacement equation isFor a sinusoidal bend, ${\partial }_z^2{\mathbf{\xi }}_{\perp }=-k_{\parallel }^2{\mathbf{\xi }}_{\perp }$. Positive pressure-anisotropy-modified magnetic tension therefore gives a restoring force, and inertia produces oscillation. In an isotropic plasma, this is precisely the positive magnetic tension $B_0^2/(4\pi )$.

Particles moving predominantly along bent magnetic field lines contribute a transverse momentum flux. Their larger parallel pressure opposes the restoring curvature force supplied by the magnetic field and perpendicular pressure. Once $p_{\parallel 0}-p_{\perp 0}>B_0^2/(4\pi )$, the net force points along the original displacement: a larger bend produces a larger outward force, giving positive feedback. Equivalently, the bending contribution to the quadratic potential energy is $\frac{1}{2}{\mathcal{T}}_0|{\partial }_z{\mathbf{\xi }}_{\perp }{|}^2$ and has become negative.

In this static, homogeneous fluid model the unstable roots areWith $e^{-i\omega t}$ the root $+i\gamma$ grows. Its real angular frequency is zero, so the unstable pattern grows without propagation in the equilibrium rest frame; its partner decays. Advection by a background flow would change that frame-dependent statement. The growth increasing without bound as $|k_{\parallel }|$ increases is a limitation of this long-wavelength model: finite gyroradius and other kinetic corrections cannot be neglected at sufficiently short scales.

Write the physical magnetic field as ${\mathbf{B}}_{\mathrm{phys}}$ and use the constant mass density ${\rho }_0$ to introduce Alfvén velocity units, $\mathbf{B}={\mathbf{B}}_{\mathrm{phys}}/\sqrt{{\mu }_0{\rho }_0}$. If $P$ is the fluid's mechanical pressure, the magnetohydrodynamic total pressure per unit mass density is $p=P/{\rho }_0+B^2/2$. The ideal magnetohydrodynamic equations then reduce toThe first equation is momentum conservation: the magnetic pressure has been included in $p$, leaving magnetic tension explicitly. The second is the ideal magnetohydrodynamic induction equation, expressing magnetic flux freezing. The last two equations impose incompressibility and the solenoidal magnetic-field constraint.

The model assumes a smooth conducting-fluid continuum with constant mass density and constant magnetic permeability, nonrelativistic velocity, and scales on which a single-fluid description is appropriate. Negligible electrical resistivity removes magnetic diffusion; negligible viscosity removes viscous momentum diffusion. Equivalently, the magnetic Reynolds number and the Reynolds number are large on the scales being retained. The low-frequency approximation neglects the displacement current; Hall and other kinetic corrections to Ohm's law are absent. Incompressibility is an imposed approximation, commonly justified when the Mach number is small. There is no independently evolved thermodynamic pressure: $p$ enforces the incompressibility condition. We omit external body forces; a conservative one could instead be absorbed into $p$.

With zero velocity and no time dependence, the magnetic second-moment tensor equation holds automatically, but the magnetohydrodynamic momentum equation still requiresThe magnetic tension must therefore be a pressure gradient. Locally, its necessary integrability condition ison a simply connected region this is also sufficient for a scalar $p_0$, subject to the global boundary conditions. In physical units the same balance is $\nabla P_0={\mathbf{J}}_0\times {\mathbf{B}}_{\mathrm{phys},0}$: the Lorentz force density is balanced by mechanical pressure, rather than necessarily vanishing. A zero mean magnetic field does not make the local magnetic tension zero, so an arbitrary tangle is not automatically a static equilibrium.

Average the equations in part b and make exactly the prescribed scale-separation approximation: the retained perturbations vary on lengths much greater than $\ell$, and perturbations at the tangle scale are omitted. Products of two retained perturbations can then be neglected. Writing $\mathbf{v}=\delta \mathbf{u}$, $m=\delta M$ and $\pi =\delta p$, the linearized equations areNo unmodelled small-scale response has been inserted into this closure.

Introduce the fluid displacement $\mathbf{\xi }$ by $\mathbf{v}={\partial }_t\mathbf{\xi }$ and choose $\nabla \cdot \mathbf{\xi }=0$. Integrating the magnetic second-moment tensor equation giveswhere $s$ is a time-independent integration tensor set by the initial stress. For a pure oscillatory normal mode, $s=0$. Its restoring force is therefore $v_A^2{\nabla }^2\mathbf{\xi }$: the isotropic magnetic tangle has the same transverse response as an incompressible elastic medium with shear modulus ${\rho }_0{v}_A^2$.

For a plane wave proportional to $e^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$, with $\omega \ne 0$, incompressibility gives $\mathbf{k}\cdot \hat{\mathbf{v}}=0$. The tensor equation and magnetohydrodynamic momentum equation becomeDotting the second equation with $\mathbf{k}$ yields $\hat{\pi }=0$ for $k\ne 0$, since $k_i{k}_j{\hat{m}}_{ij}=0$. Substitution then givesThere are two independent transverse polarizations. The phase speed is $v_A$ in every direction, and the two signs of $\omega$ give group velocities $\pm v_A\hat{\mathbf{k}}$. These magnetoelastic waves in an isotropic magnetic tangle exist because a shear deformation distorts the second moment of the tangled magnetic field, producing a restoring magnetic tension even though its mean is zero.

For comparison, a uniform magnetic field with Alfvén velocity ${\mathbf{B}}_0$ gives the incompressible Alfvén wave relation ${\omega }^2=(\mathbf{k}\cdot {\mathbf{B}}_0{)}^2$. Its frequency depends on the direction of $\mathbf{k}$, and energy propagates along ${\mathbf{B}}_0$. Here the averaged restoring response is isotropic: there is no preferred mean-field direction. Both systems resist transverse deformation through magnetic tension, rather than through compressive sound.

The propagating branch is not the entire perturbation space. For example, $\mathbf{v}=0$, $m_{ij}=f(\mathbf{r}){\delta }_{ij}$ and $\pi =f(\mathbf{r})$ solve the linearized equations without propagation, for any small, slowly varying $f$. Such a static stress perturbation in magnetoelastic wave equations balances a pressure gradient. Thus the requested wave conclusion applies to the oscillatory branch; it must not exclude stationary initial stress perturbations.

Start in an inertial frame with the inviscid magnetohydrodynamic momentum equation,Here $\rho$ is mass density, $P$ is the isotropic fluid pressure, $\mathbf{v}$ is inertial-frame velocity, $\mathbf{J}$ is electric current density, $\mathbf{B}$ is the magnetic field, and $\Phi$ is gravitational potential per unit mass. In the low-frequency approximation to Ampère's law, $\mathbf{J}=\nabla \times \mathbf{B}/{\mu }_0$. The vector calculus identityseparates the Lorentz force density into magnetic tension and a magnetic pressure gradient. Transform the material acceleration to a rigid frame rotating with constant angular velocity ${\mathbf{\Omega }}_0$, using the stated frame transformation. The rotating-frame magnetohydrodynamic momentum equation isThe final terms are the Coriolis force and centrifugal force densities; $\mathbf{u}$ is the velocity relative to the rotating frame, and $D_t={\partial }_t+\mathbf{u}\cdot \nabla$ uses that frame.

This requires a smooth conducting-fluid continuum with nonrelativistic motion, a constant magnetic permeability ${\mu }_0$, the solenoidal magnetic-field constraint, a negligible displacement current and negligible net electric-charge force. Viscous forces and anisotropic fluid stresses are omitted. A time-dependent frame would add an Euler force. Incompressible flow is not needed for this force equation. Finite electrical resistivity does not change its displayed instantaneous magnetic force; the later frozen-field calculation additionally uses ideal magnetohydrodynamics. A differentially rotating fluid does not define a single rigid frame with spatially varying ${\mathbf{\Omega }}_0$: choose a fixed local reference rate and retain the fluid's shear in $\mathbf{u}$.

The equation describes small plane-wave perturbations of a uniform, stationary plasma in ideal magnetohydrodynamics, with propagation parallel to the background magnetic field. The nonzero field defines $\hat{\mathbf{b}}={\mathbf{v}}_A/v_A$; $c_s$ is the sound speed for the assumed thermodynamic closure. Neglect viscosity, electrical resistivity, gravity and microscopic dispersive effects. A uniform background flow would instead introduce its Doppler-shifted frequency.

Decompose $\mathbf{u}={\mathbf{u}}_{\perp }+u_{\parallel }\hat{\mathbf{b}}$. The displayed matrix equation then becomesThus there are two independent transverse Alfvén wave polarizations with ${\omega }^2=k^2{v}_A^2$, restored by magnetic tension, and a longitudinal acoustic mode with ${\omega }^2=k^2{c}_s^2$, restored by pressure. The transverse modes have no first-order density compression; a strictly longitudinal displacement along the field does not bend it. These are the parallel-propagating ideal magnetohydrodynamic wave branches. In the general oblique problem, fast and slow magnetosonic modes mix compression and magnetic response; their parallel limits coincide with the faster and slower of these speeds. If $c_s=v_A$, all three velocity polarizations share the same frequency magnitude. The velocity equation alone does not enumerate nonpropagating entropy perturbations.

An Alfvén wave is a transverse disturbance restored by magnetic tension. In a uniform conducting fluid with background ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$, a small transverse displacement ${\mathbf{\xi }}_{\perp }(z,t)$ bends the magnetic field lines. Magnetic flux freezing gives ${\mathbf{b}}_{\perp }=B_0{\partial }_z{\mathbf{\xi }}_{\perp }$, and the linear transverse Lorentz force density is $(B_0/{\mu }_0){\partial }_z{\mathbf{b}}_{\perp }$. Thus the Linear Alfvén wave equations reduce toFor this shear polarization, $\delta \rho =\delta p=0$ at linear order. The disturbance transports energy along the background magnetic field; a single traveling wave has equal perturbation magnetic energy and kinetic energy, with ${\mathbf{b}}_{\perp }=\mp \sqrt{{\mu }_0\rho }{\mathbf{u}}_{\perp }$ for propagation along $\pm {\mathbf{B}}_0$. These properties distinguish it from compressive magnetosonic waves.

The ideal magnetohydrodynamics approximation requires strong electrical coupling and wavelengths large compared with microscopic scales. Finite electrical resistivity and viscosity dissipate the wave's energy. In stellar winds and other astrophysical plasmas, Alfvén waves carry disturbances and can contribute to heating when dissipated. The Lundquist liquid-metal Alfvén wave experiment provides a laboratory counterpart: a conducting liquid in a strong applied magnetic field transmits a transverse magnetic disturbance, making the field's mechanical restoring action measurable.

A local thermal imbalance can amplify a gas perturbation even without an initial displacement of an interface. Let $\mathcal{L}(\rho ,T)$ denote net cooling per unit mass, including heating with a negative sign, and suppose $\mathcal{L}({\rho }_0,T_0)=0$. When the sound-crossing time is short compared with the thermal evolution time, pressure equilibrates and an ideal gas perturbation is approximately isobaric. The linearized thermal equation isThus a negative constant-pressure derivative produces isobaric thermal instability: a cooler parcel experiences a net loss that cools it further, while a warmer parcel heats further. For example, isobaric thermal stability of a heated ideal gas with $\mathcal{L}=A\rho T^{\alpha }-H$ gives $\sigma =(1-\alpha )H/(c_p{T}_0)$ and instability for $\alpha <1$. Different heating laws and fixed-density perturbations give different criteria; thermal conduction damps sufficiently short-scale temperature contrasts. Such feedback helps a gas separate into denser cool and rarer warm regions when stable thermal branches exist.

The Rayleigh-Taylor instability instead releases gravitational potential energy when a heavier fluid lies above a lighter fluid in a downward gravitational field. For two inviscid incompressible half-spaces with no surface tension or magnetic field, a horizontal perturbation of wavenumber $k$ hasThe Atwood number measures the density contrast. A descending heavy-fluid finger and an ascending light-fluid bubble both lower the total gravitational potential energy, increasing the original displacement. Interchanging the layers gives an oscillatory stable interface. An accelerating or decelerating astrophysical shell has an analogous effective acceleration, allowing mixing at suitable ejecta interfaces in a supernova remnant. Finite interface width, viscosity, surface tension and magnetic tension modify the short-wavelength behavior; the ideal sharp-interface formula does not select a finite fastest-growing wavelength.