Gravitational instability of an astrophysical disk

Disk self-gravity can overcome pressure and rotational support, producing growing density disturbances, spiral structure, or fragmentation.

For a homogeneous inviscid razor-thin disk approximation with solid-body rotation and barotropic sound speed $c$, the compressive wave branch obeys ${\omega }^2=c^2{k}^2-2\pi G{\Sigma }_0|k|+4{\Omega }^2$. The razor-thin disk Poisson kernel supplies the self-gravity term, and the radial epicyclic frequency is $2|\Omega |$. Growing modes have ${\omega }^2<0$; a separate zero-frequency balanced mode can also exist.

At fixed ${\Sigma }_0,\Omega \ne 0$, slowly decreasing $c$ first reaches a zero of the wave-frequency minimum at $c=\pi G{\Sigma }_0/(2|\Omega |)$. The corresponding wavelength is $\ell ={\pi }^{2}G{\Sigma }_0/(2{\Omega }^2)$. It estimates the spacing and scale of growing density concentrations just below marginality; linear theory does not determine the final nonlinear clump radius.

Put $Q=2|\Omega |c/(\pi G{\Sigma }_0)$. The uniformly rotating gas-sheet dispersion relation has growing waves precisely for $Q<1$, at $q_{-}<|k|<q_{+}$ with $q_{\pm }=(\pi G{\Sigma }_0/c^2)(1\pm \sqrt{1-Q^2})$. Pressure stabilizes short waves and rotation stabilizes long waves. Equality $Q=1$ is marginal; $Q\ge 1$ is the gas Toomre stability criterion.

For an isolated cold initially resting razor-thin disk, the displayed pointwise bound makes the absolute gravity-equivalent magnetic surface density smaller than its actual surface density. The positive-kernel comparison of thin-disk fields then makes the integrand in the horizontal virial balance of a cold magnetized fluid negative above and below the disk. Consequently its radial second moment begins to decrease. This establishes initial contraction in the global virial sense, not inward acceleration of every individual fluid element or unlimited subsequent collapse.

Secular gravitational instability uses dissipation to weaken the epicyclic support that stabilizes a disk dynamically. It can therefore grow more slowly than an orbital time in parameter ranges where the inviscid disk is stable.

The local axisymmetric density/velocity system of a pressured dust layer with linear gas drag and a fixed Keplerian shearing sheet gas flow has this cubic dispersion relation. The $\Omega /2$ azimuthal velocity coefficient includes advection of the Keplerian shear. The razor-thin disk Poisson kernel supplies the $|k|$ self-gravity term. Keeping the full amplitude determinant retains the zero-frequency branch and avoids dividing out the secular mode.

For any nonzero positive drag, the cubic dust gravitational dispersion relation with gas drag is negative at $s=0$ if ${\omega }^2<{\Omega }^2$ and positive at sufficiently large positive $s$. The intermediate value theorem therefore guarantees a growing real root. This existence argument is independent of the weak-drag expansion and includes the dynamically unstable ${\omega }^2<0$ regime.

The mode originating at the neutral root grows for $0<{\omega }^2<{\Omega }^2$. At small nonzero $|k|$ this gives $s_{\sec }\sim 2\pi ϵG{\sigma }_0|k|/\Omega$. Gas drag transfers angular momentum to the fixed gas flow and permits secular gravitational instability even for a dust Toomre parameter above unity. If ${\omega }^2<0$, the small-root continuation is damped, while a separate dynamical root grows. The formula is singular at ${\omega }^2=0$.

Turbulence can raise dust random speeds, increase layer thickness and mix density enhancements by eddy diffusion, delaying secular gravitational instability. On sufficiently long wavelengths, weak-drag growth is proportional to $|k|$, whereas simple diffusive damping is proportional to $k^2$. Thus diffusion alone need not remove all long-wave growth in an infinite idealization, although finite size and lifetime can make it ineffective. Coherent turbulent concentration can also increase local surface density, so the net effect requires a stochastic transport model.

For a dust layer of extent $L$, its smallest admissible wavenumber is of order $1/L$. The secular gravitational instability band ends at $k_{\mathrm{crit}}=2\pi G{\sigma }_0/c^2=2/(QH)$, with Toomre parameter $Q=c\Omega /(\pi G{\sigma }_0)$ and $H=c/\Omega$. Requiring an admissible mode in the band gives $Q\lesssim L/H$. Boundary conditions fix the order-unity prefactor; a finite disk lifetime also limits practical growth.

For fixed ${\omega }^2={\Omega }^2-2\pi G{\sigma }_0|k|+c^2{k}^2>0$, the two oscillatory roots of the dust gravitational dispersion relation with gas drag acquire negative real parts at first order in $ϵ$. Their frequencies have no first-order shift. The expansion is not uniform near ${\omega }^2=0$, where the oscillation and damping scales can become comparable.

For a razor-thin isothermal gas disk,Axisymmetric disturbances are stable when $Q\ge 1$: pressure stabilizes short wavelengths, epicyclic motion stabilizes long wavelengths, and self-gravity drives intermediate wavelengths.

At the marginal radial wavenumber, the forced axisymmetric density mode with vortensity has zero squared frequency. Its mass density amplitude is $C_0+C_{1}t-\Omega {\Sigma }_0^2{q}^{\prime }{t}^2$. Thus there is no exponential instability but nonzero conserved vortensity can force quadratic growth, and a homogeneous generalized mode can grow linearly. The spectral boundary $Q=1$ of the Toomre stability criterion must not be mistaken for boundedness of every initial-value solution.

Linear axisymmetric compressive modes of a barotropic closure of a razor-thin disk combine radial epicyclic restoration, softened self-gravity and pressure. The off-plane razor-thin Poisson kernel supplies the exponential gravity reduction. Here $k$ is the nonnegative radial wavenumber magnitude. The full linear system also has an axisymmetric geostrophic mode.

The most unstable softened disk wavenumber minimizes the squared mode frequency. Instability is $Q<Q_c$, while $Q\ge Q_c$ is spectrally stable for this compressive pair. Since $s_{\ast }<(1+\delta {)}^{-1}$, $Q_c^2<(1+2\delta )/(1+\delta {)}^2\le 1$. At zero gravitational softening both bounds become equality. The physical path with fixed $\kappa ϵ/c_s$ obeys $\delta =(\kappa ϵ/c_s)/Q$.

With $s=Q{c}_{s}k/\kappa$ and $\delta =\kappa ϵ/(Q{c}_s)$, maximize $F(s)=2s{e}^{-\delta s}-s^2$. Its unique positive critical point satisfies the displayed equation. For positive gravitational softening, $e^{-\delta s_{\ast }}<1$ and $1-\delta s_{\ast }>0$, giving the strict bound. Unsoftened gravity has $s_{\ast }=1$.

For a razor-thin fluid disk with sound speed or effective random speed $c$, surface density $\Sigma$ and radial epicyclic frequency $\kappa$, $Q=c\kappa /(\pi G\Sigma )$ compares pressure and rotational support with self-gravity. The local axisymmetric fluid Toomre stability criterion is $Q\ge 1$. For a Keplerian shearing sheet, $\kappa =\Omega$. Collisionless stellar disks use a different numerical normalization; gas drag can permit secular growth even when this fluid parameter exceeds unity.

In a star-dominated Keplerian disk with $H\simeq c_s/\Omega$ and characteristic disk mass $M_D\sim \pi r^2\Sigma$, the Toomre stability criterion becomes $Q\sim (H/r)(M_{\ast }/M_D)$. Thus instability can occur at a disk-to-star mass ratio comparable to the small disk aspect ratio. The numerical coefficient depends on whether $M_D$ means a local annular estimate or integrated enclosed mass.

In a cooling self-gravitating disk, $Q<1$ excites gravitational turbulence whose dissipation raises the sound speed, while $Q>1$ suppresses that heating and lets cooling lower the sound speed. A sustained state can therefore regulate itself near $Q\simeq 1$.

A mixing-length estimate for gravitational turbulence takes the largest unstable wavelength $\ell$ and orbital growth rate $\Omega$, giving turbulent velocity $u_{\mathrm{turb}}\sim \Omega \ell$ and kinematic viscosity $\nu \sim u_{\mathrm{turb}}\ell \sim \Omega {\ell }^2$.

For $P=K{\rho }^{1+1/n}$, dimensionless pseudo-enthalpy $w=n(\rho /{\rho }_0{)}^{1/n}$, and $\zeta =\Omega z/c_0$, vertical hydrostatic balance and the Poisson equation reduce toThe midplane conditions are $w(0)=n$ and $w^{\prime }(0)=0$, and the first zero gives the disk surface.

The self-gravitating radius of an accretion disk is the radius at which its Toomre stability criterion reaches $Q=1$. Beyond it, a smooth thin disk is vulnerable to spiral structure, gravitational transport, or fragmentation.

The shearing sheet is a local Cartesian approximation to a differentially rotating disk. It retains Coriolis force, linearized tidal gravity, and a background linear shear.

A shearing box solves the shearing sheet equations in a finite domain with periodic azimuthal and vertical faces and a shearing-periodic boundary condition on its radial faces. It follows a small region rotating at a reference orbital frequency. The model can sustain stress and exchange energy with its imposed differential rotation, but its identified radial faces do not represent a global disk mass sink.

For zero mean velocity and magnetic field in a shearing box, the perturbation energy obeys $\dot{E}\le 2(|A|-4{\pi }^2\eta /L^2)E$ when kinematic viscosity and magnetic diffusivity agree. Bound the work of Reynolds stress and Maxwell stress tensor by $2|A|E$, and use the spectral gap of a shearing-periodic box to bound dissipation below. Thus the displayed bound on magnetic Reynolds number guarantees exponential decay and excludes sustained turbulence.

For background velocity $-2Ax{\mathbf{e}}_y$, opposite radial faces of a shearing box are identified after the displayed azimuthal shift, modulo $L_y$. Products and spatial derivatives inherit this condition. Their radial surface integrals cancel because a translation preserves an integral over a periodic azimuthal interval.

For zero-mean fields in a shearing box, Fourier modes have wavevectors $(2\pi n/L_x+2At2\pi m/L_y,2\pi m/L_y,2\pi p/L_z)$. A nonzero azimuthal or vertical index supplies at least $2\pi /L$, where $L$ is the largest box side; otherwise the nonzero radial index does. Parseval identity gives the displayed Poincaré inequality uniformly in time.

Periodic azimuthal and vertical boundary terms vanish. Radial boundary terms cancel after the azimuthal translation prescribed by the shearing-periodic boundary condition. Consequently the volume average of a divergence is zero, enabling integration by parts for the smooth shearing-periodic fields.

A stationary mass density disturbance can be balanced by a changed azimuthal velocity in a shearing sheet. The continuity and azimuthal equations give zero radial velocity, while the radial equation balances Coriolis force against pressure and gravity. This mode is separate from the density-wave pair in the softened Toomre dispersion relation and is lost if one divides every equation by frequency.

An infinite gravitating particle lattice with spacing $h$ in the azimuthal direction is a shearing-sheet equilibrium when all particles sit at $x_n=0,y_n=nh$. Opposite-neighbor accelerations cancel absolutely. The linear gravitational response to pair displacement differences has transverse and longitudinal coefficients $1$ and $-2$, divided by $h^3|j-n{|}^3$.

For $q=GmF(hk)/(h^3{\Omega }^2)$, the squared dimensionless growth rates are $s^2/{\Omega }^2=(q-1\pm \sqrt{9q^2-26q+1})/2$. Exponential growth begins for $q>(13-4\sqrt{10})/9$. Since the gravitating lattice Fourier kernel is largest at $hk=\pi$, the full infinite lattice is exponentially unstable when $Gm/(h^3{\Omega }^2)>(13-4\sqrt{10})/[9F(\pi )]$. Marginal repeated imaginary roots and zero-frequency secular modes require separate treatment.

The discrete normal modes of an infinite gravitating particle lattice involve $F(\xi )=2{\sum }_{l=1}^{\infty }(1-\cos (l\xi ))/l^3$. It is even, nonnegative and $2\pi$-periodic. Its derivative is a positive reciprocal-square sine series for $0<\xi <\pi$, so its maxima occur exactly at odd multiples of $\pi$, with $F(\pi )=(7/2)\zeta (3)$.

The steady planar velocity $\mathbf{u}=\alpha y{\mathbf{e}}_x-\beta x{\mathbf{e}}_y$ supports a bounded polytropic patch when $\alpha =({\beta }^2-3{\Omega }^2)/\beta$ and $\sqrt{3}\Omega <\beta <2\Omega$. Its polytropic enthalpy is $Q=Q_0-(2\Omega -\beta )(\beta x^2+\alpha y^2)/2$ with $Q_0>0$. Its fluid free boundary is the material ellipse $\beta x^2+\alpha y^2=2Q_0/(2\Omega -\beta )$. No additional self-gravity potential is part of this model.

In an unstratified Keplerian shearing sheet with sound speed $c$, the four linear velocity-density equations yield ${\omega }^4-({\Omega }^2+c^2{k}^2){\omega }^2+{\Omega }^2{c}^2{k}_z^2=0$. The lower-frequency branch approaches the inertial wave relation ${\omega }^2={\Omega }^2{k}_z^2/k^2$ for $ck\gg \Omega$; the upper branch is predominantly acoustic.

A dilute dust component can be approximated as a pressureless fluid with velocity $\mathbf{v}$ and surface density of a disk ${\Sigma }_d$. In a gas velocity $\mathbf{u}$, linear drag force gives acceleration $-(\mathbf{v}-\mathbf{u})/t_s$ for positive aerodynamic stopping time $t_s$. Its equations include Coriolis acceleration, the shearing-sheet tidal potential, and ${\partial }_t{\Sigma }_d+\nabla \cdot ({\Sigma }_d\mathbf{v})=0$. Neglecting gas backreaction is justified only when the prescribed-gas approximation is appropriate. Strong orbit crossing or appreciable dust velocity dispersion can invalidate the single-valued pressureless-fluid description.

Let $v_x,v_y$ and $u_x,u_y$ be the dust and gas velocity perturbations after subtracting the common background shear, suppressing primes. Put $t_s>0$ for the aerodynamic stopping time. Linear axisymmetric dust perturbations about a Keplerian shearing sheet obey $D_{ϵ}{v}_x-2\Omega v_y=ϵ\Omega u_x$ and $D_{ϵ}{v}_y+\Omega v_x/2=ϵ\Omega u_y$, with $ϵ\Omega =t_s^{-1}>0$. Eliminating $v_y$ givesThus a gas epicycle $u_x=u\cos (kx-\Omega t)$, $u_y=u\sin (kx-\Omega t)/2$ supplies both $2ϵ{\Omega }^{2}u\sin (kx-\Omega t)$ and ${ϵ}^2{\Omega }^{2}u\cos (kx-\Omega t)$. Keeping the latter is necessary for an exact linear equation beyond first order in drag. The homogeneous roots are $-ϵ\Omega \pm i\Omega$, separating damping from epicyclic frequency.

If the prescribed gas velocity is itself a free epicyclic motion solution, subtract its two momentum equations from the forced dust epicycle with aerodynamic drag equations. The velocity difference obeys a damped epicycle and decays as $e^{-ϵ\Omega t}$. Therefore dust approaches the gas velocity with identical phase and amplitude for every positive drag rate. At zero drag the difference does not decay, so the late-time limit is nonuniform as drag vanishes. A finite-wavelength inertial-acoustic wave is detuned from the free epicycle; matching its phase and amplitude in the weak-drag limit additionally requires $|\omega -\Omega |\ll ϵ\Omega$. For a gas density wave of frequency $\omega$, the exact radial transfer amplitude isobtained by solving the two dust equations and using $U_y=-i\Omega U_x/(2\omega )$. At $\omega =\Omega$ it is exactly one.

Let ${\Sigma }_{d0}$ be the uniform background dust surface density of a disk and ${\delta }_d=({\Sigma }_d-{\Sigma }_{d0})/{\Sigma }_{d0}$ its fractional perturbation. Here $k$ is the radial wavenumber, $\Omega$ is the epicyclic frequency and $u$ is a real velocity amplitude. For that uniform background and late velocity $v_x=u\cos (kx-\Omega t)$, linear dust continuity equation gives ${\partial }_t{\delta }_d=-{\partial }_x{v}_x=ku\sin (kx-\Omega t)$. Integrating yields the displayed formula. The coherent travelling-wave fractional excess is $|ku|/\Omega$, with validity requiring this number much smaller than one. The static term $C(x)$ records initial density and transient displacement: drag damps velocity, but supplies no density diffusion. The long-wavelength gas wave has the same fractional density amplitude, so entrainment alone does not produce an additional dust-to-gas ratio enhancement at this order.

The orbital shear parameter measures the fractional radial decrease in orbital angular frequency. A Kepler orbit has $q=3/2$, while rigid rotation has $q=0$. The radial epicyclic frequency obeys ${\kappa }_r^2=2(2-q){\Omega }^2$; this follows by differentiating the squared specific angular momentum $h^2=r^4{\Omega }^2$.

Expand a unit-mass particle's Lagrangian about an axisymmetric circular orbit of radius $r_0$ and frequency ${\Omega }_0$, using $r=r_0+x$ and $\phi ={\Omega }_{0}t+y/r_0$. Circular-orbit balance removes the linear radial term; total-time-derivative invariance of a Lagrangian removes $r_0{\Omega }_0\dot{y}$. To second order,Here $q$ is the orbital shear parameter and ${\Omega }_z$ is the vertical epicyclic frequency. The Euler-Lagrange equations retain the local Coriolis acceleration and tidal gravity.

The time-independent particle Lagrangian in a shearing sheet has conserved rotating-frame energyThe velocity-linear Coriolis acceleration term cancels from this expression. Up to the reference-orbit constant, it is the second-order expansion of inertial specific orbital energy minus ${\Omega }_0$ times inertial specific angular momentum. Its negative radial tidal term allows inelastic collisions to lower the total energy while increasing the radial extent of a ring.

In a Keplerian shearing sheet, an epicyclic guiding center $x_0$ carries rotating-frame energy $-3{\Omega }^2{x}_0^2/8$, plus nonnegative radial and vertical oscillation energies. Inelastic collisions dissipate the total Jacobi energy in a shearing sheet, while momentum conservation preserves $\sum x_0$. Thus $\sum x_0^2$ must grow: the ring spreads about its fixed mean radius. The effect is an angular momentum transport process, with some particles moving inward and others outward.

A Kida vortex is an exact elliptical patch of uniform vorticity embedded in a linear shear flow. In a Keplerian shearing sheet its aspect ratio controls both the circulation period and the stability of three-dimensional perturbations.

The material-boundary and vorticity conditions give $\alpha =S/[r(r-1)]$ and $\beta =Sr/(r-1)$. A pressureless particle subject to linear drag, Coriolis acceleration and tidal gravity has the drag-polynomial stability for all positive stopping rates with $K=2\Omega (2\Omega -S)$, $L=\Omega (\alpha +\beta -S)$ and $P=\alpha \beta$. In Keplerian shear, $K/{\Omega }^2=1$, $L/{\Omega }^2=3(r+1)/[2r(r-1)]$ and $P/{\Omega }^2=9/[4(r-1{)}^2]$. The exact criterion is $r\ge 3$, including the boundary $K=L>P$ at $r=3$. Attraction is local to the centre under the prescribed interior velocity field; a trajectory leaving the patch cannot be followed using that field alone.

For the incompressible velocity $u_x=\alpha y$, $u_y=-\beta x$ in a shearing sheet, advection produces $-\alpha \beta (x,y)$ and the Coriolis acceleration produces $(2\Omega \beta x,2\Omega \alpha y)$. Including the radial tide $2\Omega Sx$ gives $p/\rho =(A{x}^2+B{y}^2)/2+C$. A material ellipse has $\beta =r^2\alpha$. A vortex patch is immersed in surrounding fluid, so this pressure need not vanish on its boundary; imposing that condition would describe a different free-boundary problem.

The stratified incompressible shearing sheet adds a buoyancy displacement $\theta$ with $D\theta /Dt=u_z$ and acceleration $-N^2\theta {\mathbf{e}}_z$. Axisymmetric plane waves obeyinterpolating between inertial and buoyancy oscillations.

A Keplerian shearing sheet has background velocity ${\mathbf{u}}_0=-(3/2)\Omega x{\mathbf{e}}_y$, radial shear rate $S=3\Omega /2$, and radial epicyclic frequency ${\kappa }_r=\Omega$.

Vertical shear instability destabilizes differential rotation that varies with height, when buoyancy restoration is absent or sufficiently weakened. In the homogeneous incompressible flow model with background $U_y=-3\Omega x/2+q\Omega z$, an axisymmetric perturbation with wavevector $(k_x,0,k_z)\ne 0$ hasThe pressure-free radial/vertical momentum equation and the azimuthal equation give this result after using incompressibility. Growth requires $2q{k}_x{k}_z>k_z^2$, so a small vertical shear favors nearly radial wavevectors. Here $q$ is the vertical shear amplitude, distinct from the orbital shear parameter $3/2$. Stratification, cooling and boundaries can change this simplified criterion.

For the unstratified local vertical shear instability, maximize $s^2/{\Omega }^2=(2qR-1)/(1+R^2)$ over $R=k_x/k_z$. Its derivative vanishes when $q{R}^2-R-q=0$. For $q>0$ the maximum uses $R=(1+\sqrt{1+4q^2})/(2q)$, and substitution gives the displayed expression. For small $q$, $R\sim q^{-1}$ and $s_{\max }=\Omega q(1-q^2/2+O(q^4))$. Changing the sign of the shear changes the preferred sign of $R$ but gives the same maximum magnitude; there is no exponential growth at $q=0$.

For a perturbation ${\mathbf{u}}^{\prime }=\tilde{\mathbf{u}}f(\mathbf{k}\cdot \mathbf{x})e^{st}$ with constant $\tilde{\mathbf{u}},\mathbf{k}$ and nonconstant differentiable $f$, incompressibility forces $\mathbf{k}\cdot \tilde{\mathbf{u}}=0$. Consequently ${\mathbf{u}}^{\prime }\cdot \nabla {\mathbf{u}}^{\prime }=f{f}^{\prime }{e}^{2st}(\mathbf{k}\cdot \tilde{\mathbf{u}})\tilde{\mathbf{u}}=0$. About a shear depending only on $x,z$ and directed along $y$, an axisymmetric such perturbation therefore obeys its linear amplitude equations exactly, including at finite amplitude in this homogeneous model. For pressure $P^{\prime }=\tilde{P}g(\xi )e^{st}$, nonzero pressure requires $g^{\prime }=Cf$, and $C$ can be absorbed into $\tilde{P}$; choose $g^{\prime }=f$. This relation does not imply $g=f$ for an arbitrary profile.

For constant density ${\rho }_0$ and body force ${\Omega }^2(3x-2qz){\mathbf{e}}_x$, dotting the incompressible flow momentum equation with ${\rho }_0\mathbf{u}$ gives the kinetic energy flux $(K+P)\mathbf{u}$ and source ${\rho }_0{\Omega }^2(3x-2qz)u_x$, where $K={\rho }_0|\mathbf{u}{|}^2/2$. Coriolis acceleration does no work. Absorbing ${\Phi }_t=-3{\Omega }^2{x}^2/2$ gives the displayed balance. For nonzero $q$, the remaining force has curl $-2{\Omega }^{2}q{\mathbf{e}}_y$, so it cannot be incorporated into a scalar mechanical potential. The model can exchange energy with its maintained vertical-shear background; at $q=0$ ordinary Jacobi energy in a shearing sheet conservation is recovered under zero boundary flux.

An adverse radial specific entropy gradient can drive an oscillatory instability through the finite lag supplied by thermal conduction or relaxation. This differs from a viscous-convective instability, in which viscosity changes rotational stabilization.

For $|N^2|\ll {\Omega }^2$, this leading amplitude-growth rate is largest at thermal diffusion rate of a Fourier mode $\beta =\Omega$, giving ${\sigma }_{\max }=-N^2/(4\Omega )$ when $N^2<0$.

A horizontal-velocity perturbation depending only on height has zero vertical velocity for nonzero wavenumber under incompressibility. Its pressure amplitude then vanishes in the unstratified vertical momentum equation.

The nonoscillatory thermal branch of the radially stratified shearing sheet spectrum. Its decoupled small-buoyancy limit is a diffusing scalar perturbation.

Vortensity is absolute vorticity divided by surface density. In a two-dimensional inviscid barotropic flow, it is conserved along fluid trajectories.

Linearized vortensity is the first-order perturbation of vortensity about a background flow. For a shearing wave in a uniform Keplerian sheet, its amplitude is constant.

Linearized vortensity conservation is the perturbative form of material vortensity conservation. In an inviscid barotropic shearing sheet, the linearized vortensity obeys $D{f}^{\prime }/Dt=0$.

For a radial mode with constant nonzero wavenumber in an inviscid razor-thin disc, eliminate velocity using continuity and the conserved linearized vortensity. The resulting oscillator has ${\omega }_k^2={\kappa }^2+c_s^2{k}^2-2\pi G{\Sigma }_0|k|$. The vortensity gives a constant forcing and hence the stationary balanced component when ${\omega }_k^2>0$. It is lost by an ansatz that divides every amplitude equation by a nonzero frequency. Negative squared frequency gives exponential instability; zero frequency allows secular growth.

A shearing wave is a local disturbance whose wavevector evolves under background shear. In a Keplerian sheet, $k_x(t)=k_x(0)+(3/2)\Omega k_{y}t$.

For background shear ${\mathbf{u}}_0=-2Ax{\mathbf{e}}_y$, take ${\mathbf{u}}^{\prime }=\mathrm{Re}[\mathbf{v}(t)e^{i\mathbf{k}(t)\cdot \mathbf{r}}]$ with real $\mathbf{k}$ and $\mathbf{k}\cdot \mathbf{v}=0$. Then ${\mathbf{u}}^{\prime }\cdot \nabla {\mathbf{u}}^{\prime }=0$ pointwise, including conjugate terms. Choosing ${\dot{k}}_x=2A{k}_y$, ${\dot{k}}_y={\dot{k}}_z=0$ cancels background phase advection. The resulting linear amplitude equations are therefore exact finite-amplitude solutions of the incompressible Navier-Stokes equation in this unbounded or compatible shearing-periodic setting.

// Target: gravitational-instability-of-an-astrophysical-disk.bigb

In an incompressible shearing sheet, a single real-wavevector velocity and magnetic Fourier mode transverse to that wavevector has identically zero quadratic self-advection and cross-advection. Advecting its phase by background shear and stretching its spatially uniform field give the displayed evolution laws. Their opposite contributions keep the Alfvén frequency constant. The resulting viscous and resistive wave-amplitude equations are exact finite-amplitude solutions; physical pressure adjusts to the quadratic magnetic-pressure terms.

In an incompressible shearing sheet with background velocity $-2Ax{\mathbf{e}}_y$, take both the perturbation velocity and the entire magnetic field to be real single Fourier modes with a common real wavevector and transverse amplitudes. Then every quadratic directional derivative vanishes, including terms involving complex conjugates. There is no uniform background magnetic field and consequently no linear magnetic-tension coupling between the amplitudes. Advecting the phase requires ${\dot{k}}_x=2A{k}_y$, ${\dot{k}}_y={\dot{k}}_z=0$. The velocity amplitude obeys $\dot{\tilde{\mathbf{v}}}-2A{\tilde{v}}_x{\mathbf{e}}_y+2\mathbf{\Omega }\times \tilde{\mathbf{v}}=-i\mathbf{k}\tilde{\psi }-\nu k^2\tilde{\mathbf{v}}$; the magnetic amplitude obeys the displayed equation. The modified pressure includes magnetic pressure; the physical pressure can contain a mean and second harmonic even when the modified pressure is a single mode.

For a zero-mean magnetic shearing wave, the real wavevector satisfies $k_x=k_{x0}+2A{k}_{y}t$, with $k_y,k_z$ constant. Dotting the velocity-amplitude equation with its complex conjugate eliminates Coriolis force and pressure work, giving $d|\tilde{\mathbf{v}}{|}^2/dt=4A\mathrm{Re}({\tilde{v}}_x{\tilde{v}}_y^{\ast })-2\nu k^2|\tilde{\mathbf{v}}{|}^2$. The inequality $2|v_x{v}_y|\le |\mathbf{v}{|}^2$ proves the displayed bound; the magnetic bound has the same form with $\eta$ replacing $\nu$. NowFor $A{k}_y\ne 0$ and positive kinematic viscosity and magnetic diffusivity, the negative cubic exponent dominates the linear shear-work bound, so both amplitudes tend to zero. Transient growth is allowed. Positivity matters: an inviscid vertical velocity mode with $k_z=0$ can remain constant, and an ideal vertical magnetic mode with $k_z=0$ can also remain constant.

For $k_z=v_z=0$ and $k_y\ne 0$, define $Z=i(k_x{v}_y-k_y{v}_x)$. Uniform rotation does not enter its evolution: $\dot{Z}=-\nu k^{2}Z$, $v_x=i{k}_{y}Z/k^2$ and $v_y=-i{k}_{x}Z/k^2$. With $T=k_x/k_y$ and $\mathrm{Re}=A/(\nu k_y^2)$, the perturbation energy ratio is $E(T)/E(T_0)=[(1+T_0^2)/(1+T^2)]\exp [-(T+T^3/3-T_0-T_0^3/3)/\mathrm{Re}]$. Leading waves can gain energy as shear reduces their wavevector magnitude, before viscosity and subsequent winding damp them.

// Target: hydrodynamic-stability.bigb

For the shearing-wave energy profile, stationary points satisfy $-2\mathrm{Re}T=(1+T^2{)}^2$. At large Reynolds number, the initial minimum is at $T_{-}\sim -(2\mathrm{Re}{)}^{1/3}$ and the subsequent maximum is at $T_{+}\sim -1/(2\mathrm{Re})$. Their energy ratio is $G_{\max }\sim (2\mathrm{Re}{)}^{2/3}{e}^{-2/3}$. The Reynolds number here uses $A$, half the background shear rate; using $2A$ instead changes the coefficient convention.

// Target: astrophysics.bigb

For a solenoidal velocity amplitude $\mathbf{v}$ with $\mathbf{k}=(k_x,-\Omega t{k}_x,k_z)$ and $\dot{\mathbf{v}}+\Omega v_y\hat{\mathbf{x}}=-i\mathbf{k}q$, differentiating $\mathbf{k}\cdot \mathbf{v}=0$ gives $q=2i\Omega k_x{v}_y/K^2$, where $K^2=|\mathbf{k}{|}^2$. Thus ${\dot{v}}_y=2\Omega k_x{k}_y{v}_y/K^2$ and ${\dot{v}}_z=2\Omega k_x{k}_z{v}_y/K^2$. For the real helicity quantity $H=i{k}_x(v_y{v}_z^{\ast }-v_z{v}_y^{\ast })$, the mixed forcing terms cancel, leaving $\dot{H}=2\Omega k_x{k}_{y}H/K^2=-{\dot{K}}^{2}H/K^2$. ConsequentlyThe invariant is a consequence of inviscid linear shear dynamics, not a claim that $H$ itself is constant under the shear.

A shearing-wave ansatz in a uniform inviscid isothermal shearing sheet removes background advection when the radial wavevector grows as $k_x(t)=k_x(0)+S{k}_{y}t$. The razor-thin disk Poisson kernel reduces the pressure-plus-gravity amplitude to $[c_s^2-2\pi G{\Sigma }_0/k]{\tilde{\Sigma }}^{\prime }/{\Sigma }_0$. Continuity couples it to $i{k}_x{v}_x+i{k}_y{v}_y$, while the two momentum equations have rotational couplings $-2\Omega v_y$ and $(2\Omega -S)v_x$. The linearized vortensity amplitude remains constant, because the barotropic pressure and gravitational forces have zero curl and the background vortensity is uniform.