In asteroseismology, an r-mode (or Rossby mode) is a non-axisymmetric, predominantly toroidal oscillation of a rotating star or fluid body whose restoring force is the Coriolis force. The r-modes are the stellar analogue of the Rossby waves of terrestrial meteorology and oceanography, and they belong to the broader class of inertial modes of rotating fluids. They attracted intense interest in gravitational-wave astronomy after it was shown in 1998 that the r-modes of a relativistic star are generically unstable to the emission of gravitational waves through the Chandrasekhar–Friedman–Schutz instability (CFS) mechanism—an instability that operates at any rotation rate, however slow.^[1][2] Because of this, r-modes are considered one of the most promising mechanisms by which a rapidly rotating neutron star might radiate detectable continuous gravitational waves and might have its spin limited.^[3][4]

The oscillation modes of a non-rotating star separate into spheroidal (polar) modes—such as the pressure (p) modes, gravity (g) modes, and the fundamental (f) mode—and purely toroidal (axial) motions. For a non-rotating star the toroidal displacements are degenerate zero-frequency motions: they neither compress the fluid nor displace it radially, so they experience no buoyancy or pressure restoring force.^[5] When the star rotates, these toroidal motions acquire a finite frequency because the Coriolis force provides a restoring force, turning them into genuine, time-dependent oscillations—the r-modes.^[6]

To leading order in a slow-rotation expansion, the frequency of an r-mode in the frame co-rotating with the star is

${\displaystyle \sigma _{R}={\frac {2m\Omega }{l(l+1)}},}$

where ${\displaystyle \Omega }$ is the star's angular velocity and the integers ${\displaystyle l}$ and ${\displaystyle m}$ are the indices of the spherical harmonics that label the mode.^[5][7] This frequency is remarkable in being independent of the equation of state of the stellar matter, a property first established by Provost, Berthomieu and Rocca.^[7] Because the corotating-frame frequency is smaller than ${\displaystyle m\Omega }$, an r-mode that propagates in the retrograde direction (opposite to the rotation) as seen in the co-rotating frame is dragged forward and appears prograde to a distant inertial observer—precisely the configuration required for the CFS instability.^[7][3] For the dominant ${\displaystyle l=m=2}$ r-mode this implies an inertial-frame frequency of ${\displaystyle {\tfrac {4}{3}}\Omega }$, so the gravitational waves it emits appear at about four-thirds of the star's spin frequency.^[8]

The r-modes are a special subclass of the rotationally restored inertial modes; they are the modes that remain purely axial in the limit of slow rotation, whereas the generic inertial mode mixes axial and polar parity.^[9][10] In a rotating star the dominant toroidal velocity field of an r-mode couples weakly to a smaller spheroidal component, which produces small density and temperature perturbations and thereby renders the modes observable.^[6]

Rossby waves were first identified as large-scale meandering patterns in the Earth's atmosphere by Carl-Gustaf Rossby in 1939.^[5] The corresponding global oscillations of a rotating, self-gravitating star were named "r modes" by John Papaloizou and J. E. Pringle in 1978, who studied them in connection with the short-period oscillations of cataclysmic variables.^[11] Their basic properties in slowly and uniformly rotating stars were worked out shortly afterwards by Provost, Berthomieu and Rocca, by Smeyers and collaborators, and by Saio.^[7][12]

The modern significance of the r-modes emerged in 1998, when Nils Andersson showed by direct calculation that the r-modes of a slowly rotating relativistic star are unstable to gravitational radiation,^[1] and John L. Friedman and Sharon Morsink proved that this instability is generic—every r-mode of every rotating perfect-fluid star has negative canonical energy and is therefore CFS-unstable for arbitrarily slow rotation.^[2] The growth and damping timescales were computed almost simultaneously by Lindblom, Owen and Morsink and by Owen and collaborators, who recognised that the instability is driven by current-multipole (gravitomagnetic) gravitational radiation and could dominate the spin evolution of newly born neutron stars.^[13][14][15]

The r-mode instability is the application of the Chandrasekhar–Friedman–Schutz instability to the r-modes. Gravitational waves emitted by a mode that is retrograde in the rotating frame but prograde in the inertial frame remove positive angular momentum from the star, which drives the (negative) angular momentum of the perturbation more negative and so amplifies the mode rather than damping it.^[3] Unlike the f-mode instability, which sets in only above a critical rotation rate and first through high-${\displaystyle m}$ modes, the r-mode instability operates at any rotation rate, and the fastest-growing mode is the low-order ${\displaystyle l=m=2}$ r-mode.^[2][13]

The r-modes couple to gravitational radiation predominantly through the current quadrupole rather than the usual mass quadrupole. This coupling proved to be far stronger than had been anticipated, so strong that for a hot, rapidly rotating young neutron star it overwhelms the viscous damping forces.^[14] Lindblom, Owen and Morsink found that the instability acts primarily in the ${\displaystyle l=2}$ r-mode and, over the roughly one year needed for a newborn star to cool to about ${\displaystyle 10^{9}}$ K, can carry away most of the star's angular momentum, reducing its spin to a small fraction of the Keplerian mass-shedding rate.^[13]

Whether the instability actually develops depends on a competition between gravitational-radiation driving and viscous (and other) damping, summarised by the instability window in the plane of stellar angular velocity versus interior temperature. In the simplest "minimal" model the mode is damped by shear viscosity at low temperatures and by bulk viscosity at high temperatures, so that the window opens at intermediate temperatures and widens with increasing spin.^[4][16] The detailed shape of the window is sensitive to the microphysics of dense matter, including the rigidity of the crust, a viscous boundary layer at the crust–core interface, superfluidity and superconductivity, hyperon bulk viscosity, mutual friction, and magnetic fields.^[4][16]

Once a mode enters the instability window its amplitude grows exponentially, but the growth cannot continue indefinitely; it is halted by non-linear hydrodynamic processes. The dominant saturation mechanism is parametric coupling of the r-mode to other inertial modes, analysed using the formalism of Schenk, Arras, Flanagan, Teukolsky and Wasserman.^[17] Detailed calculations by Arras and collaborators, and large mode-network integrations by Brink, Teukolsky and Wasserman, indicate that the r-mode saturates at a dimensionless amplitude of order ${\displaystyle 10^{-4}}$ or smaller, well below unity.^[18][19] Bondarescu, Teukolsky and Wasserman incorporated this non-linear saturation into models of the coupled spin and thermal evolution of both newborn and accreting neutron stars, finding a richer variety of evolutionary trajectories than earlier single-mode treatments.^[20][21] The persisting uncertainty in the saturation amplitude is one of the principal obstacles to firm predictions of the gravitational-wave signal.^[4]

The r-mode instability has been invoked in several astrophysical contexts:

• Spin-down of newborn neutron stars. A neutron star formed rotating near break-up may be spun down by the r-mode instability during its first year, radiating an amount of energy of order one per cent of a solar rest-mass as gravitational waves and leaving the star rotating at a period comparable to the inferred initial period of the Crab Pulsar (roughly 10–20 ms). This offers a natural explanation for why young pulsars in supernova remnants are observed to rotate well below their mass-shedding limits.^[13][22]
• Spin limit of accreting neutron stars. Bildsten and, independently, Andersson, Kokkotas and Stergioulas proposed that the r-mode instability limits the spin-up of neutron stars accreting in low-mass X-ray binaries, balancing the accretion torque against gravitational-wave losses. This could explain why accreting neutron stars are observed to spin within a relatively narrow band of frequencies and why no sub-millisecond pulsars have been found.^[23][24]
• Persistent gravitational-wave sources. If the instability reaches a steady state in which accretion torque balances radiation torque, the gravitational-wave luminosity is tied to the observed X-ray luminosity, making the brightest low-mass X-ray binaries candidate sources of continuous gravitational waves; the same reasoning has been applied to hypothetical strange quark stars.^[3][4]

Extending the r-mode calculation from Newtonian gravity to general relativity proved subtle. Working in a slow-rotation, low-frequency approximation, Kojima reduced the problem for non-barotropic stars to a single second-order ordinary differential equation ("Kojima's equation") and found that relativistic frame dragging, which makes the local rotation rate depend on radius, replaces the single Newtonian r-mode frequency with a continuous band of frequencies—a continuous spectrum.^[25] The existence of this continuous spectrum was subsequently established rigorously by Beyer and Kokkotas.^[10] Lockitch, Andersson and Friedman clarified the situation by showing that Kojima's equation applies only to non-barotropic stars, that for barotropic stars axial and polar perturbations must be treated together as "hybrid" modes, and that for uniform-density models a discrete mode solution—the genuine relativistic r-mode—exists in addition to the continuous spectrum.^[26][9] Because they acquire both axial and polar character through relativistic corrections, these discrete modes are sometimes called axial-led hybrid modes.^[9]

Subsequent work has computed relativistic r-modes for realistic equations of state and incorporated additional dissipation. Kraav, Gusakov and Kantor showed that general relativity weakens the gravitational-radiation driving of slowly rotating r-modes while enhancing their dissipation through bulk viscosity and particle diffusion in superconducting matter, substantially modifying the instability window relative to Newtonian predictions.^[16][27]

No gravitational-wave signal from an r-mode has yet been detected. Because the dominant r-mode radiates near four-thirds of the spin frequency rather than at twice the spin frequency (as expected for a rotating mass "mountain"), dedicated searches must scan a different range of frequencies and frequency derivatives; Caride, Inta, Owen and Rajbhandari developed the methodology for targeting r-modes of known pulsars.^[8] Applying this method, Rajbhandari and collaborators carried out the first searches for r-mode gravitational waves from the Crab Pulsar using data from the first two observing runs of Advanced LIGO, setting upper limits on the strain amplitude near ${\displaystyle 10^{-25}}$ that beat the pulsar's spin-down limit—the first r-mode search to do so for a known pulsar.^[28] Targeted and broadband searches during the third observing run of Advanced LIGO and Advanced Virgo—including searches of the glitching X-ray pulsar PSR J0537−6910 and of accreting millisecond pulsars—have likewise reported no detection and set increasingly stringent upper limits on r-mode amplitudes.^[29][4]

Although the r-mode instability is studied mainly in neutron stars, r-modes themselves occur in any rotating star. They were originally discussed in the context of cataclysmic variables and have long been sought in the Sun, where toroidal flows consistent with r-modes have been reported.^[11][5] The clearest stellar detections came from high-precision photometry by the Kepler space telescope, which revealed r-mode signatures in the light curves of many rotating A and F main-sequence stars; these modes appear as groups of frequencies slightly below the rotation frequency in the observer's frame.^[6]

• Chandrasekhar–Friedman–Schutz instability
• Neutron-star oscillation
• Rossby wave
• Inertial wave
• Asteroseismology
• Gravitational wave
• Neutron star

• Andersson, N. (2003). "Gravitational waves from instabilities in relativistic stars". Classical and Quantum Gravity. 20 (7): R105–R144. arXiv:astro-ph/0211057. doi:10.1088/0264-9381/20/7/201.
• Stergioulas, N. (2003). "Rotating Stars in Relativity". Living Reviews in Relativity. 6 (1) 3. arXiv:gr-qc/0302034. Bibcode:2003LRR.....6....3S. doi:10.12942/lrr-2003-3. PMC 5256079. PMID 28179861.