Centrifugal Instability: Theory and Applications

• Centrifugal Instability is the tendency for rotating flows to overturn when a radially displaced fluid element experiences net forces that amplify rather than restore its motion.
• It is analyzed through criteria like the Rayleigh discriminant and is applicable to Taylor–Couette flows, stratified fluids, plasmas, and astrophysical systems.
• Nonlinear evolution of CFI involves Reynolds-stress redistribution and complex interactions with viscosity, stratification, magnetic tension, and inertial effects.

Centrifugal instability (CFI) is the tendency of a rotating flow with curved streamlines to overturn when a radially displaced fluid element experiences a net force that amplifies, rather than restores, its displacement. In its classical form, CFI is governed by the outward variation of specific angular momentum and is exemplified by Taylor–Couette flow; in later formulations it also appears in stratified, viscous, diffusive, compressible, relativistic, magnetic, plasma, and stellar settings. Across these settings, the central issue is whether centrifugal forcing dominates the restoring effects of pressure, buoyancy, magnetic tension, viscosity, or acoustic compressibility (Leclercq et al., 2016, Gourgouliatos et al., 2017, Komissarov et al., 2019, Gueroult et al., 2017, Shi et al., 2022).

1. Classical criterion and foundational quantities

Rayleigh’s original criterion states that a rotating flow is linearly unstable to axisymmetric vortices if the specific angular momentum

$L(r)=r^2\Omega(r)$

decreases outward, that is,

$\frac{dL}{dr}<0.$

An equivalent form uses the Rayleigh discriminant

$\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$

with instability where $\Phi(r)<0$ (Marcotte et al., 2016).

In Taylor–Couette flow, with inner and outer cylinders of radii $r_i,r_o$ and angular speeds $\Omega_i,\Omega_o$, the standard dimensionless parameters are

$$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$

together with the Brunt–Väisälä frequency

$N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$

the Richardson number

$Ri=\frac{N^2}{\Omega_i^2},$

and the Froude-type number

$Fr=\frac{\Omega_i}{N}.$

In the inviscid, unstratified limit, Rayleigh’s criterion becomes

$\frac{dL}{dr}<0.$0

so that the “Rayleigh line” $\frac{dL}{dr}<0.$1 marks the classical onset boundary (Leclercq et al., 2016).

The same angular-momentum logic extends beyond cylindrical annuli. In a thin spherical shell, near the equator, the same basic condition applies to the angular momentum per unit mass about the rotation axis, and a viscous-corrected criterion can be written for the spherical gap. This establishes that CFI is not tied to a single geometry, but to the centrifugal balance associated with curved rotation (Marcotte et al., 2016).

2. Taylor–Couette flow beyond the inviscid limit

In viscous, stratified Taylor–Couette flow, axisymmetric and non-axisymmetric disturbances behave very differently. For axisymmetric modes ($\frac{dL}{dr}<0.$2), the neutral condition remains exactly

$\frac{dL}{dr}<0.$3

No axisymmetric linear mode can go unstable for $\frac{dL}{dr}<0.$4 at any $\frac{dL}{dr}<0.$5 or $\frac{dL}{dr}<0.$6, and in the thin-gap limit an energy method shows that for $\frac{dL}{dr}<0.$7 the generalized energy of all streamwise-independent perturbations decays monotonically. The Rayleigh line is therefore an immutable stability boundary for $\frac{dL}{dr}<0.$8, even with stratification, viscosity, or finite disturbance amplitude (Leclercq et al., 2016).

For non-axisymmetric modes ($\frac{dL}{dr}<0.$9), the threshold is modified at finite $\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$0 and $\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$1 through the condition

$\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$2

and at large $\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$3 one obtains

$\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$4

with $\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$5 as $\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$6. A new helical $\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$7 CI-like branch appears for $\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$8 just above $\Phi(r)=r^3\frac{d(\Omega^2)}{dr}+4r^2\Omega^2,$9 when viscosity and stratification act jointly; as $\Phi(r)<0$0 it satisfies

$\Phi(r)<0$1

which are CI characteristics, but it exists at $\Phi(r)<0$2 only at finite $\Phi(r)<0$3 and $\Phi(r)<0$4 (Leclercq et al., 2016).

A central result is the continuous connection between centrifugal instability (CI), stratorotational instability (SRI), and radiative instability (RI). At high $\Phi(r)<0$5, non-axisymmetric spectra can display two local maxima of $\Phi(r)<0$6: one CI-type with $\Phi(r)<0$7, and one SRI/RI-type with finite $\Phi(r)<0$8. As $\Phi(r)<0$9 decreases from infinity, the maxima approach and coalesce at $r_i,r_o$0–$r_i,r_o$1 for $r_i,r_o$2, or even lower for $r_i,r_o$3, eliminating the jump in optimal axial wavenumber. Below that range, CI and SRI are smoothly connected and indistinguishable at onset; a similar morphing connects SRI and RI when $r_i,r_o$4 is moderate and the gap finite (Leclercq et al., 2016).

Viscosity and stratification also produce qualitative departures from inviscid theory. In unstratified viscous flow ($r_i,r_o$5), a non-axisymmetric branch localized at the outer cylinder appears and is unstable only because of viscosity. For $r_i,r_o$6, the first $r_i,r_o$7 bifurcation is oscillatory and arises from the collision of two steady Taylor-vortex branches. At strong stratification ($r_i,r_o$8) and $r_i,r_o$9, a dominant $\Omega_i,\Omega_o$0 mode with negative azimuthal phase speed $\Omega_i,\Omega_o$1 propagates against the inner-cylinder rotation. At weak stratification ($\Omega_i,\Omega_o$2) and moderate Reynolds number ($\Omega_i,\Omega_o$3), an $\Omega_i,\Omega_o$4 mode for $\Omega_i,\Omega_o$5 is generated by the merging of two CI-type local maxima from $\Omega_i,\Omega_o$6 (Leclercq et al., 2016).

Recent work on highly diffusive, stratified Taylor–Couette flow isolates the role of thermal diffusion. For $\Omega_i,\Omega_o$7, the stabilizing role of stratification is suppressed, and the dependence on $\Omega_i,\Omega_o$8 and $\Omega_i,\Omega_o$9 collapses onto the single rescaled parameter

$$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$0

For $$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$1, $$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$2, and $$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$3, the primary threshold is fitted by

$$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$4

The secondary threshold $$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$5 increases in the range $$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$6 at $$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$7, and also increases as $$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$8 rises at $$\eta=\frac{r_i}{r_o},\qquad \mu=\frac{\Omega_o}{\Omega_i},\qquad Re=\frac{r_i\Omega_i(r_o-r_i)}{\nu},$$9, indicating delayed onset of non-axisymmetric breakdown under intermediate diffusivity (Park, 9 Dec 2025).

3. Laboratory and engineering manifestations

In a thin spherical shell with aspect ratio $N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$0, centrifugal instability first appears as axisymmetric Taylor vortices in the equatorial region. Numerically, the critical gap Reynolds number is $N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$1, corresponding to $N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$2 for $N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$3. The secondary hydrodynamic instability sets in at $N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$4–$N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$5, with dominant azimuthal wavenumbers $N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$6 at $N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$7 and $N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$8 at $N=\sqrt{-\frac{g}{\rho_0}\frac{d\bar\rho}{dz}},$9. These spherical Taylor–Couette vortices drive a strongly subcritical dynamo: $Ri=\frac{N^2}{\Omega_i^2},$0 as $Ri=\frac{N^2}{\Omega_i^2},$1, the minimum is $Ri=\frac{N^2}{\Omega_i^2},$2 around $Ri=\frac{N^2}{\Omega_i^2},$3–$Ri=\frac{N^2}{\Omega_i^2},$4, and for $Ri=\frac{N^2}{\Omega_i^2},$5 the threshold levels off to a constant $Ri=\frac{N^2}{\Omega_i^2},$6. Global rotation can reduce the dynamo onset by up to a factor $Ri=\frac{N^2}{\Omega_i^2},$7 when the flow remains in the laminar or weakly wavy regime (Marcotte et al., 2016).

In fast-rotating free-surface cylinders, the relevant instability is a centrifugal–gravity resonant instability between a horizontal centrifugal edge wave at the inner interface and a vertical gravity wave at the outer upper boundary. The isolated wave frequencies are

$Ri=\frac{N^2}{\Omega_i^2},$8

The instability is obtained only from the resonance of the two counter-propagating waves $Ri=\frac{N^2}{\Omega_i^2},$9 and $Fr=\frac{\Omega_i}{N}.$0, with normal-mode frequency

$Fr=\frac{\Omega_i}{N}.$1

and growth requires

$Fr=\frac{\Omega_i}{N}.$2

Pairings that phase-lock without satisfying this condition remain neutral (Yellin-Bergovoy et al., 2017).

In the forced wake of a circular cylinder at $Fr=\frac{\Omega_i}{N}.$3, three-dimensional structure can arise below the usual three-dimensionalization threshold through a pulsed centrifugal instability of the oscillating Stokes layer at the wall. The local inviscid condition is expressed through the Rayleigh discriminant

$Fr=\frac{\Omega_i}{N}.$4

with instability where $Fr=\frac{\Omega_i}{N}.$5. At $Fr=\frac{\Omega_i}{N}.$6 the critical Taylor number is $Fr=\frac{\Omega_i}{N}.$7, corresponding to $Fr=\frac{\Omega_i}{N}.$8; at $Fr=\frac{\Omega_i}{N}.$9, $\frac{dL}{dr}<0.$00, corresponding to $\frac{dL}{dr}<0.$01. At $\frac{dL}{dr}<0.$02, $\frac{dL}{dr}<0.$03, the measured spanwise wavelength is $\frac{dL}{dr}<0.$04, and more generally $\frac{dL}{dr}<0.$05 (D'Adamo et al., 2015).

Weak rotation can also destabilize buoyant convection when centrifugal force acts against the main convective circulation. At low Prandtl number, the counteraction can split the main vortex into two counter-rotating vortices whose interaction leads to a three-dimensional centrifugal instability. At larger Prandtl number, the same counteraction steepens an unstable thermal stratification and triggers a Rayleigh–Bénard-type mechanism. In the characteristic rotating-lid problem, examples include $\frac{dL}{dr}<0.$06, $\frac{dL}{dr}<0.$07 with $\frac{dL}{dr}<0.$08, and $\frac{dL}{dr}<0.$09, $\frac{dL}{dr}<0.$10 with $\frac{dL}{dr}<0.$11 (Gelfgat, 2011).

Rapidly rotating two-layer fluids provide another limiting case in which centrifugal acceleration replaces gravity. For an inviscid, immiscible, sharp interface, the dispersion relation can be written

$\frac{dL}{dr}<0.$12

with growth for $\frac{dL}{dr}<0.$13, that is, when the inner fluid is heavier than the outer. Surface tension introduces a cutoff wavenumber, viscosity always reduces the growth rate relative to the inviscid case, and interface diffusion inhibits growth relative to a sharp jump (Scase et al., 2018).

Curved free-shear layers extend centrifugal instability into compressible open flows. In the nonlinear boundary region equations, the curvature forcing enters through the wall-normal momentum term $\frac{dL}{dr}<0.$14, which is the leading centrifugal contribution. The freestream Mach number $\frac{dL}{dr}<0.$15, the shear-layer thickness $\frac{dL}{dr}<0.$16, the disturbance amplitude $\frac{dL}{dr}<0.$17, and the relative velocity difference $\frac{dL}{dr}<0.$18 control development: higher $\frac{dL}{dr}<0.$19 tends to delay onset and move the mixing region downstream, thinner shear layers strengthen growth, larger $\frac{dL}{dr}<0.$20 seeds earlier nonlinear amplification, and larger $\frac{dL}{dr}<0.$21 strengthens the centrifugal forcing (Es-SAhli et al., 2024).

4. Nonlinear evolution, saturation, and transport

The nonlinear evolution of CFI often proceeds through Reynolds-stress redistribution of angular momentum. In the semi-linear model of an anticyclonic Gaussian vortex,

$\frac{dL}{dr}<0.$22

the mean azimuthal flow obeys an evolution equation forced by the axially averaged Reynolds stress, while the dominant axial harmonic evolves linearly on the slowly changing mean flow. The key quadratic quantity is

$\frac{dL}{dr}<0.$23

and its divergence redistributes azimuthal momentum until the mean profile becomes centrifugally stable. For $\frac{dL}{dr}<0.$24, the semi-linear model reproduces direct numerical simulation at both $\frac{dL}{dr}<0.$25 and $\frac{dL}{dr}<0.$26, and the final state approaches the homogenized angular-momentum profile predicted in the inviscid limit, smoothed by an $\frac{dL}{dr}<0.$27 viscous transition layer at finite Reynolds number (Yim et al., 2019).

In stratified and diffusive Taylor–Couette flow, nonlinear saturation can be followed by distinct secondary routes. For $\frac{dL}{dr}<0.$28, $\frac{dL}{dr}<0.$29, and $\frac{dL}{dr}<0.$30, the axisymmetric Taylor vortices saturate first, after which a highly non-axisymmetric mode with $\frac{dL}{dr}<0.$31 grows and produces a steady wavy Taylor-vortex state. For $\frac{dL}{dr}<0.$32 at the same $\frac{dL}{dr}<0.$33 and $\frac{dL}{dr}<0.$34, several azimuthal modes compete and the flow does not settle to a steady wave, instead exhibiting irregular chaotic fluctuations. Angular-momentum transport is quantified through the torque-based Nusselt number

$\frac{dL}{dr}<0.$35

and during the axisymmetric-vortex regime it follows the Di-Prima form

$\frac{dL}{dr}<0.$36

Once secondary instability or chaos sets in, the transport departs from this axisymmetric scaling (Park, 9 Dec 2025).

At relativistic cylindrical interfaces, nonlinear saturation likewise removes the destabilizing profile. In the rotating two-layer simulations, tracer diagnostics show that radial mixing occurs where $\frac{dL}{dr}<0.$37 decreases with radius, and the $\frac{dL}{dr}<0.$38-averaged $\frac{dL}{dr}<0.$39 profile flattens over time, removing the negative-gradient region. This is closely aligned with the homogenization picture obtained for the Gaussian-vortex problem (Gourgouliatos et al., 2017, Yim et al., 2019).

This suggests that, although linear onset criteria vary sharply across geometries and constitutive assumptions, nonlinear CFI often saturates by eroding the radial decrease of the quantity that originally drove the instability: specific angular momentum, its square, or the corresponding relativistic discriminant.

5. Compressible, relativistic, and magnetic generalizations

For relativistic rotating fluids, the governing equations are the continuity equation $\frac{dL}{dr}<0.$40 and energy–momentum conservation $\frac{dL}{dr}<0.$41 with

$\frac{dL}{dr}<0.$42

Angular momentum conservation becomes

$\frac{dL}{dr}<0.$43

and radial equilibrium is

$\frac{dL}{dr}<0.$44

For a discontinuous cylindrical interface, the relativistic Rayleigh discriminant is

$\frac{dL}{dr}<0.$45

and instability requires

$\frac{dL}{dr}<0.$46

For continuous profiles, the local criterion becomes

$\frac{dL}{dr}<0.$47

where $\frac{dL}{dr}<0.$48 is the relativistic rotational Mach number. Simulations of two uniformly rotating cylindrical layers follow the jump criterion exactly: unstable models develop finger-like penetration and turbulent sheaths, while stable or borderline cases remain quiescent or saturate weakly (Gourgouliatos et al., 2017).

In compressible Newtonian pressure-supported rotation, the same Mach-number dependence appears explicitly. A generalized discriminant can be defined as

$\frac{dL}{dr}<0.$49

and centrifugal instability occurs where $\frac{dL}{dr}<0.$50. Axisymmetric simulations of transonic rotating flows agree perfectly with this criterion. In gravity-dominated accretion disks, however, the relevant axisymmetric condition is equivalent to the Solberg–Høiland criterion, so the classical Rayleigh form is recovered for barotropic disks despite highly supersonic orbital rotation (Komissarov et al., 10 Jun 2026).

Magnetic tension inhibits CFI by introducing a critical wavelength. For a cylindrical interface threaded by an axial magnetic field,

$\frac{dL}{dr}<0.$51

so modes with $\frac{dL}{dr}<0.$52 are stabilized. A corresponding threshold in Alfvén Mach number is

$\frac{dL}{dr}<0.$53

and in Newtonian shell calculations with $\frac{dL}{dr}<0.$54 and $\frac{dL}{dr}<0.$55 this gives $\frac{dL}{dr}<0.$56, in excellent agreement with simulations. In relativistic jets, the analysis implies that CFI develops only for relatively low magnetization, with $\frac{dL}{dr}<0.$57 in the kinetic-energy-dominated regime (Komissarov et al., 2019).

Three-dimensional RMHD simulations of AGN jet recollimation sharpen this conclusion. A relatively weak azimuthal magnetic field can completely suppress the recollimation instability, with critical nozzle magnetization

$\frac{dL}{dr}<0.$58

In the HD and MHD1 models, disruption occurs rapidly downstream of reconfinement; in MHD2 the instability grows more slowly and leaves low-$\frac{dL}{dr}<0.$59 distortions; in MHD3 with $\frac{dL}{dr}<0.$60 the jet remains laminar to the end of the computational domain. The instability is interpreted as a variant of CFI, and the suppression threshold is consistent with magnetic CFI theory (Matsumoto et al., 2020).

A recurring misconception is that CFI in astrophysical jets is interchangeable with Kelvin–Helmholtz instability or Rayleigh–Taylor instability. The relativistic analysis distinguishes them explicitly: KHI arises from velocity shear across a straight interface, RTI is recovered as a special case of the centrifugal criterion when $\frac{dL}{dr}<0.$61 and density varies, and CFI dominates when streamline curvature is strong and shear-driven KHI is weak or compressibly stabilized (Gourgouliatos et al., 2017).

6. Plasma and stellar realizations

In collisionless, magnetized plasmas driven toward rapid azimuthal rotation by strong radial electric fields, CFI arises from the difference between ion and electron angular drift velocities. For cold ions in the fast-rotation regime, the Brillouin parameter is

$\frac{dL}{dr}<0.$62

and as $\frac{dL}{dr}<0.$63 the system approaches the Brillouin limit. In the slow-rotation limit ($\frac{dL}{dr}<0.$64), the growth rate is

$\frac{dL}{dr}<0.$65

which reduces to $\frac{dL}{dr}<0.$66 for $\frac{dL}{dr}<0.$67. In the fast-rotation limit ($\frac{dL}{dr}<0.$68, $\frac{dL}{dr}<0.$69),

$\frac{dL}{dr}<0.$70

so the growth rate increases linearly with azimuthal mode number. At sufficiently large $\frac{dL}{dr}<0.$71, electron inertia suppresses growth, with cutoff

$\frac{dL}{dr}<0.$72

This places the onset of plasma CFI in the operating range envisioned for plasma mass separation devices (Gueroult et al., 2017).

In massive rotating stars, differential rotation and viscous angular-momentum transport can excite overstable radial modes. For a spherically averaged rotating equilibrium,

$\frac{dL}{dr}<0.$73

the perturbation problem can be reduced to coupled equations for $\frac{dL}{dr}<0.$74 and $\frac{dL}{dr}<0.$75. In the low-viscosity limit, the complex frequency shift is described by a work integral, and under a WKB approximation

$\frac{dL}{dr}<0.$76

A necessary condition for overstability is

$\frac{dL}{dr}<0.$77

somewhere in the star. For realistic $\frac{dL}{dr}<0.$78 models, dimensionless growth rates on the main sequence are $\frac{dL}{dr}<0.$79–$\frac{dL}{dr}<0.$80, rise to $\frac{dL}{dr}<0.$81 while the star crosses the Hertzsprung–Russell gap, and then fall again in the red-supergiant phase (Shi et al., 2022).

These plasma and stellar examples extend CFI far beyond laboratory rotating liquids. This suggests that centrifugal destabilization is best understood not as a single cylinder-specific mechanism, but as a broader instability class in which the radial organization of angular momentum, effective inertia, and curvature-driven forcing determines whether disturbances are restored, phase-locked, magnetically suppressed, or amplified into nonlinear transport and mixing.