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A ball on the top of a hill is an unstable situation.

Instability in systems is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior.

In control theory, a system is unstable if any of the roots of its characteristic equation has real part greater than zero. This is equivalent to any of the eigenvalues of the state matrix having real part greater than zero.

In structural engineering, a structure can become unstable when excessive load is applied. Beyond a certain threshold, structural deflections magnify stresses, which in turn increases deflections. This can take the form of buckling or crippling. The general field of study is called structural stability.

Fluid instabilities

Fluid instabilities occur in liquids, gases and plasmas, and are often characterized by the shape that form; they are studied in fluid dynamics and magnetohydrodynamics. Fluid instabilities include:

Plasma instabilities

Plasma instabilities can be divided into two general groups (1) hydrodynamic instabilities (2) kinetic instabilities. Plasma instabilities are also categorised into different modes:

(azimuthal wave number)
NoteDescriptionRadial modesDescription
m=0Sausage instability:
displays harmonic variations of beam radius with distance along the beam axis
n=0Axial hollowing
n=1Standard sausaging
n=2Axial bunching
m=1Sinuous, kink or hose instability:
represents transverse displacements of the beam crosssection without change in the form or in a beam characteristics other than the position of its center of mass
m=2Filamentation modes:
growth leads towards the breakup of the beam into separate filaments.
Gives an elliptic cross-section
m=3Gives a pyriform (pear-shaped) cross-section

Source: Andre Gsponer, "Physics of high-intensity high-energy particle beam propagation in open air and outer-space plasmas" (2004)

List of plasma instabilities

  • Bennett pinch instability (also called the z-pinch instability )
  • Beam acoustic instability
  • Bump-in-tail instability
  • Buneman instability,[2] (same as Farley-Buneman instability?)
  • Cherenkov instability,[3]
  • Chute instability
  • Coalescence instability,[4]
  • Collapse instability
  • Counter-streaming instability
  • Cyclotron instabilities, including:
  • Alfven cyclotron instability
  • Electron cyclotron instability
  • Electrostatic ion cyclotron Instability
  • Ion cyclotron instability
  • Magnetoacoustic cyclotron instability
  • Proton cyclotron instability
  • Nonresonant Beam-Type cyclotron instability
  • Relativistic ion cyclotron instability
  • Whistler cyclotron instability


  1. Shengtai Li, Hui Li "Parallel AMR Code for Compressible MHD or HD Equations" (Los Alamos National Laboratory) [1]
  2. Buneman, O., "Instability, Turbulence, and Conductivity in Current-Carrying Plasma" (1958) Physical Review Letters, vol. 1, Issue 1, pp. 8-9
  3. Kho, T. H.; Lin, A. T., "Cyclotron-Cherenkov and Cherenkov instabilities" (1990) IEEE Transactions on Plasma Science (ISSN 0093-3813), vol. 18, June 1990, p. 513-517
  4. Finn, J. M.; Kaw, P. K., "Coalescence instability of magnetic islands" (1977) Physics of Fluids, vol. 20, Jan. 1977, p. 72-78. (More citations)
  5. Uhm, H. S.; Siambis, J. G., "Diocotron instability of a relativistic hollow electron beam" (1979) Physics of Fluids, vol. 22, Dec. 1979, p. 2377-2381.

External links

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