# Instabilities

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## instability

A property of the steady state of a system such that certain disturbances or perturbations introduced into the steady state will increase in magnitude, the maximum perturbation amplitude always remaining larger than the initial amplitude.

The method of small perturbations, assuming permanent waves, is the usual method of testing for instability; unstable perturbations then usually increase exponentially with time. An unstable nonlinear system may or may not approach another steady state; the method of small perturbations is incapable of making this prediction. The small perturbation may be a wave or a parcel displacement. The parcel method assumes that the environment is unaffected by the displacement of the parcel. The slice method has occasionally been used as a modification of the parcel method to gain a little information about the interaction of parcel and environment. Stability as defined above is an asymptotic concept; other definitions are possible. Precision is required of the user, and caution of the reader. The concept of instability is employed in many sciences. In meteorology the reference is usually to one of the following.

- Static instability (or hydrostatic instability) of vertical displacements of a parcel in a fluid in hydrostatic equilibrium. (
*See*conditional instability, absolute instability, convective instability, buoyant instability.) - Hydrodynamic instability (or dynamic instability) of parcel displacements or, more usually, of waves in a moving fluid system governed by the fundamental equations of hydrodynamics, to which the quasi-hydrostatic approximation may or may not apply. (
*See*Helmholtz instability, inertial instability, shearing instability, baroclinic instability, barotropic instability, rotational instability.)

The space scale of unstable waves is important in meteorology: Thus Helmholtz, baroclinic, and barotropic instability give, in general, unstable waves of increasing wavelength. The timescale is also important: A perturbation that grows for two days before dying out is effectively unstable for many meteorological purposes, but this is an initial-value problem and one cannot assume the existence of permanent waves. These meteorological types of hydrodynamic instability must not be confused with the phenomenon often referred to by mathematicians and physicists by the same term. A great deal of study has been devoted to the problem of the onset of turbulence in simple flows under laboratory conditions, and here viscosity is a source of instability.*See*computational instability.

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