# Kelvin–Helmholtz instability

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Numerical simulation of a temporal Kelvin–Helmholtz instability

The Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) can occur when there is velocity shear in a single continuous fluid, or where there is a velocity difference across the interface between two fluids. An example is wind blowing over water: The instability manifests in waves on the water surface. More generally, clouds, the ocean, Saturn's bands, Jupiter's Red Spot, and the sun's corona show this instability.

## Overview A KH instability on the planet Saturn, formed at the interaction of two bands of the planet's atmosphere

The theory predicts the onset of instability and transition to turbulent flow in fluids of different densities moving at various speeds. Helmholtz studied the dynamics of two fluids of different densities when a small disturbance, such as a wave, was introduced at the boundary connecting the fluids.

For some short enough wavelengths, if surface tension is ignored, two fluids in parallel motion with different velocities and densities yield an interface that is unstable for all speeds. Surface tension stabilises the short wavelength instability however, and theory predicts stability until a velocity threshold is reached. The theory, with surface tension included, broadly predicts the onset of wave formation in the important case of wind over water.[citation needed]

It was recently discovered that the fluid equations governing the linear dynamics of the system admit a parity-time symmetry, and the Kelvin-Helmholtz instability occurs when and only when the parity-time symmetry breaks spontaneously.

For a continuously varying distribution of density and velocity (with the lighter layers uppermost, so that the fluid is RT-stable), the dynamics of the KH instability is described by the Taylor–Goldstein equation and its onset is given by the Richardson number $Ri$ . Typically the layer is unstable for $Ri<0.25$ . These effects are common in cloud layers. The study of this instability is applicable in plasma physics, for example in inertial confinement fusion and the plasmaberyllium interface.

Numerically, the KH instability is simulated in a temporal or a spatial approach. In the temporal approach, experimenters consider the flow in a periodic (cyclic) box "moving" at mean speed (absolute instability). In the spatial approach, experimenters simulate a lab experiment with natural inlet and outlet conditions (convective instability).