Richtmyer–Meshkov instability

The Richtmyer–Meshkov instability (RMI) occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time. This is followed by a nonlinear regime with bubbles appearing in the case of a light fluid penetrating a heavy fluid, and with spikes appearing in the case of a heavy fluid penetrating a light fluid. A chaotic regime eventually is reached and the two fluids mix. This instability can be considered the impulsive-acceleration limit of the Rayleigh–Taylor instability.^[1]

Dispersion Relation

For ideal MHD

${\displaystyle (\omega ^{2}-2k_{\parallel }^{2}/\beta )(\omega ^{4}-(2/\beta +1)k^{2}\omega ^{2}+2k_{\parallel }^{2}k^{2}/\beta )=0}$

For Hall MHD

${\displaystyle {\displaystyle (\omega ^{2}-2k_{\parallel }^{2}/\beta )(\omega ^{4}-(2/\beta +1)k^{2}\omega ^{2}+2k_{\parallel }^{2}k^{2}/\beta )-2d_{s}^{2}k_{\parallel }^{2}k^{2}\omega ^{2}(\omega ^{2}-k^{2})/\beta =0}}$

For QMHD

${\displaystyle {\displaystyle {\displaystyle ((1+2/\beta c^{2})\omega ^{2}-2k_{\parallel }^{2}/\beta )((1+2/\beta c^{2})\omega ^{4}-(2/\beta +1)k^{2}\omega ^{2}+2k_{\parallel }^{2}k^{2}/\beta )-2d_{s}^{2}k_{\parallel }^{2}k^{2}\omega ^{2}(\omega ^{2}-k^{2})/\beta =0}}}$

History

R. D. Richtmyer provided a theoretical prediction,^[2] and E. E. Meshkov (Евгений Евграфович Мешков)^(ru) provided experimental verification.^[3] Materials in the cores of stars, like Cobalt-56 from Supernova 1987A were observed earlier than expected. This was evidence of mixing due to Richtmyer–Meshkov and Rayleigh–Taylor instabilities. ^[4]

Examples

During the implosion of an inertial confinement fusion target, the hot shell material surrounding the cold D–T fuel layer is shock-accelerated. This instability is also seen in magnetized target fusion (MTF).^[5] Mixing of the shell material and fuel is not desired and efforts are made to minimize any tiny imperfections or irregularities which will be magnified by RMI.

Supersonic combustion in a scramjet may benefit from RMI as the fuel-oxidants interface is enhanced by the breakup of the fuel into finer droplets. Also in studies of deflagration to detonation transition (DDT) processes show that RMI-induced flame acceleration can result in detonation.

See also

• Rayleigh–Taylor instability
• Mushroom cloud
• Plateau–Rayleigh instability
• Salt fingering
• Kármán vortex street
• Kelvin–Helmholtz instability
• Hydrodynamics