Introduction to

Fluid Instabilities

K. Yavuz Ekşi

What is Fluid Stability?

• A steady state solution of the hydrodynamic equations corresponding to a fluid configuration may not exist or may not prevail if it ever existed.

• There are always small perturbations in any system.

• If such small perturbations grow in time then the steady state solution is unstable and can not serve as a model for some realistic situation.

How are fluid instabilities relevant to astrophysical phenomena?

• What are the solar granules?• How does stars form out of gas clouds?• How does galaxies form in an expanding

universe?• Why are there spiral arms of galaxies?• Turbulence in accretion disks: How is the angular

momentum transported outwards?• Why do Dwarf Novae burst?

• The Great Red Spot of Jupiter...• The clumpiness of matter in SNRs

Stability

• Linearize the equations around the equilibrium• Check whether small perturbations grow.

linearly stable

linearly unstable

linearly stablenon-linearly unstable

Linearly unstableNon-linearly stable

Gravitational Instability

• Jeans instability is the basic reason why the matter in the Universe is not spread uniformly

• Stars and galaxies are believed to be the end products of perturbations which initially started growing due to the Jeans instability

Gravitational Instability: Jeans Criterion

• When a region of a gas is compressed, the excess pressure there tries to smoothen out the compression: acoustic waves.

• The compressed region has enhanced gravitation more gas is pulled into the compressed region.

• For typical sound waves in the air, the enhancement of gravity in the regions of compression is utterly negligible.

• For perturbations of gaseous bodies of astronomical size, the enhanced gravitation in the region of compression may overpower the expansive tendency of the excess pressure, more material pulled there, an instability triggered.

Linear Stability Analysis for GI

Equilibrium Equations

Dynamical equations

Perturb the Equilibrium

-Insert into the dynamical equations .Neglect the 2nd order terms.Subtract the equilibrium equations

Assume that all the perturbations are of the form

Using these in the equations above

where

Defining

Jeans mass

If k<kj then ! is imaginary and the perturbation growsexponentially…

And the enhanced gravitation in the region of compression may overpower the expansive tendency of the excess pressure

k<kj is equivalent to M>Mj

Assuming the interstellar matter to have 1 hydrogen atom per cm3 at temperature 100K, we obtain a Jeans mass of about 8×1038 g.

This is several orders of magnitude larger than the typical mass of a star (about 1033 g)

Presumably the interstellar matter first breaks into large chunks with masses corresponding to clusters of stars rather than individual stars.

Then somehow these contracting chunks of gas have to break further to produce stars.

The presence of angular momentum or magnetic field complicates the process.

Rayleigh-Taylor Instability (RTI)

• also called Richtmyer-Meshkov instability,

• shows the competition between surface tension and gravity.

• Occurs anytime a dense fluid is accelerated by a light fluid e.g. a heavy fluid over a light fluid.

Visualisations of RTI

http://www.itsc.com/movies/raytay.mpg

2D fingers in Rayleigh-Taylor instability

When the RTI starts, the boundary between the fingers

and the surrounding medium is subject to KHI

3D fingers in Rayleigh-Taylor instability

Crab NebulaThe heavy fluid on top of light fluid, by the principle of equivalence, is same as heavy fluid accelerated by a light fluid.

As the interface is decelerated, we can represent it by an outward gravitational field in the rest frame of the interface. Then we have the dense shell of ejected gas lying on top of the less dense gas outside. It was found by Gull (1975) that this gives rise to the Rayleigh-Taylor instability and gives rise to the clumpy appearance of material in Crab.

Simulations…

Fryxell, Müller & Arnett (1991) ApJ, 367, 619

Kelvin-Helmholtz Instability (KHI)

Kelvin-Helmholtz (inviscid) instablility:

1 1,U

2 2,U z

x

22 2 2 2 21 2 1 2 1 2x x yk U U g k k

linear stability analysis+normal mode disturbance:

KHI…

http://www.maths.man.ac.uk/~mheil/MATTHIAS/Fluid-Animations/kelvin.mpg

http://www.itsc.com/movies/kelvin.mpg

KHI in a jethttp://www.maths.man.ac.uk/~mheil/MATTHIAS/Fluid-Animations/bouyjet.mpg

http://www.itsc.com/movies/bouyjet.mpg

Kelvin Helmholtz instablility

A long rectangular tube initially horizontal is filled with water above colored brine The fluids are allowed to diffuse for about an hour, and the tube then quickly tilted six degrees, setting the fluids into motion. The brine accelerates uniformly down the slope, while the water above similarly accelerates up the slope. Sinusoidal instability of the interface occurs after a few seconds, and has here grown nonlinearly into regular spiral rolls.

Kelvin-Helmholtz (inviscid) instablility:

22 2 2 2 21 2 1 2 1 2x x yk U U g k k

~ instability due to heavy fluid on the upside

~ instability due to shear

~ instability due to an rapid downward vertical acceleration

and heavy fluid rests below

~ instability for all cases

1 1,U

2 2,U

Wall Shear Flows

~ inviscidly unconditionally stable (Rayleigh analysis)

~ viscously unstable (Orr-Sommerfeld analysis) Re 5772

UD

~ unstable in labs as Re > 2000

Kelvin-Helmholtz as seen in a cloud formation

From the National Center for Atmospheric Research.

http://venus.eng.deu.edu.tr/isilab/kelvin_helm_rollup.htm

Hiroshige Utagawa "Vortices in the Konaruto stream"

In rivers, these structures can appear when there is a sudden widening of the river bed.

Vincent Van Gogh "La Nuit Etoilee=Starry Night" 

V838 Monocerotis

Expanding halo of light around a distant star.

The illumination of interstellar dust comes from the red supergiant star at the middle of the image, which gave off a flashbulb-like pulse of light two years before the image was taken.

A KHI on Saturn, formed at the interaction of two bands of the planet's atmosphere

Image taken from the Cassini probe of NASA

Transport of solar wind into Earth's magnetosphere through rolled-up Kelvin−Helmholtz vortices

• Nature 430, 755 (12 August 2004) Hasegawa et al.

• Establishing the mechanisms by which the solar wind enters Earth's magnetosphere is one of the biggest goals of magnetospheric physics, as it forms the basis of space weather phenomena such as magnetic storms and aurorae.

• It is generally believed that magnetic reconnection is the dominant process

• However the plasma content in the outer magnetosphere increases during northward solar-wind magnetic field conditions, contrary to expectation if reconnection is dominant

• Here the authors show that during northward solar-wind magnetic field conditions—in the absence of active reconnection at low latitudes—there is a solar-wind transport mechanism associated with the nonlinear phase of the Kelvin−Helmholtz instability

Convective Instability

Sometimes we have to deal with fluids heated from below.Such flows have cold gases overlying hot. e.g. Atmosphere of the Earth.

When does such an adverse temperature gradient become unstable WRT the tendency to develope overturning motions?

Schwarzschild Criterion (1906) forConvective instability:

ds>0 in the direction of gravity

Rayleigh-Bénard convection

• Convection cells in a fluid is heated from below.

Transition to convection of an initially quiescent layer of fluid that has a vertically unstable temperature gradient.

http://www.itsc.com/movies/benard1.mpg

• Granules on the solar surface are the tops of convection cells.

Thermal Instability

• Dwarf Novae: Opacity increases with temperature => disk can not cool=>becomes even hotter

Rotational Instability

• Inviscid case: Rayleigh Criterion (1917)

• Viscous case: Taylor Criterion (1923)

• Magnetic case: Velikhov (1959)/Chandrasekhar(1961)

Couette Flow: a fluid confined between two cylinders which rotate at different rates. Results apply to accretion disks.

Rotational Instability-Rayleigh criterion

• Accretion disks (=K=(GM/r3)1/2) can not be rotatioanally unstable according to this criteria. So how can they be turbulent have anomalous viscosity?

Rayleigh Criterion for Instability

Specific angular momentum, j = r2, must decrease outwards.

Angular momentum has to be transferred outward to let the central star accrete the matter of the disk, else the matter can’t approach the star.

Turbulence

• A collective phenomenon• Unstable flows often evolve into a state

of motion called TURBULENCE, with a chaotic 3-D vorticity field with broad spectrum of small temporal and spatial scales.

Leonardo da Vinci (1452-1519) wrote “Observe the motion of the surface of the water, which resembles that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the curls. Thus the water has eddying motions, one part of which is due to the principal current, the other to random and reverse motion.”

“... the smallest eddies are almost numberless, and largethings are rotated only by large eddies and not by small ones,and small things are turned by small eddies and large.”

Reynolds Experiment

• The configuration of Reynolds experiment on flow along a pipe.

• Experiment made with 3 tubes of different diameters.

• Streakes of colored water entered the tube with clear water.

From Reynolds 1883, fig. 13

General results of the Reynolds Experiment

• (a)When the velocities were sufficiently low, the streak of color extended in a beautiful straight line through the tube.

• (b)As the velocity was increased by small stages, at some point in the tube, the color band would all at once mix up with the surrounding water, and fill the rest of the tube with a mass of colored water. (Any increase in the velocity caused the point of break down to approach the trumpet.

• (c) On viewing the tube by the light of an electric spark, the mass of color resolved itself into a mass of more or less distinct curls, showing eddies.

From Reynolds 1883, Figs 3,4 & 5

Intermittency

• Just above the critical Reynolds number, turbulence reveals its intermittent character:

• The disturbance would suddenly come on through a certain length of the tube and pass away and then come on again, giving the appereance of flashes and these flashes would often comence successively at one point in pipe.

• Such flashes are now called “turbulent spots” or “turbulent bursts.

From Reynolds 1883, Fig. 16.

Definition of Turbulence-1

T. von Karman qoutes G. I. Taylor with the following definition of turbulence:

“Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they past

solid surfaces or even when neighboring streams of the same fluid flow past or over one another.”

Definition of Turbulence-2

• J.O. Hinze,(in his book onTurbulence), offers yet another definition:

Turbulent fluid motion is an irregular condition of the flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be discerned.”

A Modern Definition of Turbulence

• Turbulence is any chaotic solution to the 3-D Navier-Stokes equations that is sensitive to initial data and which occurs as a result of successive instabilities of laminar flows as a bifurcation parameter is increased through a succession of values.

Universally Accepted Features of Turbulence

• Disorganized, chaotic, seemingly random behavior;• No repeatability (i.e., sensitivity to initial conditions);• Extremely large range of length and time scales (but

such that the smallest scales are still sufficiently large to satisfy the continuum hypothesis);

• Enhanced diffusion (mixing) and dissipation (both of which are mediated by viscosity at molecular scales);

• Three dimensionality, time dependence and rotationality (hence, potential flow cannot be turbulent because it is by definition irrotational);

• Intermittency in both space and time.

MHD Turbulence• When the fluid is conducting turbulent motions are accompanied by

B-field fluctuations.• Alfvén effect: small scale fluctuations are weakly interacting Alfvén

waves propagating along the large scale field (Iroshnikov 1964, Kraichnan 1965).

• Energy spectrum predicted to be flatter (k-3/2) than the Kolmogorov spectrum (k-5/3).

• BUT the energy spectrum of the Solar wind is closer to the Kolmogorov spectrum

• Solution: MHD turbulence is unisotropic (Goldreich & Sridhar 1995)-spectrum is more strongly developed perpendicular to the local B-field where the Alfvén effect is not operative. This explains why Kolmogorov-like dynamics is effective.

• Self-organization processes in MHD turbulence that have no hydrodynamic counterpart.

Magnetorotational Instability

• In the presence of a weak magnetic field the criterion for instability changes profoundly and Keplerian disks are unstable WRT to the new criteria.

The Great Red Spot ofJupiter

…has been observed for 300 years although the expected life time is about 1 day

How could such a large eddy manage to persist over the centuries?

Hook, R., “A spot in one of the belts of Jupiter” Phil. Trans. 1, 3 (1665)

• The turbulence in Jupiter's atmosphere does not behave similar to homogenious, isotropic turbulence in which vortices randomly merge, cascading their energies into progressively smaller scales.

.GRS swallows only eddies of its own sign and repells the opposites

.This strange behavior depends critically on the dynamics in Jupiters atmosphere being 2D in character.The GRS is a nearly 2D vortex not attached to any topographic feature

Marcus, P. S., Jupiter's Great Red Spot and other vortices, Annu. Rev. Astron. Astrophys. 31, 523-573 (1993)

References

links

• http://www.galleryoffluidmechanics.com/

• http://www.navier-stokes.net/

• http://www.fluidmech.net/• http://www.enseeiht.fr/hmf/travaux/CD0001/travaux/optmfn/hi/01pa/hyb72/index.htm

• http://perso.wanadoo.fr/laurent.nack/

• http://www.itsc.com/movies/index.htm• http://woodall.ncsa.uiuc.edu/dbock/projects/3drt/ • http://astron.berkeley.edu/~jrg/ay202/lectures.html

Other References

• “Waves in Fluids”, Lighthill (1978)

• “Hydrodynamic stability” Drazin and Reid (1981)