Lecture “Planet Formation”

Topic:

Introduction to

hydrodynamics and

magnetohydrodynamics

Lecture by: C.P. Dullemond

Equations of hydrodynamics

Hydrodynamics can be formulated as a set of conservation equations + an equation of state (EOS). Equation of state relates pressure P to density and (possibly) temperature T

In astrophysics: ideal gas (except inside stars/planets):

Sometimes assume adiabatic flow:

For typical H2/Hemixture:

For H2 (molecular): =7/5

For H (atomic): =5/3

Sometimes assume given T (this is what we will do in this lecture, because often T is fixed to external temperature)

Equations of hydrodynamics

Conservation of mass:

Conservation of momentum:

Energy conservation equation need not be solved if T is given (as we will mostly assume).

Equations of hydrodynamics

Comoving frame formulation of momentum equation:

Continuityequation

So, the change of v along the fluid motion is:

Equations of hydrodynamics

Momentum equation with (given) gravitational potential:

So, the complete set of hydrodynamics equations (with given temperature) is:

Isothermal sound wavesNo gravity, homonegeous background density (0=const).Use linear perturbation theory to see what waves are possible

So the continuity and momentum equation become:

Supersonic flows and shocks

If a parcel of gas moves with v<cs, then any obstacle ahead receives a signal (sound waves) and the gas in between the parcel and the obstacle can compress and slow down the parcel before it hits the obstacle.

If a parcel of gas moves with v>cs, then sound signals do not move ahead of parcel. No ‘warning’ before impact on obstacle. Gas is halted instantly in a shock-front and the energy is dissipated.

Chain collision on highway: visual signal too slow to warn upcoming traffic.

Shock example: isothermalGalilei transformation to frame of shock front.

Momentum conservation:

Continuity equation: (1)

(2)

Combining (1) and (2), eliminating i and o yields:

Incoming flow is supersonic: outgoing flow is subsonic:

Viscous flows

Most gas flows in astrophysics are inviscid. But often an anomalous viscosity plays a role. Viscosity requires an extra term in the momentum equation

The tensor t is the viscous stress tensor:

shear stress (the second viscosity is rarely important in astrophysics)

Navier-Stokes Equation

Magnetohydrodynamics (MHD)

• Like hydrodynamics, but with Lorentz-force added• Mostly we have conditions of “Ideal MHD”: infinite

conductivity (no resistance):– Magnetic flux freezing– No dissipation of electro-magnetic energy– Currents are present, but no charge densities

• Sometimes non-ideal MHD conditions:– Ions and neutrals slip past each other (ambipolar diffusion)– Reconnection (localized events)– Turbulence induced reconnection

Ideal MHD: flux freezing

Galilei transformation to comoving frame (’)

( infinite, but j finite)

Galilei transformation back:

Suppose B-field is static (E-field is 0 because no charges):

Gas moves along the B-field

Ideal MHD: flux freezing

More general case: moving B-field lines.

A moving B-field is (by definition) accompanied by a E-field. To see this, let’s start from a static pure magnetic B-field (i.e. without E-field). Now move the whole system with some velocity u (which is not necessarily v):

On previous page, we derived that in the comoving frame of the fluid (i.e. velocity v), there is no E-field, and hence:

(Flux-freezing)

Ideal MHD: flux freezingStrong field: matter can only move along given field lines (beads on a string):

Weak field: field lines are forced to move along with the gas:

Ideal MHD: flux freezing

Coronalloops onthe sun

Ideal MHD: flux freezing

Mathematical formulation of flux-freezing: the equation of‘motion’ for the B-field:

Exercise: show that this ‘moves’ the field lines using the example of a constant v and gradient in B (use e.g. right-hand rule).

Ideal MHD: equations

Lorentz force:

Ampère’s law: (in comoving frame)

(Infinite conductivity: i.e. no displacement current in comoving frame)

Momentum equation magneto-hydrodynamics:

Momentum equation magneto-hydrodynamics:

Ideal MHD: equations

Momentum equation magneto-hydrodynamics:

Magnetic pressure

Magnetic tension

Tension in curved field:

force

Non-ideal MHD: reconnectionOpposite field bundles close together:

Localized reconnection of field lines:

Acceleration of matter, dissipation by shocks etc.Magnetic energy is thus transformed into heat