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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