The formation of stars and planets Day 1, Topic 3: Hydrodynamics and Magneto-hydrodynamics 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 H 2 /He mixture: For H 2 (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: Continuity equation 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 waves No 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<c s, 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>c s, 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: isothermal Galilei 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 freezing Strong 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 Coronal loops on the 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:

Ideal MHD: equations Momentum equation magneto-hydrodynamics: Magnetic pressure Magnetic tension Tension in curved field: force

Non-ideal MHD: reconnection Opposite field bundles close together: Localized reconnection of field lines: Acceleration of matter, dissipation by shocks etc. Magnetic energy is thus transformed into heat

Appendix: Tools for numerics

Numerical integration of ODE An ordinary differential equation: Numerical form (zeroth order accurate, usually no good): Higher order algorithms (e.g. Runge-Kutta: very reliable): Implicit first order (fine for most of our purposes):

Numerical integration of ODE Implicit integration for linear equations: algebraic Implicit integration of non-linear equations: can require sophisticated algorithm in pathological cases. For this lecture the examples are benign, and a simple recipe works: Simple recipe: First take y i+1 = y i. Do a step, find y i+1. Now redo step with this new y i+1 to find another new y i+1. Repeat until convergence (typically less than 5 steps). Implicit integration: we don’t know y i+1 in advance...