The formation of stars and planets

Day 1, Topic 3:

Hydrodynamicsand

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):

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P = ρkT

μ mp

Sometimes assume adiabatic flow:

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T ∝ ρ γ −1

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P ∝ ρ γ €

μ =2.3

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:

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∂∂t

+∇ ⋅ ρ v( ) = 0

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∂ v( )∂t

+∇ ⋅ ρvv+ P( ) = 0

Conservation of momentum:

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

Equations of hydrodynamics

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0 =∂ ρv( )∂t

+∇ ⋅ ρvv+ P( )

Comoving frame formulation of momentum equation:

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=v∂ρ

∂t+ ρ∂v

∂t+ v∇ ⋅ ρv( ) + ρv ⋅∇v+∇P

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=v∂ρ

∂t+∇ ⋅ ρv( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥+ ρ

∂v

∂t+ v ⋅∇v

⎡ ⎣ ⎢

⎤ ⎦ ⎥+∇P

Continuityequation

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= ∂v∂t

+ v ⋅∇v+1

ρ∇P

⎡

⎣ ⎢

⎤

⎦ ⎥

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≡∂v∂t

+ v ⋅∇v

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Dv

Dt

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

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=−1

ρ∇P

Equations of hydrodynamics

Momentum equation with (given) gravitational potential:

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∂v∂t

+ v ⋅∇v = −1

ρ∇P −∇Φ

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

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∂v∂t

+ v ⋅∇v = −1

ρ∇P −∇Φ

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∂∂t

+∇ ⋅ ρ v( ) = 0

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P = ρkT

μmp

≡ ρ cs2

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

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=0 + ρ1

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v = v1

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∂v1

∂t= −

1

ρ 0

∇ ρ1cs2

( ) = −cs

2

ρ 0

∇ρ1

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∂1

∂t+∇ ⋅ ρ 0v1( ) = 0

So the continuity and momentum equation become:

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∂2ρ1

∂t 2+ ρ 0∇ ⋅

∂v1

∂t= 0

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∂2ρ1

∂t 2− cs

2∇ 2ρ1 = 0

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.

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i

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v i

€

o

€

vo

Momentum conservation:

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iv i2 + Pi = ρ ovo

2 + Po

Continuity equation:

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iv i = ρ ovo (1)

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i v i2 + cs

2( ) = ρ o vo

2 + cs2

( ) (2)

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

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v ivo = cs2 Incoming flow is supersonic:

outgoing flow is subsonic:

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v i> cs

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vo< cs

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

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∂v∂t

+ v ⋅∇v = −1

ρ∇P +

1

ρ∇ ⋅t

The tensor t is the viscous stress tensor:

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tik = ρν∂vi∂xk

+∂vk∂x i

− 23δ ik

∂vl∂x l

⎛

⎝ ⎜

⎞

⎠ ⎟+ ζ δ ik

∂vl∂x l

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 (’)

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j'=σ E'

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E'= 0( infinite, but j finite)

Galilei transformation back:

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j= j'

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E =E'−1

cv×B

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=−1

cv×B

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

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v×B = 0 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):

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Emove =Estatic −1

cu×B = −

1

cu×B

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

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−1

cv×B ≡E = Emove ≡ −

1

cu×B

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(v−u) ×B = 0 (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:

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B2

8π<< Pgas + ρ v

2

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B2

8π>> Pgas + ρ v

2

Ideal MHD: flux freezing

Coronalloops onthe sun

Ideal MHD: flux freezing

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∂B∂t

=∇ × (v×B)

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:

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fL ≡1

cj×B

Ampère’s law: (in comoving frame)

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∇×B =4π

cj+

1

c

∂E

∂t

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=1

4π∇ ×B( ) ×B ≡

1

4πB ⋅∇( )B−

1

8π∇B

2

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

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j=c

4π∇ ×B

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∂v∂t

+ ρv ⋅∇v = −∇P − ρ∇Φg +1

4πB ⋅∇( )B−

1

8π∇B

2

Momentum equation magneto-hydrodynamics:

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∂v∂t

+ ρv ⋅∇v = −∇P − ρ∇Φg + fL

Momentum equation magneto-hydrodynamics:

Ideal MHD: equations

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∂v∂t

+ ρv ⋅∇v = −∇P − ρ∇Φg +1

4πB ⋅∇( )B−

1

8π∇B

2

Momentum equation magneto-hydrodynamics:

Magnetic pressure

Magnetic tension

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PM =1

8πB

2

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

Appendix: Tools for numerics

Numerical integration of ODE

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dy(x)

dx= F(y,x)

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y i+1 − y ix i+1 − x i

= F(y i,x i)

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y i+1 = Ψ[y i−n,K ,y i;x i−n,K ,x i]

An ordinary differential equation:

Numerical form (zeroth order accurate, usually no good):

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y i+1 = y i + F(y i,x i) (x i+1 − x i)

Higher order algorithms (e.g. Runge-Kutta: very reliable):

Implicit first order (fine for most of our purposes):

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y i+1 − y ix i+1 − x i

= F y i+1/ 2,x i+1/ 2( )

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x i+1/ 2 ≡ 12 (x i + x i+1)

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y i+1/ 2 ≡ y(x i+1/ 2)

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 yi+1 = yi . Do a step, find yi+1. Now redo step with this new yi+1 to find another new yi+1. Repeat until convergence (typically less than 5 steps).

Implicit integration: we don’t know yi+1 in advance...