“small-scale dynamo: numerics vs. analytics“€¦ · “small-scale dynamo” 1. basics 1.1...

“Small-Scale Dynamo: Numerics vs. Analytics“

Jennifer Schober (Institut für Theoretische Astrophysik, Heidelberg)

Collaborators: Dominik Schleicher, Christoph Federrath, Ralf Klessen, Robi Banerjee, Simon Glover

Flash-Workshop

Content

1. Basics

1.1 Magnetic fields in the Universe 1.2 Small-Scale Dynamo: Stretch-Twist-Fold Model

2. Numerical Implementation in Flash

2.1 Critical Resolution 2.2 Growth Rate: Dependence on Forcing 2.3 Growth Rate: Dependence on Mach number

3. Comparison to Analytical Results

3.1 Kazantsev Theory 3.2 Critical Reynolds Number 3.3 Growth Rate

4. Conclusion

Jennifer Schober (ITA), February 2012 2/21

“Small-Scale Dynamo”

Magnetic Fields in the Universe“Small-Scale Dynamo”

Hubble observations of HH 30

[credit: http://www.wolaver.org/Space/HH30stellarjet.htm]

M87[credit: NASA and John Biretta (STScI/JHU)]

Sun[credit: "SOHO (ESA & NASA)"]

1. Basics

1.1 B-fields in Universe

1.2 Stretch- Twist-Fold

2. Numerics

2.1 Crit. Resolution

2.2 Growth: Forcing

2.3 Growth: Mach number

3. Analytics

3.1 Kazantsev Theory

3.2 Crit. Rm

3.3 Growth

4. ConclusionJennifer Schober (ITA), February 2012 3/21

http://www.wolaver.org/

Theory Observations/Upper Limits

Inflation: CMB:

QCD phase transition: reionisation:

Milky W ay:

[e.g. Turner&Widrow 1988] [e.g. Y amaz aki et al. 2006]

[e.g. Sigl et al. 1997] [e.g. Sc hleicher&Miniati 2011, in prep.]

Biermann-battery:[e.g. Xu et al. 2008] [e.g. Han. 2008]

How strong are the B-Fields?“The Small-Scale

Dynamo”

B≤4.7nG

B≤2−3nG

B≤3−4⋅103nG

B≈10−50−1nG

B≈10−3−1nG

B≈10−15 nG

=> Need mechanism that amplifies seed fields.

1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Motivation

predicted seed B-fields << observed B-fields

=> need amplification processes small-scale dynamo

- amplifies magnetic seed field exponentially

- uses kinetic energy

from turbulence

M16[credit: NASA, ESA, STScI, J. Hester and P.

Scowen (Arizona State University)]


1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Stretch-Twist-Fold Model

Source: Brandenburg&Subramanian 2005: “Astrophysical Magnetic Fields and Nonlinear Dynamo Theory“


1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Numerical Results for the Small-Scale Dynamo with FLASH


1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Turbulent Dynamo in FLASH

MHD in FLASH (v2.5/v4): - solution of the (non-ideal) MDH-equations - explicit viscosity and magnetic resistivity (magnetic Prandtl number )

sub- and supersonic dynamics: - 3rd order spatial reconstruction - HLL3R/HLLD Riemann solver

turbulent forcing: - forcing with stochastic Ornstein-Uhlenbeck process in momentum equation - forcing on the largest length scales => turbulence develops self-consistently on smaller scales - decomposition in solenoidal and compressive forcing possible


1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Pm=/≈1

Resolution Criterion

Federrath et al., ApJ, 2011

=> small-scale dynamo works for resolution > 32 cells per Jeans length


1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Different Forcing/Turbulence

Effect of different types of forcing on the structure of the field lines:

=> solenoidal forcing twists field lines more effectively

Federrath et al., PRL, 2011


1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Different Forcing/Turbulence

Federrath et al., PRL, 2011

growth rate growth rate (solenodial forcing) (compressive forcing)


1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


=> >

Dependence on Mach Number

Federrath et al.,PRL, 2011

Emt ∝e⋅t


1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


( : growth rate)

Comparison to Analytical Results


1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth

4. Conclusion Jennifer Schober (ITA), February 2012 13/21

MHD-Dynamos

idea: divide magnetic field into mean and turbulent component

put into induction equation:

=> evolution equations for mean and turbulent field (large-scale dynamo and small-scale dynamo)

B= B0 B

∂ B∂ t

=∇×v×B∇2 B


1. Basics



2. Numerics


2.2 Growth: Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth

4. Conclusion Jennifer Schober (ITA), February 2012 14/21

Kazantsev Theory

“Kazantsev Theory” [Kazantsev, 1968]: theory of the small-scale dynamo

correlation function of magnetic fluctuation:

with :

⟨ Bi x , t B j y , t ⟩=M ij r , t

M ij=ij− ri r j

r 2M N

r i r j

r2M L

∇⋅B=0

M N=12r

∂∂ r

r2M L


1. Basics



2. Numerics


2.2 Growth:

Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


−T r ∂2r ∂2 r

U r r =−r

put magnetic correlation function into induction equation => Kazantsev equation:

“mass”

“potential”

can be solved with WKB-approximation for large magnetic Prandtl numbers ( )

U r =U ,

/

Kazantsev Theory

M L r , t ∝r e2 t

T r =T ,

“Magnetic Energy Density”

“turbulence”

“turbulence”


1. Basics



2. Numerics


2.2 Growth:

Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Critical Mag. Reynolds Number

Reynolds number for minimal growth rate: set in Kazantsev equation and solve for ( ), [Schober et al 2012]

result (Kolmogorov turbulence):

result (Burgers turbulence):

numerical diffusivity => depends on resolution in grid code

=> higher critical Rm/resolution for higher compressibility

=0Rm Rm=VL /

Rm110

Rm2700Rm=Rm


1. Basics



2. Numerics


2.2 Growth:

Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


growth rate for large magnetic Prandtl numbers [Schober et al. 2012]:

(with slope of the turbulent velocity spectrum )

example 1: Kolmogorov turbulence ( )

example 2: Burgers turbulence ( )

=> faster growth at lower compressibility

Growth Rate

∝Re1−/1

∝Re1 /2

∝Re1/3

v l ∝l

=1 /3

=1 /2


1. Basics



2. Numerics


2.2 Growth:

Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Collapse of a Primordial Halo

Schober et al., in prep.


1. Basics



2. Numerics


2.2 Growth:

Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Numerics

Critical Resolution: 32 Cells/Jeans Length

Growth Rate: Growth Rate:- larger for lower com pressibility - larger for less com pressible types - dynam o action in sub- of turbulence and supersonic regim e - no dependence on Mach num ber- increases with increasing seen so far Mach num ber (extension to sm aller Pm ?)

Analytics [Schober et al. 2011]

Critical Mag. Reynolds Number:

[Federrath et al., A pJ, 2011] (Kolmogorov turb.) (Burgers turb.)

[Federrath et al., PRL, 2011]

Conclusion

Rm110 Rm2700


1. Basics



2. Numerics


2.2 Growth:

Forcing


3. Analytics


3.2 Crit. Rm

3.3 Growth


Thanks for your attention!

:-)

“small-scale dynamo: numerics vs. analytics“€¦ · “small-scale dynamo” 1. basics 1.1...

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