SPH +/- MHD

The state of the art in the dark arts

Daniel Price

Monash University

Melbourne, Australia

Gingold & Monaghan (1977):

Magnetic polytropes with SPH

Phillips & Monaghan (1985): SPH with MHD in

conservative form is unstable when beta < 1

Morris (1996): Stability analysis: a compromise

solution to stabilise the force equation with

“almost-conservative” form

Dolag, Bartelmann & Lesch (1999): SPH+MHD

applied to galaxy clusters (beta >> 1)

Børve, Omang & Trulsen (2001): Regularised SPH:

very nice MHD shocks by “remeshing” of particles

Screw conservative form: Gradient Particle

Magnetohydrodynamics? (Maron & Howes 2003)

Hosking & Whitworth (2004): use non-conservative

formulation but first to implement two fluid (ion/

neutral)

The “Middle Years”

Price & Monaghan 2004a (paper I): dissipative terms for MHD shocks

Price & Monaghan 2004b (paper II): we could do strong shocks

(variable smoothing length formulation)

Price & Monaghan 2005 (paper III): How to handle

the divergence constraint (using IGNORE or

CLEAN approach)

good results on

test problems...

advection of a current loop

(Gardiner & Stone 2006,

Rosswog & Price 2007)

Orszag-Tang vortex problem (PM05,

Rosswog & Price 2007)

Magnetic rotor problem (PM05)

∂B∂t

= ∇× (v ×B) ∇ · B = 0

...but didn’t work so

well for star formation

Price & Bate (2007), Rosswog & Price (2007):

Enter the Euler potentials

B = ∇α×∇β

∂B∂t

= ∇× (v ×B)

dα

dt= 0

dβ

dt= 0

(satisfies by construction: the PREVENT approach)∇ · B = 0

(advection of

magnetic field lines

by Lagrangian

particles)

(cf. Phillips & Monaghan 1985)

Price & Bate (2007): Effect of magnetic fields on single and binary star formation

(all with supercritical field strengths)

...problem forming discs in the presence of magnetic fields?see also Hennebelle & Fromang (2008), Hennebelle & Ciardi (2009), Mellon & Li (2009), Duffin & Pudritz (2009)

Price & Bate (2007): Effect on binary formation

... a fragmentation crisis?cf. also Hennebelle & Teyssier (2009), Mellon & Li (2008), Machida et al. (2008) and others

Price & Bate (2008): Effect of magnetic fields on star cluster formation

Price (2008): We got unhelpfully distracted by a

discussion on Kelvin-Helmholtz instabilities...

Price & Bate (2009): Effect of magnetic fields and radiation on star cluster formation

net effect is a very much reduced star formation rate / efficiency per t_ff

Dolag & Stasyszyn (2009):

SPH+MHD makes it’s way into GADGET

application to galaxy clusters, magnetic field evolution and dynamics

in spiral galaxies (Kotarba et al. 2009, Stasyszyn et al. 2010)

• advection of magnetic fields: no change in topology (A.B = 0)

• does not follow wind-up of magnetic fields

• difficult to model resistive effects -- reconnection processes not treated

correctly

Limitations of the Euler potentials approach(Rosswog & Price 2007, Price & Bate 2008, Brandenburg 2010)

dα

dt= 0

dβ

dt= 0

B = ∇α×∇β

Axel Brandenburg (at KITP 2007): “Why don’t you

just use the vector potential?”

is B = ∇×A a better approach?

∂A∂t

= v ×B +∇φ dAdt

= −Ai∇vi

8 Price

2.4.3 3D component

The 3D force contribution is given by substituting the first term in (34), taken with respect to δxia, into the second term of (33), giving"

δ(ρbBjb )

δxia

#

3D

=#jklΩb

X

c

mcAbi

Ωbρb

"

X

d

md∂Wbd(hb)

∂xkb(δba − δda)

#

∂Wbc(hb)

∂xlb

− #jklΩb

X

c

mcAci

Ωcρc

"

X

d

md∂Wcd(hc)

∂xkc(δca − δda)

#

∂Wbc(hb)

∂xlb. (39)

The fact that the δx is so deeply nested in the perturbation of B in the 3D case (i.e., via a summation for the curl ofA [equation 32], andvia a second summation for δA [equation 34]), as we will see below, leads to a force term which is somewhat complicated to calculate.

Nevertheless, it is a force term which preserves the basic symmetries that we asked for, namely momentum and energy conservation in the

SPMHD equations. For example, using the naive or ‘standard’ gauge choice (Equation 23) involves one fewer summations since the δx is notnested inside a derivative in the perturbation toA. However, the perturbation is not Galilean invariant and can be straightforwardly shown to

lead to a force that does not conserve momentum.

2.4.4 Equations of motion

Putting the perturbations (31) and (33) [the second term of which has been expanded into (38) and (39)] into (13) we have

Z

−madviadt

−X

b

mbΩb

"

Pbρ2b

− 32µ0

„

Bbρb

«2

+ξbρ2b

#

X

c

mc∂Wbc(hb)

∂xib(δba − δca)

− 1µ0

X

b

mbΩb

Bjbρ2b

#jklX

c

mc(Abk − Ack)

∂2Wbc(hb)

∂xib∂xlb

(δba − δca)

− 1µ0

X

b

mbΩb

BjbBjint,b

ρb

∂hb∂ρb

X

c

mc(δba − δca)∂2Wbc(hb)

∂xib∂hb

− 1µ0

X

b

mbΩb

Bjbρ2b

h

2δliBjext − δ

ji B

lext

i

X

c

mc∂Wbc(hb)

∂xlb(δba − δca)

− 1µ0

X

b

mbΩb

Bjbρ2b

#jklX

c

mcAbi

Ωbρb

"

X

d

md∂Wbd(hb)

∂xkb(δba − δda)

#

∂Wbc(hb)

∂xlb

+1µ0

X

b

mbΩb

Bjbρ2b

#jklX

c

mcAci

Ωcρc

"

X

d

md∂Wcd(hc)

∂xkc(δca − δda)

#

∂Wbc(hb)

∂xlb

)

δxiadt = 0, (40)

where we have collected the isotropic terms relating to smoothing length gradients into a single term by defining

ξb ≡1µ0

»

BjbHjb + B

jbB

jint,b

ζbΩb

–

, (41)

whereHj and ζ are defined in Appendix A. Since the perturbation δxi is arbitrary, upon simplification (40) implies that the principle of least

action is satisfied by the equations of motion in the form

dviadt

= −X

b

mb

"

Pa − 32µ0 B2a + ξa

ρ2aΩa

∂Wab(ha)∂xia

+Pb − 32µ0 B

2b + ξb

ρ2bΩb

∂Wab(hb)∂xia

#

ff

isotropic term

− 1µ0

X

b

mb

"

BjaΩaρ2a

#jkl(Aak − Abk)

∂2Wab(ha)

∂xia∂xla+

BjbΩbρ2b

#jkl(Aak − Abk)

∂2Wab(hb)

∂xia∂xla

#

ff

2D term

− 1µ0

X

b

mb

"

BjaBjint,a

Ωaρa

∂ha∂ρa

∂2Wab(ha)∂xia∂ha

+BjbB

jint,b

Ωbρb

∂hb∂ρb

∂2Wab(hb)∂xia∂hb

#

ff

2D∇h term

− 1µ0

h

2δliBjext − δ

ji B

lext

i

X

b

mb

"

BjaΩaρ2a

∂Wab(ha)∂xla

+Bjb

Ωbρ2b

∂Wab(hb)∂xla

#

ff

2.5D/Bext term

−X

b

mb

»

AaiΩaρ2a

Jka∂Wab(ha)

∂xka+

AbiΩbρ2b

Jkb∂Wab(hb)

∂xka

–

,

ff

3D term (42)

where the current Jk is defined according to

Jka ≡ −ρa#kjlµ0

X

b

mb

"

BjaΩaρ2a

∂Wab(ha)∂xla

+Bjb

Ωbρ2b

∂Wab(hb)∂xla

#

, (43)

c© 2009 RAS, MNRAS 000, 1–25

Price 2010 (paper IV): 25 pages* of pain later, we had

derived the ultimate vector potential formulation in SPH...

...it was beautiful, derived elegantly from a Lagrangian

variational principle, the method was exactly conservative,

novel, the divergence was constrained...

*in the published paper:

~80 in my notebook

...could this be the ultimate method for MHD in SPH?

MHD in SPH using the Vector Potential 19

Figure 8. Results of the Orszag-Tang Vortex evolution in 3D, using 128 × 128 × 16 particles (top, using our NDSPMHD code) 100 × 100 × 10 particles

(middle, using PHANTOM) and 200 × 200 × 20 particles (bottom, using PHANTOM). Here we have adopted the hybrid vector potential formulation that is

stable to clumping instabilities and gives good results in 2D. In 3D we observe exponential growth of the Ax and Ay components of the vector potential

(right panel, showing the maximum value as a function of time, alongside the evolution of total energy). When these components grow to the same order of

magnitude asAz , large, low density voids appear in the solution (left panels), together with an exponential divergence in total energy (right panels).

for the 2D problems, using zero artificial resistivity as in the 2D solution shown in Rosswog & Price (2007). Given our findings below, we

have also computed the solution as a consistency check using the implementation of the hybrid scheme in our PHANTOM SPH code (used in

Kitsionas et al. (2009) and Price & Federrath (2009)), which being parallelised, was also used to compute a higher resolution version. We

show results in Figure 8 using 128 × 128 × 16 particles with NDSPMHD (top row), 100 × 100 × 10 particles using PHANTOM (middle row)and 200 × 200 × 20 particles (bottom row), also using PHANTOM.

The results initially (not shown) are similar to the 2D results — and therefore quite reasonable — but only for a finite time. We find

c© 2009 RAS, MNRAS 000, 1–25

...and did it work?

1st problem:

same old

numerical

instability (in

2D and 3D)

2nd problem:

unconstrained

growth of

non-physical

components

of A in 3D

problems

“Axel, the answer is no.”

(for an interesting reason)

...namely that the magnetic Galilean limit of Maxwell’s

equations requires enforcement of div A = 0,

similar to the original constraint on the B field.

(Price 2010, note submitted to J. Comp. Phys.)

Current directions on SPH+MHD

• generalised Euler potentials method

• exact implementation of projection method for div B

• two fluid implementation (ions/neutrals)

why haven’t we finished all this yet?

B∗ = B−∇φ∇2φ = ∇ · B

was tried by PM05, but with only

approximate solution. With exact

method might be better

completely independent of the ideal MHD implementation

B = ∇α1 ×∇β1 +∇α2 ×∇β2 +∇α3 ×∇β3B = ∇α1 ×∇X0 +∇α2 ×∇Y0 +∇α3 ×∇Z0

α∗1 = α1∇β1 α∗2 = α2∇β2

[β∗1 , β∗2 , β

∗3 ] = [Xi, Yi, Zi]

Allows remapping procedure

(reconnection “by hand”):

Padoan et al. (2007):

“SPH simulations of large scale star

formation to date fail in all three fronts:

numerical diffusivity, numerical resolution,

and presence of magnetic fields. This

should cast serious doubts on the value

of comparing predictions based on SPH

simulations with observational data (see

also Agertz et al. 2006).”

“Numerical simulations can ... account for ... turbulence in ... star

formation only if they can generate an inertial range of turbulence,

which requires both low numerical diffusivity and large numerical

resolution. Furthermore... the magnetic field cannot be neglected”

Comparison of Mach 10, hydro turbulence

SPH=PHANTOM grid=FLASH

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*

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)

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Kinetic energy spectra

Burgers-like k-2 spectrum in the

kinetic energy for Mach 10 hydro

Price & Federrath (2010)

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Density-weighted energy spectra (ρ1/3v)

Confirms Kritsuk et al. (2007)

suggestion of Kolmogorov-like

k-5/3 spectrum in this variable

Price & Federrath (2010)

max density in

SPH at 1283

similar to max grid

density at 5123

Price & Federrath (2010)

Density resolution

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!()#*+,#$!"#!%!&'

-.& -/ & /

-.&

-/

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()*#$!"#!%!&'

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&.,

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&.0

1(23#-,/0

1(23#/-40

1(23#,/50

67893#-,/0

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67893#,/50

Price & Federrath (2010)

Probability Distribution Functions

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0 1 20 21 300

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2

241

3

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F F

F

F

F

F

F

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23A5$6%-78&9,/$:;$1235$=-8>23A5$6%-7/G$@877,>$./23A5$6%-7/G$>8-,&7$./

Density variance -- Mach number relation

Price, Federrath & Brunt (2010, in prep)

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0 1 20 21 300

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σ2s = ln(1 + b2M2), b = 0.5

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0 1 20 21 300

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3

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σ2s = ln(1 + b2M2), b = 0.5

σ2s = ln(1 + b2M2), b = 0.33

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0 1 20 21 300

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σ2s = ln(1 + b2M2), b = 0.33

σ2s = ln(1 + b2M2), b = 0.5 Lemaster & Stone best fit

Mach number

Std.

dev in

log rho

!"#

!$#

% & # ' (%

&%

#%

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(%

)'*+,-."/*012*3-45 67+8!"#9*:*&)'*+,-."/*)'*3-45#);'*+,-."/*012*3-45

<65*!":!$?'*+,-."/*#);'*3-45

If log normal, expect σ2ρ = eσ2s − 1

Trying to measure the (linear) density variance

s ≡ log(ρ/ρ)

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0 1 20 21 300

1

20 12345$6($7-8931:45$6($7-8923;45$6($7-89

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31:45$=8>>,9$.?$6($7-8923;45$=8>>,9$.?$6($7-8912345$6($@%->8&A,?31:45$6($@%->8&A,?23;45$6($@%->8&A,?B%)-)?

Brunt (2010)

(based on new

method for inferring

3D variance from 2D

observations)

(see Brunt, Federrath

and Price 2010)

Padoan, Nordlund

& Jones (1997)

Price, Federrath & Brunt (2010, in prep)

Comparison to observations

need COMPRESSIVE

DRIVING or GRAVITY

Conclusions

• being a television presenter is easier than getting MHD in SPH to work

• MHD in SPH would work if people stopped making unsubstantiated

swipes* at SPH

• Magnetic fields can significantly change star formation even at

supercritical field strengths, so we need MHD in SPH

• SPH and grid codes agree very well on the statistics of turbulence when

the resolutions are comparable: nparts = ncells to get similar spectra, but

SPH much better at resolving dense structures.

• The standard-deviation-- Mach number relation in supersonic turbulence

seems robust up to Mach 20, but observed density variances are much

higher than can be produced with solenoidally-driven turbulence alone

*defined as any paper where the criticism is based purely on a citation to Agertz et al. (2006)