Download Distribution and Properties of the ISM

Magnetic Fields and MHD
17 February 2003 (snow permitting)
Astronomy G9001 - Spring 2003
Prof. Mordecai-Mark Mac Low
Mestel,
Stellar
Magnetism
MHD Approximation
• Maxwell’s Equations in a gas
  E  4e
B  0
1 B
4
1 E
E  
B 
J
c t
c
c t
• This happens when thermal fluctuations
can’t separate electrons, ions.
• Balance TE to electric PE (Debye length) The displacement
1/ 2
 kT 
D  
2 
 4 nee 
1/ 2
T 
  7 cm   
 ne 
current vanishes if
electrons & ions
move together
Generalized Ohm’s Law
vB 
1

 E 
 J  B

 
c  cnee
J
Hall term
• so long as ions are not very massive (eg
dust grains) we may neglect the Hall term.
• If σ large, then E+(v/cB) = 0
Induction Equation
From Maxwell’s equations,
v  B

1 B

   E   
c t
c
B
    v  B
t
Lorentz Force
• Ampère’s law, in absence of displacement
4

current:   B 
J
c
1
B 

• The Lorentz force density:  J  B   
B
c
 4 
• Remember vector identity:
• so Lorentz force
1
   A   A   A    A   A2
2
1
1
1
2


B

B

B


B


B




4
4
8
net force always acts
perpendicular to B
magnetic
tension
magnetic
pressure
Magnetic Resistivity
• If σ finite, then we can use Ohm’s law and
Maxwell’s equations:
E  
J
v  B

1 B



4
c
c t
c
J  B
B
c
    v  B   
  B 
t
4
magnetic diffusivity λ
2
Magnetic
Reynolds #:
Rm 
vL

Flux Conservation
• If σ  , then magnetic flux through any
parcel of gas remains constant:
D
B
B  dS  
 dS   B  v  dS

Dt S
S t
C
B

 dS 
S t
 v  B  dS
C
 B

 
    v  B    dS  0

S  t
• Gas remains tied to field lines
dS
C
Flux Conservation Consequences
• Flux cannot be created or destroyed without
resistive effects (reconnection)
• So where did Galactic field come from?
• Flux carried with gas during collapse
• How come stars do not have same mass to
flux ratio as interstellar gas?
Jackson, Ch. 10
Classical
Electrodynamics
MHD Waves
• Linearize MHD equations:
B  B0  B1 (x, t )   0  1  x, t 
v  v1  x, t 
 P0
c 
0
2
s
   0  1 

   v  0 
     0  1  v1  0
t
t
1
  0  v1  0
t
v
1
    v    v  P 
B    B  
t
4
v1
1
2
0
 cs 1 
B 0     B1   0
t
4
B
B1
    v  B 
    v1  B 0 
t
t
Taking a time derivative of the momentum eqn:
 v1
B1 
 B0 
2  
 0 2  cs   1  
 
0
t
t 
 t  4 
2
 v1
B0
2
0 2  cs 0  v1 
       v1  B0   0
t
4
2




 v1
B
B
2
0
0


c



v






v




  0
s
1
1
2


t
40
4



0



2
B0
introduce the Alfven velocity v A 
, and choose
40
plane waves v1  v1 exp  ik  x  it  .
 v1   c  v
2
2
s
2
A
 k  v  k 
1
 v A  k  v A  k  v1   v A  v1  k   k  v1  v A   0
if k  v A then last term vanishes, leaving magnetosonic
waves with v  c  v , while if k v A :
2
s
2
A
v A  v1  0 
transverse Alfven
waves
2

 2
cs
2 2
2
 k vA    v1   v 2  1 k  v A  v1  v A  0
 A

MHD waves
Robert McPherron, UCLA
B1
v1
B2
v2
Mestel,
Stellar
Magnetism
MHD Shocks
• If B  v then shock jump conditions are
1v1   2v2
2
1
2
2
B
B
2
P1   v 
 P2   2v2 
8
8
2



B12 
1
B
1 P
2
1
 P1 
 v1   1u1  1v1 
 v1  ...2 , u =
8 
2
8 
 -1 


2
1 1
v1B1  v2 B2
continuity of
flux transport
MHD shock
B2  2 v1
• perpendicular shock: D 


B1 1 v2
D is found from the positive root of
2  2-  D   2 1    1 1M  2   D     1 1M  0,
2
2
1
v1
P1
2  cs2 
where M 1  , 1  2
  2.
cs1
B1 8   v A 1
 1
As M 1  , D 
 1
2
1
Oblique shocks
• Field at arbitrary angle to shock normal
• Parallel field must be conserved B1x  B2 x
B1 y
• Momentum conservation in frame w/ v1 y  v1x
B1x
– no magnetic energy flow across shock
• Momentum conservation then gives
2
2
2
2
B1 B1x
B2 B2 x
2
2
P1 

 1v1x  P2 

  2v2 x
8 4
8 4
B1x B1 y
B2 x B2 y
1v1x v1 y 
  2v2 x v2 y 
4
4
Oblique Shocks
• Three solutions (e.g. Mestel, p. 50):
slow
shock
intermediate (Alfvèn)
shock
fast
shock
v1
Partially Neutral Gas
• Only ions feel Lorentz force from B field
• Ions, neutrals couple through collisions,
adding symmetric terms to momentum eqn
 v n 
n 
    n  v n    v n  Pn  i  n  vi  v n  ,
 t 
 v i 
i 
   i  v i    v i  Pi 
 t 
1

   B  B   i n  v n  vi  ,
4
v
where the collisional coupling constant  
mi  mn
J-Shocks vs. C-Shocks
• Classical shock is a
discontinuous jump
or J-shock
• If vAi> vs>csn then
ions see continuous
compression by
magnetic precursor
• Neutrals dragged by
ions into continuous
compression: Cshock (Mullan 1971,
Draine 1980)
Smith & Mac Low 1997
Nonlinear Development
time
Log ρ
Mac Low & Smith 1997
Current Sheet Formation
• Brandenburg & Zweibel (1994, 1995) showed
that nonlinear nature of field diffusion from
ion-neutral drift produces sharp structures.
    B   B   B
B
    vi  B  
t

i in c
• Analogous to shock formation in strong
sound waves: magnetic pressure higher in
peaks, so waves spread and steepen.
• Zweibel & Brandenburg (1997) emphasized that
current sheets form, driving reconnection.
• Seems to explain numerical results well.
Next week’s assignments
• Read Slavin & Cox (1993, ApJ, 417, 187)
on the filling factor of hot gas with nonthermal pressures included
• Read Stone & Norman (1992b, ApJS, 80,
791) -- the MHD ZEUS paper
• Complete the blast exercise
Parallelization
• Additional issues:
– How to coordinate multiple processors
– How to minimize communications
• Common types of parallel machines
– shared memory, single program
• eg SGI Origin 2000, dual or quad proc PCs
– multiple memory, multiple program
• eg Beowulf Linux clusters, Cray T3E, ASCI systems
Shared Memory
• Multiple processors
share same memory
• Only one processor
can access memory
location at a time
• Synchronization by
controlling who
reads, writes shared
memory
U of Minn Supercomputing Inst.
Shared Memory
• Advantages
– Easy for user
– Speed of memory access
• Disadvantages
– Memory bandwidth limited.
– Increase of processors without increase of
bandwidth will cause severe bottlenecks
Distributed Memory
• Multiple processors with private memory
• Data shared across network
• User responsible for synchronization
U of Minn Supercomputing Inst.
Distributed Memory
• Advantages
– Memory scalable with number of processors. More
processors, more memory.
– Each processor can read its own memory quickly
• Disadvantages
– Difficult to map data structure to memory
organization
– User responsible for sending and receiving data
among processors
• To minimize overhead, data should be
transferred early and in large chunks.
Methods
• Shared memory
– data parallel
– loop level parallelization
• Implementation
– OpenMP
– Fortran90
– High Performance
Fortran (HPF)
• Examples
– ZEUS-3D
• Distributed memory
– block parallel
– tiled grids
• Implementation
– Message Passing Interface (MPI)
– Parallel Virtual Machine (PVM)
• Examples
– ZEUS-MP
– Flashcode
– GADGET
OpenMP
• Designate inner loops that can be distributed
across processors with DOACROSS command.
• Dependencies between loop instances prevent
parallelization
• Execution of each loop usually depends on
values from neighboring parts of grid.
• ZEUS-3D only parallelizes out to 8-10
processors with OpenMP
Cache Optimization
• Modern processors retrieve 64 bytes or more at
a time from main memory
– However it takes hundreds of cycles
• Cache is small amount of very fast memory on
microprocessor chip
– Retrievals from cache take only a few cycles.
• If successive operations can work on cached
data, speed much higher
– Fastest changing array index should be inner loop,
even if code rearrangement required
Parallel ZEUS-3D
• To run ZEUS-3D in parallel, set the variable
iutask = 1 in setup block, recompile.
– inserts DOACROSS directives
– compiles with parallel flags turned on if OS
supports them.
• Set the number of processors for the job
(usually with an environment variable)
• Run is otherwise similar to serial.
Use of IDL
• Quick and dirty movies
pause
for i=1,30 do begin & $
a=sin(findgen(10000.)) & $
hdfrd,f=’zhd_’+string(i,form=’(i3.3)’)+’aa’,d=d,x=x & $
plot,x,d[4].dat & end
• Scaling, autoscaling, logscaling 2D arrays
tvscl,alog(d)
tv,bytscl(d,max=dmax,min=dmin)
• Array manipulation, resizing
tvscl,rebin(d,nx,ny,/s) ; nx, ny multiple
tvscl,rebin(reform(d[j,*,*]),nx,ny,/s)
More IDL
• plots, contours
plot,x,d[i,*,k],xtitle=’Title’,psym=-3
oplot,x,d[i+10,*,k]
contour,reform(d[i,*,*]),nlev=10
• slicer3D
dp = ptr_new(alog10(d))
slicer3D,dp
• Subroutines, functions