magnetic fields and mhd 17 february 2003 (snow permitting) astronomy g9001 - spring 2003 prof....
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Magnetic Fields and MHD
17 February 2003 (snow permitting)
Astronomy G9001 - Spring 2003
Prof. Mordecai-Mark Mac Low
MHD Approximation• Maxwell’s Equations in a gas
Mestel, Stellar Magnetism
4 0
1 4 1e
c t c c t
E B
B EE B J
The displacement current vanishes if electrons & ions move together
• This happens when thermal fluctuations can’t separate electrons, ions.
• Balance TE to electric PE (Debye length)
1/ 2 1/ 2
27 cm
4De e
kT T
n e n
Generalized Ohm’s Law
1
ec cn e
J v B
E J B
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/cB) = 0
Induction Equation
1
c t c
t
v BBE
Bv B
From Maxwell’s equations,
Lorentz Force
• Ampère’s law, in absence of displacement
current:4
c
B J
• The Lorentz force density: 1
4c
BJ B B
21 1 1
4 4 8B
B B B B
• so Lorentz force
• Remember vector identity: 21
2A A A A A
magnetic tension
magnetic pressure
net force always actsperpendicular to B
Magnetic Resistivity
• If σ finite, then we can use Ohm’s law and Maxwell’s equations:
1
4c c t
c
v BJ BE
J B
2
4
c
t
B
v B B
magnetic diffusivity λ
Magnetic Reynolds #:
m
vLR
Flux Conservation
• If σ , then magnetic flux through any parcel of gas remains constant:
• Gas remains tied to field lines
0
S S C
S C
S
D
Dt t
t
t
BB dS dS B v dS
BdS v B dS
Bv B dS
C
dS
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?
MHD Waves
• Linearize MHD equations:
Jackson, Ch. 10 Classical Electrodynamics
0 1 0 1
2 01
0
( , ) ,
, s
t t
Pt c
B B B x x
v v x
10 1
0 10 1 10 0
0
t
t
t
v
v
v
210 1 0 1
10
4
1
4
sc
P
vv v
B
B B
vB
t
t
11 0tt
B
vv B BB
221 0
0 12
Taking a time derivative of the momentum eqn:
04sc
t t t
1v B B
2
21 00 0 1 1 02
04sc
t
v Bv v B
2
21 0 01 12
0 0
04 4
sct
v B Bv v
0
0
1 1
introduce the Alfven velocity , and choose 4
plane waves exp .
A
i i t
Bv
v v k x
2 2 21 1s Ac v v k v k
1 1 1 0A A A A v k v k v v v k k v v
2 2
22 2 2 2
1 12
if then last term vanishes, leaving magnetosonic
waves with , while if :
1 0
A
s A A
sA A A
A
v c v
ck v k
v
k v
k v
v v v v
1 0
transverse Alfven
waves
A v v
MHD waves
Robert McPherron, UCLA
MHD Shocks
• If B v then shock jump conditions are
v1 v2
B1 B2
1 1 2 2
2 22 21 2
1 1 1 2 2 2
2 221 1
1 1 1 1 1 1 2
1 2 2
1
1
8 8
1 1... , u =
8 2 8 -1
v B v B
v v
B BP v P v
B B PP v u v v
Mestel, Stellar Magnetism
continuity of flux transport
MHD shock
• perpendicular shock: 2 2 1
1 1 2
B vD
B v
2 2 21 1 1 1 1
21 1
1 1 2 21 1 1
1
is found from the positive root of
2 2- 2 1 2 1 0,
2where , .
8
1As ,
1
s
s A
D
D M D M
v P cM
c B v
M D
Oblique shocks
• Field at arbitrary angle to shock normal
• Parallel field must be conserved
• Momentum conservation in frame w/– no magnetic energy flow across shock
• Momentum conservation then gives
1 2x xB B1
1 11
yy x
x
B
v vB
2 2 2 22 21 1 2 2
1 1 1 2 2 2
1 1 2 21 1 1 2 2 2
8 4 8 4
4 4
x xx x
x y x yx y x y
B B B BP v P v
B B B Bv v v v
Oblique Shocks
• Three solutions (e.g. Mestel, p. 50):
v1
slowshock
fastshock
intermediate (Alfvèn)shock
Partially Neutral Gas• Only ions feel Lorentz force from B field
• Ions, neutrals couple through collisions, adding symmetric terms to momentum eqn
,
1,
4
where the collisional coupling constant
ii i i i
i n i n
i n n
nn n
i
i
i
n n n
n
P
m m
Pt
t
v vv
v v
B B
v
vv v
v v
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: C-shock (Mullan 1971, Draine 1980) Smith & Mac Low 1997
Nonlinear Development
Mac Low & Smith 1997
tim
e
Log ρ
Current Sheet Formation• Brandenburg & Zweibel (1994, 1995) showed
that nonlinear nature of field diffusion from ion-neutral drift produces sharp structures.
• 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.
i
i in
vt c
B BB
BB
Next week’s assignments
• Read Slavin & Cox (1993, ApJ, 417, 187) on the filling factor of hot gas with non-thermal 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 moviesfor 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)
pause
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