Magnetic Fields and MHD

1
Magnetic Fields and MHD

• 17 February 2003 (snow permitting)
• Astronomy G9001 - Spring 2003
• Prof. Mordecai-Mark Mac Low

2
MHD Approximation
Mestel, Stellar Magnetism

• Maxwells Equations in a gas

• This happens when thermal fluctuations cant
  separate electrons, ions.
• Balance TE to electric PE (Debye length)

The displacement current vanishes if electrons
ions move together
3
Generalized Ohms Law
Hall term

• so long as ions are not very massive (eg dust
  grains) we may neglect the Hall term.
• If s large, then E(v/c?B) 0

4
Induction Equation
From Maxwells equations,
5
Lorentz Force

• Ampères law, in absence of displacement current

net force always acts perpendicular to B
6
Magnetic Resistivity

• If s finite, then we can use Ohms law and
  Maxwells equations

7
Flux Conservation

• If s ? ?, then magnetic flux through any parcel
  of gas remains constant
• Gas remains tied to field lines

dS
C
8
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?

9
MHD Waves
Jackson, Ch. 10 Classical Electrodynamics

• Linearize MHD equations

10
(No Transcript)
11
(No Transcript)
12
MHD waves
Robert McPherron, UCLA
13
MHD Shocks
B1
B2
Mestel, Stellar Magnetism
v1
v2

• If B ? v then shock jump conditions are

continuity of flux transport
14
MHD shock

• perpendicular shock

15
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

16
Oblique Shocks

• Three solutions (e.g. Mestel, p. 50)

slow shock
intermediate (Alfvèn) shock
fast shock
v1
17
Partially Neutral Gas

• Only ions feel Lorentz force from B field
• Ions, neutrals couple through collisions, adding
  symmetric terms to momentum eqn

18
J-Shocks vs. C-Shocks

• Classical shock is a discontinuous jump or
  J-shock
• If vAigt vsgtcsn 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
19
(No Transcript)
20
Nonlinear Development
Log ?
time
Mac Low Smith 1997
21
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.

22
Next weeks 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

23
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

24
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.
25
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

26
Distributed Memory

• Multiple processors with private memory
• Data shared across network
• User responsible for synchronization

U of Minn Supercomputing Inst.
27
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.

28
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

29
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

30
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

31
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.

32
Use of IDL
pause

• Quick and dirty movies
• for i1,30 do begin
• asin(findgen(10000.))
• hdfrd,fzhd_string(i,form(i3.3))aa,dd,
  xx
• plot,x,d4.dat end
• Scaling, autoscaling, logscaling 2D arrays
• tvscl,alog(d)
• tv,bytscl(d,maxdmax,mindmin)
• Array manipulation, resizing
• tvscl,rebin(d,nx,ny,/s) nx, ny multiple
• tvscl,rebin(reform(dj,,),nx,ny,/s)

33
More IDL

• plots, contours
• plot,x,di,,k,xtitleTitle,psym-3
• oplot,x,di10,,k
• contour,reform(di,,),nlev10
• slicer3D
• dp ptr_new(alog10(d))
• slicer3D,dp
• Subroutines, functions