Kinetic Processes and MHD Models:
Can they work together?
Michael Hesse
NASA GSFC
Overview of things to come
MHD:
What is it?
What is it good for?
What is it not so good for?
Beyond MHD:
Other physical models
Extensions to MHD
Anisotropic pressure
Reconnection physics
Summary
MHD: What is it?
∂ρ
+ ∇i( ρ v ) = 0
∂t
∂
ρ v + ρ v i∇v = −∇p + j × B
∂t
Continuity
Momentum
E + v × B = 0 (= η j )
(resistive) Ohm’s law
∂p
+ v i∇p = −γ p∇iv + (γ − 1)η j 2
∂t
(
∇ × B = µ0 j
∂B
= −∇ × E
∂t
∇i B = 0
)
Equation of state
Ampere’s law
Faraday’s law
Absence of monopoles
MHD: When is it safe?
•Slow time scales: τ>>Ωci-1
no ion (or electron cyclotron effects)
•Pressure isotropy for total (ion + electron) pressure
can write total pressure force as gradient
•Neglect/approximate heat flux
use ideal/polytropic pressure law
•“Large” spatial scales: L>>c/ωpi
ignore Hall effects and electron pressure effects
•Two plasma species: Ions and electrons
obtain one, combined, momentum equation
•me/mi negligible
ignore ion pressure in Ohm’s law
•Slow typical velocities vt/c<<1
ignore displacement current
How is MHD good for you?
•MHD describes large scale structure
•MHD describes large scale transport
•MHD describes dominant large scale
communication (by Alfven waves)
•MHD describes large scale current systems, by
•MHD is “simple” (although codes are not!) and
“cheap”
•….
∇× B
What is MHD used for anyway?
Siebert and Siscoe
… it is hard to argue with success
Research: CDPS formation
Cold-dense plasma sheet formation
Li et al., 2005
Orioset et al., 2005
Reproducing S/C observations
Gombosi et al.
Reproducing field-aligned currents
Providing global descriptions and forecasts
It can also produce lively debates, e.g.,…
How long is the tail for northward IMF?
Raeder et al, 1999
Why does the PC potential saturate?
Gombosi et al, 2000
Siscoe et al, 2005
Concerns and worries: MHD limitations
• Single fluid model
• Magnetic flux transport is based on ions only
• No drift physics – no Hall term, simple energy equation
• Simple energy equation: isotropic pressure, no heat flux
• Dissipation, diffusion is (numerical) resistive (ad-hoc) only
• Reconnection may be slow only
What to do: Go beyond MHD?
MHD
l
a
c
si
y
h on
p
ge ripti
n
a sc
h
C de
M
od
ify
Hall-MHD
M
HD
Transport
“Coefficients”
Anisotropic
Pressure
Multi-fluid
Models
Reconnection
Physics
Kinetic Models
Hybrid, full PIC
…
Sub-Gridscale
Modeling
Consider Alternative Descriptions
•Hall-MHD
– include (some) drift physics, ion-electron decoupling
•Multi-fluid modeling
– include (fluid) physics of additional ion species
•Hybrid modeling
– include kinetic ion physics (e.g, shocks)
•Fully kinetic modeling
– do it all (EM waves, reconnection…)
… for global magnetosphere
MHD: Computational “Simplicity”
Rough estimate: 3D Magnetosphere
Size: 200REx60REx60RE
Cell size: dx=0.25RE
Without AMR ~ 4.6x107cells
Eight MHD state variables
~4x108 reals, 2-4G bytes
Ignored AMR, structured meshes etc.
Fastest wave mode is fast mode vf=(vA2+vS2)1/2
Time step limitation is dt < dx/vf
Wiltberger
Step up: Hall-MHD models
∂ρ
+ ∇i( ρ v ) = 0
∂t
∂
ρ v + ρ v i∇v = −∇p + j × B
∂t
1
1
E + v × B =η j +
j × B − ∇pe
en
en
∂p
+ v i∇p = −γ p∇iv + (γ − 1)η j 2
∂t
(
∇ × B = µ0 j
Continuity
Momentum
Ohm’s law
)
Equation of state
Ampere’s law
∂B
= −∇ × E
∂t
∇i B = 0
+ eqns of state for pe
Faraday’s law
Absence of monopoles
Time step killer: Whistler waves
Electron MHD part of Hall-MHD Ohm’s law:
(
)
1
1
E=
j×B =
∇× B × B
en
enµ0
 1

1
∂B
= −∇ × 
∇ × B × B ≈ −
∇ ×  ∇ × B × B


enµ0
∂t
 enµ0

(
)
(
)
Linearize:
∂B1
1
1


B0 i∇ ∇ × B1
≈−
∇ × ∇ × B1 × B0 = −


enµ0
enµ0
∂t
(
)
(
Fourier analysis:
(
1
B0 ik k × B1
ω B1 ∼
enµ0
)
)
Whistler waves: the bad part
B1 ⊥ B0 , k B0
1
B0 k 2 =
ω≈
enµ0
B0
µ0 nmi
mi 2
c 2
k = vA
k
2
e nµ 0
ω pi
Frequency goes like inverse wavelength squared!
This means that the time step scales like the inverse square
of the cell size
Whistlers become important if dx ≤
c
ω pi
Hall-MHD Computational Cost
Rough estimate: 3D Magnetosphere
Size: 200REx60REx60RE
Cell size: dx=0.25RE
Without AMR ~ 4.6x107cells
Eight MHD state variables
~4x108 reals, 2-4G bytes
Ignored AMR, structured meshes etc.
Germaschewki, Bhattacharjee
Fastest wave mode is fast mode vf=(vA2+vS2)1/2 or Whistler
Time step limitation is dt < dx/vf, dx/vph,w
Same as MHD as long as ion inertial length not resolved.
Next step: Multi-fluid models
∂ns
+ ∇i( ns vs ) = 0
∂t
Continuity
∂
ms ns vs + ms ns vs i∇vs = qs E + vs × B − ∇ps
∂t
me dve
1
E + ve × B = η j − ∇pe −
en
e dt
1 

ve =
qs ns vs − j 
∑

ene  s

1
ne = ∑ qs ns
e s
∂B
= −∇ × E
∂t
∇i B = 0
+ eqns of state
(
)
Momentum
Ohm’s law
Electron velocity
Quasi-neutrality
Faraday’s law
Absence of monopoles
Multi-Fluid Computational Cost
Rough estimate: 3D Magnetosphere
Size: 200REx60REx60RE
Cell size: dx=0.25RE
Without AMR ~ 4.6x107cells
5N+3 state variables
~7x108 reals, 3-6G bytes (for N=2)
Ignored AMR, structured meshes etc.
Winglee
Fastest wave mode is fast mode vf=(vA2+vS2)1/2 or Whistler
Time step limitation is dt < dx/vf, dx/vph,w, Ωi-1
Size same as ~NxMHD as long as ion inertial length not resolved.
But: time step limited by ion cyclotron frequency.
Kinetic Physics: Hybrid Model
ms
dus ,k
dt
(
= qs E + u s , k × B
)
Eqn of motion of particle k
of species s
ns (r ) = ∑ S ( r − rs ,k )
Density
ns (r )vs (r ) = ∑ S ( r − rs ,k )us ,k
Momentum
k
k
me dve
1
E + ve × B = η j − ∇pe −
en
e dt
1 

ve =
qs ns vs − j 
∑

ene  s

1
ne = ∑ qs ns
e s
∂B
= −∇ × E
∂t
Ohm’s law
Electron velocity
Quasi-neutrality
Faraday’s law
Computational effort: Hybrid
Rough estimate: 3D Magnetosphere
Size: 200REx60REx60RE
Cell size: dx=c/ωpi~0.02RE (~120km)
Without AMR ~ 9x1010cells
10 particles/cell, 6 coordinates
~6x1012 reals, 24-48Tbytes
Ignored AMR, structured meshes etc.
Additional cost of particle <-> grid operations
Fastest wave mode is Whistler mode vph,w=kvAc/ωpi~2πλ-1vAc/ωpi
Time step limitation is dt < dx/vph,w, Ωci-1
Omidi
Fully Kinetic Models
?
=
Not yet.
How About Transport Models?
• Anisotropic pressure (for regional model)
• Reconnection electric field from kinetic model
• Reconnection electric field from local resistivity (demo project)
… for global magnetosphere
Include Gyrotropic Pressure
∂
ρ v + ρ v i∇v = −∇i p + j × B
∂t
Momentum
2 p − p⊥ + SB p
+ ∇i( p v ) + 2 2 Bi( ∇v )i B − 2η j = −
∂t
B
τ micro
3
∂p
p
2
∂p⊥
p⊥
1 p − p⊥ + SB p
2
+ ∇i( p⊥ v ) + p⊥∇iv − 2 Bi( ∇v )i B − η j⊥ =
3
∂t
B
τ micro
Modified double-adiabatic approach
Hesse and Birn, 1992
Birn et al., 1995
Results
Birn et al., 1995
Anisotropy has a big effect on the evolution. How much is right?
Reconnection Electric Field: What is it?
z
x
Electron eqn. of motion:
E = −ve × B −
m  ∂v
1

∇ • Pe − e  e + ve • ∇ve 
ne e
e  ∂t

me  ∂ve
1
1

= −vi × B +
∇ • Pe −
+ ve • ∇ve 
j×B−

ne e
ne e
e  ∂t

Key term:
E=−
1
∇ • Pe
ne e
Express Through Ions and Electrons
Basic idea:
E = −vi × B +
1
1
j×B−
∇ • Pe
ne e
ne e
1
≈ −vi × B +
∇ • Pi
ni e
Analytic theory:
Erec
∂ve, x
1
2meTe
≈
e
∂x
Hesse et al., 1999
Can ion pressure tensor represent kinetic physics?
Test with equal ion and electron mass…
Kinetic, mi=me Simulation
Reconnection rate: Fast enough?
sizeable reconnection rate even at late times
Reconnection electric field: Where is it?
1  ∂ Pxy ∂ Pyz
+
Ey = −

ne e  ∂ x
∂z
 me
ve • ∇v y
−
 e
- Strongly localized dissipation region, scales smaller than CS length
- Expressed by ion pressure tensor
Implementation: How to make MHD fly
Two avenues:
- Use kinetic theory results to model ion pressure tensor
- Design a resistivity based on kinetic results
1. Match Kinetic Model to MHD
Y axis is || to J
nongyrotropic
orbit
gyrotropic
orbits
2d Z
dX
MHD Region
E=-[VxB]
dX
1/ 2
2 d X= 2 ρ i
(d X )
'
Ti mi / e B z )
=(V
REC
E
dZ
´ ´X
~ BZ (dX) VX (dX) ~ B ZV
d2
X
=
= ( VTi mi / e Bx' )
1/2 ∂ Vx
1
_
__
(2 mi Ti )
e
∂x
1/ 2
M. Kuznetsova
Implementation Inside BATSRUS
1) Search for reconnection sites:
X0(y) ,
X1(y)
2) Calculate spatial scales of diffusion regions and
reconnection electric field 1/2
d X 2 = (α / B´Z )
REC
E
(y)= α
(ρ )
2P
1/2
(ρ )
2P
dz
Vx
∂__
∂x
2=
(α / B´X
1/2
2P
(
) ρ )
α = ρi0 / R E
3) Add non-gyrotropic correction to the induction equation
EYNG ( x, y, z ) =
EREC (y)
cosh [(x-X0(y))/dX (y)] cosh [z /dZ (y)]
M. Kuznetsova
Simulation
Solar Wind Parameters:
N = 2 cm –3 , T = 20000 K , Vx = - 400 km/s, |B| = 10 nT
M. Kuznetsova
Detailed Comparison
Nongyrotropic Corrections
Numerical Resistivity
Kinetic Simulations
Resistive MHD Simulations
M. Kuznetsova
2. Calculate Equivalent Resistivity
Ereconnection
"η =
"
jy
µo v A L
=3
minimum (equivalent) Lundquist number: R=
η
localization: δx=25, δz=2.5
L=
c
ω pi
MHD Simulation with Local Resistivity
MHD Reconnection Rate
Fast reconnection possible with suitable resistivity
SUMMARY
- MHD will remain the tool of choice for large scale modeling
- Considering MHD approximations, MHD is a extremely successful research
and quantitative modeling tool
- Beyond changing models altogether, kinetic physics can be incorporated into
MHD models through
-- Suitable transport models
-- Modifications of the equations
- Much of this work is in experimental stage, but shows promising results
-- MHD can reproduce fast reconnection
- Other (related) avenues of success:
-- Local overriding MHD by better model: MHD/RCM coupling
-- Data assimilation (how?)
-- Sub-gridscale modeling
Adding physics exacts a price, but that price may be well worth paying.
Final Look
Engaged
..but not married - yet