fundamentals of plasma simulation (i)
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Fundamentals of Plasma Simulation (I). 核融合基礎学（プラズマ・核融合基礎学） 李継全（ 准 教授） / 岸本泰明（教授） / 今寺賢志（ D1 ） 2007.4.9 — 2007.7.13. Lecture two (2007.4 .) Part one: Basic concepts & theories of plasma physics ➣ Basic descriptions of plasma Basic plasma equations Single particle orbits - PowerPoint PPT PresentationTRANSCRIPT
1
Fundamentals of Plasma Simulation (I)
• 核融合基礎学（プラズマ・核融合基礎学）• 李継全（准教授） /岸本泰明（教授） /今寺賢志（D1）• 2007.4.9 — 2007.7.13
Lecture two (2007.4.)Part one: Basic concepts & theories of plasma physics➣ Basic descriptions of plasma Basic plasma equations Single particle orbits Plasma kinetic description Fluid equations Fluid/kinetic hybrid model Gyrofluid model MHD & reduced MHD Classification of equations (Poisson; wave; diffusion)
Reference books: F.F. Chen, Introduction of plasma physics S Ichimaru, Basic principles of plasma physics ……
2
How to describe a plasma?
Since a plasma may behave collectively or like an assembly of individual particles, so we have following three approaches to describe it
1. Single Particle Approach. (Incomplete in itself). Equations of particle motion → orbits of particles.
2. Kinetic Theory. Boltzmann Equation → statistical description → transport
coefficients
3. Fluid Model (MHD & reduced MHD). Moments of kinetic equation → macroscopic description (Density;
Velocity, Pressure (temperature), Currents, etc.)
All descriptions should be consistent. Sometimes they are only different ways to approximately look at the same thing.
Further, some approximate models have been developed such as: fluid-kinetic hybrid model; gyrofluid model.
3
Basic equations of plasma physicsElectric and magnetic fields (E & B) are generally determined by Maxwell’s equations, with corresponding boundary conditions and the sources (charges and currents).
t
E
cJ
cB
t
B
cE
B
E
14
10
4
Gauss’s law
No magnetic poles
Faraday law
Ampere’s law
),(),(),( trAtc
trtrE
),(),( trAtrB
In this case, electromagnetic field equations are written in the form
),(41 2
2
2
trtc
),(41 2
2
2
trJc
At
A
c
With Lorentz gauge 01
Atc
Sometimes E & B are expressed in terms of an electric potential φ and vector potential A:
4
The forces include two contributions from external electromagnetic field and also internal field, which is produced by other particles. The latter should be evaluated self-consistently.
Hence, the electromagnetic forces in a plasma depends on the current and charge densities which are determined by the collective particle interaction.
The motion of charged particles is determined by the electromagnetic fields through the equations of motion – Lorentz equation
)1
( Bc
Em
q
m
F
dt
di
i
i
i
ii
ii
dt
xd
Basic equations of plasma physics (cont.)
Equation of motion
5
Kinetic equations
To describe a plasma with a large number of particles, one can solve the coupled system of Maxwell’s equations and the equations of motion for each particle. This is a terrible job! However, there are more efficient methods to solve the plasma dynamics using statistical approximation – kinetic equation.
Consider the single particle distribution function f(t, r, v) which gives the density ofparticles in the six-dimensional space (r,v), The single particle distribution function satisfies the Boltzmann equation
ct
ffB
cE
m
qf
t
f
)1
(
s
ss dtrfqtr 3),,(),(
s
ss dtrfqtrJ 3),,(),(
The charge and current densities can be evaluated as
To describe a plasma, it needs only to solve Maxwell equations and kinetic equation!
6
Single Particle Approach
7
Single Particle Approach – orbits & drifts of particle in electromagnetic fields.
Although a plasma behaves collectively and the dynamics should be described by statistical approach, a lot of plasma phenomena can be helpfully understood in terms of single-particle motion. The motion of charged particles in assumed electric and magnetic field can provide insight into many important physical properties of plasmas.
Equation of motion + Maxwell equations
)],(1
),([ rtBc
rtEm
q
m
F
dt
di
i
i
i
ii
ii
dt
xd
t
rtE
cJ
crtB
t
rtB
crtE
),(14),(
),(1),(
→ Particle orbits in various given electromagnetic fields
8
Gyro motion & Larmor radius
qB
mcvv
gL
L
mc
qBg gyrogrequency gyroradius
Guiding center
tzz
tyy
txx
gg
gg
//0
0
0
)sin(
)cos(
From equation of motion, we can easily know that particle moves along the magnetic field with υ //0 and gyrates around the filed. Here the second magnetic field produced by moving charged particle is ignored!
Initial velocity of charged particle in magnetic field ),( 0//00
Charged particle is only experienced Lorentz force B
m
q
Orbit of charged particle is
Gyration is the most basic motion of charged particle in a magnetized plasma!
9
Electric field drifts (E×B drift)
The “E × B drift” of the gyrocenter is
2B
BEcd
Bmc
q
m
qE
dt
d
qEdt
dm
//
//
If E is non-uniform, it may cause a modification of Larmor radius effect
EEErEL
L
2
2
2)()(
2
22
41
B
BEcLd
Equation of motion is
Homework: Problem 3Derive the orbit of positive charged particle q with initial velocity [ ] in a constant uniform electric field (0,E┴,E//) and magnetic field (0,0,B), express the velocity of particle gyrocenter.
),( 0//00
10
Drifts due to general force F
Bmc
q
m
F
dt
d
Lorentz equation is
E×B drift can be generalized by substituting qE with a general constant force term F. The resulting particle drift generated by this constant force is
2
)(
qB
BFcF
This general force can be gravity, force due to non-uniform magnetic field (gradient or curvature)
11
Magnetic field gradient drifts
00 BBB c
Expanding the magnetic field at the location of the guiding center
Bvc
qxBv
c
q
dt
vdm cc
)0()1()1(
)(
qqB
mvv c
2
2)0(
2qB
BBV B
Equation of motion becomes
)()0()0(
cxBvc
q
dt
vdm
Magnetic gradient produces a force on the guiding center of charged particle due to the magnetic moment, i.e.,
BBB
mF
2
2
BBqB
mc
3
2
2
Gradient drift
Averaging on gyro-motion
12
Magnetic field curvature drifts
A particle which moves along a curved magnetic field line experiences a centrifugal force on its guiding center. This force is (often convenient to have this expressed in terms of the field gradients.)
BBB
mR
RmF
c
cc
)(
12
2//2
2//
When the B field lines are curved and the particle has a velocity v// along the field, another drift occurs.
])[(4
2// BBB
qB
mcc
Curvature drift velocity
In the frame of the guiding center a force appears because the plasma is rotating about the center of curvature.
Gradient and curvature drifts are related through Maxwell’s equations, which depends on the current density j. A particular case of interest is j = 0: vacuum fields. c
r R
BB )(
2222
// 2
1
c
ccB RqB
BRcmm
13
Drifts in varying electric field – polarization drift
If electric field E is time-varying, the particle experiences a acceleration,
2B
BEc
dt
d
dt
dd
In the frame of the guiding centre which is accelerating, a force is felt except for the force due to uniform electric filed.
2B
BE
dt
dmcFp
An additional drift is produced as
dt
Ed
qB
mcB
B
BE
dt
d
qB
mcp
222Polarization drift
Physical meaning of polarization drift: If electric field is constant, particle experiences E×B drift with a constant Larmor radius, when direction of E is changing with time, the radius of gyro-orbit suddenly changes and produce a polarization drift velocity.
14
These drifts have been determined by assumed electric and magnetic fields. They describe test particle motion. However, it should be noticed that the currents due to the drifts alter the fields. If these changes are small compared to the background field it is justified to apply the drift model. The derived particle drifts do not contain any collective behavior. For this reason it is a nontrivial aspect to compare particle and fluid plasma drifts.
Hence, single particle approach has ignored the interaction among charged particles, it is only suitable for enough low density plasma.
If drifts depend on the charge, a current can be produced as j=en(vi-ve). So polarization drift; magnetic field gradient and curvature drifts cause a current.
Remarks for single particle driftsAll these drift velocities and the particle orbit above can be derived directly by solving the motion of particle with an initial velocity (υ┴0, υ //0) in assumed time-varying non-uniform electric E(t, x) and magnetic fields B(t, x), i.e.,
)],(1
),([ xtBc
xtEm
q
dt
di
i
ii
),( with 0//00
ii
dt
xd
Gyromotion; electric field drift with Larmor radius modification; magnetic field gradient and curvature drifts; polarization drift; magnetic mirror.
15
Plasma Kinetic Description
16
Plasma Kinetic Theory – why need kinetic description
Many particle: For a plasma, the plasma parameter is g=1/(nλ 3d)<<1. Thus a plasma co
nsists of a very large number of particles. It is too tough work to calculate the orbits of all particles even if for assumed electric and magnetic fields.
Long-range force: The charged particles of a plasma are both responding to the electromagnetic fields and acting as their sources. This means charged particle moving under the influence of both the external fields and the fields generated by the particles themselves. Namely, the plasma behaves collectively. It is almost impossible to calculate the motion of all particles in a plasma self-consistently.
Fields as an average: Actually, the orbits of all particles are not so important in a plasma, the spatial and temporal development of statistical measurable quantities as a fluid, i.e., particle density, particle flux, temperature or pressure, heat flux, and so on, are more interesting. Because the collective behavior of the charged particles is a fundamental property of plasmas, we do not always need to know anything about the individual particles but, instead, we are interested in the average properties of the gas or fluid.
The description of these quantities is a matter of statistical physics, which is appropriately started using a kinetic description of plasma.
17
Kinetic description of plasma – Boltzmann equation
“kinetic” means it is relating to motion of particles. So a kinetic description includes the effects of motion of charged particles in a plasma.
An exact, microscopic kinetic description is based on and encompasses the motions of all the individual charged particles in the plasma. Our interest is in the average rather than individual particle properties in plasmas, so, an appropriate average process can be taken to obtain a general plasma kinetic equation—Boltzmann equation
Two ways to derive Boltzmann equation for a plasma
Klimontovich equation approach: It deals with the exact density of particles in the six-dimensional phase space (r; v) by using δ-functions.
Liouville equation approach: This approach starts with distribution functions and avoids δ-functions and ensemble averaging. (we will not talk about this approach in this lecture)
18
Klimontovich equation approach
Consider a single particle with orbit (xi(t);vi(t) ) in 6-dimensional phase space. The “density” of this particle is, i.e., the distribution function of single particle, )]([)]([),,( ttxxxtN ii
For particles in a plasma, the microscopic distribution function is the summation
N
iii
m ttxxxtN1
)]([)]([),,(
6-dimensional phase spaceSix-dimensional phase space with coordinates axes (x,y,z) and (vx,vy,vz) and volume element ∆x∆v
x
v
∆x
∆v
volume element in phase space∆x ∆v
(x(t);v(t))All particles (i=1, N) have time-dependentposition xi(t) and velocity vi(t). The particle path at subsequent times is a curve in phase space.
Here xi(t) and vi(t) are the spatial and velocity trajectories as the particles move.
Klimontovich
19
Basic equations for particle simulationHere xi(t) and vi(t) are determined by the equations of motion and Maxwell equations,
)1
( Bc
Em
q
m
F
dt
di
i
i
i
ii
ii
dt
xd
t
E
cJ
cB
t
B
cE
B
E
m
m
14
1
0
4
s
iss
mss
m xxqdtxNqtx )(),,(),( 3
The microscopic sources are determined by
s
iiss
mss
m xxqdtxNqtxJ )(),,(),( 3
These equations above establish a complete kinetic description of a plasma, which involves all information of particle motion with the self-generated fields. This description for a plasma provide a basic idea to numerically simulate the behavior of plasma – particle simulationPIC (particle-in-cell) method: Dawson; Birdsall & Langdon; ….This simulation needs a large number of particles ~10e+9 to have good statistics of collective behavior, for example, to remove “noise” problem.
N
iii
m ttxxxtN1
)]([)]([),,( with
20
Klimontovich equation
The description above yields too much detailed information than we need for practical purposes. We need to reduce it so that we can obtain some physically measurable quantities like density, temperature in a plasma.To do so, it may be convenient to have a single evolution equation for the entire microscopic distribution. Such an equation can be obtained by calculating the total time derivative of microscopic distribution:
)]([)]([1
ttxxdt
d
xdt
xd
tdt
dNi
N
ii
m
By using relations: )()()()( yxygyxxg dx
df
df
dg
dx
xfdg
))((
0)]([)]([1
ttxxdt
d
xdt
xd
tdt
dNi
N
ii
iim
Inserting equation of motion, we have
0),(1
),(
m
mmmm N
xtBc
xtEm
q
x
N
dt
xd
t
N
Klimontovich equation
21
Properties of the Klimontovich equation
Klimontovich equation together with the Maxwell’s equation and the definitions for charge and current densities also provide an exact and complete description of the plasma dynamics!
Klimontovich equation actually incorporates all particle equations of motion into one equation since its “characteristic curves” in (t,x,v) phase space are the equations of motion.
Conservation of particles (continuity):
0),,(),,(),,(),,(
xtN
dt
dxtN
dt
xd
t
xtN
dt
xtdN mmx
mm
No creation or destruction of charged particles as they move their trajectories determined by electric and magnetic fields in a plasma!
22
Since the Klimontovich distribution is a distribution of delta functions, it still requires basically to follow all individual particles. This is not feasible in typical application even on modern supercomputers.
We need an average procedure to get a smooth version of microscopic distribution.– Rigorous way: ensemble averaging over infinite number of realizations (i.e., all possible states). This is related to the statistic mechanics with the concepts like “temperature”. – Simple and more physical way: averaging over a small volume x v∆ ∆ in 6-dimensional phase space.
Conditions for average procedure: •The box size should be much larger than the mean space of inter-particles in a plasma to include many particles so that the statistical fluctuation is small•The box size should be smaller than, or of order of the Deybe length so that the collective plasma response on the Debye length scale can be included.
Hence, n-1/3<< x<∆ λD
From Klimontovich equation to Plasma kinetic equation
23
The average distribution function of Nm will be defined as the number of particles in such a small 6-dimensional phase space box divided by the volume of the box from (x,v) to (x+∆x, v+∆v)
Averaging procedure
VV
m
VV
xn
s
xn
ms
dxd
Ndxd
x
nxtNxtf
DD
33
33
33 limlim3/13/1
),,(),,(
Define the fluctuation (deviation from the averaged level) of complete microscopic distribution function Nm from the averaged one fs, i.e.,
),,(),,(),,(
xtNxtfxtN ms
m
Similar separation for the fields
),,(),,(),,(
xtExtExtEm ),,(),,(),,(
xtBxtBxtBm
We have 0),,( xtN m
The average distribution function fs represents the smoothed properties of the plasma species for ∆x >λD; the microscopic distribution δNm represents the “discrete particle” effects of individual charged particles for n-1/3
<< x<∆ λD .
0),,( xtE 0),,(
xtB
∆x ∆vDis
trib
uti
on
fu
ncti
on
fs
Nm
24
Substituting these forms into the Klimontovich equation and averaging it using the procedure above, we obtain our fundamental plasma kinetic equation:
m
mmsmmss NB
cE
m
qfB
cE
m
q
x
f
t
f 11
Fundamental plasma kinetic equation
The left side describes collective effects in the plasma, i.e., the evolution of the smoothed, average distribution function in response to the smoothed, average electric and magnetic fields.
The right side represents the small two-particle correlations between discrete charged particles within about a Debye length of each other. In fact, the term on the right represents the collisional effects, i.e., Coulomb collision effects on the average distribution function fs. Similarly averaging the microscopic Maxwell equations and charge and current density sources, we obtain corresponding average equations that have no extra correlation terms.
25
Fokker-Planck equation or Boltzmann equationRewriting the right side of the fundamental kinetic equation as (∂fs/∂t)c, a collision operator on the average distribution function fs. We can have Fokker-Planck (FP) or Boltzmann equation
c
ssss
t
ffB
cE
m
q
x
f
t
f
)
1(
t
B
cEE
1 ;4
sss dtxfqtx 3),,(),(
s
ss dtxfqtxJ 3),,(),(
With corresponding averaged Maxwell equations and charge and current densities,
This is a set of fundamental equations that provide a complete kinetic description of a plasma. All terms in equation are expressed by smoothed, average quantities. The particle discreteness effects (correlations of particles due to their Coulomb interactions within a Debye sphere) in a plasma are included in the collsion operator on the right side of Boltzmann equation.
The form of the collision term on the right side depends on the nature of collisions:– Boltzmann equation: for hard collisions and localized in space and time. – FP equation: for collision through cumulative contribution of many small angle Coulomb scatterings.
t
E
cJ
cBB
14 ;0
26
Reduced forms of Boltzmann equation
Electrostatic kinetic equation:For low pressure plasmas where the plasma currents are negligible and the magnetic field is external and constant in time, we can use an electrostatic approximation for the electric field (E =- ), Boltzmann equation becomes electrostatic kinetic equation
c
ssss
t
ffB
cm
q
x
f
t
f
)
1(
Conservative form of Boltzmann equation: Since x and v are independent, and electric and magnetic are independent of v, we can have a conservative form because (in the absence of collisions) motion (of particles or along the characteristics) is incompressible in the six-dimensional phase space
c
sss
s
t
ffB
cE
m
qf
xt
f
)1
(
Homework: problem 4Derive this conservation form of Boltzmann equation.
27
Reduced forms of Boltzmann equation – Vlasov equation
For the fluctuation with short time scale in high temperature laboratory plasmas or space plasmas, the collision is typically small, i.e., ω>>ν , we have so-called Vlasov equation
0)1
(
sss f
Bc
Em
q
x
f
t
f
Properties of Vlasov equation
1. Due to no collision, the filamentary structures in Vlasov plasma can become more contorted as time evolution. Hence, Vlasov code can follow the distribution function in physics for long time only before the numerical problem occurs.
2. Due to no collision, Vlasov equation has no discrete particle correlation (Coulomb collision) effects in it, it is completely reversible (in time) and its solutions follow the collisionless single particle orbits in the six-dimensional phase space.
3. A Vlasov plasma is stable since the stable distribution with dfs/dε<0 minimizes the kinetic energy.
4. Any free energy related to dfs/dε<0 may drive collective instability, profile non-homogeneity; velocity anisotropies; flows such as beams and currents.
28
Reduced forms of Boltzmann equation (cont.) – gyro-averaged kinetic equations
In a magnetized plasma, many plasma phenomena involve processes which are slow compared to the gyrofrequency and which vary slowly in space compared to the Larmor radius of individual ions or electrons. That is, the fluctuations in plasma are characterized by longer spatial scale compared to the gyroradii (L>>ρ g) and by slow processes compared to the gyrofrequency (ω<<ω c).
Under these limitations, it is possible to do two approximations: 1. Average the Boltzmann equation over the gyromotion angle; 2. Expand the Boltzmann equation around the guiding center with a small gyroradius. Procedure to derive gyro-averaged kinetic equations1. Change the independent phase space variables from (x; v) to phase space variables
with guiding center coordinates, energy, magnetic moment, and gyro-phase angle, i.e., (xg ;ε; μ , φ)
2. Splitting the distribution function fs into gyro-phase independent part <fs>φ and dependent part fs-<fs>φ
3. Get gyro-averaged kinetic equations by gyro-averaging Boltzmann equation – So, the dimensionality in phase space is reduced!
29
Drift-kinetic equation: This is a form of Fokker-Planck (Boltzmann or Vlasov) equation, which describes the evolution of distribution function fs under conditions where it occurs slowly in time compared to the gyro-period and the gradually in space compared to the gyro-radius of particle orbits. Actually, this is an equation of fs at the guiding center position xg. In principle, we should transform the results back from guiding center to real space coordinates after solving it. However, this procedure is usually neglected since the gyroradius is small and the effect is ignorable. The conditions for applying this model are: ω<<ω c ; k┴ ρ g<<1
Gyro-kinetic equation: This equation is similar to drift-kinetic equation, but it can describe the significant change of electromagnetic field across a Larmor radius by averaging their effect over the Larmor orbit. The conditions for applying this model are: ω<<ω c ; k┴ ρ g~1
Two typical gyro-averaged kinetic equations:
J Wesson, TOKAMAK (second edition), 1997
30
Derivation of drift kinetic equation
Change variables from to with ),,;( gx
);(
x
B
mmxx gg 2
);(2
1 ;
22//
2
Re-write kinetic equation by using new variables
c
ssss
g
sgs
t
ff
dt
df
dt
df
dt
d
x
f
dt
xd
t
f
Define with
where is small quantity with Larmor radius order
sssss fffff~~
sf~
Assuming c
sccg
g Em
q
tk
L
~ ;~ ;~ ;~ //
cct
BEs Em
q
~ ;~~
0~
sf
31
Derivation of drift kinetic equation (cont.)
Substitute all relations into rewritten kinetic equation in guiding center coordinate and performing gyro-averaging, we can get
c
gsgs
g
gsD
gs
t
xtfxtf
dt
d
x
xtf
t
xtf ),,,(),,,(),,,()(
),,,(//
The change of total kinetic energy can be subject to the gain of energy of the guiding center in the electric field and the change of the perpendicular energy due to a change of the magnetic field
gs xdEq
BdBd /
t
B
B
m
dt
xdEq
dt
d gs
2
2
velocitydrift center guiding
])([2 4
2//
3
2
2////
BBBBq
mcBB
Bq
mc
B
BEc
B
B
B
B
dt
xd
ssD
g
onconservatimoment magnetic to due 0 dt
d
This equation is used in linear and nonlinear studies of low frequency and long wavelength instabilities, in neoclassical transport theory where the contribution from Larmor gyration is not so important.
32
Gyrokinetic equation Drift-kinetic equation with the lowest order is sufficient for most applications. However, like the guiding center orbits it is based on, it is incorrect at second orderin the small gyroradius expansion. More precise and complete equation is gyrokinetic equation.In deriving gyrokinetic equation, we can still do gyro-averaging over gyro-phase angle. Instead of the assumption in drift kinetic equation, we have another small quantity where L is the equilibrium perpendicular gradient scale length. The distribution function is expended as
1 gk 1 Lg
10 sss fff
Writing the perturbed quantity as and the perturbed electric field is the perturbed distribution function can be obtained by expanding the linearized kinetic equation for isotropic f0,
)exp(~ xkti
AiE
iLss exg
fqf ),,(0
1
c
kbL
g satisfies the gyrokinetic equation
//1////0
00//
2)(ˆ B
kJ
kAq
kJ
kfbfgkigb
t
g
c
cs
ccg
In long wavelength limit and L→0, the distribution function is reduced to the result from drift kinetic equation.
J Wesson, TOKAMAK (second edition), 1997
33
References for the derivation of nonlinear gyrokinetic equation (classical and modern gyrokinetic theories, collected by T S Hahm)
1. Hazeltine and Meiss, Plasma confinement (book)
2. Frieman and Chen, Phys. Fluids 25, 502 (1982)
3. Lee, Phys. Fluids 26, 556 (1983)
4. Dubin, Krommes, Oberman, and Lee, Phys. Fluids 26, 3524
(1983)
5. Hagan and Frieman, Phys. Fluids 28, 2641 (1985)
6. Hahm, Lee, and Brizard, Phys. Fluids 31, 1940 (1988)
7. Hahm, Phys. Fluids 31, 2670 (1988)
8. Brizard, J. Plasma Phys. 41, 541 (1989)
9. Brizard, Phys. Plasmas 2, 459 (1995)
10.Hahm, Phys. Plasmas 3, 4658 (1996)
11.Brizard, Phys. Plasmas 7, 4816 (2000)
12.Sugama, Phys. Plasmas 7, 466 (2000)
13.Brizard, Phys. Plasmas 7, 3238 (2000)
14.Wang, Phys. Rev. E. 64, 056404 (2001)
15.Qin and Tang, Phys. Plasmas 11, 1052 (2004)
16.Brizard and Hahm, Foundations of nonlinear gyrokinetic
theory, Rev. Mod. Phys. 79, 1-468(2007)
34
Fluid description of plasma
35
Fluid description of plasma
Why fluid description:The single particle approach is rather complicated. We need a more statistical approach because we can’t follow each particle separately. If the evolution of distribution function in velocity space is important we have to use the Boltzmann equation. It is a kind of particle conservation equation.For many plasma applications, fluid moment (density, flow velocity, temperature) descriptions of a charged particle species in a plasma are sufficient.Advantages of fluid description:Fluid equations essentially involve 3 dimensions in geometric space. This advantage is especially important in computer simulations.Fluid description is explicit to understand the significance of fluid quantities such as density and temperature. Fluid variables are macroscopically measurable quantities in experiments. Microscopic approach is mathematically difficult and often not useful to follow the evolution of macroscopic variablesOmit some important physical processes (but describe others); Provide tractable approaches to many problems.
‘Fluid Description’ refers to simplified treatment of plasma which does not need the details of velocity dependence.
36
Fluid equations for a plasmaFluid equations are probably the most widely used equations for the description of inhomogeneous plasmas.
Two ways to derive fluid equations: 1. Derive the moment equations of the Boltzmann equation or Vlasov equation;2. Derive them by using properties like the conservation of mass, momentum, and energy of the fluid.
Definition of fluid moments
Define the 0th; 1st; 2nd moment of the integral over the distribution function fs as mass density ρ s; fluid bulk velocity vs; and pressure tensor π s
),,(),( 3
xtfdmxt sss
),,(
1),( 3
xtfd
nxtu s
ss
),,())((),(),(),( 3
���xtfuudmxtIxtpxtP sssss
All integrals are finite because the distribution function must fall off sufficiently rapidly with speed so that these low order, physical moments (such as the energy in the species) are finite. That is, we cannot have large numbers of particles at arbitrarily high energy because the energy in the species would be unrealistically large or divergent.
),,(
2),(
23
xtfdmxtq sss
37
Basic procedure to derive moment equations
Starting from Boltzmann (or Vlasov) equation and taking its nth moment (1; msv; msv2/2; …) by integrating over velocity space
c
snsssn
t
fd
fB
cE
m
q
x
f
t
fd
33 )1
(
Macroscopic quantities from fluid momentNumber density
Charge density
Momentum density
Current density
Scalar pressure
sss mxtn ),(
sssc nqxt ),(,
sss uxtP
),(
sss uqxtj
),(
)(),( ss PTrxtp�
Heat flux ),( xtqs
Temperature sss nTrxtT )(),( �
Calculating all integrations:
38
0th moment equation— continuity equation
c
ssss
t
fd
fB
cE
m
q
x
f
t
fd
33 )
1(
Considering the integration of distribution function over whole velocity space is the density, we integrate Boltzmann equation over velocity space (0th moment)
t
nfd
tt
fd s
ss
33
ssssss nufdffdfdx
fd
3333 )(
0,,,,
233
zyxi
si
i
zyxisiss
s fF
fFdfFfFdf
Fd i
i
force. Lorentzfor 0 function; ondistributifor 0 used have we Here,
i
is
Ff
i
Performing the integrations as follows
39
0th moment equation— continuity equation (cont.)
The right side becomes a source term Qn of particle number density due to collision, such as the production or annihilation of mass through ionization or recombination.
nss Qnut
n
Continuity equations for charge or mass densities can be obtained by multiplying number density equation by qs or ms, respectively.
with
utdt
dQun
dt
dnns
s By using full derivative, we have
For incompressible fluid, 0 u
40
1st moment equation— equation of motion
，
c
ss
ssss t
fmd
fB
cE
m
q
x
f
t
fmd
33 )1
(
sss
s untt
fmd
3
Performing the integrations by parts and using the properties of distribution function
)(
)())((
])][()[(
3
3
33
uunmP
uunmfuumd
fuuuumd
fmdfmd
ss
ssss
ss
ssss
�
0)( and
; ofindendent are and 0)1
(
BBB
BEBc
E
By using this relation
41
1st moment equation— equation of motion (cont.)
RBuc
EqnuunmPunt
m sssssss
)1
()(�
ss
ssi
kj ss
siskj
ss
sss
s
ss
ss
ss
s
s
ss
nBuc
Eq
nBuc
EqfBc
Em
qmdd
fBc
Em
qmdfB
cE
m
qmd
fBc
Em
qmd
fB
cE
m
qmd
zyx
)1
(
)1
()1
(
)1
()1
(
)1
()1
(
,,,
33
33
So, the momentum equation is yielded as
sc
ss R
t
fmd
3
By using continuity equation, we can get equation of motion
RBuc
EqnPuut
nmudt
dnm sssssssss
)1
(�
If the collisions are frequent enough, the pressure tensor becomes diagonal, or even isotropic, sopP
�
42
2nd moment equation— energy equation
c
ssssss
t
fmd
fB
cE
m
q
x
f
t
fmd
2)
1(
2
23
23
2
23
23
32
3
2
3
)()(2
)()(2
)()(22
umnp
fuuudt
m
fuuudt
m
fuuuudt
m
t
fmd
sss
ss
ss
ssss
ssssssss qPuuKmnQ�
2
2
1
2222
2
3
22
1ssss
ss umnpuw
mmK
))(()(2
1uuumnq sss
sAfdA 3Letting
The first term is
43
ssss
sss
sss
ss
ss
ssss
qPuuumnpx
qPufudm
xfwuwuwfu
xd
m
uwfuwx
dm
fuux
dm
fx
dm
x
fmd
�
�
2
2323
23
23232
3
2
3
2)()((
2
)(2
)(222
The second term is
uEnq
B
Bc
EfqdfBc
Eqd
fBc
Em
qmd
fB
cE
m
qmd
ss
ssss
ss
sss
s
ss
0)( parts by nintegratio to due zero
)1
(2
1)
1(
2
1
)1
(2
)1
(2
2
2323
23
23
2nd moment equation— energy equation (cont.)
The third term is
44
2nd moment equation— energy equation (cont.)
csssssssxsss QuEnqqPuuumnpumnpt
�
22
2
3
2
3
Finally we can get the energy equation
Qcs indicates the energy exchange through collision. From this equation, you can derive a temperature equation through p=nT by using the equations of continuity and motion.
Using continuity equation and momentum equation to remove the term ,
3
5 with )()(
1
1
csxsxs QquPupt
p �
tumn ss )( 2
45
Chain of moment equations
This equation chain must be truncated at somewhere and by some way. It is often made in the second order in many practical cases, either by neglecting the heat flux, or by using an equation of state instead of the energy equation. Here physical insight plays a crucial role. The treatment seems become a kind of art!
This procedure shows that low order moment equation includes higher moment, which is an infinite chain of hierarchy!
3rd moment equation— heat flux equation4th moment equation……To infinite
Similar way to derive high order equations
46
Closure momentsThe general procedure to close a hierarchy of fluid moment equations is to derive the needed closure moments, which are sometimes called constitutive relations, from integrals of the kinetic distribution function for higher order moments. The distribution function must be solved from a kinetic equation that takes account of the evolution of the lower order fluid moments. The resultant kinetic equation and procedure for determining the distribution function and closure moments are known as the Chapman-Enskog approach. For situations where collisional effects are dominant, the resultant kinetic equation can be solved asymptotically via an ordering scheme and the closure moments. This approach has been developed in detail for a collisional, magnetized plasma by Braginskii.
Chapman and Cowling, The Mathematical Theory of Non-Uniform Gases (1952).S.I. Braginskii, Transport Processes in a Plasma, in Reviews of Plasma Physics, M.A. Leontovich, Ed. (Consultants Bureau, New York, 1965), Vol. 1, p. 205.
For 3-moment fluid equations, in a Coulomb-collision-dominated plasma, the heat flux induced by a temperature gradient is usually determined by the microscopic collisional diffusion processTnq
In magnetized plasmas, the heat diffusion coefficients along perpendicular and parallel directions are very different, so it is separated as
//////// with and TnqTnq
47
Summary of moment equations
csssssssxsss QuEnqqPuuumnpumnpt
�
22
2
3
2
3Energy equation
cssssssssss RBuc
EqnPuut
nmudt
dnm
)1
(�Equation of motion
nss Qnut
n
Continuity equation
To the third moment, we have an unknown quantity, heat flux, which is the fourth order moment.
),,(
2),(
23
xtfdmxtq sss
In deriving these equations, we have ignored the details of treating with collision, which is important in plasma as a fluid. Fluid theory is valid when the phenomena of interest vary on a hydrodynamics scale length much larger than the fluid element: LH >> dr. i.e., slow variation of plasma phenomena.
In the limits of high density and lower temperature, the collision is high, the fluid theory is valid. But, a plasma is often described as a fluid even when it is far from being collision dominated !!! This condition means that the effects of collisions is negligible compared with the coherence produced by the self-consistent fields.
→ closure approximation
48
Why kinetic? Why fluid?
Plasma fluid theory is relatively simple and fluid quantities are measurable experimentally. Plasma fluid theory can describe most of basic plasma phenomena. For example, drift waves; cold plasma waves; MHD fluctuations; …... The advantage of fluid theory lies in the fact that the dynamics of neutral fluid has been extensively studied and many aspects of their behaviors are well understood. Although the motion of plasma fluid is much more complex than that in the neutral fluid, it is often useful to be able to draw analogies with the behavior of a plasma. From the viewpoint of calculation (simulation), fluid codes require relatively less CPU time compared to kinetic simulation (PIC or Vlasov codes). Kinetic description is essentially necessary for some plasma phenomena typically such as Landau damping process. For example, dispersion relation of two-stream instability,
For the first principal simulation, kinetic (or reduced kinetic) theory should be employed.
)()( with
01
22110
00
0
2
ununf
f
k
if
k
dP
km
e
kxxx
x
sx
49
Fluid/kinetic hybrid model – a mixed description
Plasma phenomena are characterized by a multiple space and time scales, primarily due to the different responses of electrons and ions to electric and magnetic fields.
Generally speaking, the fast varying and small scale physics phenomena require kinetic descriptions, slow varying and large scale processes can be described by more fluid models.
Some particularly interested processes occur on some of these scales but other processes occur usually. This can be described by a mixed kinetic/fluid model.
Hybrid model describes this plasma system by using kinetic model for one species (or part of one species) and by using fluid model for the rest.
The hybrid codes are defined as those numerical algorithms in which PIC particle or Vlasov codes are applied for the species treated by kinetic description and fluid code is for the species treated as a fluid.
D Winske, Space Science Review 42, 53-65 (1985); Computer space plasma physics (book) (1993)
50
Examples of fluid/kinetic hybrid model
Various types of hybrid codes depend on the problems. Some examples:
1. The interaction of a small, cold electron beam (kinetic) with a hot background electron population (fluid) because the unstable waves generated by the presence of the beam strongly affect it. (O'Neil et al., 1971)
2. Fast ions or electrons (kinetic) and background plasmas in magnetic fusion plasmas with various heatings or energetic Alpha particles.
3. Foreshock: it is characterized by particles (kinetic) that are leaked or reflected from the shock which stream back into the solar wind.
4. ……
51
Example for equations of hybrid model
t
B
cEE
1 ;4
3),,(),( dtxfqtx iii
3),,(),(),( dtxfqtxuqtxJ iiiii
t
E
cJ
cBB
14 ;0
As an example in the case with kinetic ions and fluid electrons, we have equations
Maxwell equations for electric and magnetic fields
eeii nqnq
eeiiei uquqJJJ
Kinetic ion equations
c
ssii
t
ffB
cE
m
q
x
f
t
f
)
1(
Fluid electron equations
ceeeeeeeeee RBuc
EqnPuut
nmudt
dnm
)1
(�
nee Qnut
n
This fluid equation chains should be truncated properly.
Particle PIC code or Vlasov code for simulation
Fluid code
52
Gyrofluid model – an approximately mixed description
One obtains a hierarchy of evolution equations for gyrocenter-fluid moments, i.e., for density, parallel velocity, pressure, etc. To obtain a closed set of these gyrofluid equations, one needs to invoke a closure approximation i.e., expressions for higher-order fluid moments in terms of lower-order fluid moments.
In the simulation community, the so-called Landau-closure approach that places emphasis on accurate linear Landau damping and the linear growth rate has been most widely adopted. In this approach, some kinetic effects such as linear Landau damping and a limited form of nonlinear Landau damping have been successfully incorporated in gyrofluid models.
To date, gyrofluid model still loses to involve properly some important physics processes. For example, it cannot describe the zonal-flow damping accurately and may overestimate the turbulence level in fusion plasmas.
Gyrofluid model is one kind of fluid model which includes some kinetic effects such as more accurate Larmor radius effects and Landau damping effects. This model uses a cleaver closure of the fluid moments of the gyro-kinetic equations that gave an excellent approximation to the kinetic effect of both Landau damping and finite Larmor radius. The case with ignoring Larmor radius effect is called as Landau fluid model.
53
Example for deriving a set of gyrofluid equationsHammett & Perkins, PRL 1990
Basic steps to get a Landau-fluid equation:1. Deriving a set of fluid moment equations from 1D Vlasov equation
0
sz
fE
m
q
z
f
t
f
z
q
z
uSpup
zt
p
))(1()(
z
SqnE
z
pumnu
zmnu
t
)()(
0
unt
n
2. Calculating linear response for electrostatic fluctuation from Vlasov equation
k
fdk
T
qR
T
qnfdn t
02
000
~)(
~~~
),,(~
)(0 ztfff zE
)exp(~ tiikz For fluctuation with form
High order quantity: Heat flux q; Two introduced undetermined quantities: dissipative momentum flux S; adjustable ratio of specific heats Г
Here
54
Example for deriving a set of gyrofluid equations (cont.)
Hammett & Perkins, PRL 1990
3. Expressing undetermined quantities q and S by lower moments
z
Tnq
z
umnS
4. Calculating the linear response R3 from the fluid equations with free constants χ ,μ ,Г );;,(33 RR5. Comparing the linear response functions from kinetic and fluid equations through asymptotic analysis, choosing the free parameters with the most approximate results. In some case, the higher moment equations are required for more accurate results.
Normalized response functions from exact kinetic model (solid); 3-moment (dashed) and 4-moment Landau fluid models.
The calculation to get a satisfactory gyrofluid model (finite Larmor radius effects and Landau damping) is much more complicated, sometimes up to 6- or 8-moment equations.
55
Simplified two-fluid equationsWe have derived fluid moment equations for each species. In the case without ionization and energy exchange collisions, it can be written down a set of simplified two-fluid equations for ions and electrons.
ciixiiiixi QquPupt
p
�
)()(1
1
niiii Qunt
n
iiiiiiiiiiii RBuc
EnquunmPunmt
)1
()()(�
ceexeeeexe QquPupt
p
�
)()(1
1
neeee Qunt
n
eeeeeeeeeeee RBuc
EnquunmPunmt
)1
()()(�
With friction term )( eieeeie uunmR
0 ie RR
eeiceci RuuQQ
)( 35
In some cases, it is also assumed that the pressure is scalar. The heat conduction term is neglected. The separate two (ions and electrons) fluids interact through collisions and through electromagnetic interaction.
56
Two-fluid equilibrium – diamagnetic drift and diamagnetic current
0 t 0 eizBB
At equilibrium, we have and collisionless limit
In slab geometry with straight B-field , assuming equilibrium fields and profile vary only in x, the momentum equation is. 0)()( BuEnquunmp ssssssss
From the equation in y direction, we have is a solution, we can have the equilibrium equation
0sxu
dx
dpBu
cEnqu
dx
dunm syxsssxsxss )
1(
sxss
sysxss Buc
nqu
dx
dunm 0
1
dx
dpBu
cEnq syxss
dx
dp
BnqB
Eu
ss
xsy
1
22 Bnq
pBc
B
BEcu
ssD
Fluid drifts: Second term is diamagnetic drift
Diamagnetic current:
222
)()()(
B
ppBc
B
ppBcnqnq
B
BEcunqunqj eiei
eeiieeeiii
Quasi-neutrality diamagnetic current
57
Single fluid model
Two fluid equations contain still considerable complexity which is not needed for many plasma systems. It is desirable to formulate a more appropriate set of equations which include most of the macroscopic properties of a plasma. This set of equations are so-called magnetohydrodynamics (MHD) equations. This model consists of a one-fluid model for the plasma and the Maxwell equations for the electromagnetic fields. In a charge neutral plasma, qi=-qe=e, define the following quantities for single fluid model
Mass density:
Mass flux velocity:
Total current density:
Total pressure: ei
ieeeeeeiii
meeeiii
eeiim
ppp
uunqunqunqj
umnumnu
mnmn
)(
/)(
It can be seen that
uqn
j
m
muu
eei
ei
eee qn
juu
58
Single fluid model – mass density equation
Multiply the electron and ion density equations by their respective masses and combine together to get a single fluid mass density (continuity) equation
Similarly, multiply the density equations by their respective charges and combine to yield the single fluid charge continuity equation.
0)(
ut mm
00 neutralityquasi jjt q
MHD plasmas are quasineutral and have no net charge density, we can not calculate the electric field from the Gauss' law Maxwell equation . However, since a plasma is a highly polarizable medium, the electric field E in MHD can be determined self-consistently from Ohm's law, Ampere's law and the charge continuity equation.
eeii nqnqE
59
Single fluid model – equation of single fluid motion
Adding the electron and ion momentum equations to derive a one-fluid momentum equation (equation of motion) for a plasma
Bjc
pjj
q
mmuu
t
u
me
eim
m�
1
)( 2
0)(1
)()()()(
Bunqunqc
EnqnquunmuunmPPunmunmt eeeiiieeiieeeeiiiieieeeiii
��
me
eieeii
ee
eeeii
ee
e
e
e
eei
e
e
eeeii
eeeeee
eei
e
eei
eiieeeeiiii
jj
q
mmuunmnmjj
qn
muunmnm
jjqn
mjuuj
q
mjj
qnm
mjuuj
q
muunmnm
qn
ju
qn
junm
qn
j
m
mu
qn
j
m
munmuunmuunm
22
22
2
)()(
)()()(
Using density equation and quasi-neutrality 0 j
pBjc
uutm
�
1
60
Single fluid model – generalized Ohm’s lawMultiplying the equation of ion or electron motion by qi/mi or qe/me and sum them, we can get generalized Ohm’s law.
Electron inertia term→0
ee
ei
i
ie
e
eei
i
ii
e
ee
i
ii
eeeeiiiiee
ei
i
ieeeiii
Rm
qR
m
qBu
cm
nqu
cm
nqE
m
nq
m
nq
uunquunqPm
qP
m
qunqunq
t
��
2222
)()(
jmq
mqBj
cp
m
qBu
cE
m
nq
nq
jjujju
t
j
ii
eeeie
e
e
e
ee
ee
1112
jjnq
mBj
cp
q
mBu
cE
nq
jjujju
t
j
q
mm
ee
eiee
me
i
eeme
ie
22
11
jBjc
pq
mBu
cE
nq
jjujju
t
j
q
mme
me
i
eeme
ie
11
2
Hall term<<1
Pressure force<<1
Generalized Ohm’s law
61
Single fluid model – generalized Ohm’s law (cont.)
In generalized Ohm’s law, the term proportional to me/mi should be small, the second order term is usually neglected. For slow temporal and large spatial scales, ignoring the electron inertia, hall effect and pressure, simplified Ohm’s law
jBuc
E
1
If the resistivity is small, we can get ideal Ohm’s law
01
Buc
E
In many space plasmas, the Hall term and the pressure term become more important, namely, the Hall MHD.
Some microscopic mechanisms, which are not due to actual inter-particle collisions, may lead to effective resistivity or viscosity at the macroscopic level. Various wave-particle interactions and microscopic instabilities tend to inhibit the current flow. The macroscopic effect of these processes looks analogous large and it is called anomalous resistivity.
62
Single fluid model – pressure equation (or state equation)
cecieixeeiieeiix QQqqupupupupt
p
)()(
1
1
Add ion and electron pressure equation, we have
2)()(1
1jqqupup
t
peixx
Like the two-fluid equations, the pressure equation depends on next order fluid moment. We can not obtain a closed set of MHD equations without some further approximations. One approximate treatment is to roughly drop out the next order term
mCp
The chain is also often cut by employing an equation of state under an adiabatic process.
2)(1
1jupup
t
px
In a 3-dimensional plasma, the specific heat ratio γ=5/3; for an isothermal process, γ=1; In the case with constant pressure, γ=0; for an isometric process, γ→∞, i.e. p→0 in a very low β plasma.
63
MHD equations
0)(
ut mm
pBjc
uutm
1
2)(1
1jupup
t
px
jBuc
E
1
Et
B
c
1
jc
B 4
These MHD equations can describes many physical processes in small gyroradius,magnetized plasmas — macroscopic plasma equilibrium and instabilities, Alfven waves, magnetic field diffusion. It is the simplest, lowest order model used in analyzing magnetized plasmas.Used approximations used in MHD equations:1. Exclude single particle effects such as gyro motion;2. Exact charge neutrality for low frequency phenomena much smaller than plasma frequency;3. The displacement current has been neglected due to the slow propagation of electromagnetic waves;4. Assuming isotropic pressure near thermal equilibrium;5. Different approximation can be taken in generalized Ohm’s law.
0 B
64
Properties of MHD equations1. Conservation laws of ideal MHD system mass; momentum and energy
0)(
ut mm
0412812
222
BEc
upucBpu
tmm
MHD energy density MHD energy flux
Kinetic energy
Internal energy
0 04
1
8
1)( 2
Tut
BBIBpuuut mmm
��
MHD stress tensorReynold stress
Isotopic pressure
Expansion and tension of magnetic field
Magnetic energy
Kinetic energy flux
Internal energy flux
electromagnetic field energy flux
0
Swt
�
From energy conservation, it can be seen that in an isolated system, total energy is constant while exchange between kinetic; internal and magnetic energy
constant812
223
BV pk
m WWWcBpu
xd
(Using Gauss’ theorem)
65
244 )()()()()(
2BBB
cBB
cBjababbababa
In deriving MHD momentum conservation, we used
In deriving MHD energy conservation, we used
)]([2
)()()(
)()()( ;)(2
BuBt
BBuBBBu
baabbaBEEBBuB
From momentum equation,
puBBuuuuut
u mm
)(4
1)(
puBuBt
Bcuuuu
uu
t mmm
)]([4
1
8)()(
22
222
puBuBuucBu
tmm
)]([4
1
282
222
Adding this equation with the energy equation,
puBuBuu
upuppcBu
tmm
)]([
4
1
2)(
1
1
182
222
)(42
)(1182
222
BEcuu
uppcBu
tmm
Properties of MHD equations (cont.) – Details of deriving conservation laws
66
Properties of MHD equations (cont.) 2. Ideal MHD frozen flux theorem
In an ideal MHD, since the resistivity disappears, , it can be proved that the magnetic flux Φ through every surface moving with the plasma is constant, i.e.
0 cBuE
0ˆ
CSSsdBSdB
tSdB
dt
d
dt
d
In ideal MHD, plasma carries the magnetic field to move, namely, it always contain the same amount of magnetic flux. We may simply understand “MHD frozen flux” as that two fluid elements are always connected by a magnetic field line if they were connected at one time by a field line, which is defined by the direction of the magnetic field at any moment in time. In other word, fluid elements flow freely along the line of magnetic force.
4. Entropy conservation The entropy is defined as , the overall entropy production rate is
)ln( mps
0
uuut
putdt
d
dt
dpp
dt
d
dt
dsm
mm
3. Magnetic diffusion When the fluid velocity is zero, from Ohm’s law we have BtB 24
The magnetic field changes by diffusing through the electrically conducting fluid, the change rate depends on the plasma characteristics. The field decays in a characteristic time with magnetic gradient length L.
4~ 2L
67
MHD equilibrium
In a plasma equilibrium, ∂/∂t ~ 0, u=0, MHD equations consist of force balance and Maxwell equations
pBjc
1 j
cB
4 0 B
Taking the scalar product of the force equilibrium equation with B and j, we have
0 pjpB
It indicates the pressure is constant on magnetic field lines and on current lines.The force equilibrium equation can be written as
048
)(4
11 2
BBB
pBBpBjc
p
08
2
B
pFor the case without field curvature
82B
p
Plasma internal energy
Magnetic energy
We can get many different equilibrium configurations in magnetic fusion:Tokamak, stellarator (helical system), Mirror; Θ-pinch, Z-pinch, …..
68
Reduced MHD equations
In space plasmas and also in magnetic fusion plasmas, MHD can be reduced further.
H R strauss, Phys. Fluids 19, 134(1976)R. Fitzpatrick, Phys. Plasmas 11, 937(2003)
Including the effects of electron viscosity, the equation of motion and Ohm’s law become
jBjc
pq
mjBu
cE e
me
i
211
uBjc
puut
umm
21
tEzuzuzBzB zzz ;ˆˆ ;ˆˆ
Define Ψ is parallel vector potential; stream function Φ, let ∂/∂z=0, and constant density, we can write
zzzmz
m uBut
u 2],[4
1],[
)]motion of equation(ˆ[ z
)motion of equation(ˆ z
zzzz
m jct
2],[1
],[
69
Reduced MHD equations (cont.) – two-field MHD
zzzz
m jct
2],[1
],[
42],[ t
When the parallel magnetic field is very strong, the perpendicular gradient is small, we get reduced two-field MHD equations
These equations are usually used to study the tearing mode and the magnetic reconnection in space plasmas and magnetic fusion plasmas.
)law sOhm'(ˆ z
42],[4
],[
zme
i Bq
m
t
)]law sOhm'([ˆ zzzz
me
izz
z Bc
Bc
jq
muB
t
B 42
22
44],[],[],[
zbabaj zz ˆ],[ ; ; 22 Here we defined
70
Classification of equations (Elliptic type—Poisson)
This is an Elliptic-type equation
02
2
2
2
yx
42
We will summarize typical equations. From Maxwell equation, in electrostatic limit we can define , so the Gauss’ law become Poisson equation
E
42
2
2
2
yxPerpendicular to magnetic
field
In plasma physics, it is often to calculate Poisson equation to solve electric field. It has been paid much attention to look for good Poisson solver in simulation.
71
Classification of equations (hyperbolic type—wave; convection)
02
2
2
2
x
u
t
u
),(),(),( trAtc
trtrE
),(),( trAtrB
In this case, Maxwell equation can written as electromagnetic field equations are written in the form
),(41 2
2
2
trtc
),(41 2
2
2
trJc
At
A
c
With Lorentz gauge 01
Atc
Sometimes E & B are expressed in terms of an electric potential φ and vector potential A:
It can be clearly seen that they are hyperbolic-type equation (for example 1D case)
Furthermore, the fluid equation system is typical hyperbolic and the equation is of convection equation
0
xu
tmm
x
p
x
uu
t
umm
The solution of such convection problem is the propagation of initial profile F(0,x).
)(),( utxFxtf
72
Classification of equations (parabolic type— diffusion)
This is of Parabolic type such as
2
2
x
T
t
T
Another common equation describes diffusion process such heat diffusion, magnetic diffusion in MHD. For the latter, when the velocity of fluid is zero, the Ohm’s law becomes
2
2
4x
B
t
B
2)(1
1jqupup
t
px
In the energy equation in fluid equations, assuming density is constant and using p=nT, further assuming heat flux , hence, we have
Tq
......2
2
x
T
x
Tu
t
T
When fluid velocity is zero, it is a diffusion (transport) problem
......2
2
x
T
t
T
When heat is isolated, it mainly describes a convection process
......
x
Tu
t
T
In general case, it describes the mixed processes of diffusion and convection.
73
Final remarks All physics system is described by equations. In magnetized plasmas, the equations are of linear second order partial difference equation in two independent variables x and y (perpendicular to magnetic field). These equations may be written as a most general form 02
22
2
2
GFuy
uE
x
uD
y
uC
yx
uB
x
uA
Here the coefficients are assumed as constant. In the simplest case with D=E=F=G =0, define characteristic curves
04 when type" hyperbolic"
04 when type" parabolic"
04 when type" elliptice"
2
2
2
ACB
ACB
ACB
Through some transformations, the second order partial difference equation can be classified as
yxyx ;
We have characteristic value condition
0 ;0 22 CBACBA
T. Tajima, computational plasma physics, Chpt. 5
Most equations in plasmas are of mixing type depending on the physics processes such as propagation; dissipation. We will introduce numerical methods to solve such equations involving time advance and spatial discretization.