Download Plasma Physics and Numerical Simulations

Plasma Physics and Numerical Simulations
W. J. Miloch
[email protected]
November 17, 2014
1
Definition of Plasma
Plasma is a partially or fully ionized gas. Ionized gas contains free electrons and ions which carry electric
charges. Thus, plasma is subject to electromagnetic forces and can react to external electric and magnetic
fields. Moreover, since plasma contains free charge carriers, their relative motion can also set internal
electric and magnetic fields, which in turn also influence their dynamics. In addition, plasma is subject
to other forces typical for gases, such as gravity or pressure gradient. Thus, the plasma dynamics is
complex and gives rise to a large variety of dynamic phenomena.
The gas gets ionized when the temperature is high enough to allow for ionization. Furthermore, to
consider the plasma particle to be a free charge carrier, its interaction energy with other plasma particles
should be smaller than its thermal energy. Thus most of plasmas are very hot, with thermal energies
above electronvolts, to provide ionizaton and prevent recombination 1 2 .
An important property of plasma is quasineutrality. Quasineutrality means that at large scales there
is the same number of negative and positive charges and the net charge is zero, and that small-scale
deviations from neutrality will be quickly restored by plasma. The plasma particles (i.e., electrons and
positive and negative ions) are mobile, and their relative motion gives rise to charge separation setting
up electric fields which can set other plasma particles into motion. Even though there can be a local
charge imbalance resulting in, for instance, plasma waves and oscillations, at large scales the system
remains neutral.
Quasineutrality is related to another basic property of plasma which is the Debye shielding. The
electric charge of individual plasma particle is collectively screened by other plasma particles. A positive
ion will attract electrons as well as negative ions which will form a ”cloud” around it, so that the electric
field due to the positive ion is reduced as compared to the vacuum. Similarly, the negative charges
will be shielded by positive ions (and will repel other electrons). This collective behavior is one of the
fundamental properties of plasma.
p The characteristic screening length is called the Debye length and
is given for electrons by: λDe = 0 kTe /e2 n0 , where 0 is the permittivity of vacuum, k - Boltzmann
constant, Te - electron temperature, e - electron charge, n0 - plasma density3 . The Debye length refers
1 Recombination
is a process in which ions capture free electrons and form neutral atoms.
one can encounter term cold plasma, which usually refers to plasma that is only weakly ionized (about 1%
of the gas), but even in cold plasmas the temperatures are typically few thousands Kelvins.
3 Debye shielding can be defined for each plasma species (i.e., electrons and different kind of ions); the total Debye length
P −1
in plasma including all species is given by λ−1
α λD,α , where the sum is over all plasma species α.
D =
2 Sometimes
1
to a distance over which the electric potential of charge q (resulting in potential Φ = 4πq 0 r in vacuum)
will be shielded in plasma by an exponential factor: Φ = 4πq 0 r exp(−r/λDe ).
Debye shielding has an important consequence: for the shielding to occur the number of particles in
the shielding cloud must be large enough. Otherwise there will be little collective effects, and one could
only talk about individual charged particles. If the Debye sphere can be defined as 43 πλ3De , then the
number of particles in the sphere is given by Np = n0 34 πλ3De . To treat particles as smooth continuum
we need to have Np 1. Np is often referred to as the plasma parameter.
To summarize:
Plasma is a partially ionized gas where collective effects dominate over collisions, i.e., Np is large. Plasma is quasineutral at large scales, thus the extent
of plasma is much larger than the Debye length. Plasma particles are subject
to electro-magnetic forces and other forces characteristic for fluids.
2
Examples of Plasma
Plasma constitutes about 99% of the visible universe4 .
For its abundance and distinct properties it is often
called the fourth state of matter in addition to solid,
liquid and gaseous states.
The wide range and types of naturally occurring and
laboratory plasmas are schematically shown with respect
to density and temperatures in Figure 1. Typical temperatures span over eight orders of magnitude while densities over 25 orders of magnitude.
In nature the least dense plasmas are found in the
solar wind (i.e., the plasma particles streaming from the
Sun towards the Earth), the Earth magnetosphere, and
distant nebulae. The hottest and densest plasmas are
found inside stars, where temperatures are so high that
allow for thermonuclear fusion reactions releasing huge
amounts of energy. Another rather dense plasmas are
produced in lightning or solar corona.
The Earth ionosphere, a part of the atmosphere
starting at about 90 km is ionized mainly due to solar
radiation. It is a relatively cold and low density plasma.
Ionosphere can couple to magnetospheric plasma, and in
particular in the northern regions this coupling is manifested by northern lights (aurora), where high energy
electrons excite the oxygen and nitrogen atoms, which
in turn emit light.
A similar process is found in fluorescent lamps, light
tubes, and neon lights, where the plasma is created in
a discharge. Plasma discharges are also widely used in
manufacturing, and so-called plasma processing is widely
used in nanotechnology, coating, or semi-conductor industry. They have also applications in medicine, such
as dentistry or healing complicated wounds. Finally,
plasma confinement devices, such as tokamaks (e.g.,
JET, ITER) or stellerators aim for controllable fusion
on Earth, which would provide mankind with the clean Figure 1:
Classification of plasmas with
energy for a foreseeable future, see Figure 2.
respect to density and temperature. Note
4 Note that the visible, or so-called baryonic matter constitute
swapped axis in both plots.
about 4.6% of the universe, while the other 23 % is attributed to
dark matter and 72 % to dark energy.
2
Figure 2: Illustration of the complex magnetic field configuration in the stellarator Wendelstein 7-X
(left), schematics of a tokamak (center), and the engineering plan of the ITER tokamak (right, note the
tiny human figure to the right showing the scale of ITER).
3
Equations for Plasma
Plasma particles are subject to electric and magnetic forces. They also need to follow the basic laws
for fluids. Depending on the level of interest, plasma can be studied at different levels of accuracy: by
following trajectories of individual particles, trajectories of ”small parcels” of plasma, or evolution of
velocity distributions.
3.1
Single Particle Motion
The single particle approach is the lowest level description of plasma. It is useful for determining the
trajectories (orbits) of individual particles in external forces. Usually in this approach the focus is on a
single or several particles, and it is difficult to say that this approach fulfills the plasma definition as it
usually does not include the collective effects. However, it is a useful way of understanding the plasma
dynamics in strong external fields. Each plasma particle is subjected to the Lorentz force:
~ + ~v × B),
~
F~ = q(E
(1)
~ = E(~
~ r, t) is the electric field, ~v = ~v (t) velocity of a single particle, B
~ = B(~
~ r, t) magnetic field,
where E
~
~
~a × b is the cross product for vectors ~a and b. In addition, the particles can experience other forces,
such as gravity F~ = m~g . The consequence of Eq. (1) and the cross product is
that the particle will rotate in the plane perpendicular to the magnetic field, and
that a time stationary magnetic field will not change the energy of particle5 . In
~ with constant
the absence of electric field, the particle will thus gyrate along B
gyroradius (Larmor radius) rg = mv⊥ /kqkB and gyrofrequency ωg = qB/m. The
particles will not travel across the magnetic field. However, they can move almost
freely along the magnetic field lines by following helicoidal orbits, see Figure 3.
The direct consequence of this constraint is that we observe northern lights at
high-lattidutes: because of the topology of the magnetic field of the Earth, high
energetic plasma particles can reach low altitudes in the polar regions and excite
neutral oxygen and nitrogen atoms that in turn emit light.
The particles in external fields can be subjected to many drift motions. One Figure 3: Example
~ ×B
~ drift, where all plasma species drift of helicoidal orbit for
of the most fundamental drifts is the E
~ × B/B
~ 2 . There are other drift an electron.
in the same direction with the drift velocity ~v = E
motions which depend on the field geometry and forces included; some of them
depend on the charge of the particle and can give rise to electric currents.
5 The time varying magnetic field will on the other hand change the energy of particle because it will change the electric
field; cf. Maxwell equations.
3
3.2
Fluid Approach
The electric and magnetic fields are related through the set of Maxwell equations:
~
∇×B
=
~
∇×E
=
~
∂E
µ0~j + 0 µ0
∂t
~
∂B
,
−
∂t
(2)
(3)
~ = µ0~j (relating the rotation of the magnetic field
where equation (2) is basically the Ampere law ∇ × B
~
to the current density j, where µ0 is permeability of vacuum) with the Maxwell displacement current.
The second equation (3) is the Faraday law, stating that time variations in the magnetic field give rise
to a rotating electric field. The following equations close the system of equations:
~
∇·B
=
~
∇·E
=
0
ξ
,
0
(4)
(5)
where the first states that the magnetic field is divergence free, thus there are no magnetic monopoles,
while the second is basically the Gauss law in its differential
Pform (the electric field through the closed
surface equals the charge in the enclosed volume; here ξ =
qα nα is the net charge density, where the
sum is over all plasma species α).
In the fluid approach, we are interested in the behavior of a small volume/parcel of plasma, for which
we have defined the temperature, density, and net velocity. This parcel contains a sufficient number
of plasma particles, so that the temperature and density are well defined. The fluid element follows
the basic fluid dynamics, but it also needs to account for Maxwell equations. Thus the following set of
equations need to be accounted for each plasma species α :
∂nα
+ ∇ · (nα~vα ) = 0
∂t
∂~vα
~ + ~v × B
~ − να,β nα mα (~vα − ~vβ )
mα nα
+ (~vα · ∇)~vα
= −∇pα + nα qα E
∂t
pα n−γ
α
=
const.
(6)
(7)
(8)
The velocity ~v refers now to the fluid element and can thus be position and time dependent ~v = ~v (~r, t).
Equation (6) is just an ordinary continuity equation; indeed plasma, just as any other fluid, needs to
conserve mass. In case of production or loss due to for instance ionization and recombination, relevant
terms should appear on the right hand side. The momentum equation (7) is the Navier-Stokes equation
that relates the force density acting on the plasma fluid element to the pressure gradient force, the
Lorentz force in its density equivalent, momentum exchange between different plasma species α and
β (where να,β is the collision frequency; collisions with other species are not considered here). It can
include other forces as well; for instance in solar plasmas the gravity force needs to be accounted for. The
momentum equation is coupled to the Maxwell equations, but the fields are on the other hand related to
density and velocity of the plasma (j~α = nα qα~vα and ξ = nα qα ), thus they need to be solved together.
To close the set of equations, the expression for pressure pα is given in the equation of state (8).
This is the basic set of equations that is applicable for medium and large scales in plasmas. The
basic assumption in this description is that plasma species have Maxwellian velocity distributions, which
allows to define the concept of temperature6 . It is thus applicable for time-scales which allow enough
collisions between plasma particles to thermalize the plasma.
This approach is often called two- or multi-fluid approach, as it can consider electron and ion species
separately. For each of the species we have interaction through collisional term in Eq. (7), as well as
through self-consistent electromagnetic fields. One can imagine that such a complex system gives rise to
a large variety of dynamical phenomena. Indeed, by linearizing equations, one can find many different
plasma wave modes, and their dispersion relations will in general depend on the orientation with respect
to the electric and magnetic fields.
6 Temperature
is related to the variance of the Maxwellian distribution function.
4
3.3
Magneto-hydrodynamics, MHD
When the time scales and spatial scales of interest are very large, it is possible to simplify the Maxwell
equations and consider plasma as a single fluid with a very high conductivity. Such an approach is called
magneto-hydrodynamics (MHD)7 .
The high conductivity implies that any charge separation will be instantaneously shortened due to
high, almost perfect conductivity. Thus, in the momentum equation the contribution from the electric
field can be neglected. With temporal analysis of Eq. (2) one can find that if the characteristic velocity
√
in the system is much smaller than the speed of light (c = 1/ µ0 0 ), it is reasonable to neglect the
~ = µ0~j. However,
Maxwell displacement current, which gives the Ampere law in its original form: ∇ × B
by neglecting the temporal derivative in Eq. (2), we are left with open set of equations, and we need a
~ To close the set of equations we can thus employ the Ohm law:
new relation for E.
~ + ~v × B
~
~j = σ E
(9)
Now we should recall that plasma is an almost perfect conductor. In the limit of the ideal conductor, we
~ = −~v × B,
~ which is
have the conductivity σ → ∞. To keep the currents finite we have to require that E
a very useful relation indicating that in the ideal MHD limit the electric fields in the plasma are induced
by the motion of plasma across magnetic field.
The assumptions and simplifications that we have made put restrictions on the applicability of the
MHD approach. The phenomena that are adequately described by MHD are the slow and large scale phenomena in terms of characteristic plasma frequencies and wavelengths. By neglecting the displacement
current, we are interested in the phenomena that are much slower as compared to the time variations
of electromagnetic perturbation having the same scale size propagating in free space. Considering slow
phenomena also assures that the plasma is Maxwellian. Obviously, the spatial scales considered are much
larger than λDe and other characteristic scales in plasma related to kinetic effects, and large enough that
plasma can be considered quasineutral. MHD approach is widely used in astrophysics, cosmology and
in studies of solar, space, and magnetosphere plasmas. It is also a common approach in fusion plasmas.
The MHD equations can be summarized as (now we use ρ for plasma density):
∂ρ
+ ∇ · (ρ~v ) =
∂t
∂~v
ρ
+ (~v · ∇)~v
=
∂t
~ =
∇×B
0
(10)
~
−∇p + ~j × B
(11)
µ0~j
(12)
~
∇×E
=
p
=
~j
=
~
∂B
∂t
F (ρ)
~ + ~v × B
~
σ E
~
∇·B
=
0
−
(13)
(14)
(15)
(16)
where the equation of state (14) is now written in a general form. For ideal MHD (σ → ∞), this set of
equations can be further rearranged as:
∂ρ
+ ∇ · (ρ~v ) =
∂t
∂~v
ρ
+ (~v · ∇)~v
=
∂t
~
∂B
=
∂t
p =
7 In
0
1 ~ ×B
~
−∇p +
∇×B
µ0
~
∇ × ~v × B
F (ρ)
the literature, the term magneto-hydrodynamics is sometimes also used for multi-fluid description.
5
(17)
(18)
(19)
(20)
One can notice that MHD equations basically relate the velocity field and the magnetic field. When
these two fields are related and in the limit of ideal MHD the frozen-in-concept applies. It states that a
small volume of plasma will have attached magnetic field to itself and will act as ”tracer” of the magnetic
field line. This concept is for example used to describe many magnetospheric phenomena.
3.3.1
Alfven waves
As an example of dynamic phenomena in plasma, we can linearize the set equations for ideal MHD, by
assuming a small perturbations to the background plasma. The unperturbed state is given by: ~v = 0,
~ =B
~ 0 = const. By assuming small perturbations, we then have, to
ρ = ρ0 = const, p = p0 = const, B
the first order fluctuations:
∂ρ1
∂t
∂ v~1
ρ0
∂t
~1
∂B
=
(21)
=
1 ~1 × B
~0
∇×B
−∇p1 +
µ0
~0
∇ × ~v1 × B
=
0
=
∂t
∇ · ~v1
0
(22)
(23)
(24)
where we account here for incompressible plasma by exchanging equation of state (20) by Eq. (24).
Assuming that plasma supports the wave like motion with variation exp(−i(ωt − ~k · ~r)) i.e., for quantity
a: a1 = a1,0 exp(−i(ωt − ~k · ~r)), we then have:
− iωρ0 v~1
=
~1
−iω B
=
i ~ ~ ~
k × B1 × B0
−i~kp1 +
µ0
~ 0 = i(~k · B
~ 0 )~v1 − i(~k · ~u1 )B
~0
i~k × ~v1 × B
i~k · ~v1
=
0
(25)
(26)
(27)
Note that ρ1 does not enter here, as from Eq. (21) we have that ρ1 is constant, and since it is a
deviation from equilibrium, it must be zero. These equations when combined give us the dispersion
relation containing famous Alfvén waves8 . The Alfvén waves are given by:
2
1 ~ ~ 2
k · B0 = VA2 ~k · b̂
(28)
ω2 =
µ0 ρ0
√
~ 0 . These are
where VA = B0 / µ0 ρ0 is the Alfvén speed, and b̂ is the unit vector in the direction of B
~
the transverse waves (~v1 ⊥ k) with the phase velocity ω/k = VA cos Θ, where Θ is the angle between
~ 0 . The
the wave vector and background magnetic field. However, the group velocity ∂ω/∂k is along B
Alfvén velocity characterizes the low frequency dynamics of many phenomena in astrophysical and magnetospherical plasmas and these waves have been observed by spacecrafts9 . It is interesting that in the
Alfvén waves the energy exchange is between the magnetic field and the kinetic particle motion. Plasma
can also support waves that are electromagnetic (energy transfer is between electric and magnetic fields),
and electrostatic waves (energy transfer is between particle motion and electric field). We derived here
the shear Alfvén waves for incompressible plasma. When allowing for compressibility, one can also find
the slow and fast magneto-acoustic modes.
3.4
Kinetic Approach
The MHD is useful approach for studying large scale and slow phenomena. However, many plasma
processes happen at small scales, where the collective motion of individual particles needs to be considered
in self-consistent fields. It is then necessary to consider the velocity-position phase space of the particles,
8 Hannes
Alfv́en was a Swedish physicist who received a Nobel prize in 1970 for his work on magnetohydrodynamics.
observations of Alfvén waves in space were by Pioneer and Explorer satellites in late 1950’s; and even earlier in
laboratory experiments in 1949.
9 First
6
and the spatio-temporal evolution of the particle distribution function f = f (~r, ~v , t). This function
depends in general on seven variables for each species. In the absence of production and loss of particles
and with no collisions, it is possible to write the generalized continuity equation in the phase-space as:
∂f
+ ∇{r,u} · (f U ) = 0,
(29)
∂t
where U = {vx , vy , vz , ax , ay , az } is a sort of ”super-velocity”, and ∇{r,u} is a six-dimensional gradient
operator acting on U . For incompressible plasma this equation can be re-written as:
F~
∂f
+ ~v · ∇f +
· ∇u f = 0,
(30)
∂t
m
which is known as the Vlasov equation. By including collisional term on the right hand side we will
recover the Boltzmann equation. The force field F~ can be given, as well as found self-consistently by
Maxwell equations.
The kinetic approach is mathematically advanced, and solving these equations is often challenging
and involved, and different species will have different evolution of velocity distributions. However, it is
the most detailed and most complete description of the plasma. For instance, the kinetic theory has
shown that the plasma waves can be damped without collisional interactions. This damping is called the
Landau damping. It is also worth noting that it is also possible to obtain the fluid equations starting
from the Vlasov equation.
4
Numerical Methods for Plasma
Theoretical description of plasma is nontrivial and to address many complex and nonlinear problems
one can consider using numerical simulations. In most of plasma systems, the number of particles
is very large, as it can be inferred from Figure 1. Ideally, we would like to study the first-principle
dynamics by following trajectories of all plasma particles, as described by the single particle motion, and
account for self-consistent electric and magnetic fields. The particle dynamics will affect the fields, and
thus these fields need to be updated during the simulation. Any charged particle interacts with all other
particles in the system, an if n is the number of plasma particles, the direct calculation of particle-particle
interactions has the complexity of O(n2 ), which is computationally very difficult (i.e., ”impossible”) even
with the computing power of modern computers. However, different computational approaches allow for
efficient numerical simulations of plasma. Different techniques are used for phenomena at different spatiotemporal scales: at the smallest scales one can use the Partricle-In-Cell method, or Vlasov simulations,
while at large scales the MHD approach can be applied. At intermediate scales the hybrid methods are
often used.
4.1
Particle-In-Cell method
The Particle-In-Cell (PIC) method is the most frequently
used approach to study numerically the plasma dynamics on the kinetic level, where we are interested in the
small spatio-temporal scales. The plasma is represented
by a finite number of ”macro-particles”, whose dynamics are followed in self-consistent fields. What makes the
PIC method computationally feasible is the spatial grid
for solving the field equations. This often reduces the
complexity of the simulation to O(n log ng ), where ng is
the number of grid points. Since ng n, the problem
depends almost linearly on the number of particles. The
use of grid can be justified, if there is a large number
of simulation particles per simulation cell, and physically
by the fact that at distances much larger than the Debye Figure 4: The schematics of the computalength the contribution to the field from the given particle tional cycle in the PIC method.
is strongly reduced. The PIC method requires some kind
of weighting procedure to build the charge and current
7
densities on the grid, and for projecting forces that are found on the grid back to the particles. The
weighting of particle quantities to the grid, solving the field equations on the grid, projecting forces
back to the particles, and advancing particle trajectories constitutes the main computational cycle in the
PIC method, see Figure 4. The field equations are solved either by iterative (e.g., Gauss-Seidel method,
multigrid method), or spectral field solvers using Fast Fourier Transform. The choice of particular solver
depends on the boundary conditions of the simulation box, where periodic, Dirichlet, von-Neumann or
mixed conditions can be used. The particle trajectories are often advanced with the first-order accuracy
and fast ”leap-frog” method, where the position and velocity mesh are staggered in time. The particles
have assigned mass and charge (usually representing a ”super-particle” representing several ions or
electrons). The leap-frog method for j-th particle yields:
~rj (t + ∆t)
= ~rj (t) + ~vj (t + ∆t/2)∆t
~vj (t + ∆t/2) = ~vj (t − ∆t/2) + f~j (t)∆t/mj ,
(31)
where ~rj , ~vj , mj are the position, velocity and mass of j-th particle respectively, f~j is the force acting
on the particle, and ∆t is the computational step. For stability reasons ∆t must be small enough to
resolve the smallest time scales in the system (usually the time period of electron oscillations). The grid
spacing also needs to be small to avoid numerical instabilities, and avoid artificial heating of plasma.
Strict requirements on the spatio-temporal spacing in explicit PIC scheme put constraints on the time
scales that this method can successfully address. On the other hand it is a very powerful method which
easily allows for including many additional physical phenomena, such as collisions, production and loss
of particles, or interaction with solid surfaces.
The number of simulation particles in the PIC method is large, but is still smaller than in the real
system. They can be considered as a representation of particles from the plasma distribution function.
Since the particles are weighted to the grid points, their charge is effectively a charge smoothed over
the grid cell, hence the name of the method. The numerical particles can thus overlap with others via
~ g ), where R
~ g is the position of the grid point, which are determined by the
shape-functions S(~rj − R
weighting routine (e.g., linear weighting gives a triangular shape). From Eq. (31) one can see that the
dynamics is basically given by Newton equations of motions. However, one needs to remember that the
particles are the macro-particles, and the forces are smoothed due to the shape-funciton (weighting).
4.2
Vlasov
Another approach is to solve the Vlasov equation (30), which is a hyperbolic partial differential equation
with characteristics referring to velocity and force fields. Solving this equation describes a Hamiltonian
flow in a phase-space. Vlasov codes are used for similar spatio-temporal scales as PIC codes. Since
they integrate the phase-space, they have basically no noise, while the noise is often a burden in the
PIC simulations due to a finite number of simulation particles. On the other hand, the phase space is
6-dimensional, which makes solving the full problem a difficult task. There are different schemes for
solving the Vlasov equation (e.g., semi-Lagrangian method, finite volume methods), and most of them
are computationally rather expensive.
Note that Vlasov and PIC methods can be considered as a different approach to the same problem.
The Vlasov simulation is the Eulerian description of the plasma10 . The Vlasov equation is discretized
on the a grid in a phase-space and the field equations on a spatial grid. In the PIC method we follow
orbits of individual macro-particles, which refers to the Lagrangian approach. In fact a collisionless
PIC method is somehow equivalent to solving the Vlasov equation by the method of characteristics (the
characteristics of the Vlasov equation are the particle trajectories). Since grid is also used in the PIC
method, some authors refer to mixed Eulerian-Lagrangian approach.
4.3
MHD
The fluid simulations of plasma a applicable to large scales, where we focus on the macroscopic plasma
quantities such as pressure and density. These equations can be solved using ”standard” methods. A
10 The Eulerian approach is often described with an analogy of watching the flow of the river from the bridge, while the
Lagrangian approach - in which we follow the behavior of the particle - is compared to sitting in a boat carried by the
water
8
simple approach is the explicit finite difference scheme on a spatio-temporal grid. In explicit schemes for
MHD, just as in explicit PIC codes, there is a constraint on the time step called the Courant (CourantFriedrichs-Levy) condition, which states that for a given grid size ∆x the time step ∆t must be small
enough that no signal can travel more than one grid cell in a single time step: v∆t/∆x ≤ 1. It is
important to look for a stable integration scheme. Another issue with For MHD simulations, we need
to solve the set of equations which include nonlinear, convective derivative terms (advective terms) of
the form ~v · ∇f (~r, t), and due to stability issues the FTCS (forward time centered scheme) can not be
used. Other schemes are more stable, such as Lax, Lax-Wendroff, leap-frog (centered in time centered
scheme), Finite Volume Methods. Explicit approach is often burden with numerical diffusion, thus one
needs to be careful when choosing the scheme. Another schemes, that are numerically more stable, but
also more difficult to code, are the implicit schemes, where we the new value of the variable is determined
by the old one and some undetermined new values, one example is the Crank-Nicholson method. Implicit
schemes are not constrained by the Courant criterion, but the results can be less accurate than in explicit
schemes, and computationally demanding. Often it is an advantage to use semi-implicit methods as a
compromise between accuracy and numerical efficiency.
4.4
Hybrid methods
It is also possible to create hybrid codes. They are often used to explore slower phenomena and speed up
the evolution of the system, by making certain assumptions. For example the hybrid simulations where
ions are considered as particles with the PIC method, while electrons are assumed to be a neutralizing
background with a Boltzmann distribution allow for studying phenomena at ion scales. However, the
assumption of Boltzmann distributed electrons needs to be justified (and often it is the case), and the
field solver has a nonlinear term.
5
Example: PIC code
Under the links below you will find two simple examples of the Particle-in-Cell codes.
http://laplace.ele.kyutech.ac.jp/PIC/PIC1Djava-e.html
Under this link you will find a simple PIC code together with the manual in the pdf version. This is
an interactive java applet. Try changing parameters to see different particle dynamics. For instance
modifying gyro and plasma frequencies ωce /ωpe , i.e., wce/wpe in the code will increase the magnetic
field and force particle to gyrate. The applet itself is under this link http://laplace.ele.kyutech.
ac.jp/PIC/PIC1D.html
http://www.particleincell.com/blog/2011/particle-in-cell-example/
This is a simple code written in Matlab, in which the plasma flows around an obstacle - biased plate.
Due to the plasma flow a wake behind the plate is formed. By changing the plate potential you will see
different dynamics of plasma particles. If the plasma is based more negative, the positive ions will be
focused behind the plate. To run the code download the files: flow around plate.m and eval 2dpot GS.m
from the website and run the first one in Matlab.
6
References and further reading
For more information on basic plasma physics you can read:
• Chen, F.F. (1984) Introduction to Plasma Physics and Controlled Fusion. Springer
• Pécseli, H.L. (2012) Waves and Oscillations in Plasmas. CRC Press
For further reading on the Particle-In-Cell method see:
• Birdsall, C. K.; A. B. Langdon (1985). Plasma Physics via Computer Simulation. McGraw-Hill
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• Hockney, R. W.; J. W. Eastwood (1988). Computer Simulation Using Particles. CRC Press
MHD simulations are for example covered in:
• J. Büchner; C.T. Dum; M. Scholer (2003). Space Plasma Simulation. Springer. (The book covers
also other methods in a tutorial style).
• H. Matsumoto; Y. Omura (1993). Computer Space Plasma Physics: Simulation Techniques and
Software, TerraPub ;The book is available online here: http://www.terrapub.co.jp/e-library/
cspp/
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