Fundamentals of Plasma Simulation (I)

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 • 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 physics Electric and magnetic fields (E & B) are generally determined by Maxwell’s equations, with corresponding boundary conditions and the sources (charges and currents). Gauss’s law No magnetic poles Faraday law Ampere’s law Sometimes E & B are expressed in terms of an electric potential φ and vector potential A: In this case, electromagnetic field equations are written in the form With Lorentz gauge

4. Basic equations of plasma physics (cont.) Equation of motion The motion of charged particles is determined by the electromagnetic fields through the equations of motion – Lorentz equation 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.

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 of particles in the six-dimensional space (r,v), The single particle distribution function satisfies the Boltzmann equation 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 → Particle orbits in various given electromagnetic fields

8. Gyro motion & Larmor radius Guiding center 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! Orbit of charged particle is Initial velocity of charged particle in magnetic field Charged particle is only experienced Lorentz force gyrogrequency gyroradius Gyration is the most basic motion of charged particle in a magnetized plasma!

9. Homework: Problem 3 Derive 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. Electric field drifts (E×B drift) Equation of motion is The “E × B drift” of the gyrocenter is If E is non-uniform, it may cause a modification of Larmor radius effect

10. Drifts due to general force F 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 This general force can be gravity, force due to non-uniform magnetic field (gradient or curvature)

11. Magnetic field gradient drifts Expanding the magnetic field at the location of the guiding center Equation of motion becomes Magnetic gradient produces a force on the guiding center of charged particle due to the magnetic moment, i.e., Averaging on gyro-motion Gradient drift

12. Magnetic field curvature drifts When the B field lines are curved and the particle has a velocity v// along the field, another drift occurs. 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.) 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. In the frame of the guiding center a force appears because the plasma is rotating about the center of curvature. Curvature drift velocity

13. Drifts in varying electric field – polarization drift If electric field E is time-varying, the particle experiences a acceleration, In the frame of the guiding centre which is accelerating, a force is felt except for the force due to uniform electric filed. An additional drift is produced as Polarization 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. Remarks for single particle drifts All 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., Gyromotion; electric field drift with Larmor radius modification; magnetic field gradient and curvature drifts; polarization drift; magnetic mirror. 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. 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.

15. Plasma Kinetic Description

16. Plasma Kinetic Theory – why need kinetic description Many particle: For a plasma, the plasma parameter is g=1/(nλ3d)<<1. Thus a plasma consists 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. volume element in phase space v ∆x ∆v ∆v (x(t);v(t)) ∆x x 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, For particles in a plasma, the microscopic distribution function is the summation Klimontovich Here xi(t) and vi(t) are the spatial and velocity trajectories as the particles move. 6-dimensional phase space Six-dimensional phase space with coordinates axes (x,y,z) and (vx,vy,vz) and volume element ∆x∆v All particles (i=1, N) have time-dependent position xi(t) and velocity vi(t). The particle path at subsequent times is a curve in phase space.

19. Basic equations for particle simulation Here xi(t) and vi(t) are determined by the equations of motion and Maxwell equations, The microscopic sources are determined by with 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 simulation PIC (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.

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: By using relations: Inserting equation of motion, we have Klimontovich equation

21. Properties of the Klimontovich equation Klimontovich equationtogether 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): No creation or destruction of charged particles as they move their trajectories determined by electric and magnetic fields in a plasma!

22. From Klimontovich equation to Plasma kinetic equation • 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

23. fs Nm Distribution function ∆x ∆v Averaging procedure The average distribution function of Nmwill 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) Define the fluctuation (deviation from the averaged level) of complete microscopic distribution function Nm from the averaged one fs, i.e., We have The average distribution function fsrepresents the smoothed properties of the plasma species for ∆x >λD; the microscopic distribution δNmrepresents the “discrete particle” effects of individual charged particles for n-1/3<< ∆x<λD. Similar separation for the fields

24. Fundamental plasma kinetic equation Substituting these forms into the Klimontovich equation and averaging it using the procedure above, we obtain our 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 equation Rewriting 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 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. 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.

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 =- Ñ), Boltzmann equation becomes electrostatic kinetic equation 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 Homework: problem 4 Derive 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 Properties of Vlasov equation • 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. • 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. • A Vlasov plasma is stable since the stable distribution with dfs/dε<0 minimizes the kinetic energy. • 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>>ρg) and by slow processes compared to the gyrofrequency (ω<<ωc). • 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 equations • 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) • Splitting the distribution function fs into gyro-phase independent part <fs>φ and dependent part fs-<fs>φ • Get gyro-averaged kinetic equations by gyro-averaging Boltzmann equation – So, the dimensionality in phase space is reduced!

29. Two typical gyro-averaged kinetic equations: 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<<ωc ; k┴ρg<<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<<ωc ; k┴ρg~1 J Wesson, TOKAMAK (second edition), 1997

30. Derivation of drift kinetic equation Change variables from to with Re-write kinetic equation by using new variables Define with where is small quantity with Larmor radius order Assuming

31. Derivation of drift kinetic equation (cont.) 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 Substitute all relations into rewritten kinetic equation in guiding center coordinate and performing gyro-averaging, we can get 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 J Wesson, TOKAMAK (second edition), 1997 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 order in 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 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, g satisfies the gyrokinetic equation In long wavelength limit and L→0, the distribution function is reduced to the result from drift kinetic equation.

33. References for the derivation of nonlinear gyrokinetic equation (classical and modern gyrokinetic theories, collected by T S Hahm) • Hazeltine and Meiss, Plasma confinement (book) • Frieman and Chen, Phys. Fluids 25, 502 (1982) • Lee, Phys. Fluids 26, 556 (1983) • Dubin, Krommes, Oberman, and Lee, Phys. Fluids 26, 3524 (1983) • Hagan and Frieman, Phys. Fluids 28, 2641 (1985) • Hahm, Lee, and Brizard, Phys. Fluids 31, 1940 (1988) • Hahm, Phys. Fluids 31, 2670 (1988) • Brizard, J. Plasma Phys. 41, 541 (1989) • Brizard, Phys. Plasmas 2, 459 (1995) • Hahm, Phys. Plasmas 3, 4658 (1996) • Brizard, Phys. Plasmas 7, 4816 (2000) • Sugama, Phys. Plasmas 7, 466 (2000) • Brizard, Phys. Plasmas 7, 3238 (2000) • Wang, Phys. Rev. E. 64, 056404 (2001) • Qin and Tang, Phys. Plasmas 11, 1052 (2004) • 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 ‘Fluid Description’ refers to simplified treatment of plasma which does not need the details of velocity dependence. 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 variables Omit some important physical processes (but describe others); Provide tractable approaches to many problems.

36. Fluid equations for a plasma Fluid 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 ρs; fluid bulk velocity vs; and pressure tensor πs 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.

37. Macroscopic quantities from fluid moment Number density Charge density Momentum density Current density Scalar pressure Temperature Heat flux 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 Calculating all integrations:

38. 0th moment equation— continuity equation Considering the integration of distribution function over whole velocity space is the density, we integrate Boltzmann equation over velocity space (0th moment) 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. By using full derivative, we have For incompressible fluid, Continuity equations for charge or mass densities can be obtained by multiplying number density equation by qs or ms, respectively.

40. 1st moment equation— equation of motion Performing the integrations by parts and using the properties of distribution function ， By using this relation

41. 1st moment equation— equation of motion (cont.) So, the momentum equation is yielded as By using continuity equation, we can get equation of motion If the collisions are frequent enough, the pressure tensor becomes diagonal, or even isotropic, so

42. 2nd moment equation— energy equation Letting The first term is

43. 2nd moment equation— energy equation (cont.) The second term is The third term is

44. 2nd moment equation— energy equation (cont.) 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 ,

45. Chain of moment equations Similar way to derive high order equations 3rd moment equation— heat flux equation 4th moment equation …… To infinite This procedure shows that low order moment equation includes higher moment, which is an infinite chain of hierarchy! 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!

46. Closure moments The 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. 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 process In magnetized plasmas, the heat diffusion coefficients along perpendicular and parallel directions are very different, so it is separated as 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.

47. Summary of moment equations Continuity equation Equation of motion Energy equation To the third moment, we have an unknown quantity, heat flux, which is the fourth order moment. → closure approximation 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.

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.

49. Fluid/kinetic hybrid model – a mixed description D Winske, Space Science Review 42, 53-65 (1985); Computer space plasma physics (book) (1993) 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.

50. Examples of fluid/kinetic hybrid model • Various types of hybrid codes depend on the problems. Some examples: • 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) • Fast ions or electrons (kinetic) and background plasmas in magnetic fusion plasmas with various heatings or energetic Alpha particles. • Foreshock: it is characterized by particles (kinetic) that are leaked or reflected from the shock which stream back into the solar wind. • ……