Description of Plasma BehaviorPlasma DescriptionsI) Conservation of particles (A = 1): Continuity EquationMacroscopic ParametersOne- and Two-Fluid MHD EquationsKallenrode, pp.30-44; Kirk, pp. 12 – 24; Kivelson&Russell, p.41-50Effect of the Magnetic Field on the Solar WindAzimuthal Structure of the Solar WindKallenrode, p. 67 - 75Kivelson&Russell, p. 330-342; Kirk, pp. 37-42Sound WavesWith magnetic fieldPhys 954, Solar Wind and Cosmic Rays I. Solar Wind and IMFE. MöbiusI.4 Interplanetary Magnetic FieldUp to here we have treated the solar wind and the corona as spherically symmetric. This is a gross negligence. Viewgraphs (measured Solar Wind Parameters)As the observation of the corona tells us, it is highly structured, the magnetic field plays an important role. With help of the Zeeman effect detailed maps of the magnetic field inthe corona are produced.Viewgraph (Magnetic fields)The first picture shows the surface of a very active sun with many sunspots. The magnetic field is highly disturbed and concentrated in the active regions. The other cases represent a more quiet sun with a significant dipole component of the magnetic field. The magnetic field points out of the surface at northern latitudes and into it at southern latitudes. In between the field is more or less across the surface. The regions with the magnetic field lines emerging from the surface are called Coronal Holes. These are the regions where the solar wind can emerge easily, as can be seen from the next figure.Viewgraph (Structure Corona)The corona presents an image of the magnetic field structure within it. The darker regions are the "Coronal Holes" where the flow of plasma can easily escape the grip of the magnetic field, whereas the bright regions mark trapped coronal plasma in closed magnetic field structures. We call these helmets and streamers. The model on the bottom panel represents a simplified quiet corona. Because the solar wind is a plasma the magnetic field exerts forces on the wind and must structure the wind. Before we start with this new element, let us first spend a little time on the treatment of a plasma in electric and magnetic fields.Description of Plasma BehaviorThe basic equations for electromagnetic fields are Maxwell’s Equations:(in MKSA units)1)  E = oPoisson Eqn.-16Phys 954, Solar Wind and Cosmic Rays I. Solar Wind and IMFE. Möbius2)  x B = µo J + 1/c2 ∂E/∂t Ampere's Law3)  x E + ∂B/∂t = 0 Faraday's Law4)  B = 0Maxwell’s Equations represent a consistent description of the electric and magnetic fields. This is a closed system in vacuum! However, this is different in the presence of charges. The charge density in the Poisson equation produces an electric field EThe electric current density J in Ampere's Law produces a magnetic field B. These quantities have to be calculated. This is not trivial in a plasma E acts on chargesB acts on currentsthrough the Lorentz Force:F = qE + q(V x B)Therefore, we have to include the behavior of the plasma into the equations. This can be performed with several different levels of complication, depending on the specific problem.Plasma Descriptions• trajectories of individual particlesThis is a test particle picture. It does not include any modification of background fields by the presence of the plasma. This is not a self-consistent approach, and therefore it is good only as long as the external forces are strong compared with the -17Phys 954, Solar Wind and Cosmic Rays I. Solar Wind and IMFE. Möbius"home-made" fields of the plasma itself. In essence, the plasma only reacts passive to fields.In the framework of our course it will be appropriate for the addition of pickup ions to the solar wind plasma and acceleration of a few particles out of a background plasma to high energies.However, this approach cannot be used for the core plasma itself.In principle we will have to try to describe the motion and the influence of all particles, but this will be an impossible task.• statistical approachUsed is only the probability for each particle to be atr: location in configuration spacev: location in velocity spaceIn a way it is a description for all particles, but with limited information. It explicitly includes dependencies on all other particles.We use the probability of an arbitrary particle to be at a certain location in phase spacef(r,v,t)which in principle depends on all the other particles. To solve the equation system for this complete distribution function f again amounts to an impossible task so that approximations are made for practical purposes. The distribution function f is used to compute a charge density distribution (which contains implicitly the information of all particles). Interaction between individual particles usually is weak compared to their response to the resulting electric field. At best collisions of particle pairs are considered. This description is used when features of the velocity distribution function f(v) are important for the behavior, as is needed for certain plasma waves.• fluid approachIn many cases the plasma can just be treated as a macroscopic fluid. I.e. only macroscopic parameters, such as , v, P; j, are used. Such a description is good for large scale phenomena, and therefore it is appropriate for the solar wind. This approach is called the Magneto-HydroDynamic approach (MHD).-18Phys 954, Solar Wind and Cosmic Rays I. Solar Wind and IMFE. MöbiusWe will add a brief discussion of the different approaches and lead with an abbreviatedderivation from a general description of the distribution function to the MHD equations.-19Phys 954, Solar Wind and Cosmic Rays I. Solar Wind and IMFE. MöbiusBoltzmann EquationBoltzmann EquationEvolution of the (single particle) distribution function f1 in phase space:∂f1/∂t + dr/dt . ∂f1/∂r + dv/dt . ∂f1/∂v = ∂f1/∂t ∂f1/∂t + v ∂f1/∂r + F/m ∂f1/∂v = ∂f1/∂t ∂f1/∂t: temporal evolution of f1 caused by other particles!v ∂f1/∂r: convective variation in configuration spaceF/m ∂f1/∂v: convective variation in velocity space(action of external forces on the distribution function) ∂f1/∂t influence of the other particles(1 denotes single particle distribution)External Forces (can be any type of force):F = q E electric fieldF = q (v x B) magnetic fieldNo influence of any other particles:-20Phys 954, Solar Wind and Cosmic Rays I. Solar