- 3.3 Coordinates: Analytical Geometry of the Torus. - 3.4 Field Line Dynamics: Hamilton Dynamics of the Magnetic Field. - 4.3 Littlejohn’s Variational Principle: Orbital Dynamics of the Guiding Center. - Figure 1.1 The ITER and the Sun. - The beginning of the universe is called the “Big Bang” [6]. - It is 110 times the radius of the Earth (R Sun D 70 10 4 km), and 320,000 times the mass of the Earth (M Sun D 2 10 30 kg). - a: radius of the Sun) is called the. - “chromosphere,” “colona,” and “prominence” outside of the surface.. - Salon: History of the Age and the Energy Source of the Sun. - where D s=c is called “proper time” and is a time in a rest frame (time coordi- nate at dx =dt D 0) of the object. - Table 1.1 shows a comparison of parameters of the ITER and the Sun. - Chapman who clarified the origin of the Aurora and the geomagnetic storm. - Figure 2.1 The fusion reaction. - The kinetic energy of the incident nuclei is distributed to nuclei in the compound nucleus. - The wave equation of the electric field Œ@ 2. - In the case where the potential is proportional to the square of the distance (harmonic oscillator: V .r/ D cr 2. - In the center of the mass system, angular momentum M D r p is conserved. - Figure 2.13 shows permeabil- ity of the Coulomb barrier P against E=E c . - Figure 2.14 Energy de- pendence of the Coulomb barrier permeability. - On the other hand, they move in a straight line in the direction of the magnetic field. - (d) Vector field through the vertex of the triangle. - is the “Euler index.” The relationship always holds irrespective of any division of the sphere by triangles.. - In other words, the Euler index of the torus is 0.. - Poincaré defined the “index” of the vector field. - In the end, the flow tangent to the side does not contribute to the index of the closed surface (this discussion holds only for closed surfaces). - 3.3 Coordinates: Analytical Geometry of the Torus 41 Figure 3.8 (a) René. - 3.4 Field Line Dynamics: Hamilton Dynamics of the Magnetic Field 43 Z. - Let be the toroidal angle of the torus and the poloidal angle (choice of and is arbitrary). - and find the orbit of the magnetic field along the toroidal angle . - This property is derived from the incompressibility of the magnetic field. - dp i =dt D @H=@x i (3.25) Here, H is the Hamiltonian, the sum of the kinetic energy T and the potential energy V (H D T C V. - Here, J is the current flowing in the plasma, B is the magnetic field, P is the pressure of the plasma. - This is the basic principle of the magnetic confinement fusion.. - In the (u. - Figure 3.10 (a) Definition of the magnetic surface and fluxes of toroidal plasma and (b) the geo- metric meaning of the Clebsch expression of the magnetic field. - Figure 3.12 shows the magnetic surface of the torus plasma in the cylindrical coor- dinate system (R. - Figure 3.13 Locus of the magnetic field in the flux coordinates. - The poloidal angle of the sequence of points f g j 0 g is given by j D 2j=q C 0 . - Figure 3.15 Locus of the magnetic field in cylindrical coordinates (R. - =2 0 C P plays the role of the effective potential energy.. - This is the origin of the naming of the annihilator of B r D J 1. - Note: Symmetry and Invariant of the Dynamical System [15]. - Aczel AD (2000) The Mystery of the Aleph Mathematics. - Orbital Dynamics of the Guiding Center. - Considering the orthogonal relation r u i @ x =@u j D ı ij (Equation 3.5), the Jacobian of the Boozer–Grad coordinates J D 1= r r ˛ r D 1=B 2 , and following vector relation,. - Below, we discuss expres- sion of the differential form [16]. - dependence of the co- ordinate transformation. - The main term of the Poisson brackets is f S 1 . - Similarly, the t component of the ". - By this “smoothness,” the possibility of the direction of time is introduced. - Phase space flow v of the probability density D in the 6N -dimensional phase space satisfies r v D 0. - Incompressibility of the phase space flow leads to an interesting. - B / is a solution of the Vlasov equation, T . - t/ is also a solution and is called the time- reversal solution (here, T is the “time reversal operator,” and requires a reversal of the magnetic field) [16]. - is called “phase mixing” since it occurs due to the phase overlapping of the wave.. - Collisionless damping of the electric field caused by the contin-. - “dispersion equation.” In the integral of the dispersion equation, u k v =k D. - The inverse operator of the linear operator L D ! k v is L 1 D P Œ1. - t/ is also a solution of the Vlasov equation and the density perturbation n 1 . - Shielding of the electric field occurs in the same way as Debye shielding in electrolytes. - the solution of the Poisson equation for is obtained as follows,. - b and the question of the validity of two-body collision remains.. - Assume that the behavior of the plasma is given by the following evolution equation.. - Magnetohydrodynamic behavior of the plasma can be formulated in the form of a variational principle using the Lagrangian. - The eigenvalue of the Hermitian operator (spectrum) is real. - In the case of force equilibrium (the first order term of the action integral with respect to the displacement D 0), the variation ıS is given by a quadratic form of the displacement. - The stability of the equilibrium can be determined by its sign. - This property of the linear ideal MHD operator F is called the Hermitian (self- adjoint). - Minimization of the energy integral with respect to V is easy in the cylindrical plasma. - term but the absence of the @V =@r term in the energy integral leads to following Euler equation [13],. - 2 c , the eigen mode decomposition of the solution is given by. - From the analytical condition of the second-order derivative @ 2 =@t@ x. - Here the group velocity of the wave packet v g D. - plays the role of the Hamiltonian and k the role of canonical momentum.. - t/ plays the role of the Hamiltonian. - The condition of the refractive index n D 0 (phase velocity D !=k D 1 ) is called the “cut-off,” and n D 1 (phase velocity D !=k D 0) is called “resonance.” From Equation 7.46. - by the time symmetry of the equation of mo- tion.. - final form of the perturbed distribution function f a1k . - Here, the properties of the Dirac delta function ı. - The wave equation in the vicinity of the resonance can be obtained by replacing n ? of Equation 7.61 to i.c=!/d=dx.x D r r A / as follows,. - Derivation of the kinetic dielec- tric tensor is lengthy. - The dispersion relation of the KAW is given as follows at .k ? i / 2 1:. - The origin of the polarization drift is as follows. - The particle moves in the direction of the electric field E first. - Budden KG (1955) Physics of the Ionosphere. - This will affect the physical properties of the plasma. - So, the orbit of the electrons satisfying B max E= is trapped in the weak magnetic field regime reflected by the magnetic mirror (trapped particle or banana orbit). - Then, there appears a discontinuity in the trapped–untrapped boundary of the velocity distribution function. - T b v T b N ba j i : (8.56) Collision frequencies D a .v/ and E a .v/ in the expression of the viscosity coeffi- cient for the Maxwellian are expressed as follows:. - Plasma current is the sum of the currents of each species as follows,. - nana width of the trapped electron be b . - of the plasma current is car- ried by the bootstrap current from the simulation. - the existence of the bootstrap current is confirmed [4].. - (8.89) The first term of the right-hand side is the drag force due to the magnetic field variation. - Rq), inhomogeneity of the. - Figure 9.4 Variation of the spatial structure of turbulence with magnetic shear in the electron channel [2]. - Fourier expansion of the electrostatic potential is given by. - In confined plasma, turbulence may develop perpendicular to the magnetic field because of the magnetic field. - Figure 9.8 (a) Schematics of break up of the cell by the radial electric shear. - About half of the carbon dioxide emissions to the atmosphere remain in the atmosphere