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6 The Transport Equation The only transport equation considered in this work is the Braginski fluid equation for electron energy, or electron heat conduction [40]. It is a parabolic partial differential equation, that is, it is time dependent but it settles to a steady state. In its simplest form it may be written as a flux divergence equal to a source term: ∇·Q = S (6.1) Flux, Q, is equal to a transport coefficient times a temperature derivative. The source term, S, may contain sources, sinks, a coupling term, a time derivative and boundary conditions. In 1D, equation 6.1 may be written as: ∂ ∂x ∂Te −κ ∂x = − 3ne ∂Te 2 ∂t (6.2) where, κ : conduction coefficient, m−1 s−1 . κ = χne χ : thermal diffusivity, m2 s−1 Te : electron temperature, eV ne : electron density, m−3 33 6 The Transport Equation 6.1 3D Rectangular form of the transport equation. In a 3D rectilinear mesh the points lie on mutually orthogonal axes and equation (6.2) may be written as: ∂ − ∂xi ∂T ∂T ∂T κ ⊥1 + κ ⊥2 + κk ∂x1 ∂x2 ∂x3 = − 3n ∂T 2 ∂t (6.3) The 3 terms on the left hand side are mutually orthogonal flux components. To model an anisotropic system such as a magnetized plasma we set the x3 term tangential to the field lines. This is the parallel term. The x1 and x2 terms are the perpendicular, or radial, terms. The conduction coefficients, κ⊥1 , κ⊥2 and κk , are dependent on temperature in the following way: κ⊥1 ∝ T −1/2 , κ ⊥2 ∝ T , κk ∝ T 5/2 (6.4) κ⊥1 and κ⊥2 are similar in magnitude and both are much smaller than κk so we regard the radial transport (on the plane of each Poincaré plot) as quasi-isotropic due to electrostatic micro-turbulence. The analytical expressions for these conduction coefficients are taken from the plasma formulary [41]. Discretizing equation 6.3 on a rectlinear mesh is straightforward, using standard techniques found in textbooks. The derivatives may be discretized in each direction uniquely, and there is no mixing of derivatives. The metric tensor is simply a 3×3 identity matrix. The mesh generated for W7-X is non-rectilinear. The points do not lie on mutually orthogonal axes and so there are mixed derivatives and the metric tensor is ’full’. Discretizing equation 6.3 on such a mesh is more challenging, and so we turn to a vector form of the transport equation. 34 6.2 Vector form of the transport equation 6.2 Vector form of the transport equation Below is the heat balance equation for an electron-ion fluid in vector form [40, 42, 33]: ∂uα 2 2 +∇· uα (V⊥α + hVkα ) − χ⊥α ∇uα − (χkα − χ⊥α )hh · ∇uα = −να uα +Sα(u) (6.5) ∂t 3 3 where, α : particle species, e or i uα : internal energy, eV m2 s−1 h : unit contravariant vector quantity, h = B/kBk V⊥α , Vkα : Perpendicular and parallel velocities, ms−1 χ⊥α , χkα : Perpendicular and parallel diffusivities, m2 s−1 να : heat loss rate, function of Coulomb energy exchange between electrons and ions (u) Sα : heat source term If we assume no net current in the plasma, then Vi = Ve . Neglecting velocity contributions (u) (V⊥α = Vkα = 0), and source terms (Sα = 0), and substituting in: ue = 3ne T, 2 χ⊥e = κ⊥e , ne χke = κke ne The heat balance equation (6.5) simplifies to the same form as the flux divergence equation 6.2, except that the transport coefficient contains more information. 2 κ⊥e ∇ ∇· 3 ne ⇒ 3ne T 2 2 + 3 κke κ⊥e − ne ne hh · ∇ 3ne T 2 ∂ =− ∂t 3ne ∂T ∇ · {κ⊥ + (κk − κ⊥ )h2 } · ∇T = − 2 ∂t 3ne T 2 (6.6) 35 6 The Transport Equation 36