Impact of Strong Magnetic Fields on Collision Mechanism for Transport of Charged Particles

Abstract

One of the main applications in plasma physics concerns the energy production through thermo-nuclear fusion. The controlled fusion is achieved by magnetic confinement i.e., the plasma is confined into a toroidal domain (tokamak) under the action of huge magnetic fields. Several models exist for describing the evolution of strongly magnetized plasmas, most of them by neglecting the collisions between particles. The subject matter of this paper is to investigate the effect of large magnetic fields with respect to a collision mechanism. We consider here linear collision Boltzmann operators and derive, by averaging with respect to the fast cyclotronic motion due to strong magnetic forces, their effective collision kernels.

Proofs of Propositions 3.5, 3.6

Proof of Proposition 3.5

1. We show that y→(ψ ₀(y),…,ψ _m−1(y)) is a change of coordinates. Indeed, if $y, \overline{y} \in\mathbb{R}^{m}$ verify $\psi_{i} (y) = \psi_{i} (\overline{y})$, i∈{0,1,…,m−1}, then $y, \overline{y}$ belong to the same characteristic. Thus denoting by y ₀ the discontinuity point of ψ ₀ on this characteristic, there are $h, \overline{h} \in[0,T_{c}(y_{0}))$, with $h \leq\overline {h}$ without loss of generality, such that $y = Y(h;y_{0}), \overline{y} = Y(\overline{h};y_{0})$. Integrating (b ⁰⋅∇ _y ψ ₀)(Y(s;y ₀))=I(Y(s;y ₀))=I(y ₀) between h and $\overline{h}$ we obtain

$$0 = \psi_0 (\overline{y}) - \psi_0 (y) = \psi_0 \bigl(Y(\overline {h};y_0)\bigr) - \psi_0\bigl(Y(h;y_0)\bigr) = (\overline{h} - h ) I(y_0). $$

Therefore $h = \overline{h}$ which implies $y = \overline{y}$. We have shown that y∈ℝ^m is uniquely determined by ψ ₀(y),…,ψ _m−1(y). Indeed, y belongs to the characteristic associated to the invariants ψ ₁(y),…,ψ _m−1(y) and, if we denote by y ₀ the discontinuity point of ψ ₀ on this characteristic, we have y=Y(s;y ₀), where the parameter s∈[0,T _c (y)) is determined by

$$\psi_0 (y) - \psi_0 (y_0) = \int _0^s \bigl(b^0 \cdot \nabla_y\psi_0\bigr) \bigl(Y(\tau ;y_0)\bigr) \mathrm{d}\tau= s I(y_0). $$

Finally, without loss of generality we suppose that ψ ₀(y ₀)=0 and thus ψ ₀(y)∈[0,T _c (y ₀)I(y ₀))=[0,[ψ ₀])=[0,S). Clearly the map y→(ψ ₀(y),…,ψ _m−1(y)) is a surjection between ℝ^m and [0,S)×D, which shows 1.

2. Notice that ∇ _y ψ ₀∉span{∇ _y ψ ₁,…,∇ _y ψ _m−1} since b ⁰⋅∇ _y ψ ₀≠0 and b ⁰⋅∇ _y ψ ₁=⋯=b ⁰⋅∇ _y ψ _m−1=0. Thus, for any i∈{1,…,m−1} there is a unique vector field b ⁱ such that

$$b^i \cdot\nabla_y\psi_j = \delta^i_j, \quad j \in\{0,1,\ldots ,m-1\}, $$

which proves 2. □

Proof of Proposition 3.6

Notice that for any y∈ℝ^m the function

$$s \to u\bigl(Y(s;y)\bigr) = w\bigl(\psi_0 \bigl(Y(s;y)\bigr), \psi_1 (y),\ldots,\psi_{m-1} (y)\bigr) $$

is continuous on ℝ. In particular this holds true for any discontinuity point y ₀ of ψ ₀. For any y=Y(s;y ₀), s∈(0,T _c (y ₀)) we can write

It remains to analyze the differentiability around the point y ₀. Without loss of generality we assume that I>0. Taking s>0 one gets

(75)

where $\partial_{\psi_{0}} w_{+}$ stands for the right derivative of w with respect to ψ ₀. Taking now s<0, using the S-periodicity of w with respect to ψ ₀, we obtain

(76)

where $\partial_{\psi_{0}} w_{-}$ stands for the left derivative of w with respect to ψ ₀. Combining (75), (76) we deduce that w is differentiable with respect to ψ ₀ and at any point y∈ℝ^m

$$b^0 \cdot\nabla_yu = I(y) \partial_{\psi_0} w \bigl(\psi_0 (y),\ldots ,\psi_{m-1} (y)\bigr). $$

Moreover, for any i∈{1,…,m−1} we have

$$b^i \cdot\nabla_yu = \sum_{j =0 }^{m-1} \partial_{\psi_j} w b^i \cdot\nabla_y \psi_j = \partial_{\psi _i} w \bigl(\psi_0 (y), \ldots,\psi_{m-1} (y)\bigr),\quad y \in\mathbb{R}^m. $$

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