Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton 12. MOMENTS OF PLASMA PRESSURE The Virial Equation Here we consider certain integral relations which allow us to determine information on the energy density within S φ . The techniques described to determine plasma position, and more generally plasma shape, needed only Maxwell's equations. Now we add the constraint of equilibrium. Generally we apply these results to S φ = S φplasma + S φvacuum : even simpler relations are obtained by identifying S φ with S φplasma . From a previously derived integral equation (Equation 8.19), we can obtain some very useful relationships. First, take B 2 = 0 (i.e. all the fields are described by B = B 1 ), and let the arbitrary vector Q be the polar vector (r = Re R + ze Z .). Then Equation 8.19 becomes the “virial theorem”: ⎛ ⎜ 3p + B2 ⎞ ⎡ ⎛ ∫ ⎟ dV = ⎜ p + B2 ⎞ ⎟ ( r • n) − ( r • B) ( B •n) ⎤ ⎝ 2µ 0 ⎠ ∫ ⎝ 2µ V ⎣ ⎢ 0 ⎠ µ 0 ⎦ ⎥ dS n 12.1 S n Note that if B were the self plasma field, and there was no other field, then the surface integral on the RHS approaches zero as the surface (where p = 0) approaches infinity, because B(self) ∝ R -3 . This contradicts the positive definiteness of the LHS volume integral, showing that equilibrium by self fields alone is impossible. Now restricting to toroidal symmetry, ∂/∂φ = 0. Therefore the poloidal (subscript p) and toroidal (subscript φ) fields satisfy B = B p + B φ B p •B φ = 0 12.2 Next assume that the surface S n is outside the plasma, where ∇xB = 0, and that the vectors Q and B 2 are purely poloidal (i.e. there is no toroidal component of B 2 ); Q = Q R e R + Q z e z B 2 = B 2R e R + B 2 z e z 12.3 Noting that the toroidal field outside the plasma is given by B φe = const/R, then we have ∫ S n 2 B φe 2 ⎛ ( Q • n)dS n = ∫ B φe ⎝ V ∇ • Q − 2Q • e R R ⎞ dV 12.4 ⎠ 94
Magnetic fields and tokamak plasmas Alan Wootton and B 1 • B 1 • ∇Q = B p1 •B p1 •∇Q + B 2 φeQ • e R R 12.5 Using these, we can express Equation 8.19 so that the toroidal field enters only as B φ 2 -B φe 2 on the LHS: ∫ V ⎡ ⎛ ⎜ ⎢ ⎣ ⎝ p + B 2 p1 ⎞ 2µ 0 ⎠ ⎟ ∇• Q + B 2 2 φ − B φe ⎛ ∇ • Q − 2 Q •e R ⎞ 2µ ⎝ 0 R ⎠ − B •∇Q • B ⎤ p1 p1 dV µ ⎥ 0 ⎦ ( )( B p1 •n) ⎡ = p + B 2 ⎛ p1 ⎞ ⎜ ⎟ ( Q • n) − B • Q ⎤ p1 ∫ ⎢ ⎣ ⎝ 2µ 0 ⎠ µ ⎥ dS n − S n 0 ∫ j • B p2 × Q ⎦ V [ ( )] dV 12.6 Now let P = p + B 2 2 φ − B φe 12.7 2µ 0 ⎛ T = p + ⎜ ⎝ B 2 p1 + B 2 2 φe − B φ 2µ 0 ⎞ ⎟ 12.8 ⎠ (Remember B 1 is the poloidal field). Then the integrand on the LHS of Equation 12.6 can be written as (from now on drop the subscript p for poloidal) ⎛ [...]= P ∂Q R ∂R + ∂Q z ⎞ ⎝ ∂z ⎠ + T R Q + B 2 2 1R − B 1z ⎛ ∂Q z R 2µ 0 ∂z − ∂Q R ⎞ ⎝ ∂R ⎠ − B B 1R 1z ⎛ ∂Q R µ 0 ∂z + ∂Q z ⎞ ⎝ ∂R ⎠ Now we have to chose something for Q. Let 12.9 u = R − R ch + iz = ρe iω 12.10 with R ch some characteristic radius, a fixed point within the plasma cross section. Then ρ 2 = ( R − R ch ) 2 + z 2 12.11 Now with this notation let Q R = F( u) Q z = −iF( u) 12.12 95
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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