Beta limit of the stable interchange mode for a toroidal current dipole magnetic configuration

Abstract A free boundary plasma equilibrium solver is used in the dipole field of a ring current system. The solver employs multi-grid technology to effectively address the issue of slow solution speeds caused by changes in the plasma boundary. A power law pressure model is utilized to investigate the beta limit of the stable interchange mode. The stability of the interchange mode is influenced by beta, with high-beta pressure profiles tending to excite interchange instabilities at the plasma edge. Under the condition of a stable interchange mode, we investigated the effects of peak pressure position <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>κ</mml:mi> </mml:math> , steepness of the pressure gradient g, and plasma radius divided by the D-coil radius <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>lim</mml:mi> </mml:mrow> </mml:msub> </mml:math> on the maximum beta β max and the volume averaged beta 〈 β 〉 limit. β max and 〈 β 〉 satisfy rapidly decaying power functions of g, such as β max ∼ g − 3.94 and 〈 β 〉 ∼ g −3.45 . <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> </mml:math> and 〈 β 〉 obey slowly increasing power functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>κ</mml:mi> </mml:math> , such as β max ∼ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>κ</mml:mi> </mml:math> 0.5 and 〈 β 〉 ∼ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>κ</mml:mi> </mml:math> 0.38 . Additionally, β max and 〈 β 〉 follow negative power functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mspace width="1em"/> <mml:mover accent="true"> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>lim</mml:mi> </mml:mrow> </mml:msub> </mml:math> , i.e. β max ∼ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>lim</mml:mi> </mml:mrow> </mml:msub> </mml:math> −1.98 and 〈 β 〉 ∼ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>lim</mml:mi> </mml:mrow> </mml:msub> </mml:math> −2.0 . <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>κ</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>lim</mml:mi> </mml:mrow> </mml:msub> </mml:math> do not significantly affect the trend <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mfenced close="〉" open="〈"> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> with increasing g.