Tokamak equilibria with non field-aligned axisymmetric divergence-free rotational flows

Abstract

Rotational ideal divergence-free magnetohydrodynamic (MHD) equations are expressed in terms of transformed variables w→*=(μρ)1/2v→ and μp* = (μp + w*2/2), where v→, p, and ρ are plasma velocity, pressure, and mass density, respectively. With divergence-free flows, ∇·v→=0, the plasma density ρ does not appear in the MHD equations written in terms of w→* and μp*. The non field-aligned rotational Grad-Shafranov equation is represented in spherical coordinates. Tokamak-like axisymmetric equilibria with v→ ⊥∇ρ are obtained analytically by solving for torus solutions under only three source functions.

Tokamak L/H mode transition

“…(23). 26,27 These contours also represent the poloidal stream lines of the density weighed velocity,w à , of Eq. (8) by W 0 ðWÞ ¼ bW.…”

“…[23][24][25] Recently, in terms of spherical coordinates, 26 the non field-aligned equilibria have been solved under a set of transformed MHD variables. 27 With these new variables, the functional dependence of the rotational Grad-Shafranov equation, plasma pressure, normal electric field, on the plasma variables are sufficiently simple and explicit to allow an understanding of the L=H mode transition through a positive feedback cycle.…”

Exact Equilibria for Ideal MHD Plasmas with Helically Symmetric Incompressible Flows and Variable Gravitational Field

“…Many researchers have achieved MHD equilibria with incompressible flows in cylindrical domains [15][16][17][18][19][20][21][22][23][24][25]. The mathematical complexity of the MHD equations has stood in the way of understanding hydromagnetic phenomena such as stellar winds [26][27][28][29], magnetic fields of solar prominence, and the confinement of laboratory plasma [30][31]. Several researchers [32][33][34] performed the derivation of the equilibrium equations of MHD plasmas in the literature.…”

General three-dimensional equilibria for gravitating ideal magnetohydrodynamics of field-aligned steady incompressible flows with an application to solar prominences

“…Magnetohydrodynamic (MHD) fluids play a vital role in kinetics and their applications have been extensively studied (Brushlinskii & Zhdanova 2004;Chkhetiani, Eidelman & Golbraikh 2006;Nickeler, Goedbloed & Fahr 2006;Frewer, Oberlack & Guenther 2007;White & Hazeltine 2009;Kuiroukidis 2010;Tsui et al 2011;Cicogna 2012;Zahid et al 2013;Cicogna & Pegoraro 2015;Kuiroukidis & Throumoulopoulos 2016;Reddy, Fauve & Gissinger 2018). An important application of the MHD model is the study of equilibrium.…”

“…Cicogna & Pegoraro (2015) derived a series of solutions to the GGS equation that describes the two-dimensional azimuthal symmetric plasma equilibrium in the presence of incompressible poloidal and toroidal plasma flows. For the Euler equation with a swirl, the symmetry properties of a particular GS equation describing the stationary, azimuthally symmetric magneto-static plasma equilibria were developed in Frewer et al (2007), White & Hazeltine (2009), Cicogna, Ceccherini & Pegoraro (2006), Cicogna, Pegoraro & Ceccherini (2010) and Cicogna (2008Cicogna ( , 2012. In Cicogna et al (2006), an analysis of the symmetry properties of a class of partial differential equations related to the GS equation was studied.…”