Volume integrals: lBBintegral, lABintegral

Collaboration diagram for Volume integrals: lBBintegral, lABintegral:

Variables
real, dimension(:), allocatable	allglobal::lbbintegral
	B.B integral.
real, dimension(:), allocatable	allglobal::labintegral
	A.B integral.
real, dimension(:), allocatable	allglobal::vvolume
	volume integral of \(\sqrt g\); computed in volume
real	allglobal::dvolume
	derivative of volume w.r.t. interface geometry

Detailed Description

• The energy functional, \(F \equiv \sum_l F_l\), where

  \begin{eqnarray} F_l \equiv \left( \int_{{\cal V}_l} \frac{p_l}{\gamma-1} + \frac{B_l^2}{2} dv \right) = \frac{P_l}{\gamma-1}V_l^{1-\gamma}+\int_{{\cal V}_l} \frac{B_l^2}{2} dv, \label{eq:energy_global} \end{eqnarray}

  where the second expression is derived using \(p_l V_l^\gamma=P_l\), where \(P_l\) is the adiabatic-constant. In Eqn. \((\ref{eq:energy_global})\), it is implicit that \({\bf B}\) satisfies (i) the toroidal and poloidal flux constraints; (ii) the interface constraint, \({\bf B}\cdot\nabla s=0\); and (iii) the helicity constraint (or the transform constraint).

• The derivatives of \(F_l\) with respect to the inner and outer adjacent interface geometry are stored in dFF(1:Nvol,0:1,0:mn+mn-1), where

  \( F_l \equiv\) dFF(l,0, 0)

  \(\partial F_l / \partial R_{l-1,j} \equiv\) dFF(ll,0, j)

  \(\partial F_l / \partial Z_{l-1,j} \equiv\) dFF(ll,0,mn j)

  \(\partial F_l / \partial R_{l ,j} \equiv\) dFF(ll,1, j)

  \(\partial F_l / \partial Z_{l ,j} \equiv\) dFF(ll,1,mn j)

• The volume integrals \(\int dv\), \(\int B^2 \; dv\) and \(\int {\bf A}\cdot{\bf B} \; dv\) in each volume are computed and saved in volume(0:2,1:Nvol).