Derivatives of multiplier and poloidal flux with respect to geometry: dmupfdx

Collaboration diagram for Derivatives of multiplier and poloidal flux with respect to geometry: dmupfdx:

Variables
real, dimension(:,:,:,:,:), allocatable	allglobal::dmupfdx
	derivatives of mu and dpflux wrt geometry at constant interface transform
logical	allglobal::lhessianallocated
	flag to indicate that force gradient matrix is allocated (?)
real, dimension(:,:), allocatable	allglobal::hessian
	force gradient matrix (?)
real, dimension(:,:), allocatable	allglobal::dessian
	derivative of force gradient matrix (?)
logical	allglobal::lhessian2dallocated
	flag to indicate that 2D Hessian matrix is allocated (?)
real, dimension(:,:), allocatable	allglobal::hessian2d
	Hessian 2D.
real, dimension(:,:), allocatable	allglobal::dessian2d
	derivative Hessian 2D
logical	allglobal::lhessian3dallocated
	flag to indicate that 2D Hessian matrix is allocated (?)
real, dimension(:,:), allocatable	allglobal::hessian3d
	Hessian 3D.
real, dimension(:,:), allocatable	allglobal::dessian3d
	derivative Hessian 3D

Detailed Description

• The information in dmupfdx describes how the helicity multiplier, \(\mu\), and the enclosed poloidal flux, \(\Delta \psi_p\), must vary as the geometry is varied in order to satisfy the interface transform constraint.
• The internal variable dmupfdx(1:Mvol,1:2,1:LGdof,0:1) is allocated/deallocated in newton(), and hesian() if selected.
• The magnetic field depends on the Fourier harmonics of both the inner and outer interface geometry (represented here as \(x_j\)), the helicity multiplier, and the enclosed poloidal flux, i.e. \({\bf B_\pm} = {\bf B_\pm}(x_j, \mu, \Delta \psi_p)\), so that

  \begin{eqnarray} \delta {\bf B_\pm} = \frac{\partial {\bf B}_\pm}{\partial x_j } \delta x_j + \frac{\partial {\bf B}_\pm}{\partial \mu } \delta \mu + \frac{\partial {\bf B}_\pm}{\partial \Delta \psi_p} \delta \Delta \psi_p. \end{eqnarray}

• This information is used to adjust the calculation of how force-balance, i.e. \(B^2\) at the interfaces, varies with geometry at fixed interface rotational transform. Given

  \begin{eqnarray} B_\pm^2 = B_\pm^2 (x_j, \mu, \Delta \psi_p), \end{eqnarray}

  we may derive

  \begin{eqnarray} \frac{\partial B_\pm^2}{\partial x_j} = \frac{\partial B_\pm^2}{\partial x_j } + \frac{\partial B_\pm^2}{\partial \mu } \frac{\partial \mu }{\partial x_j} + \frac{\partial B_\pm^2}{\partial \Delta \psi_p} \frac{\partial \Delta \psi_p}{\partial x_j} \end{eqnarray}

• The constraint to be enforced is that \(\mu\) and \(\Delta \psi_p\) must generally vary as the geometry is varied if the value of the rotational-transform constraint on the inner/outer interface is to be preserved, i.e.

  \begin{eqnarray} \left(\begin{array}{ccc} \displaystyle \frac{\partial {{\,\iota\!\!\!}-}_-}{\partial {\bf B}_-} \cdot \frac{\partial {\bf B}_-}{\partial \mu } & , & \displaystyle \frac{\partial {{\,\iota\!\!\!}-}_-}{\partial {\bf B}_-} \cdot \frac{\partial {\bf B}_-}{\partial \Delta \psi_p} \\ \displaystyle \frac{\partial {{\,\iota\!\!\!}-}_+}{\partial {\bf B}_+} \cdot \frac{\partial {\bf B}_+}{\partial \mu } & , & \displaystyle \frac{\partial {{\,\iota\!\!\!}-}_+}{\partial {\bf B}_+} \cdot \frac{\partial {\bf B}_+}{\partial \Delta \psi_p} \end{array} \right) \left(\begin{array}{c} \displaystyle \frac{\partial \mu}{\partial x_j} \\ \displaystyle \frac{\partial \Delta \psi_p}{\partial x_j} \end{array} \right) = - \left(\begin{array}{c} \displaystyle \frac{\partial {{\,\iota\!\!\!}-}_-}{\partial {\bf B}_-} \cdot \frac{\partial {\bf B}_-}{\partial x_j} \\ \displaystyle \frac{\partial {{\,\iota\!\!\!}-}_+}{\partial {\bf B}_+} \cdot \frac{\partial {\bf B}_+}{\partial x_j} \end{array} \right). \end{eqnarray}

• This \(2\times 2\) linear equation is solved in dforce() and the derivatives of the rotational-transform are given in diotadxup, see preset.f90 .
• A finite-difference estimate is computed if Lcheck==4.