Hamada coordinates: Difference between revisions

In this section, following D'haseleer et al we will translate the condition of straight current density lines into one for the '''Hamada''' coordinates Jacobian. For that we will make use of the equilibrium equation <math>\mathbf{j}\times\mathbf{B} = p'\nabla\psi </math>, which written in a magnetic coordinate system reads

Taking the [[Flux coordinates#flux surface average|flux surface average]] <math>\langle\cdot\rangle</math> of this equation we find <math>(-{I}'_{tor}{\Psi}'_{pol} + {I}'_{pol}{\Psi}'_{tor})= 4\pi^2{p}'\langle(\sqrt{g_f})^{-1}\rangle^{-1}</math>, so that we have

:<math> \mathbf{B}\cdot\nabla\tilde{\eta} = {p}'\left(\frac{\langle(\sqrt{g_f})^{-1}\rangle^{-1}}{\sqrt{g_f}}-1\right)

In a coordinate system where <math>\mathbf{j}</math> is straight <math>\tilde{\eta}</math> is a function of <math>\psi</math> only, and therefore LHS of this equation must be zero in such a system. It therefore follows that the Jacobian of the '''Hamada''' system must satisfy  

  \sqrt{g_H} = \langle\sqrt{g_H}^{-1}\rangle^{-1} = \frac{V'}{4\pi^2}~,

where the last identity follows from the [[Flux coordinates#Useful properties of the FSA|properties of the flux surface average]]. The Hamada angles are sometimes defined in 'turns' (i.e. <math>(\theta, \xi) \in [0,1)</math>) instead of radians (<math>(\theta, \xi) \in [0,2\pi)</math>)). This choice together with the choice of the volume <math>V</math> as radial coordinate makes the Jacobian equal to unity. Alternatively one can select <math>\psi = \frac{V}{4\pi^2}</math> as radial coordinate with the same effect.