Equations solved in TGYRO

Below we write the most general form of the transport equations solved by TGYRO. The calculation chain in NEO-CGYRO-TGYRO is a faithful representation of the comprehensive transport theory of Sugama [SH98]. This paper is required reading for a detailed understanding of the theoretical foundations of GACODE. Below we summarize Sugama’s form of the transport equations, with some practical simplifications and notational differences.

Density Transport

\[\frac{\partial \langle n_a \rangle}{\partial t} + \frac{1}{V^\prime}\frac{\partial}{\partial r} \left( V^\prime \Gamma_a \right) = S_{n,a} \; ,\]

where

\[S_{n,a} = S_{n,a}^{\rm beam} + S_{n,a}^{\rm wall} \; ,\]

and

\[\Gamma_a = \Gamma_a^{\rm neo} + \Gamma_a^{\rm tur} \; .\]

Density fluxes and sources

Variable	Definition	TGYRO unit
\(\Gamma_a^{\rm neo}\)	Neoclassical particle flux	\(1/s/cm^2\)
\(\Gamma_a^{\rm tur}\)	Turbulent particle flux	\(1/s/cm^2\)
\(S_{n,a}^{\rm beam}\)	Beam density source rate	\(1/s/cm^3\)
\(S_{n,a}^{\rm wall}\)	Wall density source rate	\(1/s/cm^3\)

Energy Transport

\[\frac{\partial \langle W_a \rangle}{\partial t} + \ \frac{1}{V^\prime}\frac{\partial}{\partial r} \left( V^\prime Q_a \right) + \Pi_a \frac{\partial \omega_0}{\partial\psi} = S_{W,a} \; ,\]

where

\[S_{W,a} = S_{W,a}^{\rm aux} + S_{W,a}^{\rm rad} + S_{W,a}^{\alpha} + S_{W,a}^{\rm tur} + S_{W,a}^{\rm col} \; ,\]

and

\[Q_a = Q_a^{\rm neo} + Q_a^{\rm tur} \; .\]

Energy fluxes and sources

Variable	Definition	TGYRO unit
\(Q_a^{\rm neo}\)	Neoclassical energy flux	\(erg/s/cm^2\)
\(Q_a^{\rm tur}\)	Turbulent energy flux	\(erg/s/cm^2\)
\(S_{W,a}^{\rm aux}\)	Auxiliary heating power density	\(erg/s/cm^3\)
\(S_{W,a}^{\rm rad}\)	Radiation heating power density	\(erg/s/cm^3\)
\(S_{W,a}^{\alpha}\)	Alpha heating power density	\(erg/s/cm^3\)
\(S_{W,a}^{\rm tur}\)	Turbulent exchange power density	\(erg/s/cm^3\)
\(S_{W,a}^{\rm col}\)	Collisional exchange power density	\(erg/s/cm^3\)

Momentum Transport

\[\frac{\partial}{\partial t} \left( \omega_0 \langle R^2 \rangle \sum_a m_a n_a \right) + \frac{1}{V^\prime}\frac{\partial}{\partial r} \left( V^\prime \sum_a \Pi_a \right) = \sum_a S_{\omega,a} \; ,\]

and

\[\Pi_a = \Pi_a^{\rm neo} + \Pi_a^{\rm tur} \; .\]

Momentum fluxes and sources

Variable	Definition	TGYRO unit
\(\Pi_a^{\rm neo}\)	Neoclassical angular momentum flux	\(erg/cm^2\)
\(\Pi_a^{\rm tur}\)	Turbulent angular momentum flux	\(erg/cm^2\)
\(S_{\omega,a}\)	Angular momentum density source rate	\(erg/cm^3\)

Connection of Fluxes to Powers

• Volume element

\[\begin{split}dV & = dr \, d\theta \, d\varphi \, J_r \\ & = dn \, dS\end{split}\]

• Jacobian

\[J_r = \frac{\partial(x,y,z)}{\partial(r,\theta,\varphi)}\]

• Normal to surface

\[dn = \frac{dr}{\left| \nabla r \right|}\]

• Volume

\[V(r) = \int_0^r dr \oint \frac{dS}{\left| \nabla r \right|} \quad \text{and} \quad V^\prime = \oint \frac{dS}{\left| \nabla r \right|}\]

• Flux-surface Average

\[\left\langle f \right\rangle = \frac{\oint d\theta \, d\varphi \, J_r \; f}{\oint d\theta \, d\varphi} = \frac{1}{V^\prime} \oint \frac{dS}{\left| \nabla r \right|} \, f\]

• Surface Area

\[\begin{split}S(r) &= \oint dS = \oint \left| \nabla r \right| \, d\theta \, d\varphi \, J_r \\ &= V^\prime \left\langle \left| \nabla r \right| \right\rangle\end{split}\]

• Flux-power relation

\[V^\prime Q = \int dr \, V^\prime \, S_W = \int dV \, S_W = P\]

• Flux-power relation units

\[V^\prime[\mathrm{cm}^2] Q [\mathrm{erg}/\mathrm{s}/\mathrm{cm}^2] = P [\mathrm{erg}/\mathrm{s}]\]