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

$$\frac{\partial ⟨n_a⟩}{\partial t}+\frac{1}{V^{\prime }}\frac{\partial }{\partial r}\left(V^{\prime }{\mathrm{\Gamma }}_a\right)=S_{n,a},$$

where

$$S_{n,a}=S_{n,a}^{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}+S_{n,a}^{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}},$$

and

$${\mathrm{\Gamma }}_a={\mathrm{\Gamma }}_a^{\mathrm{n}\mathrm{e}\mathrm{o}}+{\mathrm{\Gamma }}_a^{\mathrm{t}\mathrm{u}\mathrm{r}}.$$

Density fluxes and sources

Variable	Definition	TGYRO unit
${\mathrm{\Gamma }}_a^{\mathrm{n}\mathrm{e}\mathrm{o}}$	Neoclassical particle flux	$1/s/c{m}^2$
${\mathrm{\Gamma }}_a^{\mathrm{t}\mathrm{u}\mathrm{r}}$	Turbulent particle flux	$1/s/c{m}^2$
$S_{n,a}^{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}$	Beam density source rate	$1/s/c{m}^3$
$S_{n,a}^{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}$	Wall density source rate	$1/s/c{m}^3$

Energy Transport

$$\frac{\partial ⟨W_a⟩}{\partial t}+\frac{1}{V^{\prime }}\frac{\partial }{\partial r}\left(V^{\prime }{Q}_a\right)+{\mathrm{\Pi }}_a\frac{\partial {\omega }_0}{\partial \psi }=S_{W,a},$$

where

$$S_{W,a}=S_{W,a}^{\mathrm{a}\mathrm{u}\mathrm{x}}+S_{W,a}^{\mathrm{r}\mathrm{a}\mathrm{d}}+S_{W,a}^{\alpha }+S_{W,a}^{\mathrm{t}\mathrm{u}\mathrm{r}}+S_{W,a}^{\mathrm{c}\mathrm{o}\mathrm{l}},$$

and

$$Q_a=Q_a^{\mathrm{n}\mathrm{e}\mathrm{o}}+Q_a^{\mathrm{t}\mathrm{u}\mathrm{r}}.$$

Energy fluxes and sources

Variable	Definition	TGYRO unit
$Q_a^{\mathrm{n}\mathrm{e}\mathrm{o}}$	Neoclassical energy flux	$erg/s/c{m}^2$
$Q_a^{\mathrm{t}\mathrm{u}\mathrm{r}}$	Turbulent energy flux	$erg/s/c{m}^2$
$S_{W,a}^{\mathrm{a}\mathrm{u}\mathrm{x}}$	Auxiliary heating power density	$erg/s/c{m}^3$
$S_{W,a}^{\mathrm{r}\mathrm{a}\mathrm{d}}$	Radiation heating power density	$erg/s/c{m}^3$
$S_{W,a}^{\alpha }$	Alpha heating power density	$erg/s/c{m}^3$
$S_{W,a}^{\mathrm{t}\mathrm{u}\mathrm{r}}$	Turbulent exchange power density	$erg/s/c{m}^3$
$S_{W,a}^{\mathrm{c}\mathrm{o}\mathrm{l}}$	Collisional exchange power density	$erg/s/c{m}^3$

Momentum Transport

$$\frac{\partial }{\partial t}({\omega }_0⟨R^2⟩\sum _a{m}_a{n}_a)+\frac{1}{V^{\prime }}\frac{\partial }{\partial r}\left(V^{\prime }\sum _a{\mathrm{\Pi }}_a\right)=\sum _a{S}_{\omega ,a},$$

and

$${\mathrm{\Pi }}_a={\mathrm{\Pi }}_a^{\mathrm{n}\mathrm{e}\mathrm{o}}+{\mathrm{\Pi }}_a^{\mathrm{t}\mathrm{u}\mathrm{r}}.$$

Momentum fluxes and sources

Variable	Definition	TGYRO unit
${\mathrm{\Pi }}_a^{\mathrm{n}\mathrm{e}\mathrm{o}}$	Neoclassical angular momentum flux	$erg/c{m}^2$
${\mathrm{\Pi }}_a^{\mathrm{t}\mathrm{u}\mathrm{r}}$	Turbulent angular momentum flux	$erg/c{m}^2$
$S_{\omega ,a}$	Angular momentum density source rate	$erg/c{m}^3$

Connection of Fluxes to Powers

• Volume element

$$\begin{array}{rl}dV& =drd\theta d\phi J_r\\ & =dndS\end{array}$$

• Jacobian

$$J_r=\frac{\partial (x,y,z)}{\partial (r,\theta ,\phi )}$$

• Normal to surface

$$dn=\frac{dr}{\left|\mathrm{\nabla }r\right|}$$

• Volume

$$V(r)={\int }_0^{r}dr\oint \frac{dS}{\left|\mathrm{\nabla }r\right|}\text{and}{V}^{\prime }=\oint \frac{dS}{\left|\mathrm{\nabla }r\right|}$$

• Flux-surface Average

$$⟨f⟩=\frac{\oint d\theta d\phi J_{r}f}{\oint d\theta d\phi }=\frac{1}{V^{\prime }}\oint \frac{dS}{\left|\mathrm{\nabla }r\right|}f$$

• Surface Area

$$\begin{array}{rl}S(r)& =\oint dS=\oint \left|\mathrm{\nabla }r\right|d\theta d\phi J_r\\ & =V^{\prime }⟨\left|\mathrm{\nabla }r\right|⟩\end{array}$$

• Flux-power relation

$$V^{\prime }Q=\int dr{V}^{\prime }{S}_W=\int dV{S}_W=P$$

• Flux-power relation units

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