# 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

$$erg/s/cm^3$$

$$S_{W,a}^{\rm rad}$$

$$erg/s/cm^3$$

$$S_{W,a}^{\alpha}$$

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

$$erg/cm^3$$