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

\(erg/cm^3\)