On turbulent viscosity in relativistic jets and accretion disks

TL;DR

This paper develops and tests an entrainment model for turbulent viscosity in relativistic jets and accretion disks, successfully matching experimental data and deriving a novel expression for the viscosity parameter without relying on traditional turbulence models.

Contribution

It introduces a constrained entrainment model for turbulence in relativistic jets and accretion disks, aligning with experimental data and providing a new formulation for the viscosity parameter.

Findings

The model predicts the Smagorinsky constant $C_S \\approx 0.11$, consistent with experimental shear flow data.

For accretion disks, the model derives the same mass accretion rate as the classical $\alpha$-model.

The viscosity parameter $\alpha$ depends on the temperature slope and turbulent velocity, offering a new perspective.

Abstract

The mechanism of turbulent viscosity is the central question in investigations of turbulence. This is also the case in the accretion disk theory, where turbulence is considered to be responsible for the outward transport of angular momentum in the accretion disk. In turbulent flows, vortices transport momentum over their length scales providing the mechanism of viscosity that is controlled by mass entrainment. We have earlier proposed an entrainment model for the particular case of the relativistic jets in the radio galaxy 3C31. In this paper, we further constrain the model parameters. The model (in the non-relativistic part) is successfully tested versus experimental and simulation data on the Reynolds stresses of free mixing layers and predicts the Smagorinsky constant $C_\mathrm{S} \approx 0.11$, which is consistent with the experimental range for shear flows $C_\mathrm{S} \approx 0.1-0.12$. For accretion disks, the entrainment model allows us to derive the same accretion mass rate as in the Shakura \& Sunyaev's $\alpha$-model without appealing to the turbulent kinematic viscosity $\nu_\mathrm{t}$, and the viscosity parameter $\alpha$ derived in the form $\displaystyle \alpha = -\frac{8}{3} \beta s_\mathrm{T} \frac{\mathrm{v_t}^2}{c_\mathrm{s}^2}$ depends on the power $s_\mathrm{T}$ of the temperature slope along the disk radius, $T\propto r^{s_\mathrm{T}}$, and quadratically on the turbulent velocity $\mathrm{v_t}$.