Analytical solutions of turbulence energy in the turbulence transport models

Turbulence plays vital roles in many processes in the heliosphere, such as the acceleration of solar wind, the heating of pickup ions, and the scattering of energetic charged particles. However due to the very limited observations for different time and space, we need to seek the turbulence transport model to obtain the turbulent quantities in the heliosphere. Zank et al. (2017) developed very complex models, while Wang et al. (2022) showed that the turbulence energy equation has a simple form that resembles the well-known WKB transport equation. Based on the turbulence energy equation, we derive some analytical solutions for turbulence energy by parameterizing some other turbulent quantities and adopting a simple geometry. The analytical solutions provide more transparent insights on the evolution of turbulence and are easy to be applied in other problems.