# A Survey of Structure of Atmospheric Turbulence in Atmosphere and Related Turbulent Effects
## 2 Structure of Atmospheric Turbulence in Earth's Atmosphere  
### Theoretical Studies on Atmospheric Turbulence  
Viscous fluid has two flow states: laminar flow and turbulent flow. When the flow velocity of the fluid is low, the streamline is smooth, and the fluid is in a laminar flow state. When the velocity is very high, with a continuous increase in flow velocity of a fluid, the streamline is no longer smooth, and the entire fluid moves randomly and irregularly, a velocity random field is formed, and the fluid is in a turbulent flow state. Here, it should be emphasized that the turbulence mentioned above refers to velocity turbulence. In fact,  
Figure 1: Block structure. any physical quantity with uneven random change can constitute turbulence, such as temperature turbulence, humidity turbulence, and refractive-index turbulence, namely atmospheric turbulence [1]. Since turbulent motion is a random motion and turbulent field is a random function of time and space variables, statistics are required to describe it. In order to fully understand and describe the statistical properties of turbulent field, it is necessary to know all the structural functions of velocity. However, because it is difficult to obtain accurately high-order moments both theoretically and experimentally, a statistical theory of turbulence usually focuses on low-order moments, especially second-order statistical properties. In 1941, [PERSON], a former Soviet mathematician, established the statistical theory of turbulent velocity field based on the Richardson cascade model and by using a dimensional analysis method. Currently, this turbulence is also called Kolmogorov turbulence [2]. [PERSON] believed that these turbulent vortices of different scales coexisted under large Reynolds number, and the small scale turbulent vortices eventually reached statistical equilibrium in the cascade process, which formed local isotropic turbulence. At the same time, he believed that there was a uniform rate of energy dissipation in turbulent fields. According to the above assumptions, [PERSON] used a dimensional analysis method and finally provided the structural function of turbulent velocity field, which was a 2/3 power function of the space variable. In addition, spectral analysis is an important mathematical method for studying second-order statistical properties. The power spectral density of turbulence has become an important mathematical tool to describe the statistical properties of turbulent field in the statistical theory of turbulence. The power spectral density of the velocity-turbulent field is proportional to the spatial frequency to the \(-11/3\) power law [1]. As a viscous fluid, the atmosphere also has two flow states: laminar flow and turbulent flow. Normally, the atmosphere is in a state of turbulent flow. In an atmospheric turbulent field, the random change of velocity inevitably results in a random change of other physical quantities in space and time, such as temperature and refractive index. Some of these quantities are passive conservative quantities, which forms passive and conservative quantity random fields. It is proved that the structure function of passive and conservative turbulent fields should also satisfy the 2/3 law for Kolmogorov turbulence. Therefore, as one of of the passive conservative quantities, the refractive-index turbulent field should also satisfy the 2/3 law of the following form:  
\[D_{n}(r)=C_{n}^{2}r^{11/3}, \tag{1}\]  
where \(C_{n}^{2}\) is a refractive-index structure constant. Moreover, the refractive-index fluctuation power spectrum has the following form:  
\[\Phi_{n}(\kappa)=0.033\;C_{n}^{2}\;\kappa^{-11/3}, \tag{2}\]  
where \(\kappa\) is the magnitude of the spatial frequency vector in units of rad/m, which is known as the Kolmogorov power spectrum [1]. [PERSON] discussed small-scale variations of convected quantities such as temperature in turbulent fluid, which showed characteristics that are different from the conventional Kolmogorov spectrum at higher wavenumbers [3].