# Turbulence Energy Cascade Theory

To investigate physical properties of turbulent liquids, in the earliest study of turbulent flow, Reynolds used the theory “of similarity” to define a nondimensional quantity *Re = V? Hv,* called the Reynolds number [6,8,17-20], where *V*and *l* are

14 ? *Optical Waves and Laser Beams in the Irregular Atmosphere*

the characteristic velocity (in m/s) and size (in m) of the flow respectively, and *v* is the kinematic viscosity (in m^{2}/s). The transition from laminar to turbulent motion takes place at a critical Reynolds number, above which the motion is considered to be turbulent. The kinematic viscosity *v* of air is roughly 10^{-5} m^{2} s^{-1} [6,8]. Air motion is considered highly turbulent in the boundary layer and troposphere, where the Reynolds numbers Re ^ 10^{5} [6,8,14]. Richardson [8] first developed a theory of the turbulent energy redistribution in the atmosphere, which he called the energy cascade theory. It was noticed that smaller-scale motions originated as a result of the instability of larger ones. A cascade process, shown in Figure 1.1 taken from Reference 8, in which eddies of the largest size are broken into smaller and smaller ones, continues down to scales in which the dissipation mechanism turns the kinetic energy of motion into heat.

Let us denote by *l* the current size of turbulence eddies, by *L _{0}* and

*l*their outer and inner scales, and by к

_{0}_{0}= (2n/L

_{0}),

*к*= (2n/l) and

*K*= (2n/l

_{m}_{0}) the spatial wave numbers of these kinds of eddies, respectively. Using this notation, one can divide turbulences in the three regions:

These three regions induce strong, moderate, and weak spatial and temporal variations, respectively, of signal amplitude and phase, referred to in the literature as *scintillations* [6,8,20].

Figure 1.1 Richardson's cascade theory of turbulence.

Kolmogorov [44] introduced a hypothesis stating that during the cascade process the direct influence of larger eddies is lost and smaller eddies tend to have independent properties, universal for all types of turbulent flows. Following Kolmogorov, the energy cascade process consists of an energy input region, inertial subrange, and energy dissipation region, as sketched in Figure 1.1.

At large characteristic scales or eddies, a portion of kinetic energy in the atmosphere is converted into turbulent energy. When the characteristic scale reaches a specified outer scale size, L_{0}, the energy begins a cascade that forms a continuum of eddy size for energy transfer from a macroscale L_{0} to a microscale *l _{0}.* The scale sizes

*l*bounded above by L

_{0}and below by l

_{0}form the inertial subrange.

Kolmogorov proposed that in the inertial subrange, where L_{0} > *l >* l_{0}, turbulent motions are both homogeneous and isotropic and energy may be transferred from eddy to eddy without loss, that is, the amount of energy that is being injected into the largest structure must be equal to the energy that is dissipated as heat [44].

The term homogeneous is analogous to stationary process [6,8,20] and implies that the statistical characteristics of the turbulent flow are independent of position within the flow field. The term isotropic requires that the second and higher order statistical moments depend only on the distance between any two points in the field.

The inertial subrange is dominated by inertial forces and the average properties of the turbulent flow are determined only by the average dissipation rate ? (in units m^{2}/s^{3}) of the turbulent kinetic energy. When the size of a decaying eddy reaches l_{0}, the energy is dissipated as heat through viscosity processes. It was also hypothesized that the motion associated with the small-scale structure l_{0} is uniquely determined by the kinematic viscosity *v* and ?, where l_{0} *^r) =* (v^{3}/e)^{1/4} is the Kolmogorov microscale [19,44—47]. The Kolmogorov microscale defines the eddy size dissipating the kinetic energy. The turbulent process, shown schematically in Figure 1.1 according to the simple theory of Richardson, was then summarized by Kolmogorov and Obukhov (called in the literature the Kolmogorov—Obukhov turbulent cascade process [5]) as follows: the average dissipation rate ? of the turbulent kinetic energy will be distributed over the spatial wavelength к-range as [6]:

where, as above, *L*_{0}, *l*, and *l*_{0} are the initial (outer), current, and inner turbulent eddy sizes.

In general, turbulent flow in the atmosphere is neither homogeneous nor isotropic. However, it can be considered locally homogeneous and isotropic in small subregions of the atmosphere.