# 11.11: Turbulent Eddies - A Cascade of Energy

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If we start with the equation:

\[\frac{K E}{\rho}=\frac{1}{2}\left(u^{2}+v^{2}+w^{2}\right) \label{11.1}\]

and we write each term as its mean and turbulent parts and then multiply all these terms out and take the Reynolds average, we can then apply the rules of averaging in Equation \ref{11.1}, and only two terms survive:

\[\frac{\overline{M K E}}{\rho}=\frac{1}{2}\left(\bar{u}^{2}+\bar{v}^{2}+\bar{w}^{2}\right)\]

\[\bar{e}=\frac{1}{2}(\overline{u^{\prime 2}}+\overline{v^{\prime 2}}+\overline{w^{\prime 2}})\]

The first term is simply the kinetic energy associated with the mean wind. The second term is the kinetic energy associated with the turbulent wind and is called the ** turbulent kinetic energy, or TKE**, for short.

Which size eddies have the most energy? We can look at the relative intensity of the different scales of wind by considering the energy associated with motions of different sizes. Remember that by the Taylor hypothesis, the size of the eddy and the period of the eddy are related so that large eddies have longer periods and smaller eddies have smaller periods.

So, relative spectral intensity is just the amount of kinetic energy associated with that size eddy and the eddy size is associated with a period required for the eddy to pass over a sensor (see figure below).

- The peak in energy at 100 hours is from fronts and weather systems as they pass over a location. The spatial scale of these phenomena is relatively large and is called the synoptic scale.
- The smaller peak at about 24 hours is the diurnal cycle of wind speed, which increases during the day and then decreases at night.
- There is often a mininum in energy (called a spectral gap) at a time scale of an hour, where the circulations or eddies are relatively weak.
- This smaller peak at about 0.1 to 0.01 hours is called the “turbulent scale." This peak is caused by production of turbulent kinetic energy by buoyancy production (i.e., convection) and shear production (i.e., viscous interaction of air masses with different velocities). These eddies have the time scales of minutes and the size of the PBL.
- As the period of the eddies decreases below about 0.01 hours (about a minute), the strength of the eddies decreases.
- Eventually, on the subsecond time scale, the eddies have very little energy indeed.

So what is happening? Energy is flowing from the larger-scale eddies to the smaller-scale eddies. Eventually, the energy is dissipated through viscosity, which is a molecular-scale process. So, the energy of the larger eddies is transferred into smaller eddies, and eventually that energy is lost to viscosity, which in turn generates heating.

Figure 1: Relative spectral energy (i.e., energy per unit frequency) as a function of eddy cycling frequency or time period. As eddies generated at the PBL scale by convection and shear break down into smaller eddies, the energy is dissipated. Credit: W. Brune, after R. B. Stull's An Introduction to Boundary Layer Meteorology (1988), from a study of I. Van der Hoven (1957).

Lewis Richardson wrote a poem about this process for whorls (a.k.a. eddies) in 1922:

Big whorls have little whorls,

Which feed on their velocity;

And little whorls have lesser whorls,

And so on to viscosity

(in the molecular sense).