### 11.4 Frozen: The Taylor Hypothesis

We would like to be able to take snapshots of the eddies in three dimensions and measure all their sizes each instant. Unfortunately, we do not have a good way to do this. Instead, we can simply measure the fluctuations of a variable such as wind speed, specific humidity, or temperature with a sensor at one location for a period of time. In this way, we watch the eddies drift by the sensor. But the eddies could be changing size and shape as they drift by the sensor. Let’s put this physical concept in the context of the total derivative.

Take a variable like temperature, *T*. We know that the change in *T* with time at any location (such as where a sensor might be placed) is the sum of the total derivative and the temperature advection:

$\frac{\partial T}{\partial t}=\frac{DT}{Dt}-\overrightarrow{U}\xb7\overrightarrow{\nabla}T$

The temperature advection is the change in temperature at the sensor due to the advection of warmer or colder air past the sensor. The total derivative is the change in temperature of an air parcel moving past the sensor. Such a temperature change may be caused by any number of processes, such as the absorption or emission of radiation, condensation or evaporation (latent heating or cooling), or compression and expansion. **Taylor’s hypothesis** says that we can assume that the turbulent eddies (which we can think of as the circulation of air parcels) do not significantly evolve, or are "frozen", as they advect past the sensor and thus the change in temperature within each eddy is negligible:

$\frac{DT}{Dt}~0$

so that:

$\frac{\partial T}{\partial t}=-\overrightarrow{U}\xb7\overrightarrow{\nabla}T$

Local temperature gradients, which might be present from one side of an eddy to another, are advected across the sensor by the mean wind without the eddy changing.

When is this condition valid? Experiments suggest that this hypothesis is valid when the variation of the wind speed due to turbulence is less than ½ of the mean wind speed.

Why is this hypothesis, or maybe more precisely stated, this approximation, useful? It is commonly used to infer the sizes of turbulent structures from observations of turbulence taken over a period of time. If this approximation hold true, the size of the turbulent eddy can be estimated by multiplying the time it takes an eddy to pass by a sensor by the mean wind speed that is "carrying" this "frozen" eddy. Thus eddy size equals eddy advection time multiplied by the mean wind. This approach has proven to be very useful for understanding the structure of turbulence.

We start this study with methods to separate wind motion driven by larger scale processes, such as gradient flow or geostrophic flow, from turbulence.

#### Quiz 11-1: Boundary layer behavior.

- Please note:
**there is no practice quiz**for Quiz 11-1. - When you feel you are ready, take
**Quiz 11-1**in Canvas. You will be allowed to take this quiz only**once**. Good luck!