Taylor’s Hypothesis: Frozen Turbulence (Stull, 1988, p. 5) Turbulent flow can be viewed as a collection of eddies that evolve in time and space as they float along in the mean flow. (Reynolds decomposition) The structure, size and distribution of these eddies determine the mixing efficiency of turbulence and thus the importance of turbulent transport. Being turbulent, this distribution is likely very irregular in space and in time and is subject to ongoing change. How then can we begin to measure the size and distribution of these eddies that define turbulence? It is impractical (or even impossible) to have such high instrument density that every eddy is covered all the time. Even spatial “snapshots” are difficult and expensive. The Problem

In the discussion of Reynolds-averaging, we have come to the conclusion that under certain conditions (stationarity, homogeneity) an average over a time-series of observations in one place is an adequate representation of the ensemble average (incorporating all possible eddies). However, a time series essentially samples only those eddies that happen to float past a given sensor. If we can determine the average of all eddies that way, can we also determine their size? One difficulty is that such observations are not a pure time-series: Because the eddies evolve in time (change shape, fall apart etc.), we sample a mixture of a time series and a spatial transect through the flow. The Solution Taylor’s Hypothesis: Frozen Turbulence

This suggestion became known as Taylor’s Hypothesis and assumes that the rate of change of an eddy is small compared to the velocity of the mean flow, so that it changes only negligibly over the time it takes to float by the sensor. A practical suggestion for an escape from this dilemma came 1938 from one of the founders of modern turbulence theory in Cambridge (U.K.), Geoffrey I. Taylor : In certain circumstances, turbulence can be considered as “frozen”, as it passes by a sensor. Geoffrey Ingram Taylor * 7 March 1886 in St John's Wood, London, England  27 June 1975 in Cambridge, England

5 °C 10 °C 100 m 10 m/s (a) t 0 = 0 5 °C 10 °C 10 m/s (b) t 1 = t 0 +  t An idealized eddy of 100 m horizontal dimension contains a temperature difference of 5 °C from one end to the other (  T = 5 °C). At this point, the sensor measures a temperature of 10 °C. The same eddy has now floated past the sensor with a mean wind velocity of 10 m s -1. The sensor now measures 5 °C, assuming the structure of the eddy has not changed. Of course, turbulence always evolves and is in reality never frozen. However Taylor’s Hypothesis assumes that the time it takes the eddy to float past (  t = 10 s.in this case) is too small for the eddy to change noticeably. In other words: it appears to be frozen. Taylor’s Hypothesis

 Control Volume: nothing in nothing out Lagrangian framework Formally, the local rate of temperature measured by the sensor is ∂T/∂t = -0.5 K/s Thus, according to Taylor’s hypothesis the temperature change caused by advection past a point is written: This equation expresses the relationship between the temporal and spatial structure of the frozen eddy in a Eulerian frame of reference. In a Lagrangian framework, Taylor’s hypothesis for any conserved field variable, , is just: where (d /dt) denotes the total derivative.

How valid is Taylor’s Hypothesis? The better developed the turbulence, the more energy is being channeled through the energy cascade, and thus the faster the eddies evolve. A guideline for the validity of Taylor’s hypothesis can be formulated by requiring a limit in turbulence intensity (i T ): where s is the standard deviation of wind velocity and the denominator is the absolute wind speed magnitude. Experience shows that this criterion is often valid, so that the simplifications that follow from Taylor’s hypothesis can be applied.