Getting to know turbulence - Turbulence in the Atmosphere - page 35

Similarly, if u is homogeneous in x ,sothat U = U(t) , we intuitively expect this

spatial average in a single realization to converge to the ensemble average as the
averaging distance L increases:

U L (x,t,T lim

L

; m) = U(t). (2.26)

→∞

If u is both homogeneous and stationary, so that U does not depend on x or t ,we

expect both the time and spatial averages to converge to the ensemble average U .
The property that the time average of a stationary random variable and the space
average of a homogeneous random variable converge to the ensemble average is
called ergodicity. Physically, ergodicity means that any unbiased average of a vari-
able converges to the ensemble average. We routinely determine the ensemble mean
of a stationary, time-varying signal at a point in space through time averaging.

˜

2.4 The convergence of averages
The ensemble average has ideal properties: it commutes with linear operations, a
second application has no effect (i.e., the average of an average is the average),
and the average of a fluctuation is zero. But it can be very tedious to use with
observational data, and as discussed in Section 2.3 experimentalists typically use a
time average in stationary conditions instead.
Mechanically driven turbulent flows are often stationary by design. The atmo-
spheric boundary layer is inherently nonstationary, however, because of the diurnal
cycle and the changing synoptic conditions. But one can often find quasi-stationary
periods of up to a few hours length, which could be long enough in some problems.
This raises the question: How long must we average in time to get acceptably close
to the ensemble average? We'll outline the general answer here using statistical
concepts that we'll discuss more fully in Part III .
To answer this question † we begin with the time average of a stationary function
of time u(t) over an interval T , ˜

t 0 + T

1
T

u T u(t )dt , = ˜ (2.27)

t 0

the initial time t 0 being arbitrary. Here u(t) could be the time series of the streamwise

velocity component at a point in a turbulent flow. We shall call T the averaging
time. Now let

˜

u(t) , the sum of ensemble-mean and fluctuating parts.
By our stationarity assumption U does not depend on time. The difference between
the time and ensemble means of

u(t) ˜ = U +

u is then ˜

†

This discussion is adapted from Lumley and Panofsky ( 1964 ).