169
For t very small, R(T) ~ 1 and < y2(t) >:::::i< v 2 > t 2 , while for t large,
J R(T)dT = TI so that < y2(t) >= 2TI < v 2 > t and the rms distance from
o
zero, Yrms ex: Vi.
The analogy to coin tossing puts v as the rapid variable (analogous to
Pn ) and y as the integral of v (analogous to X) : just as in coin tossing,
after a very long time the sum y (analogous to X) is unlikely to return to
zero. While the spectrum of the coin tossing is white, the spectrum of the
sum is clearly red. As the record gets longer and longer, the likelihood of
X (N) returning to zero for large N becomes vanishingly small.
3.1.2 The variance of sample means
Consider a stationary time series with variance (72 and with correlation
time TI . Consider means over a time T »TI . The variance of the means
over time T is given by:
(7f = ~ where N = ~ is the number of correlation times in the interval
over which the mean is taken which is equivalent to the number of independent samples. We see that (7T cx: .}r so that it takes a long time to reduce
the variance of the long term means. As an example, if we consider the
annual mean temperature at a point in the presence of diurnal and other
variations, if the correlation time is of order of a week, then the annual
mean temperature will have a variance only a seventh of the variance of
the original temperature time series. We will use this later when the variance of decadal sample means is discussed in coupled general circulation
models.
3.2 The Hasselmann Mechanism
Hasselmann (1976) proposed a stochastic climate model that can most
simply be understood in terms of the coin tossing example given above
(also see Wunsch, 1992). Consider a model thermal equation of the form:
dT =Q
dt
where the fluxes Q are taken to be some sort of a random forcing on the
rate of change of temperature (this is like the forcing of a mixed layer by
random fluxes of heat at the surface). The spectra of the forcing and the
response is given by:
For t very small, R(T) ~ 1 and < y2(t) >:::::i< v 2 > t 2 , while for t large,
J R(T)dT = TI so that < y2(t) >= 2TI < v 2 > t and the rms distance from
o
zero, Yrms ex: Vi.
The analogy to coin tossing puts v as the rapid variable (analogous to
Pn ) and y as the integral of v (analogous to X) : just as in coin tossing,
after a very long time the sum y (analogous to X) is unlikely to return to
zero. While the spectrum of the coin tossing is white, the spectrum of the
sum is clearly red. As the record gets longer and longer, the likelihood of
X (N) returning to zero for large N becomes vanishingly small.
3.1.2 The variance of sample means
Consider a stationary time series with variance (72 and with correlation
time TI . Consider means over a time T »TI . The variance of the means
over time T is given by:
(7f = ~ where N = ~ is the number of correlation times in the interval
over which the mean is taken which is equivalent to the number of independent samples. We see that (7T cx: .}r so that it takes a long time to reduce
the variance of the long term means. As an example, if we consider the
annual mean temperature at a point in the presence of diurnal and other
variations, if the correlation time is of order of a week, then the annual
mean temperature will have a variance only a seventh of the variance of
the original temperature time series. We will use this later when the variance of decadal sample means is discussed in coupled general circulation
models.
3.2 The Hasselmann Mechanism
Hasselmann (1976) proposed a stochastic climate model that can most
simply be understood in terms of the coin tossing example given above
(also see Wunsch, 1992). Consider a model thermal equation of the form:
dT =Q
dt
where the fluxes Q are taken to be some sort of a random forcing on the
rate of change of temperature (this is like the forcing of a mixed layer by
random fluxes of heat at the surface). The spectra of the forcing and the
response is given by: