167
2.4 Other External Forcings
In the 1600's, the entire part of the US east of the Rockies was covered
by virgin forest. By the early 1900's, essentially all of this forest was
removed to provide agricultural land and urban living space. Over these
very long time scales, the ground albedo of the eastern US was severely
changed. Similarly, population increases in other parts of the world have
impacted the nature of the land surface and changed the nature of the
surface vegetation. Quantifying changes in land cover over time on a global
scale is fraught with difficulties (e.g. Williams, 1994) and there is no general
agreement even on the current area covered by forest.
The time scale for land surface changes can be decadal in local areas but
is centennial and longer on global scales. These changes are clearly human
induced (although climate variations are affected by means of species replacements and through the influence of forest fires). While these human
induced changes can alter natural variability, their study is at such an early
stage that they will not be discussed further.
3 Stochastic Forcing
The atmosphere is constantly in motion and, while we tend not to think of
atmospheric motions as decadal in nature, it is now well understood that
rapid atmospheric motions can induce decadal and longer motions in the
more slowly varying systems that couple to the atmosphere. This section
will describe some, by now, well known results.
3.1 Some Simple Statistical Preliminaries
3.1.1 Coin Tossing
Lets start with the simplest of all possible situations: coin tossing. Let us
assume the coin is unbiased so that the probability of heads or tails on the
n'th toss is always 50%. We might assume that the sum of the number
of heads minus tails would stay close to zero but this is not true (Feller,
1968). Fig. 4 shows the result of a coin tossing experiment of 5000 tosses.
Let the outcome of a toss be Pn = ±1 and let us define a sum of the
number of heads and tails as:
N
X(N) = LPn
n=l
2.4 Other External Forcings
In the 1600's, the entire part of the US east of the Rockies was covered
by virgin forest. By the early 1900's, essentially all of this forest was
removed to provide agricultural land and urban living space. Over these
very long time scales, the ground albedo of the eastern US was severely
changed. Similarly, population increases in other parts of the world have
impacted the nature of the land surface and changed the nature of the
surface vegetation. Quantifying changes in land cover over time on a global
scale is fraught with difficulties (e.g. Williams, 1994) and there is no general
agreement even on the current area covered by forest.
The time scale for land surface changes can be decadal in local areas but
is centennial and longer on global scales. These changes are clearly human
induced (although climate variations are affected by means of species replacements and through the influence of forest fires). While these human
induced changes can alter natural variability, their study is at such an early
stage that they will not be discussed further.
3 Stochastic Forcing
The atmosphere is constantly in motion and, while we tend not to think of
atmospheric motions as decadal in nature, it is now well understood that
rapid atmospheric motions can induce decadal and longer motions in the
more slowly varying systems that couple to the atmosphere. This section
will describe some, by now, well known results.
3.1 Some Simple Statistical Preliminaries
3.1.1 Coin Tossing
Lets start with the simplest of all possible situations: coin tossing. Let us
assume the coin is unbiased so that the probability of heads or tails on the
n'th toss is always 50%. We might assume that the sum of the number
of heads minus tails would stay close to zero but this is not true (Feller,
1968). Fig. 4 shows the result of a coin tossing experiment of 5000 tosses.
Let the outcome of a toss be Pn = ±1 and let us define a sum of the
number of heads and tails as:
N
X(N) = LPn
n=l