Update: also see posts 382, 460, 462, 460.
We often hear a distinction made between "climate" and "weather". It may surprise people that the famous mathematician, Benoit Mandelbrot, thought about this problem with completely opposite conclusions to realclimate. Mandelbrot is a prolific author who invented and popularized the concept of fractals . His popular book, The Fractal Geometry of Nature, is beautifully illustrated and well worth reading. Mandelbrot and Wallis  not only considers the distinction between climate and weather, but considers earlier versions of many tree ring series in the infamous North American tree ring network. (This consideration seems to be completely lost to dendrochronologists.)
Here is an extended quote from Mandelbrot and Wallis :
Among the classical dicta of the philosophy of science is Descartes’ prescription to “divide every difficulty into portions that are easier to tackle than the whole…. This advice has been extraordinarily useful in classical physics because the boundaries between distinct sub-fields of physics are not arbitrary. They are intrinsic in the sense that phenomena in different fields interfere little with each other and that each field can be studied alone before the description of the mutual interactions is attempted.
Subdivision into fields is also practised outside classical physics. Consider for example, atmospheric science. Students of turbulence examine fluctuations with time scales of the order of seconds or minutes, meteorologists concentrate on days or weeks, specialists whom one might call macrometeorologists concentrate on periods of a few years, climatologists deal with centuries and finally paleoclimatologists are left to deal with all longer time scales. The science that supports hydrological engineering falls somewhere between macrometeorology and climatology.
The question then arises whether or not this division of labour is intrinsic to the subject matter. In our opinion, it is not in the sense that it does not seem possible when studying a field in the above list, to neglect its interactions with others, We therefore fear that the division of the study of fluctuations into distinct fields is mainly a matter of convenient labelling and is hardly more meaningful than either the classification of bits of rock into sand, pebbles, stones and boulders or the classification of enclosed water-covered areas into puddles, ponds, lakes and seas,
Take the examples of macrometeorology and climatology. They can be defined as the sciences of weather fluctuations on time scales respectively smaller and longer than one human lifetime. But more formal definitions need not be meaningful. That is, in order to be considered really distinct, macrometeorology and climatology should be shown by experiment to be ruled by clearly separated processes, In particular there should exist at least one time span on the order of one lifetime that is both long enough for micrometeorological fluctuations to be averaged out and short enough to avoid climate fluctuations…
It is therefore useful to discuss a more intuitive example of the difficulty that is encountered when two fields gradually merge into each other. We shall summarize the discussion in M1967s of the concept of the length of a seacoast or riverbank. Measure a coast with increasing precision starting with a very rough scale and dividing increasingly finer detail. For example walk a pair of dividers along a map and count the number of equal sides of length G of an open polygon whose vertices lie on the coast. When G is very large the length is obviously underestimated. When G is very small, the map is extremely precise, the approximate length L(G) accounts for a wealth of high-frequency details that are surely outside the realm of geography. As G is made very small, L(G) becomes meaninglessly large. Now consider the sequence of approximate length that correspond to a sequence of decreasing values of G. It may happen that L(G) increases steadily as G decreases, but it may happen that the zones in which L(G) increases are separated by one or more “shelves” in which L(G) is essentially constant. To define clearly the realm of geography, we think that it is necessary that a shelf exists for values of G near àŽ». where features of interest to the geographer satisfy G>=àŽ» and geographically irrelevant features satisfy GF<ï〢½ ï〢½/ If a shelf exists, we call G(àŽ») a coast length”⤍
After this preliminary, let us return to the distinction between macrometeorology and climatology. It can be shown that to make these fields distinct, the spectral density of the fluctuations much have a clear-cut “dip” in the region of wavelengths near àŽ» with large amounts of energy located on both sides, “⤠But in fact no clear-cut dip is ever observed.
When one wishes to determine whether or not such distinct regimes are in fact observed, short hydrological records of 50 or 100 years are of little use. Much longer records are needed; thus we followed Hurst in looking for very long records among the fossil weather data exemplified by varve thickness and tree ring indices. However even when the R/s diagrams are so extended, they still do not exhibit the kinds of breaks that identifies two distinct fields.
In summary the distinctions between macrometeorology and climatology or between climatology and Paleoclimatology are unquestionably useful in ordinary discourse. But they are not intrinsic to the underlying phenomena.
Mandelbrot calculates Hurst indices and 3rd and 4 th moments for 12 varve series, 27 tree ring series from western U.S. (no bristlecones), 9 precipitation series, 1 earthquake frequency series, 11 river series and 3 Paleozoic sediment series.
Mandelbrot and Wallis, 1969. Global dependence in geophysical records, Water Resources Research 5, 321-340.