By Leonid I. Piterbarg, Alexander G. Ostrovskii (auth.)
This e-book originated from our curiosity in sea floor temperature variability. Our preliminary, although fullyyt pragmatic, target was once to derive enough mathemat ical instruments for dealing with definite oceanographic difficulties. ultimately, even though, those concerns went a long way past oceanographic purposes partially simply because one of many authors is a mathematician. We came upon that many theoretical problems with turbulent delivery difficulties were again and again mentioned in fields of hy drodynamics, plasma and strong topic physics, and arithmetic itself. There are few monographs interested by turbulent diffusion within the ocean (Csanady 1973, Okubo 1980, Monin and Ozmidov 1988). whereas deciding upon fabric for this publication we concentrated, first, on theoretical matters that may be necessary for figuring out blend methods within the ocean, and, sec ond, on our personal contribution to the matter. Mathematically all the matters addressed during this e-book are targeted round a unmarried linear equation: the stochastic advection-diffusion equation. there is not any try to derive common facts for turbulent circulation. as a substitute, the focal point is on a statistical description of a passive scalar (tracer) less than given speed information. As for functions, this booklet addresses just one phenomenon: shipping of sea floor temperature anomalies. optimistically, even though, our major ways are acceptable to different subjects.
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Additional resources for Advection and Diffusion in Random Media: Implications for Sea Surface Temperature Anomalies
Thus the parameter z characterizes the rate of decorrelation . 22) These quantities are plotted in Fig. 2 and Fig . 3. 1) with a general velocity field. Namely, let us consider the Cauchy problem with an isotropic Gaussian velocity field whose statistics are fully determined by parameters (Ttl, Ttl, and Ltl and the initial field Co (r) is determined by the single length scale, La . In doing so we assume that Co (r) is either random or deterministic. 1 (continued) e) c< 10-4 = 3, Z 1 d; = 2. 5).
42) Btl where Ue r/) I I j3(c) (tl (t ,r ) = a(c) u a(c)' j3(c) , "-e j3(c)2 = a(c) "-. 43) Such a transformation of variables is called the rescaling or renormalization. It is clear that in this situation all the time scales are functions of c only and hence any scale separation completely determines the asymptotical behavior of the mean tracer. 39) can be obtained by choosing a(c) = c 2, 37 Scale Classification j3(c) = c. 44) is the following statement on homogenization. 27). This assertion was proved in a rigorous mathematical way by different authors (Kozlov 1983, Varadhan and Papanickolaou 1982, Avellaneda and Majda 1990).
However this expression drastically depends on the relationship between the correlation time and the turnover time. If they are of the same order then the turbulent diffusivity depends on the molecular diffusivity which can essentially intensify the mixing of the fluid. In contrast, if the turnover time is much bigger then the correlation time but still much less than the observation time we arrive at the classical Fokker-Planck equation where the effective diffusivity depends on the statistics of the velocity field only.
Advection and Diffusion in Random Media: Implications for Sea Surface Temperature Anomalies by Leonid I. Piterbarg, Alexander G. Ostrovskii (auth.)