Advection and Diffusion in Random Media: Implications for by Leonid I. Piterbarg, Alexander G. Ostrovskii (auth.) PDF

By Leonid I. Piterbarg, Alexander G. Ostrovskii (auth.)

This booklet originated from our curiosity in sea floor temperature variability. Our preliminary, although totally pragmatic, aim was once to derive sufficient mathemat­ ical instruments for dealing with definite oceanographic difficulties. ultimately, even if, those issues went a ways past oceanographic purposes in part simply because one of many authors is a mathematician. We stumbled on that many theoretical problems with turbulent delivery difficulties have been many times mentioned in fields of hy­ drodynamics, plasma and sturdy subject physics, and arithmetic itself. There are few monographs involved in turbulent diffusion within the ocean (Csanady 1973, Okubo 1980, Monin and Ozmidov 1988). whereas identifying fabric for this e-book we centred, first, on theoretical matters which may be priceless for figuring out blend approaches within the ocean, and, sec­ ond, on our personal contribution to the matter. Mathematically all the concerns addressed during this publication are centred round a unmarried linear equation: the stochastic advection-diffusion equation. there is not any try and derive common statistics for turbulent circulation. in its place, the point of interest is on a statistical description of a passive scalar (tracer) below given speed records. As for purposes, this ebook addresses just one phenomenon: shipping of sea floor temperature anomalies. confidently, besides the fact that, our major methods are appropriate to different subjects.

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In case (D) the particle trapped by an eddy completes a number of full revolutions before the eddy dies. Finally in case (E), the number of complete revolutions can be very large. From an analytical viewpoint cases (A) ,(B) are not difficult. For model (A) we have Fickian equation for the mean concentration with effective diffusivity (Kubo 1963) J 00 D ~ 6"2 Ru(t)dt. 62) Scale Classification 41 In case (B) we have a similar result J 00 D '" c2 Ru(t, O)dt. 63) o In both cases the effective diffusivity is of the order c 2 and hence is much smaller than molecular diffusi vi ty (TD '" 1).

Bij ( r, r )ac(t,r) ax . j (c' (t, r ).. ) . "V(c) "V. ,s 1 = 1 2'''V. Bu>. 24) + "V. (r,r) and Bi,s(r,r) respectively. 48 CHAPTER 3 Now we show another derivation of this equation. 48). 67). For derivation of the equation for the mean tracer we need to extend this expansion up to the order r2. 26) r dWj(u)du + o(r2), Jo where as before de(r) = et+T,r(t) - r, dw(r) = w(t + r) - w(t) and the variable subscripts refer to the corresponding components of the vectors. 26) is of order r 1 / 2 , the order of the fourth term is r3/2.

Among these three types of assumptions, in our opinion the scale constraints approach is most preferable due to two advantages. First, the relation between the scales may be stated in a rigorous mathematical form (later we will demonstrate this). Second, the scale relations can be often verified in practice via experimental data. As for correlation constraints, the essential disadvantage is that of referring to the unknown field c'(t,r) which must be found from the original problem. Hence the constraints can be checked only after the problem is solved.

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