Marburg, Germany

Dust, aerosoles, smoke and other air pollutions spread, in still air, by
molecular diffusion. In addition there can be advection by atmospheric flow,
either well directed in jets or winds, or nondirected in air turbulence.
Diffusional spreading by turbulence is extraordinary effective. Instead
of the well known linear growth of the mean square particle distance with time

grows much faster under turbulent advection, according to a cubic dependence

This was discovered by Lewis Fry Richardson (1926, 1929) when he performed his ingenious study of the turbulent diffusivity . He presented data on impressively many scales. A bulk of later measurements confirmed his results, as e.g. balloon campaignes, cf. Lundgren (1981).

The anomalously fast particle distance growth could be explained from fluid dynamics (Navier-Stokes equations) by Grossmann and Procaccia (1984), Effinger and Grossmann (1984), Grossmann (1990) in mean field approximation. New results on dynamical Lagrangian time correlation decay (numerical, Grossmann and Wiele 1997 and analytical, Daems et al. 1999) stimulated efforts to even determine the absolute magnitude as characterized by the prefactor in addition to the scaling exponent (Grossmann 2002).

In particular the memory effects in the time correlation decay have turned out to be very important. And turbulent intermittency implies additional scale dependence of the turbulent diffusivity (Grossmann 2002). A survey on the past development and on the most recent surprising findings is offered in the lecture.

**References**

Daems D, Grossmann S, L'vov V and Procaccia I 1999 *Phys Rev* **E60** 6656

Effinger H and Grossmann S 1984 *Phys Rev Lett* **50** 442

Grossmann S and Procaccia I 1984 *Phys Rev* **A29** 1358

Grossmann S 1990 *Annalen der Physik (Leipzig)* **47** 577

Grossmann S and Wiele C 1997 *Z Phys* **B103** 469

Grossmann S 2002 *to be published*

Lundgren T S 1981 *J Fluid Mech* **111** 27

Richardson L F 1926 *Proc Roy Soc (London)* **A110** 709

Richardson L F 1929 *Beitr Phys Atm* **15** 24

Marburg, Germany

The appropriate tool to study the **dynamics** of a statistical
system is its (stationary) time correlation
function. This has also proven to be true for
turbulent fluid flow. The observable quantities of
interest are here the Lagrangian, scale dependent
eddies , i.e., the Eulerian velocity
differences
. These objects
are statistically time and space independent in
stationary and homogeneous turbulence, but depend
besides on scale on the time lapse
between two observations of an -eddy at times
and . By means of a continued fraction expansion the time correlation
function can be uniquely expressed in terms of the static, stationary, time independent
structure functions; but all orders of those are needed.

Analysis of the dynamical time correlation function
is
presented. was first studied in 1-pole approximation (Grossmann and
Thomae 1982), because no estimate of higher order stationary structure functions was
available then. Since Grossmann and Wiele (1997) provided numerical
data for large Reynolds number, highly turbulent flow, it became clear
that the memory effects surprisingly *reduce* the decorrelation
time. The additional effects of turbulent intermittency were elucidated
analytically in Daems et al. (1999). The main, rather unexpected results
are:

i. The static multifractality of turbulent flow destroys dynamical scaling despite good scaling of stationary moments, i.e., of power laws in of all order structure functions. ii. The deviations from dynamical scaling are a direct measure of the strength of intermittency. iii. The scale dependence of the correlation decay rate can be expressed approximately in terms of the turbulent structure function of order (Grossmann 2002).

**References**

Daems D, Grossmann S, L'vov V and Procaccia I 1999 *Phys Rev* **E60** 6656

Grossmann S and Thomae S 1982 *Z Phys* **B49** 253

Grossmann S and Wiele C 1997 *Z Phys* **B103** 469

Grossmann S 2002 *to be published*

Marburg, Germany

Turbulent heat transport in fluid layers heated from below is one
of the most intensely studied fluid flow problems. In 1900
Bénard detected interesting pattern formation in this system,
in 1916 Rayleigh calculated the underlying instability. If the
heating is increased further, chaotic motion and finally
turbulence is observed in the bulk , surrounded by Blasius type
boundary layers near the plates and walls. In the 50*th* and
60*th* power laws for the heat transport as a function of
the thermal driving were suggested for the Rayleigh-Bénard
system in accordance with other scaling behavior in turbulence,

The Nusselt number is the effective heat current including turbulent advection, nondimensionalised with the molecular heat current. The Rayleigh number measures the thermal driving due to the temperature difference across the fluid layer of height , originating from buoyancy in the gravitational field by thermal expansion , which is counteracted by the fluid's kinematic viscosity and its thermal diffusivity ,

The Prandtl number weighs the ratio of the space and time scales according to

**References**

Ahlers G and Xu X 2001 *Phys Rev Lett* **86** 3320

Castaing B, Gunaratne G, Heslot F, Kadanoff L, Libchaber A, Thomae S, Wu X, Zaleski S and
Zanetti G 1989 *J Fluid Mech* **204** 1

Chavanne X, Chilla F, Castaing B, Hebral B, Chaboud B and Chaussy J 1997 *Phys Rev Lett*
**79** 3648

Cioni S, Ciliberto S and Sommeria J 1997 *J Fluid Mech* **335** 111

Grossmann S and Lohse D 2000 *J Fluid Mech* **407** 27

Grossmann S and Lohse D 2001 *Phys Rev Lett* **86** 3316

Grossmann S and Lohse D 2002 *to be published*

Niemela J, Skrbek L, Sreenivasan K R and Donnelly R 2000 *Nature* **404** 837

Xia K Q, Lam S and Zhou S Q 2002 *Phys Rev Lett* **88** 064501

Xu X, Bajaj K M S and Ahlers G 2000 *Phys Rev Lett* **84** 4357