Tuck A.F. Atmospheric Turbulence: A Molecular Dynamics Perspective

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Atmospheric Turbulence

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Atmospheric

Turbulence:

A Molecular Dynamics

Perspective

Adrian F. Tuck

Great Clarendon Street, Oxford OX2 6DP

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Dedication

The author dedicates this book to his mentors P. G. Ashmore, B. A. Thrush,

K. R. Wilson, and R. J. Murgatroyd, who in their different ways were

everything scientists should be.

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Preface

This book grew out of a series of four invited lectures given at the

Meteorology Department, University of Reading, in February 2006, at the

instigation of Professor Alan O’Neill, director of the NERC Data Assimila-

tion Research Centre. The questions asked there served to sharpen some of

the points which had been presented. The material in those lectures drew on

work published in the literature during the previous seven years, applying

Schertzer and Lovejoy’s theory of generalized scale invariance to a large

body of airborne data which clearly showed atmospheric variability well

above instrumental noise levels but which could not be described adequately

by Gaussian PDFs and second moment power spectra.

The atmosphere is molecules in motion, but a lacuna exists as regards

explicit discussion or treatment of this fact in the meteorological literature

and among standard texts on dynamic meteorology, ﬂuid mechanics, tur-

bulence, multifractals, non-equilibrium statistical mechanics, and kinetic

molecular theory. While texts on atmospheric chemistry of course deal in

molecular behaviour, the step from kinetic molecular theory to atmospheric

motion is made often without comment, usually via application of the law

of mass action on scales many orders of magnitude larger than the mean

free path and on which true diffusion cannot be dominant. Discussions of

the progression from molecular to ﬂuid motion are found mainly in the

statistical mechanics literature but with no consideration of complicated

anisotropic large-scale ﬂows within morphologically irregular boundaries,

such as those the atmosphere exhibits. There are few examples of attempts

to examine the molecular roots of turbulence. This book aims to point out

the need to address this situation, and to offer suggestions about how to

proceed.

The central point of this book is that molecular dynamics, via the gener-

ation of vorticity in the presence of anisotropies such as gravity, planetary

rotation, and the solar beam, inﬂuences the structure of turbulence, temper-

ature, radiative transfer, and chemistry in the atmosphere. Because energy

is deposited in the atmosphere by molecules absorbing photons, energy

must propagate upward from the smallest scales. Analyses are presented of

observations by the statistical multifractal methods developed by Schertzer

and Lovejoy, which show generalized scale invariance in the atmosphere.

The need to unite molecular dynamics, turbulence theory, ﬂuid mechan-

ics, and non-equilibrium statistical mechanics is reinforced by the fact that

viii Preface

core wind speeds in jet streams can exceed one third of the most prob-

able velocity of air molecules, a breach of the conditions under which

standard derivations of the Navier–Stokes equation are made. Note that

in saying this I do not intend to imply that continuum ﬂuid mechanics

needs major reformulation in the context of the meteorological simulation

of the large-scale ﬂow by numerical process on computers for weather fore-

casting; the enterprise is too demonstrably successful for that to be the

case. Indeed, some understanding of the success of this operation emerges

naturally from analysing high-resolution observations in a statistical mul-

tifractal framework. However, for representing the smaller scales, and for

accurate accounting of the detailed energy distribution in the atmosphere,

required for climate prediction, turbulence must be properly understood

and formulated. It is my contention that it will not be achieved without

explicit recognition of the fact that ﬂuid mechanical behaviour emerges

spontaneously in the molecular dynamics simulation of a population of

Maxwellian molecules subject to an anisotropy (Alder and Wainwright

1970): turbulence has molecular roots.

A word of justiﬁcation is necessary for the heavy emphasis on aircraft

data in this book, and on the use of dropsondes deployed from aircraft in

approaching the vertical behaviour of atmospheric scaling. These sorts of

observations reached, in the late 1980s, a level of accuracy, response, sensi-

tivity, and continuity over a large enough range of scales that fairly reliable

calculation of probability distributions and the moments involved in statis-

tical multifractal analysis, especially of the rather robust H exponent, was

possible. In a smaller number of cases, the scaling exponents characteriz-

ing the intermittency and the Lévy distribution also could be calculated.

It is still true, however, that there is only a very small number of platforms

which have produced observations of the requisite quality. We can look

forward to the time when satellite data can sustain such analysis, when

numerical models of the atmosphere can simulate it, and when unmanned,

autonomous aircraft can circle the globe to extend it. We are not there yet,

however, and in the meantime the reader’s indulgence must be sought if this

book seems a little like an account of the ER-2’s history of deployments for

stratospheric ozone research. The fact is that the platform-instrument com-

bination achieved a level of performance in acquiring in situ data that has

not yet been matched in our context, even though it was far from perfect.

I defend what has been done because the implications may be far-reaching,

both conceptually and practically.

In my experience, few meteorologists have much deep acquaintance

with molecules and their behaviour, while at the same time most physical

chemists have limited appreciation of continuum ﬂuid mechanics, partic-

ularly in its unique and complicated atmospheric form. The two meet in

the persons of those who seek to include chemistry in global atmospheric

computer models, but often in such a way that the scales important to our

argument here are missing. While it is almost a cliché to say that chemistry,

Preface ix

radiation, and dynamics are coupled to produce the atmospheric state, the

point is that all three of these are coupled at very short time and space

scales in such a way that the concept of local thermodynamic equilibrium is

undermined. The long tail in the PDF of molecular speeds is kept overpop-

ulated by its mutually sustaining interaction with vortices induced by the

fast molecules themselves in reacting to anisotropy, such as that inherent

in gravity, the solar beam, planetary rotation, the surface topography, or

larger-scale winds, particularly the fast moving cores of jet streams. Trans-

lationally hot fragments from photodissociating ozone molecules therefore

do not recoil into a Maxwellian bath of equilibrated air molecules, but into

pre-existing vorticity structures. The resulting sustained overpopulation of

fast molecules will thus affect the shapes of the pressure-broadened spectral

lines of water vapour and carbon dioxide so central in climate, the rates of

chemical reactions central to the ozone and aerosol distributions, the phase

changes of water, and also the very interpretation of what atmospheric

temperature itself means. An attempt has been made where possible, in

the spirit of Berry, Rice, and Ross’s Physical Chemistry, to explain from

basic principles what is involved in each argument before dealing with the

complexities. However, that book sets a high standard to which to adhere

and a size of over a thousand pages is not justiﬁed for the present work.

References have been provided for the reader willing to branch out into

unfamiliar territory.

In order to keep the text compact, the equations have been stated without

derivation or proof, giving references and bibliographic sources instead. For

the same reason, subjects such as acoustics, aerodynamics, or shock waves

have not been considered. The arguments if correct will have considerable

implications for the phase transitions of water underlying cloud physics and

aerosol behaviour, but only the possibility that long-tailed velocity PDFs of

water molecules will affect the interaction of vapour phase molecules with

liquid droplets and ice crystals has been pointed out.

The use of the phrase ‘atmospheric turbulence’ in the title is deliberate,

in that it emphasizes the point that true laminar ﬂow is never observed in

the atmosphere—speed and directional wind shears are ubiquitous. Thus

the phrase should be interpreted in the context of this book as mean-

ing ‘the ubiquitous, omnipresent, scale invariant, ﬂuctuating structure of

atmospheric quantities’. It is not limited to the episodes intermittently

encountered by aircraft that are severe enough to discommode the aircrew

and passengers, if any.

During the writing of this book, I came to appreciate Peter Hobbs’s

remark in the Preface to Ice Physics, that authors do not ﬁnish books,

they merely abandon them.

It remains to acknowledge my debt to the many colleagues whose efforts

provided the observational and analytical foundations upon which this

book is built. Primary among these is Susan Hovde, without whose mathe-

matical and data analysis skills the work in this book would not have been