## 1. Introduction

While it is a physical fact that the atmosphere is molecules in motion, it is a fact which is absent in an explicit form from formulations and simulations of the atmosphere on all scales, whether these are from the large scales down or, more rarely, from the small scales up. Thereby, all meteorologically necessary molecular knowledge is assumed to be implicitly embodied in the gas constant, with local thermodynamic equilibrium being a universally applied approximation up to the stratopause at 50 km. An important reason for the absence is that both the Navier–Stokes equation and the Boltzmann H-equation, with or without its quantum mechanical developments, are nonlinear, with no useful analytical solutions respectively known for the macroscopic and microscopic cases. Numerical solutions by digital computer are therefore essential, but are handicapped by the immensity of the atmospheric problem in spanning the 15 orders of magnitude in length scale from the molecular mean free path at sea level to a great circle. Top-down solutions using the Navier–Stokes equation together with its thermodynamic and radiative accompaniments are of course commonplace, in the form of the global models used for weather forecasting and climate simulation. To the author's knowledge, there have been no attempts at a bottom-up, molecular dynamics solution for a population of air molecules via the H-equation or elaborations thereof. This review covers investigations in the last two decades which shed some observational light on the problem, with appeals to two areas of theory in an attempt to explain some unexpected correlations. The two theories are of molecular dynamics (Alder and Wainwright, 1970) and of generalized scale invariance (Schertzer and Lovejoy, 1985).

The Alder and Wainwright (1970) result was achieved by allowing a directional molecular flux to impinge upon an equilibrated population of Maxwellian molecules (‘billiard balls’) by direct numerical simulation in a computer. A striking phenomenon was observed: the emergence of ring currents on scales of 10^{−12} seconds and 10^{−8} metres at standard temperature and pressure (STP). These ring currents are what a meteorologist would call vortices, and in general they signify the emergence of fluid mechanical behaviour in a randomized molecular population, and in particular they indicate the central importance of vorticity. Potentially, this result has many atmospheric consequences via its embodiment of an overpopulation of high-speed molecules in the speed probability distribution function, relative to a Maxwell–Boltzmann distribution; the ring currents and the overpopulation of fast molecules are mutually sustaining, and prevent relaxation to a thermalized population. These consequences range from the impossibility of true Einstein–Smolukowski (isotropic) diffusion to the definition of temperature on a molecular basis. Also affected are the spectroscopic line shapes of water vapour, carbon dioxide, ozone, methane and nitrous oxide that largely determine the transfer of terrestrial radiation, particularly in the wings of the stronger lines. Chemical reactions are of course profoundly affected by the velocities of the colliding reactant molecules. Perhaps most importantly, the turbulent structure of the wind field is affected, on all scales from that of a small aerosol particle to at least that of an Earth radius and probably out to a great circle. This scale-invariant, turbulent structure imprints itself on the abundances of chemical species, including absorbers and emitters of radiation, meaning that energy is deposited in the atmosphere on all scales. That result vitiates the concept of energy deposition on the scale of a sunlit hemisphere, with a conservative cascade downscale to dissipation. It follows from the preceding that vorticity generation is central, on all scales. It represents the emergence of organized flow from small scales, rather than its dissipation from large scales. Because of the Alder–Wainwright ring current mechanism, turbulence will of necessity need to be viewed in the same framework as vorticity.

The question of variability as a function of scale lies at the heart of the work of Schertzer and Lovejoy (1985), resulting in the formulation of generalized scale invariance, the predictions of which are characterized by scaling exponents, *H*, of 1/3 in the horizontal and 3/5 in the vertical, along with expectations of long-tailed probability distribution functions (PDFs), accompanied by linear log-log plots of variability against scale. Intermittency is important and so are departures from monofractality; both are described by exponents, *C*_{1} and α respectively. The atmosphere's turbulent structure is neither two-dimensional (2D) or 3D, but 23/9, 2 + (1/3 ÷ 3/5), since the relationship of *H* to the spectral exponent is β = 2*H* + 1. The values for *H* of 1/3 in the horizontal and 3/5 in the vertical arise from dimensional analysis of the horizontal energy flux and the vertical buoyancy variance flux, respectively; see section 2.

It can be seen therefore that two approaches as different as molecular dynamics and generalized scale invariance predict that isotropic turbulence or diffusion is not to be expected in the atmosphere on any scale. The Alder–Wainwright mechanism generates vorticity at scales near the mean free path, meaning that isotropy in the atmosphere fails even on the smallest ‘dissipative’ scales. This review covers observational approaches to the problem, largely involving *in situ* airborne and balloon-borne instruments, and it will argue that certain correlations are explicable by appealing to both theories. However, while both theories are held to be relevant and to have physically causal properties, there are yet quantitative questions that remain unanswered.

Two commonly asked questions about the adopted approach are discussed at the end of this review. These questions are ‘Where is the physics in fractal theories?’ and ‘What can modellers do about the molecular scale generation of vorticity?’ A third might be ‘Can the ocean be approached like this?’, to which an immediate answer can be given: while the reaction from a meteorologist might be ‘good idea’, that from a physical chemist is more likely to be ‘good luck’, given the molecular complexities of liquids in general and water in particular—ice floats, for example.

This review is not intended to be comprehensive, but to summarize the author's views and results since first detecting scale invariance in atmospheric data in 1997. The emphasis is on the physical picture and diagrams, rather than on mathematics and equations. There is a large literature on other atmospheric aspects, particularly in the rainfall and hydrological literature stemming from the publication of Schertzer and Lovejoy (1987). Surface temperature (Koscielny-Bunde *et al*, 1998; Syroka and Toumi, 2001) has been examined, as has ozone column density (Toumi *et al*, 2001; Varotsos, 2005). The subject of the interaction of radiative transfer with clouds, vital both for climate simulation and remote sounding, has been subject to extensive investigation by multifractal methods; see for example Marshak *et al*(1997) and Davis *et al*(1997). Cloud physics has been and continues to be examined in a turbulent context (Bartlett and Jonas, 1972; Jonas, 1996; Falkovich *et al*, 2002; Falkovich and Pumir, 2007) for example, and very recently by Bodenschatz *et al*(2010). The importance of molecular-scale processes has been investigated by Ghosh *et al*(2007), who used a Lennard-Jones intermolecular potential to model the diffusion of water vapour to condensed phase surfaces. Note that all these processes involve either the condensation of molecules or their interaction with radiation in both absorption and emission, and are hence in principle subject to simulation by molecular dynamics. None of the so-called ‘different sorts of physics’ that are sometimes held to determine atmospheric evolution are separable into phenomenologically perceived or scale-separated compartments; physics is physics and in the atmosphere starts with air molecules and photons, continuing on through statistical mechanics, radiative transfer, continuum approximations and thus to fluid mechanics. A book-length account may be found in Tuck (2008). Kleidon and Lorenz (2005) discuss recent work on non-equilibrium statistical thermodynamics. In what follows we start from the small, fundamental scale of the constituent particles, first developing the ideas of molecular dynamics, section 3, asymmetric probability distribution functions, section 4, and generalized scale invariance, section 5, and then moving to the scaling of atmospheric observations, section 6. Before that, however, it is necessary to review briefly the present state of theories of atmospheric turbulence, a notoriously incomplete subject.