The Dimension and Intermittency of Atmospheric Dynamics D. Schertzer and S. Lovejoy EERM/CRMD, Meteorologie Nationale, 2, Avenue Rapp, 75007 Paris, France Abstract We question the existence of a dimensional transition separating quasi-two dimensional and quasi-three dimensional atmospheric motions, i.e. large- and small-scale dynamics. We insist upon the fact that no matter how this transition should occur, it would have drastic consequences for atmospheric dynamics, consequences which have not been observed in spite of many recent experiments. An alternative simpler hypothesis is proposed: that· small scale structures are continuously de- formed - flattened - at larger and larger scales by a scale invariant process. This continuous deformation may be characterised by defIning an intermediate fractal dimension Del that we call an elliptical dimension. We show both theoretically and empirically that Del = 23/9 - 2.56. Atmospheric structures are therefore never "flat" (Del = 2), nor isotropic (Del = 3), but always display aspects of both. Larger structures are on the average, more stratifIed as a result of a well-defIned stochastic process. In this scheme, the intermittency must be quite strong in order to produce the well-known meteoro- logical "animals" such as storms, fronts, etc. We propose that intermittency is characterised by hyper- bolic probability distributions with exponents ex. This possibility was fIrst suggested by Mandelbrot (1974a) for the rate of turbulent energy transfer (6). We investigate intermittency for the wind (v) the potential temperature (0), and 6 in terms of this hyperbolic intermittency. In particular, we fInd exv = 5, ex, = 5/3, ex ln6 = 10/3, which show that the fIfth moment of v, the second moment of 6, and the fourth moment of In 0 diverge. We re-examine Mandelbrot's model of intermittency and generalize it for anisotropic turbulence. We stress that it cannot be characterised by a single parameter, the of the support of turbulence: we show that, except in a trivial case, several fractal dimensions intervene. We exhibit, for instance, a two-parameter model depending on the fractal dimension of the very active regions. Finally, we sketch a direction for future work to assess this 23/9 dimensional scheme of atmospheric dynamics with hyperbolic intermittency. 1. Introduction The classical approach to the analysis of atmospheric motions (e.g. Monin 1972), considers the large scale as two-dimensional, and the small scale as three-dimensional. In this view, a transition, which for obvious reasons we call a "dimensional transition", is expected to occur in the meso-scale, possibly in association with a "meso-scale gap" (Van der Hoven 1957). This scheme favours the simplistic idea that at planetary scales the atmosphere looks like a thin envelope, whereas at human scales, it looks more like an isotropic volume. A dimensional transition, if it were to occur, would be likely to have fairly drastic consequences because of the significant qualitative difference of turbulence in two and three dimensions (Fjortoft 1953; Kraichnan 1967; Batchelor 1969). Two-dimensional turbulence is very special since in the vorticity equation, there is no source term, and it is therefore conserved. Mathematically, it introduces a second quadratic invariant (the enstrophy, or mean square vorticity), and physically, the all important stretching and folding of vortex tubes cannot occur. Since the 50's, there has been a wide debate over the effective dimension of atmospheric turbulence, due in particular to the extension of two-dimensional results to the case of quasi-geostrophy (Charney 1971; Herring 1980). 7 L. J. S. Bradbury et al. (eds.), Turbulent Shear Flows 4 © Springer-Verlag Berlin Heidelberg 1985