Scale Invariance, Symmetries, Fractals, and Stochastic Simulations of Atmospheric Phenomena

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Advances in remote sensing and in situ measurement techniques have revealed the full continuum of atmospheric motions and have underlined the importance of mesoscale processes. This paper examines the implications of three observed characteristics of mesoscale circulations: 1) the energy spectrum of the horizontal wind in the horizontal is of the form k−βh with βh ~ 5/3, (k is a wave-number); 2) the corresponding spectrum in the vertical direction is of the same scaling form, but with a very different slope (βv ~ 11/5); and 3) the variability is extreme.

Some recent work in turbulence, physics, and meteorology, that is relevant to systems with extreme variability over a wide range of scales is reviewed. The concepts of scaling, intermittency, and fractals, are briefly introduced to show how they can be used to understand the physics of both homogeneous and intermittent energy cascades in isotropic atmospheres. These concepts may be generalizable (with a formalism called generalized scale invariance), to account for atmospheric intermittency and especially for anisotropy.

Finally, it is shown how to construct fractal models.

These models are useful because they produce realizations of random fields that are broadly of the same sort as those that may be allowed by the equations, while at the same time depending on empirically determined parameters. This enables them to retain close links with both the data and the physics. Finally, possible applications in mesoscale modeling, sampling problems, remote sensing, nowcasting, hydrology, and numerical weather prediction (NWP) systems are briefly discussed.

1 Physics Department, McGill University, 3600 University St., Montreal, Quebec H3A2T8, Canada.

2 EERM/CRMD, Meteorologie Nationale, 2 avenue Rapp, 75007 Paris, France. Current address: Physics Department, McGill University, 3600 University St., Montreal, Que. H3A2T8, Canada.

Advances in remote sensing and in situ measurement techniques have revealed the full continuum of atmospheric motions and have underlined the importance of mesoscale processes. This paper examines the implications of three observed characteristics of mesoscale circulations: 1) the energy spectrum of the horizontal wind in the horizontal is of the form k−βh with βh ~ 5/3, (k is a wave-number); 2) the corresponding spectrum in the vertical direction is of the same scaling form, but with a very different slope (βv ~ 11/5); and 3) the variability is extreme.

Some recent work in turbulence, physics, and meteorology, that is relevant to systems with extreme variability over a wide range of scales is reviewed. The concepts of scaling, intermittency, and fractals, are briefly introduced to show how they can be used to understand the physics of both homogeneous and intermittent energy cascades in isotropic atmospheres. These concepts may be generalizable (with a formalism called generalized scale invariance), to account for atmospheric intermittency and especially for anisotropy.

Finally, it is shown how to construct fractal models.

These models are useful because they produce realizations of random fields that are broadly of the same sort as those that may be allowed by the equations, while at the same time depending on empirically determined parameters. This enables them to retain close links with both the data and the physics. Finally, possible applications in mesoscale modeling, sampling problems, remote sensing, nowcasting, hydrology, and numerical weather prediction (NWP) systems are briefly discussed.

1 Physics Department, McGill University, 3600 University St., Montreal, Quebec H3A2T8, Canada.

2 EERM/CRMD, Meteorologie Nationale, 2 avenue Rapp, 75007 Paris, France. Current address: Physics Department, McGill University, 3600 University St., Montreal, Que. H3A2T8, Canada.

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