Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 2 1 1995 10.5194/npg-2-16-1995 http://www.nonlin-processes-geophys.net/2/16/1995/ http://www.nonlin-processes-geophys.net/2/16/1995/npg-2-16-1995.html http://www.nonlin-processes-geophys.net/2/16/1995/npg-2-16-1995.pdf 16 22 0000-00-00 The <i>l</i>^1/2 law and multifractal topography: theory and analysis S. Lovejoy D. Lavallée D. Schertzer P. Ladoy Physics Dept., McGill University, 3600 University St., Montreal, Quebec H3A 2T8, Montreal Earth-Space Research Group, Institute for Computational Earth System Science, CRSEO-Ellison Hall, University of California, Santa Barbara, CA 93106-3060, USA Laboratoire de Météorologie Dynamique, Univ. Pierre et Marie Curie, 4 Pl. Jussieu, 75252 Paris Cedex 05, France Météorologie Nationale, 2 Avenue Rapp, Paris 75007, France Over wide ranges of scale, orographic processes have no obvious scale; this has provided the justification for both deterministic and monofractal scaling models of the earth's topography. These models predict that differences in altitude (Δh) vary with horizontal separation (<i>l</i>) as Δh ≈ <i>l</i>^H. The scaling exponent has been estimated theoretically and empirically to have the value H=1/2. Scale invariant nonlinear processes are now known to generally give rise to multifractals and we have recently empirically shown that topography is indeed a special kind of theoretically predicted &quot;universal&quot; multifractal. In this paper we provide a multifractal generalization of the<i> l</i>^1/2 law, and propose two distinct multifractal models, each leading via dimensional arguments to the exponent 1/2. The first, for ocean bathymetry assumes that the orographic dynamics are dominated by heat fluxes from the earth's mantle, whereas the second - for continental topography - is based on tectonic movement and gravity. We test these ideas empirically on digital elevation models of Deadman's Butte, Wyoming.