PHYSICAL MATHEMATICS SEMINAR
TOPIC: DEVIATIONS FROM SCALING IN GEOMORPHOLOGY AND DEVIANT
SCALING IN BIOLOGY
SPEAKER: PETER DODDS
Department of Mathematics
(Department of Earth, Atmospheric and Planetary Sciences)
Massachusetts Institute of Technology
ABSTRACT:
The structure of river networks is thought to be described by power law
statistics. We examine in particular Hack's law which relates main stream
length and drainage basin area. Ranges of exponents for Hack's law have
been reported with no definitive values standing out. We conjecture that
the reason for this variation is due to the existence of several scaling
regimes interpolated by (potentially long) crossover regions. The major
scaling regimes are related to hill slopes (non-convergent flow), random
networks and geologic constraints. We examine real data (Kansas River,
the Mississippi, and Spain/Portugal). We find clear variation in
exponents and good support for this more general view of Hack's law.
We then move over to biology where we consider Kleiber's law, the
dependency of basal metabolic rate on organismal mass. This is believed to
be a power law relationship with an exponent of 3/4 and that it holds for
all organisms (e.g., mammals, birds, bacteria and plants). We show data
for mammals with mass less than 10 kg and for birds of all masses have an
exponent indistinguishable from 2/3 which is the expected value from
dimensional analysis. The prefactor in the scaling law is well estimated
using the Stefan-Boltzmann law of blackbody radiation. We also briefly
refute recent theoretical derivations of the 3/4 law.
DATE: TUESDAY, SEPTEMBER 28, 1999
TIME: 2:30 PM
LOCATION: Building 2, Room 338
Refreshments will be served at 3:30 PM in Building 2, Room 349
Massachusetts Institute of Technology
Department of Mathematics
Cambridge, MA 02139