# University of Minnesota

Aerospace Engineering and Mechanics

**Spring 1998 Seminar Series**

*The Inverse Turbulent Cascade and its Consequences in
Laboratory and Astrophysical Flows*

*Abstract*

### One way of determining if one truly understands a physical process is to
consider how that process acts under atypical conditions. Here we are interested
in ³two-dimensional² turbulence in a three-dimensional world. In
particular we interested in finding if the familiar concepts of a inverse energy
cascade can be used to explain the large-scale structures of the atmospheres of
Jupiter and Saturn which are dominated by long-lived east-west (zonal) flows.

Little
is known about what sets their velocity scales or their length scales (i.e., the
number of zones on each planet). Typically, the energy-containing modes in a
turbulent flow span a range of scales, and in the rare cases that there are
coherent features and that they all have the same size, their lengths are
usually determined directly by the boundaries or the forcing length scales, e.g.
the diameters of turbulent Taylor vortices in a Couette apparatus are determined
by the width of the apparatus. Even turbulence in geophysical flows show this
trait: the scale of granulation on the Sun (due to turbulent convection cells)
is set by the depth of the convective zone; Jupiter's long-lived vortices,
such as the Red Spot, are set by the widths of the local zonal flows in which
they are situated.

By examining a simple forced/dissipated flow we show
that the widths of zonal flows on a b-plane are determined by a subtle
combination of the forcing and dissipation and not set by boundary conditions or
by the length scale of the forcing.

We show that under a wide variety
of conditions a turbulent flow without east-west winds forms via a inverse
energy cascade and that zonal flows (with a single dominant length scale) form
only for a small set of parameters.

We present a simple theory which
determines these parameter values and which also provides scaling laws for the
zones' velocities and widths. Thus we are able to adjust the widths and
strengths of the zonal flows by changing the forcing and dissipation rates. We
show that the coherent and the incoherent parts of the energy spectrum obey
different scaling laws and we explain why. We show how two different effects
block the energy cascade and lead to departures from Kolmogorov scaling. We
discuss the implications for Jupiter, compare the numerical experiments with
similar ones carried out previously by others, and show how one could build a
laboratory experiment that would form jovian--like winds with two easily
adjustable control parameters that determine their widths and strengths.

### Friday, April 17, 1998

209 Akerman Hall

2:30-3:30
p.m.

### Refreshments served after the seminar in
227 Akerman Hall.

Disability accomodations provided upon request.

Contact
Audrey Stark-Evers, Senior Secretary,
625-8000.