We have used flow-visualization and spectral techniques to study the spatial and
temporal properties of the flow that precedea the onset of weak turbulence in a fluid
contained between concentric cylinders with the inner cylinder rotating (the circular
Couette system). The first three flow regimes encountered as the Reynolds number is
increased from zero are well-known – Couette flow, Taylor-vortex flow, and wavy vortex
flow. The present study concerns the doubly periodic regime that follows
the (singly periodic) wavy-vortex-flow regime. Wavy-vortex flow is characterized
by a single frequency fit which is the frequency of traveling azimuthal waves
passing a point of observation in the laboratory. The doubly periodic regime was
discovered in studies of power spectra several years ago, but the fluid motion corresponding
to the second frequency fa WM not identified. We have found that fe corresponds
to a modulation of the azimuthal waves; the modulation can be observed
visually as a periodic flattening of the wavy-vortex outflow boundaries. Moreover,
in addition te the previously observed doubly periodic flow state, we have discovered
11 more doubly periodic flow states. Each state can be labeled with two integers
m and k, which are simply related to physical characteristics of the flow: m is the
number of azimuthal waves, and k is related to the phase angle between the modulation
of successive azimuthal waves by A$ = 2nk/m. This expression for the phase
angle was first conjectured from the flow-visualization measurements and then
tested to an accuracy of 0.01n in spectral measurements. Recently Rand (1981)
has used dynamical-systems concepts and symmetry considerations to derive
predictions about the space-time symmetry of doubly periodic flows in circularly
symmetric systems. He predicted that only flows with certain space-time symmetries
should be allowed. The observed flow states are in agreement with this theory.