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.