  Previous power-spectrum analyses of the Homestake and GALLEX-GNO
radiochemical solar neutrino data have shown evidence of the effects of solar
rotation and also indications of oscillations with two periods that are well
known in solar physics: one at about 154 days period (the Rieger periodicity)
and another at about 52 days period. We here analyze both data sets, taken
together, to test the hypothesis that the above periodicities, and a related
periodicity with a period of about 77 days, are due to r-mode oscillations with
spherical harmonic indices l = 3 and m = 1, 2, and 3. These would have
frequencies nuR/6, nuR/3, and nuR/2, where nuR is the sidereal rotation rate
where the waves are excited. We first form a statistic that combines
information from power spectra formed from Homestake and GALLEX-GNO data. From
this statistic, we form another statistic to search for a sidereal rotation
frequency that leads simultaneously to four peaks in the Homestake-GALLEX-GNO
spectrum: one corresponding to the synodic rotation frequency nuR - 1, and the
other three corresponding to the l = 3 r-modes. We find that, in the range 0 to
20 cycles per year, there is one notable peak at 13.88 cycles per year; This
falls in the rotation frequency range of the radiative zone. The corresponding
synodic rotation period is 28.4 days, and the corresponding r-mode periods are
157, 78, and 52 days. By applying the shuffle test to the data, we find that
the probability of obtaining by chance a peak this big or bigger in the
rotation-frequency band of the radiative-zone is less than 0.1%. The modulation
of the neutrino flux is attributed to the poloidal motion of a magnetic
structure that moves in and out of the narrow beam of neutrinos that originate
in the solar core and are detected on Earth.
