The Thousand-Pulsar-Array programme on MeerKAT -- XVIII. Complex sub-pulse modulation patterns, bi-drifting and mode changing of nine radio pulsars

We present a detailed analysis of sub-pulse modulations in nine pulsars which show evidence of changes in sub-pulse drift direction as a function of pulse longitude in the Thousand Pulsar Array single pulse survey with MeerKAT. We confirm that all of these are consistent with persistent drift direction changes. These 'bi-drifting' pulsars present a challenge to the classical carousel model for sub-pulse drifting. In general, bi-drifting in this expanded sample is less clear than some of the previously published cases, which we attribute to narrower profile widths ($<20$ degrees) or smaller $P_3$ values (close to 2$P$). However, given the broad variety of pulse shapes and drift behaviours across the pulsar population, it is unsurprising that the phenomenon is not limited to only those where it can most easily be detected. Four of our samples show at least two emission modes with different profile shapes and drift properties, which seems to be a relatively common feature of bi-drifting pulsars. We also find jumps in sub-pulse phase between adjacent components in two pulsars. In addition to our MeerKAT L-band data, we used GMRT observations for four, and MeerKAT UHF observations for two of these pulsars to investigate the frequency dependence of sub-pulse drift. We find subtle changes in the drift as a function of frequency, but no clear overall pattern. Looking at the distribution of bi-drifting pulsars over $P$, $\dot{P}$ and $P_3$ suggests they are consistent with the underlying population of all drifting pulsars.