As a femtosecond terawatt laser pulse propagates along a positive density gradient within a stratified plasma column, it drives a low-frequency electromagnetic wake wave, the period of electron fluid oscillations in the wake gradually shrinking. Phase velocity of the wake promptly exceeds the vacuum speed of light, setting in near-forward emission of terahertz Cherenkov radiation. As the wake accelerates further, the Cherenkov emission ray rotates by 180 degrees. Emission from a given plasma locality is sustained for a finite interval of time, in the middle of which the wake experiences a ‘reversal,’ its phase velocity becoming singular and changing sign (Zhang C J et al 2017 Phys. Rev. Lett. 119 064801.) At this instant, the electromagnetic energy flows at a right angle, the emission power reaching its peak. After the reversal, the wake keeps radiating into the rear hemisphere until its phase velocity becomes subluminal. Experimentally capturing evolution of the Cherenkov signal may thus shed light on the plasma wake dynamics. Far away from the plasma, the radiation fills an expanding, almost spherical shell, the shell thickness increasing with an increase in the observation angle. The length of the terahertz signal sampled in the wave zone thus ranges from zero (forward emission) to a few tens of picoseconds (backward emission.) The signal is positively chirped, its frequency increasing from the Langmuir frequency at the foot of the column to the Langmuir frequency at the top. Theoretical estimates for the regimes involving 10 TW-class drive pulses promise a few-kW emission power; the energy conversion efficiency, from optical to terahertz, of order 10^−7; and an MV/m-scale electric field strength centimeters away from the plasma.