This paper discusses numerical experiments in which an initially uniform columnar vortex is subject to several types of axisymmetric forcing that mimic the strain field of a turbulent flow. The mean value of the strain along the vortex axis is in all cases zero, and the vortex is alternately stretched and compressed. The emphasis is on identifying the parameter range in which the vortex survives indefinitely. This extends previous work in which the effect of steady single-scale non-uniform strains was studied. In a first series of experiments the effect of the unsteadiness of the forcing is analysed, and it is found that the vortex survives as a compact object if the ratio between the oscillation frequency and the strain itself is low enough. A theoretical explanation is given which agrees with the numerical results. The strain is then generalized to include several spatial scales and oscillation frequencies, with characteristics similar to those in turbulent flows. The largest velocities are carried by the large scales, while the highest gradients and faster time scales are associated with the shorter wavelengths. Also in these cases ‘infinitely long’ vortices are obtained which are more or less uniform and compact. Vorticity profiles averaged along their axes are approximately Gaussian. The radii obtained from these profiles are proportional to the Burgers' radius of the r.m.s. (small-scale) axial strain, while the azimuthal velocities are proportional to the maximum (large-scale) axial velocity differences. The study is motivated by previous observations of intense vortex filaments in turbulent flows, and the scalings found in the present experiments are consistent with those found in the turbulent simulations.