  In a magnetised plasma on scales well above ion kinetic scales, any
constant-magnitude magnetic field, accompanied by parallel Alfv\'enic flows,
forms a nonlinear solution in an isobaric, constant-density background. These
structures, which are also known as spherically polarised Alfv\'en waves, are
observed ubiquitously in the solar wind, presumably created by the growth of
small-amplitude fluctuations as they propagate outwards in the corona. Here, we
present a computational method to construct such solutions of arbitrary
amplitude with general multi-dimensional structure, and explore some of their
properties. The difficulty lies in computing a zero-divergence,
constant-magnitude magnetic field, which leaves a single, quasi-free function
to define the solution, while requiring strong constraints on any individual
component of the field. Motivated by the physical process of wave growth in the
solar wind, our method circumvents this issue by starting from low-amplitude
Alfv\'enic fluctuations dominated by a strong mean field, then "growing"
magnetic perturbations into the large-amplitude regime. We present example
solutions with nontrivial structure in one, two, and three dimensions,
demonstrating a clear tendency to form very sharp gradients or discontinuities,
unless the solution is one dimensional. As well as being useful as an input for
other calculations, particularly the study of parametric decay, our results
provide a natural explanation for the extremely sharp field discontinuities
observed across magnetic-field switchbacks in the low solar wind.
