Beltrami fields in time-dependent spatially homogeneous velocity gradient background flows

The time evolution of Beltrami fields in the presence of a time-dependent background flow with spatially homogeneous velocity gradient is analysed using the barotropic vorticity equation. For backgrounds comprising a time-dependent isotropic expansion/contraction and a time-dependent solid-body rotation, we show that every scalar Laplacian eigenfunction generates an unsteady solution of the nonlinear vorticity equation in which the non-background component remains a time-dependent Beltrami field. We derive the evolution law for the background angular velocity in the presence of time-dependent deviatoric strain and velocity divergence, and we generalise the Chandrasekhar–Kendall construction to obtain unsteady Beltrami velocity fields. When the background deformation is a similarity (vanishing deviatoric strain), the Beltrami field is frozen into an advecting flow that differs from the background only by a spatially homogeneous, time-dependent drift. In general, deviatoric strain breaks the Beltrami property, but in regimes where departures are small, we introduce a ‘Beltrami field approximation’. Because the background velocity gradient has nine time-dependent degrees of freedom, three of which are constrained by the vorticity equation, six remaining functions may be prescribed to drive the Beltrami field. We illustrate the approach by describing elastic scattering of Beltrami fields by a background-flow pulse.