Sustained axisymmetric intrusions in a rotating system

We analyse the effects of rotation on the propagation of an axisymmetric intrusion through a linearly stratified ambient fluid, arising from a sustained source at the level of neutral buoyancy. This scenario occurs during the horizontal spreading of a volcanic ash cloud, which occurs after the plume has risen to its neutral buoyancy level. A simple and well-accepted approximation for the flow at late times is that inertial effects are negligible. This leads to a lens-shaped intrusion governed by a balance between Coriolis accelerations and horizontal pressure gradients, with the radius scaling with time as rN∼t1/3. However, we show using a shallow-layer model that inertial forces cannot be neglected until significant times after the beginning of the influx. These inertial forces result in the flow forming two distinct domains, separated by a moving hydraulic jump: an outer 'head' region in which the radial velocity and thickness vary with time, and a thinner 'tail' region in which the flow is steady. Initially, the flow expands rapidly and this tail region occupies most of the flow. After about one half-revolution of the system, Coriolis accelerations halt the advance of the front, and the hydraulic jump separating the two regions propagates back towards the source of the intrusion. Only after approximately one and a half rotations of the system does inertia become insignificant and the Coriolis lens solution, with rN∼t1/3, become established. Importantly, this means that neither inertia nor Coriolis accelerations can be neglected when modelling intrusions from volcanic eruptions. We exploit the two-region flow structure to construct a new hybrid model, comprising just two ordinary differential equations for the intrusion radius and location of the hydraulic jump. This hybrid model is much simpler than the shallow-layer model, but nonetheless accurately predicts flow properties such as the intrusion radius at all stages of motion, without requiring fitted or adjustable parameters.