And finally, stay tuned for a lot of follow up work in the pipeline :)
11/11

This can have some real consequences for planet formation. SPH isn't very sensitive to instabilities, the fact we see WInDI so robustly here suggests the instability is very strong and doesn't mind a bit of viscosity (unlike most other dust instabilities)
10/11

This is shown very nicely in this particle plot in a cut in the disc for the smallest and largest stokes simulations
9/11

WInDI happens for all our simulations at different stokes, but the interpretation gets a bit messy for high stokes, because the warp evolves in the gas faster than the dust can follow, so the dust no longer feels the sloshing (breathing) horizontally (vertically)
8/11

The rings of high dust density seen in these rings are caused by the combination of the sloshing and breathing motions of the gas acting on the dust. Indeed, the breaks in those rings happen at the phase where the sloshing is minimum
7/11

Next, we see if this instability also occurs globally. We run 3D global SPH simulations with different Stokes number. WInDI is triggered globally and grows VERY quickly. Here's what it looks like for Stokes=0.1
6/11

This is shown nicely in this schematic (based on values from the calculation). Top panel shows the thickness of gas (blue) and dust (red). While the gas thickness oscillates, the dust gets compressed. This effect increases with the stokes number and warp magnitude, Figs 3, 4
5/11

We add dust to the equations and build a simple 1D (vertical) model to calculate the gas+dust evolution in this warped frame. We immediately find that, while the gas just oscillates, the dust gets compressed significantly
4/11

A horizontal mode (dubbed 'sloshing' by Dullemond+22) and a vertical 'breathing' mode. These oscillations are due to the fact that the vertical pressure gradient has an oscillating horizontal component in the presence of a warp
3/11

We start with a bit of theory, building on the seminal work of Ogilvie & Latter 2013 who introduced a warped shearing box coordinate frame to study the local motion of gas in the presence of a disc warp. This allowed to uncover 2 oscillation modes in the gas
2/11