## Applications Using Dedalus

This gallery contains images and videos from research using Dedalus. Visit our vimeo page to see more of these and other applications.

If you've used Dedalus in your work and would like to contribute to the gallery, please let us know!

### MRI dynamo in quasi-Keplerian flow

This animation shows a finite-amplitude magnetic seed triggering a subcritical dynamo instability through the magnetorotational instability (MRI) in a quasi-Keplerian shear flow. The volume rendering shows the streamwise (azimuthal) component of the magnetic field. The cross section shows surface line integrals highlighting the structure of the transverse (meridional) magnetic fields. The initial seed field, being essentially transverse and streamwise invariant, is rapidly wound up by the base flow's shear, generating a large-scale streamwise magnetic field. The latter destabilises upon exceeding a critical amplitude, giving rise to a self-sustained, fully 3D dynamo state. Further details including source code and initial conditions are given in (https://arxiv.org/abs/2112.11376).

Contributed by Paul Mannix.

### Full-sphere rotating Boussinesq convection

This simulation shows rotating Boussinesq convection in a full sphere, which occurs in the molten cores of young planets and the cores of massive stars. Both panels show the axial vorticity. The left panel shows the surface of the sphere, and the right panel shows a slice through the equatorial plane. When rotation is strong, as in this simulation, flows become elongated along the axis of rotation, and the same convective patterns can be seen at the poles (left) as at the center of the star (right). In this simulation, convection creates strong mean flows which eventually become strong enough to shut down the convection. Once the convection is no longer feeding it, this mean flow decays until convection can outburst again, and these two phenomena cycle with each other.

Contributed by Evan Anders

### Equatorial Ocean Waves

Atmospheric storms near the equator force the ocean at regular intervals. The ocean responds with internal waves of its own. The Coriolis force from the rotation of the Earth influences the ocean's waves significantly. Modellers in the past tended to simplify the mathematics of the Coriolis effect, only taking the vertical component into account (traditional). The argument is that the horizontal component is negligible. But sometimes small effects produce outsized consequences. Near the equator, the full Coriolis effect (non-traditional) can lead to sharp wave beams which can focus and dissipate energy quite differently from what is often assumed.

Contributed by Ryan Holmes

### Convection-driven melting in a full sphere

This simulation shows convection-driven melting in a full spherical domain using a phase field method. The simulation begins with a liquid core surrounded by a thick ice shell. Internal heating is applied at the center of the sphere which drives convection in the liquid. This convection transports heat radially outward, due to the spheres self gravity, which dynamically melts the surrounding ice shell. The techniques used in this toy problem may be applicable to frozen solar system bodies, etc.

Contributed by Daniel Lecoanet & Eric Hester

### Active Matter on the Sphere

A simulation of a phenomenological model of a fluid driven by active stresses, such as collective microbial swimming, on the surface of the sphere. The model is a generalized form of the incompressible Navier-Stokes equations with a linear active stress given by a composition of Laplacians, which injects energy over a finite bandwidth. The simulation utilizes spin-weighted spherical harmonics to maintain sparsity and correctly capture the regularity of different tensors at the poles. See Mickelin et al. 2018 (doi.org/10.1103/PhysRevLett.120.164503).

Contributed by Keaton Burns

### Orszag-Tang Vortex

This is a simulation of an Orzang-Tang vortex, a common test problem for magnetohydrodynamic codes. The compressible MHD equations are being solved on a doubly-periodic 2D domain with a 4096 modes in each dimension. The simulation includes dissipation, allowing the spectral method to regularize and resolve the resulting shocks.

Contributed by Daniel Lecoanet

### Obstacles are no obstacle

This is a simulation of an incompressible flow around an ellipsoidal obstacle at moderate Reynolds number (Re ~ 100) in two dimensions. The flow is visualized by advecting a passive tracer concentration field which is continually released from a localized source on the left side of the domain. The obstacle is implemented with a volume-penalized immersed boundary method.

Contributed by Eric Hester

### Buckling Elastic Sheet

The buckling instability of a thin elastic sheet. In a certain limit, this is also equivalent to the dynamics of the Magnetorotational Instability (the MRI common in astrophysics). A particular interesting things about this system is that the nonlinear feedback is completely non-local; through an integral of the solution over the whole domain. See Vasil 2015 (doi.org/10.1098/rspa.2014.0699).

Contributed by Geoff Vasil

### Nonlinear SchrÃ¶dinger Network

Nonlinear SchrÃ¶dinger equation on a network, simulated by coupling the boundaries of different fields on a 1D Chebyshev segment. Continuity of the solution and Kirchhoff's law are imposed at each vertex. A soliton initially isolated to one segment scatters at the vertices and fills the network over time. The grey coloring shows |psi|.

Contributed by Keaton Burns & Geoff Vasil.

### Three Shocks

Three different shocks with similar equations. All three cases have the same initial conditions and are simulated on periodic intervals. The equations differ only in the way they regularize sharp features. The Burgers equation uses dissipation. The KdV equation uses dispersion. The Benjamin-Ono equation uses a different kind of non-local dispersion involving the Hilbert transform (H). The comparison shows that small changes in the way a system accounts for small scales can have large qualitative consequences.

Contributed by Geoff Vasil

### Strong-Field Magnetoconvection

A simulation of magnetoconvection looking down from above. The entire domain is permeated by a strong background magnetic field. The field controls and suppresses convection in a similar manner as the field at the centre of a sunspot.

Contributed by Geoff Vasil

### Compressible Convection in Dedalus + Athena

2D compressible convection simulation, in which the top half of the box is simulated using the Athena code (https://trac.princeton.edu/Athena/), and the bottom half of the box is simulated using Dedalus.

The smooth features in the image demonstrate the two codes, using very different numerical techniques, can be stitched together to run a single simulation.

Contributed by Daniel Lecoanet

### Internal Wave Beams

Internal waves generated in 2D Boussinesq hydrodynamics. The domain is the cross section of a cylindrical annulus lying horizontally, so the r-theta plane is a remapping of the x-z plane. Plotted is the perturbation to the buoyancy, around a constant, stably-stratified background. The inner and outer cylinders are no-slip. The outer buoyancy is fixed to the background, but the inner buoyancy includes oscillatory heating (oscillatory in time, constant around the cylinder) at fixed frequency, resulting in thermally-forced IGWs radiating from the cylinder. An interesting thing about this plot is that it shows the non-specular reflection of IGWs off of boundaries: the reflected waves maintain their angle with respect to the vertical (direction of gravity), and not their angle with respect to the wall.

Contributed by Keaton Burns

### Spiral-Defect Chaos

A planform view of the vertically-integrated buoyancy perturbation in 3D Rayleigh-Benard convection. The horizontal dimensions (x,y) are discretized using parity bases to enforce stress-free boundary conditions on the side walls (notice how the rolls preferentially align perpendicular to the walls). The vertical dimension (z) is discretized using Chebyshev polynomials with no-slip boundary conditions. This run is between the intermittent and full spiral-defect chaos regimes of Rayleigh-Benard convection.

Contributed by Keaton Burns