Vortex Cores: Quantum Vortex Dynamics in Superfluid Helium

Introduction

This codebase simulates the behavior of quantum vortex cores in superfluid helium. It provides a comprehensive framework for studying vortex dynamics under various conditions, including external fields and thermal effects. The simulation uses both CPU and GPU acceleration options for performance.

Underlying Physics

Quantum vortices are topological defects in superfluid systems where the circulation is quantized. In superfluid helium, these vortices exhibit several unique properties:

Fundamental Physical Properties

1. Quantized Circulation: Unlike classical fluids, circulation in superfluid vortices is quantized in units of κ = h/m (where h is Planck's constant and m is the mass of helium-4 atom).

2. Zero Core: The vortex core has effectively zero density as the superfluid order parameter vanishes at the center.

3. Non-Local Dynamics: Vortex motion is governed by the Biot-Savart law, creating long-range interactions between vortex segments.

Motion Equations

Vortex filament motion is described by:

• Local Induction Approximation (LIA): The self-induced velocity of a vortex filament is approximated as:

  v = β κ (log(L/a)) s' × s''
  

  where β is a constant, L is a characteristic length scale, a is the core radius, s' is the tangent to the vortex line, and s'' represents the curvature.

• Full Biot-Savart Law: For more accurate non-local interactions, the complete integral form is:

  v(r) = (κ/4π) ∫ (ds × (r - s))/|r - s|³
  

  where integration is performed along all vortex lines.

• Mutual Friction: At finite temperatures (T > 0), the interaction between the normal fluid and superfluid components introduces a drag force:

  v_drag = -α (s' × (s' × v)) - α' (s' × v)
  

  where α and α' are temperature-dependent friction coefficients.

Computational Methods

Our simulation implements several advanced numerical techniques:

Vortex Discretization

• Vortices are represented as connected discrete points in 3D space
• Adaptive point distribution ensures higher resolution in regions of high curvature
• Closed vortex rings and open vortex lines are supported

Velocity Calculation

1. Local Contribution: The local induction approximation is used for self-induced motion
2. Non-Local Contribution: The Biot-Savart integral is computed for interactions between different vortex segments
3. External Fields: Various external fields can be applied:
   • Uniform flow
   • Solid body rotation
   • Oscillatory flow
   • Counterflow

Time Integration

• Fourth-order Runge-Kutta method for accurate time evolution
• Adaptive time-stepping based on the maximum velocity
• Boundary conditions enforce containment within a cylindrical domain

GPU Acceleration

• The computationally intensive Biot-Savart calculations are offloaded to GPU using WGPU
• WGSL shader code performs parallel computation of velocity contributions
• Dynamic GPU selection for multi-GPU systems

Implementation Details

The codebase is structured with several key modules:

• simulation.rs: Core simulation logic and vortex representation, output to VTK files
• physics.rs: Physical constants and equations
• compute.rs: GPU acceleration module
• extfields.rs: External field implementations

Usage

GPU Selection and Listing

List all available GPUs:

cargo run -- single --list-gpus

Select a specific GPU by name fragment:

cargo run -- single --gpu --select-gpu "nvidia" --radius 0.5 --height 1.0

Single Simulations

Uniform x-axis external flow, velocity 0.1 cm/s:

cargo run -- single --gpu --radius 0.5 --height 1.0 --temp 1.5 --steps 1000 \
  --ext-field uniform --ext-value 0.1,0,0

Rotation field with z-axis rotation, centered at (0, 0, 0.5), angular velocity 1.0 rad/s:

cargo run -- single --gpu --radius 0.5 --height 1.0 --temp 1.5 --steps 1000 \
  --ext-field rotation --ext-value 0,0,1.0 --ext-center 0,0,0.5

Oscillatory flow along y-axis:

cargo run -- single --gpu --radius 0.5 --height 1.0 --temp 1.5 --steps 1000 \
  --ext-field oscillatory --ext-value 0,0.1,0 --ext-freq 2.0 --ext-phase 0.0

Parameter Studies

Run simulations across parameter ranges:

cargo run -- study --gpu --rmin 0.2 --rmax 1.0 --rsteps 4 --tmin 1.0 --tmax 2.1 \
  --tsteps 5 --steps 500 --ext-field rotation --ext-value 0,0,1.0 --output study_results

Checkpointing and Restart

The simulation includes a comprehensive checkpoint system that allows you to:

• Save simulation state automatically at regular intervals (every 100 steps)
• Manually save checkpoints using the save_checkpoint method
• Resume simulations from checkpoints without losing state
• Switch between CPU and GPU computation when resuming
• Modify simulation parameters when resuming

Working with Checkpoints

Checkpoints are stored as JSON files containing:

• Simulation parameters (radius, height, temperature)
• Current simulation time
• Complete vortex line geometry
• Statistical data collected during simulation

Listing available checkpoints:

cargo run -- list-checkpoints

Loading a checkpoint with the standard command:

cargo run -- single --load-checkpoint checkpoint_500.json --gpu --steps 1000 --output resumed.vtk

Using the dedicated resume command:

cargo run -- resume --gpu --load-checkpoint checkpoint_500.json --steps 1000 --output resumed.vtk

Changing external field when resuming:

cargo run -- resume checkpoint_500.json --gpu --ext-field rotation --ext-value 0,0,2.0 --steps 500

Assumptions and Limitations

This simulation makes the following key assumptions:

1. Core Structure: The vortex core is treated as a 1D filament with zero thickness
2. Incompressible Flow: The superfluid is treated as incompressible
3. Mutual Friction Model: A simplified two-fluid mutual friction model based on temperature-dependent coefficients
4. Reconnection Physics: Vortex reconnections are handled using a distance-based criterion
5. Boundary Conditions: Simple reflective boundary conditions at container walls

Requirements

• Rust compiler (2024 edition or later)
• wgpu-compatible GPU for hardware acceleration
• Optional: ParaView for visualization of VTK output files

Future Work

• Implement more sophisticated reconnection algorithms
• Add support for turbulent normal fluid profiles
• Incorporate quantum turbulence statistics collection
• Improve performance through advanced tree-code algorithms for Biot-Savart integration

References

1. Donnelly, R.J. (1991). Quantized Vortices in Helium II. Cambridge University Press.
2. Schwarz, K.W. (1985). Three-dimensional vortex dynamics in superfluid ⁴He: Homogeneous superfluid turbulence. Physical Review B, 38(4), 2398-2417.
3. Barenghi, C.F., Skrbek, L., & Sreenivasan, K.R. (2014). Introduction to quantum turbulence. Proceedings of the National Academy of Sciences, 111(Supplement 1), 4647-4652.

To Do

• Fix handle_reconnections to ensure that the reconnection is performed when operating in GPU mode