Quantum Turbulence

Quantum turbulence is the name given to the turbulent flow of quantum fluids, such as superfluid Helium, which have been cooled to temperatures close to absolute zero.
Turbulence in normal fluids can be viewed as an ensemble of vortices; indeed quantum turbulence is no different.
However in quantum fluids, vortices have a fixed cross-sectional area, and have a fixed circulation, known as the quantum of circulation,

(1)
\begin{align} \oint_{C} \mathbf{v}\cdot\,d\mathbf{l} = \frac{2\pi\hbar}{m}n. \end{align}

This phenomina is directly due to the quantum effects, which are only observable on a visible scale at very low temperatures.
Quantum turbulence is a 'tangle' of these quantised vortices.
Schwarz [1] pioneered a numerical technique to model the evolution of quantised vortices, which is still used today.
Essentially one discretises each vortex into particles, which are then evolved according to their velocity, determined by the Biot-Savart integral,

(2)
\begin{align} \mathbf{u}(\mathbf{x})=\frac{1}{4\pi}\nabla \times \int_V \frac{\boldsymbol{\omega}(\mathbf{y})}{|\mathbf{x}-\mathbf{y}|} \; d \mathbf{y}. \end{align}

When two vortices (which are not parallel) become very close they are reconnected using a numerical algorithm. Schwarz did this based on intuition alone, however recent experiments appear to show that quantised vortices do indeed reconnect, click here for more details.
Vortex reconnections also lead to the devolpment of a Kelvin wave cascade, which eventually dissipates energy in the form of sound.

Quantum turbulence driven by thermal counterflow. I have used this type of turbulent forcing in a number of works, see the following article for details.

## tree approximations in n-body codes

Beginning with the pioneering work of Barnes and Hut [2] tree-code approaches have played a significant role in the success of n-body codes in astrophysics.

The essence of the technique is to rely on the fact that although each "particle" in your simulation contributes to the equation of motion of all the other particles; the rate at which this contribution reduces is fast.
So for example the gravitational attraction between two bodies is inversely proportional to the square of their separation.
This is also true of the contribution a segment of a vortex makes to the motion of a vortex point, in the vortex filament method.
Therefore we have recently included tree algorithms in our numerical simulations of quantum turbulence, with of course and fantastic increase in speed.

The tree mesh around a single vortex loop is shown above.

## qvort

The above simulations are the result of the quantum vortex code qvort, this contains many features including:-

• Velocity - Biot-Savart or Local Induction Approximation (LIA)
• Tree approximation - Barnes and Hut style tree approximation to Biot-Savart
• Normal fluid - act as damping or drive the flow with specific velocity fields (counter-flow/ABC/KS)
• Spectra - energy spectra of the superfluid or normal fluid
• Visualisation - MATLAB or VAPOR
• Timestep - explicit: 3rd order Adams-Bashforth
• Forcing - sinusoidal forcing on a boundary (Kelvin wave cascade) if desired
• Derivatives - (variable mesh spacing) finite-difference schemes
• Initial conditions - variety of initial set-ups available, can easily be added to
• Boundary conditions - Periodic/closed/open
• Quasi particles - Integrate quasi-particle trajectories subject to vortices