Tree methods, and the detection of vortical structures in the vortex

filament method

Andrew Baggaley, Carlo Barenghi, Jason Laurie, Lucy

Sherwin, Yuri Sergeev.

Vortex filament methodBiot-Savart Integral

Model reconnections algorithmically ‘cut and paste’

Mutual friction

Normal viscous fluid coupled to inviscid superfluid via mutual friction.

Superfluid component extracts energy from normal fluid component via Donelly-Glaberson instability, amplification of Kelvin waves.

Counterflow Turbulence

Tree algorithmso Introduced by Barnes &

Hut, (Nature, 1986).o De-facto method for

astrophysical simulations where gravity is important (e.g. galaxy formation).

o Relatively easy to implement numerically.

o Acceptable loss of accuracy when compared to full BS integral (AWB & Barenghi, JLTP, 2011).

o Significant improvement in speed of code O(NlogN) vs O(N2)

Sensitivity to reconnection algorithm

Coherent structures• In classical turbulence

vorticity is concentrated in vortical ‘worms’ (She & al, Nature, 1990 ; Goto, JFM, 2008)

• Are there vortex bundles in quantum turbulence ?

• Would allow a mechanism for vortex stretching, i.e. stretch the bundle.

Generation of bundles at finite temperatures

Vortex Locking - Morris, Koplik & Rouson, PRL, 2008Gaussian normal fluid vortex – Samuels, PRB, 1993

Reconnections:Bundles

remain intact

Alamri, Youd & Barenghi, PRL, 2008

Some questions…• What are the role of these structures

in QT?• Transfer energy? Allow vortex

stretching.• How can we detect these structures

(aside from our eyes)• How are structures generated?

Detecting structures

The importance of vortex bundles

AWB, PoF, 2012

A surprising result

Roche et al., EPL, 2007

• Fluctuations of vortex line density scale as .

• If we interpret L as a measure of the rms superfluid vorticity.

• Contradiction of the classical scaling of vorticity expected from K41.

• Roche & Barenghi (EPL, 2008) - vortex line density field is decomposed into a polarised component, and a random component.

• Random component advected as a passive scalar explaining scaling.

Quantum turbulence at finite temp.

Drive turbulence in superfluid component to a steady state with imposed normal ‘fluid turbulence’.

Decompose tangle into a polarised and random component.

Measure frequency spectrum of these 2 components, and their contribution to 3D energy spectrum.

Decomposition of the tangle

AWB, Laurie & Barenghi, PRL, 2012

Numerical results

AWB, Laurie & Barenghi, PRL, 2012

Left, frequency spectra (red polarised ; black total), right energy spectrum, upper random component, lower polarised component.

Thermally vs Mechanically Driven

Multi-scale flow, summation of random Fourier modes with imposed Kolmogorov spectrum.

AWB, Sherwin, Barenghi, Sergeev, PRB, 2012.

Generation of bundles via shear flow

Kelvin-Helmholtz rollup

The End