Master thesis defence by Nicholas Rathmann

A helical shell model of turbulence

Abstract Shell models of turbulence are highly instructive one-dimensional isotropic spectral models able to re-produce the rich and complicated time-dependent behaviour of turbulent cascades.

Although their Navier--Stokes-like features are impressive they have several unresolved shortcomings, amongst these the representation of the inviscid quadratic invariant helicity known to play a central role in blocking the energy cascade and in triggering intermittency.

This thesis attempts to improve the realism of shell models by re-casting the Sabra shell model in an alternate form using a helical decomposition, thus allowing helicity to be properly represented.

Within this decomposition the Sabra model is split into four helical interaction types.

Since these four interaction types occur as a weighted sum in the helically decomposed Navier--Stokes equation, so should they in a helical shell model consisting of all four interaction types.

Therefore, this study additionally attempts to determine such weights using the helically decomposed Navier--Stokes equation. In the process a new generalised helical shell model is uncovered which resembles the helical Sabra model, but unlike it is not restricted to interactions being local in spectral space.

Finally, because of the improve helicity representation this study also investigates the existence of the proposed helicity dissipation scale by numerical calculations of the full helical shell model as well as each separate interaction type, finding the dissipation scale indeed exists.

Supervisor: Peter Ditlevsen, Centre for Ice and Climate