Elucidating the bidirectional energy cascade of geophysical turbulence in time, space, and scale

Turbulent fluid motions are responsible for closing the energy budget in Earth’s atmosphere and on many astrophysical bodies, dictating their long-term evolution and climate. However, geophysical turbulence, even in the simplest contexts, remains an open problem.

Past work has shown that the theory for homogeneous and isotropic turbulence (HIT) breaks down in a fluid subject to rotation, stratification, or large aspect ratios.

Particularly affected is the central insight from the study of HIT, stating that energy moves to smaller scales through an ‘energy cascade’. In geophysical flows, energy can flow both to larger and smaller scales through a ‘bidirectional’ cascade.

The fraction of energy going to large scales depends on the value of the relevant geophysical parameter, becoming nonzero at an apparent critical point.

The goal of this project is to identify the spatial and spectral signatures of the bidirectional cascade, understand their role in the cascade’s sudden onset, and develop a quantitative theory for its subsequent development.

We plan to use a combination of numerical and statistical methods to explore the bidirectional cascade in space, time, and scale.

This analysis will be done through two complementary perspectives, investigating turbulent structures in physical space and in spectral space.

In the former, the turbulent cascade manifests itself as individual structures which break up, merge, or clump together, depending on the regime. A statistical view of these interactions will provide insight on how the nature of the flow changes.

On the other hand, in spectral space, the phases of the complex velocity amplitudes are known to be responsible for the exchange of energy among different length scales.

We will look into a possible partial synchronization between these phases in our simulations, and attempt to model the transition to a bidirectional cascade using tools from the rich field of synchronization in complex networks.