Eddy Mean Flow Interaction in Idealized Jet Using a Geometric Framework to Parameterize Eddy Fluxes

Eddy properties are analyzed using geometric characteristics of the eddy stress tensor. In this framework, one can define a ‘variance ellipse' that describes the mean eddy shape and tilt, allowing for an insightful investigation of the eddy-mean flow feedback. We first study simple idealized flows such as piecewise linear jet profiles, and perform high-resolution eddy-permitting simulations. For these simple flows, analytic solutions can be found and the dynamics are easily interpreted, forming building blocks for the understanding of more complex flows. We find a direct relation between the eddy tilt given by the geometric description and that given by the phase difference of a normal mode solution, which gives a straightforward interpretation in terms of classical stability theory. We next revisit the problem of a barotropic jet on a beta plane, where the new framework sheds light on important mechanisms such as jet stabilization and jet sharpening. The initially unstable jet gives rise to eddies which are tilted ‘against the shear'; hence perturbations can grow and extract energy from the mean flow. However, since the jet becomes gradually weaker, at a given moment it can no longer satisfy the Rayleigh condition for instability. Once the jet stabilizes, eddies abruptly become tilted ‘with the shear' therefore all perturbations decay as they return their energy to the mean flow, resulting in jet strengthening. The geometric parameters allow us to identify and describe the transition in the eddy-mean flow interaction, where down-gradient eddy fluxes suddenly become up-gradient and the behaviour of the system changes. Finally, we also use the Ray Tracing theory to investigate the eddy propagation within the jet. An analytic solution for the eddy orientation can be found assuming a plane wave solution, for which an interesting relation between the group velocity, the EP flux, and the eddy tilt exists. The ray tracing results capture the essence of the eddy propagation observed, showing how energy is radiated into the jet core in the unstable regime and outward in the stable regime.