A Geometric Interpretation of Eddy Reynolds Stresses in Barotropic Ocean Jets

Abstract

Barotropic eddy fluxes are analysed through a geometric decomposition of the eddy stress tensor. Specifically, the geometry of the eddy variance ellipse, a two-dimensional visualization of the stress tensor describing the mean eddy shape and tilt, is used to elucidate eddy propagation and eddy feedback on the mean flow. Linear shear and jet profiles are analysed and theoretical results are compared against fully nonlinear simulations. For flows with zero planetary vorticity gradient, analytic solutions for the eddy ellipse tilt and anisotropy are obtained that provide a direct relationship between the eddy tilt and the phase difference of a normal mode solution. This allows a straightforward interpretation of the eddy-mean flow interaction in terms of classical stability theory: the initially unstable jet gives rise to eddies which are tilted "against the shear" and extract energy from the mean flow; once the jet stabilises, eddies become tilted "with the shear" and return their energy to the mean flow. For a nonzero planetary vorticity gradient, ray-tracing theory is used to predict ellipse geometry and its impact on eddy propagation within a jet. An analytic solution for the eddy tilt is found for a Rossby wave on a constant background shear. The ray tracing results broadly agree with the eddy tilt diagnosed from a fully nonlinear simulation.

A Geometric Interpretation of Southern Ocean Eddy Form Stress

“…The strength and anisotropy of the eddy stresses are expressed through the size and eccentricity of the ellipses, respectively, and the ellipse orientation with respect to the background velocity shear determines whether the eddies extract or surrender energy to the mean flow. In the case of eddy Reynolds stress, the energy transfer associated with the ellipse tilt is consistent with expectations from classical stability theory (Tamarin et al 2016), and the geometric stress representation has previously been used to diagnose eddymean flow interactions in, for example, the atmospheric midlatitude westerlies (Hoskins et al 1983), the Southern Ocean (Morrow et al 1994), and in an idealized model of a western boundary current extension (Waterman and Hoskins 2013;Waterman and Lilly 2015). These studies, together with the attractive properties of the geometrically informed eddy transfer coefficient outlined in the preceding paragraph, suggest that a geometric perspective is useful to understand and parameterize eddy-mean flow interactions.…”

A Geometric Interpretation of Zonostrophic Instability

“…The reason is that symmetry considerations require an angle of zero or π/2 right at the jet core (Tamarin et al 2016). In contrast, we have significant changes in the anisotropy parameter γ 1 over all latitudes.…”

“…The geometric representation of the ellipse becomes evident be rewriting the tensor as (Marshall et al 2012;Tamarin et al 2016):…”

“…In that vein, Marshall et al (2012) applied the framework to the Eady model (Eady 1949) and showed that if the ellipse tilt is consistent with eddy growth, i.e leaning against the shear, then the framework yields a correct order of magnitude for the growth rate based on dimensional grounds only. Tamarin et al (2016) applied the framework to barotropic instability and obtained analytic solutions relating the eddy ellipse geometry with the unstable modes. In this work we seek to extend these efforts to the simplified model of forced-dissipative beta-plane turbulence.…”

Rossby waves and zonal momentum redistribution induced by localised forcing in the rotating shallow-water model

“…The eddy geometry parameters are useful due to their clear physical interpretation and because they can be analytically related to PV and momentum fluxes. Furthermore, the parameters can be used to aid in understanding Rossby wave propagation characteristics (Waterman & Hoskins 2013;Tamarin et al 2016). We complement the eddy geometry analysis by invoking eigenmode decompositions Rossby waves and zonal momentum redistribution 885 A43-3 of the plunger-induced flow response.…”