Finite-Amplitude Wave Activity and Diffusive Flux of Potential Vorticity in Eddy–Mean Flow Interaction

Noboru Nakamura Department of Geophysical Sciences, University of Chicago, Chicago, Illinois

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Da Zhu Department of Geophysical Sciences, University of Chicago, Chicago, Illinois

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Abstract

An exact diagnostic formalism for finite-amplitude eddy–mean flow interaction is developed for barotropic and quasigeostrophic baroclinic flows on the beta plane. Based on the advection–diffusion–reaction equation for potential vorticity (PV), the formalism quantifies both advective and diffusive contributions to the mean flow modification by eddies, of which the latter is the focus of the present article. The present theory adopts a hybrid Eulerian–Lagrangian-mean description of the flow and defines finite-amplitude wave activity in terms of the areal displacement of PV contours from zonal symmetry. Unlike previous formalisms, wave activity is readily calculable from data and the local Eliassen–Palm relation does not involve cubic or higher-order terms in eddy amplitude. This leads to a natural finite-amplitude extension to the local nonacceleration theorem, as well as the global stability theorems, in the inviscid and unforced limit. The formalism incorporates mixing with effective diffusivity of PV, and the diffusive flux of PV is shown to be a sink of wave activity. The relationship between the advective and diffusive fluxes of PV and its implications for parameterization are discussed in the context of wave activity budget. If all momentum associated with wave activity were returned to the zonal-mean flow, a balanced eddy-free flow would ensue. It is shown that this hypothetical flow uREF is unaffected by the advective PV flux and is driven solely by the diffusive PV flux and forcing. For this reason, uREF, rather than the zonal-mean flow, is proposed as a diagnostic for the diffusive mean-flow modification. The formalism is applied to a freely decaying beta-plane turbulence to evaluate the contribution of the diffusive PV flux to the jet formation.

Corresponding author address: Noboru Nakamura, Department of the Geophysical Sciences, University of Chicago, 5734 S. Ellis Avenue, Chicago, IL 60637. Email: nnn@uchicago.edu

Abstract

An exact diagnostic formalism for finite-amplitude eddy–mean flow interaction is developed for barotropic and quasigeostrophic baroclinic flows on the beta plane. Based on the advection–diffusion–reaction equation for potential vorticity (PV), the formalism quantifies both advective and diffusive contributions to the mean flow modification by eddies, of which the latter is the focus of the present article. The present theory adopts a hybrid Eulerian–Lagrangian-mean description of the flow and defines finite-amplitude wave activity in terms of the areal displacement of PV contours from zonal symmetry. Unlike previous formalisms, wave activity is readily calculable from data and the local Eliassen–Palm relation does not involve cubic or higher-order terms in eddy amplitude. This leads to a natural finite-amplitude extension to the local nonacceleration theorem, as well as the global stability theorems, in the inviscid and unforced limit. The formalism incorporates mixing with effective diffusivity of PV, and the diffusive flux of PV is shown to be a sink of wave activity. The relationship between the advective and diffusive fluxes of PV and its implications for parameterization are discussed in the context of wave activity budget. If all momentum associated with wave activity were returned to the zonal-mean flow, a balanced eddy-free flow would ensue. It is shown that this hypothetical flow uREF is unaffected by the advective PV flux and is driven solely by the diffusive PV flux and forcing. For this reason, uREF, rather than the zonal-mean flow, is proposed as a diagnostic for the diffusive mean-flow modification. The formalism is applied to a freely decaying beta-plane turbulence to evaluate the contribution of the diffusive PV flux to the jet formation.

Corresponding author address: Noboru Nakamura, Department of the Geophysical Sciences, University of Chicago, 5734 S. Ellis Avenue, Chicago, IL 60637. Email: nnn@uchicago.edu

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