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Long-Range Dynamics of a Shallow Water Triad: Renormalization, Modulation, and Cyclogenesis

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  • 1 Department of Mathematics, Case Western Reserve University, Cleveland, Ohio
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Abstract

Long-time-range atmospheric variability has many sources, and nonlinear mode interactions could play an important role. This paper sets out to explore such internal mechanisms of the long-range variability for the rotating shallow water (RSW) system, at its lowest spectral resolution (triad), but for long (decadal) timescales, about 105 Coriolis units. A small Rossby–Froude number ε sets multiple timescales for the system, ranging from the fast gravity–oscillation scale t to the slow eddy-turnover scale τ = εt and yet slower (climatological) scales. A two-step averaging/renormalization procedure is developed to study underlying mechanisms of long-range variability. It explains the basic phases (relaxation, intensification) and the role of wave–vortex interactions, and reveals a novel mechanism of long-range variability, due to modulation of vortex motions [potential vorticity (PV) modes] by gravity waves. The long cycles [on timescales O(104)] are characterized by relative value of the zonal PV component of the flow. They include longer phases of high zonal PV, intermittent with shorter transitional “blocking” phases, dominated by nonzonal modes. Theoretical results are compared with numeric simulations to assess validity and predictive skill of the renormalized systems.

Corresponding author address: David Gurarie, Dept. of Mathematics, Case Western Reserve University, Cleveland, OH 44106. Email: dxg5@po.cwru.edu

Abstract

Long-time-range atmospheric variability has many sources, and nonlinear mode interactions could play an important role. This paper sets out to explore such internal mechanisms of the long-range variability for the rotating shallow water (RSW) system, at its lowest spectral resolution (triad), but for long (decadal) timescales, about 105 Coriolis units. A small Rossby–Froude number ε sets multiple timescales for the system, ranging from the fast gravity–oscillation scale t to the slow eddy-turnover scale τ = εt and yet slower (climatological) scales. A two-step averaging/renormalization procedure is developed to study underlying mechanisms of long-range variability. It explains the basic phases (relaxation, intensification) and the role of wave–vortex interactions, and reveals a novel mechanism of long-range variability, due to modulation of vortex motions [potential vorticity (PV) modes] by gravity waves. The long cycles [on timescales O(104)] are characterized by relative value of the zonal PV component of the flow. They include longer phases of high zonal PV, intermittent with shorter transitional “blocking” phases, dominated by nonzonal modes. Theoretical results are compared with numeric simulations to assess validity and predictive skill of the renormalized systems.

Corresponding author address: David Gurarie, Dept. of Mathematics, Case Western Reserve University, Cleveland, OH 44106. Email: dxg5@po.cwru.edu

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