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Exact instantaneous optimals in the non-geostrophic Eady problem and the detrimental effects of discretization

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Abstract

We derive exact analytical expressions for flow configurations that optimize the instantaneous growth rate of energy in the linear Eady problem, along with the associated growth rates. These optimal perturbations are relevant linear stability analysis, but, more importantly, they are relevant for understanding the energetics of fully nonlinear baroclinic turbulence. The optimal perturbations and their growth rates are independent of the Richardson number. The growth rates of the optimal perturbations grow linearly as the horizontal wavelength of the perturbation decreases. Perturbation energy growth at large scales is driven by extraction of potential energy from the mean flow, while at small scales it is driven by extraction of kinetic energy from the mean shear. We also analyze the effect of spatial discretization on the optimal perturbations and their growth rates. A second-order energy-conserving discretization on the Arakawa B grid generally has too weak growth rates at small scales and is less accurate than two second-order discretizations on the Arakawa C grid. The two C-grid discretizations, one that conserves energy and another that conserves both energy and enstrophy, yield very similar optimal perturbation growth rates that are significantly more accurate than the B-grid discretization at small scales.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Colorado, Boulder, CO, 80309, USA

    William Barham & Ian Grooms

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  1. William Barham
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  2. Ian Grooms
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Correspondence to Ian Grooms.

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Communicated by William Dewar.

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WB is supported by the US National Science Foundation grant DMS 1407340.

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Barham, W., Grooms, I. Exact instantaneous optimals in the non-geostrophic Eady problem and the detrimental effects of discretization. Theor. Comput. Fluid Dyn. 33, 125–139 (2019). https://doi.org/10.1007/s00162-019-00488-w

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