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Kelvin–Helmholtz Instability of Potential Vorticity Layers: A Route to Mixing

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  • 1 Department of Mathematics, University College London, London, United Kingdom
  • | 2 Department of Applied Physics and Applied Mathematics, and Department of Earth and Environmental Sciences, Columbia University, New York, New York
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Abstract

The linear and nonlinear dynamics of layers of anomalously high potential vorticity (PV) are studied in detail. It is well known that PV layers are subject to slow, balanced, mixed barotropic–baroclinic instabilities. In this paper, it is shown that, in addition, PV layers are subject to a Kelvin–Helmholtz instability, operating on much smaller spatial and faster temporal scales.

For simplicity, spatially infinite layers of uniform anomalous PV are considered. Such layers are characterized by two key parameters: the ratio Δq of their anomalous PV to the background PV, and the angle α between the layer and the direction of the ambient stratification gradient (in suitably scaled coordinates). It is found that Kelvin–Helmholtz appears, for certain values of α, whenever Δq > 8.

Of notable interest is the case of an initially vertical PV layer embedded in a weak ambient shear flow: for sufficiently large Δq, once the PV layer is tilted past a critical angle, Kelvin–Helmholtz instability becomes possible. It is argued that the breakdown of PV layers due to a Kelvin–Helmholtz instability induced by ambient shear might be an important systematic mechanism leading to irreversible mixing during stratosphere–troposphere exchange events. This is discussed in the context of an example of Kelvin–Helmholtz instability observed near a tropopause fold.

Corresponding author address: J. G. Esler, Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, United Kingdom. Email: gavin@math.ucl.ac.uk

Abstract

The linear and nonlinear dynamics of layers of anomalously high potential vorticity (PV) are studied in detail. It is well known that PV layers are subject to slow, balanced, mixed barotropic–baroclinic instabilities. In this paper, it is shown that, in addition, PV layers are subject to a Kelvin–Helmholtz instability, operating on much smaller spatial and faster temporal scales.

For simplicity, spatially infinite layers of uniform anomalous PV are considered. Such layers are characterized by two key parameters: the ratio Δq of their anomalous PV to the background PV, and the angle α between the layer and the direction of the ambient stratification gradient (in suitably scaled coordinates). It is found that Kelvin–Helmholtz appears, for certain values of α, whenever Δq > 8.

Of notable interest is the case of an initially vertical PV layer embedded in a weak ambient shear flow: for sufficiently large Δq, once the PV layer is tilted past a critical angle, Kelvin–Helmholtz instability becomes possible. It is argued that the breakdown of PV layers due to a Kelvin–Helmholtz instability induced by ambient shear might be an important systematic mechanism leading to irreversible mixing during stratosphere–troposphere exchange events. This is discussed in the context of an example of Kelvin–Helmholtz instability observed near a tropopause fold.

Corresponding author address: J. G. Esler, Department of Mathematics, University College London, 25 Gower Street, London WC1E 6BT, United Kingdom. Email: gavin@math.ucl.ac.uk

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