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Spirals in Potential Vorticity. Part I: Measures of Structure

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  • 1 Department of Meteorology, University of Reading, Reading, United Kingdom
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Abstract

Advection in weather systems results in filamentary and spiral structures in tracers, whose complexity increases as stirring progresses. Characterizations of fine-scale structures in chemical tracers, which are typically unresolved in atmospheric analyses or models, may enable a treatment of mixing between air masses that is very different from a simple diffusion. In addition, filaments in Ertel potential vorticity (PV) and other active tracers can have a direct influence on the surrounding flow that will depend to some extent upon their spatial arrangement as well as internal structure. Here attention is focused on a particular baroclinic wave life cycle that is distinguished by the existence of an exceptionally persistent, synoptic-scale, cyclonic vortex. In this region the PV field exhibits a spiral-shaped filament that is eventually disrupted by vortex rollup due to the nonlinear development of barotropic instability. Similar spirals have been observed in satellite imagery. In this paper the characterization of the structure of PV spirals by a geometrical measure and by a spectral measure and the relationship between the two is considered.

The scale-invariant nature of a spiral can be characterized geometrically by the Kolmogorov capacity (or box-counting dimension) of the set of points of intersection between the spiral and a cut through it (DK). The capacity of the spiral in the baroclinic wave is found to be almost constant (DK ≈ 0.4) during a period when the number of turns increases from 2 to 5. The constancy of DK results from the steadiness of the radial dependence of angular velocity. Another, more traditional, measure of tracer structure is the power spectrum, which might be expected to be related to Kolmogorov capacity in the scale-invariant subrange. However, total wavenumber spectra for PV in the life cycle show two subranges with very different spectral slopes, neither of which relate to the value of capacity. It is hypothesized that the observed atmospheric kinetic energy spectrum is also not directly related to accumulating discontinuities in PV because the scale-invariant subrange of PV structures, from synoptic scales to mesoscales, is too narrow.

In conclusion, the Kolmogorov capacity is a more useful measure of structures formed by advection. For instance, the capacity of PV spirals is used as the basis for an investigation of their stability in Part II. The characterization of tracer structure with geometrically based measures, like Kolmogorov capacity, could also be helpful in studies of mixing.

Corresponding author address: Dr. John Methven, Department of Meteorology, University of Reading, P.O. Box 243, Earley Gate, Reading RG6 6BB, United Kingdom.

Email: J.Methven@reading.ac.uk

Abstract

Advection in weather systems results in filamentary and spiral structures in tracers, whose complexity increases as stirring progresses. Characterizations of fine-scale structures in chemical tracers, which are typically unresolved in atmospheric analyses or models, may enable a treatment of mixing between air masses that is very different from a simple diffusion. In addition, filaments in Ertel potential vorticity (PV) and other active tracers can have a direct influence on the surrounding flow that will depend to some extent upon their spatial arrangement as well as internal structure. Here attention is focused on a particular baroclinic wave life cycle that is distinguished by the existence of an exceptionally persistent, synoptic-scale, cyclonic vortex. In this region the PV field exhibits a spiral-shaped filament that is eventually disrupted by vortex rollup due to the nonlinear development of barotropic instability. Similar spirals have been observed in satellite imagery. In this paper the characterization of the structure of PV spirals by a geometrical measure and by a spectral measure and the relationship between the two is considered.

The scale-invariant nature of a spiral can be characterized geometrically by the Kolmogorov capacity (or box-counting dimension) of the set of points of intersection between the spiral and a cut through it (DK). The capacity of the spiral in the baroclinic wave is found to be almost constant (DK ≈ 0.4) during a period when the number of turns increases from 2 to 5. The constancy of DK results from the steadiness of the radial dependence of angular velocity. Another, more traditional, measure of tracer structure is the power spectrum, which might be expected to be related to Kolmogorov capacity in the scale-invariant subrange. However, total wavenumber spectra for PV in the life cycle show two subranges with very different spectral slopes, neither of which relate to the value of capacity. It is hypothesized that the observed atmospheric kinetic energy spectrum is also not directly related to accumulating discontinuities in PV because the scale-invariant subrange of PV structures, from synoptic scales to mesoscales, is too narrow.

In conclusion, the Kolmogorov capacity is a more useful measure of structures formed by advection. For instance, the capacity of PV spirals is used as the basis for an investigation of their stability in Part II. The characterization of tracer structure with geometrically based measures, like Kolmogorov capacity, could also be helpful in studies of mixing.

Corresponding author address: Dr. John Methven, Department of Meteorology, University of Reading, P.O. Box 243, Earley Gate, Reading RG6 6BB, United Kingdom.

Email: J.Methven@reading.ac.uk

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