Abstract
The permanent precession of a baroclinic geophysical vortex is reproduced, under the quasi-geostrophic approximation, using three potential vorticity anomaly modes in spherical geometry. The potential vorticity modes involve the spherical Bessel functions of the first kind $\text{j}_{l}(\unicode[STIX]{x1D70C})$ and the spherical harmonics $\text{Y}_{l}^{m}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ , where $l$ is the degree, $m$ is the order, and $(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ are the spherical coordinates. The vortex precession is interpreted as the horizontal and circular advection by a large-amplitude background flow associated with the spherical mode $c_{0}\text{j}_{0}(\unicode[STIX]{x1D70C})$ of the small-amplitude zonal mode $c_{2,0}\text{j}_{2}(\unicode[STIX]{x1D70C})\text{Y}_{2}^{0}(\unicode[STIX]{x1D703})$ tilted by a small-amplitude mode $c_{2,1}\text{j}_{2}(\unicode[STIX]{x1D70C})\text{Y}_{2}^{1}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ , where $\{c_{0},c_{2,0},c_{2,1}\}$ are constant potential vorticity modal amplitudes. An approximate time-dependent, closed-form solution for the potential vorticity anomaly is given. In this solution the motion of the potential vorticity field is periodic but not rigid. The vortex precession frequency $\unicode[STIX]{x1D714}_{0}$ depends linearly on the amplitudes $c_{0}$ and $c_{2,0}$ of the modal components of order 0, while the slope of the precessing axis depends on the ratio between the modal amplitude $c_{2,1}$ and $\unicode[STIX]{x1D714}_{0}$ .
Highlights
In this work we have provided a simple mathematical model, based on three potential vorticity anomaly modes in spherical geometry, which explains, under the quasi-geostrophic approximation, the permanent precession of geophysical vortices
The precession of this new coherent vortex structure is interpreted as the horizontal and circular advection by the background flow associated with the spherical mode 0j0(ρ) of the mode 2,0j2(ρ)Y02(ρ, θ ) vertically tilted by the mode 2,1j2(ρ)Y12(ρ, θ, φ)
We may associate with the spherical QG vortex, whose potential vorticity anomaly is given only by the mode-0 ˆ 0j0(ρ), an intrinsic frequency ω0 = − ˆ 0j0(ρ1)/3
Summary
Geophysical vortices, such as those in the atmosphere and oceans, are observed in a dynamical state that often departs, if only slightly, from a symmetric flow rotating around a vertically oriented axis (for example, Boulanger, Meunier & Le Dizès 2007). A. Viúdez alignment of vortices could be substantially improved with a mathematical model able to describe in a simple way the permanent precession of vortices with distributed potential vorticity anomaly. This work provides, under the quasi-geostrophic (QG) approximation, a vortex model with distributed potential vorticity able to sustain, under inviscid and adiabatic conditions, a permanent precession of its vertical axis. This approximate solution (x, t) is periodic but not rigid, and addresses both the precession frequency and precession axis slope of the vortex.
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