A stable precessing quasi-geostrophic vortex model with distributed potential vorticity

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 $\unicode[STIX]{x1D714}_{0}$.

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.
The basic QG dynamics required to describe the precessing vortex model is first briefly introduced in § 2, including the fundamental equation expressing the material conservation of QG potential vorticity anomaly (x, t) (PVA) by the horizontal geostrophic flow u(x, t). In the next section § 3 we introduce a PVA distribution (x, t 0 ) at the initial time, say, t 0 = 0, involving only three modes, comprising spherical Bessel functions of the first kind j l (ρ), and spherical harmonics Y m l (θ , ϕ), where (ρ, θ, ϕ) are the spherical coordinates. The modes used are the spherical zero-degree modeˆ 0 j 0 (ρ), and the second-degree modesˆ 2,0 j 2 (ρ)Y 0 2 (θ ) andˆ 2,1 j 2 (ρ)Y 1 2 (θ, ϕ). It is postulated that this PVA distribution (x, t 0 ) evolves, subjected to the QG dynamics, as a stable precessing vortex as long as the modal amplitudes |ˆ 2,0 | and |ˆ 2,1 | are smaller than the spherical vortex amplitude |ˆ 0 | (that is, as long as the modes of degree 2 are small perturbations to the spherical mode). The stable precession is verified using three-dimensional numerical simulations. Then a closed-form PVA field˜ (x, t) is obtained in § 4 as an approximate solution to the unsteady (x, t). This approximate solution˜ (x, t) is periodic but not rigid, and addresses both the precession frequency and precession axis slope of the vortex. Finally, concluding remarks are given in § 5.

Basic QG dynamics
The inviscid adiabatic QG flow is governed by the conservation of QG PVA (x, t), by the horizontal geostrophic flow u(x, t) ≡ −∇ h × (φe z ), scaled by f −1 0 , where f 0 is the constant background vorticity, or Coriolis parameter, and φ(x, t) is the geopotential anomaly field. The QG PVA (x, t) is the sum of the dimensionless (scaled by f −1 0 ) vertical component of geostrophic vorticity ζ (x, t) = ∇ 2 h φ and the dimensionless vertical stratification anomaly S(x, t) = −∂D(x, t)/∂z = ∂ 2 φ/∂ẑ 2 , where D is the vertical displacement of isopycnals. Aboveẑ ≡ (N 0 /f 0 )z, where N 0 is the constant background Brunt-Väisälä frequency. Henceforth we omit the hat symbol (ˆ) inẑ and will always work in the QG space, now simply denoted as (x, y, z). The QG PVA equals, in the vertically stretched QG space (x, y, z), the Laplacian of φ(x, t), In terms of the geopotential φ(x, t) the QG PVA conservation (2.1) is where J {A, B} ≡ ∂A/∂x∂B/∂y − ∂A/∂y∂B/∂x is the Jacobian operator. Steady-state solutions to (2.3), with separation of variables in spherical coordinates (ρ, θ , ϕ) and regular at the origin, are the product of the spherical Bessel functions of the first kind j l (ρ) with the spherical harmonics Y m l (θ, ϕ), of degree l and order m, which satisfy the Helmholtz equation ∇ 2 ( j l (ρ)Y m l (θ, ϕ)) = −j l (ρ)Y m l (θ , ϕ). These solutions, with distributed PVA, are used to describe the precessing QG vortex in the next section.

The three modes vertically precessing QG vortex
In this section we describe the precessing QG vortex directly from the initial QG PVA distribution (x, t 0 ). The purpose is to show that, for some range of the modal vortex amplitudes, this initial condition leads to a stable precessing vortex. Those readers already familiar with the QG PV dynamics will rapidly understand the vortex precession directly from the geometry of the modal components of the vortex configuration. The more rigorous mathematical justification of this vortex configuration is postponed to the next section.
Mode 2,1 (ρ, θ, ϕ) (figure 1b) is, by itself, baroclinically unstable or, better expressed, is the baroclinic instability, in the sense that it consists of two baroclinic 890 R1-3 dipoles, one above the other, travelling horizontally in opposite directions along straight trajectories. These baroclinic dipoles would experience vertical shear since, for symmetry reasons, the flow velocity vanishes at the mid-depth z = 0. However, the vortex may be stable due to, again, the addition of the spherical mode 0 (ρ), which adds curvature to the 2,1 dipole trajectories, as occurs with the radial and dipolar modes in the two-dimensional Chaplygin-Lamb vortex (Chaplygin 1903;Flierl, Stern & Whitehead 1983;Meleshko & van Heijst 1994), or as happens with the spherical and dipolar modes, which depend on j 0 (ρ) and j 1 (ρ)Y 1 1 (θ , ϕ) in baroclinic QG dipoles (Viúdez 2019). If this curvature radius is much smaller than the vortex radius (roughly if z|ˆ 2,1 |/|ˆ 0 | zρ 2 -that is, if |ˆ 0 | |ˆ 2,1 |), the inner vortex remains stable and oscillates due to the presence of the two dipoles of the 2,1 mode.

A. Viúdez
The specification of the PVA in the intermediate transition domain shell ρ 1 < ρ < ρ 2 is not, however, unique. Several possibilities, different from that in (3.4), seem to be possible. Extending the mode-0 domain to ρ ρ 2 is another option; although in this case the spherical mode becomes unstable in the intermediate region ρ 1 < ρ ρ 2 , since the gradient of j 0 (ρ) changes sign at ρ = ρ 1 . Another option is to stretch the radial variable of the spherical Bessel function j 0 (k 0 ρ), with k 0 ≡ ρ 1 /ρ 2 , and extend the inner domain to ρ ρ 2 , so that j 0 (k 0 ρ 2 ) = j 0 (ρ 1 ). However, in this case the vortex becomes unsteady in the inner domain ρ ρ 2 , since the advective cross-terms of modes {0} and {2, 1} no longer cancel out (that is, Thus we have used (3.4) as the least unsteady among the different possibilities considered (being aware, however, that other solutions may be possible).
Numerical simulations with a relatively low resolution (128 3 and 256 3 grid points) showed that the vortex configuration is stable and displays a precessing PVA field. Only two particular cases are described here, in which the perturbation amplitudes |ˆ 2,0 | = |ˆ 2,1 | = |ˆ 0 |/2 are relatively large, in order to shown more clearly the vortex precession and that, even in these cases, the vortex remains stable. The initial PVA distributions are shown in figure 2 and their time evolution in movies 1, 2 and 3 (supplementary material available at https://doi.org/10.1017/jfm.2020.130). The precessing vortex may be conceptually regarded as a family of horizontal, two-dimensional Chaplygin-Lamb dipoles, parameterized by the depth z, whose trajectories are horizontal circles of radius proportional to the depth z and centred along the vertical axis e z . The initial points of these Chaplygin-Lamb dipole trajectories are located along a straight vertically tilted axis, which corresponds to the vortex precession axis.
When the modal PVA amplitudesˆ 0 andˆ 2,0 have different sign (initial PVA in figure 2(a) and time evolution in movies 1 and 2) this family of Chaplygin-Lamb dipoles is visualized from the two tilted spherical caps of PVA anomaly, with sign opposite to that of the vortex core, located above and below the vortex core. In this particular simulation the precessing vortex experienced n p = 11 anticlockwise precessions in a time period t 4860, which corresponds to an angular velocity ω 0 = 2πn p / t 0.0142.
When the zonal modal PVA amplitudesˆ 0 andˆ 2,0 have the same sign (initial PVA in figure 2(b) and time evolution in movie 3) the family of Chaplygin-Lamb dipoles 890 R1-5 A. Viúdez is easily inferred from the tilted torus of PVA anomaly, with sign opposite to that of the vortex core, located around the vortex core at mid-depth z 0. In this particular simulation the precessing vortex experienced n p = 10 anticlockwise precessions in a time period t 4950, which corresponds to an angular velocity ω 0 = 2πn p / t 0.0127. We note that, in the two cases described above, the PVA distributions of opposite sign to that of the vortex core, having a geometry similar to two spherical caps and one tilted torus, though of small amplitude in comparison with that of the vortex core, are necessary to maintain the vortex precession.
We conclude this section by asserting that the precession of a baroclinic vortex may be interpreted as the horizontal and circular advection by a large-amplitude spherical mode 0 (ρ) of the small-amplitude vertical mode 2,0 (ρ, θ ) tilted by a small-amplitude mode 2,1 (ρ, θ, ϕ). The next section provides the mathematical justification of this assertion.
The background flowū(x) ≡ −∇ × (φ(x) e z ) is horizontal and is the sum of the background modal velocity fields The positionr(X, t) of particles X = (X, Y, Z) moving with the background flowū(x) = u 0 (r) +ū 2,0 (r) +ū 2,1 (z) therefore satisfies the equation ∂r ∂t (X, t) =ū 0 (r(X, t)) +ū 2,0 (r(X, t)) +ū 2,1 (z(X, t)) = ω 0 e z ×r(X, t) + ξ 0 e z ·r(X, t)e y = ω 0 (r(X, t)e ϕ + γ 0 Ze y ), (4.24) where, obviously,z(X, t) = z = Z = e z ·r(X, t), sinceū(x) is horizontal, and we have defined the angular velocity ω 0 , vertical shearing ξ 0 , and their ratio γ 0 as assuming, in the definition of the ratio γ 0 , that ω 0 = 0. The solution to (4.24) is where R[α] is the two-dimensional rotation tensor that rotates a vector (x, y) counterclockwise through an angle α. The background motion is therefore a horizontal rotation of the particles (X, Y, Z) by an angle ω 0 t around the point (−γ 0 Z, 0, Z). The background motion is not a rigid motion in the three-dimensional space, due to the constant γ 0 , but it might be thought of as a continuous family, parameterized by the vertical coordinate z, of changes of reference of horizontal two-dimensional frames rotating around the points (−γ 0 Z, 0, Z) = (−γ 0 , 0, 1)Z, where γ 0 defines the tangent of the axis comprising the centres of rotation at every depth z. More importantly, since the streamlines of the horizontal background flow are circles, the background flow has no diffluence, no bifurcation points, and therefore will not largely distort the modal distributions, as long as the vertical shear ξ 0 remains small or the ratio |γ 0 | 1. This property is characteristic of modes j l (ρ)Y 0,±1 l (θ , ϕ), where the azimuthal wavenumber is 0 or ±1.

Concluding remarks
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 modeˆ 0 j 0 (ρ) of the modeˆ 2,0 j 2 (ρ)Y 0 2 (ρ, θ ) vertically tilted by the modê 2,1 j 2 (ρ)Y 1 2 (ρ, θ, ϕ). Unlike what happens in the case of the quasi-geostrophic 890 R1-10