# Eady Baroclinic Instability of a Circular Vortex

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*Symmetry*)

## Abstract

**:**

## 1. Introduction

## 2. Surface Quasi-Geostrophic Model and Equations

## 3. Mean Flow Calculation

#### 3.1. Preliminaries about Fourier Decomposition in Cylindrical Coordinates

- $\varphi $ representing an angle, then f is $2\pi $-periodic in $\varphi $, so can be decomposed in Fourier modes:$$f(r,\varphi ,z)=\sum _{n\in \mathbf{N}}\tilde{f}(r,n,z){e}^{in\varphi},$$$$\tilde{f}(r,n,z)=\frac{1}{2\pi}{\int}_{0}^{2\pi}f(r,\varphi ,z){e}^{-in\varphi}\mathrm{d}\varphi .$$
- For every fixed $n\in \mathbf{N}$ and $z\in {\mathbf{R}}^{+}$, the functions $r\mapsto \tilde{f}(r,n,z)$ can be written as inverse Hankel transforms (kind of Fourier transform for radial functions):$$\tilde{f}(r,n,z)={\int}_{0}^{\infty}\widehat{f}(\rho ,n,z){J}_{n}\left(\rho r\right)\rho \mathrm{d}\rho ,$$$$\widehat{f}(\rho ,n,z)={\int}_{0}^{\infty}\tilde{f}(r,n,z){J}_{n}\left(\rho r\right)r\mathrm{d}r.$$

**Remark**

**1.**

#### 3.2. Application to SQG Flows

**Remark**

**2.**

#### 3.3. Basic State: Two Top-Hat Vortices

**Remark**

**3.**

**Remark**

**4.**

## 4. Linear Evolution of the Vortex Boundaries under Perturbations

#### 4.1. Framework

**Remark**

**5.**

#### 4.2. Calculation of the Perturbed Quantities

#### 4.3. Dynamics of the Perturbations

## 5. Results

#### 5.1. Preliminaries: Study of the Integrals ${I}_{n}-{I}_{1}$ and ${M}_{n}$

#### 5.2. Normal Modes

**Remark**

**6.**

**Remark**

**7.**

**Remark**

**8.**

- In the upper part of each panel, where $\sigma $ is larger than a threshold ${\sigma}_{\mathrm{critic}}$ depending on n. We recover here the results of [13,14]. They found that a top-hat vortex, alone in an SQG model, is stable. In this area, the system is linearly stable for barotropic (horizontal shear) instability. They are sufficiently far from the other ($\sigma $ is proportional to H), so we could neglect the interactions. The two-layer SQG model is then viewed as two one-layer SQG models where there are two independent top-hat vortices. We define ${\sigma}_{\mathrm{critic}}$ as the critical value of $\sigma $ leading to instability, all other parameters being fixed. ${\sigma}_{\mathrm{critic}}$ is a decreasing function with respect to n. The stability of high mode perturbations is reached for a smaller distance between vortices than low mode perturbations. This is due to the relation between horizontal and vertical wave numbers in the SQG model.
- In the bottom left and the bottom right sides of each panel, the system is also stable. In these areas, the vortices are close to each other, but have very different intensities. An interpretation of this situation could be that perturbations on one of the vortices have very different phase speeds around the contour than for the other vortex. The impossibility for these two (Rossby) waves to phase lock and resonate stabilizes the whole system.

- For small $\sigma $, the two vortices are too close to each other for the wave to grow; thus, short-wave cut-off (usual for the Eady model) can be explained by the absence of phase locking between waves.
- For intermediate $\sigma $, the distance between the two vortices allows the mode to grow (phase locking with the proper phase shift is possible), and then, the system is unstable. The smaller $\sigma $ is, the shorter the most unstable waves are.
- For large $\sigma $, the two vortices are far from each other, wave–wave interaction is weak, and the mode is stable.

#### 5.3. Singular Modes

## 6. Conclusions and Perspectives

- Investigate the effect of different radii for the two vortices;
- Shift one vortex with respect to the other and study the evolution of tilted vortices;
- Consider two different modes ${n}_{\mathrm{s}}$ and ${n}_{\mathrm{b}}$ of perturbation for the two vortices;
- Consider other radial shapes for the vortices (Gaussian, etc.).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of the System Descibing the Dynamics of the Perturbations

**Remark**

**A1.**

## Appendix B. Proof of Convergence and Numerical Method to Compute I_{n} − I_{1} and M_{n}

- For x in a neighborhood of 0, ${J}_{n}\left(x\right)\underset{0}{\sim}\frac{{x}^{n}}{{2}^{n}n!}$ so for $n>1$:$$\begin{array}{cc}\hfill {f}_{n}\left(x\right)& \underset{0}{\sim}-\frac{{J}_{1}{\left(x\right)}^{2}}{tanh\left(\sigma x\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {f}_{n}\left(x\right)& \underset{0}{\sim}-\frac{x}{4\sigma}\hfill \end{array}$$$${g}_{n}\left(x\right)\underset{0}{\sim}\frac{{x}^{2n-1}}{{4}^{n}{(n!)}^{2}\sigma}.$$Therefore, the two functions are integrable in 0.
- For x in a neighborhood of $+\infty $,$${J}_{n}\left(x\right)\underset{\infty}{=}\sqrt{\frac{2}{\pi x}}sin\left(x-\frac{n\pi}{2}+\frac{\pi}{4}\right)-\frac{4{n}^{2}-1}{4\sqrt{2\pi}{x}^{\frac{3}{2}}}sin\left(x-\frac{n\pi}{2}-\frac{\pi}{4}\right)+o\left(\frac{1}{{x}^{\frac{3}{2}}}\right)$$The computation gives for $n=2p>1$:$${f}_{2p}\left(x\right)\underset{\infty}{\sim}\frac{2sin\left(2x\right)}{\pi x}$$$${f}_{2p+1}\left(x\right)\underset{\infty}{\sim}\frac{1-{\left(2p+1\right)}^{2}}{\pi {x}^{2}}cos\left(2x\right).$$We can quickly conclude for the odd case because ${f}_{2p+1}=O\left(\frac{1}{{x}^{2}}\right)$ is absolutely convergent in $+\infty $. The even case is a modified integral sine, so that it converges.For every $n\in \mathbf{N}$, we have a quick convergence in $+\infty $ for ${g}_{n}$:$${g}_{n}\left(x\right)\underset{\infty}{\sim}\frac{4}{\pi x}{sin}^{2}\left(x-\frac{n\pi}{2}+\frac{\pi}{4}\right){e}^{-\sigma x}.$$

**Figure A1.**The integrands ${f}_{n}$ and ${g}_{n}$ are in the solid line; the asymptotes are plotted with crosses; the envelops for the top right panel are in dashed lines. Notice that for the bottom right panel, the asymptote depends only on the parity of n. This explains why there are only two asymptotes plotted. Here, we take the parameter $\sigma =1$.

- For ${f}_{n}$ in 0:$${\int}_{0}^{\epsilon}{f}_{n}\left(x\right)\mathrm{d}x\simeq -\frac{{\epsilon}^{2}}{8\sigma}$$
- For ${f}_{2p}$ in $+\infty $:$$\begin{array}{cc}\hfill {\int}_{A}^{\infty}{f}_{2p}\left(x\right)\mathrm{d}x& \simeq \frac{2}{\pi}{\int}_{2A}^{\infty}\frac{sint}{t}\mathrm{d}t\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \simeq 1-\frac{2}{\pi}Si\left(2A\right)\hfill \end{array}$$
- For ${f}_{2p+1}$ in $+\infty $:$$\begin{array}{cc}\hfill {\int}_{A}^{\infty}{f}_{2p+1}\left(x\right)\mathrm{d}x& \simeq \frac{1-{(2p+1)}^{2}}{\pi}\left(-{\int}_{A}^{\infty}\frac{2sin\left(2x\right)}{x}\mathrm{d}x-{\left[\frac{cos2x}{x}\right]}_{A}^{\infty}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \simeq \frac{1-{(2p+1)}^{2}}{\pi}\left(2Si\left(2A\right)-\pi +\frac{cos2A}{A}\right)\hfill \end{array}$$
- For ${g}_{n}$ in 0:$${\int}_{0}^{\epsilon}{g}_{n}\left(x\right)\mathrm{d}x\simeq \frac{{\epsilon}^{2n}}{2n{(n!)}^{2}{4}^{n}\sigma}$$
- For ${g}_{n}$ in $+\infty $:$$\begin{array}{cc}\hfill {\int}_{A}^{\infty}{g}_{n}\left(x\right)\mathrm{d}x& \simeq \frac{4}{\pi}{\int}_{A}^{\infty}\frac{{sin}^{2}\left(x-\frac{n\pi}{2}+\frac{\pi}{4}\right)}{x}{e}^{-\sigma x}\mathrm{d}x\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \le \frac{4}{\pi}{\int}_{A}^{\infty}\frac{1}{x}{e}^{-\sigma x}\mathrm{d}x\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \le \frac{4}{\pi}\frac{{e}^{-\sigma A}}{\sigma A}\hfill \end{array}$$Therefore, if we take $\sigma A$ sufficiently large (in practice, we take $\sigma A\simeq 20$), this part can be neglected.

${\mathit{f}}_{2\mathit{p}}$ | ${\mathit{f}}_{2\mathit{p}+1}$ | ${\mathit{g}}_{\mathit{n}}$ | |
---|---|---|---|

${\int}_{0}^{\epsilon}$ | $-\frac{{\epsilon}^{2}}{8\sigma}$ | $-\frac{{\epsilon}^{2}}{8\sigma}$ | $\frac{{\epsilon}^{2n}}{2n{(n!)}^{2}{4}^{n}\sigma}$ |

${\int}_{A}^{\infty}$ | $1-\frac{2}{\pi}Si\left(2A\right)$ | $\frac{1-{(2p+1)}^{2}}{\pi}\left(2Si\left(2A\right)-\pi +\frac{cos2A}{A}\right)$ | 0 |

## Appendix C. Nomenclature

q | Potential Vorticity (PV) |

J | Jacobian operator |

H | the distance between the two vortices or the Heaviside function |

L | the horizontal scale of the dynamics |

$\psi $ | stream function of the total or perturbed flow (from Remark 2) |

b | buoyancy $b={f}_{0}{\partial}_{z}\psi $ |

${N}_{0}$ | Brunt–Vaïsala frequency |

${f}_{0}$ | Coriolis frequency |

${R}^{\mathrm{s}},{R}^{\mathrm{b}}$ | the radii of the vortices |

$\frac{\mathrm{D}}{\mathrm{D}t}$ | horizontal Lagrangian derivative: $\frac{\mathrm{D}}{\mathrm{D}t}={\partial}_{t}+U\xb7\nabla $ |

$\sigma ={N}_{0}H/{f}_{0}L$ | square root of the Burger number |

${B}_{0}^{\mathrm{s}},{B}_{0}^{\mathrm{b}}$ | intensities of the two steady vortices |

K | horizontal Fourier variable |

$(r,\varphi ,z)$ | radial, angular, and vertical coordinates in the cylindrical system |

${J}_{n}$ | Bessel functions of the first kind |

$\widehat{f}$ | Fourier transform of any function f |

${\psi}^{\mathrm{s}},{\psi}^{b}$ | total or perturbed stream functions at the two levels |

${\Psi}^{\mathrm{s}},{\Psi}^{\mathrm{b}}$ | stream functions of the basic state at the two levels |

${B}^{\mathrm{s}},{B}^{\mathrm{b}}$ | buoyancies of the basic state at the two levels |

${B}_{0}^{\mathrm{s}},{B}_{0}^{\mathrm{b}}$ | intensities of the buoyancies of the basic state |

${U}_{r}^{\mathrm{s}},{U}_{r}^{\mathrm{b}}$ | radial velocities of the basic state at the two levels |

${U}_{\varphi}^{\mathrm{s}},{U}_{\varphi}^{\mathrm{b}}$ | angular velocities of the basic state at the two levels |

${I}_{n},{M}_{n}$ | integrals defined in Equation (19); from Equation (34), applied in $r=1$ |

${\eta}^{\mathrm{s}},{\eta}^{\mathrm{b}}$ | perturbation of the vortices’ radii |

${\delta}_{1}$ | the Dirac mass in 1 |

${\mu}^{\mathrm{s}},{\mu}^{\mathrm{b}}$ | multiplicative constants of the normal modes |

${\omega}_{n}={a}_{n}+i{b}_{n}$ | the normal mode of the perturbation |

${A}_{n}$ | matrix defined in Equation (35) |

${\chi}_{n}$ | characteristic polynomial of ${A}_{n}$: ${\chi}_{n}\left(X\right)=det\left(X\phantom{\rule{4pt}{0ex}}{\mathrm{Id}}_{2}-{A}_{n}\right)$ |

${\Delta}_{n}$ | discriminant of the second-order polynomial ${\chi}_{n}$ |

${f}_{n}$ and ${g}_{n}$ | respectively, integrands of ${I}_{n}-{I}_{1}$ and ${M}_{n}$ |

$\epsilon $ | size of the perturbation $\epsilon \ll 1$ |

$O\left(\epsilon \right)$ | Landau notation; design any function bounded by $\epsilon $ when $\epsilon \ll 1$ |

$f\left(x\right)\underset{0}{\sim}g\left(x\right)$ | f is similar to g in $x\to 0$ if $g\left(x\right)\ne 0$ and $\frac{f\left(x\right)}{g\left(x\right)}\underset{x\to 0}{\to}1$ |

$n!$ | factorial n: $n!=n\times (n-1)\times (n-2)\times \cdots \times 2\times 1$ |

Si | $Si\left(x\right)={\int}_{0}^{x}\frac{sin\left(t\right)}{t}\mathrm{d}t$ is the integral sine function |

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**Figure 2.**Graphs of ${I}_{1}$, ${M}_{1}$, and ${I}_{1}-{M}_{1}$ as functions of r, for $\sigma =1$ constant.

**Figure 4.**${b}_{n}$ for $n=2,3,4$, and 5, with respect to $\frac{{B}_{0}^{\mathrm{s}}}{{B}_{0}^{\mathrm{b}}}$ and $\sigma $. Be aware that the color bar differs for each panel.

**Figure 5.**Singular mode for $n=2$ for different times $t=1,2,5,10,50$. Here, ${B}_{0}^{\mathrm{s}}=0.25$. The bottom right panel represents the growth rates for normal mode $n=2$. We can note the convergence of singular modes to normal modes.

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**MDPI and ACS Style**

Vic, A.; Carton, X.; Gula, J.
Eady Baroclinic Instability of a Circular Vortex. *Symmetry* **2022**, *14*, 1438.
https://doi.org/10.3390/sym14071438

**AMA Style**

Vic A, Carton X, Gula J.
Eady Baroclinic Instability of a Circular Vortex. *Symmetry*. 2022; 14(7):1438.
https://doi.org/10.3390/sym14071438

**Chicago/Turabian Style**

Vic, Armand, Xavier Carton, and Jonathan Gula.
2022. "Eady Baroclinic Instability of a Circular Vortex" *Symmetry* 14, no. 7: 1438.
https://doi.org/10.3390/sym14071438