# Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Three-Layer Model

_{1}= 600 m, H

_{2}= 1000 m, and H

_{3}= 2400 m, respectively. So, dimensionless thicknesses of these layers are 0.15, 0.25, and 0.6, respectively. The first and second deformation radii calculated from the average density stratification correspond to the horizontal scale where the Coriolis and buoyancy effects are similar in magnitude. These deformation radii are as follows: Rd

_{1}= 32 km and Rd

_{2}= 15 km [67].

## 3. Numerical Modeling of Vortex Interaction

#### 3.1. Cyclonic Surfer Vortex and Anticyclonic Intrathermocline Lenses

#### 3.2. Interaction of a Surface Vortex with Two Middle Layer Vortices

#### 3.2.1. Collinear Initial Configuration

#### 3.2.2. Impact of the Two External Intrathermocline Vortices on the Surface Cyclone

## 4. Discussion and Conclusions

- If the cyclone of the upper layer and the anticyclonic lens of the middle layer are separated by some distance, then such a two-layer vortex can either move forward (when its total effective vorticity is zero) or rotate relative to the center of vorticity (when its total effective vorticity is nonzero). In any case, both vortices can move far enough from the original location (Section 3.1).
- If two middle layer (intrathermocline) vortices of opposite signs are initially located on different sides relative to the central surface vortex, then (a) if they are separated far enough, all three vortices move inside individual coaxial annular regions; (b) if the distance between them is small, after a temporary bounded stage of movement, they leave the vicinity of the surface vortex (Section 3.2.1).
- If the intrathermocline vortices make up a pair running into the surface vortex, then two regimes are possible: (a) the pair passes under the surface vortex, changing its direction in its vicinity; (b) the dipole is delayed in the vicinity of the surface vortex, and at this intermediate stage, all three vortices move within a bounded region, after which it is freed from the influence of the cyclone and carried away from it. Such intermediate stages can have different durations which do not regularly depend on the initial distance between the vortices of the pair (Section 3.2.2).
- If the intrathermocline vortices have different intensities (which is a more realistic situation), then the vortices of the middle layer always move along loop-like trajectories in the vicinity of the surface vortex. For certain initial distances between the intrathermocline vortices, their movements have a periodic character (Section 3.2.2).

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The translational velocity $V\text{}\mathrm{vs}.\text{}L$ of the vortex structure composed of the point vortices (4) in the upper and middle layers. The coordinates of the markers correspond to the parameters of the numerical experiments shown in Figure 2a.

**Figure 2.**The initial portions of trajectories of two-layer point vortex structures. The red and blue lines depict the trajectories of cyclonic eddies of the upper layer and anticyclonic eddies of the middle layer, respectively. (

**a**) Translations with velocity (2) at $L=0.56$ (middle pair), $L=0.28$ (inner pair) and $L=0.84$ (outer pair) for the same duration (cf. the three markers on the green curve in Figure 1); (

**b**) rotations with angular velocity (4) relative to the center of vorticity (6) for $L=2.00$ and ${\Lambda}_{2}^{1}/{\Lambda}_{1}^{1}=-1/2;$ (

**c**) same as (

**b**) but for ${\Lambda}_{2}^{1}/{\Lambda}_{1}^{1}=-1/4$. Red and blue markers identify the initial positions of the vortices of the upper and middle layers, respectively.

**Figure 3.**The trajectories of the initial collinear tripolar two-layer point vortex structures at ${\Lambda}_{2}^{1}/{\Lambda}_{1}^{1}=1/2$: (

**a**) $L=1.00$, (

**b**) $L=1.19,$ (

**c**) $L=1.20,$ and (

**d**) $2.00$. The green/blue line is the trajectory of the cyclonic/anticyclonic vortex in the middle layer. The red line is the trajectory of the upper layer cyclone. Markers indicate the initial positions of the vortices. All vortices begin their motion from the initial positions in the cyclonic direction.

**Figure 4.**The trajectories of the two-layer point vortex structures when an intrathermocline pair runs into a surface cyclone at $B=-16$; ${\Lambda}_{2}^{1}/{\Lambda}_{1}^{1}=1/2;$ and (

**a**) $L=1.0$, (

**b**) $L=2.0,$ (

**c**) $L=2.4,$ (

**d**) $L=3.0,$ (

**e**) $L=4.0,$ (

**f**) $L=4.3,$ (

**g**) $L=5.0,$ and (

**h**) $L=\mathrm{6.0.}$

**Figure 5.**Some properties of the finite motion of vortices at ${\Lambda}_{2}^{1}/{\Lambda}_{1}^{1}=1/2:$ (

**a**) The average period T (violet line) of the vortex structure rotation and the number of rotations N (brown line) of the vortices in the closed region vs. the distance L. The curves are plotted according to calculations carried out at values of L with a step of 0.1 inside the interval [0.1; 6.0]. (

**b**) Dependences of y-coordinates for the cyclone of the upper layer (red line) and for the cyclone of the middle layer (green line) at $L=4.8$ in a time interval slightly exceeding the interval of finite motion of the vortices $t\in \left[17.54;373.12\right]$.

**Figure 6.**The trajectories of the two-layer point vortex structure when an two middle layer vortices run into a surface cyclone in the time intervals equal to about 4 loop-like rotations (

**a**) and about 40 loop-like rotations (

**b**) at $L=5,$ $B=-16,$ ${\Lambda}_{2}^{1}/{\Lambda}_{1}^{1}=1/8$, and ${\Lambda}_{2}^{1}/{\Lambda}_{2}^{2}=-1/2$. All vortices begin their motion from the initial positions in the cyclonic direction.

**Figure 7.**Examples of the choreographies at $B=-16;$ ${\Lambda}_{2}^{1}/{\Lambda}_{1}^{1}=1/4;$ ${\Lambda}_{2}^{1}/{\Lambda}_{2}^{2}=-1/2$; and (

**a**) ${L}_{1}=10.7250,$ (

**b**) ${L}_{2}=6.1786,$ (

**c**) ${L}_{3}=4.4957,$ (

**d**) ${L}_{4}=3.6150,$ (

**e**) ${L}_{5}=3.0708,$ and (

**f**) ${L}_{6}=\mathrm{2.6490.}$ All vortices begin their motion from the initial positions in the cyclonic direction.

**Figure 8.**The values of ${L}_{n}\left(n=1,2,\dots ,6\right)$ for the choreographies of Figure 7 (blue markers), i.e., at ${\Lambda}_{2}^{1}/{\Lambda}_{1}^{1}=1/4,$ and the same at ${\Lambda}_{2}^{1}/{\Lambda}_{1}^{1}=1/8$ (brown markers) and ${\Lambda}_{2}^{1}/{\Lambda}_{1}^{1}=1/2$ (green markers).

**Figure 9.**Illustration of the transition from one-mode to two-mode choreography of the Figure 7 with increasing of distance $L=i\xb7\Delta :$ $i=0,1,\dots ,5$ for panels (

**a**), (

**b**), …, (

**f**). Numerals indicate the numbers of successively formed loops. All vortices begin their motion from the initial positions in the cyclonic direction.

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**MDPI and ACS Style**

Sokolovskiy, M.A.; Carton, X.J.; Filyushkin, B.N. Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach. *Mathematics* **2020**, *8*, 1228.
https://doi.org/10.3390/math8081228

**AMA Style**

Sokolovskiy MA, Carton XJ, Filyushkin BN. Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach. *Mathematics*. 2020; 8(8):1228.
https://doi.org/10.3390/math8081228

**Chicago/Turabian Style**

Sokolovskiy, Mikhail A., Xavier J. Carton, and Boris N. Filyushkin. 2020. "Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach" *Mathematics* 8, no. 8: 1228.
https://doi.org/10.3390/math8081228