# Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 2: Finite-Core-Vortex Approach and Oceanographic Application

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## Abstract

**:**

## 1. Introduction

- (1)
- Lagrangian conservation: in unforced, non-dissipative flows, each fluid column conserves its potential vorticity along its motion.
- (2)
- Invertibility: potential vorticity is invertible into stream-function via the inversion of a generalized Laplacian operator. Horizontal velocity derives from stream-function via a skew-symmetric matrix and a gradient operator.
- (3)
- The “impermeability” theorem: in a fluid layer without the time-change of total mass, nor of total momentum, the volume integral of potential vorticity is conserved [18].

## 2. Model

_{1}= 600 m, H

_{2}= 1000 m and H

_{3}= 2400 m, respectively (see Figure 1). 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 equal to: Rd

_{1}= 32 km and Rd

_{2}= 15 km [20]. The time scale T * is chosen as the rotational period of an isolated circular vortex patch. Assuming that the characteristic horizontal scale R = Rd

_{1}and that the maximum velocity of a vortex of unit radius is 40 cm/s, then we get T * ≈ 9 days. This allows us to render our model dimensionless.

#### 2.1. Mathematical Model

^{2}H

^{2}/f

^{2}R

^{2}), where N is the buoyancy frequency (static stability frequency) and H is the vertical scale of motion. In the quasi-geostrophic model, Ro is small, and Bu is of order unity. Here, we will also assume that R is much smaller than the scale of planetary variations of f, so that f is constant (f-plane approximation). Therefore, the fluid is hydrostatic, and the motion is governed at first order by the balance of the Coriolis acceleration and of the horizontal pressure gradient, in the horizontal momentum equations. A stream-function is then obtained by dividing the dynamic pressure by density and by the Coriolis parameter. The curl of the Euler momentum equations on a rotating planet provide a vorticity equation. In a reasonable approximation of the ocean, we assume that the model stratification is layer wise (layers of constant density). Then, the horizontal velocity is uniform vertically in each layer. Finally, we obtain that the quasi-geostrophic model is governed by the conservation of potential vorticity in each layer:

**Ψ**). We also used the formal vector

**Π**with 3 components ${\Pi}_{i}$.

**Z**and

**Φ**:

_{1}and F

_{2}here and in (2) are expressed in terms of the eigenvalues λ

_{i}:

_{1}= 600 m, H

_{2}= 1000 m and H

_{3}= 2400 m; and R = Rd

_{1}= 32 km, Rd

_{2}= 15 km we obtain from (5) and (6) F

_{1}= 0.14 and F

_{2}= 0.7378 with h

_{1}= 0.15, h

_{2}= 0.25, h

_{3}= 0.6.

**Φ**from (3) using the Green’s function ${G}_{i}:$

#### 2.2. Numerical Models

_{i}is the number of vortex patches in the ith layer and ${\Pi}_{i}^{j}$ are the potential vorticities of vortex patches, which are nonzero constants inside compact areas ${S}_{i}^{j}\left(i=1,2,3;j=1,\dots ,{N}_{i}\right)$. Thus, instead of (7), we can write:

#### 2.3. Physical Model of the Present Problem

_{1}, with the convexity directed upward/downward. The cyclonic/anticyclonic vortex of the middle layer will generate a configuration of η

_{1}and η

_{2}in the form of double concave/double convex lens. As an example, Figure 2 shows a vertical section of these surfaces’ shapes, calculated by the Equation (14) for y = 0 and t = 0 for the case ${N}_{1}=1,{\Pi}_{1}^{1}>0$, ${N}_{2}=2,{\Pi}_{2}^{1}=-{\Pi}_{2}^{2}>0$ inside the initially circular regions; and ${N}_{3}=0$. Note that in the figure, the vertical scale is significantly increased for clarity. The dimensionless radii of the vortices in the figure are as follows: ${R}_{1}^{1}=4$ and ${R}_{2}^{1}={R}_{2}^{2}=1$.

- If the surface cyclone over a large intrathermocline vortex forms a stable structure.
- If the surface cyclone remains coherent and/or if it sheds filaments, small eddies, or small eddy pairs.
- The evolution of the intrathermocline vortices and their ability to deform the upper cyclone with a characteristic surface signature.
- If the intra-layer or inter-layer interactions dominate and in particular, if horizontal vortex pairs or vertical vortex pairs (hetons) or the vertical alignment of like-signed vorticity patches, are the prevalent mechanism.

## 3. Interaction of a Surface Vortex with a Single Intrathermocline Lens

#### 3.1. Vertically Aligned Vortices

#### 3.2. Vertically Shifted Vortices

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

#### 4.1. Collinear Initial Configuration

_{1}= 6 (i.e., 192 km and 32 km). Figure 9b shows that no qualitative changes compared with Figure 9a were observed. In this case, 91.1% of the original volume remains in the central core. Note, that in Figure 9b, for convenience, the dimensions of the surface vortices are shown on the same scale as in Figure 9a, but their actual increase by 1.5 times is visible by the corresponding decrease in the size of the intrathermocline vortices.

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

#### 4.2.1. Case of Intrathermocline Dipole

#### 4.2.2. Case of Asymmetric Middle Layer Vortices

## 5. Discussion and Conclusions

- A two-layer vortex with a vertical axis, consisting of a large-scale cyclone of the upper layer and a sufficiently large anticyclone of the middle layer, can be unstable with respect to small perturbations of the shape of the vortex patches and decompose into smaller two-layer vortex structures. In the case when the intratermocline anticyclone is of the order of Rd
_{1}, the two-layer vortex behaves stably (Section 3.1). Note that such a vortex structure consisting of point vortices is also always stable. - If the centers of the cyclone of the upper layer and the anticyclonic lens of the middle layer are spaced, 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). These properties are also characteristic of point vortices. For finite-core vortices, in addition, there is a certain interval of distances between their centers, in which the motion of the vortex patches is accompanied by a partial destruction of the surface cyclone and a deviation from the trajectory predicted by the theory of point vortices occurs (Section 3.2).
- If intrathermocline dipole is initially located below the surface vortex core, the formation of a such new type of vortex structure occurs when most of the surface cyclone core is left behind and two of its fragments pair up with the lenses and drift away. The two-layer structure including the lenses has a complex form: one part consists of two cyclonic vortex patches in the upper and middle layers, and the second part has a hetonic nature (Section 4.1).
- If the intrathermocline dipole is running onto the surface vortex, then (like in the point vortex case [19]) we have two regimes: (a) the pair passes under the surface vortex, changing its direction in its vicinity and capturing part of its mass; (b) the cyclonic vortex of the intrathermocline dipole is captured by the surface cyclone, and the anticyclonic lens rotates at the periphery of the newly formed bilayer cyclonic structure; and, unlike the case of point vortices, the dipole of the middle layer cannot leave the vicinity of the surface cyclone. (Section 4.2.1).
- If an asymmetry in the distribution of potential vorticity is observed in a vortex structure incident on a surface cyclone, then it makes loop-like movements and, at the same time, rotates around the cyclone of the upper layer. It is striking that in the case of finite-core vortices (depending on the initial distance between them), states can be formed that resemble the modal structures obtained for point vortices (19) as quasi-stationary solutions (Section 4.2.2).

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The average annual vertical density profile (solid magenta line) characteristic for the Atlantic [21] and its piecewise constant approximation (solid black line). Dashed lines indicate undisturbed interfaces between layers.

**Figure 2.**A plane section of the vortex patches which cuts the vertical axes of all three cylindrical vortices for the case when the vortices of the middle layer dipole are located under a large-scale surface vortex. The cyclone of the upper layer is colored red, and the cyclonic and anticyclonic vortices of the middle layer are green and blue, respectively. The dashed lines depict the cross-section of the deformable interface (14) between the layers under the influence of vorticity, as well as solid upper and lower surfaces.

**Figure 3.**Neutral stability curve of the two-layer axisymmetric structure composed of the synoptic surface cyclonic vortex and the anticyclonic middle-layer lens for the mode m = 2 on the plane of variables $\left({r}_{2};{\Pi}_{2}\right)$ with ${r}_{1}=4$ and ${\Pi}_{1}=0.25$. The designations S and U correspond to stable and unstable regions. The round marker corresponds to the parameters used in the experiment shown in Figure 4.

**Figure 4.**Horizontal projections of the synchronous configurations of the vortex-patch contours of the upper (red) and middle (blue) layers at the dimensionless moments indicated on each subplot, for $\left({r}_{1};{\Pi}_{1}\right)=\left(4;0.25\right)$ and $\left({r}_{2};{\Pi}_{2}\right)=\left(3;-0.60\right)$, i.e., a ratio of “effective” vorticity, ${{\rm P}}_{1}/{{\rm P}}_{2}=-0.4444$. The round marker in the region of instability in Figure 3 corresponds to this experiment.

**Figure 5.**The translational velocity $V\text{}\mathrm{vs}.\text{}L$ of the vortex structure composed of the point vortices in upper and middle layers (green line) [19] and finite-area vortex patches in the same layers (black-red line) at ${r}_{1}=4;{{\rm P}}_{1}=0.6$ and ${r}_{2}=1;{{\rm P}}_{2}=-0.6$ on the distance L between vortices in the first case and their centers in the second case. The part of the curve in which the surface vortex loses its compactness and partially collapses is marked in red. Markers on the black ($L=0.71$, $L=7.00$) and red ($L=1.00,L=6.50$) lines correspond to the experiments presented in Figure 6.

**Figure 6.**Gallery of the superposed vortex contours in the upper layer (red lines) and middle layer (blue lines) for ${r}_{1}=4,{r}_{2}=1$ and ${{\rm P}}_{1}=-{{\rm P}}_{2}=0.6$ at times $t=0,1,2,\dots ,24$, and different distances between the vortices: (

**a**) $L=0.71$; (

**b**) $L=1.0$; (

**c**) $L=6.5$; (

**d**) $L=7.0$ (the markers in Figure 5 correspond to these values of L). Vortex patches at $t=0$ are filled. The time evolution goes from bottom to top in each panel.

**Figure 7.**Motion of a two-layer heton (vertical dipolar vortex), with the same parameters ${r}_{1},{r}_{2},{{\rm P}}_{1},{{\rm P}}_{2}$, as in Figure 6, but with $L=4$ at the indicated dimensionless times.

**Figure 8.**This figure is like the last panel of Figure 8 (for which ${{\rm P}}_{2}=-0.6$). Here we have (

**a**) ${{\rm P}}_{2}=-0.3,$ (

**b**) ${{\rm P}}_{2}=-0.15,$ (

**c**) ${{\rm P}}_{2}=-0.075$.

**Figure 9.**Interaction of a surface cyclonic vortex with ${P}_{1}=0.6$ and the pair of vortex patches located in the middle layer with ${r}_{2}^{1}={r}_{2}^{2}=1$, ${{\rm P}}_{2}^{1}=-{{\rm P}}_{2}^{2}=-0.01875$ at $L=1$. The contour of the cyclonic/anticyclonic vortex in the middle layer is in green/blue: (

**a**) ${r}_{1}=4$, (

**b**) ${r}_{1}=6$.

**Figure 10.**An intrathermocline dipole, approaching the surface cyclone at $P1=0.6$, ${r}_{1}=6$, ${r}_{2}^{1}={r}_{2}^{2}=1$, ${{\rm P}}_{2}^{1}=-{{\rm P}}_{2}^{2}=0.3,$ ${y}_{2}^{1}={y}_{2}^{2}=-16$ at $t=1,2,\dots ,12:$ (

**a**) $L=1,$ (

**b**) $L=6$.

**Figure 11.**Sets of superimposed vortex patches contour at r

_{1}=6, ${r}_{2}^{1}={r}_{2}^{2}=1,$ ${{\rm P}}_{2}^{1}/{{\rm P}}_{2}^{2}=-1/2,$ ${{\rm P}}_{2}^{1}/{{\rm P}}_{1}^{1}=1/2,{y}_{2}^{1}={y}_{2}^{2}=-16,$ and (

**a**) $L=2,$ (

**b**) $L=4$.

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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 2: Finite-Core-Vortex Approach and Oceanographic Application. *Mathematics* **2020**, *8*, 1267.
https://doi.org/10.3390/math8081267

**AMA Style**

Sokolovskiy MA, Carton XJ, Filyushkin BN. Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 2: Finite-Core-Vortex Approach and Oceanographic Application. *Mathematics*. 2020; 8(8):1267.
https://doi.org/10.3390/math8081267

**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 2: Finite-Core-Vortex Approach and Oceanographic Application" *Mathematics* 8, no. 8: 1267.
https://doi.org/10.3390/math8081267