# Volume Transport by a 3D Quasigeostrophic Heton

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## Abstract

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## 1. Introduction

## 2. Equations of Motion

#### 2.1. Point Vortex Solutions to the Quasi-Geostrophic Potential Vorticity Equation

#### 2.2. Heton Motion

#### 2.3. The Passive Scalar Motion Induced by an Isolated 3D Heton

## 3. The Structure of Heton-Induced Transport

#### 3.1. Vertically Aligned Heton: $Y=0$, $Z\ne 0$

#### 3.2. Horizontally Aligned Heton: $Y=1$, $Z=0$

#### 3.3. Tilted Heton: $Y=1$, $Z\ne 0$

- A heteroclinic regime, $\left|z\right|<{z}_{p}=\sqrt{3}$. This regime is characterized by the presence of four fixed points, two stable and two unstable. The unstable fixed points are connected by heteroclinic orbits, which bound the trapping region in each horizontal plane.
- A homoclinic regime, ${z}_{p}\le \left|z\right|\le {z}_{sn}\left(Z\right)$. This regime is characterized by the presence of two fixed points, one stable and one unstable. The unstable fixed point has an associated homoclinic orbit, which bounds the trapping region in each horizontal plane.
- A drift regime, $\left|z\right|>{z}_{sn}\left(Z\right)$. This regime has no fixed points and no trapping.

## 4. Volume Trapped by a Tilted Heton, $\mathit{Z}\ne \mathbf{0}$

#### 4.1. Scaling Theory for $Z\gg 1$

- The scaling of the streamfunction determines the scaling of the trapped volume;
- The vortex and drift components of the streamfunction scale uniformly, ${\psi}_{v}$∼${\psi}_{d}$;
- The scaling is isotropic, $\mathbf{x}$∼${Z}^{\alpha}$ with $\alpha >0$.

#### 4.2. Numerical Calculation of the Trapped Volume

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of a three-dimensional heton with an anticyclonic point vortex (circulation $-\Gamma $) initially located at $(0,Y,Z)$ and a cyclonic vortex (circulation $\Gamma $) initially located at $(0,-Y,-Z)$.

**Figure 2.**Phase portraits of the flow induced by a heton with $Z=0$ on horizontal planes with (

**a**) $z=-2$, and (

**b**) $z=0$. These plots show the normalized planar velocity field (grey arrows), the heteroclinic separatrix $\psi =0$ (black) that bounds the trapping region, and stable and unstable fixed points (blue and red dots, respectively). In panel (

**a**), at `large’ distances from the heton, no fluid is trapped. Closer to the heton, as in panel (

**b**), a trapping region exists.

**Figure 3.**Bifurcation diagram (

**a**) and contour slices (

**b**) for the $Z=0$ case. Panel (

**a**) illustrates the generation/destruction of stable (solid blue curves) and unstable (dashed red curves) fixed points. At $z=-{z}_{sn}=-\sqrt{3}$, a double saddle-node bifurcation leads to the creation of two pairs of fixed points. The stable pair is on the $x=0$ plane, and the unstable pair is on the $y=0$ plane. These fixed points are destroyed by another double saddle-node bifurcation at $z={z}_{sn}=\sqrt{3}$. The contour slices of panel (

**b**) illustrate the three-dimensional volume trapped by the heton at five equally spaced planes between $z=-\sqrt{3}$ and $z=0$.

**Figure 4.**Phase portraits of the flow induced by a heton with $Z=1$ on horizontal planes with (

**a**) $z=-4$, (

**b**) $z=-2$, (

**c**) $z=-1$, and (

**d**) $z=0$. The legend remains the same as in Figure 2, except that the separatrix is given by $\psi ={\psi}_{u}$, where ${\psi}_{u}$ is the value of the streamfunction at an unstable fixed point. No fluid is trapped sufficiently far away from the heton (

**a**), but a finite trapping region bounded by a homoclinic separatrix is seen to exist upon increasing z (

**b**). Further increase in z shows the presence of a heteroclinically bound trapping region (

**c**). Finally, at $z=0$, the trapping region is seen to be symmetric about the x and y axes, as in the $Z=0$ case (

**d**).

**Figure 5.**Bifurcation diagram (

**a**) and contour slices (

**b**) for the $Z=1$ case. Panel (

**a**) illustrates the generation/destruction of stable (solid blue curves) and unstable (dashed red curves) fixed points. At $z=-{z}_{sn}\approx -3.18$, a saddle-node bifurcation leads to the creation of a saddle and a center, both on the plane $x=0$. At $z=-{z}_{p}=-\sqrt{3}$, a pitchfork bifurcation causes the saddle to give way to a center on $x=0$ and two saddles on $y=-z$. Similar bifurcations occur for $z>0$, destroying the fixed points. The contour slices of panel (

**b**) illustrate the three-dimensional volume trapped by the heton at five equally spaced planes between $z=-3.18$ and $z=0$.

**Figure 6.**Same as Figure 5 but for $Z=10$. Panel (

**a**) illustrates the generation/destruction of stable (solid blue curves) and unstable (dashed red curves) fixed points. The saddle-node bifurcation is now at $z=-{z}_{sn}\approx -42.96$. The pitchfork bifurcation remains at $z=-{z}_{p}=-\sqrt{3}$. The contour slices of panel (

**b**) illustrate the three-dimensional volume trapped by the heton at five equally spaced planes between $z=-42.96$ and $z=0$.

**Figure 7.**Schematic illustrating the geometry of the streamfunction values and the structure of the invariant manifolds at a fixed height z defining the trapped volume in the (

**a**) homoclinic and (

**b**) heteroclinic regime for $z<0$. For $z>0$, the structures are flipped across $y=0$, and regions inside heteroclinic separatrices display the same pattern of streamfunction values, but regions inside homoclinic separatrices have $\psi >{\psi}_{u}$. Here, ${\psi}_{u}$ is the streamfunction value at an unstable fixed point. Note that when $Z=0$, there is no homoclinic regime, the trapped volume is symmetric about all three coordinate planes, and ${\psi}_{u}=0$.

**Figure 8.**Logarithmic plots showing the variation of the (

**a**) saddle-node bifurcation point ${z}_{sn}$ and (

**b**) trapped volume ${V}_{T}$ with Z (solid line). Both plots conform to the expected scaling (dashed line).

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Sivakumar, A.; Weiss, J.B.
Volume Transport by a 3D Quasigeostrophic Heton. *Fluids* **2022**, *7*, 92.
https://doi.org/10.3390/fluids7030092

**AMA Style**

Sivakumar A, Weiss JB.
Volume Transport by a 3D Quasigeostrophic Heton. *Fluids*. 2022; 7(3):92.
https://doi.org/10.3390/fluids7030092

**Chicago/Turabian Style**

Sivakumar, Adhithiya, and Jeffrey B. Weiss.
2022. "Volume Transport by a 3D Quasigeostrophic Heton" *Fluids* 7, no. 3: 92.
https://doi.org/10.3390/fluids7030092