# Volume Transport by a 3D Quasigeostrophic Heton

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 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

- Gryanik, V.M.; Doronina, T.N.; Olbers, D.J.; Warncke, T.H. The theory of three-dimensional hetons and vortex-dominated spreading in localized turbulent convection in a fast rotating stratified fluid. J. Fluid Mech.
**2000**, 423, 71–125. [Google Scholar] [CrossRef] - Carton, X. Hydrodynamic Modeling of Oceanic Vortices. Surv. Geophys.
**2001**, 22, 179–263. [Google Scholar] [CrossRef] - Chelton, D.B.; Schlax, M.G.; Samelson, R.M.; de Szoeke, R.A. Global observations of large oceanic eddies. Geophys. Res. Lett.
**2007**, 34, L15606. [Google Scholar] [CrossRef] - Chelton, D.B.; Schlax, M.G.; Samelson, R.M. Global observations of nonlinear mesoscale eddies. Prog. Oceanogr.
**2011**, 91, 167–216. [Google Scholar] [CrossRef] - Zhang, Z.; Wang, W.; Qiu, B. Oceanic mass transport by mesoscale eddies. Science
**2014**, 345, 322–324. [Google Scholar] [CrossRef] - Koshel, K.V.; Ryzhov, E.A.; Carton, X.J. Vortex Interactions Subjected to Deformation Flows: A Review. Fluids
**2019**, 4, 14. [Google Scholar] [CrossRef][Green Version] - Helmholtz, H. LXIII. On Integrals of the hydrodynamical equations, which express vortex-motion. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1867**, 33, 485–512. [Google Scholar] [CrossRef] - Aref, H. Point vortex dynamics: A classical mathematics playground. J. Math. Phys.
**2007**, 48, 065401. [Google Scholar] [CrossRef][Green Version] - Morikawa, G.K. Geostrophic Vortex Motion. J. Atmos. Sci.
**1960**, 17, 148–158. [Google Scholar] [CrossRef][Green Version] - Charney, J.G. Numerical experiments in atmospheric hydrodynamics. In Experimental Arithmetic, High Speed Computing and Mathematics, Proceedings of the Fifteenth Symposium in Applied Mathematics of the American Mathematical Society, Chicago, IL, USA, 12–14 April 1962 and Atlantic City, NJ, USA, 16–19 April 1962; Metropolis, N.C., Taub, A.H., Todd, J., Tompkins, C.B., Eds.; American Mathematical Society: Providence, RI, USA, 1963; pp. 289–310. [Google Scholar]
- Gryanik, V.M. Dynamics of localized vortex perturbations “vortex charges” in a baroclinic fluid. Izv. Atmos. Ocean. Phys.
**1983**, 19, 347–352. [Google Scholar] - Hogg, N.G.; Stommel, H.M. The Heton, an Elementary Interaction Between Discrete Baroclinic Geostrophic Vortices, and Its Implications Concerning Eddy Heat-Flow. Proc. R. Soc. London. Ser. A Math. Phys. Sci.
**1985**, 397, 1–20. [Google Scholar] - Young, W.R. Some interactions between small numbers of baroclinic, geostrophic vortices. Geophys. Astrophys. Fluid Dyn.
**1985**, 33, 35–61. [Google Scholar] [CrossRef] - Gryanik, V.M.; Sokolovskiy, M.A.; Verron, J. Dynamics of Heton-like Vortices. Regul. Chaotic Dyn.
**2006**, 11, 383. [Google Scholar] [CrossRef] - Petersen, M.R.; Julien, K.; Weiss, J.B. Vortex cores, strain cells, and filaments in quasigeostrophic turbulence. Phys. Fluids
**2006**, 18, 026601. [Google Scholar] [CrossRef] - Baker-Yeboah, S.; Flierl, G.R.; Sutyrin, G.G.; Zhang, Y. Transformation of an Agulhas eddy near the continental slope. Ocean Sci.
**2010**, 6, 143–159. [Google Scholar] [CrossRef][Green Version] - Rogachev, K.A.; Carmack, E.C.; Foreman, M.G.G. Bowhead whales feed on plankton concentrated by estuarine and tidal currents in Academy Bay, Sea of Okhotsk. Cont. Shelf Res.
**2008**, 28, 1811–1826. [Google Scholar] [CrossRef] - Sokolovskiy, M.A.; Verron, J. The Introductory Chapter. In Dynamics of Vortex Structures in a Stratified Rotating Fluid; Sokolovskiy, M.A., Verron, J., Eds.; Atmospheric and Oceanographic Sciences Library; Springer International Publishing: Cham, Switzerland, 2014; pp. 1–36. [Google Scholar] [CrossRef]
- Legg, S.; Marshall, J. A Heton Model of the Spreading Phase of Open-Ocean Deep Convection. J. Phys. Oceanogr.
**1993**, 23, 1040–1056. [Google Scholar] [CrossRef][Green Version] - Legg, S.; Jones, H.; Visbeck, M. A Heton Perspective of Baroclinic Eddy Transfer in Localized Open Ocean Convection. J. Phys. Oceanogr.
**1996**, 26, 2251–2266. [Google Scholar] [CrossRef][Green Version] - Reinaud, J.N.; Carton, X. The interaction between two oppositely travelling, horizontally offset, antisymmetric quasi-geostrophic hetons. J. Fluid Mech.
**2016**, 794, 409–443. [Google Scholar] [CrossRef][Green Version] - Sokolovskiy, M.A.; Koshel, K.V.; Dritschel, D.G.; Reinaud, J.N. N-symmetric interaction of N hetons. I. Analysis of the case N = 2. Phys. Fluids
**2020**, 32, 096601. [Google Scholar] [CrossRef] - Gryanik, V.M. Dynamics of singular geostrophic vortices in a two-layer model of the atmosphere (ocean). Izv. Atmos. Ocean. Phys.
**1983**, 19, 171–179. [Google Scholar] - Gryanik, V.M.; Doronina, T.N. Advective Transport of Dynamically Passive Additives by Baroclinic Singular Geostrophic Vortices in the Atmosphere (Ocean). Izv. Akad. Nauk SSSR Fiz. Atmos. I Okeana
**1990**, 26, 1011–1026. [Google Scholar] - Gryanik, V.M.; Doronina, T.N. The Interaction between Intense Baroclinic Quasi-Geostrophic Vortices in Flows with Vertical and Horizontal Velocity Shears. Izv. Atmos. Ocean. Phys.
**1997**, 33, 155–166. [Google Scholar] - Pedlosky, J. Geophysical Fluid Dynamics; Springer: Berlin/Heidelberg, Germany, 1987; Volume 710. [Google Scholar]
- Vallis, G.K. Atmospheric and Oceanic Fluid Dynamics; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Weiss, J.B.; Knobloch, E. Mass transport and mixing by modulated traveling waves. Phys. Rev. A
**1989**, 40, 2579–2589. [Google Scholar] [CrossRef][Green Version] - Solomon, T.H.; Weeks, E.R.; Swinney, H.L. Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. Phys. Rev. Lett.
**1993**, 71, 3975–3978. [Google Scholar] [CrossRef] - Weiss, J.B. Point-vortex dynamics in three-dimensional ageostrophic balanced flows. J. Fluid Mech.
**2022**, 936, A19. [Google Scholar] [CrossRef]

**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).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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