# A Geometric Perspective on the Modulation of Potential Energy Release by a Lateral Potential Vorticity Gradient

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Numerical Simulations and Results

^{−1}. The strength of the background PV gradient is measured by defining the nondimensional variable ${\beta}^{*}=\beta {L}_{d}^{2}/{U}_{1}$. Five simulations were run, spanning choices of ${\beta}^{*}=\left\{0,0.5,1.0,1.5,2.0\right\}$.

## 4. Conclusions

## Supplementary Materials

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Dritschel, D.G.; McIntyre, M.E. Multiple jets as PV staircases: The Phillips effect and the resilience of eddy-transport barriers. J. Atmos. Sci.
**2008**, 65, 855–874. [Google Scholar] [CrossRef] - Greenslade, M.D.; Haynes, P.H. Vertical transition in transport and mixing in baroclinic flows. J. Atmos. Sci.
**2008**, 65, 1137–1157. [Google Scholar] [CrossRef] - Ferrari, R.; Nikurashin, M. Suppression of eddy diffusivity across jets in the Southern Ocean. J. Phys. Oceanogr.
**2010**, 40, 1501–1519. [Google Scholar] [CrossRef][Green Version] - Smith, K.S. Tracer transport along and across coherent jets in two-dimensional turbulent flow. J. Fluid Mech.
**2005**, 544, 133–142. [Google Scholar] [CrossRef][Green Version] - Rhines, P.B. Waves and turbulence on a beta-plane. J. Fluid Mech.
**1975**, 69, 417–443. [Google Scholar] [CrossRef][Green Version] - Sukoriansky, S.; Dikovskaya, N.; Galperin, B. On the arrest of inverse energy cascade and the Rhines scale. J. Atmos. Sci.
**2007**, 64, 3312–3327. [Google Scholar] [CrossRef] - Dritschel, D.G.; Scott, R.K. Jet sharpening by turbulent mixing. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2011**, 369, 754–770. [Google Scholar] [CrossRef] - Williams, G.P. Planetary circulations: 1. Barotropic representation of Jovian and terrestrial turbulence. J. Atmos. Sci.
**1978**, 35, 1399–1426. [Google Scholar] [CrossRef][Green Version] - Lian, Y.; Showman, A.P. Deep jets on gas-giant planets. Icarus
**2008**, 194, 597–615. [Google Scholar] [CrossRef] - Beron-Vera, F.J.; Brown, M.G.; Olascoaga, M.J.; Rypina, I.I.; Kocak, H.; Udovydchenkov, I.A. Zonal jets as transport barriers in planetary atmospheres. J. Atmos. Sci.
**2008**, 65, 3316–3326. [Google Scholar] [CrossRef] - Berloff, P.; Kamenkovich, I.; Pedlosky, J. A mechanism of formation of multiple zonal jets in the oceans. J. Fluid Mech.
**2009**, 628, 395–425. [Google Scholar] [CrossRef][Green Version] - Peng, M.S.; Jeng, B.F.; Williams, R.T. A numerical study on tropical cyclone intensification. Part I: Beta effect and mean flow effect. J. Atmos. Sci.
**1999**, 56, 1404–1423. [Google Scholar] [CrossRef] - Kidston, J.; Frierson, D.M.W.; Renwick, J.A.; Vallis, G.K. Observations, simulations, and dynamics of jet stream variability and annular modes. J. Clim.
**2010**, 23, 6186–6199. [Google Scholar] [CrossRef] - Stamper, M.A.; Taylor, J.R.; Fox-Kemper, B. The growth and saturation of submesoscale instabilities in the presence of a barotropic jet. J. Phys. Oceanogr.
**2018**, 48, 2779–2797. [Google Scholar] [CrossRef] - Galperin, B.; Sukoriansky, S.; Dikovskaya, N. Zonostrophic turbulence. Phys. Scr.
**2008**, 2008, 014034. [Google Scholar] [CrossRef] - Galperin, B.; Sukoriansky, S.; Dikovskaya, N. Geophysical flows with anisotropic turbulence and dispersive waves: Flows with a β-effect. Ocean Dyn.
**2010**, 60, 427–441. [Google Scholar] [CrossRef] - Farrell, B.F.; Ioannou, P.J. Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci.
**2007**, 64, 3652–3665. [Google Scholar] [CrossRef] - Srinivasan, K.; Young, W. Zonostrophic instability. J. Atmos. Sci.
**2012**, 69, 1633–1656. [Google Scholar] [CrossRef][Green Version] - Constantinou, N.C.; Farrell, B.F.; Ioannou, P.J. Emergence and equilibration of jets in beta-plane turbulence: Applications of stochastic structural stability theory. J. Atmos. Sci.
**2014**, 71, 1818–1842. [Google Scholar] [CrossRef][Green Version] - Tobias, S.; Marston, J. Direct statistical simulation of out-of-equilibrium jets. Phys. Rev. Lett.
**2013**, 110, 104502. [Google Scholar] [CrossRef][Green Version] - Bakas, N.A.; Ioannou, P.J. A theory for the emergence of coherent structures in beta-plane turbulence. J. Fluid Mech.
**2014**, 740, 312–341. [Google Scholar] [CrossRef][Green Version] - Bakas, N.A.; Constantinou, N.C.; Ioannou, P.J. S3T stability of the homogeneous state of barotropic beta-plane turbulence. J. Atmos. Sci.
**2015**, 72, 1689–1712. [Google Scholar] [CrossRef][Green Version] - Ioannou, P.; Lindzen, R.S. Baroclinic instability in the presence of barotropic jets. J. Atmos. Sci.
**1986**, 43, 2999–3014. [Google Scholar] [CrossRef] - James, I.N. Suppression of baroclinic instability in horizontally sheared flows. J. Atmos. Sci.
**1987**, 44, 3710–3720. [Google Scholar] [CrossRef][Green Version] - Nakamura, N. An illustrative model of instabilities in meridionally and vertically sheared flows. J. Atmos. Sci.
**1993**, 50, 357–376. [Google Scholar] [CrossRef][Green Version] - Panetta, R.L. Zonal jets in wide baroclinically unstable regions: Persistence and scale selection. J. Atmos. Sci.
**1993**, 50, 2073–2106. [Google Scholar] [CrossRef][Green Version] - Taylor, J.R.; Bachman, S.D.; Stamper, M.A.; Hosegood, P.; Adams, K.; Sallee, J.B.; Torres, R. Submesoscale Rossby waves on the Antarctic Circumpolar Current. Sci. Adv.
**2018**, 4, eaao2824. [Google Scholar] [CrossRef][Green Version] - Nakamura, N. Momentum flux, flow symmetry, and the nonlinear barotropic governor. J. Atmos. Sci.
**1993**, 50, 2159–2179. [Google Scholar] [CrossRef] - Green, J.S.A. A problem in baroclinic stability. Q. J. R. Meteorol. Soc.
**1960**, 86, 237–251. [Google Scholar] [CrossRef] - Lindzen, R.S. The Eady problem for a basic state with zero PV gradient but β ≠ 0. J. Atmos. Sci.
**1994**, 51, 3221–3226. [Google Scholar] [CrossRef] - Vallis, G.K. Atmospheric and Oceanic Fluid Dynamics; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Farrell, B.F.; Ioannou, P.J. Formation of jets by baroclinic turbulence. J. Atmos. Sci.
**2008**, 65, 3353–3375. [Google Scholar] [CrossRef][Green Version] - Klocker, A. Opening the window to the Southern Ocean: The role of jet dynamics. Sci. Adv.
**2018**, 4, eaao4719. [Google Scholar] [CrossRef][Green Version] - Eady, E.T. Long waves and cyclone waves. Tellus
**1949**, 1, 33–52. [Google Scholar] [CrossRef][Green Version] - Thorpe, A.J.; Hoskins, B.J.; Innocentini, V. The parcel method in a baroclinic atmosphere. J. Atmos. Sci.
**1989**, 46, 1274–1284. [Google Scholar] [CrossRef][Green Version] - Taylor, J.R.; Ferrari, R. Buoyancy and wind-driven convection at mixed layer density fronts. J. Phys. Oceanogr.
**2010**, 40, 1222–1242. [Google Scholar] [CrossRef][Green Version] - Heifetz, E.; Alpert, P.; Da Silva, A. On the parcel method and the baroclinic wedge of instability. J. Atmos. Sci.
**1998**, 55, 788–795. [Google Scholar] [CrossRef] - Maddison, J.; Marshall, D. The Eliassen-Palm flux tensor. J. Fluid Mech.
**2013**, 729, 69. [Google Scholar] [CrossRef] - Burns, K.J.; Vasil, G.M.; Oishi, J.S.; Lecoanet, D.; Brown, B.P. Dedalus: A flexible framework for numerical simulations with spectral methods. Phys. Rev. Res.
**2020**, 2, 023068. [Google Scholar] [CrossRef][Green Version] - Charney, J.G. The dynamics of long waves in a baroclinic westerly current. J. Meteorol.
**1947**, 4, 136–162. [Google Scholar] [CrossRef] - Kuo, H.L. Baroclinic instabilities of linear and jet profiles in the atmosphere. J. Atmos. Sci.
**1979**, 36, 2360–2378. [Google Scholar] [CrossRef][Green Version] - Haine, T.W.N.; Marshall, J. Gravitational, symmetric, and baroclinic instability of the ocean mixed layer. J. Phys. Oceanogr.
**1998**, 28, 634–658. [Google Scholar] [CrossRef] - Cai, M.; Huang, B. A new look at the physics of Rossby waves: A mechanical–Coriolis oscillation. J. Atmos. Sci.
**2013**, 70, 303–316. [Google Scholar] [CrossRef] - Blumsack, S.L.; Gierasch, P. Mars: The effects of topography on baroclinic instability. J. Atmos. Sci.
**1972**, 29, 1081–1089. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Values of (

**a**) ${C}_{b}$, (

**b**) ${C}_{q}$, and (

**c**) the component terms of ${C}_{b}$ in (16), across the interior model layers. Mean values for each curve are shown by the dots on the left side of each panel.

**Figure 2.**(

**a**) Diagnosed growth rates, (

**b**) behavior of the terms in (16), (

**c**) values of ${C}_{b}$, and (

**d**) values of ${C}_{b}$ weighted by growth rate, as functions of the nondimensional wavenumber $\mu $. The gray dashed lines indicate the most unstable Eady wave at $\mu =1.61$. The legend for these plots is the same as in Figure 1. (

**e**) Schematic showing the steepening of the buoyancy flux slope when meridional parcel excursions $\delta y$ are limited as ${\beta}^{*}$ increases.

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Bachman, S.D. A Geometric Perspective on the Modulation of Potential Energy Release by a Lateral Potential Vorticity Gradient. *Fluids* **2020**, *5*, 142.
https://doi.org/10.3390/fluids5030142

**AMA Style**

Bachman SD. A Geometric Perspective on the Modulation of Potential Energy Release by a Lateral Potential Vorticity Gradient. *Fluids*. 2020; 5(3):142.
https://doi.org/10.3390/fluids5030142

**Chicago/Turabian Style**

Bachman, Scott D. 2020. "A Geometric Perspective on the Modulation of Potential Energy Release by a Lateral Potential Vorticity Gradient" *Fluids* 5, no. 3: 142.
https://doi.org/10.3390/fluids5030142