# American Institute of Mathematical Sciences

doi: 10.3934/jcd.2021028
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## An algebraic approach to the spontaneous formation of spherical jets

 CRM Ennio De Giorgi, Collegio Puteano, Scuola Normale Superiore Piazza dei Cavalieri, 3, Pisa, I-56100, Italy

Received  April 2021 Revised  October 2021 Early access December 2021

The global structure of the atmosphere and the oceans is a continuous source of intriguing challenges in geophysical fluid dynamics (GFD). Among these, jets are determinant in the air and water circulation around the Earth. In the last fifty years, thanks to the development of more and more precise and extensive observations, it has been possible to study in detail the atmospheric formations of the giant-gas planets in the solar system. For those planets, jets are the dominant large scale structure. Starting from the 70s, various theories combining observations and mathematical models have been proposed in order to describe their formation and stability. In this paper, we propose a purely algebraic approach to describe the spontaneous formation of jets on a spherical domain. Analysing the algebraic properties of the 2D Euler equations, we give a characterization of the different jets' structures. The calculations are performed starting from the discrete Zeitlin model of the Euler equations. For this model, the classification of the jets' structures can be precisely described in terms of reductive Lie algebras decomposition. The discrete framework provides a simple tool for analysing both from a theoretical and and a numerical perspective the jets' formation. Furthermore, it allows to extend the results to the original Euler equations.

Citation: Milo Viviani. An algebraic approach to the spontaneous formation of spherical jets. Journal of Computational Dynamics, doi: 10.3934/jcd.2021028
##### References:
 [1] R. V. Abramov and A. J. Majda, Statistically relevant conserved quantities for truncated quasigeostrophic flow, Proc. Nat. Acad. Sci. (USA), 100 (2003), 3841-3846.  doi: 10.1073/pnas.0230451100.  Google Scholar [2] V. I. Arnold, Sur la géometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique del fluids parfaits, Ann. Inst. Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.  Google Scholar [3] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998.  Google Scholar [4] M. Bordemann, J. Hoppe, P. Schaller and M. Schlichenmaier, $\mathfrak {gl}(\infty)$ and geometric quantization, Comm. Math. Phys., 138 (1991), 209-244.  doi: 10.1007/BF02099490.  Google Scholar [5] M. Bordemann, E. Meinrenken and M. Schlichenmaier, Toeplitz quantization of Kähler manifolds and $\mathfrak {gl}(n), n\to\infty$ limits, Comm. Math. Phys., 165 (1994), 281-296.  doi: 10.1007/BF02099772.  Google Scholar [6] F. Bouchet and A. Venaille, Statistical mechanics of two-dimensional and geophysical flows, Phys. Rep., 515 (2012), 227-295.  doi: 10.1016/j.physrep.2012.02.001.  Google Scholar [7] J. Burzlaff, E. DeLoughry and P. Lynch, Generation of zonal flow by resonant rossby-haurwitz wave interactions, Geophys. Astrophys. Fluid Dyn., 102 (2008), 165-177.  doi: 10.1080/03091920701491576.  Google Scholar [8] R. Fjørtoft, On the changes in the spectral distribution of kinetic energy for two-dimensional, nondivergent flow, Tellus, 5 (1953), 225-230.  doi: 10.3402/tellusa.v5i3.8647.  Google Scholar [9] B. Galperin and P. Read, Zonal Jets: Phenomenology, Genesis, and Physics, Cambridge University Press, 2019.  doi: 10.1017/9781107358225.  Google Scholar [10] B. Haurwitz, The motion of atmospheric disturbances on the spherical earth, J. Mar. Res., 3 (1940), 254-267.   Google Scholar [11] J. Hoppe, Ph.D. thesis MIT cambridge, 1982. Google Scholar [12] J. Hoppe and S.-T. Yau, Some properties of matrix harmonics on S2, Comm. Math. Phys., 195 (1998), 66-77.  doi: 10.1007/s002200050379.  Google Scholar [13] A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser, 1996. doi: 10.1007/978-1-4757-2453-0.  Google Scholar [14] R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluid., 10 (1967), 1417-1423.  doi: 10.1063/1.1762301.  Google Scholar [15] A. Majda and X. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511616778.  Google Scholar [16] J. Marsden and A. Weinstein, Co-adjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.  Google Scholar [17] J. Miller, P. B. Weichman and M. C. Cross, Statistical mechanics, euler's equation, and jupiter's red spot, Phys. Rev. A, 45 (1992), 2328-2359.   Google Scholar [18] K. Modin and M. Viviani, A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics, J. Fluid Mech., 884 (2020), 27pp. doi: 10.1017/jfm.2019.944.  Google Scholar [19] K. Modin and M. Viviani, Canonical scale separation in two-dimensional incompressible hydrodynamics, arXiv, 2021. Google Scholar [20] K. Obuse and M. Yamada, Three-wave resonant interactions and zonal flows in two-dimensional rossby-haurwitz wave turbulence on a rotating sphere, Phys. Rev. Fluids, 4 (2019), 024601.  doi: 10.1103/PhysRevFluids.4.024601.  Google Scholar [21] G. M. Reznik, L. I. Piterbarg and E. A. Kartashova, Nonlinear interactions of spherical Rossby modes, Dynamics of Atmospheres and Oceans, 18 (1993), 235-252.   Google Scholar [22] P. Rhines, Observations of the energy-containing oceanic Eddies, and theoretical models of waves and turbulence, Boundary-Layer Meteorology, 345 (1973), 345-360.  doi: 10.1007/BF02265243.  Google Scholar [23] P.-M. Rios and E. Straume, Symbol Correspondences for Spin Systems, Springer, 2014. doi: 10.1007/978-3-319-08198-4.  Google Scholar [24] M. Viviani, A minimal-variable symplectic method for isospectral flows, BIT, 60 (2020), 741-758.  doi: 10.1007/s10543-019-00792-1.  Google Scholar [25] G. P. Williams, Jupiter's atmospheric circulation, Nature, 257 (1975). doi: 10.1038/257778a0.  Google Scholar [26] V. Zeitlin, Finite-mode analogues of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure, Phys. D, 49 (1991), 353-362.  doi: 10.1016/0167-2789(91)90152-Y.  Google Scholar [27] V. Zeitlin, Self-Consistent-Mode Approximation for the Hydrodynamics of an Incompressible Fluid on Non rotating and Rotating Spheres, Physical Review Letters, 93 (2004), 353-362.   Google Scholar

show all references

##### References:
 [1] R. V. Abramov and A. J. Majda, Statistically relevant conserved quantities for truncated quasigeostrophic flow, Proc. Nat. Acad. Sci. (USA), 100 (2003), 3841-3846.  doi: 10.1073/pnas.0230451100.  Google Scholar [2] V. I. Arnold, Sur la géometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique del fluids parfaits, Ann. Inst. Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.  Google Scholar [3] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, 125. Springer-Verlag, New York, 1998.  Google Scholar [4] M. Bordemann, J. Hoppe, P. Schaller and M. Schlichenmaier, $\mathfrak {gl}(\infty)$ and geometric quantization, Comm. Math. Phys., 138 (1991), 209-244.  doi: 10.1007/BF02099490.  Google Scholar [5] M. Bordemann, E. Meinrenken and M. Schlichenmaier, Toeplitz quantization of Kähler manifolds and $\mathfrak {gl}(n), n\to\infty$ limits, Comm. Math. Phys., 165 (1994), 281-296.  doi: 10.1007/BF02099772.  Google Scholar [6] F. Bouchet and A. Venaille, Statistical mechanics of two-dimensional and geophysical flows, Phys. Rep., 515 (2012), 227-295.  doi: 10.1016/j.physrep.2012.02.001.  Google Scholar [7] J. Burzlaff, E. DeLoughry and P. Lynch, Generation of zonal flow by resonant rossby-haurwitz wave interactions, Geophys. Astrophys. Fluid Dyn., 102 (2008), 165-177.  doi: 10.1080/03091920701491576.  Google Scholar [8] R. Fjørtoft, On the changes in the spectral distribution of kinetic energy for two-dimensional, nondivergent flow, Tellus, 5 (1953), 225-230.  doi: 10.3402/tellusa.v5i3.8647.  Google Scholar [9] B. Galperin and P. Read, Zonal Jets: Phenomenology, Genesis, and Physics, Cambridge University Press, 2019.  doi: 10.1017/9781107358225.  Google Scholar [10] B. Haurwitz, The motion of atmospheric disturbances on the spherical earth, J. Mar. Res., 3 (1940), 254-267.   Google Scholar [11] J. Hoppe, Ph.D. thesis MIT cambridge, 1982. Google Scholar [12] J. Hoppe and S.-T. Yau, Some properties of matrix harmonics on S2, Comm. Math. Phys., 195 (1998), 66-77.  doi: 10.1007/s002200050379.  Google Scholar [13] A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser, 1996. doi: 10.1007/978-1-4757-2453-0.  Google Scholar [14] R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluid., 10 (1967), 1417-1423.  doi: 10.1063/1.1762301.  Google Scholar [15] A. Majda and X. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511616778.  Google Scholar [16] J. Marsden and A. Weinstein, Co-adjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.  Google Scholar [17] J. Miller, P. B. Weichman and M. C. Cross, Statistical mechanics, euler's equation, and jupiter's red spot, Phys. Rev. A, 45 (1992), 2328-2359.   Google Scholar [18] K. Modin and M. Viviani, A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics, J. Fluid Mech., 884 (2020), 27pp. doi: 10.1017/jfm.2019.944.  Google Scholar [19] K. Modin and M. Viviani, Canonical scale separation in two-dimensional incompressible hydrodynamics, arXiv, 2021. Google Scholar [20] K. Obuse and M. Yamada, Three-wave resonant interactions and zonal flows in two-dimensional rossby-haurwitz wave turbulence on a rotating sphere, Phys. Rev. Fluids, 4 (2019), 024601.  doi: 10.1103/PhysRevFluids.4.024601.  Google Scholar [21] G. M. Reznik, L. I. Piterbarg and E. A. Kartashova, Nonlinear interactions of spherical Rossby modes, Dynamics of Atmospheres and Oceans, 18 (1993), 235-252.   Google Scholar [22] P. Rhines, Observations of the energy-containing oceanic Eddies, and theoretical models of waves and turbulence, Boundary-Layer Meteorology, 345 (1973), 345-360.  doi: 10.1007/BF02265243.  Google Scholar [23] P.-M. Rios and E. Straume, Symbol Correspondences for Spin Systems, Springer, 2014. doi: 10.1007/978-3-319-08198-4.  Google Scholar [24] M. Viviani, A minimal-variable symplectic method for isospectral flows, BIT, 60 (2020), 741-758.  doi: 10.1007/s10543-019-00792-1.  Google Scholar [25] G. P. Williams, Jupiter's atmospheric circulation, Nature, 257 (1975). doi: 10.1038/257778a0.  Google Scholar [26] V. Zeitlin, Finite-mode analogues of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure, Phys. D, 49 (1991), 353-362.  doi: 10.1016/0167-2789(91)90152-Y.  Google Scholar [27] V. Zeitlin, Self-Consistent-Mode Approximation for the Hydrodynamics of an Incompressible Fluid on Non rotating and Rotating Spheres, Physical Review Letters, 93 (2004), 353-362.   Google Scholar
Example of a decomposition of $\mathfrak{D}(7, 3)$ as explained in Step 1, where the symbols $\lbrace*, x, = \rbrace$ denote the only possible non-zero entries
Ratio of the energy of the diagonal components and total energy of $W$ at $t = t_0, (t_{end}+t_0)/2, t_{end}$. This values clearly show the evolution of the fluid into a quasi-zonal state
">Figure 3.  Values of the variables appearing in the inequalities (13) at $t = t_0, (t_{end}+t_0)/2, t_{end}$, for the same simulation of Figure 5
">Figure 4.  Values of the variables appearing in the inequalities (14) at $t = t_0, (t_{end}+t_0)/2, t_{end}$, for the same simulation of Figure 5
The vorticity fields $W, W_s$ at $t = t_0$ and $t = t_{end}$, in $\mathfrak{D}(257, 3)$
The vorticity fields $W, W_s$ at $t = t_0$ and $t = t_{end}$, in $\mathfrak{D}(257, 7)$
The vorticity fields $W, W_s$ at $t = t_0$ and $t = t_{end}$, in $\mathfrak{D}(257, 11)$
The vorticity fields $W, W_s$ at $t = t_0$ and $t = t_{end}$, in $\mathfrak{D}(257, 17)$
Illustration of Theorem 4.2 and Remark 3 in $\mathfrak{D}(257, 3)$ for vorticity fields $W_{86, 1}, W_{86, 2}, W_{85, 1}$ at $t = t_0$ and $t = t_{end}$
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