TY - JOUR

T1 - Stratospheric Planetary Flows from the Perspective of the Euler Equation on a Rotating Sphere

AU - Constantin, A.

AU - Germain, P.

N1 - Funding Information:
The authors thank Vladimir Šverak for a very helpful communication on the prescribed Gauss curvature equation on the sphere. They are also grateful for the comments and suggestions made by the referee. While working on this project, PG was visiting the University of Vienna, to which he is very grateful. PG was supported by the NSF grant DMS-1501019, by the Simons collaborative grant on weak turbulence, and by the Center for Stability, Instability and Turbulence (NYUAD). AC was supported by the Austrian Science Foundation (FWF) "Wittgenstein-Preis Z 387-N".
Funding Information:
The authors thank Vladimir Šverak for a very helpful communication on the prescribed Gauss curvature equation on the sphere. They are also grateful for the comments and suggestions made by the referee. While working on this project, PG was visiting the University of Vienna, to which he is very grateful. PG was supported by the NSF grant DMS-1501019, by the Simons collaborative grant on weak turbulence, and by the Center for Stability, Instability and Turbulence (NYUAD). AC was supported by the Austrian Science Foundation (FWF) "Wittgenstein-Preis Z 387-N".
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/7

Y1 - 2022/7

N2 - This article is devoted to stationary solutions of Euler’s equation on a rotating sphere, and to their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system and in polar regions of the Earth. For the Euler equation, under appropriate conditions, rigidity results are established, ensuring that the solutions are either zonal or rotated zonal solutions. A natural analogue of Arnold’s stability criterion is proved. In both cases, the lowest mode Rossby–Haurwitz stationary solutions (more precisely, those whose stream functions belong to the sum of the first two eigenspaces of the Laplace-Beltrami operator) appear as limiting cases. We study the stability properties of these critical stationary solutions. Results on the local and global bifurcation of non-zonal stationary solutions from classical Rossby–Haurwitz waves are also obtained. Finally, we show that stationary solutions of the Euler equation on a rotating sphere are building blocks for travelling-wave solutions of the 3D system that describes the leading order dynamics of stratospheric planetary flows, capturing the characteristic decrease of density and increase of temperature with height in this region of the atmosphere.

AB - This article is devoted to stationary solutions of Euler’s equation on a rotating sphere, and to their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system and in polar regions of the Earth. For the Euler equation, under appropriate conditions, rigidity results are established, ensuring that the solutions are either zonal or rotated zonal solutions. A natural analogue of Arnold’s stability criterion is proved. In both cases, the lowest mode Rossby–Haurwitz stationary solutions (more precisely, those whose stream functions belong to the sum of the first two eigenspaces of the Laplace-Beltrami operator) appear as limiting cases. We study the stability properties of these critical stationary solutions. Results on the local and global bifurcation of non-zonal stationary solutions from classical Rossby–Haurwitz waves are also obtained. Finally, we show that stationary solutions of the Euler equation on a rotating sphere are building blocks for travelling-wave solutions of the 3D system that describes the leading order dynamics of stratospheric planetary flows, capturing the characteristic decrease of density and increase of temperature with height in this region of the atmosphere.

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U2 - 10.1007/s00205-022-01791-3

DO - 10.1007/s00205-022-01791-3

M3 - Article

AN - SCOPUS:85131078502

VL - 245

SP - 587

EP - 644

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 1

ER -