# American Institute of Mathematical Sciences

April  2021, 20(4): 1385-1412. doi: 10.3934/cpaa.2021025

## Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows

 1 School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China 2 Department of Earth and Atmospheric Sciences, Indiana University, Bloomington, IN 47405, USA 3 College of Mathematics, Sichuan University, Chengdu, Sichuan, 610065, China

* Corresponding author

Received  September 2020 Revised  December 2020 Published  March 2021

Fund Project: The research of the corresponding author was supported by National Science Foundation of China (NSFC) grant 11901408. The research of C. Kieu was supported by ONR Awards N000141812588 and N000142012411

This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.

Citation: Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1385-1412. doi: 10.3934/cpaa.2021025
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##### References:
Illustration of the steady-state flow $\psi_{S}$ driven by the Kolmogorov forcing, which is obtained from the QG model (2.1). The blue dashed curve represents the horizontal profile of the mean flow, while the black arrows represent the direction of the mean flow on the horizontal domain $\Omega_a$
Marginal stability curves $\text{R}_{m}^*(a)$ determined by equation $\text{Re}\rho_{m, 1}( \text{R}) = 0$ for $0.1\leq a\leq0.35$
Marginal stability curves $\text{R}^*(a)$ determined by $\text{R}^*(a) = \min\{ \text{R}: \text{Re}\rho_{m, 1}( \text{R}) = 0, m = 1, 2, \cdots, k\}$ for $0.1\leq a\leq0.35$
Illustration of the first critical wave number $n$ for a range of the aspect ratio parameter $0.1\leq a\leq0.35$ and $E = 0.05$
The topological structure of the continuous transition
The topological structure of the catastrophic transition and a separation of periodic orbits
The case $\delta \text{Re}(\rho_{n, 1}) < \text{Re}(\rho_{n+1, 1})$ in Theorem 4.4
The case $\delta \text{Re}(\rho_{n, 1}) > \text{Re}(\rho_{n+1, 1})$ in Theorem 4.4
The regions in the $\text{Re}(\rho_{n, 1})$--$\text{Re}(\rho_{n+1, 1})$ plane with different dynamical behaviours according to Theorem 4.4. In region $\texttt{IV}$, the basic steady state is locally asymptotically stable. In regions $\texttt{I}, \texttt{III}$ and $\texttt{V}$, the system undergoes a supercritical Hopf bifurcation. In region $\texttt{II}$, the system will tend to a double time periodic solution. This solution corresponds to an invariant 2D torus
Transition number with $0.1\leq a\leq0.35$ and $\epsilon = 0.3$
The observed periodic solution and $\epsilon = 0.3$ with $E = 0.05$ at $\text{R} = 3.8717> \text{R}^* = 3.8517$, whose period is T = 2.776. The real period is 3.2116 days
Coefficients in equations (4.6) with various $E$ and $\epsilon = 0.3$ at the critical $\text{R}^*$
 $a$ $E$ $n$ $\text{R}^*$ $A_{1}$ $B_{1}$ $C_{1}$ $D_{1}$ 0.11934208 0.01 4(5) 2.9824 -0.5545 -1.6917 -1.8423 -1.1788 0.11935935 0.03 4(5) 3.4699 -0.2779 -0.8562 -0.8955 -0.5052 0.09768743 0.01 5(6) 2.9518 -0.6002 -1.7218 -1.8647 -1.1083 0.09771521 0.03 5(6) 3.4344 -0.2983 -0.8626 -0.8986 -0.4841 0.08267975 0.01 6(7) 2.9345 -0.6330 -1.7385 -1.8711 -1.0615 0.08271008 0.03 6(7) 3.4141 -0.3127 -0.8655 -0.8982 -0.4697
 $a$ $E$ $n$ $\text{R}^*$ $A_{1}$ $B_{1}$ $C_{1}$ $D_{1}$ 0.11934208 0.01 4(5) 2.9824 -0.5545 -1.6917 -1.8423 -1.1788 0.11935935 0.03 4(5) 3.4699 -0.2779 -0.8562 -0.8955 -0.5052 0.09768743 0.01 5(6) 2.9518 -0.6002 -1.7218 -1.8647 -1.1083 0.09771521 0.03 5(6) 3.4344 -0.2983 -0.8626 -0.8986 -0.4841 0.08267975 0.01 6(7) 2.9345 -0.6330 -1.7385 -1.8711 -1.0615 0.08271008 0.03 6(7) 3.4141 -0.3127 -0.8655 -0.8982 -0.4697
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