Stability of two-dimensional Taylor–Green vortices in rotating stratified fluids

Abstract

The linear stability of the two-dimensional Taylor–Green vortices, which is a spatially periodic array of vortices, in rotating stratified fluids is investigated by local and modal stability analysis. Five types of instability appear in general: the pure hyperbolic instability, the strato-hyperbolic instability, the rotational-hyperbolic instability, the centrifugal instability and the elliptic instability. The condition for each instability and the estimate of the growth rate, which are useful in interpreting numerical results, are obtained in the framework of local stability analysis. Realizability of an instability is introduced to predict whether an unstable mode corresponding to an unstable region found in the local stability analysis exists at finite Reynolds numbers. In the absence of stratification, the pure hyperbolic instability is dominant for weak rotation; it is stabilized for strong rotation. For strong anti-cyclonic rotation, the elliptic instability or the centrifugal instability becomes dominant depending on the parameter values; further stronger rotation stabilizes both instabilities. For strong cyclonic rotation, the rotational-hyperbolic instability or the elliptic instability becomes dominant, although the growth rate is smaller than the anti-cyclonic cases. Strong stratification changes the stability properties. The strato-hyperbolic instability occurs for weak rotation. The rotational-hyperbolic instability and the elliptic instability are weakened under cyclonic rotation, while the latter survives and extends the unstable range under anti-cyclonic rotation. The pure hyperbolic instability and the centrifugal instability are less affected by stratification. The mode structures of each instability are in good agreement with the corresponding solution to local stability equations, confirming the physical mechanism of the instability.

Stability of Stuart vortices in rotating stratified fluids

“…One of the important characteristics of the arrays of vortices is that there exist hyperbolic points, which add the hyperbolic instability to the above list of instabilities. The hyperbolic instability can be further classified into the pure-hyperbolic instability (Friedlander & Vishik 1991;Lifschitz & Hameiri 1991;Sipp & Jacquin 1998;Pralits, Giannetti & Brandt 2013), the strato-hyperbolic instability (Suzuki, Hirota & Hattori 2018;Hattori et al 2021) and the rotational-hyperbolic instability (Sipp, Lauga & Jacquin 1999;Godeferd, Cambon & Leblanc 2001;Hattori & Hirota 2023).…”

“…In our previous work (Hattori et al 2021), the linear stability of a periodic array of vortices in non-rotating stratified fluids has been investigated in detail; the effects of rotation were studied in Hattori & Hirota (2023), while the base flow was fixed to the 2-D Taylor-Green vortices. The latter work revealed several important aspects of the stability of a periodic array of vortices in rotating stratified fluids.…”

“…We clarify how the growth rate and other characteristics of the instability depend on rotation, stratification and vorticity distribution. There are important reasons for studying the stability of the Stuart vortices after our detailed stability analysis of the 2-D Taylor-Green vortices (Hattori & Hirota 2023). First, the Stuart vortices are often regarded as a model of vortices that develop in a mixing layer as a result of the Kelvin-Helmholtz instability; understanding the stability of the Stuart vortices should be important because mixing layers are observed frequently in the atmosphere and the oceans.…”

“…Next, it is important to investigate how the opposite-signed vortices, which exist in the 2-D Taylor-Green vortices but do not in the Stuart vortices, affect the stability properties. Finally, the instability condition and the estimate of the growth rate derived for each instability in Hattori & Hirota (2023) has been shown to be effective only for the 2-D Taylor-Green vortices; they should be tested for a different base flow that has different magnitudes of strain rates and vorticity, to show their applicability to general base flows. Although there are several works on the stability of the Stuart vortices with or without rotation (Pierrehumbert & Widnall 1982;Leblanc & Cambon 1998;Potylitsin & Peltier 1999;Godeferd et al 2001), there is no work on the effects of stratification on the stability of the Stuart vortices.…”

“…This paper is organized as follows. In § 2 the problem is formulated; concise expressions for the instability conditions and the estimates of the growth rates, which have been introduced in Hattori & Hirota (2023) in the framework of local stability analysis, are also summarized briefly. The methods of the numerical stability analysis are explained in § 3.…”