# American Institute of Mathematical Sciences

March  2009, 2(1): 215-229. doi: 10.3934/krm.2009.2.215

## Three-dimensional instabilities in non-parallel shear stratified flows

 1 Department of Mathematics and Statistics, Center for Environmental Fluid Dynamics, Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-1804, United States, United States 2 N/A, United States

Received  November 2008 Revised  November 2008 Published  January 2009

The instabilities of non-parallel flows ($\overline{U}(x_3)$, $\overline{V}(x_3), 0)$ ($\overline{V} \ne 0$) such as those induced by polarized inertia-gravity waves embedded in a stably stratified environment are analyzed in the context of the 3D Euler-Boussinesq equations. We derive a sufficient condition for shear stability and a necessary condition for instability in the case of non-parallel velocity fields. Three dimensional numerical simulations of the full nonlinear equations are conducted to characterize the respective modes of instability, their topology and dynamics, and subsequent breakdown into turbulence. We describe fully three-dimensional instability mechanisms, and study spectral properties of the most unstable modes. Our stability/instability criteria generalizes that in the case of parallel shear flows ($\bar{V}=0$), where stability properties are governed by the Taylor-Goldstein equations previously studied in the literature. Unlike the case of parallel flows, the polarized horizontal velocity vector rotating with respect to the vertical coordinate ($x_3$) excites unstable modes that have different spectral properties depending on the orientation of the velocity vector. At each vertical level, the horizontal wave vector of the fastest growing mode is parallel to the local vector ($d\overline{U}(x_3)/dx_3$, $d \overline{V}(x_3)/dx_3)$. We investigate three-dimensional characteristics of the unstable modes and present computational results on Lagrangian particle dynamics.
Citation: Alex Mahalov, Mohamed Moustaoui, Basil Nicolaenko. Three-dimensional instabilities in non-parallel shear stratified flows. Kinetic and Related Models, 2009, 2 (1) : 215-229. doi: 10.3934/krm.2009.2.215
 [1] Anna Geyer, Ronald Quirchmayr. Weakly nonlinear waves in stratified shear flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2309-2325. doi: 10.3934/cpaa.2022061 [2] Olof Heden, Faina I. Solov’eva. Partitions of $\mathbb F$n into non-parallel Hamming codes. Advances in Mathematics of Communications, 2009, 3 (4) : 385-397. doi: 10.3934/amc.2009.3.385 [3] Mário Bessa, Jorge Rocha. Three-dimensional conservative star flows are Anosov. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 839-846. doi: 10.3934/dcds.2010.26.839 [4] Mats Gyllenberg, Ping Yan. On the number of limit cycles for three dimensional Lotka-Volterra systems. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 347-352. doi: 10.3934/dcdsb.2009.11.347 [5] Eric S. Wright. Macrotransport in nonlinear, reactive, shear flows. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2125-2146. doi: 10.3934/cpaa.2012.11.2125 [6] Christopher Logan Hambric, Chi-Kwong Li, Diane Christine Pelejo, Junping Shi. Minimum number of non-zero-entries in a stable matrix exhibiting Turing instability. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021128 [7] Hong Cai, Zhong Tan. Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1847-1868. doi: 10.3934/dcds.2016.36.1847 [8] Xiaoli Li, Boling Guo. Well-posedness for the three-dimensional compressible liquid crystal flows. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1913-1937. doi: 10.3934/dcdss.2016078 [9] Li Liu. Unique subsonic compressible potential flows in three -dimensional ducts. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 357-368. doi: 10.3934/dcds.2010.27.357 [10] Calin I. Martin. On three-dimensional free surface water flows with constant vorticity. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2415-2431. doi: 10.3934/cpaa.2022053 [11] Biswajit Basu. On the nonlinear three-dimensional models in equatorial ocean flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2271-2290. doi: 10.3934/cpaa.2022085 [12] Xu Zhang. Sinai-Ruelle-Bowen measures for piecewise hyperbolic maps with two directions of instability in three-dimensional spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2873-2886. doi: 10.3934/dcds.2016.36.2873 [13] V. Torri. Numerical and dynamical analysis of undulation instability under shear stress. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 423-460. doi: 10.3934/dcdsb.2005.5.423 [14] Anna Geyer, Ronald Quirchmayr. Shallow water models for stratified equatorial flows. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4533-4545. doi: 10.3934/dcds.2019186 [15] Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031 [16] Weiping Yan. Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1359-1385. doi: 10.3934/dcds.2015.35.1359 [17] Biswajit Basu. On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4783-4796. doi: 10.3934/dcds.2019195 [18] Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations and Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007 [19] Myoungjean Bae, Hyangdong Park. Three-dimensional supersonic flows of Euler-Poisson system for potential flow. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2421-2440. doi: 10.3934/cpaa.2021079 [20] Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237

2020 Impact Factor: 1.432