Abstract
Asymptotic techniques are used to model the quasi-steady state vortices that have been observed in two-dimensional simulations of vortex roll-up in stratified shear layers. A time-independent nonlinear critical layer analysis is used to find a family of steady-state finite amplitude vortices in the Holmboe model of an inviscid stratified shear layer, with the vorticity inside closed streamlines based on the Stuart vortex. The vortices are compared to results of simulations and also an alternative model where the vorticity was constant inside closed streamlines.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. P. Klaassen, and W. R. Peltier, The effect of Prandtl number on the evolution and stability of Kelvin-Helmholtz billows. Geophys. Astrophys. Fluid Dyn. 32 (1985) 23–60.
G. P. Klaassen, and W. R. Peltier, The influence of stratification on secondary instability in free shear layers. J. Fluid Mech. 227 (1991) 71–106.
P. C. Patnaik, F. S. Sherman and G. M. Corcos, A numerical study of Kelvin-Helmholtz waves of finite amplitude. J. Fluid Mech. 73 (1976) 215–240.
J. Sommeria, C. Staquet and R. Robert, Final equilibrium state of a two-dimensional shear layer. J. Fluid Mech. 233 (1991) 661–689.
C. Staquet, Two-dimensional secondary instabilities in a strongly stratified shear layer. J. Fluid Mech. 296 (1995) 71–106.
L.-P. Wang and M. R. Maxey, Kinematical descriptions for mixing in stratified or homogeneous shear flows. In: J. M. Redondo and O. Métais (eds.) Mixing in Geophysical Flows Barcelona: CIMNE (1995) pp. 14–34.
H. Tanaka, Quasi-linear and non-linear interactions of finite amplitude perturbations in a stably stratified fluid with hyperbolic shear. J. Meteorol. Soc. Japan 53 (1975) 1–31.
J. T. Stuart, On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29 (1967) 417–440.
D. J. Benney and R. T. Bergeron, A new class of nonlinear waves in parallel flows. Stud. Appl. Math. 48 (1969) 181–204.
R. E. Davis, On the high Reynolds number flow over a wavy boundary. J. Fluid Mech. 36 (1969) 337–346.
G. K. Batchelor, On steady laminar flow with closed streamlines at large Reynolds numbers. J. Fluid Mech. 1 (1956) 177–190.
R. E. Kelly and S. A. Maslowe, The nonlinear critical layer in a slightly stratified shear layer. Stud. Appl. Math. 49 (1970) 301–332.
S. A. Maslowe, The generation of clear air turbulence by nonlinear waves. Stud. Appl. Math. 51 (1972) 1–16.
S. A. Maslowe, Finite-amplitude Kelvin-Helmholtz billows. Boundary-Layer Meteorology 5 (1973) 43–52.
S. N Brown and K. Stewartson, The evolution of the critical layer of a Rossby wave. Part II. Geophys. Astrophys. Fluid Dyn. 10 (1978) 1–18.
R. Haberman, Critical layers in parallel flows. Stud. Appl. Math. 51 (1972) 139–161.
R. Haberman, Wave-induced distortions of a slightly stratified shear flow: a nonlinear critical layer effect. J. Fluid Mech. 58 (1973) 727–735.
K. Stewartson, Theory of nonlinear stability. Fluid Dyn. Trans. (Poland) 7 (1974) 101–120.
K. Stewartson, The evolution of the critical layer of a Rossby wave. Geophys. Astrophys. Fluid Dyn. 9 (1978) 185–200.
L. M. Mack, A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73 (1976) 497–520.
J. W. Murdock and K. Stewartson, Continuous spectra of the Orr-Sommerfeld equation. Phys. Fluids 20 (1978) 1404–1411.
P. B. Rhines and W. R. Young, How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133 (1983) 133–145.
J. Holmboe, Unpublished lecture notes. University of California at Los Angeles, Los Angeles CA USA (1960).
J. W. Miles, On the stability of heterogeneous shear flows. Part 2. J. Fluid Mech. 16 (1963) 209–227.
J. W. Miles, On the stability of heterogeneous shear flows. J. Fluid Mech. 10 (1961) 495–508.
G. I. Taylor, Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. London A132 (1931) 499–523.
R. Mallier, On disturbances to a stratified mixing layer. Stud. Appl. Math. 107 (2001) 185–206.
I. G. Shukhman and S. M. Churilov, Effect of slight stratification on the nonlinear spatial evolution of a weakly unstable wave in a free shear layer. J. Fluid Mech. 343 (1997) 197–233.
A. Barcilon and P. G. Drazin, Nonlinear waves of vorticity. Stud. Appl. Math. 106 (2001) 437–474.
R. Mallier, Stuart vortices in a stratified shear layer.I: the Garcia model. Geophys. Astrophys. Fluid Dyn. 74 (1994) 73–97.
R. Mallier, Stuart vortices on a beta-plane. Dyn. Atmosph. Oceans 22 (1995) 213–238.
T. B. Gatski, Vortex motion in a wall-bounded viscous flow. Proc. R. Soc. London A397 (1985) 397–414.
L. N. Howard, Note on a paper of John W. Miles. J. Fluid Mech. 10 (1961) 509–512.
L. N. Howard and S. A Maslowe, Stability of stratified shear flows. Boundary-Layer Meteorology 4 (1973) 511–523.
R. Mallier, Weakly Nonlinear Waves in Mixing Layers. Ph.D. Thesis. Brown University, Providence RI USA (1991) 396 pp.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mallier, R. Stuart vortices in a stratified mixing layer: the Holmboe model. Journal of Engineering Mathematics 47, 121–136 (2003). https://doi.org/10.1023/A:1025890823745
Issue Date:
DOI: https://doi.org/10.1023/A:1025890823745