Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 566
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Alford, Matthew H. MacKinnon, Jennifer A. Simmons, Harper L. and Nash, Jonathan D. 2016. Near-Inertial Internal Gravity Waves in the Ocean. Annual Review of Marine Science, Vol. 8, Issue. 1, p. 95.

    Benage, M. C. Dufek, J. and Mothes, P. A. 2016. Quantifying entrainment in pyroclastic density currents from the Tungurahua eruption, Ecuador: Integrating field proxies with numerical simulations. Geophysical Research Letters, Vol. 43, Issue. 13, p. 6932.

    Brumer, Sophia E. Zappa, Christopher J. Anderson, Steven P. and Dugan, John P. 2016. Riverine skin temperature response to subsurface processes in low wind speeds. Journal of Geophysical Research: Oceans, Vol. 121, Issue. 3, p. 1721.

    Buban, Michael S. and Ziegler, Conrad L. 2016. The Formation of Small-Scale Atmospheric Vortices via Horizontal Shearing Instability. Journal of the Atmospheric Sciences, Vol. 73, Issue. 5, p. 2061.

    Cyr, Frédéric and van Haren, Hans 2016. Observations of Small-Scale Secondary Instabilities during the Shoaling of Internal Bores on a Deep-Ocean Slope. Journal of Physical Oceanography, Vol. 46, Issue. 1, p. 219.

    Donda, J. M. M. van Hooijdonk, I. G. S. Moene, A. F. van Heijst, G. J. F. Clercx, H. J. H. and van de Wiel, B. J. H. 2016. The maximum sustainable heat flux in stably stratified channel flows. Quarterly Journal of the Royal Meteorological Society, Vol. 142, Issue. 695, p. 781.

    Hirota, Makoto and Morrison, Philip J. 2016. Stability boundaries and sufficient stability conditions for stably stratified, monotonic shear flows. Physics Letters A, Vol. 380, Issue. 21, p. 1856.

    Hoecker-Martínez, Martín S. Smyth, William D. and Skyllingstad, Eric D. 2016. Oceanic Turbulent Energy Budget using Large-Eddy Simulation of a Wind Event during DYNAMO. Journal of Physical Oceanography, Vol. 46, Issue. 3, p. 827.

    Jurisa, Joseph T. Nash, Jonathan D. Moum, James N. and Kilcher, Levi F. 2016. Controls on Turbulent Mixing in a Strongly Stratified and Sheared Tidal River Plume. Journal of Physical Oceanography, Vol. 46, Issue. 8, p. 2373.

    Knox, John A. Black, Alan W. Rackley, Jared A. Wilson, Emily N. Grant, Jeremiah S. Phelps, Stephanie P. Nevius, David S. and Dunn, Corey B. 2016. Aviation Turbulence.

    Kundu, Pijush K. Cohen, Ira M. and Dowling, David R. 2016. Fluid Mechanics.

    Lin, Yuh-Lang 2016. Aviation Turbulence.

    Liu, Zhiyu 2016. On instability and mixing on the UK Continental Shelf. Journal of Marine Systems, Vol. 158, p. 72.

    Liu, Chuanyu Köhl, Armin Liu, Zhiyu Wang, Fan and Stammer, Detlef 2016. Deep-reaching thermocline mixing in the equatorial pacific cold tongue. Nature Communications, Vol. 7, p. 11576.

    Mazzini, Piero L. F. and Chant, Robert J. 2016. Two-dimensional circulation and mixing in the far field of a surface-advected river plume. Journal of Geophysical Research: Oceans, Vol. 121, Issue. 6, p. 3757.

    Prat, V. Guilet, J. Viallet, M. and Müller, E. 2016. Shear mixing in stellar radiative zones. Astronomy & Astrophysics, Vol. 592, p. A59.

    Rahn, David A. Parish, Thomas R. and Leon, David 2016. Observations of Large Wind Shear above the Marine Boundary Layer near Point Buchon, California. Journal of the Atmospheric Sciences, Vol. 73, Issue. 8, p. 3059.

    Barkan, Roy Winters, Kraig B. and Llewellyn Smith, Stefan G. 2015. Energy Cascades and Loss of Balance in a Reentrant Channel Forced by Wind Stress and Buoyancy Fluxes. Journal of Physical Oceanography, Vol. 45, Issue. 1, p. 272.

    Chowdhury, Mijanur R. Wells, Mathew G. and Cossu, Remo 2015. Observations and environmental implications of variability in the vertical turbulent mixing in Lake Simcoe. Journal of Great Lakes Research, Vol. 41, Issue. 4, p. 995.

    Cyr, Frédéric Bourgault, Daniel and Galbraith, Peter S. 2015. Behavior and mixing of a cold intermediate layer near a sloping boundary. Ocean Dynamics, Vol. 65, Issue. 3, p. 357.


On the stability of heterogeneous shear flows

  • John W. Miles (a1)
  • DOI:
  • Published online: 01 March 2006

Small perturbations of a parallel shear flow U(y) in an inviscid, incompressible fluid of variable density ρ0(y) are considered. It is deduced that dynamic instability of statically stable flows ($\rho ^{\prime}_0 (y)\; \textless \; 0$) cannot be other than exponential, in consequence of which it suffices to consider spatially periodic, travelling waves. The general solution of the resulting differential equation is considered in some detail, with special emphasis on the Reynolds stress that transfers energy from the mean flow to the travelling wave. It is proved (as originally conjectured by G. I. Taylor) that sufficient conditions for stability are $U^{\prime}(y) \not= 0$ and $J(y)\; \textgreater \frac {1} {4}$ throughout the flow, where $J(y) = -g \rho^{\prime}_0(y)|\rho (y)U^{\prime 2}(y)$ is the local Richardson number. It also is pointed out that the kinetic energy of a normal mode in an ideal fluid may be infinite if $0 \; \textless \; J(y_c) \; \textless \; \frac {1}{4}$, where $U(y_c)$ is the wave speed.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *