Self-similarity of strongly stratified inviscid flows

It is well-known that strongly stratified flows are organized into a layered pancake structure in which motions are mostly horizontal but highly variable in the vertical direction. However, what determines the vertical scale of the motion remains an open question. In this paper, we propose a scaling law for this vertical scale Lv when no vertical lengthscales are imposed by initial or boundary conditions and when the fluid is strongly stratified, i.e., when the horizontal Froude number is small: Fh=U/NLh≪1, where U is the magnitude of the horizontal velocity, N the Brunt–Vaisala frequency and Lh the horizontal lengthscale. Specifically, we show that the vertical scale of the motion is Lv=U/N by demonstrating that the inviscid governing equations in the limit Fh→0, without any a priori assumption on the magnitude of Lv, are self-similar with respect to the variable zN/U, where z is the vertical coordinate. This self-similarity fully accounts for the layer characteristics observed in recent studies reportin...

[1]  J. Weinstock Saturated and Unsaturated Spectra of Gravity Waves and Scale-Dependent Diffusion , 1990 .

[2]  J. Chomaz,et al.  Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid , 2000, Journal of Fluid Mechanics.

[3]  P. Orlandi,et al.  Layer formation and transitions in stratified circular Couette flow , 1996 .

[4]  R. Kraichnan Inertial Ranges in Two‐Dimensional Turbulence , 1967 .

[5]  J. Chomaz,et al.  Theoretical analysis of the zigzag instability of a vertical columnar vortex pair in a strongly stratified fluid , 1998, Journal of Fluid Mechanics.

[6]  E. J. Hopfinger,et al.  Turbulence in stratified fluids: A review , 1987 .

[7]  Douglas K. Lilly,et al.  Stratified Turbulence and the Mesoscale Variability of the Atmosphere , 1983 .

[8]  E. Dewan,et al.  Saturation and the “universal” spectrum for vertical profiles of horizontal scalar winds in the atmosphere , 1986 .

[9]  C. Sidi,et al.  Experimental probability density functions of small-scale fluctuations in the stably stratified atmosphere , 2000, Journal of Fluid Mechanics.

[10]  F. Browand,et al.  The behavior of a turbulent front in a stratified fluid: Experiments with an oscillating grid , 1987 .

[11]  Susumu Kato,et al.  Observational evidence of a saturated gravity wave spectrum in the troposphere and lower stratosphere , 1988 .

[12]  M. Gregg,et al.  Diapycnal mixing in the thermocline: A review , 1987 .

[13]  C. W. Atta,et al.  Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid , 1983, Journal of Fluid Mechanics.

[14]  J. Chomaz,et al.  Three-dimensional stability of a vertical columnar vortex pair in a stratified fluid , 2000, Journal of Fluid Mechanics.

[15]  G. Spedding,et al.  Energy dissipation and vortex structure in freely decaying, stratified grid turbulence , 1996 .

[16]  J. W. Kaufman,et al.  Spectral Analysis of Detailed Vertical Wind Speed Profiles , 1969 .

[17]  E. Hopfinger,et al.  Stability and bifurcations in stratified Taylor–Couette flow , 2000, Journal of Fluid Mechanics.

[18]  Chester S. Gardner,et al.  Seasonal variability of gravity wave activity and spectra in the mesopause region at Urbana , 1991 .

[19]  Yoshifumi Kimura,et al.  Diffusion in stably stratified turbulence , 1996, Journal of Fluid Mechanics.

[20]  Ann E. Gargett,et al.  The scaling of turbulence in the presence of stable stratification , 1988 .

[21]  J. Herring,et al.  Numerical experiments in forced stably stratified turbulence , 1989, Journal of Fluid Mechanics.

[22]  G. Spedding The evolution of initially turbulent bluff-body wakes at high internal Froude number , 1997, Journal of Fluid Mechanics.

[23]  J. Chomaz,et al.  Vertical diffusion of the far wake of a sphere moving in a stratified fluid , 1993 .

[24]  K. N. Helland,et al.  The evolution of grid-generated turbulence in a stably stratified fluid , 1986, Journal of Fluid Mechanics.

[25]  J. Herring,et al.  Numerical simulations of freely evolving turbulence in stably stratified fluids , 1989, Journal of Fluid Mechanics.

[26]  Steve Smith,et al.  Evidence for a Saturated Spectrum of Atmospheric Gravity Waves. , 1987 .

[27]  H. Widdel,et al.  Further study of a saturated gravity wave spectrum in the mesosphere , 1991 .

[28]  Owen S. Lee,et al.  INTERNAL WAVES IN THE OCEAN , 1962 .

[29]  J. Riley,et al.  Fluid Motions in the Presence of Strong Stable Stratification , 2000 .

[30]  A. Gargett,et al.  A Composite Spectrum of Vertical Shear in the Upper Ocean , 1981 .

[31]  Jung-Tai Lin,et al.  Wakes in Stratified Fluids , 1979 .

[32]  J. Lumley,et al.  A First Course in Turbulence , 1972 .

[33]  C. Hines,et al.  The Saturation of Gravity Waves in the Middle Atmosphere. Part II: Development Of Doppler-Spread Theory , 1991 .

[34]  F. Browand,et al.  Turbulence, similarity scaling and vortex geometry in the wake of a towed sphere in a stably stratified fluid , 1996, Journal of Fluid Mechanics.

[35]  B. Boubnov Stratified circular Couette flow: instability and flow regimes , 1996 .

[36]  Kelvin K. Droegemeier,et al.  Stratified Turbulence in the Atmospheric Mesoscales , 1998 .