Stratified Turbulence

Stratified Flow

When you hear on the weather forecast that there is a thermal inversion it means that light (warm) air is sitting above heavier (cold) air.  The air is stratified.  Gravity impedes air moving vertically and traps pollution near the ground.  The stratosphere gets its name because the air in that region is stratified all the time.  This is why airplane contrails persist for so long.  Large parts of the ocean are stratified as is the lowest region of the atmosphere over land at night. 

Stratified Turbulence

The renowned research meteorologist D. K. Lilly coined the term Stratified Turbulence for fully three-dimensional turbulence in which turbulent motion in the vertical is constrained by gravity.  A question central to research into this flow regime is how much of classical turbulence theory applies to stratified turbulence.   In predicting weather, for instance, we'd like to be able to use existing turbulence models, but they are not always applicable.

Click on the thumbnails below to see the effect that buoyancy has on turbulence. The stratified case has regions of intense turbulence and regions of almost no turbulence that are particularly visible in the plots of dissipation rate. The images were generated by the direct numerical simulations reported in hebert06b.pdf.   See Image Gallery for additional images.

Vertical velocity in unstratified turbulence
Vertical velocity in unstratified turbulence

Vertical velocity in stratified turbulence
Vertical velocity in stratified turbulence

Log of the dissipation rate in unstratified turbulence
Log of the dissipation rate in unstratified turbulence

Log of the dissipation rate in stratified turbulence
Log of the dissipation rate in stratified turbulence

Importance of High Buoyancy Reynolds Number

Turbulence is inherently 3D, but gravity impedes vertical motion of turbulent eddies larger than a given size called the Ozmidov length.  At very small lengths (smaller than the Kolmogorov length), turbulent motion is suppressed by viscosity.  So in important ratio for stratified turbulence is the Ozmidov length / Kolmogorov length, which is related to the buoyancy Reynolds number, Rb.  Rb must be above some value, typically taken to be 25 - 30, for stratified turbulence to be in interesting.  If the flow is strongly stratified as it is in many geophysical flows, at least 500 billion grid points are needed for a 3D simulation at high enough Rb.

Kelvin-Helmholtz Instability

Several mechanisms cause turbulence in stratified flows.  One is the Kelvin-Helmholtz instability visible in movies below from the direct numerical simulations reported in debk03a.pdf.


Vertical velocity on a horizontal plane. Red indicates upward flow and blue downward flow.

Density field on a vertical plane. The magenta bar corresponds to black bar in the lefthand figure. Kelvin-Helmholtz rollups can be seen triggering the turbulence.



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