TY - JOUR
T1 - A prediction for the optimal stratification for turbulent mixing
AU - Tang, Wenbo
AU - Caulfield, C. P.
AU - Kerswell, R. R.
N1 - Funding Information:
C. P. Caulfield and W. Tang would like to thank the National Science Foundation for support under the Collaborations in Mathematical Geosciences (CMG) initiative (ATM-0222104). C. P. Caulfield would also like to acknowledge the support of the EU Marie Curie International Reintegration Grant MIRG-CT-2005-016563. We also gratefully acknowledge the hospitality of the 2008 GFD Program at Woods Hole Oceanographic Institution and the ‘Nature of High Reynolds Turbulence’ programme at the Isaac Newton Institute for Mathematical Sciences.
PY - 2009/9
Y1 - 2009/9
N2 - By identifying the stratification which leads to maximal buoyancy flux in a stably-stratified plane Couette flow, we make a prediction of what bulk stratification (as a function of the shear) is optimal for turbulent mixing. A previous attempt to do this (Caulfield, Tang & Plasting, J. Fluid Mech., vol. 498, 2004, p. 315) failed due to an unexpected degeneracy in the variational problem. Here, we overcome this issue by parameterizing the variational problem implicitly with the overall mixing efficiency which is then optimized across to return a rigorous upper bound on the buoyancy flux. We find that the bulk Richardson number quickly approaches 1/6 in the asymptotic limit of high shear with the associated mixing efficiency tending to 1/3. The predicted mean profiles associated with the bound appear to have a layered structure, with the gradient Richardson number being low both in the interior, and in boundary layers near the walls, with a global maximum, also equal to 1/6, occurring at the edge of the boundary layers.
AB - By identifying the stratification which leads to maximal buoyancy flux in a stably-stratified plane Couette flow, we make a prediction of what bulk stratification (as a function of the shear) is optimal for turbulent mixing. A previous attempt to do this (Caulfield, Tang & Plasting, J. Fluid Mech., vol. 498, 2004, p. 315) failed due to an unexpected degeneracy in the variational problem. Here, we overcome this issue by parameterizing the variational problem implicitly with the overall mixing efficiency which is then optimized across to return a rigorous upper bound on the buoyancy flux. We find that the bulk Richardson number quickly approaches 1/6 in the asymptotic limit of high shear with the associated mixing efficiency tending to 1/3. The predicted mean profiles associated with the bound appear to have a layered structure, with the gradient Richardson number being low both in the interior, and in boundary layers near the walls, with a global maximum, also equal to 1/6, occurring at the edge of the boundary layers.
UR - http://www.scopus.com/inward/record.url?scp=76249127304&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=76249127304&partnerID=8YFLogxK
U2 - 10.1017/S0022112009990711
DO - 10.1017/S0022112009990711
M3 - Article
AN - SCOPUS:76249127304
SN - 0022-1120
VL - 634
SP - 487
EP - 497
JO - journal of fluid mechanics
JF - journal of fluid mechanics
ER -