Layering, Instabilities, and Mixing in Turbulent Stratified Flows

Abstract

Understanding how turbulence leads to the enhanced irreversible transport of heat and other scalars such as salt and pollutants in density-stratified fluids is a fundamental and central problem in geophysical and environmental fluid dynamics. This review discusses recent research activity directed at improving community understanding, modeling, and parameterization of the subtle interplay between energy conversion pathways, instabilities, turbulence, external forcing, and irreversible mixing in density-stratified fluids. The conceptual significance of various length scales is highlighted, and in particular, the importance is stressed of overturning or scouring in the formation and maintenance of layered stratifications, i.e., robust density distributions with relatively deep and well-mixed regions separated by relatively thin interfaces of substantially enhanced density gradient. Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 53 is January 7, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Instability of vertically stratified horizontal plane Poiseuille flow

“…Of interest for further studies is the interplay between wave-related instabilities, turbulence and mixing (Caulfield 2021). As described in our study, the unstable flow saturates at a finite-amplitude meandering layered pattern.…”

Non-Modal Three-Dimensional Optimal Perturbation Growth in Thermally Stratified Mixing Layers

“…where * denotes the complex conjugate. In the following, the inner product of two complex vectors and associates with the energy norm (7), (8) and is given by…”

“…The phenomenon appears to be so complicated and to pose so many questions that in spite of hundreds of studies published, it remains the subject of many current researches [3][4][5][6]. For the details, the reader is referred to review papers [7][8][9].…”

“…energy norm(7),(8) and is given by〈 , 〉 = 〈 , M 〉 = ∫ [ * , * , * ] M [ that ( , , ) = ‖ ‖ 2 is the energy amplification at time = , or growth func-tion, is defined as ([38-40]) for ( ) in (10), with assuming (0) = 1, and recalling that ( ) = (ℒ ) (0) = ( ) (0), yields ( , , ) = max (0)≠0 〈 ( ) (0), ( ) (0)〉 = max (0)≠0 〈 (0), * ( ) ( ) (0)〉 = ‖ * ( ) ( )‖ (11) where * ( ) is the adjoint of the operator ( ) (see Section 2.2). The operator * ( ) ( ) is self-adjoint and normal, its eigenvalues are real and non-negative, and ‖ * ( ) ( )‖ = max = is the largest eigenvalue that corresponds to the maximum possible relative growth attainable at the time τ.…”

“…. The energy norm(7) of an arbitrary pe can be connected with the Euclidean norm of the vector ( ) defined by decomposition of the matrix : matrix, p is the eigenvector, and = 〈 , 〉 is th which is the complex-valued Hermitian matrix. Then, using factorization definite Gram matrix = , the maximum energy growth at time ca as the value of the 2-norm of the matrix ( ) −1 and is equal to t maximal singular value.…”