Advances in Rapid Distortion Theory: From Rotating Shear Flows to the Baroclinic Instability

2005 ◽  
Vol 73 (3) ◽  
pp. 449-460 ◽  
Author(s):  
Aziz Salhi ◽  
Claude Cambon
Keyword(s):  
Stability Analysis ◽  
Reynolds Stress ◽  
Shear Flows ◽  
Baroclinic Instability ◽  
Vertical Direction ◽  
Vertical Shear ◽  
Stratified Medium ◽  
Turbulence Structure ◽  
Rapid Distortion Theory ◽  
System Rotation

The essentials of rapid distortion theory (RDT) are briefly recalled for homogeneous turbulence subjected to rotational mean flows, including its linkage to stability analysis. The latter “linkage” is of particular importance from our viewpoint, since it also attracted the attention of Charles Speziale, resulting in at least two papers [Speziale, C. G., Abid, R., and Blaisdell, G. A., 1996, “On the Consistency of Reynolds Stress Turbulence Closures With Hydrodynamic Stability Theory,” Phys. Fluids, 8, pp. 781–788 and Salhi, A., Cambon, C., and Speziale, C. G., 1997, “Linear Stability Analysis of Plane Quadratic Flows in a Rotating Frame,” Phys. Fluids, 9(8), pp. 2300–2309] with particular emphasis on rotating flows. New analytical solutions and related RDT results are presented for shear flows including buoyancy forces, with system rotation or mean density stratification. Finally, combining shear, rotation and stratification, RDT is shown to be pertinent to revisiting the baroclinic instability. This instability results from the tilting of mean isopycnal surfaces under combined effects of vertical shear and system rotation, in a vertically (stably) stratified medium rotating around the vertical direction. In addition, the challenge of reproducing RDT dynamics in single-point closure models is briefly discussed, from the viewpoint of structure-based modeling [Cambon C., Jacquin, L., and Lubrano, J.-L., 1992, “Towards a New Reynolds Stress Model for Rotating Turbulent Flows,” Phys. Fluids A, 4, pp. 812–824 and Kassinos, S. C., Reynolds, W. C., and Rogers, M. M., 2000, “One-Point Turbulence Structure Tensors,” J. Fluid Mech., 428, pp. 213–248.

Advances in Rapid Distortion Theory: From Rotating Shear Flows to the Baroclinic Instability

2003 ◽  
Author(s):  
Claude Cambon ◽  
Aziz Salhi
Keyword(s):  
Shear Flows ◽  
Single Point ◽  
Baroclinic Instability ◽  
Vertical Direction ◽  
Rotating Flows ◽  
Vertical Shear ◽  
Stratified Medium ◽  
Rapid Distortion Theory ◽  
System Rotation ◽  
Rotating Shear Flows

The essentials of Rapid Distortion Theory (RDT) are recalled for homogeneous turbulence subjected to rotational mean flows, including its linkage to stability analysis. The latter ‘linkage’ is of particular importance from our viewpoint, since it also attracted the attention of Charles Speziale, resulting in at least two papers (Speziale et al., 1996; Salhi et al., 1997) with particular emphasis on rotating flows. New analytical solutions and related RDT results are presented for shear flows possibly including buoyancy forces, with system rotation or mean density stratification. Finally, combining shear, rotation and stratification, RDT is shown to be pertinent to revisiting the baroclinic instability. This instability results from the tilting of mean isopycnal surfaces under combined effects of vertical shear and system rotation, in a vertically (stably) stratified medium rotating around the vertical direction. In conclusion, the challenge of reproducing RDT dynamics in single-point closure models is briefly discussed, from the viewpoint of structure-based modeling (Cambon et al., 1992; Kassinos et al., 2000).

scholarly journals Stability Analysis of the Labrador Current

2014 ◽  
Vol 44 (2) ◽  
pp. 445-463 ◽  
Author(s):  
Sören Thomsen ◽  
Carsten Eden ◽  
Lars Czeschel
Keyword(s):  
Stability Analysis ◽  
Growth Rates ◽  
Baroclinic Instability ◽  
Cross Flow ◽  
Vertical Shear ◽  
Maximal Growth ◽  
Phase Velocities ◽  
Labrador Current ◽  
Lateral Mixing ◽  
Weak Stratification

Abstract Mooring observations and model simulations point to an instability of the Labrador Current (LC) during winter, with enhanced eddy kinetic energy (EKE) at periods between 2 and 5 days and much less EKE during other seasons. Linear stability analysis using vertical shear and stratification from the model reveals three dominant modes of instability in the LC: 1) a balanced interior mode with along-flow wavelengths of about 30–45 km, phase velocities of 0.3 m s−1, maximal growth rates of 1 day−1, and surface-intensified but deep-reaching amplitudes; 2) a balanced shallow mode with along-flow wavelengths of about 0.3–1.5 km, phase velocities of 0.55 m s−1, about 3 times larger growth rates, but amplitudes confined to the mixed layer (ML); and 3) an unbalanced symmetric mode with the largest growth rates, vanishing phase speeds, and along-flow structure, and very small cross-flow wavelengths, also confined to the ML. Both balanced modes are akin to baroclinic instability but operate at moderate-to-small Richardson numbers Ri with much larger growth rates as for the quasigeostrophic limit of Ri ≫ 1. The interior mode is found to be responsible for the instability of the LC during winter. Weak stratification and enhanced vertical shear due to local buoyancy loss and the advection of convective water masses from the interior result in small Ri within the LC and up to 3 times larger growth rates of the interior mode in March compared to summer and fall conditions. Both the shallow and the symmetric modes are not resolved by the model, but it is suggested that they might also play an important role for the instability in the LC and for lateral mixing.

Modelling of rapid pressure—strain in Reynolds-stress closures

1994 ◽  
Vol 269 ◽  
pp. 143-168 ◽  
Author(s):  
Arne V. Johansson ◽  
Magnus Hallbäck
Keyword(s):  
Strain Rate ◽  
Reynolds Stress ◽  
Sudden Change ◽  
Model Parameters ◽  
Rapid Distortion Theory ◽  
Flow Parameters ◽  
Turbulent Reynolds Number ◽  
Principal Axes ◽  
Anisotropy Tensor ◽  
Rapid Pressure

The most general form for the rapid pressure—strain rate, within the context of classical Reynolds-stress transport (RST) closures for homogeneous flows, is derived, and truncated forms are obtained with the aid of rapid distortion theory. By a classical RST-closure we here denote a model with transport equations for the Reynolds stress tensor and the total dissipation rate. It is demonstrated that all earlier models for the rapid pressure—strain rate within the class of classical Reynolds-stress closures can be formulated as subsets of the general form derived here. Direct numerical simulations were used to show that the dependence on flow parameters, such as the turbulent Reynolds number, is small, allowing rapid distortion theory to be used for the determination of model parameters. It was shown that such a nonlinear description, of fourth order in the Reynolds-stress anisotropy tensor, is quite sufficient to very accurately model the rapid pressure—strain in all cases of irrotational mean flows, but also to get reasonable predictions in, for example, a rapid homogeneous shear flow. Also, the response of a sudden change in the orientation of the principal axes of a plane strain is investigated for the present model and models proposed in the literature. Inherent restrictions on the predictive capability of Reynolds-stress closures for rotational effects are identified.

scholarly journals Steep Shelf Stabilization of the Coastal Bransfield Current: Linear Stability Analysis

2014 ◽  
Vol 44 (2) ◽  
pp. 714-732 ◽  
Author(s):  
F. J. Poulin ◽  
A. Stegner ◽  
M. Hernández-Arencibia ◽  
A. Marrero-Díaz ◽  
P. Sangrà
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Baroclinic Instability ◽  
Shear Instability ◽  
Coastal Current ◽  
Horizontal Shear ◽  
Shelf Slope ◽  
Topographic Parameter ◽  
Selection Of

Abstract In situ measurements obtained during the 2010 COUPLING cruise were analyzed in order to fully characterize the velocity structure of the coastal Bransfield Current. An idealized two-layer shallow-water model was used to investigate the various instability processes of the realistic current along the coastal shelf. Particularly studied is how the topographic parameter To (ratio between the shelf slope and the isopycnal slope of the surface current) impacts the growth and the wavelength of the unstable perturbations. For small bottom slopes, when the evolution of the coastal current is controlled by the baroclinic instability, the increase of the topographic parameter To yields a selection of smaller unstable wavelengths. The growth rates increase with small values of To. For larger values of To (To ≳ 10, which is relevant for the coastal Bransfield Current), the baroclinic instability is strongly dampened and the horizontal shear instability becomes the dominant one. In this steep shelf regime, the unstable growth rate and the wavelength selection of the baroclinic coastal current remains almost constant and weakly affected by the amplitude of the bottom velocity or the exact value of the shelf slope. Hence, the linear stability analysis of an idealized Bransfield Current predicts a typical growth time of 7.7 days and an alongshore scale of 47 km all along the South Shetland Island shelf. The fact that these large growth times are identical to the typical transit time of water parcels along the shelf may explain why the current does not exhibit any unstable meanders.

scholarly journals The linear instability of the stratified plane Couette flow

2018 ◽  
Vol 853 ◽  
pp. 205-234 ◽  
Author(s):  
Giulio Facchini ◽  
Benjamin Favier ◽  
Patrice Le Gal ◽  
Meng Wang ◽  
Michael Le Bars
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Couette Flow ◽  
Vertical Direction ◽  
Stability Diagram ◽  
Linear Instability ◽  
Plane Couette Flow ◽  
Horizontal Shear ◽  
The Stability

We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonal to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the streamwise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number $Re$ and the Froude number $Fr$, which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e. $Fr\sim 1$) and above a moderate value of the Reynolds number $Re\gtrsim 700$. The instability results from a wave resonance mechanism already known in the context of channel flows – for instance, unstratified plane Couette flow in the shallow-water approximation. The result is confirmed by fully nonlinear direct numerical simulations and, to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water, linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold values of $Fr$ and $Re$ indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although slightly modified due to streamwise confinement.

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