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Stability of the anabatic Prandtl slope flow in a stably stratified medium

Published online by Cambridge University Press:  18 December 2019

Cheng-Nian Xiao Open the ORCID record for Cheng-Nian Xiao [Opens in a new window]  and
Inanc Senocak Open the ORCID record for Inanc Senocak [Opens in a new window]
Cheng-Nian Xiao
Affiliation:
Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 3700 O’Hara St., Pittsburgh, PA15261, USA
Inanc Senocak*
Affiliation:
Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 3700 O’Hara St., Pittsburgh, PA15261, USA
*
†Email address for correspondence: senocak@pitt.edu
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Abstract

In the Prandtl model for anabatic slope flows, a uniform positive buoyancy flux at the surface drives an upslope flow against a stable background stratification. In the present study, we conduct linear stability analysis of the anabatic slope flow under this model and contrast it against the katabatic case as presented in Xiao & Senocak (J. Fluid Mech., vol. 865, 2019, R2). We show that the buoyancy component normal to the sloped surface is responsible for the emergence of stationary longitudinal rolls, whereas a generalised Kelvin–Helmholtz (KH) type of mechanism consisting of shear instability modulated by buoyancy results in a streamwise-travelling mode. In the anabatic case, for slope angles larger than $9^{\circ }$ to the horizontal, the travelling KH mode is dominant whereas, at lower inclination angles, the formation of the stationary vortex instability is favoured. The same dynamics holds qualitatively for the katabatic case, but the mode transition appears at slope angles of approximately $62^{\circ }$. For a fixed slope angle and Prandtl number, we demonstrate through asymptotic analysis of linear growth rates that it is possible to devise a classification scheme that demarcates the stability of Prandtl slope flows into distinct regimes based on the dimensionless stratification perturbation number. We verify the existence of the instability modes with the help of direct numerical simulations, and observe close agreements between simulation results and predictions of linear analysis. For slope angle values in the vicinity of the junction point in the instability map, both longitudinal rolls and travelling waves coexist simultaneously and form complex flow structures.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 885 , 25 February 2020 , A13
Copyright
© 2019 Cambridge University Press

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References

Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327–342.CrossRefGoogle Scholar
Banta, R. M. 1984 Daytime boundary-layer evolution over mountainous terrain. Part 1. Observations of the dry circulations. Mon. Weath. Rev. 112 (2), 340–356.2.0.CO;2>CrossRefGoogle Scholar
Beare, R. J., Macvean, M. K., Holtslag, A. A. M., Cuxart, J., Esau, I., Golaz, J.-C., Jimenez, M. A., Khairoutdinov, M., Kosovic, B., Lewellen, D. et al. 2006 An intercomparison of large-eddy simulations of the stable boundary layer. Boundary-Layer Meteorol. 118 (2), 247–272.CrossRefGoogle Scholar
Candelier, J., Le Dizès, S. & Millet, C. 2011 Shear instability in a stratified fluid when shear and stratification are not aligned. J. Fluid Mech. 685, 191–201.CrossRefGoogle Scholar
Candelier, J., Le Dizès, S. & Millet, C. 2012 Inviscid instability of a stably stratified compressible boundary layer on an inclined surface. J. Fluid Mech. 694, 524–539.CrossRefGoogle Scholar
Carpenter, J. R., Balmforth, N. J. & Lawrence, G. A. 2010 Identifying unstable modes in stratified shear layers. Phys. Fluids 22 (5), 054104.CrossRefGoogle Scholar
Carpenter, J. R., Tedford, E. W., Heifetz, E. & Lawrence, G. A. 2011 Instability in stratified shear flow: review of a physical interpretation based on interacting waves. Appl. Mech. Rev. 64 (6), 060801.CrossRefGoogle Scholar
Chen, J., Bai, Y. & Le Dizès, S. 2016 Instability of a boundary layer flow on a vertical wall in a stably stratified fluid. J. Fluid Mech. 795, 262–277.CrossRefGoogle Scholar
Chen, T. S. & Tzuoo, K.-L. 1982 Vortex instability of free convection flow over horizontal and inclined surfaces. Trans. ASME J. Heat Transfer 104 (4), 637–643.CrossRefGoogle Scholar
Clever, R. M. & Busse, F. H. 1977 Instabilities of longitudinal convection rolls in an inclined layer. J. Fluid Mech. 81 (1), 107–127.CrossRefGoogle Scholar
Coleman, G. N., Ferziger, J. H. & Spalart, P. R. 1990 A numerical study of the turbulent Ekman layer. J. Fluid Mech. 213, 313–348.CrossRefGoogle Scholar
Defant, F. 1949a Zur Theorie der Hangwinde, nebst Bemerkungen zur Theorie der Berg-und Talwinde. Archiv für Meteorologie, Geophysik und Bioklimatologie, Serie A 1 (3-4), 421–450.CrossRefGoogle Scholar
Defant, F. 1949b Zur Theorie der Hangwinde, nebst Bemerkungen zur Theorie der Berg- und Talwinde (A theory of slope winds, along with remarks on the theory of mountain winds and valley winds). Archiv für Meteorologie, Geophysik und Bioklimatologie, Serie A (Theoretical and Applied Climatology) 1 (3–4), 421–450; (English translation: Whiteman, C. D., and Dreiseitl, E., 1984: Alpine meteorology: Translations of classic contributions by A. Wagner, E. Ekhart and F. Defant. PNL-5141/ASCOT-84-3. Pacific Northwest Laboratory, Richland, Washington, 121 pp).Google Scholar
Deloncle, A., Chomaz, J.-M. & Billant, P. 2007 Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid. J. Fluid Mech. 570, 297–305.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Eaves, T. S. & Balmforth, N. J. 2019 Instability of sheared density interfaces. J. Fluid Mech. 860, 145–171.CrossRefGoogle Scholar
Facchini, G., Favier, B., Le Gal, P., Wang, M. & Le Bars, M. 2018 The linear instability of the stratified plane Couette flow. J. Fluid Mech. 853, 205–234.CrossRefGoogle Scholar
Fedorovich, E. & Shapiro, A. 2009 Structure of numerically simulated katabatic and anabatic flows along steep slopes. Acta Geophys. 57 (4), 981–1010.CrossRefGoogle Scholar
Fedorovich, E. & Shapiro, A. 2017 Oscillations in Prandtl slope flow started from rest. Q. J. R. Meteorol. Soc. 143 (703), 670–677.CrossRefGoogle Scholar
Fernando, H. J. S., Pardyjak, E. R., Di Sabatino, S., Chow, F. K., De Wekker, S. F. J., Hoch, S. W., Hacker, J., Pace, J. C., Pratt, T., Pu, Z. et al. 2015 The MATERHORN: unraveling the intricacies of mountain weather. Am. Meteorol. Soc. B 96 (11), 1945–1967.CrossRefGoogle Scholar
Fernando, H. J. S. & Weil, J. C. 2010 Whither the stable boundary layer? A shift in the research agenda. Am. Meteorol. Soc. B 91 (11), 1475–1484.CrossRefGoogle Scholar
Giometto, M. G., Katul, G. G., Fang, J. & Parlange, M. B. 2017 Direct numerical simulation of turbulent slope flows up to Grashof number Gr = 2. 1 × 1011. J. Fluid Mech. 829, 589–620.CrossRefGoogle Scholar
Görtler, H. 1959 Über eine analogie zwischen den instabilitäten laminarer grenzschichtströmungen an konkaven wänden und an erwärmten wänden. Ing.-Arch. 28 (1), 71–78.CrossRefGoogle Scholar
Grisogono, B. & Oerlemans, J. 2001a Katabatic flow: analytic solution for gradually varying eddy diffusivities. J. Atmos. Sci. 58 (21), 3349–3354.2.0.CO;2>CrossRefGoogle Scholar
Grisogono, B. & Oerlemans, J. 2001b A theory for the estimation of surface fluxes in simple katabatic flows. Q. J. R. Meteorol. Soc. 127 (578), 2725–2739.CrossRefGoogle Scholar
Haaland, S. E. & Sparrow, E. M. 1973 Vortex instability of natural convection flow on inclined surfaces. Intl J. Heat Mass Transfer 16 (12), 2355–2367.CrossRefGoogle Scholar
Iyer, P. A. & Kelly, R. E. 1974 The stability of the laminar free convection flow induced by a heated inclined plate. Intl J. Heat Mass Transfer 17 (4), 517–525.CrossRefGoogle Scholar
Jacobsen, D. A. & Senocak, I. 2013 Multi-level parallelism for incompressible flow computations on GPU clusters. Parallel Comput. 39 (1), 1–20.CrossRefGoogle Scholar
Kosović, B. & Curry, J. A. 2000 A large eddy simulation study of a quasi-steady, stably stratified atmospheric boundary layer. J. Atmos. Sci. 57 (8), 1052–1068.2.0.CO;2>CrossRefGoogle Scholar
Kundu, P. K., Cohen, I. M. & Dowling, D. R. 2016 Fluid Mechanics, 6th edn. Elsevier.Google Scholar
Le Dizès, S. & Billant, P. 2009 Radiative instability in stratified vortices. Phys. Fluids 21 (9), 096602.CrossRefGoogle Scholar
Lin, M.-H. 2001 Numerical study of formation of longitudinal vortices in natural convection flow over horizontal and inclined surfaces. Intl J. Heat Mass Transfer 44 (9), 1759–1766.CrossRefGoogle Scholar
Lloyd, J. R. & Sparrow, E. M. 1970 On the instability of natural convection flow on inclined plates. J. Fluid Mech. 42 (3), 465–470.CrossRefGoogle Scholar
Mahrt, L. 1998 Stratified atmospheric boundary layers and breakdown of models. Theor. Comput. Fluid Dyn. 11 (3-4), 263–279.CrossRefGoogle Scholar
Mahrt, L. 2014 Stably stratified atmospheric boundary layers. Annu. Rev. Fluid Mech. 46, 23–45.CrossRefGoogle Scholar
Mason, P. J. & Derbyshire, S. H. 1990 Large-eddy simulation of the stably-stratified atmospheric boundary layer. Boundary-Layer Meteorol. 53 (1-2), 117–162.CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10 (4), 496–508.CrossRefGoogle Scholar
Monin, A. S. & Obukhov, A. M. 1954 Basic laws of turbulent mixing in the atmosphere near the ground. Tr. Akad. Nauk SSSR Geofiz. Inst 24 (151), 163–187.Google Scholar
Monti, P., Fernando, H. J. S., Princevac, M., Chan, W. C., Kowalewski, T. A. & Pardyjak, E. R. 2002 Observations of flow and turbulence in the nocturnal boundary layer over a slope. J. Atmos. Sci. 59 (17), 2513–2534.2.0.CO;2>CrossRefGoogle Scholar
Nieuwstadt, F. T. M. 1984 The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci. 41 (14), 2202–2216.2.0.CO;2>CrossRefGoogle Scholar
Pera, L. & Gebhart, B. 1973 Natural convection boundary layer flow over horizontal and slightly inclined surfaces. Intl J. Heat Mass Transfer 16 (6), 1131–1146.CrossRefGoogle Scholar
Prandtl, L. 1942 Führer durch die Strömungslehre. Vieweg und Sohn.Google Scholar
Prandtl, L. 1952 Essentials of Fluid Dynamics: With Applications to Hydraulics, Aeronautics, Meteorology and other Subjects. Blackie & Son.Google Scholar
Salehipour, H., Caulfield, C. P. & Peltier, W. R. 2016 Turbulent mixing due to the Holmboe wave instability at high Reynolds number. J. Fluid Mech. 803, 591–621.CrossRefGoogle Scholar
Sandu, I., Beljaars, A., Bechtold, P., Mauritsen, T. & Balsamo, G. 2013 Why is it so difficult to represent stably stratified conditions in numerical weather prediction (NWP) models? J. Adv. Model. Earth Sy. 5 (2), 117–133.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Schumann, U. 1990 Large-eddy simulation of the up-slope boundary layer. Q. J. R. Meteorol. Soc. 116 (493), 637–670.CrossRefGoogle Scholar
Serafin, S., Adler, B., Cuxart, J., De Wekker, S. F. J., Gohm, A., Grisogono, B., Kalthoff, N., Kirshbaum, D. J., Rotach, M. W., Schmidli, J. et al. 2018 Exchange processes in the atmospheric boundary layer over mountainous terrain. Atmosphere 9 (3), 102.CrossRefGoogle Scholar
Shah, S. K. & Bou-Zeid, E. 2014 Direct numerical simulations of turbulent Ekman layers with increasing static stability: modifications to the bulk structure and second-order statistics. J. Fluid Mech. 760, 494–539.CrossRefGoogle Scholar
Shakespeare, C. J. 2019 Spontaneous generation of internal waves. Phys. Today 72 (6), 34–39.CrossRefGoogle Scholar
Shakespeare, C. J. & Taylor, J. R. 2014 The spontaneous generation of inertia–gravity waves during frontogenesis forced by large strain: theory. J. Fluid Mech. 757, 817–853.CrossRefGoogle Scholar
Shapiro, A. & Fedorovich, E. 2004 Unsteady convectively driven flow along a vertical plate immersed in a stably stratified fluid. J. Fluid Mech. 498, 333–352.CrossRefGoogle Scholar
Sparrow, E. M. & Husar, R. B. 1969 Longitudinal vortices in natural convection flow on inclined plates. J. Fluid Mech. 37 (2), 251–255.CrossRefGoogle Scholar
Steeneveld, G.-J. 2014 Current challenges in understanding and forecasting stable boundary layers over land and ice. Front. Env. Sci. Engng 2, 41.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223 (605–615), 289–343.Google Scholar
Turner, J. S. 1979 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar
Umphrey, C., DeLeon, R. & Senocak, I. 2017 Direct numerical simulation of turbulent katabatic slope flows with an immersed boundary method. Boundary-Layer Meteorol. 164 (3), 367–382.CrossRefGoogle Scholar
Whiteman, C. D. 1990 Observations of thermally developed wind systems in mountainous terrain. In Atmospheric Processes Over Complex Terrain, pp. 5–42. Springer.CrossRefGoogle Scholar
Whiteman, C. D. 2000 Mountain Meteorology: Fundamentals and Applications. Oxford University Press.Google Scholar
Xiao, C. & Senocak, I. 2019 Stability of the Prandtl model for katabatic slope flows. J. Fluid Mech. 865, R2.CrossRefGoogle Scholar
Zardi, D. & Whiteman, C. D. 2013 Diurnal mountain wind systems. In Mountain Weather Research and Forecasting, pp. 35–119. Springer.CrossRefGoogle Scholar