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Stability of the anabatic Prandtl slope flow in a stably stratified medium. (English) Zbl 1460.76324

Summary: 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 [the authors, ibid. 865, Paper No. R2, 14 p. (2019; Zbl 1429.86005)]. 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.

MSC:

76E20 Stability and instability of geophysical and astrophysical flows
76F45 Stratification effects in turbulence
76E05 Parallel shear flows in hydrodynamic stability

Citations:

Zbl 1429.86005
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[1] Baines, P. G. & Mitsudera, H.1994On the mechanism of shear flow instabilities. J. Fluid Mech.276, 327-342. · Zbl 0889.76019
[2] Banta, R. M.1984Daytime boundary-layer evolution over mountainous terrain. Part 1. Observations of the dry circulations. Mon. Weath. Rev.112 (2), 340-356.
[3] 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.2006An intercomparison of large-eddy simulations of the stable boundary layer. Boundary-Layer Meteorol.118 (2), 247-272.
[4] Candelier, J., Le Dizès, S. & Millet, C.2011Shear instability in a stratified fluid when shear and stratification are not aligned. J. Fluid Mech.685, 191-201. · Zbl 1241.76171
[5] Candelier, J., Le Dizès, S. & Millet, C.2012Inviscid instability of a stably stratified compressible boundary layer on an inclined surface. J. Fluid Mech.694, 524-539. · Zbl 1250.76066
[6] Carpenter, J. R., Balmforth, N. J. & Lawrence, G. A.2010Identifying unstable modes in stratified shear layers. Phys. Fluids22 (5), 054104.
[7] Carpenter, J. R., Tedford, E. W., Heifetz, E. & Lawrence, G. A.2011Instability in stratified shear flow: review of a physical interpretation based on interacting waves. Appl. Mech. Rev.64 (6), 060801.
[8] Chen, J., Bai, Y. & Le Dizès, S.2016Instability of a boundary layer flow on a vertical wall in a stably stratified fluid. J. Fluid Mech.795, 262-277. · Zbl 1359.76108
[9] Chen, T. S. & Tzuoo, K.-L.1982Vortex instability of free convection flow over horizontal and inclined surfaces. Trans. ASME J. Heat Transfer104 (4), 637-643.
[10] Clever, R. M. & Busse, F. H.1977Instabilities of longitudinal convection rolls in an inclined layer. J. Fluid Mech.81 (1), 107-127. · Zbl 0361.76044
[11] Coleman, G. N., Ferziger, J. H. & Spalart, P. R.1990A numerical study of the turbulent Ekman layer. J. Fluid Mech.213, 313-348.
[12] Defant, F.1949aZur Theorie der Hangwinde, nebst Bemerkungen zur Theorie der Berg-und Talwinde. Archiv für Meteorologie, Geophysik und Bioklimatologie, Serie A1 (3-4), 421-450.
[13] Defant, F.1949bZur 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).
[14] Deloncle, A., Chomaz, J.-M. & Billant, P.2007Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid. J. Fluid Mech.570, 297-305. · Zbl 1105.76025
[15] Drazin, P. G. & Reid, W. H.2004Hydrodynamic Stability, 2nd edn. Cambridge University Press.
[16] Eaves, T. S. & Balmforth, N. J.2019Instability of sheared density interfaces. J. Fluid Mech.860, 145-171. · Zbl 1415.86006
[17] Facchini, G., Favier, B., Le Gal, P., Wang, M. & Le Bars, M.2018The linear instability of the stratified plane Couette flow. J. Fluid Mech.853, 205-234. · Zbl 1415.76225
[18] Fedorovich, E. & Shapiro, A.2009Structure of numerically simulated katabatic and anabatic flows along steep slopes. Acta Geophys.57 (4), 981-1010.
[19] Fedorovich, E. & Shapiro, A.2017Oscillations in Prandtl slope flow started from rest. Q. J. R. Meteorol. Soc.143 (703), 670-677.
[20] 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.2015The MATERHORN: unraveling the intricacies of mountain weather. Am. Meteorol. Soc. B96 (11), 1945-1967.
[21] Fernando, H. J. S. & Weil, J. C.2010Whither the stable boundary layer? A shift in the research agenda. Am. Meteorol. Soc. B91 (11), 1475-1484.
[22] Giometto, M. G., Katul, G. G., Fang, J. & Parlange, M. B.2017Direct numerical simulation of turbulent slope flows up to Grashof number Gr = 2. 1 × 10^11. J. Fluid Mech.829, 589-620. · Zbl 1460.86026
[23] Görtler, H.1959Über eine analogie zwischen den instabilitäten laminarer grenzschichtströmungen an konkaven wänden und an erwärmten wänden. Ing.-Arch.28 (1), 71-78. · Zbl 0087.41201
[24] Grisogono, B. & Oerlemans, J.2001aKatabatic flow: analytic solution for gradually varying eddy diffusivities. J. Atmos. Sci.58 (21), 3349-3354.
[25] Grisogono, B. & Oerlemans, J.2001bA theory for the estimation of surface fluxes in simple katabatic flows. Q. J. R. Meteorol. Soc.127 (578), 2725-2739.
[26] Haaland, S. E. & Sparrow, E. M.1973Vortex instability of natural convection flow on inclined surfaces. Intl J. Heat Mass Transfer16 (12), 2355-2367. · Zbl 0275.76018
[27] Iyer, P. A. & Kelly, R. E.1974The stability of the laminar free convection flow induced by a heated inclined plate. Intl J. Heat Mass Transfer17 (4), 517-525.
[28] Jacobsen, D. A. & Senocak, I.2013Multi-level parallelism for incompressible flow computations on GPU clusters. Parallel Comput.39 (1), 1-20.
[29] Kosović, B. & Curry, J. A.2000A large eddy simulation study of a quasi-steady, stably stratified atmospheric boundary layer. J. Atmos. Sci.57 (8), 1052-1068.
[30] Kundu, P. K., Cohen, I. M. & Dowling, D. R.2016Fluid Mechanics, 6th edn. Elsevier. · Zbl 1315.76002
[31] Le Dizès, S. & Billant, P.2009Radiative instability in stratified vortices. Phys. Fluids21 (9), 096602.
[32] Lin, M.-H.2001Numerical study of formation of longitudinal vortices in natural convection flow over horizontal and inclined surfaces. Intl J. Heat Mass Transfer44 (9), 1759-1766. · Zbl 1008.76083
[33] Lloyd, J. R. & Sparrow, E. M.1970On the instability of natural convection flow on inclined plates. J. Fluid Mech.42 (3), 465-470.
[34] Mahrt, L.1998Stratified atmospheric boundary layers and breakdown of models. Theor. Comput. Fluid Dyn.11 (3-4), 263-279. · Zbl 0948.76029
[35] Mahrt, L.2014Stably stratified atmospheric boundary layers. Annu. Rev. Fluid Mech.46, 23-45. · Zbl 1297.76077
[36] Mason, P. J. & Derbyshire, S. H.1990Large-eddy simulation of the stably-stratified atmospheric boundary layer. Boundary-Layer Meteorol.53 (1-2), 117-162.
[37] Miles, J. W.1961On the stability of heterogeneous shear flows. J. Fluid Mech.10 (4), 496-508. · Zbl 0101.43002
[38] Monin, A. S. & Obukhov, A. M.1954Basic laws of turbulent mixing in the atmosphere near the ground. Tr. Akad. Nauk SSSR Geofiz. Inst24 (151), 163-187.
[39] Monti, P., Fernando, H. J. S., Princevac, M., Chan, W. C., Kowalewski, T. A. & Pardyjak, E. R.2002Observations of flow and turbulence in the nocturnal boundary layer over a slope. J. Atmos. Sci.59 (17), 2513-2534.
[40] Nieuwstadt, F. T. M.1984The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci.41 (14), 2202-2216.
[41] Pera, L. & Gebhart, B.1973Natural convection boundary layer flow over horizontal and slightly inclined surfaces. Intl J. Heat Mass Transfer16 (6), 1131-1146. · Zbl 0258.76034
[42] Prandtl, L.1942Führer durch die Strömungslehre. Vieweg und Sohn.
[43] Prandtl, L.1952Essentials of Fluid Dynamics: With Applications to Hydraulics, Aeronautics, Meteorology and other Subjects. Blackie & Son. · Zbl 0048.42801
[44] Salehipour, H., Caulfield, C. P. & Peltier, W. R.2016Turbulent mixing due to the Holmboe wave instability at high Reynolds number. J. Fluid Mech.803, 591-621.
[45] Sandu, I., Beljaars, A., Bechtold, P., Mauritsen, T. & Balsamo, G.2013Why is it so difficult to represent stably stratified conditions in numerical weather prediction (NWP) models?J. Adv. Model. Earth Sy.5 (2), 117-133.
[46] Schmid, P. J. & Henningson, D. S.2001Stability and Transition in Shear Flows. Springer. · Zbl 0966.76003
[47] Schumann, U.1990Large-eddy simulation of the up-slope boundary layer. Q. J. R. Meteorol. Soc.116 (493), 637-670.
[48] 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.2018Exchange processes in the atmospheric boundary layer over mountainous terrain. Atmosphere9 (3), 102.
[49] Shah, S. K. & Bou-Zeid, E.2014Direct 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.
[50] Shakespeare, C. J.2019Spontaneous generation of internal waves. Phys. Today72 (6), 34-39.
[51] Shakespeare, C. J. & Taylor, J. R.2014The spontaneous generation of inertia-gravity waves during frontogenesis forced by large strain: theory. J. Fluid Mech.757, 817-853. · Zbl 1416.76337
[52] Shapiro, A. & Fedorovich, E.2004Unsteady convectively driven flow along a vertical plate immersed in a stably stratified fluid. J. Fluid Mech.498, 333-352. · Zbl 1134.76463
[53] Sparrow, E. M. & Husar, R. B.1969Longitudinal vortices in natural convection flow on inclined plates. J. Fluid Mech.37 (2), 251-255.
[54] Steeneveld, G.-J.2014Current challenges in understanding and forecasting stable boundary layers over land and ice. Front. Env. Sci. Engng2, 41.
[55] Taylor, G. I.1923Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A223 (605-615), 289-343.
[56] Turner, J. S.1979Buoyancy Effects in Fluids. Cambridge University Press. · Zbl 0443.76091
[57] Umphrey, C., DeLeon, R. & Senocak, I.2017Direct numerical simulation of turbulent katabatic slope flows with an immersed boundary method. Boundary-Layer Meteorol.164 (3), 367-382.
[58] Whiteman, C. D.1990Observations of thermally developed wind systems in mountainous terrain. In Atmospheric Processes Over Complex Terrain, pp. 5-42. Springer.
[59] Whiteman, C. D.2000Mountain Meteorology: Fundamentals and Applications. Oxford University Press.
[60] Xiao, C. & Senocak, I.2019Stability of the Prandtl model for katabatic slope flows. J. Fluid Mech.865, R2.
[61] Zardi, D. & Whiteman, C. D.2013Diurnal mountain wind systems. In Mountain Weather Research and Forecasting, pp. 35-119. Springer.
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