Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T19:56:13.764Z Has data issue: false hasContentIssue false
  • English
  • Français

Instabilities and small-scale waves within the Stewartson layers of a thermally driven rotating annulus

Published online by Cambridge University Press:  21 February 2018

Thomas von Larcher Open the ORCID record for Thomas von Larcher [Opens in a new window] ,
Stéphane Viazzo ,
Uwe Harlander ,
Miklos Vincze  and
Anthony Randriamampianina
Thomas von Larcher*
Affiliation:
Institute of Mathematics, Freie Universität Berlin, Arnimallee 9, D-14195 Berlin, Germany
Stéphane Viazzo
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, M2P2, Marseille, France
Uwe Harlander
Affiliation:
Department of Aerodynamics and Fluid Mechanics, Brandenburg University of Technology Cottbus-Senftenberg, Siemens-Halske-Ring 14, D-03046 Cottbus, Germany
Miklos Vincze
Affiliation:
MTA-ELTE Theoretical Physics Research Group, Budapest, H-1117, Hungary
Anthony Randriamampianina
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, M2P2, Marseille, France
*
†Email address for correspondence: larcher@math.fu-berlin.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We report on small-scale instabilities in a thermally driven rotating annulus filled with a liquid with moderate Prandtl number. The study is based on direct numerical simulations and an accompanying laboratory experiment. The computations are performed independently with two different flow solvers, that is, first, the non-oscillatory forward-in-time differencing flow solver EULAG and, second, a higher-order finite-difference compact scheme (HOC). Both branches consistently show the occurrence of small-scale patterns at both vertical sidewalls in the Stewartson layers of the annulus. Small-scale flow structures are known to exist at the inner sidewall. In contrast, short-period waves at the outer sidewall have not yet been reported. The physical mechanisms that possibly trigger these patterns are discussed. We also debate whether these small-scale structures are a gravity wave signal.

JFM classification

Geophysical and Geological Flows: Baroclinic flows Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 841 , 25 April 2018 , pp. 380 - 407
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abide, S. & Viazzo, S. 2005 A 2D compact fourth-order projection decomposition method. J. Comput. Phys. 206, 252–276.Google Scholar
Abrahamson, S. D., Eaton, J. K. & Koga, D. J. 1989 The flow between shrouded corotating disks. Phys. Fluids A 1 (2), 241–251.Google Scholar
Barcilon, V. & Pedlosky, J. 1967 Linear theory of rotating stratified fluid motions. J. Fluid Mech. 29, 1–17.CrossRefGoogle Scholar
Billant, P. & Gallaire, F. 2005 Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. J. Fluid Mech. 542, 365–379.Google Scholar
von Böckh, P. & Wetzel, T. 2014 Wärmeübertragung. Springer Vieweg.Google Scholar
Borchert, S., Achatz, U. & Fruman, M. D. 2014 Gravity wave emission in an atmosphere-like configuration of the differentially heated rotating annulus experiment. J. Fluid Mech. 758, 287–311.Google Scholar
Castrejón-Pita, A. A. & Read, P. L. 2007 Baroclinic waves in an air-filled thermally driven rotating annulus. Phys. Rev. E 75, 026301.Google Scholar
Chomaz, J. M., Ortiz, S., Gallaire, F. & Billant, P. 2010 Stability of quasi two-dimensional vortices. In Lecture Notes in Physics: Fronts, Waves and Vortices in Geophysical Flows (ed. Flór, J.-B.), Lecture Notes in Physics, vol. 805, pp. 35–59. Springer.Google Scholar
Dettinger, M. D., Ghil, M., Strong, C. M., Weibel, W. & Yiou, P. 1995 Software expedites singular-spectrum analysis of noisy time series. EOS Trans. AGU 76 (2), 12, 14, 21.Google Scholar
Fein, J. S. & Pfeffer, R. L. 1976 An experimental study of the effects of Prandtl number on thermal convection in a rotating, differentially heated cylindrical annulus of fluid. J. Fluid Mech. 75, 81–112.Google Scholar
Flór, J.-B., Scolan, H. & Gula, J. 2011 Frontal instabilities and waves in a differentially rotating fluid. J. Fluid Mech. 685, 532–542.Google Scholar
Fritts, D. C. & Alexander, M. J. 2003 Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys. 41, 1003.Google Scholar
Früh, W.-G. 2015 Amplitude vacillation in baroclinic flows. In Modeling Atmospheric and Oceanic Flows: Insights from Laboratory Experiments and Numerical Simulations (ed. von Larcher, Th. & Williams, P. D.), pp. 61–84. John Wiley & Sons.Google Scholar
Früh, W.-G. & Read, P. L. 1997 Wave interactions and the transition to chaos of baroclinic waves in a thermally driven rotating annulus. Phil. Trans. R. Soc. Lond. A 355, 101–153.CrossRefGoogle Scholar
Fultz, D., Long, R. R., Owens, G. V., Bohan, W., Kaylor, R. & Weil, J. 1959 Studies of thermal convection in a rotating cylinder with some implications for large-scale atmospheric motions. Meteorological Monographs. vol. 4, pp. 1–104. American Meteorological Society.Google Scholar
Ghil, M., Allen, M. R., Dettinger, M. D., Ide, K., Kondrashov, D., Mann, M. E., Robertson, A. W., Saunders, A., Tian, Y., Varadi, F. et al. 2002 Advanced spectral methods for climatic time series. Rev. Geophys. 40 (1), 3.1–3.41.Google Scholar
Gill, A. 1982 Atmosphere–Ocean Dynamics. Academic Press.Google Scholar
Goldstein, D., Handler, R. & Sirovich, L. 1993 Modeling a no-slip flow boundary with an external force field. J. Comput. Phys. 105 (2), 354–366.Google Scholar
Harlander, U., Larcher, Th., Wang, Y. & Egbers, C. 2011 PIV- and LDV-measurements of baroclinic wave interactions in a thermally driven rotating annulus. Exp. Fluids 51 (1), 37–49.Google Scholar
Harlander, U., von Larcher, Th., Wright, G. B., Hoff, M., Alexandrov, K. & Egbers, C. 2015 Orthogonal decomposition methods to analyze PIV, LDV and thermography data of a thermally driven rotating annulus laboratory experiment. In Modeling Atmospheric and Oceanic Flows: Insights from Laboratory Experiments and Numerical Simulations (ed. von Larcher, Th. & Williams, P. D.), pp. 315–336. John Wiley & Sons.Google Scholar
Hart, J. E. & Kittelman, S. 1996 Instabilities of the sidewall boundary layer in a differentially driven rotating cylinder. Phys. Fluids 8, 692–696.Google Scholar
Hide, R. 1958 An experimental study of thermal convection in a rotating fluid. Phil. Trans. R. Soc. Lond. A 250, 441–478.Google Scholar
Hide, R. & Mason, P. J. 1975 Sloping convection in a rotating fluid. Adv. Phys. 24, 47–99.Google Scholar
Hien, S., Rolland, J., Borchert, S., Schoon, L., Zülicke, C. & Achatz, U. 2018 Spontaneous inertia–gravity wave emission in the differentially heated rotating annulus experiment. J. Fluid Mech. 838, 5–41.Google Scholar
Hignett, P. 1985 Characteristics of amplitude vacillation in a differentially heated rotating fluid annulus. Geophys. Astrophys. Fluid Dyn. 31, 247–281.Google Scholar
Hunt, J. C. R. 1987 Vorticity and vortex dynamics in complex turbulent flows. Trans. Can. Soc. Mech. Engng 11, 21–35.Google Scholar
Hunt, J. C. R., Wray, A. A. & Moin, P. 1988 Eddies, streams, and convergence zones in turbulent flows. In Studying Turbulence Using Numerical Simulation Databases, Proceedings of the 1988 Summer Program, pp. 193–208. Center for Turbulence Research, Stanford University.Google Scholar
Jacoby, T. N. L., Read, P. L., Williams, P. D. & Young, R. M. B. 2011 Generation of inertia–gravity waves in the rotating, thermal annulus by a localised boundary layer instability. Geophys. Astrophys. Fluid Dyn. 105, 161–181.Google Scholar
James, I. N., Jonas, P. R. & Farnell, L. 1980 A combined laboratory and numerical study of fully developed steady baroclinic waves in a cylindrical annulus. Q. J. R. Meteorol. Soc. 107, 51–78.Google Scholar
von Larcher, Th. & Dörnbrack, A. 2015 Numerical simulations of baroclinic driven flows in a thermally driven rotating annulus using the immersed boundary method. Meteorol. Z. 23, 599–610.Google Scholar
von Larcher, Th. & Egbers, C. 2005 Experiments on transitions of baroclinic waves in a differentially heated rotating annulus. Nonlinear Process. Geophys. 12, 1033–1041.Google Scholar
Leppiler, V., Goharzadeh, A., Prigent, A. & Mutabazi, I. 2008 Weak temperature gradient effect on the stability of the circular Couette flow. Eur. Phys. J. B 61, 445–455.CrossRefGoogle Scholar
Lopez, J. M. & Marques, F. 2010 Sidewall boundary layer instabilities in a rapidly rotating cylinder driven by a differentially corotating lid. Phys. Fluids 22, 114109.Google Scholar
Lovegrove, A. F., Read, P. L. & Richards, C. J. 2000 Generation of inertia–gravity waves in a baroclinically unstable fluid. Q. J. R. Meteorol. Soc. 126, 3233–3254.Google Scholar
Lu, H.-I. & Miller, T. L. 1997 Characteristics of annulus baroclinic flow structure during amplitude vacillation. Dyn. Atmos. Oceans 27, 485–503.Google Scholar
McBain, G. D., Armfield, S. W. & Desrayaud, G. 2007 Instability of the buoyancy layer on an evenly heated vertical wall. J. Fluid Mech. 587, 453–469.Google Scholar
Morita, O. & Uryu, M. 1989 Geostrophic turbulence in a rotating annulus of fluid. J. Atmos. Sci. 46, 2349–2355.Google Scholar
Oguic, R., Viazzo, S. & Poncet, S. 2015 A parallelized multidomain compact solver for incompressible turbulent flows in cylindrical geometries. J. Comput. Phys. 300, 710–731.Google Scholar
Ohlsen, D. R. & Hart, J. E. 1989 Nonlinear interference vacillation. Geophys. Astrophys. Fluid Dyn. 45 (3–4), 213–235.Google Scholar
O’Sullivan, D. & Dunkerton, T. J. 1995 Generation of inertia–gravity waves in a simulated life-cycle of baroclinic instability. J. Atmos. Sci. 52, 3695–3716.Google Scholar
Pedlosky, J. 1970 Finite-amplitude baroclinic waves. J. Atmos. Sci. 27, 15–30.Google Scholar
Pfeffer, R. L., Applequist, S. R., Kung, R., Long, C. & Buzyna, G. 1997 Progress in characterizing the route to geostrophic turbulence and redesigning thermally driven rotating annulus. Theor. Comput. Fluid Dyn. 9, 253–267.Google Scholar
Plougonven, R. & Snyder, C. 2005 Gravity waves excited by jets: propagation versus generation. Geophys. Res. Lett. 32, L18802.Google Scholar
Plougonven, R. & Snyder, C. 2007 Inertia–gravity waves spontaneously excited by jets and fronts. Part I: different baroclinic life cycles. J. Atmos. Sci. 64, 2502–2520.Google Scholar
Plougonven, R., Teitelbaum, H. & Zeitlin, V. 2003 Inertia gravity wave generation by the tropospheric midlatitude jet as given by the fronts and Atlantic storm-track experiment radio sounding. J. Geophys. Res. 108, 4686.Google Scholar
Prusa, J. M., Smolarkiewicz, P. K. & Wyszogrodzki, A. A. 2008 EULAG, a computational model for multiscale flows. Comput. Fluids 37, 1193–1207.Google Scholar
Randriamampianina, A. 2013 Inertia gravity wave characteristics within a baroclinic cavity. C. R. Méc. 341, 547–552.Google Scholar
Randriamampianina, A. & Crespo del Arco, E. 2015 Inertia–gravity waves in a liquid-filled, differentially heated, rotating annulus. J. Fluid Mech. 782, 144–177.Google Scholar
Randriamampianina, A., Früh, W.-G., Read, P. L. & Maubert, P. 2006 Direct numerical simulations of bifurcations in an air-filled rotating baroclinic annulus. J. Fluid Mech. 561, 359–389.Google Scholar
Rayleigh, Lord 1917 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93 (648), 148–154; http://rspa.royalsocietypublishing.org/content/93/648/148.full.pdf.Google Scholar
Read, P. L. 1992 Applications of singular systems analysis to ‘baroclinic chaos’. Physica D 58, 455–468.Google Scholar
Read, P. L., Bell, M. J., Johnson, D. W. & Small, R. M. 1992 Quasi-periodic and chaotic flow regimes in a thermally-driven, rotating fluid annulus. J. Fluid Mech. 238, 599–632.Google Scholar
Read, P. L., Maubert, P., Randriamampianina, A. & Früh, W.-G. 2008 Direct numerical simulation of transitions towards structural vacillation in an air-filled, rotating, baroclinic annulus. Phys. Fluids 20, 044107.CrossRefGoogle Scholar
Read, P. L., Perez, E. P., Moroz, I. M. & Young, R. M. B. 2015 General circulation of planetary atmospheres: insights from rotating annulus and related experiments. In Modeling Atmospheric and Oceanic Flows: Insights from Laboratory Experiments and Numerical Simulations (ed. von Larcher, Th. & Williams, P. D.), pp. 9–44. John Wiley & Sons.Google Scholar
Smolarkiewicz, P. K. 1991 On forward-in-time differencing for fluids. Mon. Weath. Rev. 119, 2505–2510.Google Scholar
Smolarkiewicz, P. K. & Margolin, L. G. 1997 On forward-in-time differencing for fluids: an Eulerian/semi-Lagrangian non-hydrostatic model for stratified flows. Atmos-Ocean Special 35, 127–157.Google Scholar
Smolarkiewicz, P. K. & Margolin, L. G. 1998 MPDATA: a positive definite solver for geophysical flows. J. Comput. Phys. 140, 459–480.Google Scholar
Smolarkiewicz, P. K., Sharman, R., Weil, J., Perry, S. G., Heist, D. & Bowker, G. 2007 Building resolving large-eddy simulations and comparison with wind tunnel experiments. J. Comput. Phys. 227 (1), 633–653.Google Scholar
Synge, J. L. 1933 The stability of heterogeneous liquid. Trans. R. Soc. Can. 27, 1–18.Google Scholar
Tollmien, W. 1935 Ein allgemeines Kriterium der Instabilität laminarer Geschwindigkeitsverteilungen. Nachr. Ges. Wiss. Göttingen, Math. Phys. Klasse NF 1, 79–114.Google Scholar
Vanneste, J. 2013 Balance and spontaneous wave generation in geophysical flows. Annu. Rev. Fluid. Mech. 45, 147–172.Google Scholar
Vautard, R., Yiou, P. & Ghil, M. 1992 Singular-spectrum analysis: a toolkit for short, noisy chaotic signals. Physica D 58, 95–126.Google Scholar
Viazzo, S. & Poncet, S. 2014 Numerical simulation of the flow stability in a high aspect ratio Taylor–Couette system submitted to a radial temperature gradient. Comput. Fluids 101, 15–26.Google Scholar
Vincze, M., Borchert, S., Achatz, U., von Larcher, Th., Baumann, M., Hertel, C., Remmler, S., Alexandrov, K., Egbers, C., Fröhlich, J. et al. 2015 Benchmarking in a rotating annulus: a comparative experimental and numerical study of baroclinic wave dynamics. Meteorol. Z. 23, 611–635.Google Scholar
Vincze, M., Borcia, I., Harlander, U. & Le Gal, P. 2016 Double-diffusive convection and baroclinic instability in a differentially heated and initially stratified rotating system: the barostrat instability. Fluid Dyn. Res. 48, 061414.Google Scholar
Viùdez, A. & Dritschel, D. G. 2006 Spontaneous generation of inertia–gravity wave packets by balanced geophysical flows. J. Fluid Mech. 553, 107–117.Google Scholar
Williams, P. D., Haine, T. W. N. & Read, P. L. 2008 Inertia–gravity waves emitted from balanced flow: observations, properties, and consequences. J. Atmos. Sci. 65, 3543–3556.Google Scholar
Zhang, F. 2004 Generation of mesoscale gravity waves in upper-tropospheric jet-front systems. J. Atmos. Sci. 61, 440–457.Google Scholar

von Larcher et al. supplementary movie 1

Thermally driven rotating annulus: time sequence of the Q-criterion (isosurface $Q=10^(-4) s^(-2)$) displayed in the unfolded cylinder. Orientation view in the ($\Phi,z$)-plane and viewpoint from the inner sidewall. The time period of the animation in physical units 106s (corresponding nearly to a quarter of the drift period).

Download von Larcher et al. supplementary movie 1(Video)
Video 7.5 MB

von Larcher et al. supplementary movie 2

Thermally driven rotating annulus: time sequence of the Q-criterion (isosurface $Q=10^(-4) s^(-2)$) displayed in the unfolded cylinder. Orientation view in the ($\Phi,z$)-plane and viewpoint from the outer sidewall. The time period of the animation in physical units 106s (corresponding nearly to a quarter of the drift period).

Download von Larcher et al. supplementary movie 2(Video)
Video 7.6 MB

von Larcher et al. supplementary movie 3

Tracks of particles in the thermally driven rotating annulus: 4 millions of particles released in a ($\phi,r$) plane located at the top of the cavity (in the Ekman boundary layer). Particles are colored by the local temperature. The particles are tracked during 266.5s (the drift period is equal to 380s).

Download von Larcher et al. supplementary movie 3(Video)
Video 11.2 MB

von Larcher et al. supplementary movie 4

Tracks of particles in the thermally driven rotating annulus: 4 millions of particles released in a ($\phi,z$) plane inside the Stewartson inner sidewall boundary layer (at radius $r=45.15 mm$). Particles are colored by the local temperature. The particles are tracked during 266.5s (the drift period is equal to 380s).

Download von Larcher et al. supplementary movie 4(Video)
Video 14.9 MB