Papers
Topics
Authors
Recent
Search
2000 character limit reached

Edge-wave phase-shifts versus normal-mode phase-tilts in an Eady problem with a sloping boundary

Published 18 Apr 2024 in physics.ao-ph and physics.flu-dyn | (2404.12417v1)

Abstract: One mechanistic interpretation of baroclinic instability is that of mutual constructive interference of Rossby edge-waves. While the two edge-waves and their relative phase-shifts are invoked as part of the mechanistic interpretation, the phase-tilts of the related normal modes are often presented instead. Here we highlight the differences between edge-wave phase-shifts and normal-mode phase-tilts, in the context of an Eady problem modified by the presence of a sloping boundary. We argue and present evidence that the normal-mode phase-tilt is potentially a misleading quantity to use, and edge-wave phase-shifts should be the ones that are mechanistically relevant. We also provide a clarification for the mechanistic rationalization for baroclinic instability in the presence of slopes (such as suppression of growth rates) that is valid over all parameter space, in contrast to previous attempts. We further present evidence that there is a strong correlation between quantities diagnosed from the GEOMETRIC framework with the edge-wave phase-shifts, but not the normal-mode phase-tilts. The result is noteworthy in that the geometric framework makes no explicit reference to the edge-wave structures in its construction, but the correlation suggests that in problems where edge-wave structures are not so well-defined or readily available, the GEOMETRIC framework should still capture mechanistic and dynamical information. Some implications for parameterization of baroclinic instability and relevant eddy-mean feedbacks are discussed. For completeness, we also provide an explicit demonstration that the linear instability problem of the present modified Eady problem is parity-time symmetric, and speculate on some suggestive links between parity-time symmetry, shear instability, and the edge-wave interaction mechanism.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (68)
  1. G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics (Cambridge University Press, 2006).
  2. Y. Kaspi and G. R. Flierl, Formation of jets by baroclinic instability on gas planet atmospheres, J. Atmos. Sci. 64, 3177 (2007).
  3. D. W. Hughes, R. Rosner, and N. O. Weiss, The solar tachocline (Cambridge University Press, 2007).
  4. R. J. Teed, C. A. Jones, and R. Hollerbach, Rapidly rotating plane layer convection with zonal flow, Geophys. Astrophys. Fluid Dyn. 104, 457 (2010).
  5. I. Polichtchouk and J. Y. Cho, Baroclinic instability on hot extrasolar planets, Mon. Not. R. Astron. Soc. 424, 1307 (2012).
  6. P. A. Gilman and M. Dikpati, Baroclinic instability in the solar tachocline, Astrophys. J. 787, 60 (1402).
  7. P. A. Gilman, Effect of toroidal fields on baroclinic instability in the solar tachocline, Astrophys. J. 801, 22 (2015).
  8. R. Yellin-Bergovoy, O. M. Umurhan, and E. Heifetz, A minimal model for vertical shear instability in protoplanetary accretion disks, Geophys. Astrophys. Fluid Dyn. 115, 674 (2021).
  9. J. G. Charney, Dynamics of long waves in a baroclinic westerly current, J. Meteor. 4, 135 (1947).
  10. E. T. Eady, Long waves and cyclone waves, Tellus 1, 33 (1949).
  11. N. A. Phillips, The general circulation of the atmosphere: a numerical experiment, Q. J. Roy. Met. Soc. 82, 123 (1956).
  12. J. S. A. Green, A problem in baroclinic instability, Q. J. Roy. Met. Soc. 86, 237 (1960).
  13. J. G. Charney and M. E. Stern, On the stability of internal baroclinic jets in a rotating atmosphere, J. Atmos. Sci. 19, 159 (1962).
  14. J. Pedlosky, The stability of currents in the atmosphere and the ocean: Part I, J. Atmos. Sci. 21, 201 (1964a).
  15. J. Pedlosky, The stability of currents in the atmosphere and the ocean: Part II, J. Atmos. Sci. 21, 342 (1964b).
  16. T. G. Shepherd, Nonlinear saturation of baroclinic instability: part I: the two-layer model, J. Atmos. Sci. 45, 2014 (1990b).
  17. T. G. Shepherd, Nonlinear saturation of baroclinic instability: part II: continuous stratified fluid, J. Atmos. Sci. 46, 888 (1990c).
  18. C. D. Thorncroft, B. J. Hoskins, and M. E. McIntyre, Two paradigms of baroclinic-wave life-cycle behaviour, Q. J. Roy. Met. Soc. 119, 17 (1993).
  19. V. D. Larichev and I. M. Held, Eddy amplitudes and fluxes in a homogeneous model of fully developed baroclinic instability, J. Phys. Oceanogr. 25, 2285 (1995).
  20. M. A. Spall and D. C. Chapman, On the efficiency of baroclinic eddy heat transport across narrow fronts, J. Phys. Oceanogr. 28, 2275 (1998).
  21. A. F. Thompson and W. R. Young, Two-layer baroclinic eddy heat fluxes: Zonal flows and energy balance, J. Atmos. Sci. 64, 3214 (2007).
  22. J. G. Esler, The turbulent equilibration of an unstable baroclinic jet, J. Fluid Mech. 599, 241 (2008).
  23. S. D. Bachman and B. Fox-Kemper, Eddy parametrization challenge suite I: Eady spindown, Ocean Modell. 64, 12 (2013).
  24. C. Chang and I. M. Held, A scaling theory for the diffusivity of poleward eddy heat transport based on Rhines scaling and the global entropy budget, J. Atmos. Sci. 79, 1743 (2022).
  25. J. S. A. Green, Transfer properties of the large-scale eddies and the general circulation of the atmosphere, Q. J. Roy. Met. Soc. 96, 157 (1970).
  26. P. H. Stone, A simplified radiative-dynamical model for the static stability of rotating atmospheres, J. Atmos. Sci. 29, 405 (1972).
  27. P. D. Killworth, On the parameterization of eddy transfer, Part I. Theory, J. Mar. Res. 55, 1171 (1997).
  28. P. D. Killworth, On the parameterization of eddy transfer, Part II. Tests with a channel model, J. Mar. Res. 56, 349 (1998).
  29. C. Eden, A closure for meso-scale eddy fluxes based on linear instability theory, Ocean Modell. 39, 362 (2011).
  30. D. P. Marshall, J. R. Maddison, and P. S. Berloff, A framework for parameterizing eddy potential vorticity fluxes, J. Phys. Oceanogr. 42, 539 (2012).
  31. J. R. Maddison and D. P. Marshall, The Eliassen–Palm flux tensor, J. Fluid Mech. 729, 69 (2013).
  32. S. K. Blumsack and P. Gierasch, Mars: The effects of topography on baroclinic instability, J. Atmos. Sci. 29, 1081 (1972).
  33. C. R. Mechoso, Baroclinic instability of flows along sloping boundaries, J. Atmos. Sci. 37, 1393 (1980).
  34. P. E. Isachsen, Baroclinic instability and eddy tracer transport across sloping bottom topography: How well does a modified Eady model do in primitive equation simulations?, Ocean Modell. 39, 183 (2011).
  35. K. Brink, Baroclinic instability of an idealized tidal mixing front, J. Mar. Res. 70, 661 (2012).
  36. C. Chen and I. Kamenkovich, Effects of topography on baroclinic instability, J. Phys. Oceanogr. 43, 790 (2013).
  37. P. E. Isachsen, Baroclinic instability and the mesoscale eddy field around the Lofoten Basin, J. Geophys. Res. Oceans 120, 2884 (2015).
  38. J. Pedlosky, Baroclinic instability over topography: Unstable at any wave number, J. Mar. Res. 74, 1 (2016).
  39. G. E. Manucharyan, A. F. Thompson, and M. A. Spall, Eddy-Memory mode of multi-decadal variability in residual-mean ocean circulations with application to the Beaufort Gyre, J. Phys. Oceanogr. 47, 855 (2017).
  40. R. D. Hetland, Suppression of baroclinic instabilities in buoyancy-driven flow over sloping bathymetry, J. Phys. Oceanogr. 47, 49 (2017).
  41. M. Trodahl and P. E. Isachsen, Topographic influence on baroclinic instability and the mesoscale eddy field in the northern North Atlantic Ocean and the Nordic seas, J. Phys. Oceanogr. 48, 2593 (2018).
  42. Y. Wang and A. L. Stewart, Eddy dynamics over continental slopes under retrograde winds: Insights from a model inter-comparison, Ocean Modell. 121, 1 (2018).
  43. G. E. Manucharyan and P. E. Isachsen, Critical role of continental slopes in halocline and eddy dynamics of the Ekman-driven Beaufort Gyre, J. Geophys. Res. Oceans 124, 2679 (2019).
  44. S. Chen, C. Chen, and J. A. Lerczak, On baroclinic instability over continental shelves: Testing the utility of Eady-type models, J. Phys. Oceanogr. 50, 3 (2020).
  45. Y. Tanaka, Stability of a flow over bottom topography: A general condition and a linear analysis in a two-layer quasi-geostrophic model with a possible application to a Kuroshio meander, J. Geophys. Res. Oceans. 126, e2021JC017849 (2021).
  46. H. Wei and Y. Wang, Full-depth scalings for isopycnal eddy mixing across continental slopes under upwelling-favorable winds, J. Adv. Model. Earth. Syst. 13, e2021MS002498 (2021).
  47. H. Wei, Y. Wang, and J. Mak, Parameterizing eddy buoyancy fluxes across prograde shelf/slope fronts using a slope-aware GEOMETRIC closure, J. Phys. Oceanogr. 54, 359 (2024).
  48. A. Nummelin and P. E. Isachsen, Parameterizing mesoscale eddy buoyancy transport over sloping topography, ESS Open Archive 10.22541/essoar.168394750.04852652/v2 (2024).
  49. F. P. Bretherton, Baroclinic instability and the short wavelength cut-off in terms of potential vorticity, Q. J. Roy. Met. Soc. 92, 335 (1966).
  50. B. J. Hoskins, M. E. McIntyre, and A. W. Robertson, On the use and significance of isentropic potential vorticity maps, Q. J. Roy. Met. Soc. 111, 877 (1985b).
  51. E. Heifetz, C. H. Bishop, and P. Alpert, Counter-propagating Rossby waves in the barotropic Rayleigh model of shear instability, Q. J. Roy. Met. Soc. 125, 2835 (1999).
  52. N. Harnik and E. Heifetz, Relating overreflection and wave geometry to the counter-propagating Rossby wave perspective: Toward a deeper mechanistic understanding of shear instability, J. Atmos. Sci. 64, 2238 (2007).
  53. E. Heifetz and A. Guha, Normal form of synchronization and resonance between vorticity waves in shear flow instability, Phys. Rev. E 100, 043105 (2019).
  54. C. M. Bender, PT Symmetry: In Quantum And Classical Physics, 1st ed. (World Scientific, 2018).
  55. R. Zhang, H. Qin, and J. Xiao, PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability, J. Math. Phys. 61, 012101 (2020).
  56. T. W. David, P. Delplace, and A. Venaille, How do discrete symmetries shape the stability of geophysical flows?, Phys. Fluids 34, 056605 (2022).
  57. H. C. Davies and C. H. Bishop, Eady edge waves and rapid development, J. Atmos. Sci. 51, 1930 (1994).
  58. I. M. Held, Pseudomomentum and the orthogonality of modes in shear flows, J. Atmos. Sci. 42, 2280 (1985).
  59. E. Heifetz, N. Harnik, and T. Tamarin, Canonical hamiltonian representation of pseudoenergy in shear flows using counter-propagating rossby waves, Q. J. Roy. Met. Soc. 135, 2161 (2009).
  60. E. Heifetz and A. Guha, A generalized action-angle representation of wave interaction in stratified shear flows, J. Fluid Mech. 834, 220 (2017).
  61. S. Waterman and J. M. Lilly, Geometric decomposition of eddy feedbacks in barotropic systems, J. Phys. Oceanogr. 45, 1009 (2015).
  62. C. Meng and Z. Guo, Vorticity wave interaction and exceptional points in shear flow instabilities (2023), arXiv:2303.11842 [physics.flu-dyn] .
  63. E. Heifetz and J. Methven, Relating optimal growth to counter-propagating Rossby Waves in shear instability, Phys. Fluids 17, 064107 (2005).
  64. I. N. James, Suppression of baroclinic instability in horizontally sheared flows, J. Atmos. Sci. 44, 3710 (1987).
  65. N. Nakamura, An illustrative model of instabilities in meridionally and vertically sheared flows, J. Atmos. Sci. 50, 357 (1993b).
  66. N. Harnik, D. G. Dritschel, and E. Heifetz, On the equilibration of asymmetric barotropic instability, Q. J. Roy. Met. Soc. 140, 2444 (2014).
  67. R. A. Cairns, The role of negative energy waves in some instabilities of parallel flows, J. Fluid Mech. 92, 1 (1979).
  68. A. Mostafazadeh, Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 43, 205 (2002).

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

Sign Up for Free