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Transient energy growth in the ageostrophic Eady model

Published online by Cambridge University Press:  27 December 2019

Varvara E. Zemskova Open the ORCID record for Varvara E. Zemskova [Opens in a new window] ,
Pierre-Yves Passaggia  and
Brian L. White
Varvara E. Zemskova*
Affiliation:
Department of Physics, University of Toronto, Toronto, ON, M5R 2M8Canada
Pierre-Yves Passaggia
Affiliation:
Department of Physics, University of Toronto, Toronto, ON, M5R 2M8Canada University of Orléans, INSA-CVL, PRISME, EA 4229, 45072, Orléans, France
Brian L. White
Affiliation:
Department of Physics, University of Toronto, Toronto, ON, M5R 2M8Canada
*
†Email address for correspondence: barbara.zemskova@utoronto.ca
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Abstract

The problem of optimal initial disturbances in thermal wind shear is revisited and extended to include non-hydrostatic effects. This systematic study compares transient and modal growth rates of submesoscale instabilities over a large range of zonal and meridional wavenumbers, aspect ratios and different Richardson number regimes. Selection criteria were derived to remove spurious and unresolved instability modes that arise from the eigenvalue problem and we generalize the study of the hydrostatic Eady problem by Heifetz & Farrell (J. Atmos. Sci., vol. 60, 2003; J. Atmos. Sci., vol. 64 (12), 2007, pp. 4366–4382; Q. J. R. Meteorol. Soc., vol. 134 (635), 2008, pp. 1627–1633) to thin fronts, characterized by large aspect ratios. Such fronts are commonly found at the early stages of frontogenesis, for example, in the ocean mesoscale eddies and near the eye wall of hurricanes. In particular, we show that transient energy growth rates are up to two orders of magnitude larger than modal counterparts for a wide range of Richardson number and that the effects of transient energy gain become even greater when non-hydrostatic effects become important and/or for large Richardson numbers. This study also compares the dominant energy pathways contributing to the energy growth at short and long times. For symmetric modes, we recover the inertia–gravity instability described in Xu et al. (J. Atmos. Sci., vol. 64 (6), 2007, pp. 1764–1781). These mechanisms are shown to be the most powerful mediator of vertical transport when compared with the fastest growing baroclinic and symmetric modes. These results highlight the importance of transient processes in the ocean and the atmosphere.

JFM classification

Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 885 , 25 February 2020 , A29
Copyright
© The Author(s), 2019. Published by Cambridge University Press

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References

von Appen, W.-J., Wekerle, C., Hehemann, L., Schourup-Kristensen, V., Konrad, C. & Iversen, M. H. 2018 Observations of a submesoscale cyclonic filament in the marginal ice zone. Geophys. Res. Lett. 45 (12), 6141–6149.Google Scholar
Arobone, E. & Sarkar, S. 2015 Effects of three-dimensionality on instability and turbulence in a frontal zone. J. Fluid Mech. 784, 252–273.CrossRefGoogle Scholar
Bakas, N. A. & Farrell, B. F. 2009a Gravity waves in a horizontal shear flow. Part I: growth mechanisms in the absence of potential vorticity perturbations. J. Phys. Oceanogr. 39 (3), 481–496.CrossRefGoogle Scholar
Bakas, N. A. & Farrell, B. F. 2009b Gravity waves in a horizontal shear flow. Part II: interaction between gravity waves and potential vorticity perturbations. J. Phys. Oceanogr. 39 (3), 497–511.CrossRefGoogle Scholar
Boccaletti, G., Ferrari, R. & Fox-Kemper, B. 2007 Mixed layer instabilities and restratification. J. Phys. Oceanogr. 37 (9), 2228–2250.CrossRefGoogle Scholar
Brandt, L. 2014 The lift-up effect: the linear mechanism behind transition and turbulence in shear flows. Eur. J. Mech. (B/Fluids) 47, 80–96.CrossRefGoogle Scholar
Brannigan, L., Marshall, D.P., Naveira Garabato, A.C., Nurser, A.J.G. & Kaiser, J. 2017 Submesoscale instabilities in mesoscale eddies. J. Phys. Ocean. 47 (12), 3061–3085.CrossRefGoogle Scholar
Callies, J., Ferrari, R., Klymak, J. M. & Gula, J. 2015 Seasonality in submesoscale turbulence. Nat. Comm. 6, 6862.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 1961 Hydromagnetic and Hydrodynamic Stability. Clarendon.Google Scholar
Drobinski, P. & Foster, R. C. 2003 On the origin of near-surface streaks in the neutrally-stratified planetary boundary layer. Boundary-Layer Meteorol. 108 (2), 247–256.CrossRefGoogle Scholar
Eady, E. T. 1949 Long waves and cyclone waves. Tellus 1 (3), 33–52.CrossRefGoogle Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18 (4), 487–488.CrossRefGoogle Scholar
Ellis, R. & Businger, S. 2010 Helical circulations in the typhoon boundary layer. J. Geophys. Res. 115 (D6), D06205.CrossRefGoogle Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31 (8), 2093–2102.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993a Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5 (6), 1390–1400.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993b Stochastic dynamics of baroclinic waves. J. Atmos. Sci. 50 (24), 4044–4057.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993c Stochastic forcing of perturbation variance in unbounded shear and deformation flows. J. Atmos. Sci. 50 (2), 200–211.2.0.CO;2>CrossRefGoogle Scholar
Foster, R. 2013 Signature of large aspect ratio roll vortices in synthetic aperture radar images of tropical cyclones. Oceanography 26 (2), 58–67.CrossRefGoogle Scholar
Gardner, D. R., Trogdon, S. A. & Douglass, R. W. 1989 A modified tau spectral method that eliminates spurious eigenvalues. J. Comput. Phys. 80 (1), 137–167.CrossRefGoogle Scholar
Gary, J. & Helgason, R. 1970 A matrix method for ordinary differential eigenvalue problems. J. Comput. Phys. 5 (2), 169–187.CrossRefGoogle Scholar
Gnanadesikan, A., Slater, R. D., Swathi, P. & Vallis, G. K. 2005 The energetics of ocean heat transport. J. Clim. 18 (14), 2604–2616.CrossRefGoogle Scholar
Grisouard, N. 2018 Extraction of potential energy from geostrophic fronts by inertial-symmetric instabilities. J. Phys. Oceanogr. 48 (5), 1033–1051.CrossRefGoogle Scholar
Grisouard, N. & Thomas, L. N. 2015 Critical and near-critical reflections of near-inertial waves off the sea surface at ocean fronts. J. Fluid Mech. 765, 273–302.CrossRefGoogle Scholar
Grisouard, N. & Thomas, L. N. 2016 Energy exchanges between density fronts and near-inertial waves reflecting off the ocean surface. J. Phys. Oceanogr. 46 (2), 501–516.CrossRefGoogle Scholar
Heifetz, E. & Farrell, B. 2003 Generalized stability of nongeostrophic baroclinic shear flow. Part I: large Richardson number regime. J. Atmos. Sci. 60, 2083–2100.2.0.CO;2>CrossRefGoogle Scholar
Heifetz, E. & Farrell, B. F. 2007 Generalized stability of nongeostrophic baroclinic shear flow. Part II: intermediate Richardson number regime. J. Atmos. Sci. 64 (12), 4366–4382.CrossRefGoogle Scholar
Heifetz, E. & Farrell, B. F. 2008 Non-normal growth in symmetric shear flow. Q. J. R. Meteorol. Soc. 134 (635), 1627–1633.CrossRefGoogle Scholar
Lorenz, E. N. 1955 Available potential energy and the maintenance of the general circulation. Tellus 7 (2), 157–167.CrossRefGoogle Scholar
Manning, M. L., Bamieh, B. & Carlson, J.2007 Descriptor approach for eliminating spurious eigenvalues in hydrodynamic equations. arXiv:0705.1542.Google Scholar
Molemaker, M. J., McWilliams, J. C. & Yavneh, I. 2005 Baroclinic instability and loss of balance. J. Phys. Oceanogr. 35 (9), 1505–1517.CrossRefGoogle Scholar
Morrison, I., Businger, S., Marks, F., Dodge, P. & Businger, J. A. 2005 An observational case for the prevalence of roll vortices in the hurricane boundary layer. J. Atmos. Sci. 62 (8), 2662–2673.CrossRefGoogle Scholar
Nakamura, N. 1988 Scale selection of baroclinic instability – effects of stratification and nongeostrophy. J. Atmos. Sci. 45 (21), 3253–3268.2.0.CO;2>CrossRefGoogle Scholar
Nolan, D. S., Dahl, N. A., Bryan, G. H. & Rotunno, R. 2017 Tornado vortex structure, intensity, and surface wind gusts in large-eddy simulations with fully developed turbulence. J. Atmos. Sci. 74 (5), 1573–1597.CrossRefGoogle Scholar
Omand, M. M., DAsaro, E. A., Lee, C. M., Perry, M. J., Briggs, N., Cetinić, I. & Mahadevan, A. 2015 Eddy-driven subduction exports particulate organic carbon from the spring bloom. Science 348 (6231), 222–225.CrossRefGoogle ScholarPubMed
Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I. A perfect liquid. Proc. R. Irish Acad. Sec. A 27, 9–68.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50 (4), 689–703.CrossRefGoogle Scholar
Park, J., Billant, P. & Baik, J.-J. 2017 Instabilities and transient growth of the stratified Taylor–Couette flow in a Rayleigh-unstable regime. J. Fluid Mech. 822, 80–108.CrossRefGoogle Scholar
Passaggia, P.-Y. & Ehrenstein, U. 2013 Adjoint based optimization and control of a separated boundary-layer flow. Eur. J. Mech. (B/Fluids) 41, 169–177.CrossRefGoogle Scholar
Passaggia, P.-Y., Meunier, P. & Le Dizès, S. 2014 Response of a stratified boundary layer on a tilted wall to surface undulations. J. Fluid Mech. 751, 663–684.CrossRefGoogle Scholar
Passaggia, P.-Y., Scotti, A. & White, B. 2017 Transition and turbulence in horizontal convection: linear stability analysis. J. Fluid Mech. 821, 31–58.CrossRefGoogle Scholar
Ramachandran, S., Tandon, A., Mackinnon, J., Lucas, A. J., Pinkel, R., Waterhouse, A. F., Nash, J., Shroyer, E., Mahadevan, A., Weller, R. A. et al. 2018 Submesoscale processes at shallow salinity fronts in the Bay of Bengal: observations during the winter monsoon. J. Phys. Oceanogr. 48 (3), 479–509.CrossRefGoogle Scholar
Sarkar, S., Pham, H. T., Ramachandran, S., Nash, J. D., Tandon, A., Buckley, J., Lotliker, A. A. & Omand, M. M. 2016 The interplay between submesoscale instabilities and turbulence in the surface layer of the bay of bengal. Oceanography 29 (2), 146–157.CrossRefGoogle Scholar
Schmid, P. J. & Brandt, L. 2014 Analysis of fluid systems: stability, receptivity, sensitivity: lecture notes from the FLOW-NORDITA summer school on advanced instability methods for complex flows, Stockholm, Sweden, 2013. Appl. Mech. Rev. 66 (2), 024803.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2012 Stability and Transition in Shear Flows, vol. 142. Springer Science & Business Media.Google Scholar
Scotti, A. & Passaggia, P.-Y. 2019 Diagnosing diabatic effects on the available energy of stratified flows in inertial and non-inertial frames. J. Fluid Mech. 861, 608–642.CrossRefGoogle Scholar
Solberg, H. 1936 Le mouvement dinertie de latmosphere stable et son role dans la theorie des cyclones. In Proces-Verbaux des séances de lUnion International de Géodésie et Géophysique (IUGG), pp. 66–82. IUGG.Google Scholar
Stamper, M. A. & Taylor, J. R. 2017 The transition from symmetric to baroclinic instability in the Eady model. Ocean Dyn. 67 (1), 65–80.CrossRefGoogle Scholar
Stone, P. 1966 On non-geostrophic baroclinic stability. J. Atmos. Sci. 23, 390–400.2.0.CO;2>CrossRefGoogle Scholar
Stone, P. 1970 On non-geostrophic baroclinic stability: part II. J. Atmos. Sci. 27, 721–726.2.0.CO;2>CrossRefGoogle Scholar
Stone, P. 1971 Baroclinic stability under non-hydrostatic conditions. J. Fluid Mech. 45, 659–671.CrossRefGoogle Scholar
Taylor, J. R. & Ferrari, R. 2009 On the equilibration of a symmetrically unstable front via a secondary shear instability. J. Fluid Mech. 622, 103–113.CrossRefGoogle Scholar
Taylor, J. R. & Ferrari, R. 2010 Buoyancy and wind-driven convection at mixed layer density fronts. J. Phys. Oceanogr. 40 (6), 1222–1242.CrossRefGoogle Scholar
Taylor, J. R. & Ferrari, R. 2011 Shutdown of turbulent convection as a new criterion for the onset of spring phytoplankton blooms. Limnol. Oceanogr. 56 (6), 2293–2307.CrossRefGoogle Scholar
Thomas, L. N., Taylor, J. R., Ferrari, R. & Joyce, T. M. 2013 Symmetric instability in the gulf stream. Deep-Sea Res. 91, 96–110.Google Scholar
Vallis, G. K. 2017 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Vasavada, A. R. & Showman, A. P. 2005 Jovian atmospheric dynamics: an update after Galileo and Cassini. Rep. Prog. Phys. 68 (8), 1935.CrossRefGoogle Scholar
Walters, R. A. & Carey, G. F. 1983 Analysis of spurious oscillation modes for the shallow water and Navier–Stokes equations. Comput. Fluids 11 (1), 51–68.CrossRefGoogle Scholar
Wolfe, C., Cessi, P., McClean, J. & Maltrud, M. 2008 Vertical heat transport in eddying ocean models. Geophys. Res. Lett. 35 (23), L23605.CrossRefGoogle Scholar
Worsnop, R. P., Lundquist, J. K., Bryan, G. H., Damiani, R. & Musial, W. 2017 Gusts and shear within hurricane eyewalls can exceed offshore wind turbine design standards. Geophys. Res. Lett. 44 (12), 6413–6420.CrossRefGoogle Scholar
Xu, Q. 2007 Modal and nonmodal symmetric perturbations. Part I: completeness of normal modes and constructions of nonmodal solutions. J. Atmos. Sci. 64 (6), 1745–1763.CrossRefGoogle Scholar
Xu, Q., Lei, T. & Gao, S. 2007 Modal and nonmodal symmetric perturbations. Part II: nonmodal growths measured by total perturbation energy. J. Atmos. Sci. 64 (6), 1764–1781.CrossRefGoogle Scholar
Young, W. 1994 The subinertial mixed layer approximation. J. Phys. Oceanogr. 24 (8), 1812–1826.2.0.CO;2>CrossRefGoogle Scholar
Zemskova, V. E., White, B. L. & Scotti, A. 2015 Available potential energy and the general circulation: partitioning wind, buoyancy forcing, and diapycnal mixing. J. Phys. Oceanogr. 45 (6), 1510–1531.CrossRefGoogle Scholar