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A perturbation approach to understanding the effects of turbulence on frontogenesis

Published online by Cambridge University Press:  25 November 2019

Abigail S. Bodner
Department of Earth, Environmental and Planetary Sciences, Brown University, Providence,RI02912, USA
Baylor Fox-Kemper
Department of Earth, Environmental and Planetary Sciences, Brown University, Providence,RI02912, USA
Luke P. Van Roekel
Theoretical Division, Fluid Dynamics and Solid Mechanics, Los Alamos National Laboratory, Los Alamos, NM87545, USA
James C. McWilliams
Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles,CA90095-1565, USA
Peter P. Sullivan
National Center for Atmospheric Research, Boulder, CO80307-3000, USA
E-mail address:


Ocean fronts are an important submesoscale feature, yet frontogenesis theory often neglects turbulence – even parameterized turbulence – leaving theory lacking in comparison with observations and models. A perturbation analysis is used to include the effects of eddy viscosity and diffusivity as a first-order correction to existing strain-induced inviscid, adiabatic frontogenesis theory. A modified solution is obtained by using potential vorticity and surface conditions to quantify turbulent fluxes. It is found that horizontal viscosity and diffusivity tend to be readily frontolytic – reducing frontal tendency to negative values under weakly non-conservative perturbations and opposing or reversing front sharpening, whereas vertical viscosity and diffusivity tend to only weaken frontogenesis by slowing the rate of sharpening of the front even under strong perturbations. During late frontogenesis, vertical diffusivity enhances the rate of frontogenesis, although perturbation theory may be inaccurate at this stage. Surface quasi-geostrophic theory – neglecting all injected interior potential vorticity – is able to describe the first-order correction to the along-front velocity and ageostrophic overturning circulation in most cases. Furthermore, local conditions near the front maximum are sufficient to reconstruct the modified solution of both these fields.

JFM Papers
© 2019 Cambridge University Press

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Bachman, S. D., Fox-Kemper, B. & Bryan, F. O. 2015 A tracer-based inversion method for diagnosing eddy-induced diffusivity and advection. Ocean Model. 86, 114.CrossRefGoogle Scholar
Bachman, S. D., Fox-Kemper, B., Taylor, J. R. & Thomas, L. N. 2017 Parameterization of frontal symmetric instabilities. Part I. Theory for resolved fronts. Ocean Model. 109, 7295.CrossRefGoogle Scholar
Barcilon, V. 1998 On the barotropic ocean with bottom friction. J. Mar. Res. 56 (4), 731771.CrossRefGoogle Scholar
Barkan, R., Molemaker, J. M., Srinivasan, K., McWilliams, J. C. & D’Asaro, E. A. 2019 The role of horizontal divergence in submesoscale frontogenesis. J. Phys. Oceanogr 49 (6), 15931618.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 2013 Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer Science & Business Media.Google Scholar
Benthuysen, J. & Thomas, L. N. 2012 Friction and diapycnal mixing at a slope: boundary control of potential vorticity. J. Phys. Oceanogr. 42 (9), 15091523.CrossRefGoogle Scholar
Blumen, W. 1978 Uniform potential vorticity flow. Part I. Theory of wave interactions and two-dimensional turbulence. J. Atmos. Sci. 35 (5), 774783.2.0.CO;2>CrossRefGoogle Scholar
Bond, N. A. & Fleagle, R. G. 1985 Structure of a cold front over the ocean. Q. J. R. Meteorol. Soc. 111 (469), 739759.CrossRefGoogle Scholar
Boutle, I. A., Belcher, S. E. & Plant, R. S. 2015 Friction in mid-latitude cyclones: an Ekman-PV mechanism. Atmos. Sci. Lett. 16 (2), 103109.CrossRefGoogle Scholar
Bretherton, F. P. 1966 Critical layer instability in baroclinic flows. Q. J. R. Meteorol. Soc. 92 (393), 325334.CrossRefGoogle Scholar
Callies, J., Flierl, G., Ferrari, R. & Fox-Kemper, B. 2016 The role of mixed-layer instabilities in submesoscale turbulence. J. Fluid Mech. 788, 541.CrossRefGoogle Scholar
Capet, X., Klein, P., Hua, B. L., Lapeyre, G. & McWilliams, J. C. 2008a Surface kinetic energy transfer in surface quasi-geostrophic flows. J. Fluid Mech. 604, 165174.CrossRefGoogle Scholar
Capet, X., McWilliams, J. C., Molemaker, J. M. & Shchepetkin, A. F. 2008b Mesoscale to submesoscale transition in the California current system. Part I. Flow structure, eddy flux, and observational tests. J. Phys. Oceanogr. 38 (1), 2943.CrossRefGoogle Scholar
Capet, X., Roullet, G., Klein, P. & Maze, G. 2016 Intensification of upper-ocean submesoscale turbulence through Charney baroclinic instability. J. Phys. Oceanogr. 46 (11), 33653384.CrossRefGoogle Scholar
Charney, J. G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28 (6), 10871095.2.0.CO;2>CrossRefGoogle Scholar
Chavanne, C. P. & Klein, P. 2016 Quasigeostrophic diagnosis of mixed layer dynamics embedded in a mesoscale turbulent field. J. Phys. Oceanogr. 46 (1), 275287.CrossRefGoogle Scholar
Cooper, I. M., Thorpe, A. J. & Bishop, C. H. 1992 The role of diffusive effects on potential vorticity in fronts. Q. J. R. Meteorol. Soc. 118 (506), 629647.CrossRefGoogle Scholar
Crowe, M. N. & Taylor, J. R. 2018 The evolution of a front in turbulent thermal wind balance. Part 1. Theory. J. Fluid Mech. 850, 179211.CrossRefGoogle Scholar
Dijkstra, H. A. & Molemaker, J. M. 1997 Symmetry breaking and overturning oscillations in thermohaline-driven flows. J. Fluid Mech. 331, 169198.CrossRefGoogle Scholar
Eady, E. T. 1949 Long waves and cyclone waves. Tellus 1 (3), 3352.CrossRefGoogle Scholar
Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 41, 253282.CrossRefGoogle Scholar
Fox-Kemper, B., Danabasoglu, G., Ferrari, R., Griffies, S. M., Hallberg, R. W., Holland, M. M., Maltrud, M. E., Peacock, S. & Samuels, B. L. 2011 Parameterization of mixed layer eddies. Part III. Implementation and impact in global ocean climate simulations. Ocean Model. 39, 6178.CrossRefGoogle Scholar
Fox-Kemper, B. & Ferrari, R. 2009 An eddifying Parsons model. J. Phys. Oceanogr. 39 (12), 32163227.CrossRefGoogle Scholar
Fox-Kemper, B., Ferrari, R. & Hallberg, R. W. 2008 Parameterization of mixed layer eddies. Part I. Theory and diagnosis. J. Phys. Oceanogr. 38 (6), 11451165.CrossRefGoogle Scholar
Gent, P. R., McWilliams, J. C. & Snyder, C. 1994 Scaling analysis of curved fronts. Validity of the balance equations and semigeostrophy. J. Atmos. Sci. 51 (1), 160163.2.0.CO;2>CrossRefGoogle Scholar
Gill, A. E. 1982 Atmosphere-Ocean Dynamics (International Geophysics Series). Academic Press.Google Scholar
Griffies, S. M. 1998 The Gent–McWilliams skew flux. J. Phys. Oceanogr. 28 (5), 831841.2.0.CO;2>CrossRefGoogle Scholar
Håkansson, M. 2002 A two-dimensional numerical study of effects of vertical diffusion in frontal zones. Q. J. R. Meteorol. Soc. 128 (585), 24392467.CrossRefGoogle Scholar
Hamlington, P. E., Van Roekel, L. P., Fox-Kemper, B., Julien, K. & Chini, G. P. 2014 Langmuir–submesoscale interactions: descriptive analysis of multiscale frontal spindown simulations. J. Phys. Oceanogr. 44 (9), 22492272.CrossRefGoogle Scholar
Harnik, N. & Heifetz, E. 2007 Relating overreflection and wave geometry to the counterpropagating Rossby wave perspective: toward a deeper mechanistic understanding of shear instability. J. Atmos. Sci. 64 (7), 22382261.CrossRefGoogle Scholar
Haynes, P. H. & McIntyre, M. E. 1987 On the evolution of vorticity and potential vorticity in the presence of diabatic heating and frictional or other forces. J. Atmos. Sci. 44 (5), 828841.2.0.CO;2>CrossRefGoogle Scholar
Hoskins, B. J. 1975 The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci. 32 (2), 233242.2.0.CO;2>CrossRefGoogle Scholar
Hoskins, B. J. 1982 The mathematical theory of frontogenesis. Annu. Rev. Fluid Mech. 14 (1), 131151.CrossRefGoogle Scholar
Hoskins, B. J. 1991 Towards a PV-𝜃 view of the general circulation. Tellus 43 (4), 2736.Google Scholar
Hoskins, B. J. & Bretherton, F. P. 1972 Atmospheric frontogenesis models: mathematical formulation and solution. J. Atmos. Sci. 29 (1), 1137.2.0.CO;2>CrossRefGoogle Scholar
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteorol. Soc. 111 (470), 877946.CrossRefGoogle Scholar
Klein, P., Lapeyre, G., Roullet, G., Le Gentil, S. & Sasaki, H. 2011 Ocean turbulence at meso and submesoscales: connection between surface and interior dynamics. Geophys. Astrophys. Fluid Dyn. 105 (4–5), 421437.CrossRefGoogle Scholar
Kurgansky, M. V. & Pisnichenko, I. A. 2000 Modified ertels potential vorticity as a climate variable. J. Atmos. Sci. 57 (6), 822835.2.0.CO;2>CrossRefGoogle Scholar
LaCasce, J. H. & Mahadevan, A. 2006 Estimating subsurface horizontal and vertical velocities from sea-surface temperature. J. Mar. Res. 64 (5), 695721.CrossRefGoogle Scholar
LaCasce, J. H. & Wang, J. 2015 Estimating subsurface velocities from surface fields with idealized stratification. J. Phys. Oceanogr. 45 (9), 24242435.CrossRefGoogle Scholar
Lapeyre, G. & Klein, P. 2006 Dynamics of the upper oceanic layers in terms of surface quasigeostrophy theory. J. Phys. Oceanogr. 36 (2), 165176.CrossRefGoogle Scholar
Lapeyre, G., Klein, P. & Hua, B. L. 2006 Oceanic restratification forced by surface frontogenesis. J. Phys. Oceanogr. 36 (8), 15771590.CrossRefGoogle Scholar
Large, W. G., McWilliams, J. C. & Doney, S. C. 1994 Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization. Rev. Geophys. 32 (4), 363403.CrossRefGoogle Scholar
Li, Q., Reichl, B. G., Fox-Kemper, B., Adcroft, A. J., Belcher, S., Danabasoglu, G., Grant, A., Griffies, S. M., Hallberg, R. W., Hara, T. et al. 2019 Comparing ocean boundary vertical mixing schemes including Langmuir turbulence. J. Adv. Model. Earth Syst. (JAMES) (in press).CrossRefGoogle Scholar
Mahadevan, A. 2016 The impact of submesoscale physics on primary productivity of plankton. Annu. Rev. Mar. Sci. 8, 161184.CrossRefGoogle ScholarPubMed
Mahadevan, A. & Archer, D. 2000 Modeling the impact of fronts and mesoscale circulation on the nutrient supply and biogeochemistry of the upper ocean. J. Geophys. Res. 105 (C1), 12091225.CrossRefGoogle Scholar
Marshall, J. C. & Nurser, A. G. 1992 Fluid dynamics of oceanic thermocline ventilation. J. Phys. Oceanogr. 22 (6), 583595.2.0.CO;2>CrossRefGoogle Scholar
McWilliams, J. C. 2016 Submesoscale currents in the ocean. Proc. R. Soc. A 472, 20160117.CrossRefGoogle ScholarPubMed
McWilliams, J. C. 2017 Submesoscale surface fronts and filaments: secondary circulation, buoyancy flux, and frontogenesis. J. Fluid Mech. 823, 391432.CrossRefGoogle Scholar
McWilliams, J. C., Gula, J., Molemaker, J. M., Renault, L. & Shchepetkin, A. F. 2015 Filament frontogenesis by boundary layer turbulence. J. Phys. Oceanogr. 45 (8), 19882005.CrossRefGoogle Scholar
McWilliams, J. C., Molemaker, M. & Olafsdottir, E. 2009 Linear fluctuation growth during frontogenesis. J. Phys. Oceanogr. 39 (12), 31113129.CrossRefGoogle Scholar
McWilliams, J. C., Sullivan, P. P. & Moeng, C.-H. 1997 Langmuir turbulence in the ocean. J. Fluid Mech. 334, 130.CrossRefGoogle Scholar
Moeng, C.-H. 1984 A large-eddy-simulation model for the study of planetary boundary-layer turbulence. J. Atmos. Sci. 41 (13), 20522062.2.0.CO;2>CrossRefGoogle Scholar
Molemaker, J. M., McWilliams, J. C. & Capet, X. 2010 Balanced and unbalanced routes to dissipation in an equilibrated Eady flow. J. Fluid Mech. 654, 3563.CrossRefGoogle Scholar
Nagai, T., Tandon, A. & Rudnick, D. L. 2006 Two-dimensional ageostrophic secondary circulation at ocean fronts due to vertical mixing and large-scale deformation. J. Geophys. Res. 111, C09038.CrossRefGoogle Scholar
Nakamura, N. 1994 Nonlinear equilibration of two-dimensional Eady waves: simulations with viscous geostrophic momentum equations. J. Atmos. Sci. 51 (7), 10231035.2.0.CO;2>CrossRefGoogle Scholar
Nakamura, N. & Held, I. M. 1989 Nonlinear equilibration of two-dimensional Eady waves. J. Atmos. Sci. 46 (19), 30553064.2.0.CO;2>CrossRefGoogle Scholar
Olita, A., Capet, A., Claret, M., Mahadevan, A., Poulain, P. M., Ribotti, A., Ruiz, S., Tintoré, J., Tovar-Sánchez, A. & Pascual, A. 2017 Frontal dynamics boost primary production in the summer stratified Mediterranean sea. Ocean Dyn. 67 (6), 767782.Google Scholar
Parsons, A. T. 1969 A two-layer model of Gulf Stream separation. J. Fluid Mech. 39 (3), 511528.CrossRefGoogle Scholar
Pearson, B. & Fox-Kemper, B. 2018 Log-normal turbulence dissipation in global ocean models. Phys. Rev. Lett. 120 (9), 094501.CrossRefGoogle ScholarPubMed
Pedlosky, J. 1982 Geophysical Fluid Mechanics. Springer.Google Scholar
Pham, H. T. & Sarkar, S. 2018 Ageostrophic secondary circulation at a submesoscale front and the formation of gravity currents. J. Phys. Oceanogr. 48 (10), 25072529.CrossRefGoogle Scholar
Pollard, R. T. & Regier, L. A. 1992 Vorticity and vertical circulation at an ocean front. J. Phys. Oceanogr. 22 (6), 609625.2.0.CO;2>CrossRefGoogle Scholar
Renault, L., McWilliams, J. C. & Gula, J. 2018 Dampening of submesoscale currents by air–sea stress coupling in the californian upwelling system. Sci. Rep. 8 (1), 13388.Google ScholarPubMed
Rhines, P. B. 1986 Vorticity dynamics of the oceanic general circulation. Annu. Rev. Fluid Mech. 18 (1), 433497.CrossRefGoogle Scholar
Rotunno, R., Skamarock, W. C. & Snyder, C. 1994 An analysis of frontogenesis in numerical simulations of baroclinic waves. J. Atmos. Sci. 51 (23), 33733398.2.0.CO;2>CrossRefGoogle Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.Google Scholar
Shakespeare, C. J. & Taylor, J. R. 2013 A generalized mathematical model of geostrophic adjustment and frontogenesis: uniform potential vorticity. J. Fluid Mech. 736, 366413.CrossRefGoogle Scholar
Siedler, G., Griffies, S. M., Gould, J. & Church, J. 2013 Ocean Circulation and Climate: A 21st Century Perspective. Academic Press.Google Scholar
Smith, K. M., Hamlington, P. E. & Fox-Kemper, B. 2016 Effects of submesoscale turbulence on ocean tracers. J. Geophys. Res. 121 (1), 908933.CrossRefGoogle Scholar
Stamper, M. A., Taylor, J. R. & Fox-Kemper, B. 2018 The growth and saturation of submesoscale instabilities in the presence of a barotropic jet. J. Phys. Oceanogr 48 (11), 27792797.CrossRefGoogle Scholar
Sullivan, P. P. & McWilliams, J. C. 2018 Frontogenesis and frontal arrest of a dense filament in the oceanic surface boundary layer. J. Fluid Mech. 837, 341380.CrossRefGoogle Scholar
Suzuki, N., Fox-Kemper, B., Hamlington, P. E. & Van Roekel, L. P. 2016 Surface waves affect frontogenesis. J. Geophys. Res. 121 (5), 35973624; Gulf Oil Spill special section.Google Scholar
Taylor, J. R. & Ferrari, R. 2011 Ocean fronts trigger high latitude phytoplankton blooms. Geophys. Res. Lett. 38, L23601.CrossRefGoogle Scholar
Thomas, L. N. 2005 Destruction of potential vorticity by winds. J. Phys. Oceanogr. 35 (12), 24572466.CrossRefGoogle Scholar
Thomas, L. N. & Ferrari, R. 2008 Friction, frontogenesis, and the stratification of the surface mixed layer. J. Phys. Oceanogr. 38 (11), 25012518.CrossRefGoogle Scholar
Thompson, L. 2000 Ekman layers and two-dimensional frontogenesis in the upper ocean. J. Geophys. Res. 105 (C3), 64376451.CrossRefGoogle Scholar
Twigg, R. D. & Bannon, P. R. 1998 Frontal equilibration by frictional processes. J. Atmos. Sci. 55 (6), 10841087.2.0.CO;2>CrossRefGoogle Scholar
Wenegrat, J. O., Thomas, L. N., Gula, J. & McWilliams, J. C. 2018 Effects of the submesoscale on the potential vorticity budget of ocean mode waters. J. Phys. Oceanogr. 48 (9), 21412165.CrossRefGoogle Scholar
Xu, Q., Gu, W. & Gao, J. 1998 Baroclinic Eady wave and fronts. Part I. Viscous semigeostrophy and the impact of boundary condition. J. Atmos. Sci. 55 (24), 35983615.2.0.CO;2>CrossRefGoogle Scholar

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