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A generalized mathematical model of geostrophic adjustment and frontogenesis: uniform potential vorticity

Published online by Cambridge University Press:  06 November 2013

Callum J. Shakespeare  and
J. R. Taylor
Callum J. Shakespeare
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
J. R. Taylor*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

Fronts, or regions with strong horizontal density gradients, are ubiquitous and dynamically important features in the ocean and atmosphere. In the atmosphere, fronts are associated with some of the most severe weather events, while in the ocean, fronts are associated with enhanced turbulence, water mass transformation and biological activity. Here, we examine the dynamics involved in the formation of fronts, or frontogenesis, in detail using a generalized mathematical framework. This extends previous work which has generally revolved around two limiting cases: fronts generated through forcing due to a convergent large-scale flow, and fronts generated spontaneously during the geostrophic adjustment of an initially unbalanced flow. Here, we introduce a new generalized momentum coordinate to simultaneously describe forced and spontaneous frontogenesis. The nonlinear, inviscid, Boussinesq, hydrostatic governing equations for uniform PV flow are solved for arbitrary Rossby and Froude number. The solution is then examined in three distinct cases. Firstly, for a zero potential vorticity (PV) flow bounded by rigid lids, a general solution is derived for the transient response of the fluid to an arbitrary initial mass imbalance and deformation field. The deformation frontogenesis solution of Hoskins & Bretherton (J. Atmos. Sci., vol. 29, 1972, pp. 11–37) and the mass imbalance solution of Blumen (J. Phys. Oceanogr., vol. 30, 2000, pp. 31–39) emerge as two limits of this general solution. Secondly, the problem of geostrophic adjustment of an initial mass imbalance (no deformation field) is considered for uniform PV flow bounded by rigid lids. The general solution is derived, composed of an adjusted state and a transient component describing the propagation of inertia–gravity waves. The criteria for the occurrence of a frontal discontinuity is determined in terms of the Rossby and Froude numbers. The uniform PV solution reduces identically to the zero PV solution of Blumen in the limit of vanishing background stratification. Thirdly, we examine the more general case of uniform PV flow with a deformation field and either balanced or unbalanced initial conditions. In this case the solution is composed of a time-varying mean state – matching the Hoskins & Bretherton solution in the limit of small strain – and an inertia gravity wave field, the dynamics of which are examined in detail. Our analysis provides a unifying framework capable of describing frontal formation and geostrophic adjustment in a wide variety of settings.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Geophysical and Geological Flows: Ocean circulation Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 736 , 10 December 2013 , pp. 366 - 413
Copyright
©2013 Cambridge University Press 

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References

Blumen, W. 1972 Geostrophic adjustment. Rev. Geophys. Space Phys. 10, 485–528.CrossRefGoogle Scholar
Blumen, W. 1990 A semigeostrophic Eady-wave frontal model incorporating momentum diffusion. Part 1. Model and solutions. J. Atmos. Sci. 47, 2890–2902.2.0.CO;2>CrossRefGoogle Scholar
Blumen, W. 1997 A model of inertial oscillations with deformation frontogenesis. J. Atmos. Sci. 54, 2681–2692.2.0.CO;2>CrossRefGoogle Scholar
Blumen, W. 2000 Inertial oscillations and frontogenesis in a zero potential vorticity model. J. Phys. Oceanogr. 30, 31–39.2.0.CO;2>CrossRefGoogle Scholar
Blumen, W., Gamage, N., Grossman, R. L., Lemone, M. A. & Miller, L. J. 1996 The low-level structure and evolution of a dry arctic front over the central United States. Part 2. Comparison with theory. Mon. Weath. Rev. 124, 1676–1691.2.0.CO;2>CrossRefGoogle Scholar
Blumen, W. & Williams, R. T. 2001 Unbalanced frontogenesis. Part 1. Zero potential vorticity. J. Atmos. Sci. 58, 2180–2195.2.0.CO;2>CrossRefGoogle Scholar
Blumen, W. & Wu, R. 1995 Geostrophic adjustment: frontogenesis and energy conversion. J. Phys. Oceanogr. 25, 428–438.2.0.CO;2>CrossRefGoogle Scholar
Boccaletti, G., Ferrari, R. & Fox-Kemper, B. 2007 Mixed layer instabilities and restratification. J. Phys. Oceanogr. 37, 2228–2250.CrossRefGoogle Scholar
Bouchut, F., Sommer, J. & Zeitlin, V. 2004 Frontal geostrophic adjustment and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 2. High-resolution numerical simulations. J. Fluid Mech. 514, 35–63.CrossRefGoogle Scholar
Bretherton, F. P. & Garrett, C. J. R. 1969 Wave trains in inhomogeneous moving media. Proc. R. Soc. A 302, 529–554.Google Scholar
Buhler, O. & McIntyre, M. E. 2005 Wave capture and wave-vortex duality. J. Fluid Mech. 534, 67–95.CrossRefGoogle Scholar
Cullen, M. & Purser, R. 1984 An extended theory of semigeostrophic frontogenesis. J. Atmos. Sci. 41, 1477–1497.2.0.CO;2>CrossRefGoogle Scholar
Davies, H. C. & Muller, J. C. 1988 Detailed description of deformation-induced semi-geostrophic frontogenesis. Q. J. R. Meteorol. Soc. 114, 1201–1219.Google Scholar
Eliassen, A. 1959 On the formation of fronts in the atmosphere. In The Atmosphere and Sea in Motion (ed. Bolin, B.), Rockefeller Institute Press.Google Scholar
Eliassen, A. 1962 On the vertical circulation in frontal zones. Geophys. Publ. 24 (4), 147–160.Google Scholar
Ferrari, R. 2011 A frontal challenge for climate models. Science 332 (6027), 316–317.CrossRefGoogle ScholarPubMed
Hoskins, B. J. 1982 The mathematical theory of frontogenesis. Annu. Rev. Fluid Mech. 14, 131–151.CrossRefGoogle Scholar
Hoskins, B. J. & Bretherton, F. P. 1972 Atmospheric frontogenesis models: mathematical formulation and solution. J. Atmos. Sci. 29, 11–37.2.0.CO;2>CrossRefGoogle Scholar
Koshyk, J. N. & Cho, H. 1992 Dynamics of a mature front in a uniform potential vorticity semigeostrophic model. J. Atmos. Sci. 49 (6), 497–510.2.0.CO;2>CrossRefGoogle Scholar
Mahadevan, A., D’Asaro, E., Lee, C. & Perry, M. J. 2012 Eddy-driven stratification initiates North Atlantic spring phytoplankton blooms. Science 337 (6090), 54–58.CrossRefGoogle ScholarPubMed
McWilliams, J. C. & Molemaker, M. J. 2009 Linear fluctuation growth during frontogenesis. J. Phys. Oceanogr. 39, 3111–3129.CrossRefGoogle Scholar
McWilliams, J. C. & Molemaker, M. J. 2011 Baroclinic frontal arrest: a sequel to unstable frontogenesis. J. Phys. Oceanogr. 41, 601–619.CrossRefGoogle Scholar
Neves, A. 1996 Unbalanced frontogenesis with constant potential vorticity. Master’s thesis, Naval Postgraduate School, Monterey, California.Google Scholar
Ostdiek, V. & Blumen, W. 1997 A dynamic trio: inertial oscillation, deformation frontogenesis, and the Ekman–Taylor boundary layer. J. Atmos. Sci. 54, 1490–1502.2.0.CO;2>CrossRefGoogle Scholar
Ou, H. W. 1984 Geostrophic adjustment: a mechanism for frontogenesis. J. Phys. Oceanogr. 14, 994–1000.2.0.CO;2>CrossRefGoogle Scholar
Plougonven, R. & Zeitlin, V. 2005 Lagrangian approach to geostrophic adjustment of frontal anomalies in stratified fluid. Geophys. Astrophys. Fluid Dyn. 9, 101–135.CrossRefGoogle Scholar
Rossby, C. G. 1938 On the mutual adjustment of pressure and velocity distributions in certain simple current systems. Part 2. J. Mar. Res. 1, 239–263.CrossRefGoogle Scholar
Sawyer, J. S. 1956 On the vertical circulation at meteorological fronts and its relation to frontogenesis. Proc. R. Soc. Lond. Ser. A 234, 346–362.Google Scholar
Snyder, C., Skamarock, W. & Rotunno, R. 1993 Frontal dynamics near and following frontal collapse. J. Atmos. Sci. 50, 3194–3211.2.0.CO;2>CrossRefGoogle Scholar
Tandon, A. & Garrett, C. 1994 Mixed layer restratification due to a horizontal density gradient. J. Phys. Oceanogr. 24 (6), 1419–1424.2.0.CO;2>CrossRefGoogle Scholar
Taylor, J. R. & Ferrari, R. 2011 Ocean fronts trigger high latitude phytoplankton blooms. Geophys. Res. Lett. 38, L23601.CrossRefGoogle Scholar
Thomas, L. N. 2012 On the effects of frontogenetic strain on symmetric instability and inertia–gravity waves. J. Fluid Mech. 711, 620–640.CrossRefGoogle Scholar
Thomas, L. N. & Joyce, T. M. 2010 Subduction on the northern and southern flanks of the gulf stream. J. Phys. Oceanogr. 40 (2), 429–438.CrossRefGoogle Scholar
Thomas, L. N., Tandon, A. & Mahadevan, A. 2008 Submesoscale processes and dynamics. In Geophysical Monograph Series 177: Ocean Modelling in an Eddying Regime. American Geophysical Union.Google Scholar
Thomas, L. N., Taylor, J. R., Ferrari, R. & Joyce, T. M. 2013 Symmetric instability in the gulf stream. Deep-Sea Res. II 91, 96–110.Google Scholar
Twigg, R. D. & Bannon, P. R. 1998 Frontal equilibration by frictional processes. J. Atmos. Sci. 55, 1084–1087.2.0.CO;2>CrossRefGoogle Scholar
Wu, R. & Blumen, W. 1995 Geostrophic adjustment of a zero potential vorticity flow initiated by a mass imbalance. J. Phys. Oceanogr. 25, 439–445.2.0.CO;2>CrossRefGoogle Scholar