Final Report Summary - CONVECT (A new approach to modelling turbulent planetary circulations)
One of the key scientific questions in atmospheric and oceanic fluid dynamics concerns the origin and maintenance of global circulation patterns and specifically high-speed localised streams of 'jets' (Rhines, 1975; Rhines, 1994; Dritschel & McIntyre, 2008; Dritschel & Scott, 2011). The Earth's atmosphere exhibits not only a 'jet stream' but also other similar flow features like the 'subtropical jet'. Both these phenomena have a major influence on weather, since low pressure systems responsible for most severe weather are guided along these jets. The oceans also exhibit jets, which are indeed much more numerous, including not only the gulf stream, Kuroshio and other major currents, but also a plethora of smaller jets (Maximenko et al, 2008). On the other hand, all major gas giant planets are carpeted by a pattern of alternating jets (Marcus, 1993). Jets are important because they transport heat, chemicals and biological tracers over long distances.
In addition, though this aspect is much less well studied, these flows certainly exhibit a wide array of coherent eddies or vortices. A low pressure 'cyclone' is an example from the Earth's atmosphere and the oceans are also now known to be 'eddy dominated', i.e. filled with vortices at all depths and over a wide range of scales, extending from less than a kilometre to a couple of hundred kilometres (Ebbesmeyer et al, 1986). Moreover, the great red spot on Jupiter is but one of many known intense vortices in the atmospheres of the gas giant planets. Vortices are also important for tracer transport, as they can efficiently trap tracers and transport them over long distances.
Jets and vortices are, to this day, poorly understood because they result from highly nonlinear fluid-dynamical processes. This renders any mathematical analysis immensely challenging, and, at present, we have no commonly accepted theory for the origin and maintenance of jets and vortices. This project aimed to partially redress this issue by studying simplified models of jets and vortices in order to better understand their fundamental causes and quantify their properties as a function of key external parameters.
The first part of the project involved developing a novel computational method capable of efficiently studying jets and vortices under a wide range of external influences, such as forcing, damping, rotation, stratification etc. A major breakthrough was achieved in this direction, allowing for the simulation of complex geophysical fluid flows like turbulence at unprecedented resolution (Fontane & Dritschel, 2009; Dritschel & Fontane, 2010a; Dritschel & Fontane, 2010b). We firstly applied this method to a classical problem in geophysical turbulence, namely the evolution of two-dimensional flows subject to 'narrow-band' forcing, which was near a particular wavelength. This problem had its origins 40 years ago, in Kraichnan (1967), who proposed a now widely accepted theory for the inverse cascade of energy that was spreading to large scales. Kraichnan argued that such a cascade would proceed by local spectral interactions, stepping from one mode to the next at larger scales and implying an energy spectrum, as a function of wavenumber k, with a minus five thirds power law. Our results challenged Kraichnan's theory. We found that the cascade did not occur only locally, but also by long-range interactions arising from a population of vortices that spontaneously emerged. The vortices, furthermore, steepened the energy spectrum to a minus two power law, a difference that we verified to be significant (Fontane, Scott & Dritschel, 2011).
The novel computational method was subsequently applied to study the emergence of jets and vortices in flows with a background planetary vorticity gradient, induced by a planet's rotation. We firstly quantified the way in which turbulence sharpened a pre-existing broad jet as a function of:
1. the level of background turbulence, and
2. the Rossby deformation length, i.e. the scale at which rotation and stratification had comparable effects.
We then studied how jets and vortices emerged in forced flows starting from rest. This was also a classical problem in the field, with many relevant previous studies. Yet, to date, virtually all studies considered narrow-band spectral forcing, which was unrealistic for the Earth's and other planetary atmospheres. We contrasted narrow-band and broad-band forcing, the latter created by adding small-scale dipolar vortices, which represented a simple model of the effects of unresolved convection, and showed that the latter led to much more intense and irregular jets. Unexpectedly, we found that, even for very weak forcing and damping, the emerging jets were not straight, but wavy. The waves were induced by small-scale vortices crossing through the jets during microwave breaking events. The vortices then remained trapped near the jets, leaving the potential vorticity weakly non-monotonic. Two further papers regarding this issue were in preparation by the time of the project completion.
The list of relevant publications and documentation is as follows:
1. Dritschel, D.G. and Fontane, J., 'The HyperCASL algorithm', IUTAM Symposium on Turbulence in the Atmosphere and Oceans, (Ed. David Dritschel), Springer (2010).
2. Dritschel, D.G. and Fontane, J., 'The combined lagrangian advection method', Journal of Computational Physics 229(14), 5408-5417 (2010).
3. Dritschel, D.G. and McIntyre, M. E., 'Multiple jets as PV staircases: the Phillips effect and the resilience of eddy-transport barriers', Journal of the Atmospheric Sciences 65, 855-874 (2008).
4. Dritschel, D.G. and Scott, R.K. 'Jet sharpening by turbulent mixing', Phil. Trans. Roy. Soc. A 369, 754-770 (2011).
5. Ebbesmeyer, C.C. Taft, B.A. McWilliams, J.C. Shen, C.Y. Riser, S.C. Rossby, H.T. Biscaye, P.E. and Ostlund, H.G. 'Detection, structure and origin of extreme anomalies in a western Atlantic oceanographic section', Journal of physical oceanography 16, 591-612 (1986).
6. Fontane, J. and Dritschel, D.G. 'The HyperCASL algorithm: a new approach to the numerical simulation of geophysical flows', Journal of Computational Physics 228(17), 6411-6425 (2009).
7. Fontane, J., Scott, R.K. and Dritschel, D.G. 'Vortical control of forced two-dimensional turbulence', Journal of Fluid Mechanics, submitted in 2011.
8. Kraichnan, R.H. 'Inertial ranges in two-dimensional turbulence', Physics of Fluids 10(7), 1417-1423 (1967).
9. Marcus, P.S. 'Jupiter's great red spot and other vortices', Annual Review of Astronomy and Astrophysics 31, 523-573 (1993).
10. Maximenko, N., Melnichenko, O., Niiler, P. and Sasaki, H., 'Stationary mesoscale jet-like features in the ocean', Geophysical Research Letters 35, L08603 (2008).
11. Rhines, P.B. 'Waves and turbulence on a beta-plane', Journal of Fluid Mechanics 69, 417-443 (1975).
12. Rhines, P. B., 'Jets', Chaos 4, 313-339 (1994).
In addition, though this aspect is much less well studied, these flows certainly exhibit a wide array of coherent eddies or vortices. A low pressure 'cyclone' is an example from the Earth's atmosphere and the oceans are also now known to be 'eddy dominated', i.e. filled with vortices at all depths and over a wide range of scales, extending from less than a kilometre to a couple of hundred kilometres (Ebbesmeyer et al, 1986). Moreover, the great red spot on Jupiter is but one of many known intense vortices in the atmospheres of the gas giant planets. Vortices are also important for tracer transport, as they can efficiently trap tracers and transport them over long distances.
Jets and vortices are, to this day, poorly understood because they result from highly nonlinear fluid-dynamical processes. This renders any mathematical analysis immensely challenging, and, at present, we have no commonly accepted theory for the origin and maintenance of jets and vortices. This project aimed to partially redress this issue by studying simplified models of jets and vortices in order to better understand their fundamental causes and quantify their properties as a function of key external parameters.
The first part of the project involved developing a novel computational method capable of efficiently studying jets and vortices under a wide range of external influences, such as forcing, damping, rotation, stratification etc. A major breakthrough was achieved in this direction, allowing for the simulation of complex geophysical fluid flows like turbulence at unprecedented resolution (Fontane & Dritschel, 2009; Dritschel & Fontane, 2010a; Dritschel & Fontane, 2010b). We firstly applied this method to a classical problem in geophysical turbulence, namely the evolution of two-dimensional flows subject to 'narrow-band' forcing, which was near a particular wavelength. This problem had its origins 40 years ago, in Kraichnan (1967), who proposed a now widely accepted theory for the inverse cascade of energy that was spreading to large scales. Kraichnan argued that such a cascade would proceed by local spectral interactions, stepping from one mode to the next at larger scales and implying an energy spectrum, as a function of wavenumber k, with a minus five thirds power law. Our results challenged Kraichnan's theory. We found that the cascade did not occur only locally, but also by long-range interactions arising from a population of vortices that spontaneously emerged. The vortices, furthermore, steepened the energy spectrum to a minus two power law, a difference that we verified to be significant (Fontane, Scott & Dritschel, 2011).
The novel computational method was subsequently applied to study the emergence of jets and vortices in flows with a background planetary vorticity gradient, induced by a planet's rotation. We firstly quantified the way in which turbulence sharpened a pre-existing broad jet as a function of:
1. the level of background turbulence, and
2. the Rossby deformation length, i.e. the scale at which rotation and stratification had comparable effects.
We then studied how jets and vortices emerged in forced flows starting from rest. This was also a classical problem in the field, with many relevant previous studies. Yet, to date, virtually all studies considered narrow-band spectral forcing, which was unrealistic for the Earth's and other planetary atmospheres. We contrasted narrow-band and broad-band forcing, the latter created by adding small-scale dipolar vortices, which represented a simple model of the effects of unresolved convection, and showed that the latter led to much more intense and irregular jets. Unexpectedly, we found that, even for very weak forcing and damping, the emerging jets were not straight, but wavy. The waves were induced by small-scale vortices crossing through the jets during microwave breaking events. The vortices then remained trapped near the jets, leaving the potential vorticity weakly non-monotonic. Two further papers regarding this issue were in preparation by the time of the project completion.
The list of relevant publications and documentation is as follows:
1. Dritschel, D.G. and Fontane, J., 'The HyperCASL algorithm', IUTAM Symposium on Turbulence in the Atmosphere and Oceans, (Ed. David Dritschel), Springer (2010).
2. Dritschel, D.G. and Fontane, J., 'The combined lagrangian advection method', Journal of Computational Physics 229(14), 5408-5417 (2010).
3. Dritschel, D.G. and McIntyre, M. E., 'Multiple jets as PV staircases: the Phillips effect and the resilience of eddy-transport barriers', Journal of the Atmospheric Sciences 65, 855-874 (2008).
4. Dritschel, D.G. and Scott, R.K. 'Jet sharpening by turbulent mixing', Phil. Trans. Roy. Soc. A 369, 754-770 (2011).
5. Ebbesmeyer, C.C. Taft, B.A. McWilliams, J.C. Shen, C.Y. Riser, S.C. Rossby, H.T. Biscaye, P.E. and Ostlund, H.G. 'Detection, structure and origin of extreme anomalies in a western Atlantic oceanographic section', Journal of physical oceanography 16, 591-612 (1986).
6. Fontane, J. and Dritschel, D.G. 'The HyperCASL algorithm: a new approach to the numerical simulation of geophysical flows', Journal of Computational Physics 228(17), 6411-6425 (2009).
7. Fontane, J., Scott, R.K. and Dritschel, D.G. 'Vortical control of forced two-dimensional turbulence', Journal of Fluid Mechanics, submitted in 2011.
8. Kraichnan, R.H. 'Inertial ranges in two-dimensional turbulence', Physics of Fluids 10(7), 1417-1423 (1967).
9. Marcus, P.S. 'Jupiter's great red spot and other vortices', Annual Review of Astronomy and Astrophysics 31, 523-573 (1993).
10. Maximenko, N., Melnichenko, O., Niiler, P. and Sasaki, H., 'Stationary mesoscale jet-like features in the ocean', Geophysical Research Letters 35, L08603 (2008).
11. Rhines, P.B. 'Waves and turbulence on a beta-plane', Journal of Fluid Mechanics 69, 417-443 (1975).
12. Rhines, P. B., 'Jets', Chaos 4, 313-339 (1994).