CHAPTER 6 Planetary Solitary Waves

CHAPTER 6 Planetary solitary waves J.P. Boyd University of Michigan, Ann Arbor, MI, USA. Abstract This chapter describes solitary waves in the atmosphere and ocean with diameters from perhaps a hundred kilometers to scales larger than the earth: vortices embedded in a shear zone, Rossby solitons and equatorial Kelvin solitary waves among others. One theme is that while nonlinear coherent structures are readily identified in both observations and numerical models, it is difficult to determine when these have the balance of dispersion and nonlinear frontogenesis that is central to classical soliton theory, and also unclear whether the quest for this balance is illuminating. A second theme is that periodic trains of vortices should be regarded as solitary waves, but usually are not, because the vortices are dynamically isolated even though they are geographically close. A third theme is the generalization to radiating or weakly nonlocal solitary waves: all solitons slowly decay by viscosity (in nature if not always in theory), so it is foolish to exclude coherent structures that have in addition an even slower decay through radiation. No observation of a large nonlinear coherent structure has been universally accepted as a soliton large enough in scale to feel Coriolis forces. The reasons may have more to do with overly narrow concepts of a ‘solitary wave’ than to a lack of data. The Great Red Spot of Jupiter and Legeckis eddies in the terrestrial equatorial ocean are well-known vortices for which an appropriately broad notion of solitons is probably useful. In contrast, hurricanes, although nonlinear coherent structures, are so strongly forced and damped that it is these mechanisms and also the complex feedback from the cloud scale to the vortex scale and back that are rightly the focus, and not the ideas of soliton theory. 1 Introduction Persistent, spatially localized structures are a ubiquitous feature of the ocean and atmosphere and of the atmospheres of other planets. Vortices with diameters of a WIT Transactions on State of the Art in Science and Engineering, Vol 9, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/978-1-84564-157-3/06 126 Solitary Waves in Fluids couple of thousand kilometers or so are the heart of large-scale weather patterns in the midlatitudes. Gulf Stream rings are intense ocean vortices, lasting for many months, or even years, that are spawned by the Gulf Stream, the oceanic equivalent of hurricanes. The Great Red Spot of Jupiter is an anticyclonic vortex that has been spinning for at least three centuries, so vast that Earth would fit comfortably within it. Long-lived, spatially localized structures of large scale or planetary scale are clearly of great importance. Consequently, so much has been written about Rossby vortices and coherent structures, as catalogued in the books and reviews of Table 1, that it is impossible to give a complete treatment in a brief space. Indeed, an entire monograph has Table 1: Selected reviews, monographs and special issues on Rossby solitary waves and coherent vortices. Topics, always including Rossby Reference solitons and vortices Miles (1980) Solitary waves including Rossby solitons Malanotte Rizzoli (1982) Rossby solitons in channels Ho and Huerre (1984) Perturbed free shear layers McWilliams (1985) Small-scale (submesoscale) ocean vortices Maslowe (1986) Critical layers in shear flows Flierl (1987) Ocean vortices Boyd (1989a) Polycnoidal waves; Rossby solitons Nihoul and Jamart (1989) Conference proceedings on ocean vortices Hopfinger and van Heijst (1991) Laboratory experiments on solitons and modons Boyd (1991a) Nonlinear equatorial waves Boyd (1991b) Weakly nonlocal solitary waves Boyd and Haupt (1991) Polycnoidal waves Olson (1991) Observations of ocean vortices Sommeria et al. (1991) Laboratory experiments Meleshko and van Heijst (1992) Chaplygin and Lamb’s discovery of modons Nezlin and Snezhkin (1993) Monograph on Rossby vortices and solitons Nezlin and Sutyrin (1994) Vortices on gas giant planets Marcus (1993) Great Red Spot of Jupiter Chaos, vol. 4 (1994) Special issue on coherent structures Korotaev (1997) Weakly nonlocal Rossby vortex solitons Boyd (1998b) Nonlocal, radiating solitary waves – one chapter on Rossby solitons Ingersoll (1999, 2002) Good images of outer planet vortices Barcilon and Drazin (2001) Nonlinear vorticity waves Carton (2001) Ocean vortices Vasavada and Showman (2005) Jovian dynamics; Great Red Spot WIT Transactions on State of the Art in Science and Engineering, Vol 9, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Solitary Waves in Fluids 127 already been devoted to the subject (Nezlin and Snezhkin 1993). There are literally hundreds of interesting articles that cannnot be discussed here because of space limitations. Instead, we will concentrate on a handful of major themes, important both to solitary wave theory in general and to Rossby solitons in particular. In a long review focused entirely on Rossby solitons, Paola Rizzoli wrote in 1982: ...unambiguous experimental evidence has not yet been produced for the exis- tence of solitary Rossby waves, either in the ocean or in the atmosphere. Nearly a quarter of a century later, this is still true: there are no universally-accepted observations of Rossby solitary waves. There are, however, many promising candi- dates as discussed below. In part, the lack of accepted soliton sightings is because the historical development of ‘soliton theory’ has led to definitions and mental models that are too narrow. In the next three sections, we elaborate on this theme. 2 Coherent structures and solitons Solitary waves are a subset of a much larger family of ‘coherent structures’. The formal definitions of each are as follows: Definition 1 (Coherent structure): a steadily translating, finite-amplitude, spatially localized disturbance that does not evolve in time, or evolves only slowly. Definition 2 (Solitary wave): a solitary wave, also known as a ‘soliton’, is a coherent structure that does not evolve in time because of a perfect balance between the steepening effects of nonlinearity and the spreading effects of wave dispersion. The crucial distinction is that in the classical theories of solitary waves, such as those of the Korteweg–de Vries equation, the longevity and persistence of amplitude, speed and shape are attributed to a specific mechanism: the balance between nonlinearity and dispersion. The complication is that vortices and other coherent structures can have many of the characteristics of solitary waves without sharing this steepening-balances-dispersion mechanism. Two-dimensional turbulence in an incompressible fluid without Coriolis forces is perhaps the clearest illustration. McWilliams (1984) showed that an initial state, which is filled with many small vortices of random amplitudes and phases will spontaneously evolve to a state with a few vortices, separated by large gulfs free of organized vortices, through repeated vortex mergers. Twenty years later, he observed (McWilliams 2004: p. 38): The most important change [in our understanding of turbulence] is the realization that the generic behaviour of turbulence is the development of coherent structures at high-Re values. The case for this was first and best made in 2D turbulence because of its computational affordability; however, this phenomenon has since been confirmed in many turbulent regimes, albeit with different typical flow patterns in different regimes. WIT Transactions on State of the Art in Science and Engineering, Vol 9, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 128 Solitary Waves in Fluids Two-dimensional incompressible turbulence is an extreme illustration of coherent- structures-that-are-not-solitary-waves because the equations of motion are com- pletely wave-free. There are no sound waves because the fluid is incompressible, no gravity waves because there are no buoyancy forces, and no Rossby waves because there are no Coriolis forces. Nevertheless, coherent structures – McWilliam’s ‘long- lived, isolated, axisymmetric monopole’ vortices – spontaneously emerge as illus- trated in Fig. 1. Table 2 catalogues the many similarities between disc-shaped or elliptical- shaped vortices (in two dimensions) or cylindrical or elliptic-cylindrical vortices (in three dimensions), and solitary waves. Even more dramatically, hurricanes and typhoons have similarities to solitons: they are strongly nonlinear, emerge spontaneously from initial conditions very different from the final structure, are Initial t=50 2 2 y 0 y 0 -2 -2 -2 0 2 -2 0 2 x x Figure 1: Contour plots of the vorticity for typical two-dimensional turbulence at the initial time (left) and at t = 50 (right). Positive-valued isolines are solid; negative contours are dashed. Table 2: Properties of vortices and solitary waves. Property Solitary wave Coherent vortices Spatially localized Yes Yes No time evolution except for Yes Yes steady translation and collisions and near-collisions Form spontaneously from highly Yes Yes nonsolitonic or noncoherent structure initial conditions Nonlinearity essential Yes Yes Balance between nonlinearity and Yes Sometimes dispersion WIT Transactions on State of the Art in Science and Engineering, Vol 9, © 2007 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Solitary Waves in Fluids 129 spatially localized and are immortal in the absence of interactions. (For hurricanes, landfall is fatal because this chokes off the supply of moist surface air that sus- tains the storm; on an ‘aqua planet’, free of land, a hurricane could spin and roar forever.) Why aren’t hurricanes classed as solitary waves? How can Rizzoli’s claim of ‘unambiguous experimental evidence has not yet been produced’be made without laughing? The answer is partly pragmatism and partly history. The pragmatism is that although there indeed have been a few papers that have identified hurricanes as solitons, or at least linked the storm with the concept (Zhao et al. 2001), a hurricane is both strongly forced and strongly dissipative. The latent heat forcing is what makes hurricanes so powerful; the equally strong damping is why hurricanes die quickly after landfall when the forcing is choked off.

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