Jupiter’S Great Red Spot

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Encyclopedia of Nonlinear Science

JUPITER’S GREAT RED SPOT

Jupiter’s Great Red Spot is a large swirling cloud mass of reddish-brown appearance (see figure in color plate section). Situated in Jupiter’s southern hemisphere, it straddles the south tropical zone and, to the north of this, the south equatorial belt. The Great Red Spot (GRS) is roughly elliptical in shape, with the semi-major axis zonally aligned (east-west) and with dimensions approximately 22, 000 km (twice the diameter of the Earth) by 11, 000 km. The atmospheric motions associated with the GRS are visible in the cloud layer near the tropopause. It is generally agreed to be a vortex (Mitchell et al., 1981); and Smith et al. (1979a) give an estimate of the vorticity. This vortex is anticyclonic (rotating in the opposite sense to that induced by the planetary rotation), that is, anticlockwise, but with a weakly counter-rotating, or possibly quiescent inner region. The GRS is at high pressure and low temperature relative to its surroundings. A striking feature associated with the GRS is the turbulent oscillating cloud system to the northwest.

In 1664, Robert Hooke observed a small dark spot in one of the southern belts of Jupiter, reported in Hooke (1665). This is considered by some contemporary authors to be an early manifestation of the GRS (see also Rogers, 1995). By observation of this feature, Hooke was able to make a good estimate of Jupiter’s rate of rotation. Many further observations of this and other features of Jupiter were made by Hooke and by Giovanni Cassini over the next five years (e.g., Cassini, 1672), and intermittently by others over the next 50 years (see Denning, 1899). However, it was not until 1831 that Samuel H.Schwabe identified what can be clearly recognized as a feature called the Hollow, in his drawing reproduced in Rogers (1995). As the Hollow is clearly apparent above the GRS in contemporary observations, this is taken to imply the presence of the GRS. Continuous observations began in 1872 (Peek, 1958) and from these it appears that the GRS was considerably larger in the late 1800s than it is now (Beebe & Youngblood, 1979; Rogers, 1995). Very much more detailed observations became available once space probes reached Jupiter; first was Pioneer 10 in 1973, then Pioneer 11 in 1974, Voyagers 1 and 2 in 1979, Galileo in 1996, and then most recently Cassini from October 1, 2000 through until March 22, 2001. The Voyager and Galileo missions (Smith et al., 1979a, b; Vasavada et al., 1998) provided very high resolution still images, while Cassini (Porco et al., 2003) gave more dynamic information. From 1994, after corrections were made to faulty optics by the Endeavour crew, the Hubble telescope has been providing good images of Jupiter and the GRS. These, along with the Cassini sequences, clearly show vortex interactions between the GRS and smaller, intermittent vortices, thus allowing a much better understanding of the underlying dynamical processes (Morales-Juberías et al., 2002).

Attempts have been made since 1961 to model the GRS and to understand the processes giving rise to the motions observed, beginning with Hide (1961), who suggested that the GRS was the visible manifestation of a Taylor column. The quantity of relevance to these calculations is the potential vorticity q. As the atmosphere of Jupiter is stably stratified and there are significant rotational effects at the scale of the GRS, the most commonly used model is the quasi-geostrophic approximation (a simplification of the complete atmospheric dynamic equations, giving a balance of horizontal pressure gradient forces and horizontal Coriolis forces due to Jupiter’s rotation). Here q is defined as

(1)

where ψ(x, y, z, t) is the stream function for the flow and (x, y, z) are local cartesian coordinates, with x and y being the zonal and meridional directions, respectively, and z being a measure of the depth. The basic state density profile is ρ₀(z) and the effects of Jupiter’s rotation are introduced by the beta-plane approximation, the assumption that we can treat part of Jupiter as a plane, over which the Coriolis parameter f varies linearly: f=f₀+βy, where f₀ is the reference value at the latitude of the GRS and β is the ambient potential vorticity gradient. The buoyancy frequency N(z), the frequency with which a parcel of fluid displaced from equilibrium will oscillate, is determined by the vertical temperature variation. The potential vorticity is materially conserved, according to

(2)

where the horizontal velocity υ=(−∂ψ/∂y, ∂ψ/∂x).

Many researchers have modeled the vortical motions associated with the GRS by assuming a thin upper layer of constant density (of order 100 km; Bowling & Ingersoll, 1989) above a much deeper lower layer of slightly higher density in which there is a uniform steady zonal flow, for example, Ingersoll and Cuong (1981), Marcus (1988), and Marcus et al. (1990). The surface of the upper layer is above the visible cloud level. These models assume barotropic motions, that is, uniform in depth with no horizontal component of vorticity, in the upper layer where isolated vortices are assumed to be generated by mechanisms such as zonal shear flow instability. Merger of vortices can lead to the formation of larger vortices, and the relevant issues are then the stability and lifetime of isolated vortices such as the GRS in the observed zonal shear. A statistical mechanics approach to such equilibria is provided in Michel and Robert (1994).

There have also been discussions on the role of vertical density stratification, allowing baroclinic instability, but generally limited to the quasi-geostrophic case of (1) and (2). For example, Achterberg and Ingersoll (1994) studied the barotropic and the first two baroclinic modes, and Cho et al. (2001) used as many as 60 vertical modes. The high horizontal resolution used in Cho et al. (2001) allowed the authors to generate features such as the counter-rotating core (Vasavada et al., 1998, Figure 7, reporting Galileo data and similar Voyager data), but as in all other studies to date, the turbulent cloud system to the northwest of the spot does not appear.

A deficiency of the quasigeostrophic model is that vortices with radius significantly greater than the Rossby radius of deformation, L_R, tend to decay by radiation of Rossby waves in the absence of any external forcing. (L_R is a measure of the scale at which Coriolis forces become significant.) This would imply a lifetime for the GRS much shorter than the observed period, which is certainly greater than 150 years. In addition, the quasigeostrophic equations have the disadvantage that neither anticyclonic nor cyclonic vortices are preferentially created or destroyed, whereas observations show that the GRS is anticyclonic, as are 90% of the other (typically much smaller) vortices observed on Jupiter (Mac Low & Ingersoll, 1986).

For two-layer models, the zonal flows in the bottom layer can be incorporated in a reduced gravity single-layer shallow water model with meridionally varying topography and a free surface, which is taken to be above the visible cloud layer (Dowling & Ingersoll, 1989). There are various levels of approximation to the shallow-water equations beyond the quasi-geostrophic approximation. The intermediate geostrophic (IG) approximation (Williams & Yamagata, 1984; Nezlin & Sutyrin, 1994) arises when there is significant nonlinearity giving rise to finite free-surface distortion, as is to be expected for vortices with a scale much greater than the Rossby radius, such as the GRS. A more complete discussion of the asymptotic treatment of the nonlinear terms, along with higher-order approximations, is given in Stegner and Zeitlin (1996).

An alternative approach to understanding the GRS as a solitary wave was first introduced by Maxworthy and Redekopp (1976a, b) and Redekopp (1977). They considered the quasigeostrophic equations (1) and (2) and were able to find elementary soliton solutions in the limit of small amplitude disturbance, which obeyed either the Korteweg-de Vries equation (KdV) or the modified KdV equation. These balance zonal shear and dispersion to give isolated disturbances that could be regarded as vortices. However, as noted by several authors (e.g., Ingersoll & Cuong, 1981), these soliton solutions of the KdV equation pass through one another unchanged in form (elastic collisions). This is contrary to the behavior of vortices generally, and more particularly the GRS, where merger with smaller vortices is observed (Morales-Juberías et al., 2002). However, Petviashvili (1980), by going beyond quasigeostrophy, was able to derive solitary axisymmetric vortex solutions, without an imposed zonal shear, for a curved shallow layer, with anticyclones living longer. Nezlin et al. (1990) refer to this anticyclonic vortex as a Rossby soliton, with weak nonlinearity balanced by Rossby wave dispersion. These solitary solutions undergo inelastic collisions and were later thought to be found experimentally in parabolic water tank experiments (Nezlin et al., 1990), although Nycander (1993) and Stegner and Zeitlin (1996) demonstrated that this weakly nonlinear balance could not in fact exist in the parabolic geometry and that stronger nonlinearity was required to explain the laboratory observations. Stegner and Zeitlin (1996) go on to show that axisymmetric soliton-like solutions of a 2-d KdV equation are found for spherical geometry, corresponding to the surface of Jupiter, using a nonlinear model. Furthermore, only an anticyclonic vortex can be supported in this case, which is in accord with the observed rotation of the GRS. However, this cannot explain the preferential formation of anticyclonic vortices smaller than the Rossby radius, where the quasigeostrophic approximation remains appropriate. Finally, as explained by Stegner and Zeitlin (1996), their 2-d KdV solutions do not apply directly to the GRS as neither the elliptical shape nor the counter-rotating inner core can be accounted for.

C.MACASKILL AND T.M.SCHAERF