Simple Dynamical Models of Neptune's Great Dark Spot

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Simple Dynamical Models of Neptune's Great Dark Spot Simple Dynamical Models of Neptune's Great Dark Spot L. M. POLVANI,* J. WISDOM, E. DEJONG, A. P. INGERSOLL radio rotation period of Neptune is 16.11 hours. The large amplitude oscillations ofthe shape ofNeptune's We fit ellipses to the boundary of the Great Dark Spot with the Great Dark Spot are well reproduced by simple dynamical aid of the SAOimage software, which allows an ellipse to be models of an isolated vortex embedded in a background superimposed on the image and, with a pointer, to adjust the center, shear flow. From the time series of the aspect ratio and minor axis, major axis, and orientation of that ellipse. In most inclination of the vortex values are estimated for the images we find that the shape of the Great Dark Spot is well background shear and the mean vorticity of the Great represented by an ellipse (see, for instance, Fig. 1). In a few cases a Dark Spot, and a lower bound is placed on the value of tail is also present, but an elliptical core is still easily identified (as the Rossby deformation radius. These models imply the illustrated in Fig. 2); in these cases the elliptical core is taken to existence ofa planetary-scale zone ofdeterministic chaotic represent the vortex. Only in two images, during the ejection of the advection in the atmosphere of Neptune. large tail, is the representation of the Great Dark Spot as an ellipse ambiguous. Table 1 gives the parameters of the fit ellipses. We define X(O < A < 1) to be the ratio of the minor to major axis, the on June 26, 2014 inverse of the aspect ratio, and specify the orientation of the ellipse T 1 HE RECENT ENCOUNTER OF THE VOYAGER 2 SPACECRAFT by the angle (p between the major axis and the zonal direction, with Neptune has revealed the presence of a large coherent measured counterclockwise. structure in its atmosphere, dubbed the Great Dark Spot (1), Before settling on the manual elliptical fits, we tried a variety of which undergoes dramatic variations in its shape. We propose that different techniques to objectively extract the boundary ofthe Great the observed oscillations of the Great Dark Spot are the natural Dark Spot from the images: clipping, gradient, laplacian, smooth- oscillations of a vortex in a background shear flow. ing, all of these in various combinations. The boundaries deter- In view of the paucity of atmospheric features on Neptune the mined by these methods were judged to be unsatisfactory represen- vorticity cannot be measured directly. By fitting the observed tations ofthe Great Dark Spot as one sees it in the images. The main www.sciencemag.org oscillations in shape to those of sheared elliptical vortices we obtain problem is caused by the presence ofbright clouds which overlie the estimates of the background shear and the anomalous vorticity of Great Dark Spot. These distort the boundaries determined by the the Great Dark Spot without using any knowledge of the velocity simple objective methods. We did not try to develop a more fields on Neptune. The vorticity is found to be larger than that of sophisticated fitting algorithm which could automatically account the ambient flow, as is the case for the Great Red Spot on Jupiter. for the presence of the clouds. Both vortices are anticyclonic. Because we had no objective method for fitting the boundary of We find that the latitudinal variation of the Coriolis parameter the Great Dark Spot, we estimated the errors in our fits by repeating Downloaded from (the 1B effect) has little influence on the dynamics ofthe system. Our each fit independently several times (see Table 1). These error dynamical models suggest that the Rossby radius of deformation is estimates rigorously only represent our ability to reproduce our own larger than about two thirds ofthe diameter ofthe Great Dark Spot. fits. The true errors are likely to be larger. For example, we did not Last, our models imply the existence of a planetary-scale zone of correct for the fact that the projection of the images distorts the deterministic chaotic fluid motion in the atmosphere of Neptune. shape of the Great Dark Spot. Some uncertainty is also introduced Voyager image analysis. We analyzed 28 projected Voyager by the presence ofthe bright clouds near the boundary ofthe Great images, which were taken from the same images as figure 4A of Dark Spot (as can be seen from Fig. 2). Smith et al. (1). The projection used was linear in latitude and We note that the area of the Great Dark Spot as determined by longitude, and centered at -19.6° latitude. These data span 31 our fit ellipses is nearly constant (Fig. 3), even though our fitting Great Dark Spot rotation periods. The Great Dark Spot rotation procedure was not geared toward an accurate determination of the period, namely the rotation period of the planet as defined by the area. This would have required more careful photometry. Rather, Great Dark Spot, is approximately 18.3 hours. For comparison, the our fitting procedure was geared solely to determining the aspect ratio and orientation of the Great Dark Spot. The diameter of a circle with the same area as the Great Dark Spot, as determined by L. M. Polvani is in the Department ofMathematics and J. Wisdom is in the Department our fit ellipses, is D = 9842 + 57 km, excluding the first image and of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute ofTechnology, Cambridge, MA 02139. E. Dejong and A. P. Ingersoll are in the Division ofGeological the last two images. The fit to the first image appears anomalous, and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125. both in the area of the vortex and in the orientation angle. The last *Present address: Department of Applied Physics, Columbia University, New York, two points show a marked decrease in the area which is associated NY 10027. with the ejection of a large tail. Despite these misgivings, all of the 21 SEPTEMBER 1990 RESEARCH ARTICLES 1393 data points were used in the fits wthedynai in this paper. Dynamical models. We compae the.Voyager i-a hicralr of simple models of the dynamics-ofthe Great D these the Great Dark Spot is taketp be a sin background shear flow.: her assume piecewiseconst outside the vortex. U i i served in numerical simlio In all of the models the dya isgivsix.I conservation ofgeostrophic p i vortii f DIT with the material derivative D_a Dt aIt where]j is the. JaoDbian oper;atoa 4. V.* streanfifnction, ,with the thecomponent due totethe, tial vorticities are 11v and H11,nd the qv aand qB W . (XI, y), comoving with the zonalo Spot, with positive x eastward andpositive y- n ndilipticat core' the sranfinction the velocity (u,-) cond one firo.m is) (1).: ay ax Model 1-batropic f-plane. The simplest mo4 dtlstream un~tk~a 4,and lB, associated with the dynamics ofa single layer ofro g constan d y satisfies motons are two dimensional andq - :4 known as the br a :*;= IT5(4) planetary vorticity, the. PO a vortct iS gi plyerlaiveoricities qv and qB. We which is simply the relative vorftcty.- d11a bacwi an&sear flow to be constant in timoe and uniformi ic., atsEsociated streanifinon given by: *4,000B = SI2 (5)F d-tuund shearand is also equal to the negative of Oiticity qB. The background velocity UB is a simple SUBY (6) cal tkha in this simplest Cas, aln clliptical vortex patch is an _ IlOnhilKar dine-dependent solution of Eq. 1, widely known as- _ida vortex (7). The state ofthe system is entirely specified by X, 4>eiawcrse ofte aspect ratio, and p, the orientation ofthe vortex. lIsevolve according to: ^- - k + s2l](5 H~~~~~~t> -~~~~~~~~~~~~~~~sW- dt- --,IsqV-K.-fI'-[-1 QIK-qvXI(l + X)2 is the angular velocity of the ellipt ortex patch in the absence of shear (a Kirchhoff vortex), and A.(1, + X2)I(1 - X2). +v The vorate orientation angle c.an rotate or oscillate d _ tbe value off sliq, and the initial values of andX. We K_n ,(8) that lr zones of te ticv chaotic * ion can exist around such v s.h chaotic zones produce rapid nixing in the surroinding fluid. iih fit- 00Model 2-barotropic P-plane. The next model alsol takesie to Fig. 1. A repri of the Coriolis p elipse supennq account .the laitudinal varianon elipse. This in :cor,m,sponas tn-ruxwm tnc scnes prcs fitm -= 2ml sin {, where 21/A is the Great Dark Spotr 4) i d 4A of Smith et di. (1). and Is tie latitude, which was ap 9. 1394 : . D 7 -.;SC.,NGE, VL Neptune encounter. The potential vorticity in this model is given by Fig. 3. The area of the 1 00 11 = V2+ where ,1 = 2Q cos O/RNeptune is the derivative of Great Dark Spot versus +py, time, as determined fwith respect to northward distance. This system is known as the t from our fit The 80 i T _ T 1 7 i I quasi-geostrophic (-plane model. ellipses. error bars in I - Figs. 3, 4, I, ~ Under the assumption ofpiecewise constant potential vorticity fl, and 5 correspond to the I I I I I I the components tlv and B respectively satisfy uncertainties given m 60 i Table 1. ti Co V2V = IIV V2B + 1Y = fIB = -S (8) From the second ofthese equations we obtain the expression for the 0ca 40 streamfunction associated with the background shear: 20 2 1B= - Sy -11Y (9) O whereas tv has to be determined numerically.
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