Baroclinic Model of Jupiter’s Great Red Spot

Abstract

This paper proposes a quasi-geostrophic baroclinic model of Jupiter’s Great Red Spot (GRS) as a localized eddy formation in a continuously stratified rotating atmosphere under the action of a horizontal shear flow in the f-plane approximation. On the basis of the theory of ellipsoidal vortices, an analytical relationship is obtained between the geometric dimensions of the vortex, the potential vorticity of the vortex core, and the characteristics of the background flow. Measurements of a number of characteristics of both the vortex and the background flow in the Voyager 1 (1979), Galileo (1996), and Cassini (2000) missions were used. Based on the theory, the vertical size of the GRS was calculated, which turned out to be close to the same characteristic measured in the Voyager 1 (1979) mission.

APPENDIX

1.1 SIGNIFICANCE ASSESSMENT OF THE β-EFFECT IN THE DESCRIPTION OF THE GRS

In our theory, we used the law of conservation of potential vorticity (2) without taking into account the β effect. If we consider the β effect, then the expression for the potential vorticity will take the form

$$\sigma = {{\Delta }_{h}}\psi + \frac{\partial }{{\partial z}}\frac{{{{f}^{2}}}}{{{{N}^{2}}}}\frac{{\partial \psi }}{{\partial z}} + ~\beta y,$$

(A1)

where y is the deviation of the liquid particle in the northern direction from its initial position. Parameter value $\beta = 0.4742\,\, \times \,\,{{10}^{{ - 11}}}$ (m s)^–1 for latitude 22°. The maximum deviation from a latitude of 22° to the north in different periods is $b = \left( {6{\kern 1pt} - {\kern 1pt} 7} \right)\,\, \times \,\,{{10}^{3}}$ km. Hence, it follows that the change in potential vorticity due to the $\beta$ effect in the extreme case, when the particle is deflected from a latitude of 22° to a distance $b = 6\,\, \times \,\,{{10}^{3}}$ km to the north, will be

$${{\left( {\delta \sigma } \right)}_{{{\text{max}}}}} = \beta b = 2.84\,\, \times \,\,{{10}^{{ - 5}}}\,\,{{{\text{s}}}^{{ - 1}}}{\text{.}}$$

(A2)

In this case, we estimate the relative change in the potential vorticity in the core as

$$\frac{{{{{\left( {\delta \sigma } \right)}}_{{{\text{max}}}}}}}{{\sigma + {{\Gamma }}}} = \frac{{\beta b}}{{\sigma + {{\Gamma }}}} = \frac{{0.284\,\, \times \,\,{{{10}}^{{ - 4}}}~}}{{1.05\,\, \times \,\,{{{10}}^{{ - 4}}}}} = 0.27.$$

(A3)

Here, $\left( {\sigma + {{\Gamma }}} \right) = 1.05\,\, \times \,\,{{10}^{{ - 4}}}$ s^–1 is the experimental estimation of the potential vorticity of particles of the vortex core. This estimate is valid only for one particle that has shifted from a latitude of 22° and reached the northernmost point of the vortex core. On the average, over the volume of the ellipsoid, the particles deviate to the north by an amount of $\frac{3}{8}b$ (this is the position of the center of gravity of the semiellipsoid). Then the average error over the volume of the nucleus

$$\left| {\delta \sigma } \right| = \frac{3}{8}\beta b = 1.07\,\, \times \,\,{{10}^{{ - 5}}}\,\,{{{\text{s}}}^{{ - 1}}}$$

(A4)

and the relative error in determining the value of the potential vorticity of the core when neglecting β effect on average will be the value

$$\frac{{\left| {\delta \sigma } \right|}}{{\sigma + {{\Gamma }}}} = \frac{{1.07\,\, \times \,\,{{{10}}^{{ - 5}}}}}{{1.05\,\, \times \,\,{{{10}}^{{ - 4}}}}} = 0.1.$$

(A4)

As we see, neglecting the $\beta$ effect in terms of potential vorticity will give a maximum error of 27% and an average of 10% for vorticity. This is a significant amount. Therefore, all of the above GRS models use a more accurate approximation theory of $\beta$ planes. The error of the applied theory in both the f-plane approximation and in the $\beta$-plane approximation is on the order of the Rossby number, the estimates of which at different times were in the range from 0.15 to 0.30. Thus, the error of both theories covers the additional effect from the refinement of the potential vorticity in the approximation β-plane. As a result, almost due to the error, it equalizes the approaches to the f- and $\beta$-planes. However, in fairness, we note that the $\beta$-plane approximation from a physical point of view describes a richer spectrum of phenomena, for example, the westward drift of a vortex and the Rossby waves that occur when flowing around a vortex. Neglecting the $\beta$ effect, we discard these phenomena as well.

It should also be noted that with time the vortex decreases in size and intensifies in terms of potential vorticity values, and this process is accompanied by an increase in the Rossby number.