## 1. Introduction

[2] Despite decades of investigation, Jupiter's atmospheric circulation is still not well understood. Major questions revolve around Jupiter's dynamical structure, in particular, how its zonal winds are forced (from below by internal heat or from above by solar heating) and how deeply they extend into the interior. Many authors have attempted to better constrain and understand Jupiter's dynamics based on various approaches (typically deep convection models vs. shallow atmospheric models) and different numerical methods [e.g., *Williams*, 1978; *Cho and Polvani*, 1996; *Nozawa and Yoden*, 1997; *Huang and Robinson*, 1998; *Williams*, 2003; *Aurnou and Heimpel*, 2004; *Heimpel et al.*, 2005; *Vasavada and Showman*, 2005; *Showman et al.*, 2006; *Heimpel and Aurnou*, 2007; *Lian and Showman*, 2008].

[3] Deep convection models of the Jovian zonal winds [e.g., *Busse*, 1976; *Heimpel et al.*, 2005; *Heimpel and Aurnou*, 2007] suggest that the bottom boundary of the deep convection is at about 0.9 R_{J} (R_{J} is the radius of Jupiter). *Liu et al.* [2008] discuss the importance of Lorentz forces in constraining the depth of the Jovian circulation. They argue that the observed zonal winds on Jupiter cannot extend more deeply than 0.96 R_{J}. Shallow forcing models typically argue that the zonal winds are driven by moist convection and are a surface phenomenon, i.e., a ‘weather layer’ [e.g., *Ingersoll and Cuzzi*, 1969; *Ingersoll et al.*, 2004].

[4] The penetration depth of Jupiter's zonal winds also has implications for the planet's internal structure. If the zonal winds are ‘deep’ and involve a non-negligible amount of mass, interior models that typically assume solid-body (SB) rotation must be modified, and corrections to the gravitational coefficients due to dynamics must be included [*Hubbard*, 1982, 1999]. A recent interior model of Jupiter by *Militzer et al.* [2008] suggests that Jupiter's measured *J*_{4} value can only be fit when differential rotation is considered because no interior models that fit Jupiter's *J*_{4} could be found under the assumption of solid-body rotation. The interior model presented by *Militzer et al.* [2008], however, consists of 2-layers(a heavy-element core surrounded by a hydrogen-helium envelope), while standard 3-layer Jupiter models with solid-sbody rotation can typically fit Jupiter's gravitational coefficient (*J*_{2}, *J*_{4}, *J*_{6}) [e.g., *Saumon and Guillot*, 2004; *Nettelmann et al.*, 2008]. As the question of whether Jupiter's interior structure is consistent with solid-body rotation remains open, it is clear that a better determination of the depth of Jupiter's zonal winds is desirable for both dynamical and interior models.

[5] Jupiter's internal dynamics, or more precisely, its rotation profile, can be constrained by measurements of its high-order gravitational coefficients. Hubbard and collaborators [e.g., *Hubbard*, 1982, 1999; *Kaspi et al.*, 2010] showed that a departure from solid-body rotation is detectable in gravitational coefficients larger than about degree 10 [*Kaspi et al.*, 2010]. The Juno mission to Jupiter [*Bolton*, 2005] is designed to provide accurate measurements of Jupiter's gravitational field to high order and can therefore constrain its dynamical structure. Gravitational anomalies can also be used directly to infer Jupiter's density anomalies and therefore its rotation profile [see *Kaspi et al.*, 2010, and references therein]. In this paper we use an equipotential theory to derive Jupiter's shape when differential rotation on cylinders is included and suggest that shape data, i.e., occultation radii, can provide an independent method of constraining its internal rotation profile.