Scaling laws for convection and jet speeds in the giant planets

Abstract

Three dimensional studies of convection in deep spherical shells have been used to test the hypothesis that the strong jet streams on Jupiter, Saturn, Uranus, and Neptune result from convection throughout the molecular envelopes. Due to computational limitations, these simulations must be performed at parameter settings far from jovian values and generally adopt heat fluxes 5–10 orders of magnitude larger than the planetary values. Several numerical investigations have identified trends for how the mean jet speed varies with heat flux and viscosity in these models, but no previous theories have been advanced to explain these trends. Here, we show using simple arguments that if convective release of potential energy pumps the jets and viscosity damps them, the mean jet speeds split into two regimes. When the convection is weakly nonlinear, the equilibrated jet speeds should scale approximately with F/ν, where F is the convective heat flux and ν is the viscosity. When the convection is strongly nonlinear, the jet speeds are faster and should scale approximately as (F/ν)^1/2. We demonstrate how this regime shift can naturally result from a shift in the behavior of the jet-pumping efficiency with heat flux and viscosity. Moreover, both Boussinesq and anelastic simulations hint at the existence of a third regime where, at sufficiently high heat fluxes or sufficiently small viscosities, the jet speed becomes independent of the viscosity. We show based on mixing-length estimates that if such a regime exists, mean jet speeds should scale as heat flux to the 1/4 power. Our scalings provide a good match to the mean jet speeds obtained in previous Boussinesq and anelastic, three-dimensional simulations of convection within giant planets over a broad range of parameters. When extrapolated to the real heat fluxes, these scalings suggest that the mass-weighted jet speeds in the molecular envelopes of the giant planets are much weaker—by an order of magnitude or more—than the speeds measured at cloud level.

Introduction

At the cloud levels near ∼1 bar pressure, numerous east–west (zonal) jet streams dominate the meteorology of the giant planets Jupiter, Saturn, Uranus, and Neptune, but the depth to which these jets extend into the interior remains unknown. Endpoint theoretical scenarios range from weather-layer models where the jets are confined to a layer several scale heights deep to models where the jets extend throughout the molecular envelope (∼10⁴ km thick) on cylinders parallel to the rotation axis (for a review see Vasavada and Showman, 2005). For Jupiter, in situ observations by the Galileo probe at 7°N latitude show that the equatorial jet extends to at least ∼20 bars (∼150 km below the visible clouds) (Atkinson et al., 1997), and indirect inferences suggest that the jets at other latitudes extend to at least ∼5–10 bars pressure (e.g., Dowling, 1995, Morales-Juberias and Dowling, 2005, Legarreta and Sánchez-Lavega, 2008, Sánchez-Lavega et al., 2008). At Neptune, gravity data suggest that the fast jets are confined to the outermost few percent of the planet’s mass (Hubbard et al., 1991). Comparable data are currently lacking for Jupiter and Saturn but will be obtained by NASA’s Juno and Cassini missions, respectively, in coming years.

Three-dimensional (3D) numerical simulations of convection in rotating spherical shells have been performed by several groups to investigate the possibility that the jets on the giant planets result from convection in the interior. Both free-slip and no-slip momentum boundary conditions at the inner and outer boundaries have been explored; of these, the free-slip case—which allows the development of strong zonal flows—is most relevant to giant planets. So far, such studies have neglected the high electrical conductivity and coupling to magnetic fields expected to occur in the deep (≳1 Mbar) planetary interior.

In this line of inquiry, most studies to date make the Boussinesq assumption in which the basic-state density, thermal expansivity, and other background properties are assumed constant with planetary radius; the convection is driven by a constant temperature difference imposed between the bottom hot boundary and top cold boundary (Aurnou and Olson, 2001, Christensen, 2001, Christensen, 2002, Aurnou and Heimpel, 2004, Heimpel et al., 2005, Heimpel and Aurnou, 2007, Aurnou et al., 2008). These studies show that convection with free-slip boundaries can drive multiple jets with speeds greatly exceeding the convective speeds. Studies using thick shells tend to produce ∼3–5 jets (Aurnou and Olson, 2001, Christensen, 2001, Christensen, 2002). When the shell thickness is only ∼10% of the planetary radius, then at least under some parameter combinations, ∼15–20 jets can occur, similar to the number observed on Jupiter and Saturn (Heimpel et al., 2005, Heimpel and Aurnou, 2007). Nevertheless, many factors in addition to shell thickness can affect the number of jets.

In real giant planets, the density and thermal expansivity each vary by several orders of magnitude from the cloud layer to the deep interior, and a new generation of convection models is emerging to account for this strong radial variation in basic-state properties. Using the anelastic approximation, which accounts for this layering, Evonuk and Glatzmaier, 2006, Evonuk and Glatzmaier, 2007, Evonuk, 2008, Glatzmaier et al., 2009 present idealized two-dimensional (2D) simulations in the equatorial plane exploring hypothetical basic-state density profiles, with density varying by up to a factor of 55 across the convection zone. In contrast, Jones and Kuzanyan (2009) present 3D simulations using an idealized basic-state density structure, with density varying by a factor of up to 148, while Kaspi et al., 2009, Kaspi et al., 2010 present 3D simulations with a realistic jovian interior structure, with density varying by nearly a factor of 10⁴ from the deep interior to the 1-bar level. These anelastic studies likewise suggest that the jets could penetrate deeply through the molecular envelope.

A challenge with all of the above-described simulations is that, for computational reasons, they must be performed using heat fluxes and viscosities that differ greatly from those on Jupiter, Saturn, Uranus, and Neptune (Fig. 1). Thus, while simulations can produce jets with speeds similar to the observed values of ∼100–200 m s⁻¹ for some combinations of parameters (e.g., Christensen, 2001, Christensen, 2002, Aurnou and Olson, 2001, Heimpel et al., 2005, Heimpel and Aurnou, 2007, Aurnou et al., 2007, Aurnou et al., 2008, Kaspi et al., 2009, Jones and Kuzanyan, 2009), this does not imply that convection in the interior of real giant planets would necessarily produce jets with such speeds. In fact, depending on the parameter combinations, simulations with free-slip boundary conditions can produce jets that equilibrate to mean speeds² ranging over many orders of magnitude, from arbitrarily small (less than 1 m s⁻¹) to 1000 m s⁻¹ or more. Assessing the likely wind speeds in the molecular envelopes of the real giant planets—and determining whether their observed jets can be pumped by convection in their interiors—requires the development of a theory that can be extrapolated from the simulation regime to the planetary regime.

Currently, however, there is no published theory that can explain the jet speeds obtained in simulations with free-slip boundaries nor their dependence on heat flux, viscosity, and other parameters. Several investigations have presented scaling laws describing how the mean zonal-wind speeds vary with control parameters when no-slip boundary conditions are used, as potentially relevant to Earth’s outer core (e.g., Aurnou et al., 2003, Aubert, 2005). These are essentially theories for the magnitude of wind shear in the fluid interior for cases where the zonal velocity is pinned to zero at the boundaries. However, these scaling laws are not applicable to the giant planets, where the fluid at the outer boundary can move freely, lacks a frictional Ekman layer, and exhibits strong jets. Several attempts have also been made to quantify how the convective velocities scale with parameters, but—regardless of the boundary conditions—these relationships cannot be applied to the jet speeds because the convective and jet speeds can differ greatly and their ratio may depend on the heat flux and other parameters.

The goal of this paper, therefore, is to develop scaling laws for how the jet speeds depend on heat flux and viscosity that explain the simulated behavior within the simulated regime and, ideally, allow an extrapolation to real planets. The simulations themselves make a number of simplifications (e.g., ignoring magnetohydrodynamics in the deep interior at pressures exceeding ∼1 Mbar), but in our view building a theory of this idealized case is a prerequisite for understanding more realistic systems.

We first quantify the degree of overforcing in current studies, since this issue has received little attention in the literature (Section 2). Next, we quantify the dependence of the convective speeds on planetary parameters and compare them to results from an anelastic general circulation model from Kaspi et al. (2009) (Section 3). Armed with this information, we construct simple scalings for the characteristic jet speeds in three regimes. In the first two regimes (Section 4), the viscosity is large enough so that viscous damping of the jets provides the dominant kinetic-energy loss mechanism. Christensen (2002) suggested the existence of a third regime where the jet speeds become independent of the viscosity; we construct possible scalings for this regime in Section 5. In Section 6, we combine the three regimes and discuss extrapolations to Jupiter. Section 7 concludes.