Convection is the predominant mechanism by which energy and angular momentum are transported in the outer portion of the Sun, and the resulting overturning motions are also the primary energy source for the solar global magnetic field. An accurate model of the solar dynamo therefore requires a complete description of the convective motions, but these motions remain poorly understood. Studying stellar convection numerically remains challenging since it occurs within a parameter regime that is extreme by computational standards. The fluid properties of the convection zone are characterized in part by the Prandtl number Pr=ν/κ, where ν is the kinematic viscosity and κ is the thermal diffusion; in the Sun and other stars, the Prandtl number is extremely low, Pr≈ 10-7. The influence of the Prandtl number on the convective motions at the heart of the dynamo is not well understood since most numerical studies are limited to using Pr≈1. Here we systematically vary the Prandtl number over a broad range of thermal forcing to explore its influence on the convective dynamics. For sufficiently large thermal driving, the simulations reach a so-called convective free-fall state where diffusion no longer plays an important role in the interior dynamics. In general, we find that simulations with a lower Prandtl number generate faster convective flows and a broader range of scales for equivalent levels of thermal forcing. However, the characteristics of the spectral distribution of the velocity remain largely insensitive to changes in the Prandtl number. Importantly, we find that the Prandtl number plays a key role in determining when the free-fall regime is reached by controlling the thickness of the thermal boundary layer.

Initial study

Our initial study was focused on examining how the kinetic energy behaved as the Prandtl number and Rayleigh number were varied. The final results were published in the following paper:

Prandtl-number Effects in High-Rayleigh-number Spherical Convection [ads]
Orvedahl et al. 2018, ApJ, 856

Figure 1 shows the dimensional kinetic energy as a function of the Rayleigh number for various values of the Prandtl number. After a certain cutoff, the kinetic energy saturates and does not grow as the Rayleigh number is further increased. Also shown is the Reynolds number as a function of a rescaled Rayleigh number.

 

KE and Re vs Ra

Figure 1: Dimensional kinetic energy vs Rayleigh number and Reynolds number as a function of a rescaled Rayleigh number.

The large-scale (mean) force balance in convection-driven dynamos in a spherical geometry is analyzed, as relevant to the geodynamo. The forces are analyzed with numerical simulation data and asymptotic theory. In agreement with previous work, the mean force balance is shown to be thermal wind (Coriolis, pressure gradient, buoyancy) in the meridional plane and Coriolis-Lorentz in the zonal direction. Particular emphasis is given to determining the asymptotic size (with the small parameter being the Ekman number, Ek) of the forces, and the velocity and magnetic fields. The thermal wind balance requires that the mean zonal velocity scales as O(Ek-1/3), whereas the meridional circulation is asymptotically smaller. The mean Lorentz force appears to be O(Ek1/6) weaker than the mean buoyancy force. A consequence of this asymptotic ordering in the forces is that Taylor's constraint is satisfied to accuracy O(Ek1/6), despite the absence of a leading-order magnetostrophic balance. Such dynamos might be referred to as semi-magnetostrophic since the Lorentz force enters the leading order force balance only in a single dimension. The consequences of the force balance are discussed with respect to torsional oscillations.

We investigate the process of large-scale (axisymmetric) magnetic field generation in rotating, spherical dynamo models. Within the parameter space sampled, we find that, as a function of the Rayleigh number, Ra, the mean magnetic energy of a dynamo initially grows, then reaches a saturated value at which point it remains constant with increasing Ra. In all models the saturation mechanism is consistent with the Malkus-Proctor scenario in which the mean magnetic field becomes comparable in strength to the meridional circulation. Upon saturating, a further increase in Ra results in the dominance of the fluctuating (nonaxisymmetric) magnetic field in a semi-magnetostrophic (MS) force balance, where the mean Lorentz force enters only one component of the primary mean momentum equation balance. This saturation is robust across varying Ekman numbers so long as the Rossby number remains small. The characteristic field strength (both mean and fluctuating) is shown to be nearly independent of the Ekman number and is explained via the primary force balance. All of the low Rossby number dynamos can be characterized as so-called α2Ω mean field dynamos since both small-scale induction and the shear resulting from the mean flow are important in the field generation process. The asymptotic scaling behavior of the various terms in the mean induction equation is also investigated.

Highlight of Some Results

Our initial study was focused on examining the leading-order force balance that is established within planetary-type dynamo models. We found that the force balance is predominantly thermal wind, i.e., a three way balance between Coriolis, pressure gradient, and buoyancy. The Lorentz force does not show up in the leading-order force balance, it is weaker by about O(Ek1/6).

The second study looked into the mean (axisymmetric) magnetic field and its behavior as the Rayleigh number is varied. The magnetic Prandtl number was also varied, but did not show a significant difference over the cases. The main result is that the mean dipolar magnetic energy saturates as the Rayleigh number is increased beyond some cutoff. This result was observed across all Ekman numbers that were used in the study, indicating that it is a robust feature of rapidly rotating convection.

Figure 1 shows the rms forces in the radial direction for all simulations that used Ra = 10 times critical. The forces are labeled as 'a' for advection, 'c' for Coriolis, 'v' for viscous, 'p' for pressure gradient, 'l' for Lorentz, and 'b' for buoyancy. The Lorentz force is almost exactly balanced by the leading-order force balance, indicating the Lorentz force is of second order. The particular slopes as a function of Ekman number have implications for extrapolating the results down to planetary core conditions.

 

Force vs Ek

Figure 1: Rms forces in the radial direction as a function of Ekman number. All cases shown have Ra=10x and Pm=2.

Figure 2 shows the mean dipolar magnetic field as a function of Rayleigh number. There is a threshold at Ra=10x that denotes a significant change in scaling behavior. Beyond this threshold, the magnetic energy no longer grows as the Rayleigh number is further increased.

 

Mean dipolar ME vs Ra

Figure 2: Mean dipolar magnetic energy as a function of Rayleigh number. After a certain threshold in Ra, the magnetic energy saturates. This occurs for all Ekman numbers shown. Different colors indicate different Pm, with red corresponding to Pm=2. The linestyles/symbols indicate various Ek: circles are high, triangles are lower, diamonds are slightly smaller, and stars are the lowest Ek.