ENVIRONMENT AND ENERGY

Buoyancy driven flows, also called convective flows, are ubiquitous in geophysical and astrophysical objects. Deep inside the Earth, the geomagnetic field is generated by the inductive effect of turbulent convective motions in its outer, liquid iron core. Convection also occurs in giant planets like Jupiter and Saturn, and may be involved in generating the strong zonal winds observed on their surface. Because of the large spatial extent and the rapid rotation of these objects, the flows are typically highly turbulent, but nevertheless they are strongly constrained by Coriolis forces. As this situation is characteristic for many geophysical and astrophysical objects, the goal of this project is to develop a deep physical understanding of turbulent, rotationally constrained convection.

While numerical simulations of the geophysical systems described above have become a common research tool, all current models lack the resolution to represent the key physical processes involved in a realistic fashion. This is only partly due to the usual problems associated with resolving small-scale turbulence. Rapid rotation represents another fundamental obstacle because it induces extremely small length and time scales that characterize even the primary convective instabilities that drive the flow.

A common way to deal with this problem is to reduce the strength of rotation in the simulations by many orders of magnitude as compared to realistic values - far enough to guarantee that the leading order dynamics occurs entirely within the limited range of explicitly resolvable scales. However, this procedure is problematic in a number of ways. Relating such simulations to natural systems requires certainty that both share a common dynamical regime, such that the simulation results can safely be extrapolated over many orders of magnitude to planetary rotation rates. The scaling laws used in such extrapolations are currently largely empirical and lack a solid theoretical foundation. Finally, the turbulence level realizable in simulations with moderate rotation rates is fairly limited because increasing the flow amplitude quickly leads to an unrealistic loss of rotational influence on the turbulent eddies.

The problems described above reflect a general lack of physical understanding of rapidly rotating convective systems. The goal of this project is therefore to complement models that are tailored to reproduce geophysical observations as closely as possible with research efforts focusing on the basic physical processes involved. Conceptually simple, plane layer models are studied because (i) they allow for extreme parameter values and (ii) the simple geometry facilitates theoretical advances. Furthermore, (iii) laboratory studies of rotating convection are usually confined to a similar setup.

So far, reliable data that can be used to test theoretical ideas or to guide theoretical work is extremely sparse. Experiments on rapidly rotating convective flows have proven to be difficult to conduct. While complex, large-scale facilities are currently build that will hopefully improve the situation in the future, simulations also provide a powerful tool to explore previously uncharted territory and remain the only way to reveal the complex three-dimensional flow structure. Such simulations also allow for an inclusion of magnetic fields and strong compressibility effects, which are hard to study in the laboratory, but are clearly relevant in natural systems.

Model and Methods

Our simulation code supports a variable background density in order to allow for compression or expansion of fluid particles as they travel between different depth levels. This effect can be strong in geophysical systems, as the mass of the overlying material typically causes a large pressure increase with depth. In Jupiter for example, density is expected to vary by four orders of magnitude from the 1 bar level to the bottom of the outer convection zone. The so-called anelastic approximation is used to account for these effects in our model. Different thermal diffusion processes ranging from the molecular Fourier law to parameterized turbulent entropy diffusion are supported. Simulations using the classical Boussinesq approximation can also be performed to facilitate comparisons with laboratory experiments. The code can solve the full magneto-hydrodynamic equations to compute flow-induced magnetic fields and to study their effect on the fluid flow.

Technically, the momentum density and the magnetic field are represented by poloidal and toroidal potentials, while the energy equation is formulated in terms of either entropy or temperature deviations from an adiabatic background. The code supports a mixed pseudo-spectral / high order finite difference discretization or a fully spectral method based on an integral preconditioned Chebyshev approach, depending on the application scenario. Time stepping is done by an AB/BDF IMEX method with second or third order accuracy. Memory requirements are proportional to the number of grid points, with the number of floating point operations per time step exhibiting this proportionality up to a small logarithmic factor. NetCDF is used for I/O.

Results

The computing time provided has allowed us to run extensive simulations in previously unexplored regions of parameter space. In a first series of computations, background density stratifications and magnetic fields were neglected in order to allow detailed comparisons with experiments and existing theory. Figure 1 shows a visualization of the temperature field in one simulation out of a systematic parameter study. The flow is turbulent, yet strongly influenced by Coriolis forces down to small scales.

Simulations like the one illustrated in figure 1 are sufficiently far within the rapidly rotating regime to allow for detailed comparisons with existing theoretical models based on an asymptotic reduction of the governing equations. Our simulations have shown that in contrast to the expectations, this theory only holds for very idealized mechanical boundary conditions [1]. For realistic, laboratory-style boundary conditions, no quantitative agreement between the theory and simulations could be found. The problem has been traced to active Ekman boundary layers which, although covering less than one thousandth of the layer depth, have a leading order influence on the flow field [1]. Guided by our simulations, the theory has been revised [2], and the improved version reproduces the simulation results very well [3,4]. An example is shown in figure 2, where power spectra obtained from direct numerical simulations are compared with results obtained using the refined asymptotic model equations. Excellent agreement is evident in this figure.