Rapidly Rotating Convection in Geophysical Systems
Stephan Stellmach and Ulrich Hansen
Institut für Geophysik, Westfälische Wilhelms-Universität Münster
Local Project ID:
HPC Platform used:
JUQUEEN of JSC
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.
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 . 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 . Guided by our simulations, the theory has been revised , 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.
A systematic parameter study, covering varying flow amplitudes and fluid properties, has demonstrated that the theory works well over large parameter ranges. The confirmation of the validity of the asymptotic model equations is a key result of our work, as these can be used to investigate the system behavior for parameter values many orders of magnitude beyond the capabilities of DNS .
The simulations provide further insight into the nature of turbulence in rapidly rotating convection. In particular, a strong, non-local inverse cascade of barotropic kinetic energy has been found [1,3]. If the boundaries do not support lateral shear stresses, this results in the formation of a coherent, depth-independent dipole condensate . For rigid no-slip boundaries, Ekman friction prevents the formation of long-lived, coherent large-scale vortices . Instead, the upscale kinetic energy transport results in erratic and intermittent large-scale structures which frequently break up, a process visible in figure 1. The presence of upscale transport in turbulent rotating convection is an important result in the geophysical context, and one of the key predictions to test in future laboratory experiments.
Simulations involving background density stratifications have also been performed, and we have been able to demonstrate that the interplay of rotation with the compression and expansion of fluid parcels can give rise to the formation of multiple jets and to potential vorticity layering. Furthermore, the scaling of the jet thickness has been shown to be compatible with simple theoretical arguments . The underlying effect may be involved in driving the zonal winds observed on Jupiter and other giant planets. As a by-product of this work, we have also been able to systematically test different modeling approaches for sub-sonic, compressible convective flows frequently used in geo- and astrophysics .
Currently, the topology and dynamical influence of a flow-induced magnetic field is studied in detail. Future simulations will also shed light on the important dynamical regime where the rotational influence on the small flow scales is lost, while the large-scale dynamics is still dominated by Coriolis forces. This parameter regime remains largely unexplored to date, but certainly has geophysical relevance.
 Stellmach, S., Lischper, M., Julien, K., Vasil,G., Cheng, J., Ribeiro, A., King, E., Aurnou, J.M.: Approaching the asymptotic regime of rapidly rotating convection: Boundary layers versus interior dynamics. Physical Review Letters 113 (25), 254501 doi: 10.1103/Phys- RevLett.113.254501 (2014)
 Julien K., Aurnou J., Calkins M., Knobloch E., Marti P., Stellmach S., Vasil G. A nonlinear model for rotationally constrained convection with Ekman pumping. Journal of Fluid Mechanics, 798, pp. 50-87. doi: 10.1017/jfm.2016.225 (2016)
 Plumley M., Julien K., Marti P., Stellmach S.: The effects of Ekman pumping on quasi-geostrophic Rayleigh-Bénard convection. Journal of Fluid Mechanics, 803, pp. 51-71. doi: 10.1017/jfm.2016.452 (2016)
 Plumley M., Julien K., Marti P., Stellmach S.: Sensitivity of rapidly rotating Rayleigh-Bénard convection to Ekman pumping. Phys. Rev. Fluids 2, 094801, doi: 10.1103/PhysRevFluids.2.094801 (2017)
 Verhoeven J., Stellmach S. The compressional beta effect: A source of zonal winds in planets? Icarus, 237, pp. 143-158. doi: 10.1016/j.icarus.2014.04.019 (2014)
 Verhoeven J., Wiesehöfer T., Stellmach S. Anelastic Versus Fully Compressible Turbulent Rayleigh-Bénard Convectio. Astrophysical Journal, 805 (1). doi: 10.1088/0004-637X/805/1/62 (2015)
Dr. Stephan Stellmach
Institut für Geophysik
Westfälische Wilhelms-Universität Münster
Corrensstr. 24, D-48149 Münster (Germany)
e-mail: stellma [@] uni-muenster.de
JSC project ID: chms15