Philippe Marti

Performance benchmarks for a next generation numerical dynamo model

Numerical simulations of the geodynamo have successfully represented many observable characteristics of the geomagnetic field, yielding insight into the fundamental processes that generate magnetic fields in the Earths core. Because of limited spatial resolution, however, the diffusivities in numerical dynamo models are much larger than those in the Earths core, and consequently, questions remain about how realistic these models are. The typical strategy used to address this issue has been to continue to increase the resolution of these quasi-laminar models with increasing computational resources, thus pushing them toward more realistic parameter regimes. We assess which methods are most promising for the next generation of supercomputers, which will offer access to O(106) processor cores for large problems. Here we report performance and accuracy benchmarks from 15 dynamo codes that employ a range of numerical and parallelization methods. Computational performance is assessed on the basis of weak and strong scaling behavior up to 16,384 processor cores. Extrapolations of our weak-scaling results indicate that dynamo codes that employ two-dimensional or three-dimensional domain decompositions can perform efficiently on up to ∼106 processor cores, paving the way for more realistic simulations in the next model generation.

The effects of Ekman pumping on quasi-geostrophic Rayleigh–Bénard convection

Numerical simulations of 3D, rapidly rotating Rayleigh-Benard convection are performed using an asymptotic quasi-geostrophic model that incorporates the effects of no-slip boundaries through (i) parameterized Ekman pumping boundary conditions, and (ii) a thermal wind boundary layer that regularizes the enhanced thermal fluctuations induced by pumping. The fidelity of the model, obtained by an asymptotic reduction of the Navier-Stokes equations that implicitly enforces a pointwise geostrophic balance, is explored for the first time by comparisons of simulations against the findings of direct numerical simulations and laboratory experiments. Results from these methods have established Ekman pumping as the mechanism responsible for significantly enhancing the vertical heat transport. This asymptotic model demonstrates excellent agreement over a range of thermal forcing for Pr ~1 when compared with results from experiments and DNS at maximal values of their attainable rotation rates, as measured by the Ekman number (E ~ 10^{-7}); good qualitative agreement is achieved for Pr > 1. Similar to studies with stress-free boundaries, four spatially distinct flow morphologies exists. Despite the presence of frictional drag at the upper and/or lower boundaries, a strong non-local inverse cascade of barotropic (i.e., depth-independent) kinetic energy persists in the final regime of geostrophic turbulence and is dominant at large scales. For mixed no-slip/stress-free and no-slip/no-slip boundaries, Ekman friction is found to attenuate the efficiency of the upscale energy transport and, unlike the case of stress-free boundaries, rapidly saturates the barotropic kinetic energy. For no-slip/no-slip boundaries, Ekman friction is strong enough to prevent the development of a coherent dipole vortex condensate. Instead vortex pairs are found to be intermittent, varying in both time and strength.

A nonlinear model for rotationally constrained convection with Ekman pumping

A reduced model is developed for low-Rossby-number convection in a plane layer geometry with no-slip upper and lower boundaries held at fixed temperatures. A complete description of the dynamics requires the existence of three distinct regions within the fluid layer: a geostrophically balanced interior where fluid motions are predominantly aligned with the axis of rotation, Ekman boundary layers immediately adjacent to the bounding plates, and thermal wind layers driven by Ekman pumping in between. The reduced model uses a classical Ekman pumping parameterization to alleviate the need to resolve the Ekman boundary layers. Results are presented for both linear stability theory and a special class of nonlinear solutions described by a single horizontal spatial wavenumber. It is shown that Ekman pumping (which correlates positively with interior convection) allows for significant enhancement in the heat transport relative to that observed in simulations with stress-free boundaries. Without the intermediate thermal wind layer, the nonlinear feedback from Ekman pumping would be able to generate heat transport that diverges to infinity at finite Rayleigh number. This layer arrests this blowup, resulting in finite heat transport at a significantly enhanced value. With increasing buoyancy forcing, the heat transport transitions to a more efficient regime, a transition that is always achieved within the regime of asymptotic validity of the theory, suggesting that this behaviour may be prevalent in geophysical and astrophysical settings. As the rotation rate increases, the slope of the heat transport curve below this transition steepens, a result that is in agreement with observations from laboratory experiments and direct numerical simulations.

The breakdown of the anelastic approximation in rotating compressible convection: implications for astrophysical systems

The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e. incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilize the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convective instability. In combination, these effects increase the relative magnitude of the time derivative of the density perturbation contained in the conservation of mass equation to non-negligible levels; the resulting convective instabilities occur in the form of compressional quasi-geostrophic oscillations. We find that the anelastic equations, which neglect this term, cannot capture these instabilities and possess spuriously growing eigenmodes in the rapidly rotating, low Prandtl number regime. It is shown that the Mach number for rapidly rotating compressible convection is intrinsically small for all background states, regardless of the departure from adiabaticity.

Shear-driven parametric instability in a precessing sphere

The present numerical study aims at shedding light on the mechanism underlying the precessional instability in a sphere. Precessional instabilities in the form of parametric resonance due to topographic coupling have been reported in a spheroidal geometry both analytically and numerically. We show that such parametric resonances can also develop in spherical geometry due to the conical shear layers driven by the Ekman pumping singularities at the critical latitudes. Scaling considerations lead to a stability criterion of the form |Po| > O(E4/5), where Po represents the Poincare number and E represents the Ekman number. The predicted threshold is consistent with our numerical simulations as well as previous experimental results. When the precessional forcing is supercriticial, our simulations show evidence of an inverse cascade, i.e., small scale flows merging into large scale cyclones with a retrograde drift. Finally, it is shown that this instability mechanism may be relevant to precessing celestial bodi...

Onset of rotating and non-rotating convection in compressible and anelastic ideal gases

A linear stability analysis for compressible convection in a plane layer geometry both with and without the influence of rotation is presented. For the rotating cases we employ the tilted -plane geometry that allows for varying angles between the rotation and gravity vectors. The stability criteria for compressible and anelastic ideal gases is compared. As expected, the critical parameters for the compressible equations approach those of the anelastic equations as the background stratification approaches the adiabatic (anelastic) limit. For the rotating cases, we observe asymptotic scaling behavior in the critical parameters in both compressible and anelastic fluids as the Taylor number becomes large. In contrast to the incompressible limit, finite tilt angles between the gravity and rotation vectors result in propagating compressible Rossby waves as the most unstable eigenmode and the critical parameters are established for a range of stratification levels and Taylor numbers; all wave orientations are found to propagate in prograde and equatorward directions for non-isothermal background states. We also compare the linear stability of the thermodynamically rigorous anelastic equations with an anelastic model that replaces thermal diffusion with an entropy diffusion-like term in the energy equation; it is shown that the linear stability of the entropy diffusion model yields qualitatively similar results for the critical parameters in comparison to the full anelastic set. We show that a thermodynamically rigorous alternative to the entropy diffusion model is the isothermal adiabatic background state in which temperature and entropy become equivalent thermodynamic quantities and viscous heating becomes subdominant in the energy equation; the stability characteristics of this model are also presented.

Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers: Single mode solutions

The onset of dynamo action is investigated within the context of a newly developed low Rossby, low magnetic Prandtl number, convection-driven dynamo model. This multiscale model represents an asymptotically exact form of an α^{2} mean field dynamo model in which the small-scale convection is represented explicitly by finite amplitude, single mode solutions. Both steady and oscillatory convection are considered for a variety of horizontal planforms. The kinetic helicity is observed to be a monotonically increasing function of the Rayleigh number. As a result, very small magnetic Prandtl number dynamos can be found for sufficiently large Rayleigh numbers. All dynamos are found to be oscillatory with an oscillation frequency that increases as the strength of the convection is increased and the magnetic Prandtl number is reduced. Kinematic dynamo action is strongly controlled by the profile of the helicity; single mode solutions which exhibit boundary layer behavior in the helicity show a decrease in the efficiency of dynamo action due to the enhancement of magnetic diffusion in the boundary layer regions. For a given value of the Rayleigh number, lower magnetic Prandtl number dynamos are excited for the case of oscillatory convection in comparison to steady convection. With regard to planetary dynamos, these results suggest that the low magnetic Prandtl number dynamos typical of liquid metals are more easily driven by thermal convection than by compositional convection.

A fully spectral methodology for magnetohydrodynamic calculations in a whole sphere

Abstract We present a fully spectral methodology for magnetohydrodynamic (MHD) calculations in a whole sphere. The use of Jones–Worland polynomials for the radial expansion guarantees that the physical variables remain infinitely differentiable throughout the spherical volume. Furthermore, we present a mathematically motivated and systematic strategy to relax the very stringent time step constraint that is present close to the origin when a spherical harmonic expansion is used for the angular direction. The new constraint allows for significant savings even on relatively simple solutions as demonstrated on the so-called full sphere benchmark, a specific problem with a very accurately-known solution. The numerical implementation uses a 2D data decomposition which allows it to scale to thousands of cores on present-day high performance computing systems. In addition to validation results, we also present three new whole sphere dynamo solutions that present a relatively simple structure.

A computationally efficient spectral method for modeling core dynamics

An efficient, spectral numerical method is presented for solving problems in a spherical shell geometry that employs spherical harmonics in the angular dimensions and Chebyshev polynomials in the radial direction. We exploit the three-term recurrence relation for Chebyshev polynomials that renders all matrices sparse in spectral space. This approach is significantly more efficient than the collocation approach and is generalizable to both the Galerkin and tau methodologies for enforcing boundary conditions. The sparsity of the matrices reduces the computational complexity of the linear solution of implicit-explicit timestepping schemes to

Three-dimensional quasi-geostrophic convection in the rotating cylindrical annulus with steeply sloping endwalls

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