Jörg Schmalzl

Mixing in vigorous, time‐dependent three‐dimensional convection and application to Earth's mantle

An understanding of the mechanism of mixing in highly viscous convecting fluids is of crucial importance in explaining the observed geochemically heterogeneous nature of Earths mantle. Using constant viscosity numerical experiments, we describe the mixing mechanism of time-dependent Rayleigh-Benard convection with an infinite Prandtl number in a three-dimensional (3-D) rectangular container. Mixing is observed by following the positions of passive tracers advected by the flow. The major mixing mechanisms may be described in terms of the within-cell mixing and the cross-cell mixing. The flow structure in which tracers move on toroidal surfaces, that was previously observed in steady state 3-D convection systems is perturbed by boundary layer instabilities in the time-dependent experiments. This flow structure allows a very efficient exchange of mass between the boundary layers and the core of the convection cell even in the absence of time dependence. We compare this result with calculations carried out in two spatial dimensions. In similar two-dimensional (2-D) experiments, exchange of mass between boundary layers and core of the convection cell is entirely effected by the boundary layer instabilities. Mixing between neighboring cells appears much slower in three dimensions than in similar 2-D experiments, perhaps because the 3-D cell structure is more stable relative to the boundary layer instabilities. The inferred mixing rates are observed to be relatively insensitive to initial tracer location, but the timescale for mixing, tm, decreases with increasing Rayleigh number (tm goes approximately as Ra(−3/2)). The timescale of mixing is an important constraint on the large scale structure of Earth, because large-scale geochemical heterogeneities persist to the present day, implying that the mantle is not well mixed.

Mixing properties of three‐dimensional (3‐D) stationary convection

Passive tracers in steady‐state three‐dimensional (3‐D) convective flows with infinite Prandtl number, which is relevant for the Earth’s mantle, show a remarkable flow structure. Individual flowlines as shown by Poincare sections of the tracer paths lie on a two‐dimensional (2‐D) surface with distorted toroidal topology. Furthermore, the space occupied by the convecting fluid is filled by a set of these toroidal surfaces nested one within another. The small radius of the innermost toroidal surface approaches zero, defining a closed streamline whose location we have determined in specific cases using numerical solutions. The outermost of the toroidal surfaces coincides with the upper and lower surfaces of the layer and with vertical symmetry planes which separate the flow between neighboring cells. Both square and hexagonal convection planforms show a triangular cellular structure with triangles defined by (π/2,π/4,π/4) and (π/2,π/6,π/3), respectively. The outer toroidal surface is closed by a horizontal f...

Mixing the Earth's mantle by thermal convection: A scale dependent phenomenon

We have investigated the mixing properties of 2D Rayleigh Benard convection in infinite Prandtl number flows. The mixing properties were monitored by injecting tracer particles into previously calculated flow fields. A particle correlation function H(r) and the corresponding correlation dimension have been used to characterize the mixing efficiency of the flow as function of its vigor and its structure. We demonstrate that the chosen method captures the mixing process in a detailed manner. Our study suggests that mixing properties of the flow depend on the spatial scale. At Rayleighnumbers 106 < Ra < 108 heterogeneities within one circulation cell are destroyed rapidly while two adjacent cells can remain unmixed substantially longer.

The Influence of the Prandtl Number on the Style of Vigorous Thermal Convection

The effect of the Prandtl number on convection in a planar three-dimensional geometry is investigated in this study. We have employed a numerical scheme to integrate the governing equations. Differently from previous studies we have chosen stress-free boundaries. Experiments have been performed at a Rayleigh number of Ra = 10 6 for Prandtl numbers (Pr) ranging from 0.025 to 100. We have further conducted one experiment in the limiting case of infinite Prandtl number. Despite the differences in the geometry and the boundary conditions, as compared to other studies, we find a similar transition in the dynamics of the flow when the Prandtl number is increased. While the velocity and the temperature structure show diffusive character at low Pr, sharp thermal boundary layers form at high Pr. The heat transport efficiency increases with Pr until a transition value is reached, from there on Nu behaves almost asymptotically. The transition can not be caused by a change in hierarchies between velocity and thermal boundary layers, as suggested in other studies. Due to the stress-free boundaries, a velocity boundary layer does not exist. We observe that the toroidal part of the flow is strong at low Pr and looses its strength with increasing Pr, thus it is likely to be responsible for the transition. In a further chapter we demonstrate that due to the neglect of the toroidal part in two-dimensional calculations at low Pr results are obtained which are misleading, even in a qualitative sense. Infinite Pr results from 2D calculations closely resemble the dynamics of fully 3D flows.

A fully implicit model for simulating dynamo action in a Cartesian domain

Abstract We present a fully implicit numerical method to solve the incompressible MHD equations in a strongly rotating Cartesian domain. The equations are solved in a primitive variable formulation using a finite volume discretization. In order to use massively parallel computers, we applied a domain decomposition approach in space. The performance of this model is compared with an earlier model, which treated the convective terms of the equations in an explicit manner. Our results indicate that although the fully implicit method needs about three times the memory of the implicit–explicit method, it is superior in terms of computational efficiency. As an application of this model, we investigated the influence of the Prandtl number in the range of 0.01–1000 on the dynamics of the dynamo.

Using standard image compression algorithms to store data from computational fluid dynamics

Three-dimensional numerical modeling of fluid flows is an important research tool to understand many fluid dynamical effects observed in nature. With the strong growth of available computational resources the use of such models has greatly increased over the last years. Because the available mass storage has not increased in the same order as the CPU speed many researchers nowadays face the problem of how to store and transfer the large data sets produced by the model calculations for post-processing. The use of lossy wavelet-based compression techniques on this data has been investigated in several publications. These techniques are often specialized to one problem and are not easy to implement. In the area of digital media, however, advances have been made for still image (JPEG, JPEG-2000) and motion image (MPEG) compression. In this paper we investigate the usefulness of these image compression algorithms for the storage of data from computational fluid dynamics on regular cartesian grids. We analyze both the compression ratios achieved and the error introduced by these lossy compression schemes. We found that, for our purposes, the JPEG compression scheme allows an easy-to-use, portable, robust, and computationally efficient lossy compression. For the easy use of these compression algorithms we present a simple wrapper library.

Variable quality compression of fluid dynamical data sets using a 3‐D DCT technique

In this paper we present a data compression scheme that is especially suited for the compression of data sets resulting from computational fluid dynamics (CFD). By adopting the concept of the JPEG image compression standard and extending the approach of Schmalzl (2003), we employ a three-dimensional discrete cosine transform of the data. The resulting frequency components are rearranged, quantized, and, finally, stored using standard variable length integer codes. The compression ratio and also the introduced loss of accuracy can be adjusted over a wide range by means of two compression parameters to give the desired compression profile. Using the proposed technique, compression ratios of more than 60:1 are possible with a mean error of the compressed data of less than 0.05%.

Using an optical mouse sensor to track geophysical field measurements

To position arbitrary outdoor measurements, the most common methods are the use of global positioning system (GPS) technology or distance tracking from reference locations. The system to be introduced here consists of a combination of an optical distance tracking system and a civil, autonomous GPS receiver. Through the application of a recursive filter, both components support and correct each other mutually. The system is intended as an autonomous, real-time and low-cost alternative to expensive differential GPS solutions in geophysical field surveying. The combination approach features a very good relative and decent absolute positioning precision. First measurements using the two-coil geo-electromagnetics instrument Geonics EM31 in combination with the developed positioning system show promising results.

The Dynamics of Subcritical Double-Diffusive Convection in the Southern Ocean: An Application to Polynyas

We investigated the nature of subcritical double diffusive convection in the southern ocean with a time dependent two-dimensional finite-element method based on stream-function, compositional and temperature fields. The initial and boundary conditions are chosen with special respect to open-water polynyas which play an important role in the heat budget and in the gas exchange of the antarctic ocean and the polar atmosphere. The temperature and salinity profiles from the Mikhail Somov measurements in the Weddell gyre (1981) have been used in our calculations. They showed cold fresh water overlaying a warm layer with a higher salt concentration. Input of freshwater has been neglected in our model. We have investigated the evolution of overturning convection from an initially layered state and its dependence on the thermal and chemical Rayleigh number. Our results indicate that the initially layered period is important for the transport of heat and salt from the lower to the upper, cold and fresh layer. Even under conditions which are stable in the static sense, we observed overturning convection with a high heat transport rate.