High Performance Computing in Science and Engineering’ 04

Scientific computing on super-computers has become one of the standard methods of research in physics. This is clearly demonstrated by the articles in this section, which present a selection of projects related to physical re-search currently running at the HLRS. The presented work does not only cover a wide range of physics, it is also dominated by long-term projects which continuously have progressed over several years applying well estab-lished numerical methods.

We investigate the chain conformations and phase separation in binary polymer blends. Using large scale semi-grandcanonical Monte Carlo simulations and finite size scaling, we investigate the molecular extension and the intermolecular paircorrelation function in thin films with hard, non-preferentially adsorbing surfaces. The interplay between chain conformations, demixing and the validity of mean field theory is investigated for a large variation of chain lengths 16 ≤

N

≤ 512. Three regimes of film thickness

D

can be distinguished: (i) For film thicknesses much larger than the unperturbed chain extension

R

e

, bulk behavior is observed, i.e., the critical temperature of demixing

T

c

increases linearly with chain length, and the mean field theory becomes asymptotically correct for large

N

. (ii) For

D

∼

R

e

, the critical temperature scales linearly,

T

c

∼

N

, but the mean field theory overestimates the prefactor even in the limit

N

→ ∞ (iii) For ultrathin films, the chain conformations are quasi-two-dimensional,

T

c

∼ √

N

and mean field theory completely fails.

Lattice Monte-Carlo simulations of quantum chromodyanmics (QCD) have shown that strongly interacting matter undergoes a phase transition at some temperature

T

c

[

1

]. Above the transition temperatures

T

c

hadrons (the experimentally observed strongly interacting particles) cease to exist and a new state of matter, the so-called quark-gluon plasma (QGP) was predicted to exist [

1

]. One of the most prominent features of QGP is the screening of static chromoelectric fields. At large distances (i.e. distances much larger than the inverse temperature) the screening is exponential and can be parametrized by a temperature dependent chromoelectric screening mass. Chromoelectric screening masses have been extensively studied by us in the last few years [

2

–

3

] in the framework of our project (Nr. 11725). More recently chromoelectric screening has been studied in terms of the free energy of static quark anti-quark pair [

4

,

5

,

6

,

7

]. Moreover we developed a technique to separate energy and entropy contributions to the free energy [

8

,

9

].

Simulating the flow of suspensions is an extremely difficult and demanding problem. Suspensions are mixtures of fluid and granular materials, and each component alone is a challenge to numerical modelers. When they are combined in a suspension, neither component can be neglected, so all the difficulties of both fluids and grains must be solved, in addition to the new problem of describing the interaction between the two.

In the current project, we simulate thermal convection and generation of magnetic fields inside a fully convective star, like an M-dwarf or a T-tauri star. The numerical resolution required to reasonably well describe convection properties inside the star is 256

3

to 512

3

.

Symbiotic systems consist of a red giant undergoing strong mass loss and a white dwarf. More than hundred symbiotic stars are known, but only about ten systems show jet emission. The most famous systems are R Aquarii, CH Cyg and MWC 560. While the first two objects are seen at high inclinations — a fact which makes it possible to study the morphology and structure of jets of symbiotic stars — the jet axis in MWC 560 is practically parallel to the line of sight. This special orientation provides the opportunity to observe the outflowing gas as line absorption in the source spectrum (Schmid et al. 2001). Therefore MWC 560 can be used to probe the short term evolution and the propagation of the gas outflow in jets from white dwarfs.

We discuss image potential states on an insulator surface, LiF(001)-(1×1), within ab-initio many-body perturbation theory. The image potential states originate from the interaction of electrons outside the surface with polarization charges inside the substrate. They are responsible for characteristic features in the electron energy loss spectrum of the material. The onset of excitation energies is at 9.2 eV, which is several eV lower than the bulk excitations.

We have used the vectorised and parallelised magnetohydrodynamics code NIRVANA on the NEC SX-5 and the new SX-6 installation in parallel mode to simulate the interaction of jets with a galactic wind that might be typical for the star-bursting radio-galaxies of the early universe.

In the following we give a summary of the computational physics articles presented in the frame of solid-state physics. In the project of Prof. P. Nielaba from the physics department in Konstanz new insights into the electronic transport in nano-wires, elastic constants of model colloids, pore condensates and phase transitions in nano-systems in ex-ternal potentials and reduced geometry have been obtained. Despite the fact that many new experimental techniques have studied these properties in sys-tems of the size of the a few nanometers, the theoretical investigations are still in an initial stage. In this field computer simulations have become more and more important since the nano-systems in reduced geometry contain typically between 10 and 10.000 particles. This size is nearly ideal for the application of computer simulation methods. The summary given by P. Nielaba then de-scribes in detail the computational efficiency implemented at the HLRS in Stuttgart and the results obtained in the projects in 2003.

Elastic and structural properties of model colloids have been studied with particular emphasis on the effect of quenched impurities and of external fields. The structural and electronic properties of atomic wires has been analysed. In the following sections an overview is given on the results of our recent computations on quantum effects, structures and phase transitions in such systems.

Resolving the atomic and electronic structures of nanoclusters represents an important preliminary for their controlled use in future nanotechnologies. Here we show through the comparison of density-functional calculations with high-resolution photoelectron spectroscopy that 1.4 nm nanoparticles of silver (negatively charged clusters of 53 to 58 atoms) are icosahedral-based structures displaying a perfect icosahedral-induced electronic shell structure for Ag

55

−

and slightly perturbed shell structures for the neighboring cluster sizes. At variance, 55-atom gold clusters exhibit several isomeric structures of low symmetry, with a largely diminished electronic shell structure. This surprising qualitative difference is attributed to strong relativistic bonding effects in gold.

Large scale molecular dynamics (MD) computer simulations are used to study the amorphous alkali silicates (Li

2

O)(2·SiO

2

) [LS2], (K

2

O)(2·SiO

2

) [KS2], and (0.5·Li

2

O)(0.5·K

2

O)(2·SiO

2

) [LKS2]. These systems are characterized by a fast alkali ion motion in a relatively immobile Si-0 matrix. We investigate the so-called mixed alkali effect (MAE) which is reflected as a significant decrease of the alkali ion diffusion constants in LKS2 as compared to the corresponding binary systems LS2 and KS2. We show that the subtle interplay between the structure on intermediate length scales and the alkali diffusion is important to understand the microscopic origin of the MAE.

The adsorption on methylchloride (CH

3

Cl) on the Si(001) surface is studied by

first-principles

calculations using the gradient-corrected density-functional theory (DFT-GGA) together with ultrasoft pseudopotentials and a plane-wave basis set. The energetically favoured adsorption geometries are examined with respect to their bandstructures, surface dipoles and charge transfer characteristics.

Quantum Monte Carlo simulations are used to study the dynamics and the critical properties of strongly correlated systems relevant to the fields of cold quantum gasses and high-

T

c

superconductivity. Recent advances in cooling techniques of quantum gasses allow to reach the degenerate regime for fermionic samples. Loading these systems on optical lattices can bring the gas to a strongly correlated regime. We analyze the properties of trapped degenerate Fermi gasses on optical lattices and show that they display quantum critical behavior and universality at the boundaries between metallic and Mott insulating phases. On our other field of interest, high-

T

c

superconductivity, a Quantum Monte Carlo algorithm we developed recently is used to study the dynamics of the nearest-neighbor (n.n)

t-J

model relevant to the low energy properties of the copper oxides materials. We show that antiholons identified in the supersymmetric inverse squared (ISE)

t-J

model are generic excitation of the n.n. model since they are clearly visible in the single-particle spectral function of the n.n.

t-J

model in the whole Luttinger-liquid regime. We have further shown that even the analysis of the two-particle spectral functions of the n.n.

t-J

model can be based on the elementary excitations of the ISE

t-J

model.

We present a numerical study of the doping dependence of the spectral function of the n-type cuprates. Using cluster-perturbation theory and the self-energy-functional approach, we calculate the spectral function of the Hubbard model with next-nearest neighbor electronic hopping amplitude

t′

= −0.35

t

and on-site interaction

U

= 8

t

at half filling and doping levels ranging from

x

= 0.077 to

x

= 0.20. We show that a comprehensive description of the single particle spectrum of the electron doped cuprates is only possible within a strongly correlated model. Weak coupling approaches that are based upon a collapse of the Mott gap by vanishing on-site interaction

U

are ruled out.

Numerical simulation of complex flows has always demanded the biggest com-puters both in storage capacity and in performance that were available on the market. This situation is still going on. The following paragraph repre-sents a selection of papers that were submitted as yearly demanded progress reports to the HLRS. Although most of the reports revealed a very high sci-entific standard those papers were preferably selected for publication that clearly demostrated the unalterable usage of high performance computers (HFC) for the solution of the problem.

Investigations on laminar-turbulent transition for high-speed flows at hypersonic Mach-numbers will be presented. Dissociation takes place above a temperature of T>2000K within the boundary layer, a temperature which is reached easily at Mach-numbers above M=5. Additional degrees of freedom for the energy must be taken into account by employing a vibrational energy equation. Chemical reactions take place which are modeled by a 5-species model proposed by Park [

Par89

]. Further details on the chemical modeling can be found in [

Ste02

,

Ste03

].

Controlled disturbances can be introduced by means of a disturbance strip at the wall which is also capable to model point source disturbances. Results will be shown for free-flight conditions at an altitude of H=50Km and at a speed of M=20. Experiments for qualitative validation of the results are available in [

MM00

].

By means of spatial direct numerical simulations (DNS) based on the complete Navier-Stokes equations the effect of three-dimensional discrete suction on the spatial development of a laminar boundary-layer flow generic for the front part of a swept-back airliner wing has been investigated. The baseflow is an accelerated Falkner-Skan-Cooke boundary layer, on a swept wedge with semi-opening angle of 45° (Hartree parameter

β

H

= 0.5) which is mainly characterised by crossflow instability. The simulations of the microscale phenomena confirm that 3-d suction at the wall can excite unstable crossflow disturbances that have to be minimised by using either slot arrays or hole arrays with high porosity, otherwise the stabilising (2-d) effect of suction is compromised. Premature transition through oversuction could be identified as a convective secondary instability of the flow field deformed by strong steady crossflow vortices emerging from the suction panel.

Shock-wave/turbulent-boundary-layer interaction compression-ramp flow is a canonical test configuration for statistical turbulence modeling. Extensive experimental data are available, whereas computational data focus mainly on Reynolds-averaged computations employing a wide range of turbulence models. In figure 1 basic flow features are sketched [Zhe91]. The undisturbed incoming turbulent boundary layer interacts with the shock wave, for suffi-ciently large deflection angles resulting in a separation region near the com-pression corner, and a A-shock system containing the separation region. Sub-sequently the disturbed boundary layer passes through the Prandtl-Meyer expansion near the decompression corner and finally relaxes towards a devel-oped zero-pressure-gradient boundary layer.

In this work the large-eddy simulation (LES) is used to investigate incompressible flow around a sphere with trip wire. The sphere is located in a channel with square cross-section, and the bulk Reynolds number is

Re

= 50 000. The computational effort implied by demands for sufficient spatial and temporal resolution of the flow structures requires parallel runs on a high-performance computer. The numerical results are compared to the experimental ones in order to provide reliable data for testing, calibrating and improvement of statistical turbulence models. The time-averaged LES-results and the measured data obtained by the laser-Doppler-anemometry (LDA) for the velocity and the Reynolds-stress components are in reasonable agreement. Accuracy of the predicted mean-flow velocity component is particularly good. Comparison of the Reynolds stresses shows certain deviations in the far wake, agreement is however acceptable from the qualitative point of view.

Large Eddy Simulations of passive heat transfer in a turbine cascade with and without oncoming wakes have been performed. The computational geometry is chosen in accordance with previous experiments performed by Liu and Rodi in Karlsruhe. Oncoming wakes are found to influence the heat transfer between the outer flow and the turbine blades. The computations were carried out using 64 processors on the Hitachi-SR8000 F1.

To control the reaction progress in supersonic combustors the fuel/air mixing has to be optimized which is investigated numerically in this paper. The mixing process is strongly influenced by the design of the fuel strut injector. Optimization studies may help to improve the mixing efficiency of real size scramjet (supersonic combustion ramjet) engines. The strut design used in this paper is the result of previous experimental and numerical investigations [

1

,

2

,

3

]. It has been verified that the use of lobed strut injectors improves the mixing by generation of streamwise vortices in the core region of the combustion chamber. The present study compares two different nozzle exit designs using basically the same strut shape. 295 K cold hydrogen is injected with Mach 2.0 into a 1300 K hot Mach 2.0 supersonic air flow.

The aerodynamic performance, maneuverability and flight stability of air-crafts operating in the transonic regime are highly dependent on the deformation of their wings under aerodynamic loads. The accurate prediction of aeroelastic properties, such as aeroelastic equilibrium configurations under cruise conditions, is therefore crucial in early design stages. Due to increasing computer power and further development of numerical methods, direct numerical aeroelastic simulation, in which the governing equations for the fluid and the structure are solved consistently in time, has become feasible.

The project NHLRes ([

1

], [

2

]) is concerned with the simulation of aircraft aerodynamics and thus belongs to the research field of computational fluid dynamics (CFD) for aerospace applications. NHLRes comprises the numerical simulation of the viscous flow around transport aircraft high lift configurations based on the solution of the Reynolds-averaged Navier-Stokes equations. The project NHLRes consists of five parts representing the analysis of complex 3D-flow features, wake vortex simulation, optimization for three-dimensional high lift flow, aerodynamic interactions between the propeller and high lift wings and finally the usage of large eddy simulation (LES) of the flow around high lift configurations.

The present annual report summarises ongoing investigations performed at the Institut für Industrielle Fertigung und Fabrikbetrieb Universität Stuttgart (IFF) on the numerical simulation of spray painting in the automotive industry. Two examples, i.e. powder coating simulation using a corona spray gun and spray-painting simulation with high-speed rotary bell are showed. Numerical models for electrostatically supported painting processes were implemented in Fluent, a commercial CFD code based on an unstructured finite volume mesh. These models account for all important effects involved in the relevant physical processes, being able to predict the film thickness distribution and the paint transfer efficiency on the work piece. The calculations were carried out using VOLVOX-Cluster in the High Performance Computing Center Stuttgart.

In the last decades, tremendous progress has been made in the area of numerical methods and computer technology but also new experimental techniques evolved and have been transferred to new application areas. This article describes the combination of two recent and innovative techniques. On the numerical side, the lattice Boltzmann method (LBM) is used for detailed simulations of the flow in complicated 3-D structures. On the experimental side, the principles of nuclear magnetic resonance (NMR) are exploited to scan the 3-D structure of arbitrary objects (e.g. random packings of spheres) with a resolution of about 0.1 mm or better (magnetic resonance imaging, MRI) and to obtain information about the velocity of the fluid in selected planes of the same object. The combination of both methods allows for the first time with justifiable effort to investigate in 3-D and on a local level exactly the same arbitrarily complicated structures experimentally and numerically. This can be utilized first to validate the methods and results mutually, second to detect artifacts, but also third to replace or complement experimental investigations by “numerical experiments” on high performance computers which can provide a larger amount of detailed 3-D information with less effort.

The 3D Parallel-Multiblock URANUS code has been extended by models for radiative exchange between the surface elements and for heat conduction within the TPS (Thermal

P

rotection

S

ystem). The coupling of the newly developed models with catalytic effects for the real TPS, predicted by a global catalysis model, and with temperature dependent emissivity leads to significant differences in surface temperature distribution. Results for the X-38 re-entry vehicle will be discussed in some detail. Large memory and computational time requirements arise in order to solve the non-equilibrium Navier-Stokes equations on 1.02 million cells coupled with the surface models.

Chemistry is a science of change. At the heart of a chemical reaction, chemical change happens through the formation of new bonds (and breaking of of old ones). Properties such as the electronic and geometric structure of molecules and their energies are relevant to chemistry, but these are still static proper-ties. Chemical reactions on the other hand are dynamic, that is, something happens.

The iron(III) catalyzed Michael reaction works fine with simple enones, but other Michael acceptors such as acrylic acid methyl ester did not show any reactivity in the experiments done so far. Therefore we performed quantum chemical computations to assess the reactivity of various quite different Michael acceptors. Since previous studies showed that the C-C bone forming step most likely occurs at a mononuclear iron center with two dionato ligands, the barrier heights of such steps have been calculated with hybrid density functional methods. A mixed anhydride of acrylic acid and trifluoroacetic acid was identified as a very promising candidate to carry out further experiments.

Quantum chemical calculations at the MP2/[aug]-cc-pVDZ level were used to generate a two-dimensional potential energy surface for an unusual double proton transfer reaction in which the region around the transition state is characterized by a plateau of almost constant energy. A cut of the first electronically excited singlet state potential energy surface along the ground-state reaction path was computed using time-dependent density functional theory. In addition, solvent effects which lead to significant changes of the surface were studied using a self-consistent reaction field approach.

Reactive scattering is one of the fundamental processes in atomic and molecular collision dynamics. Reactions of hydrogen systems are of a particular interest in this respect, because they are amenable to the most rigorous theoretical treatment and thus represent ideal prototype cases for a detailed comparison of theory and experiment. This is best exemplified by the neutral hydrogen system H + H

2

and F + H

2

or the ionic system He + H

2

+

. Quantum chemistry has provided a very accurate potential energy surface for these systems, especially for H + H

2

. The collision dynamics is treated by quasi-classical trajectory or by rigorous, fully converged quantum methods. There is a considerable number of very detailed experimental results on which theory can be tested.

We present ab initio molecular dynamics (MD) simulations of the simplest amino acid, glycine, at the water / pyrite interface under extreme pressure / temperature conditions. These simulations are aimed to contribute to the discussion of the “iron-sulfur world” (ISW) scenario, an intriguing proposal in the controversial field of “Origin of Life” research. The simulations show that glycine easily desorbs from a water / pyrite interface through hydrogen-bond assistance. The retention time is only of the order of a picosecond and the surface bonding is best understood as a relatively weak electrostatic interaction. However, we have found indications of glycine activation due to the interaction with the surface, and thus for a possible reaction with a suitable anchor molecule.

Two papers in this volume deal with more theoretical aspects of computer sci-ence related to supercomputing. Both focus on the problem of benchmarking for high performance computer systems. This emphasises both the impor-tant role and the extreme difficulty of finding adequate methods to evaluate the performance of high speed systems. Both papers deal with performance aspects of parallel programs.

As the successor of

parb_ana

the project

parbana1

was started to find some answers to questions the first project left open. The main problem encountered in examining SX-4 and SX-5 at the

High Performance Computing Center Stuttgart (HLRS)

was the impossibility to test these machines with all planed trials concerning mixed workloads due to software hangups. To illustrate the problem as well as the undertaken efforts to solve it, we will divide this report into three parts. First,

PARbench

the used benchmark system is described briefly. Thereafter, the results and problems investigating the SX vector computers are shown. In the final part the changes made to

PARbench

as well as the new findings are discussed.

SKaMPI is now an established benchmark for MPI implementations. The development of SKaMPI-5 strives for improvements in several directions: (i) extension of the benchmark to cover more functionality of MPI, (ii) construction of a collection of collective algorithm kernels which are not supported by core MPI collective operations. (iii) a redesign of the SKaMPI benchmark allowing it to be extended more easily (thus matching requests from SKaMPI users).

In the present paper we give an overview of the extension of SKaMPI for the evaluation of virtual topologies, describe the foundations of new algorithms for fast all-to-all communication specifically tailored for the case of differing message sizes, and give a first impression of what SKaMPI-5 will look like, for which we now have a prototype running.

Earth sciences belong to those scientific disciphnes which have to rely on high-performance computing not only because of the complexity of the prob-lems this discipline has to cope with but also for other reasons. Since the geometric structures are in general three-dimensional, the storage capacities required are large, and because of the large amount of data to be processed, high computing speeds are also required. For the second time the transac-tions contain two articles on this subject in this chapter. They clearly confirm that application of high-performance computing in analyses of the earth sci-ences is an absolute must. Needless to say, that the articles reported here are concerned with key problems of the subject.

This paper is concerned with numerical considerations of fluid effects on wave propagation. The focus is on effective elastic properties (i.e. velocities) in different kinds of dry and fluid-saturated fractured media. We apply the so-called rotated staggered finite-difference grid (RSG) technique. Using this modified grid it is possible to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without explicit boundary conditions and without averaging elastic moduli. Therefore the RSG allows an efficient and precise numerical study of effective velocities in fractured structures. This is also true for structures where theoretically it is only possible to predict upper and lower bounds. We simulate the propagation of plane P- and S-waves through three kinds of randomly cracked 3D media. Each model realization differs in the porosity of the medium and is performed for dry and fluid-saturated pores. The synthetic results are compared with the predictions of the well known Gassmann equation and the Biot velocity relations. Although we have a very low porosity in our models, the numerical calculations showed that the Gassmann equation cannot be applied for isolated pores (thin penny-shaped cracks). For Fontainebleau sandstone we observe with our dynamic finite-difference approach the exact same elastic properties as with a static finite-element approach. For this case the Gassmann equation can be checked successfully. Additionally, we show that so-called open-cell Gaussian random field models are an useful tool to study wave propagation in fluid-saturated fractured media. For all synthetic models considered in this study the high-frequency limit of the Biot velocity relations is very close to the predictions of the Gassmann equation. However, using synthetic rock models saturated with artificial “heavy” water we can roughly estimate the corresponding tortuosity parameter.

This is a report on first steps for a combination of two numerical models of the evolution of the Earth’s mantle: The first one, K3, is a new 2-D convection-fractionation model that simulates the growth of continents and of the geochemically complementary depleted mantle reservoir. The second model shows the 3-D generation of oceanic lithospheric plates and subducting sheet-like downwellings in a spherical-shell mantle. Based on the abundances of the present-day geochemical reservoirs of Hofmann (1988) we developed a numerical dynamical model of convection and of chemical differentiation in the Earth’s mantle. It is shown that a growing and additionally laterally moving continent and a growing depleted mantle evolved from an initially homogeneous primordial mantle. The internal heat production density of the evolving mantle depends on the redistribution of the radioactive elements by fractionation and convection. The fractionation generates separate geochemical reservoirs. However, the convection blurs the reservoirs by mixing. Although we take into account also the effects of the two phase transitions in 410 and 660 km depth, it is essentially the dependence of the viscosity on radius which guarantees the conservation of the major geochemical reservoirs. This model has no internal compulsory conditions. The principal idea of this first model is to compute the relative viscosity variations as a function of depth from observable quantities. We develop a self-consistent theory using the Helmholtz free energy, the Ullmann-Pan’kov equation of state, the free volume Grüneisen parameter and Gilvarry’s formulation of Lindemann’s law. In order to receive the relative variations of the radial factor of the viscosity, we insert the pressure,

P

, the bulk modulus,

K

, and

∂K/∂P

from PREM. For mantle layers deeper than 771 km we used the perovskite melting curve by Zerr and Boehler (1993, 1994) in order to estimate the relative viscosity. For the calibration of the viscosity we have chosen the standard postglacial-uplift viscosity beneath the continental lithosphere. Furthermore, we took into account the dependence of the viscosity on temperature and on the degree of depletion of volatiles. An essential first new result of this paper is a

high-viscosity transition layer

and a

second low-viscosity

layer below it. Although our model mantle is essentially heated from within, we assume additionally a small heat flow at the CMB. This is necessary because of the dynamo theory of the outer core. The second main result of this first model is a

more distinct bipartition of the mantle

in a depleted upper part and a lower part rich in incompatible elements, yet. This result is rather insensitive to variations of the Rayleigh number and of the thermal boundary condition at CMB. The different parts of this paper are closely connected by the algorithm. The continuation of the first finding leads to a 3-D, up to now purely thermal model of mantle evolution and plate generation. This second model was used to carry out a series of three-dimensional compressible spherical-shell convection calculations with another new, but related viscosity profile, called

eta3

, that is derived from PREM and mineral physics, only. Here, the Birch-Murnaghan equation was used to derive the Grüneisen parameter as a function of depth. Adding the pressure dependence of the thermal expansion coefficient of mantle minerals, we derived the specific heats,

c

p

and

c

v

, too. Using the Gilvarry formulation, we found a new melting temperature of the mantle and the new viscosity profile,

eta3

. The features of

eta3

are a high-viscosity transition layer, a second low-viscosity layer beginning under the 660-km discontinuity, and a strong viscosity maximum in the central parts of the lower mantle. The rheology is Newtonian but it is supplemented by a viscoplastic yield stress,

∂

y

. A viscosity-level parameter,

r

n

, and

∂

y

have been varied. For a medium-sized Rayleigh-number-yield-stress area,

eta3

generates a stable, plate-tectonic behavior near the surface and simultaneously thin sheet-like downwellings in the depth. Outside this area three other types of solution were found. The presence of

two

internal low-viscosity layers and of ∂

y

is obviously conducive for plateness and thin sheet-like downwellings. The distribution of the downwellings is more Earth-like if the yield stress is added. The outlines of a combination of the two models have been discussed.