J. Fluid Mech. (2013), . 735, pp. doi:10.1017/jfm.2013.507 Sheet … · 2020. 1. 24. · 650 B. Futterer et al. To rearrange all the latter results in our context, we ﬁrst refer

J Fluid Mech (2013) vol 735 pp 647ndash683 ccopy Cambridge University Press 2013 647doi101017jfm2013507

Sheet-like and plume-like thermal flow in aspherical convection experiment performed

under microgravity

B Futterer12dagger A Krebs1 A-C Plesa3 F Zaussinger1 R Hollerbach4D Breuer3 and C Egbers1

1Chair of Aerodynamics and Fluid Mechanics Brandenburg University of Technology CottbusSiemens-Halske-Ring 14 03046 Cottbus Germany

2Institute of Fluid Dynamics and Thermodynamics Otto von Guericke Universitat MagdeburgUniversitatsplatz 18 39106 Magdeburg Germany

3Institute of Planetary Research German Aerospace Center Rutherfordstrasse 212489 Berlin Germany

4Institute of Geophysics Earth and Planetary Magnetism Group ETH Zurich Sonneggstrasse 58092 Zurich Switzerland

(Received 8 February 2013 revised 21 August 2013 accepted 19 September 2013first published online 29 October 2013)

We introduce in spherical geometry experiments on electro-hydrodynamic drivenRayleighndashBenard convection that have been performed for both temperature-independent (lsquoGeoFlow Irsquo) and temperature-dependent fluid viscosity properties(lsquoGeoFlow IIrsquo) with a measured viscosity contrast up to 15 To set up a self-gravitating force field we use a high-voltage potential between the inner and outerboundaries and a dielectric insulating liquid the experiments were performed undermicrogravity conditions on the International Space Station We further run numericalsimulations in three-dimensional spherical geometry to reproduce the results obtainedin the lsquoGeoFlowrsquo experiments We use Wollaston prism shearing interferometry forflow visualization ndash an optical method producing fringe pattern images The flowpatterns differ between our two experiments In lsquoGeoFlow Irsquo we see a sheet-likethermal flow In this case convection patterns have been successfully reproducedby three-dimensional numerical simulations using two different and independentlydeveloped codes In contrast in lsquoGeoFlow IIrsquo we obtain plume-like structuresInterestingly numerical simulations do not yield this type of solution for the lowviscosity contrast realized in the experiment However using a viscosity contrast oftwo orders of magnitude or higher we can reproduce the patterns obtained in thelsquoGeoFlow IIrsquo experiment from which we conclude that nonlinear effects shift theeffective viscosity ratio

Key words Benard convection geophysical and geological flows nonlinear dynamicalsystems

dagger Email address for correspondence futterertu-cottbusde

648 B Futterer et al

1 IntroductionConvection in spherical shells under the influence of a radial buoyancy force is of

considerable interest in geophysical flows such as mantle convection (eg Schubert ampBercovici 2009) Laboratory experiments involving this configuration are complicatedby the fact that gravity is then vertically downwards rather than radially inwardsOne alternative is to conduct the experiment in microgravity thereby switchingoff the vertically downward buoyancy force Imposing a voltage difference betweenthe inner and outer shells and using the temperature dependence of the fluidrsquosdielectric properties can further create a radial force The first such thermo-electro-hydrodynamic (TEHD) experiment was designed by Hart Glatzmaier amp Toomre(1986) using a hemispherical shell and was accomplished on the space shuttleChallenger in May 1985

A later experiment called lsquoGeoFlowrsquo not only involved a full shell instead ofa hemisphere but also was designed to be performed on the International SpaceStation (ISS) where considerably longer flight times are possible (Egbers et al 2003)Two versions of lsquoGeoFlowrsquo have now been accomplished lsquoGeoFlow Irsquo involving anessentially constant-viscosity fluid was processed from August 2008 to January 2009(Futterer et al 2010) lsquoGeoFlow IIrsquo involving a fluid with pronounced temperature-dependent viscosity was processed from March 2011 to May 2012 (Futterer et al2012) The purpose of this paper is to present an integrative comparison of the twoexperiments with one another with accompanying numerical results and with relatedresults in the literature

The arrangement of this paper is as follows With the review in sect 2 we present anextended literature study related to thermo-viscous RayleighndashBenard convection Therewe also address the differences between Cartesian and spherical geometries Thenin sect 3 we go through the physical basis for our TEHD experiment The discussionof the list of properties involved in our experiment is the basis for linking thelsquoclassicalrsquo spherical RayleighndashBenard system (driven by Archimedean buoyancy) withour TEHD driven RayleighndashBenard experiment The final objective in this sectionis the calculation of the electric Rayleigh number from the experimental data andreordering it into domains available from the literature The lsquoGeoFlowrsquo experimentseries uses a very specific flow visualization the Wollaston schlieren interferometry orWollaston prism shearing interferometry The observed patterns of convection deliverfringe images which requires additional knowledge on interpretation possibilities Thisis demonstrated with a generic example in sect 4 Then in sectsect 5 and 6 we discussthe experimentally observed flow regimes and corresponding numerical models forboth experiment series lsquoGeoFlow Irsquo and lsquoGeoFlow IIrsquo Based on further numericalsimulations the deliberation of sect 7 is on heat transfer properties related to differentconvective patterns We summarize and give conclusions in sect 8 involving a discussionon the nonlinear effects considered in our numerical models and an outlook forrecently performed experiment series lsquoGeoFlow IIbrsquo

2 Thermally driven flow in liquids of temperature-dependent viscosityIn mantle dynamics the basics of fluid flow movement and its interaction with

tectonic plates are still the object of study As extensively reviewed in Schubert ampBercovici (2009) research on those topics is focused on theoretical numerical andexperimental materials and methods and moreover on the geophysical observationsthemselves Especially laboratory experiments have the character of lsquoexploring newphysics and testing theoriesrsquo (Davaille amp Limare 2009) There authors summarizethe results from a huge series of tank experiments which are designed to understand

Thermal flow in spherical convection experiments under microgravity 649

RayleighndashBenard convection phenomena and its relations to overall mantle dynamicsThey focus on the generation of plumes in highly viscous Newtonian and non-Newtonian liquids with a strong temperature dependence of the viscosity Moreoverthey describe their thermophysical and geometrical properties and present theirinteraction with the thermal boundaries Finally the authors relate their results togeophysical globally observed convectively driven zones in the Earthrsquos mantle

Generally in simple mantle dynamics (eg neglecting internal heating andthermochemical or non-Newtonian convection) we consider the thermal driving ofa RayleighndashBenard system filled with a highly viscous liquid by the Rayleigh number

Ra= αg1Td3

νrefκ (21)

using the following symbols α coefficient of volume expansion g acceleration due togravity 1T = (Thot minus Tref ) temperature difference between the geometrical boundarieswhich are heated to Thot and cooled at a reference value Tref d characteristic lengthscale of the geometry (eg outer radius of a spherical shell system minus inner radiusd = ro minus ri) νref reference kinematic viscosity and κ thermal diffusivity From nowon we will consider that the reference index ref refers to the cool boundary of thesystem where the various parameters are taken With the Prandtl number

Pr = νref

κ(22)

treated as infinite Pr rarrinfin we ignore inertial effects Typical values for the Earthrsquosmantle are νrarr O(1016) and κ rarr O(10minus7) and lead to this assumption (Schubert ampBercovici 2009) In this section we discuss results for thermally driven flow in fluidswith temperature-dependent viscosity This property is assessed with the viscosity ratio

γ = νref

ν(Tref+1T)= νcold

νhot (23)

which delivers the ratio between the reference viscosity at the cool boundary and theactual lowest possible viscosity at the hot boundary due to a specific temperaturedifference in the system Further temperature-dependent properties eg thermalexpansion seem to have a rather low impact on the dynamics (Ogawa 2008) comparedto the viscosity effects Complex double-diffusion convection models have beenpresented by for example Hansen amp Yuen (1994) However owing to our usedhomogeneous experimental fluid chemical aspects are not regarded here

During the past decades various studies have addressed the topic of temperature-dependent viscosity mostly for Cartesian geometries (Booker 1976 Nataf amp Richter1982 Stengel Oliver amp Booker 1982 Richter Nataf amp Daly 1983 Morris ampCanright 1984 Busse amp Frick 1985 White 1988 Christensen amp Harder 1991 OgawaSchubert amp Zebib 1991 Hansen amp Yuen 1993 Davaille amp Jaupart 1994 Solomatov1995 Tackley 1996 Kameyama amp Ogawa 2000) While numerical simulations andlaboratory experiments consider the RayleighndashBenard system in a rectangular boxnaturally the spherical shell convection is treated by means of numerical simulationsonly for constant viscosity (Baumgardner 1985 Bercovici Schubert amp Glatzmaier1989a Bercovici et al 1989b Bercovici Schubert amp Glatzmaier 1991 1992Schubert Glatzmaier amp Travis 1993) and for varying viscosity (Ratcliff Schubertamp Zebib 1996 Kellogg amp King 1997 Ratcliff et al 1997 Zhong et al 2000Yanagisawa amp Yamagishi 2005 Stemmer Harder amp U Hansen 2006 Hernlund ampTackley 2008 Huttig amp Stemmer 2008b Zhong et al 2008 Huttig amp Breuer 2011)

To rearrange all the latter results in our context we first refer to Solomatov (1995)who introduces scaling relations for three main viscosity contrasts in rectangulargeometry ie the mobile-lid sluggish-lid and stagnant-lid convection The mobile-liddomain is characterized by convection in isoviscous systems and systems with smallviscosity contrast where the surface layers are mobile and convection cells reach theouterupper cold boundary of the sphericalCartesian box domain The sluggish-liddomain also known as the transitional domain is reached for larger viscosity contrastsγ isin (102 104) resulting in a reduction of the velocity at the surface In systemswith viscosity contrasts γ gt 104 convection develops below an immobile lid throughwhich heat is transported only by conduction In this regime stagnant-lid convectionoccurs Ratcliff et al (1997) and Huttig amp Breuer (2011) find a low-degree regime forbottom-heated and purely internally heated convection in a three-dimensional sphericalshell respectively This regime has been found to lie between the mobile-lid andstagnant-lid domains and is characterized by long wavelengths Low degrees forbottom-heated convection in a three-dimensional Cartesian box were also observedby Tackley (1993)

Moreover Solomatov (1995) describes the mobile-lid domain behaving as aconstant-viscosity domain with lsquotwo thermal boundary layers of approximately equalthicknessrsquo The limit between the mobile-lid and sluggish-lid domains not only isdefined by the viscosity ratio itself but also considers a different energetic balanceof the cold and hot boundary layers This is shown in figure 2 of Androvandiet al (2011) where the authors collect all the relevant numerical and experimentalcontributions and describe the obtained convection pattern as a function of the top andbottom Rayleigh numbers in a three-dimensional Cartesian box geometry with

Rahot = γ Racold (24)

when Ratop = Racold = Ra (equation (21)) In figure 2 of Androvandi et al (2011)the transition between mobile-lid and sluggish-lid domains is markedly analogous tothe situation in Solomatov (1995) for a top Rayleigh number Racold = 106 at γ = 102whereas for Racold = 105 the threshold is already at γ = 101 Thus a sluggish-liddomain can be assessed by lower Rayleigh numbers and smaller viscosity contrastswhereas for higher Ra a higher viscosity ratio is needed to reach it

Androvandi et al (2011) collect experimental data from rigid boundary conditionsin the same diagram together with numerical data from free-slip numerical simulationsIn addition the experiments investigate slightly different parameters in comparison tothe numerical simulation This might be related to the fact that frequently experimentsrun in a higher domain than numerical simulations achieve However experimentscapture nonlinear effects and associated instabilities at most (Schubert amp Olson2009) It is also obvious that there are still huge regimes not fully assessed eitherwith experiments or with numerical simulations Nevertheless the generic regimes fordifferent fluid flow behaviours are visible in figure 2 of Androvandi et al (2011) Inthe mobile-lid domain if either the top or the bottom Ra is increased above a criticalvalue a transition from steady state to time-dependent convection can be observedFor supercritical Ra plumes are the specific convective pattern not only for the nearlyconstant viscosity contrast but also for the stagnant-lid domain In the domains inbetween we have a stabilization ie transition from steady state to time-dependentbehaviour occurs only at higher critical Ra But the planform of convection becomesrather complex Especially for intermediate viscosity ratio and high Ra the authorscapture the coexistence of several scales of convection ie slabs and plumes

Steady mobile-lid ndash tetrahedralcubic patternie sheet-like downwellingand cylindrical upwellingfor bottom heatedndash sheet-like upwelling andcylindrical downwellingfor pure internally heated

Steady sluggish-lid

ndash low-degree convection with

cylindrical downwellings

ndash chains of plumes upwelling

Steady stagnant-lidndash plume-like upwelling and sheet-likedownwelling for bottom heatedndash plume-like downwelling and sheet-likeupwellings for pure internally heated

Time-dependent

Mobile-lid

Sluggish-lid

Stagnant-lid

Baumgardner (1985)

Bercovici et al (1989b)

Bercovici et al (1992)

Bercovici et al (1989a)

Schubert et al (1993)

Ratcliff et al (1996)

Stemmer et al (2006)

Hernlund amp Tackley (2008)

Zhong et al (2008)

10ndash1107 108 109106105104

1010103

FIGURE 1 (Colour online) Spatio-temporal behaviour of incompressible NewtonianBoussinesq convection in spherical shells as a function of cool outer Rayleigh and hotinner Rayleigh numbers related via the viscosity ratio by Rahot = γRacold with Racold = Ra =(αg1Td3)(νrefκ) The graphical illustration follows the idea from figure 2 in Androvandiet al (2011) who introduced it for the Cartesian box With the viscosity ratio γ wedistinguish the mobile-lid (γ lt 102) the sluggish-lid and stagnant-lid (γ gt 104) domainsThe description of the convective planform in the steady flow cases for all viscosity domainsgives evidence on the influence of γ ie isolated plumes are only induced via increasingthe viscosity ratio Considering internal heating reverses the direction of the thermally drivenflow eg from upwelling to downwelling in comparison to bottom heated spherical shellconvection The transition to time dependence in the upper parameter domain is not describedin the literature In addition it is not clearly discussed by means of periodicity or chaotic timeseries and therewith only assessed qualitatively

For the spherical geometry we present the different convecting regimes found inthe literature in a similar way to Androvandi et al (2011) for Cartesian geometry(figure 1) All the numerical simulations which are of relevance here are performedfor a radius ratio of inner to outer spherical radius η = riro = 055 and infinite Prandtlnumber Pr rarrinfin with free-slip boundary conditions assuming an incompressibleNewtonian Boussinesq fluid The authors vary the Rayleigh number up to Ra12 = 107

and consider the viscosity ratio up to γ lt 105 via the FrankndashKamenetskii viscosityapproximation ie a linear approximation of the exponent in the Arrhenius law

micro= a eminus ln(γ )(TminusTref ) (25)

Commonly they choose Ra12 which references the viscosity at a non-dimensionaltemperature of 05 (note that the non-dimensional temperature is between 0 and 1in the studied domain) Stemmer et al (2006) give a general expression relating thedifferent Rayleigh numbers Ra12 Racold and Rahot for our case to

Rahot = γ 05 Ra12 = γ Racold (26)

Especially for the control of experiments it is more convenient to use either the inneror outer Rayleigh number at the hot or cool boundaries respectively In this paper ourreference is always the lower Rayleigh number at the outer cool boundary

By recalculating the specific Rayleigh numbers from (26) we collect the literatureresults from numerical experiments analogously to Androvandi et al (2011) whichresults in our figure 1 for incompressible Newtonian Boussinesq fluids The convectiveregimes are again divided into domains with low transitional and high viscositycontrast equivalent to mobile- sluggish- and stagnant-lid domains Note that it isalso possible to use the term lsquonon-Boussinesqrsquo related to such a strong temperaturedependence of the kinematic viscosity of the fluid which breaks the symmetrybetween the top and bottom boundary layers (Zhang Childress amp Libchaber 1997)To the best of our knowledge a scaling for the domains as presented in Solomatov(1995) is not available for three-dimensional spherical geometry apart from for purelyinternally heated convection (Huttig amp Breuer 2011) which is not under considerationhere Nevertheless in figure 1 we observe for the constant and nearly constantviscosity domains the transition from steady to time-dependent flow states Howeverthe authors describe the time dependences for the planform of convection solelyqualitatively It is the planform of steady-state convection that differs clearly from therectangular box The spherical symmetry delivers the coexistence of two patterns atetrahedral and a cubic mode which are both characterized by sheet-like downwellingsand plume-like upwellings The increase of the viscosity ratio induces the planform ofcylindrical downwellings at the pole and a chain of plumes upwelling at the equator(Ratcliff et al 1996) In the stagnant-lid domain thermal up- and downwellings arein the form of plumes only From the literature it is not possible to compare time-dependent patterns between the Cartesian and spherical geometries as they are notfully assessed Only Ratcliff et al (1997) present both systems concurrently Forthe mobile-lid domain they describe the upwelling to be sheet-like in the box butplume-like in the sphere

Besides the onset of convection for constant-viscosity fluids (γ = 1) which isRahot = Racold = Ra = 712 (Bercovici et al 1989a Ratcliff et al 1996) detailedcritical Rayleigh numbers for the transitions between different convective regimesare not available That value for the onset is below the well-known value of Ra= 1708in an infinite plane layer (Hebert et al 2010) It is not possible to compare the twovalues directly as for the spherical geometry the critical value depends on both thecurvature δ = (ro minus ri)ri which is related to the radius ratio by δ = 1η minus 1 andthe critical mode (Chandrasekhar 1981) If the radius ratio is increasing we get intothe limit of the plane layer reaching the value of Ra = 1719 (Zhang Liao amp Zhang2002) If we have a lower η with higher δ the critical Ra is decreasing (eg downto 712 for η = 055) Thus it is obvious that an increase of the curvature decreasesthe dimension of the hot boundary layer in comparison to the cold boundary layerThis imbalance might lead to a more unstable situation already at a smaller criticaltemperature difference in addition to the thickness of the fluid layer increasing

Generally the viscosity contrast also has an influence on the critical Rayleighnumber for the onset of convection It decreases with increasing viscosity contrast ifthe reference is defined at the cold outer boundary Ratcliff et al (1996) present atη = 055 eg for γ = 103 Ra12 = 2196 (corresponding to Racold = 69) as the criticalRayleigh number for self-sustaining convection with most unstable wavenumberlcrit = 2 and for γ = 104 Ra12 = 3400 with lcrit = 4 (corresponding to Racold = 34)If however the reference Rayleigh number is defined at the lower boundary layer weexpect a competition between sphericity and temperature-dependent viscosity for the

onset of convection (Bercovici et al 1989a chapter 2 p 77) This seems to be validonly if we consider the Rayleigh number at the hot boundary which indeed increasesanalogously to (26) and in comparison to the values above

With this summarized knowledge identifying also unresolved issues we have toquantify the contribution of our experiment ie the description of our parameterdomains Subsequently the main objective is to calculate the specific Rayleigh numberand viscosity ratio to reorder our data into a comparable regime set-up This leadsto the fact that our experiment will deliver the description of flow patterns in thelower-viscosity-contrast domain for several orders of the Rayleigh number As alreadystated previously a direct comparison may not be possible The objective here isto compare the generic flow behaviour in specific parameter domains For this wecontinue with the physical basics of TEHD driven flow in spherical shells

3 Thermo-electro-hydrodynamic driven flow in spherical shellsWe consider a fluid-filled spherical shell set-up with the inner and outer boundaries

maintained at different temperatures The crucial challenge in spherical experimentsis the set-up of a self-gravitating force field (Busse 2002) as in an Earth-basedlaboratory the gravitational field is aligned with a vertically directed unit vector gezTo set up a radially directed buoyancy ger we apply an electric field E by a voltagepotential V0 with E=minusnablaV0 Then we consider the electric body force

fE = fC + fDEP + fES (31)

formed by the electrophoretic (Coulomb) force fC and the dielectric force fD which iscomposed of the dielectrophoretic force fDEP and the electrostrictive force fES In detailwe have from Landau Lifshitz amp Pitaevskii (1984) that

fE = qEminus 12

E2nablaε +nabla(

12ρE2 partε

partρ

) (32)

where E = |E| Applying direct current (dc) electric fields exert Coulomb forceswhereas in alternating current (ac) electric fields of high frequency f 1τE (withthe charge relaxation time τE) the dielectrophoretic force fDEP is dominant In generalwe can neglect the gradient force fES as it has no contribution to the flow field(Yoshikawa Crumeyrolle amp Mutabazi 2013) With the use of a high-frequency achigh-voltage potential V0 between the spherical boundaries we drop Coulomb effectsin the electric body force leading to

fE = fDEP =minus 12 E2nablaε (33)

Finally it is only this dielectrophoretic force that acts on a dielectric insulatingliquid of temperature-dependent permittivity ε To derive further relations we needinformation on (i) the electric field E and (ii) the temperature-dependent variation ofthe permittivity ε Here we just mention both relations for more details the interestedreader is referred to Yavorskaya Fomina amp Balyaev (1984) and Hart et al (1986) forapplication in the spherical shell and to Jones (1979) and more recently Yoshikawaet al (2013) for the cylindrical annulus For the calculation of the electric field wetreat the spherical shell as a spherical capacitor with

E(r)= riro

ro minus ri

1r2er V0 (34)

For the variation of the permittivity we use

ε = ε1[1minus αE(T minus Tref )] ε1 = ε0εr(Tref ) (35)

with the relative permittivity εr and the vacuum permittivity ε0 The coefficient ofdielectric expansion αE is related to the permittivity by partεpartT = minusε0εrαE Thiscorresponds to the coefficient of volume expansion as partρpartT = minusρ0α This isin analogy to the lsquoclassicalrsquo Boussinesq approximated RayleighndashBenard convectionwhich is mainly maintained by the temperature dependence of the density ρ with

f = αTg ρ = ρ0[1minus α(T minus Tref )] ρ0 = ρ(Tref ) (36)

To summarize from Hart et al (1986) and Yoshikawa et al (2013) we have for theelectric buoyancy force

fE = αETgE ε = ε1[1minus αE(T minus Tref )] ε1 = ε0εr(Tref ) (37)

In this formulation we get an acceleration gE defined by

gE = 2ε0εr

ro minus ri

1r5er (38)

It is obvious that this lsquoelectricrsquo gravity varies with the depth ie at the innerboundary of the spherical shell we have a higher acceleration Up to the outerboundary the functional relation follows 1r5 resulting in a lower gravity potentialat that boundary

The magnitude of the vector gE reaches the order of up to 10minus1 m sminus2 at the outerspherical radius of our experiment (Futterer et al 2010) This induces the need formicrogravity conditions in order to get a pure radially directed electric buoyancy fieldnot superimposed by the vertically directed Archimedean buoyancy forces (equation(36)) We consider therefore a TEHD convection in microgravity (equation (37))which is driven by the temperature dependence of the electric permittivity ε Nextwe have to discuss the lsquodifferencersquo between the Archimedean buoyancy and electricbuoyancy mainly present in the radial dependence of the electric gravity and thecoefficient of dielectric expansion Thus in the following we present details on thedielectrophoretically induced self-gravitating force field in the equational set-up andlater on further physical properties of our spherical TEHD RayleighndashBenard system

31 Different radial variations of radial gravityWe consider the Boussinesq approximation to be valid for the description of thefluid motion since the velocities appear to be small and the density fluctuations(and permittivity variations respectively) are linear (Chandrasekhar 1981 Sugiyamaet al 2007 Futterer et al 2012) The following relations are the basis for rescalinglength r time t temperature T the effective pressure P and the viscosity micror = (ro minus ri)rlowast = drlowast t = d2κtlowast T minus Tref = 1T Tlowast P = κνd2Plowast and micro = microrefmicro

lowastAbandoning the lowast the equations for mass and momentum conservation as well as thetemperature become

nabla middotu= 0 (39)

Prminus1

[partupartt+ (u middotnabla)u

]=minusnablaP+nabla middot [micro(nablau+ (nablau)T)] + RaETrminus5er (310)

partt+ u middotnablaT =nabla middotnablaT (311)

with the inner radius ri the outer radius ro and the unit vector in radial direction erTherewith the thermal drive is now weighted by an electric Rayleigh number

RaE = αE1TgEd3

νrefκ

)minus5

With the depth-dependent gravity (38) we obtain different values at the hot and coolboundaries for the electric Rayleigh number (prop gE) when a specific 1T is regardedat a constant viscosity level When we rewrite this equation by taking into accountconstants geometrical aspects temperature-dependent physical properties referencephysical properties and variational parameters we obtain for the cold outer Rayleighnumber

RaEcold = 2ε0V20︸︷︷︸

r2i r7

o︸︷︷︸geometry

εr(Tcold)αE(Tcold)︸︷︷︸temp dep

1microcoldκcold︸︷︷︸

references

1T︸︷︷︸variable

and for the hot inner Rayleigh number

RaEhot = 2ε0V20

r2i r7

εr(Thot)αE(Thot)1

microhotκhot1T (314)

Finally we have

RaEhot = γ(

)5εr(Thot)αE(Thot)

εr(Tcold)αE(Tcold)RaEcold (315)

With η = 05 the initial difference between the cold and hot boundaries in their RaE

amounts to a factor of 32 The range of the ratio resulting from relative permittivityεr and dielectric expansion coefficient αE at the hot and cold boundaries depends onthe experimental fluid and will be discussed later on A subsequent question for theinterpretation of our experimental data is whether these additional factors lead to anenhancement of the viscosity ratio for the fluid flow behaviour But first we have tochoose RaEcold or RaEhot in our equation (310) consistently in order to avoid involvingthe electric Rayleigh number as a function and not as a parameter From now on wewill always refer the electric Rayleigh number to the outer spherical radius This isconsistent with choosing the cool boundary as reference

Now we assume that the TEHD driven convection has the property of a depth-dependent gravity and we seek comparable situations In the mantles of planetarybodies however gravity is taken to be constant whereas hydrodynamic convectivemodes in the Earthrsquos liquid outer core are considered with a linear dependence (Busse2002) In addition for the Earthrsquos atmosphere the planetrsquos gravitational potentialacts with 1r2 but here convective modes are mainly driven by rotational effectsin differential heating at one surface A classical spherical shell convection modelas regarded here especially with rigid boundary conditions and isothermally heatedis not considered here (Cullen 2007) Also there is a clear tendency for settingup for example baroclinic situations in cylindrical geometry The different radialdependences seem to become visible only in the planform of convection Busse(1975) and Busse amp Riahi (1982) regard the symmetry-breaking bifurcations in thespherical shell convection for the linear dependence of gravity They already derive thegeneric nature of the branches with polyhedral modes In Bercovici et al (1989b) wefind the confirmation of tetrahedral and cubic patterns as the dominating flow mode

for the constant-gravity case In those flow structures the lsquocornersrsquo of the patternsare the cylindrically formed upwellings Feudel et al (2011) show that the 1r5

dependence induces axisymmetric and five-fold symmetries besides the cubic modeThe tetrahedral mode does not occur In their case the cubic mode is dominated bysheet-like upwellings near the hot boundary layer besides the cylindrically formedupwellings This is in contrast to the cubic mode of Bercovici et al (1989b) forexample where no lsquoridgesrsquo occur in between This is a first hint that lsquoclassicalrsquo plumesare a specific property of the constant-gravity case

The objective to tend into a specific planform of convection seems to be a reductionof shear between upwellings and downwellings (Bercovici et al 1989a) Owing toan increase of the rising or sinking velocity the gravitational potential energy isreleased more efficiently Bercovici et al (1989a) claim that lsquoin three-dimensionalconvection with any geometry a cylindrical upwelling or downwelling will necessarilybe surrounded by a sheet-like return flowrsquo The differences between planar andspherical geometries ie the asymmetry of up- and downwellings in the sphere resultfrom geometrical effects Thus the specific radial dependence only results in differentconvection cell wavelengths The whole scenario remains generic

Therewith we can compare the classical and TEHD RayleighndashBenard system Forthis we need to rescale our Rayleigh numbers due to the fact that (i) in mantleconvection we have the buoyancy term RaTer and (ii) in lsquoGeoFlowrsquo we have thebuoyancy term RaETrminus5er in the momentum equation The non-dimensional radii arerlowasti = 1 and rlowasto = 2 and the real dimensions of our experiments are ri = 135 mmand ro = 27 mm To compare (i) and (ii) when running mantle convection tests(ie without rminus5 in the buoyancy term) we take the reference at the top multiplied bythe radial dependence leading to REcold(rlowasto)

minus5 For example if we have a Rayleighnumber in lsquoGeoFlowrsquo with RaEcold = 2 times 104 then a simulation run takes theRayleigh number Racold = 2times 104times 2minus5 = 2times 10432= 625 to meet the correspondingmantle convection domain With an additional test on the mean temperature profileand root mean squared velocity between the different buoyancy terms we observea clear difference in the magnitudes of the order of 10 in the velocity if thefactor (rlowast)minus5 = 132 is not considered Finally this leads to the Rayleigh domainfor the lsquoGeoFlowrsquo experiment series as depicted in figure 2 For our numericalsimulation we perform both (i) the mantle set-up with constant gravity as well as(ii) the lsquoGeoFlowrsquo set-up with variational gravity For the mantle set-up we simulateadditional parameters in the neighbourhood of the experimental set-up

We have already stated that the general symmetry of convective patterns does notdepend on the functional form of the radial gravity Now we can compare the criticalRayleigh numbers for the different cases In a mantle model with η = 055 the onset ofconvection for the constant-viscosity case (γ = 1) occurs at Rahot = Racold = Ra = 712but in our own simulation owing to the no-slip boundary conditions we observe theonset with Ra isin (21times103 22times103] If we introduce a viscosity ratio eg γ = 32 weget the onset with Racold isin (475 500] Thus the viscosity contrast decreases the onsetof convection

For lsquoGeoFlow Irsquo with η = 05 we observe it at RaE|ro = 2491 (Travnikov Egbersamp Hollerbach 2003 Feudel et al 2011) Their Rayleigh number is related to theouter sphere and the equations are scaled besides others with the outer sphericalradius ro which is in contrast to choosing the lsquoclassicalrsquo gap width of the researchcavity as characteristic length as introduced above for obtaining RaE|d = RaEcold Ifwe rescale their equations we get RaEcold = 9964 as critical onset But to compareit with the constant-gravity case we have to divide this value by 32 and we

Cubicridge

Cubic star(5 plumes onthe equator)

Cubic tetrahedralstar

Mobile-lid

Sluggish-lid

Time-dependent Stagnant-lid

Only cubic Cubictetrahedral5 plumes

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GeoFlow II WE II exp

GeoFlow II WE III exp

GAIA num

FIGURE 2 (Colour online) lsquoGeoFlowrsquo experimental (lsquoexprsquo) and numerical (lsquonumrsquo)parameter domains of incompressible Newtonian TEHD Boussinesq convection in sphericalshells as a function of cool outer Rayleigh and hot inner Rayleigh numbers related via theviscosity ratio by RahotGeoFlow = γ 32αprimeRacoldGeoFlow For lsquoGeoFlowrsquo we use the analogouselectric Rayleigh number RacoldGeoFlow = RaEcold32 = [(αEgE|ro1Td3)(νrefκ)(dro)

minus5]32with the electric acceleration gE|ro = (2ε0εr)ρref [riro(ro minus ri)]2V2

0 rminus5 related to the outerboundary r = ro For the development of the parameters related to TEHD effects refer tosect 3 lsquoGeoFlow IIrsquo is performed in two different working environments lsquoWE IIrsquo and lsquoWEIIIrsquo which differ in their reference viscosity Specific lsquoGeoFlowrsquo patterns of convection aredescribed in sectsect 51 and 61 From the illustration in figure 1 the classical convection regimesie stagnant-lid sluggish-lid and mobile-lid related to γ are also included In additionthe numerically simulated parameters Rahot = γRacold for incompressible Newtonian mantleBoussinesq convection in spherical shells are depicted (lsquoGAIA numrsquo) Again increasing theviscosity ratio gives rise to plumes The convective planform of stationary flow is preciselyallocated Unsteady time-dependent flow is characterized by plumes

will have RaEcold32 = 312 for lsquoGeoFlow Irsquo It is worth mentioning that Travnikovet al (2003) find the onset of isoviscous convection for decreasing radius ratio tooη = 04 RaE|ro = 1898 η = 03 RaE|ro = 1483 η = 02 RaE|ro = 1162 and η = 01RaE|ro = 837 This shows a decrease for the critical value if the curvature increasesand corresponds to the general trend as reviewed in sect 2 above This is a further reasonto treat the TEHD set-up as an analogue for a RayleighndashBenard system

Up to now we have focused the discussion on the specific functional form ofthe gravity in the buoyancy term But the physical properties of a real liquid arealso the basis for the calculation of the Rayleigh number resulting in figure 2 Wesummarize the physical properties in tables 1 and 2 and their consideration in eithernon-dimensional parameters or equations in tables 3 and 4 Following the idea of(315) the two following subsections will discuss the temperature-dependent viscosityand the details of dielectric expansion The objective is to clarify the magnitude of allthree contributions

Parameter Unit Value (at 25 C)

Viscosity νref m2 sminus1 500times 10minus6

Thermal conductivity λ W Kminus1 mminus1 116times 10minus1

Specific heat c J kgminus1 Kminus1 163times 103

Density ρref kg mminus3 920times 102

Thermal diffusivity κ = λ(ρref c) m2 sminus1 774times 10minus8

Thermal coefficient ofvolume expansion

α 1K 108times 10minus3

Relative permittivity εr 27Thermal coefficient ofdielectric expansion

αE =(Yavorskaya et al 1984Hart et al 1986)

α(εrminus 1)(εr+ 2)(3εr) 1K 107times 10minus3

TABLE 1 Physical properties of the silicone oil M5 which is the experimental fluid oflsquoGeoFlow Irsquo We list the values of the parameters used in the experimental framework Asa well-defined liquid all data are available from the data sheet (Bayer Leverkusen 2002)

Parameter Value (at 20 C) Value (at 30 C)

Viscosity νref 142times 10minus5 973times 10minus6

Thermal conductivity λ 163times 10minus1 158times 10minus1

Specific heat c 247times 103 247times 103

Density ρref 829times 102 822times 102

Thermal diffusivity κ = λ(ρref c) 794times 10minus8 776times 10minus8

α 945times 10minus4 945times 10minus4

Relative permittivity εr 86 883Thermal coefficient ofdielectric expansion

α(εr minus 1)(εr + 2)(3εr) 295times 10minus3 246times 10minus3

(new) minus(1εr)(partεrpartT) 995times 10minus3 971times 10minus3

TABLE 2 Physical properties of the alkanol 1-nonanol which is the experimental fluidof lsquoGeoFlow IIrsquo We list the values of the parameters for the two thermal workingenvironments of the experiment lsquoWE IIrsquo and lsquoWE IIIrsquo Alkanols are liquids whosephysical properties are not always clearly quantified especially in comparison to siliconeoils The cosmetics industry mainly employs 1-nonanol because of its lemon fragrance Werefer to data sheets from several companies (Merck 2003 FIZ Chemie 2010) and to thedata collection from Lide (2008) The units are the same as in table 1 and are not listeddue to limited space

32 Temperature-dependent viscosityIn geodynamic models the temperature-dependent dynamic viscosity micro as part of thestress tensor usually follows the FrankndashKamenetskii approximation of the Arrheniuslaw as in (25) If the liquid properties are temperature-independent as for the siliconeoil used in lsquoGeoFlow Irsquo the relation is not of relevance and the viscosity is regardedas constant (Futterer et al 2012) In (310) the viscous term becomes simpler withnabla2u In lsquoGeoFlow IIrsquo we use an alkanol as experimental fluid with a pronouncedvariability of the viscosity with temperature leading to γ 6 15 In Futterer et al

Parameter Thermal behaviour Physical model

νref Linear with small slopeasymp const (measured at BTU)

Simple nablau in (310)

λ Constant Linear Fourierrsquos law ofthermal conduction in (311)

c Estimated to be constant mdashρref Linear (measured at BTU) Boussinesq and Rayleigh

number Ra and RaE

κ = λ(ρref c) Calculated from λ ρ c Reference Prandtl numberPr ref

α Constant Boussinesq and Rayleighnumber Ra and RaE not ofrelevance in microgravity

εr Constant Analogue Boussinesq andRayleigh number RaE

αE = α(εrminus1)(εr+2)(3εr) Calculated from εr Analogue Boussinesq andRayleigh number RaE

TABLE 3 lsquoGeoFlow Irsquo experimental fluid silicone oil M5 Overview of the generalthermal behaviour and its consideration in a physical model eg as approximation orequation Silicone oils are liquids whose physical properties are clearly and accuratelyquantified (table 1) This is mainly due to their broad field of application Therewith manydata are available from the data sheet (Bayer Leverkusen 2002) In addition we measuredproperties with significant experimental relevance (viscosity and density) at BrandenburgUniversity of Technology (BTU)

(2012) we demonstrate the validity of (25) for our used experimental liquid InFutterer et al (2012) we present test cases of the Earth laboratory set-up wherethe fluid-filled spherical shell is regarded as a heated inner sphere and cooled outersphere corresponding to the fluid physics of natural convection in spherical enclosures(Futterer et al 2007 Scurtu Futterer amp Egbers 2010)

In the Earth laboratory the gravitational potential is no longer parallel to the heatingand cooling surfaces This results in a clear distinction from a RayleighndashBenardsystem by a base flow at infinitesimal small temperature difference induced bybaroclinic vorticity (Dutton 1995 Bahloul Mutabazi amp Ambari 2000) Here theapplication of a high-voltage potential induces a slight enhancement of heat transfervia the created convective flow pattern It acts as a possibility for flow control butthe variations of the permittivity are negligible due to the clear dominance of thedensity changes This is the reason for conducting the experiment under microgravityconditions where gravity and hence Archimedean buoyancy driven convection dueto a unidirectional gravity field is absent In Futterer et al (2012) we only presentexperiments In the recent work we are able to reproduce those patterns by anumerical model following (39)ndash(311) and (25) For the preparatory studies weinterpret the delivered patterns of convection to be induced by the temperaturedependence of the viscosity

33 Thermal expansion in microgravityFor the calculation of the Rayleigh number RaE in the microgravity experimentswe have to include a temperature dependence of the thermal expansivity αE if we

νref Power law (measured atBTUAP)

Arrhenius law

λ Not available (implicit fromLide (2008))

Linear Fourierrsquos law ofthermal conduction in (311)

c Not available (implicit fromLide (2008))

ρref Linear (measured at BTU) Boussinesq and Rayleighnumber Ra and RaE

κ = λ(ρref c) Calculated from λ ρ c Prandtl number Pr

α Constant (data sheet fromMerck)

Boussinesq and Rayleighnumber Ra and RaE not ofrelevance in microgravity

εr Polynomial fit (data fromFIZ)

Analogue Boussinesq andRayleigh number RaE

αE =minus(1εr)(partεrpartT) Calculated from εr Analogue Boussinesq andRayleigh number RaE

TABLE 4 lsquoGeoFlow IIrsquo experimental fluid alkanol 1-nonanol Overview of the generalthermal behaviour and its consideration in a physical model eg as approximation orequation In addition to referring to data sheets from several companies (Merck 2003 FIZChemie 2010) and to the data collection from Lide (2008) we performed measurementsof properties with significant experimental relevance (viscosity and density) at BrandenburgUniversity of Technology (BTU) and at the company Anton Paar GmbH (AP)

use 1-nonanol instead of M5 It is a very specific property of silicone oils to betemperature-independent in all physical properties and they are often only specified bytheir different viscosities The decision to use 1-nonanol in the same experimental set-up is promoted through the overall comparable technical behaviour and the variabilityof the viscosity In addition we take into account the temperature dependence of thepermittivity in (37) this being focused upon in the microgravity environment as asubstitute for density in (36)

Up to now the calculation of the TEHD expansion coefficient αE has been derivedvia the relation from Yavorskaya et al (1984) and Hart et al (1986)

αE = α (εr minus 1)(εr + 2)3εr

Behind this we assume is the following statement lsquoequivalent to that usually madefor a Boussinesq liquid that ε is assumed constant except where multiplied by theelectrostatic gravityrsquo (Hart et al 1986 p 523) Regarding the liquid properties ofM5 (Bayer Leverkusen 2002) this assumption is absolutely valid But for 1-nonanola measurement curve for εr with variation of the temperature T (FIZ Chemie 2010)and its polynomial fit with εr = f (T) = 00005T2 minus 01063T + 10595 requires furtherconsideration here With

αE =minus 1εr

partεr

partT=minus10minus6T2 + 5times 10minus5T + 00096 (317)

we get for the specific reference temperature and temperature difference a factor thatis approximately 10 times higher than with (316) and therewith resulting in a differentRaE But in the working environment of the experiments it remains a linear behaviourlike the natural volume expansion coefficient and will not need a further equationUsing the information from the physical properties the contribution

αprime = εr(Thot)αE(Thot)

εr(Tcold)αE(Tcold)gt 087 (318)

remains of the order of oneIn order to investigate nonlinear effects in addition to focusing on the temperature

dependence we test an extended standard Boussinesq approximation The mainidea is based on an extension of the electric buoyancy force fE in (37) by aquadratic term βE(T minus Tref )

2 This results in an altered Rayleigh number RaE2 =RaE[1 minus (βEαE)(T minus Tref )] By choosing βE sufficiently small hence focusing mainlyon the linear temperature dependence the influence of the second-order term vanishesOwing to the fact that measurements of the working fluidrsquos density indicate onlya minor quadratic dependence on temperature the numerical test is performedwith small values of RaE2 The overall solution does not change under the givenassumptions Hence the first-order Boussinesq approximation is an adequate and validtruncation reproducing the correct fluid flows in the depicted parameter space Herewe refer to a comprehensive review on nonlinear properties of convection given byBusse (1978)

34 Effects from Prandtl number radius ratio and boundary conditionsBesides the impact of the radial dependence in the buoyancy term and the calculationof a suitable Rayleigh number the left side of (310) requires a short comment onthe influence of the Prandtl number Pr = νref κ Indeed in our experiments wehave a finite Prandtl number with PrGeoFlow I asymp 65 and PrGeoFlow II = 125 and 200When comparing the solutions with finite and infinite Pr a benchmark with our twoavailable codes (Hollerbach 2000) and GAIA (Huttig amp Stemmer 2008a) adaptedfor the TEHD model delivers differences in the Nusselt number only of the orderof 10minus2 and reproduces the planform of convection with a very good agreement(Futterer et al 2009) Additionally both codes are well established for spherical shellconvection issues in geo- and astrophysical research therefore we refer the reader tothe literature In this present work the goal is to apply the codes to our experimentalset-up As a conclusion from the simulations the Prandtl number might be disregardedin the numerical simulation and is considered also to have a negligible impact in theexperiment

Further tests have been done to address the differences between our experimentand state-of-the-art mantle convection models due to the radius ratio η which is 05for lsquoGeoFlowrsquo and 055 for the mantle If we use 05 instead of 055 we observe aminimal influence on the velocity and temperature (both velocity and temperature arenegligibly smaller than in the 055 case)

A final remark has to be made on the boundary conditions Naturally ourclosed experiment has no-slip conditions but the mantle studies perform numericalsimulations with free-slip conditions Our tests deliver lower velocities for no-slipconditions ie the convection is less vigorous than in free-slip cases and highertemperatures but no additional convection patterns with new symmetry classes wereobtained

35 Summary

From the previous discussion on the depth-dependent gravity the temperature-dependent viscosity ratio and the thermal expansion in microgravity we can concludefor (315) that

RaEhot|max = 15times 32times 09times RaEcold (319)

Considering the factors we assume an accuracy of one digit after the decimalpoint as relevant impact for this equation For the temperature-independent siliconeoil in lsquoGeoFlow Irsquo we derive γ isin (10 11) and αprime = 10 whereas with the useof the temperature-dependent alkanol in lsquoGeoFlow IIrsquo we have γ isin (10 15) andαprime isin (09 10) The influence of the depth-dependent gravity remains constant with(1η)5 = 32 in both experiments Considering also the scaling down on both sides ofthis equation with 132 we revisit figure 2 This demonstrates the enhancement ofthe viscosity contrast by the dominating factor of 32 in which γ is initially assessedonly via the liquid itself to be in the range between 1 and 2 The contribution fromdielectric thermal expansion seems even to damp down slightly the small viscositycontrast Finally we conclude at this stage that both experiments mainly differ intheir Rayleigh number regime lsquodistributedrsquo along an enhanced viscosity ratio of 32Referring figure 2 to figure 1 we expect to be able to describe the dynamics ofthe nearly constant-viscosity domain with lsquoGeoFlow Irsquo With lsquoGeoFlow IIrsquo we mightcapture domains of transitional dynamics of the higher viscosity contrast domains

In figure 3 we depict the variation of the cold Rayleigh number related to the liquid-related viscosity contrast This image does not take into account the enhancement viathe factor 32 But this is necessary for the experimental protocol as the pumping ofthe liquid in the cooling circuit is related to its viscosity too It is assumed that this isthe most important parameter which is varied according to experimental protocol via(i) choosing a reference environment to be either Tref = 20 C or 30 C for the viscosityvariation up to γ asymp 15 and (ii) increasing the temperature difference between the innerand outer spherical shell boundaries up to 1T = 10 K The high-voltage potential V0

is set at the beginning of the experiment as this produces the electro-hydrodynamicbuoyancy in the microgravity environment For lsquoGeoFlow Irsquo it is V0 = 10 kV inlsquoGeoFlow IIrsquo a lower value with V0 = 65 kV is sufficient due to the higher relativepermittivity of 1-nonanol

4 A generic example for flow visualizationIn the lsquoGeoFlowrsquo experiment series we use an optical method for flow visualization

(Merzkirch 1987) as delivered by the Optical Diagnostics Module (ODM) of theFluid Science Laboratory more precisely the Wollaston prism shearing interferometer(WSI) introduced by Dubois et al (1999) The method detects variations in thefluid refractive index which reflects changes in the density (and the permittivity) ofthe fluid Refractive index gradients are visualized either by a deflection of light orby phase shift disturbances which are measured by such interferometers (Merzkirch1987) At the least the WSI yields images with fringe patterns of convection Then thedistances between the fringe lines measured in the resolving direction correspond withthe magnitude of the directional derivative of the temperature field The resolution isrelated to the polarization angle ω of the Wollaston prism the main optical componentof the WSI set-up The following generic example explains how the fringe pattern canbe used to discriminate between sheet-like and plume-like flow upwellings

(times 104)

011 12 13 1410 15

C (non active GeoFlow I) C (GeoFlow II)

C (GeoFlow II)

FIGURE 3 (Colour online) Rayleigh number as a function of the viscosity ratio in lsquoGeoFlowrsquoexperiments at spherical shells of radius ratio η = 05 filled either with silicone oil (N)or alkanol (4copy) The experimental protocol is as follows set-up of desired voltage V0reference image recording (V0 = desired value 1T = 0 n = 0008 Hz) loop on temperaturedifference 1T with different start values and fixed increments including set-up of desired1T wait time corresponding for thermal time scale τ = d2κref for stabilization scientificpictures acquisition to be repeated at next 1T (Mazzoni 2011) During lsquoGeoFlow Irsquo thecooling loop kept its temperature in the vicinity of the ISS FSL temperature of 25 C Theheat loss of the system to the environment was assumed to be negligible During lsquoGeoFlowIIrsquo the cooling loop was controlled actively at either 20 C or 30 C in order to control thereference viscosity In this illustration the lsquoGeoFlow Irsquo and lsquoIIrsquo domains differ clearly Theviscosity contrast here is related only to the behaviour of the real liquid

We consider a fluid flow event given in the Cartesian box [minus1 1]2timesdprime where dprime gt 0is small in comparison with the extension of the horizontal square [minus1 1]2 As themeasurement technique is an integrative method along the measurement section weassume the temperature distribution of the fluid to be sufficiently well represented bythe mean temperature taken in the approximate path of the laser light beam ie thevertical direction (z goes from 0 to dprime)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

Therefore it remains to consider the two-dimensional temperature distribution H(x)on the [minus1 1] square We generate comparable temperature sheet-like and plume-likeflow upwellings using the following one-dimensional temperature flow profile functionh Rgt0rarr Rgt0 given by

h(u) = 14

(2minus u)3 minus 4(1minus u)3 for u 6 1(2minus u)3 for 1lt u 6 20 for 2lt u

FIGURE 4 Flow visualization by means of Wollaston prism shearing interferometry genericexample Column 1 contour plots of the sheet-like HS (top) and the plume-like HPtemperature distributions (bottom) Columns 2ndash4 related fringe patterns of convection ininterferograms for three directions of polarization of the Wollaston prism ω = 90 (vertical)30 and 0 (horizontal)

We obtain the plume-like temperature field HP by rotating this cubic spline around thevertical axis

HP(x) = h( 13 |x|) (43)

The sheet-like temperature field HS is yielded by shifting the profile orthogonal to itsplane

HS(x) = h( 13 |x2|) (44)

The two temperature fields are presented by the contour plots in the left column offigure 4

Assuming that the distances of the interferometry curves depend linearly on themagnitude of the directional derivative of the temperature field we compute thecorresponding fringe patterns for three directions of the polarization The secondcolumn shows the patterns for the vertical direction (ω = 90) the last column for thehorizontal direction (ω = 0) and the third column for the direction with an angle ofω = 30 to the horizontal axis We use sinΛ |partHpartn| as fringe generator with a laserwavelength of Λ= 532 nm

Using the generic example we obtain the following characterization of the sheet-and plume-like temperature upwellings

(a) The interferogram of a sheet-like upwelling consists of parallel lines The numberof black lines reaches a maximum when the direction of polarization is orthogonalto the temperature isolines and decreases when the angle between the direction ofpolarization and the lsquosheetrsquo direction tends to zero Therefrom we cannot detectsheet-like upwellings which run parallel to the direction of polarization (figure 4row 1 column 4)

(b) The interferogram of a plume-like upwelling is characterized by a double ringstructure The symmetry axis in between the ring pair is orthogonal to the directionof polarization With it we possess a rule of thumb to determine the direction ofpolarization on unclassified interferograms of plume-like upwellings

5 Sheet-like upwellings in a self-gravitating spherical shell experiment51 Experimental observation

To navigate on the sphere we use a spherical coordinate system with radial razimuthal ϕ and meridional θ directions The rotation axis in the centre of the spherecorresponds to an angle θ = 0 The optical axis of the FSL camera used to obtainthe interferograms of the lsquoGeoFlowrsquo experiment intersects the spherical centre with anangle of θ = 30 The angle of beam spread from the WSI covers 88 on the sphereTherewith the images taken from the sphere range from θ = minus14 beyond the northpole down to θ = 74 near the south pole In this way we can map the position of aninterferogram pixel onto the sphere

As we want to collect as much image information as possible of the spherersquossurface we rotate the sphere with the slowest possible rotation frequency ofn = 0008 Hz and take a photo every ϕi = 60 of the rotation ie with i = 6 onephoto every [1(0008 sminus1)]6 = 2083 s Neglecting the overlapping regions of theseimages we use one photo to compute a sphere segment interferogram that sweepsthe azimuthal from minus30 up to 30 of the meridian containing the optical axis Thenorthndashsouth extension of this segment goes from the north pole down to the parallelθ = 74 near the equator The montage of six interferograms is presented in figure 5The six positions of the optical axes are marked by black crosses The white meridiansare the boundaries between the separate interferograms The direction of interpolationis given by the grey parallel lines in one separate interferogram The white rings showthe parallels with θ = 15 30 45 and 60 counted from the north pole The equator ismarked by the bold black line

The lowest experimentally available RaEcold32= 518times 102 at 1T = 03 K deliversimages identical to the reference images which are collected before setting upthe parameters at 1T = 0 (figure 5d) At the next Rayleigh number RaEcold32 =644 times 103 we observe a very clear singular pattern of fine lines (figure 5c) AtRaEcold32 = 111 times 104 the gradients increase with an increase of line density andexperience turning into hooks (figure 5b) At the experimentally observable limitof RaEcold32 = 146 times 104 the patterns form sets of broader thermal upwellingsin addition to some finer remaining fringe lines (figure 5a) All fringe patterns ofconvection correlate with sheet-like upwellings

Considering the onset of convection in the stability analysis at RaEcold32 =312 times 102 in comparison to the assessed Rayleigh domain we would expect to bealways above the onset of convection There are two possible reasons for observing nofringes at RaEcold32 = 518 times 102 On the one hand there might be a shift betweenstability analysis and experimental measurement On the other hand interferometryrequires a critical temperature difference which is necessary to produce fringes Forthe lowest parameter domain RaEcold32 isin (2 times 102 6 times 103) we have no more dataavailable due to technical problems on orbit Thus we are not able to detect the onsetof convection in the experiment precisely related to the specific critical temperaturedifference

All samples are snapshots of time-dependent flows For the real experimentalvelocity we can estimate the time frame of one turn in order to get back to the

FIGURE 5 The TEHD spherical shell convection experiment lsquoGeoFlow Irsquo with temperature-independent physical properties of the liquid (silicone oil with Pr = 65) Sheet-likeupwelling fringe patterns of convection for variation of the Rayleigh number RaEcold32at (a) 146 times 104 (b) 111 times 104 and (c) 644 times 103 Flow without specific pattern or evenbelow onset of convection at (d) 519 times 102 Increase of thermal driving enhances sheet-likeconvective planform and increases the global velocity which is not of high amplitude

same image position again which is t = 1(0008 Hz) = 125 s During that time ofapproximately 2 min the patterns at a specific position on the image move onlyslightly in general but not at the same rate over the whole image itself To collect the

spatial change of the whole pattern in between two images an automated procedureof image processing is a requirement which is not available for those compleximage data Neither commercial software nor self-developed tools have fulfilled ourobjectives That makes it impossible to assume one single velocity and requires anextended review of available methods However this is not the focus of this work Weconclude here that we observe sheet-like upwellings with long-term time-dependentbehaviour

52 Numerical simulationIn sect 3 we present the governing equations of the TEHD driven RayleighndashBenardconvection which govern the fluid dynamics of the lsquoGeoFlowrsquo experiment in particularThis theory is used to compute the temperature distribution of the fluid in the sphericalshell We apply the numerical code from Hollerbach (2000) ie a spectral methodfor the three-dimensional numerical simulation of the equations (39)ndash(311) We referthe reader to Hollerbach (2000) for a detailed description of the spectral solution inthe magnetohydrodynamic framework The no-slip boundary conditions on the velocityfield and the fixed temperature values at the inner and outer shell close our setof equations In Travnikov et al (2003) Futterer et al (2010) and Feudel et al(2011) we have extensively applied the code to ideal isoviscous fluids with γ = 1We summarize the most important results of Futterer et al (2010) with a focus onthe agreement with the experimental data and those of Feudel et al (2011) with thebifurcation scenario resulting from path following analysis in numerical simulations

In the numerical simulation we observe the onset of convection at RaEcold32 =312 times 102 On increasing the Rayleigh number up to RaEcold32 = 313 times 103 theresults depict a steady-state convection with the coexistence of cubic axisymmetricand five-fold symmetry classes The transition to time-dependent states occurssuddenly the time evolution delivers a very slow motion comparable to those observedin the experimental data The available experimental data agree with the numericaldata in the time-dependent domain For this domain we choose to present data fromthe non-stationary regime and the agreement with experimental data in figure 6

The time-dependent pattern of the TEHD driven analogous RayleighndashBenardBoussinesq convection combines the properties of all steady-state patterns from thelower Rayleigh domain In the chosen snapshot in figure 6 we observe the cubicpattern produced by sheet-like upwellings in the radial velocity field illustrationThis is also present in the temperature field data when it is visualized on aspherical surface To represent the details of the thermal upwellings we plot alsothe temperature field in a cut and highlight an arbitrarily chosen isosurface of thetemperature There are no distinct plumes but broadly merged thermal flow ie sheet-like upwellings

In order to compare the numerical simulations with the experiment we computethe fringe patterns for a given temperature distribution T(ϕ θ r) with the approach ofsect 4 Again we compute the mean of the temperature distribution in the approximatedirection of light ie in the radial direction from the inner to outer radius of theexperiment geometry Thus we reduce the complexity of the temperature distributionto the following distribution to a surface of a sphere

H(ϕ θ) = 1ro minus ri

int ro

T(ϕ θ r) dr (51)

Let us consider the experimental interferogram given in figure 6 The 60 meridiansare visible via their intersection at the north pole Again the black cross marks the

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

FIGURE 6 Numerical simulation of lsquoGeoFlow Irsquo Boussinesq TEHD model with constantviscosity (a) snapshot of sample at RaEcold32 = 625 times 103 with non-dimensionaltemperature field distributed in the sphere and highlighted isosurface of T = 05 (b) radialvelocity component in a mid-spherical surface r = 05 Fringe patterns of convection withsheet-like thermal upwelling (c) from numerical simulation of lsquoGeoFlow Irsquo BoussinesqTEHD model with constant viscosity at RaEcold32 = 625 times 103 and (d) observed in araw image in the experiments at RaEcold32= 644times 103

centre of the circle-shaped image it is equivalently the intersection of the spherewith the optical axis The grey parallels depict the direction of polarization of theinterferogram We compute the gradient nablaϕθH from the numerical data by takingthe central differences and obtain partH(ϕ θ)partn via the multidimensional chain rule ofdifferentiation applied to the rotational transformations Again we use sinΛ |partHpartn|as fringe generator The fringe pattern convection calculated from the temperature fieldobtained in the numerical simulations is in very good agreement with the observedexperimental data We conclude here that we are able to validate the experimentallyobserved images from the lsquoGeoFlow Irsquo experiment (sheet-like thermal flow) with thenumerical model of an ideal isoviscous TEHD driven analogous RayleighndashBenardBoussinesq convection system as presented in sect 3

6 Plume-like upwellings in a self-gravitating spherical shell experiment withenhanced viscosity ratio and at higher Rayleigh numbers

61 Experimental observationIn the following we discuss the observed fringe patterns of convection in the lsquoGeoFlowIIrsquo parameter domain The experiment itself is performed in two different set-ups Firstthe reference cooling circuit is fixed to the lower temperature regime of Tref min = 20 C

Therewith the initial viscosity of the fluid is higher Second the situation is reversedwith the upper limit of Tref max = 30 C and a lower initial viscosity of the fluid Inthis case higher Rayleigh numbers are reached Figure 3 shows the set points forthe Rayleigh number as a function of the viscosity ratio This ratio results from theproperties of the used liquids and remains below 2 for both experiment series In thisillustration with only one Rayleigh number RaEcold32 related to the viscosity ratio thepossible enhancement by a factor of 32 is not visible The observation will have tosupport the illustration in figure 2

In lsquoGeoFlow IIrsquo an increment of 1T = 01 K delivers RaEincrrefmin32 = 862 times 102

and RaEincrrefmax32 = 3115 times 103 resulting in a dense scan of the possible domainFor lsquoGeoFlow Irsquo a flow visualization is only available at the marked symbols but thewhole domain is assessed also via numerical simulation In addition in lsquoGeoFlow Irsquothe cooling circuit is not controlled actively ie it is assumed to be the general roomtemperature of the ISS Again from the illustration in figure 2 it becomes more clearthat the two experiment series assess different domains only the upper lsquoGeoFlow Irsquoand lower lsquoGeoFlow IIrsquo data can superimpose

Regarding now the hydrodynamic stability related to figure 1 we do not expectto track the transition from the conductive to the convective flow state above acritical temperature difference and Rayleigh number respectively This is visible forthe experimental data already at the lowest assessed set points In general in thelower Ra domain every single image shows pairs of elliptically formed rings whichare distributed irregularly in the space between the polar and equatorial parts of thespherical research cavity This fringe pattern of convection clearly corresponds to theplume-like upwelling type Again as in the lsquoGeoFlow Irsquo case figure 7 combinesmultiple images Those spots of plume-like upwelling are depicted in figure 7 for asample at (d) RaEcold32 = 614 times 103 and (c) RaEcold32 = 128 times 104 Furthermorepatterns are not constant in space and time changing between measured frames Thisis also observed for the mid range of the parameter domain However the amount ofthose time-dependent lsquohot spotsrsquo increases with viscosity This has also been observedin numerical models of mantle convection (Huttig amp Breuer 2011)

The overall temporal behaviour of the fluid flow is observed at all stages of theperformed set points Only the specific planforms experience a transition of mergedpatterns The plumes are no longer separated from each other but are connected viaadditional fringes RaEcold32 = 255 times 104 (figure 7b) and RaEcold32 = 381 times 104

(figure 7a) No significant difference in flow patterns is observed between the twodifferent viscosity domains One might discuss the sharper contrast of the plumes inthe higher viscous working environment depicted in the right column of figure 7 Fromthe viewpoint of flow visualization it is a question of refractive index behaviour Therefractive index is a function of the relative permittivity and the relative permeabilityIn the case of higher viscosity (higher Pr) the permittivity is also higher Subsequentlythe general refractive index level might be higher and lead to more deflection resultingin more contrast in the images But there is also another possibility to relate thiswith a Prandtl number effect In the real experiment the two liquids differ in theirreference Prandtl number Starting from lsquoGeoFlow Irsquo we can calculate Pr ref = 65and in lsquoGeoFlow IIrsquo we get Pr ref WE II = 200 and Pr ref WE III = 125 At this stageone might identify an influence of the Prandtl number with higher values inducingsharper contrasts in plume-like upwellings Clear evidence and an explanation for thisinfluence requires further study eg with further experiments increasing the Prandtlnumber In this work here we will refer to numerical simulations varying the Prandtlnumber for different models below

FIGURE 7 The TEHD spherical shell convection experiment lsquoGeoFlow IIrsquo with temperature-dependent physical properties of the liquid (liquid alkanol with PrTref=30C = 125 andPrTref=20C = 200) and pronounced variation of the viscosity contrast γ in two referenceworking environments WE III (Tref = 30 C) left column and WE II (Tref = 20 C) rightcolumn Fringe patterns of convection for variation of the Rayleigh number RaEcold32 atdifferent levels of γ (a) 381 times 104 (left) (b) 255 times 104 (left) and 256 times 104 (right)(c) 128 times 104 and 128 times 104 and (d) 614 times 103 and 624 times 103 The upper right panelremains empty because in that lower viscosity regime of WE II this high Rayleigh numberis not reached Parameters below the onset of convection are not observed The thermal driveproduces a plume-like convective planform the increase of it delivers merged patterns withlsquoconnectionsrsquo between the plumes All patterns are snapshots of highly irregular flow

If we summarize now the route for the parameter domain of both experiment seriesas depicted in figure 2 we observe sheet-like thermal flow in the lower domain Thenwe capture plume-like flow with single sharp spots connected with broad-banded flowin the upper domain Especially the plumes are known in simple mantle dynamics withsignificant viscosity ratios in the stagnant-lid domain If we observe a plume in ourexperiment we can state here that the depth-dependent gravity enhances the viscosityratio in (315) The next step is to validate this result with numerical simulation

62 Numerical simulationIn an initial step we only consider the thermoviscous related terms of our numericalTEHD model in the range of γ below 2 within the non-parallelized spectral code(Hollerbach 2000) Certainly the TEHD model involves the general factorial differenceof 32 between the cold and hot Rayleigh numbers This does not lead to a changein flow behaviour relative to the constant-viscosity scenario The spatio-temporalproperties of the convective patterns remain comparable to the lsquoGeoFlow Irsquo casesie sheet-like However the lsquoGeoFlow IIrsquo experimental data show a convectionstructure dominated by plume-like upwellings which may be achieved by increasingthe effective viscosity ratio Therefore we use the code GAIA which can handle highviscosity contrasts (six orders of magnitude between neighbouring grid points and upto 40 system-wide) With GAIA we were already able to validate the results fromlsquoGeoFlow Irsquo with a focus on the influence of the Prandtl number For the lsquoGeoFlowIIrsquo case we can increase the parameters even above the effective γ of 32 We alsobalance the influence of the boundary conditions (no-slip versus free-slip) and thedifferent buoyancy magnitudes (constant linear and 1r5)

In mantle convection models free-slip boundary conditions are considered realisticfor simulations of the interior dynamics of planetary bodies However in a closedexperiment the boundary conditions are automatically set to no-slip When comparingfor example the magnitude of velocity between the free-slip and no-slip boundariesfor a simulation with a cold Rayleigh number Racold = 625 times 102 and a viscositycontrast γ = 32 we obtain a velocity that is enhanced by a factor of 24 in thefree-slip case relative to the no-slip case This difference is caused by the zerovelocity at the boundaries in the no-slip case while in the free-slip run the lateralvelocity can develop freely This influences the temperature field resulting in a lowermean temperature for the free-slip case since heat is transported more efficientlythan in the no-slip model Additionally the upper thermal boundary layer is locatedcloser to the surface in the free-slip case meaning that thermal upwellings rise to ashallower depth compared to the no-slip run All these differences hinder the directquantitative comparison between the lsquoGeoFlowrsquo experiments and published mantleconvection models

Therefore using no-slip boundary conditions in GAIA we have performed a seriesof numerical simulations (filled diamonds in figure 2) whose snapshots are shown infigure 8 The critical Rayleigh number was found to be Racoldcr isin (21times 103 22times 103]for an isoviscous mantle system with constant gravity With a viscosity contrastγ = 32 we obtain Racoldcr isin (475 500] Performing a numerical simulation forlsquoGeoFlowrsquo requires a radial dependence This calculation with GAIA results in acritical onset of RaEcoldcr32 isin (34375 375] for lsquoGeoFlow Irsquo By increasing eitherthe Rayleigh number or the viscosity contrast the convection changes from subcriticalto steady state and then to time dependent A time-dependent behaviour is obtainedfor the simulations plotted above the black border in figure 8 All the simulations infigure 8 are done using a random initial perturbation as appropriate for the comparison

104 105 106 107

FIGURE 8 (Colour online) Numerical simulation of the mantle Boussinesq model withvariation of the viscosity contrast γ samples at Racold = 103 104 105 106 for γ =1 103 resulting in Rahot = 104 107 Patterns of convection are illustrated via anarbitrarily chosen non-dimensional isosurface of the temperature field The increase of γleads to plumes which have a narrower channel of thermal upflow in their stem whena higher parameter domain and thus a higher thermal driving via Racold is consideredAdditionally there are small ridges of thermal upwellings from the hot boundary layer Atime-dependent behaviour is obtained for the simulations plotted above the black borderRefer also to figure 2 for the transition from steady state to unsteady flow regimes

with the experimental data Obviously the picture changes if the initial perturbation isset to cubical for example The cubical initial conditions result in six upwellings onthe axes which remain stable also for higher Rayleigh numbers (eg Racold = 1 times 104

and Rahot = 1times 106)In figure 8 we observe the increase of the number of thermal upwellings with

increasing viscosity contrast This behaviour was also observed in purely internallyheated systems with insulating bottom boundary as discussed in Huttig amp Breuer(2011) A low-degree regime as described in Huttig amp Breuer (2011) and Ratcliff et al(1997) has not been found in our simulations This is most likely to be due to theno-slip boundary conditions which would prevent a long-wavelength solution (Tackley1993) It shows once more the importance of the boundary treatment Additionallywith increasing viscosity ratio we clearly observe plumes These plumes have anarrower channel of thermal upflow and a broader head with increasing γ Onincreasing the Rayleigh number the plumes become interconnected by thin sheet-likeupwellings along the hot boundary layer

104 105 106 107

FIGURE 9 Fringe patterns of convection for numerical simulation of the mantle Boussinesqmodel with variation of the viscosity contrast γ samples at Racold = 103 104 105 106 forγ = 1 103 resulting in Rahot = 104 107 Plumes from the lower Ra domain resultin clear plume-like fringes with a pair of rings plumes from the higher Ra domain are notclearly separated as a pair of rings but even merge with sheet-like patterns resulting from theridges produced in the thermal boundary layer

Using the flow visualization techniques presented in sect 3 we can compute the fringepatterns for the convection structure shown in figure 8 Figure 9 shows the ellipticallyformed pair of rings representing a plume-like and the continuous lines that indicatea sheet-like upwelling in the upper domain The stationary regime does not clearlydeliver either plume-like flow or sheet-like fringe patterns of convection At this stageof analysis this is related to the way a temperature profile moves into the opticalsegment In the numerical reconstruction of the fringe patterns we are able to varythe position of the optical axis By this means we can meet the real situation in theexperiment where the optical axis is fixed and the pattern is moving freely Figure 10shows a match of the experimental data with fringe patterns Here we conclude thatthe lsquoGeoFlow IIrsquo domain reflects fluid flow dynamics from geodynamic models TheTEHD model seems to be valid only for the higher domain In the lower regime onlythe mantle model is able to validate the observed patterns

If we briefly summarize that we observe sheets in lsquoGeoFlow Irsquo and plumes inlsquoGeoFlow IIrsquo one might ask whether there is an influence of the Prandtl numberas described in Schmalzl Breuer amp Hansen (2002) and Breuer et al (2004) Forvery low Pr isin (01 10) they capture a rather low-degree convection pattern with

(a) (c)

(b) (d)

FIGURE 10 Fringe patterns of convection with plume-like thermal upwelling fromnumerical simulation (top) and observed in a raw image in the experiments (bottom) Thelower domain of the experiment exhibits plume-like thermal upwelling the higher domain issuperimposed by sheet-like patterns related to ridges of thermal upwelling in the boundarylayer While the lower domain is only reproduced by a pure mantle model the higher domainis assessable via both the mantle as well as the TEHD model

bulgy structures interconnected by sheets in Cartesian box models In the lsquoGeoFlowrsquoexperiments the Pr numbers used are 6464 125 and 200 A first benchmark forlsquoGeoFlow Irsquo when using an infinite Pr and Pr = 6464 as in the experiment hasshown that in this range the Prandtl number can be assumed to be infinite In thelsquoGeoFow IIrsquo experiment Pr is about two times larger than in lsquoGeoFlow Irsquo and thesame argument should hold

7 Heat transfer properties of convective patternsIn table 5 we show output quantities of the numerical simulations performed

in this work For steady-state simulations three different initial spherical harmonicperturbations of the temperature field have been used cubic (promotes six upwellingson the axes) tetrahedral (promotes four upwellings distributed in the form of atetrahedron) or random Beside the root mean square velocity average temperature andNusselt number we show also the weighted and dominant degree of the convectivepattern (Huttig amp Breuer 2011) The latter two offer additional information aboutthe structural complexity being directly correlated with the number of up- anddownwellings In general for sheet-like thermal structures (for all cases using thelsquoGeoFlowrsquo configuration) we find that the heat transfer is more effective than forplume-dominated flow as indicated by both higher Nusselt numbers and higher rootmean square velocities (table 5) In order to better understand the difference in theconvective structure and the cooling behaviour we select two simulations where

Case Rahot Set-up IC State vrms Tavg Nu Wmode Modemax

Racold = 1times 103

1 1times 104 Mantle Cubic s 5224 0287 144 4068 42 1times 104 Mantle Tetra s 4925 0290 141 4492 43 1times 104 Mantle Random s 5228 0287 143 4079 4

4 32times104 Mantle Random s 9825 0299 193 4690 55 32times104 GeoFlow Random s 12162 0309 219 4821 4

6 5times 104 Mantle Cubic s 11813 0300 204 4091 47 5times 104 Mantle Tetra s 11241 0296 190 3233 38 5times 104 Mantle Random s 11655 0304 209 5077 59 5times 104 GeoFlow Random s 14183 0318 235 4300 4

26 1times 107 Mantle Random t 94201 0473 478 9129 827 1times 107 GeoFlow Random t 106808 0542 495 8524 6

Racold = 1times 104

37 32times105 Mantle Random s 47386 0302 397 6440 5

TABLE 5 (Continued on next page)

38 32times105 GeoFlow Random t 54575 0346 389 4441 3

39 5times 105 Mantle Cubic s 54335 0307 394 5384 440 5times 105 Mantle Tetra s 54163 0312 422 6701 541 5times 105 Mantle Random s 53272 0318 433 7389 642 5times 105 GeoFlow Random t 62396 0371 420 4659 3

Racold = 1times 105

51 1times 105 Mantle Random t 71098 0230 392 586 3

54 32times106 Mantle Random t 165579 0327 720 8411 755 32times106 GeoFlow Random t 203597 0409 743 9230 6

Racold = 1times 106

TABLE 5 Results for the cases presented in this work Rahot and Racold are the Rayleighnumbers at the inner and outer boundary The set-up shows either the use of a pure mantlemodel or the lsquoGeoFlowrsquo TEHD model Initial starting conditions (IC) vary between cubictetrahedral or random In the fourth column of the table we show whether the simulationsreached a steady-state (s) or a time-dependent state (t) Further quantities are the rootmean square velocity vrms average temperature Tavg and Nusselt number Nu respectivelyThe output parameters Wmode and Modemax give information about the spherical harmonicsdegrees While Wmode is a weighted spherical harmonics degree Modemax is the dominantdegree reached

Thermal flow in spherical convection experiments under microgravity 677R

Rayleigh number

104 105

(a) (b)

FIGURE 11 Influence of the self-gravitating configuration on the variation of the buoyancyterm and the convective patterns (a) Variation of the buoyancy term caused by eithera temperature-dependent viscosity (grey lower full line maximum grey upper full lineminimum black line volume averaged) or 1r5 as used in the lsquoGeoFlowrsquo experiments(dashed-dotted line) (b) Convection patterns obtained for Racold = 104 and Rahot = 32 times 105

(the buoyancy term increases towards the inner boundary) left the buoyancy term changeswith 1r5 right the buoyancy term changes due to the temperature-dependent viscosity

the Rayleigh number varies either (i) locally due to temperature-dependent viscosity(viscosity decreases with increasing temperature) according to micro(T) = exp(minusγT) withγ = ln(32) or (ii) as 1r5 (lsquoGeoFlowrsquo configuration and with constant viscosity) TheRayleigh numbers at the inner and outer boundary in these simulations are the same(Racold = 104 and Rahot = 32times105) but vary with depth as in figure 11(a) The volume-averaged Rayleigh number is 343 times 104 for the temperature-dependent viscosity caseand 624 times 104 for the lsquoGeoFlowrsquo configuration Thus the increase of the Nusseltnumber for the sheet-like thermal structures are related to a higher effective Rayleighnumber

In the case of the lsquoGeoFlowrsquo configuration the lower thermal boundary layer istherefore more unstable than in the case where the Rayleigh number changes solelydue the temperature-dependent viscosity The first scenario shows sheet-like structureswhile the latter produces a plume-like flow (figure 11b) However on increasingthe Rayleigh number at the outer boundary above 104 both the sheet-like and theplume-like structures change to a structure that shows plumes interconnected by sheetsThis structure is independent of the depth dependence of the Rayleigh number Aninterconnection between individual plumes has also been observed by Houseman(1990) in three-dimensional Cartesian box models with stress-free boundaries andwas associated with convection patterns at high Rayleigh numbers in at least partiallybottom-heated systems

8 Summary discussion and outlookIn this paper we present results from our lsquoGeoFlowrsquo spherical convection

experiments performed under microgravity which use an analogy of TEHD convectionto reflect a RayleighndashBenard system in a spherical experiment In the first experimentlsquoGeoFlow Irsquo we use a silicone oil (M5) whose physical properties are consideredto be temperature-independent In the second experiment lsquoGeoFlow IIrsquo we keepthe experimental design and set-up and fill the research cavity with an alkanol (1-nonanol) whose properties are comparable to the silicone oil except for a pronounced

variability of the viscosity with temperature This allows us to capture the initialbasis of hydrodynamic effects in a spherical experiment motivated by Earthrsquos mantleconvection phenomena The experimental domain captures several orders of thethermal driving parameter ie the Rayleigh number along with an enhanced viscosityratio of 32 To the best of our knowledge this has been treated by means of numericalsimulations only (figure 1) or in rather complex experiments in rectangular geometries(Androvandi et al 2011) After a discussion of the governing equations and specificproperties of the TEHD convection we present the results from both experimentseries and benchmark them against numerical simulation We observe a sheet-likethermal flow (visible in our flow visualization as interferograms of fringes) whenthe physical properties of the fluid do not vary with temperature ndash a result fromlsquoGeoFlow Irsquo When we use a liquid with temperature-dependent viscosity (lsquoGeoFlowIIrsquo) we observe a plume-like dominated flow (visible in the interferograms as pairsof concentrically organized ellipsoidal rings) For the lsquoGeoFlow Irsquo situation we areable to reproduce our sheet-like fringe patterns of convection Both available numericalcodes Hollerbach (2000) and GAIA describe the equations of the system with justa TEHD related buoyancy term Hence we prove experimentally that the symmetryclasses for patterns of convection in a spherical shell system are more or less genericOnly the details depend on the specific functional form of the gravitational potentialUsing the example of a cubic symmetry a mantle model with constant gravity deliversplume patterns of convection called cubic if four plumes are upwelling from theequator (Bercovici et al 1989a) A cubic symmetry formed by four upwellingsconnected with sheets in the upper and lower hemisphere is discussed in Busse (1975)and Busse amp Riahi (1982) for a linear dependence of the gravitational potential Herewe describe it for a 1r5 dependence resulting from the TEHD model All patterns inthe lsquoGeoFlow Irsquo domain are independent of the Prandtl number (Pr gt 50) Thus theexperiment can be considered as a Stokes flow

For the lsquoGeoFlow IIrsquo case the situation is rather different as we depict in figure 10Dividing the plume-like fringe patterns of convection into two mainly observed typesthe single plumes in the lower domain and the superimposed ridges in the higherdomain we are able to reproduce them with some additional conditions For the singleplumes which are not interconnected by sheet-like upwellings we have an agreementwith a mantle model ie without 1r5 dependence of the buoyancy term For thehigher domain both the mantle as well as the TEHD model deliver the same result Inboth cases we see evidence that depth-dependent gravity acts via an enhancement ofthe viscosity contrast Thus if using a pure TEHD model γ is directly comparableHowever if instead a mantle model is used then γ needs to be higher Figure 12shows how the patterns obtained with the numerical simulations correlate to thestructures obtained in the lsquoGeoFlowrsquo experiments and demonstrates the inconsistencybetween the numerical and the lsquoGeoFlow IIrsquo experiment for low surface Rayleighnumber It still remains to be clarified why for the temperature-dependent viscosity(lsquoGeoFlow IIrsquo) the convection pattern changes from sheet-like to plume-like To thisend more experimental data (expected from a subsequent lsquoGeoFlow IIbrsquo mission) andnumerical simulations are needed

When we compare the combination of experimental and numerical work for theTEHD issues we have the results from Hart et al (1986) where a very goodagreement between experimental and numerical data is found But the authors usea temperature equation different from ours They include on the right side of ourequation (311) a term related to the dissipative heating and in addition they add a

Obtained patternsGeoFlow II

Plumes Sheets Sheets

Expected and obtainedpatterns GeoFlow I

Expected and obtained patternsGeoFlow II

Expected patternsGeoFlow II

GeoFlow I

GeoFlow IIInterconnected plumes

FIGURE 12 Parameter space and corresponding structures obtained in the numerical studyThe pattern domains obtained in the two GeoFlow experiments are shown in different greyshades

term lsquodepending on E that represents the work done by electric forcesrsquo (Hart et al1986 p 522)

partε

parttT

Dt2ρc (81)

In our pre-studies for lsquoGeoFlow Irsquo we omitted this term as a result of an extendeddiscussion related to its effective contribution to the flow (Travnikov et al 2003)Our lsquoGeoFlow Irsquo results support this approach Considering the TEHD in the verticalannulus as in Malik et al (2012) and Yoshikawa et al (2013) we have a comparablediscussion on the so-called feedback effect ie via the Gauss equation nabla middot (εE) = 0taking into account the influence lsquoon the temperature disturbances on the electricpermittivityrsquo (Malik et al 2012) Thus they keep the same temperature equation as inour model While an extended numerical study would deliver an answer as to whichof the other two TEHD equation models fit our experimental observation we mightassume that in the lower domain we have such feedback effects producing the singleplumes not reflected in our recent TEHD model

It has to be mentioned again that we have an agreement also in our selectedtests in the Earth laboratory But in the microgravity the permittivity replaces thedensity For 1-nonanol the liquid of lsquoGeoFlow IIrsquo the feedback effect seems to playa more important role than for M5 the liquid of lsquoGeoFlow Irsquo Owing to technicalpre-studies we act on the assumption that our used liquids do not suffer from fatigueNevertheless we summarize that we are able to reproduce plume-like flow in aspherical experiment by means of TEHD experiments

In December 2012 we started the mission lsquoGeoFlow IIbrsquo which will support morehighly resolved analysis related to the time resolution of the flow modes In additionthe transient behaviour will be reviewed Both require long-term observations whichwere only authorized after the second mission lsquoGeoFlow IIrsquo which lasted fromMarch 2011 until May 2012 To clarify the feedback effects we plan to performadditional parameter sets on orbit with a variation of the high-voltage potential Thiswill distinguish the temperature effects on the permittivity as we vary the electricenergy

AcknowledgementsThe lsquoGeoFlowrsquo project is funded by ESA (grant no AO-99-049) and by the German

Aerospace Center DLR (grant nos 50 WM 0122 and 50 WM 0822) The authorswould also like to thank ESA for funding the lsquoGeoFlowrsquo Topical Team (grant no1895005NLVJ) The scientists also thank the industrial companies involved for theirsupport namely Astrium GmbH Friedrichshafen Germany the User Support andOperations Center MARS Telespazio Naples Italy and E-USOC Madrid Spain

During the main part of this work the first author (BF) was supported by theBrandenburg Ministry of Science Research and Culture (MWFK) as part of theInternational Graduate School at Brandenburg University of Technology (BTU) From1 October 2012 BF holds a one-year visiting professorship at the Otto von GuerickeUniversitat Magdeburg funded through the Saxony-Anhalt Ministry of Economicsand Research (MWW) RH is supported at ETH Zurich by the European ResearchCouncil (grant no 247303 (MFECE))

Furthermore fruitful discussions with Anne Davaille Laboratoire FASTCNRSUPMC Universite Paris Sud Orsay France and Olivier Crumeyrolle HarunoriYoshikawa and Innocent Mutabazi Laboratoire Ondes et Milieux Complexes UMR6294 CNRS Universite du Havre France are gratefully acknowledged

We thank the Northern German Network for High-Performance Computing (HLRN)for the computer resources used for the parallelized calculations

R E F E R E N C E S

ANDROVANDI S DAVAILLE A LIMAREA A FOUCQUIERA A amp MARAIS C 2011 At leastthree scales of convection in a mantle with strongly temperature-dependent viscosity PhysEarth Planet Inter 188 132ndash141

BAHLOUL A MUTABAZI I amp AMBARI A 2000 Codimension 2 points in the flow inside acylindrical annulus with a radial temperature gradient Eur Phys J Appl Phys 9 253ndash264

BAUMGARDNER J P 1985 Three-dimensional treatment of convective flow in the Earthrsquos mantleJ Stat Phys 39 501ndash511

BAYER LEVERKUSEN (now GE Bayer Silicones Germany) 2002 Bayer Silicones Baysilone FluidsM Technical Data Sheet 11112002 (delivered with the liquids)

BERCOVICI D SCHUBERT G amp GLATZMAIER G A 1989a Three-dimensional spherical modelsof convection in the Earthrsquos mantle Science 244 950ndash955

BERCOVICI D SCHUBERT G amp GLATZMAIER G A 1991 Modal growth and coupling inthree-dimensional spherical convection Geophys Astrophys Fluid Dyn 61 149ndash159

BERCOVICI D SCHUBERT G amp GLATZMAIER G A 1992 Three-dimensional convection of aninfinite-Prandtl-number compressible fluid in a basally heated spherical shell J Fluid Mech239 683ndash719

BERCOVICI D SCHUBERT G GLATZMAIER G A amp ZEBIB A 1989b Three-dimensionalthermal convection in a spherical shell J Fluid Mech 206 75ndash104

BOOKER J R 1976 Thermal convection with strongly temperature-dependent viscosity J FluidMech 76 741ndash754

BREUER M WESSLING S SCHMALZL J amp HANSEN U 2004 Effects of inertia inRayleighndashBenard convection Phys Rev E 69 026302

BUSSE F H 1975 Patterns of convection in spherical shells J Fluid Mech 72 67ndash85BUSSE F H 1978 Non-linear properties of thermal convection Rep Prog Phys 41 1930ndash1967BUSSE F H 2002 Convective flows in rapidly rotating spheres and their dynamo action Phys

Fluids 14 1301ndash1313BUSSE F H amp FRICK H 1985 Square-pattern convection in fluids with strongly temperature-

dependent viscosity J Fluid Mech 150 451ndash465BUSSE F H amp RIAHI N 1982 Patterns of convection in spherical shells Part 2 J Fluid Mech

182 283ndash301

CHANDRASEKHAR S 1981 Hydrodynamic and Hydromagnetic Stability DoverCHRISTENSEN U amp HARDER H 1991 Three-dimensional convection with variable viscosity

Geophys J Intl 104 213ndash226CULLEN M 2007 Modelling atmospheric flows Acta Numerica 16 67ndash154DAVAILLE A amp JAUPART C 1994 Onset of thermal convection in fluids with temperature-

dependent viscosity application to the oceanic mantle J Geophys Res 99 19 843ndash19 866DAVAILLE A amp LIMARE A 2009 Laboratory studies of mantle convection In Mantle Dynamics

(ed G Schubert amp D Bercovici) Treatise on Geophysics 7 pp 89ndash165 ElsevierDUBOIS F JOHANNES L DUPONT O DEWANDEL J L amp LEGROS J C 1999 An integrated

optical set-up for fluid physics experiments under microgravity conditions Meas Sci Technol10 934ndash945

DUTTON J A 1995 Dynamics of Atmospheric Motion DoverEGBERS C BEYER W BONHAGE A HOLLERBACH R amp BELTRAME P 2003 The

GEOFLOW-experiment on ISS (Part I) Experimental preparation and design Adv Space Res32 171ndash180

FEUDEL F BERGEMANN K TUCKERMAN L EGBERS C FUTTERER B GELLERT M ampHOLLERBACH R 2011 Convection patterns in a spherical fluid shell Phys Rev E 83046304

FIZ CHEMIE (Fachinformationszentrum Chemie GmbH Berlin) 2010 Relative permittivity at zerofrequency 2010-03-29 ID 1972shkgal0 httpwwwfiz-chemiedeinfotherm

FUTTERER B BRUCKS A HOLLERBACH R amp EGBERS C 2007 Thermal blob convection inspherical shells Intl J Heat Mass Transfer 50 4079ndash4088

FUTTERER B DAHLEY N KOCH S SCURTU N amp EGBERS C 2012 From isoviscousconvective experiment lsquoGeoFlow Irsquo to temperature-dependent viscosity in lsquoGeoFlow IIrsquo ndash Fluidphysics experiments on-board ISS for the capture of convection phenomena in Earthrsquos outercore and mantle Acta Astronaut 71 11ndash19

FUTTERER B EGBERS C DAHLEY N KOCH S amp JEHRING L 2010 First identification ofsub- and supercritical convection patterns from GeoFlow the geophysical flow simulationexperiment integrated in Fluid Science Laboratory Acta Astronaut 66 193ndash200

FUTTERER B SCURTU N EGBERS C PLESA A-C amp BREUER D 2009 Benchmark onPrandtl number influence for GeoFlow II a mantle convection experiment in spherical shellsIn Proceedings of the 11th International Workshop on Modelling of Mantle Convection andLithospheric Dynamics Braunwald Switzerland

HANSEN U amp YUEN D A 1993 High Rayleigh number regime of temperature-dependentviscosity convection and the Earthrsquos nearly thermal history Geophys Res Lett 202191ndash2194

HANSEN U amp YUEN D A 1994 Effects of depth-dependent thermal expansivity on the interactionof thermal chemical plumes with a compositional boundary Phys Earth Planet Inter 86205ndash221

HART J E GLATZMAIER G A amp TOOMRE J 1986 Space-Laboratory and numerical simulationsof thermal convection in a rotating hemispherical shell with radial gravity J Fluid Mech 173519ndash544

HEBERT F HUFSCHMID R SCHEEL J amp AHLERS G 2010 Onset of RayleighndashBenardconvection in cylindrical containers Phys Rev E 81 046318

HERNLUND J W amp TACKLEY P J 2008 Modelling mantle convection in the spherical annulusEarth Planet Sci Lett 274 380ndash391

HOLLERBACH R 2000 A spectral solution of the magneto-convection equations in sphericalgeometry Intl J Numer Meth Fluids 32 773ndash797

HOUSEMAN G A 1990 The thermal structure of mantle plumes axisymmetric or triple junctionGeophys J Intl 102 15ndash24

HUTTIG C amp BREUER D 2011 Regime classification and planform scaling for internally heatedmantle convection Phys Earth Planet Inter 186 111ndash124

HUTTIG C amp STEMMER K 2008a Finite volume discretization for dynamic viscosities on Voronoigrids Phys Earth Planet Inter 171 137ndash146

HUTTIG C amp STEMMER K 2008b The spiral grid a new approach to discretize the sphere and itsapplication to mantle convection Geochem Geophys Geosyst 9 Q02018

JONES T B 1979 Electrohydrodynamically enhanced heat transfer in liquids ndash a review Adv HeatTransfer 14 107ndash148

KAMEYAMA M amp OGAWA M 2000 Transitions in thermal convection with stronglytemperature-dependent viscosity in a wide box Earth Planet Sci Lett 180 355ndash367

KELLOGG L H amp KING S D 1997 The effect of temperature dependent viscosity on thestructure of new plumes in the mantle results of a finite element model in a sphericalaxisymmetric shell Earth Planet Sci Lett 148 13ndash26

LANDAU L D LIFSHITZ E M amp PITAEVSKII L D 1984 Course of Theoretical Physicsndash Electrodynamics of Continuous Media Butterworth-Heinemann

LIDE D R 2008 Handbook of Chemistry and Physics CRC PressMALIK S V YOSHIKAWA H N CRUMEYROLLE O amp MUTABAZI I 2012 Thermo-electro-

hydrodynamic instabilities in a dielectric liquid under microgravity Acta Astronaut 81563ndash569

MAZZONI S 2011 GeoFlow II experiment scientific requirements RQ 3 European Space AgencyESA European Space Research and Technology Centre ESTEC Noordwijk The Netherlandsreference SCI-ESA-HSF-ESR-GEOFLOW II

MERCK (Merck KGaA Darmstadt Germany) 2003 1-Nonanol zur Synthese Safety Data Sheet806866 28052003 (delivered with the liquids)

MERZKIRCH W 1987 Flow Visualization Academic PressMORRIS S amp CANRIGHT D R 1984 A boundary-layer analysis of Benard convection with

strongly temperature-dependent viscosity Earth Planet Sci Lett 36 355ndash377NATAF H C amp RICHTER F M 1982 Convection experiments in fluids with highly

temperature-dependent viscosity and the thermal evolution of the planets Phys Earth PlanetInter 29 320ndash329

OGAWA M 2008 Mantle convection a review Fluid Dyn Res 40 379ndash398OGAWA M SCHUBERT G amp ZEBIB A 1991 Numerical simulations of three-dimensioanl

thermal convection a fluid with strongly temperature-dependent viscosity J Fluid Mech 233299ndash328

RATCLIFF J T SCHUBERT G amp ZEBIB A 1996 Effects of temperature-dependent viscosity onthermal convection in a spherical shell Physica D 97 242ndash252

RATCLIFF J T TACKLEY P J SCHUBERT G amp ZEBIB A 1997 Transitions in thermalconvection with strongly variable viscosity Phys Earth Planet Inter 102 201ndash212

RICHTER F M NATAF H C amp DALY S F 1983 Heat transfer and horizontally averagedtemperature of convection with large viscosity variations J Fluid Mech 129 173ndash192

SCHMALZL J BREUER M amp HANSEN U 2002 The influence of the Prandtl number on the styleof vigorous thermal convection Geophys Astrophys Fluid Dyn 96 381ndash403

SCHUBERT G amp BERCOVICI D 2009 Mantle Dynamics 7 Treatise on Geophysics ElsevierSCHUBERT G GLATZMAIER G A amp TRAVIS B 1993 Steady three-dimensional internally

heated convection Phys Fluids A 5 1928ndash1932SCHUBERT G amp OLSON P 2009 Treatise on Geophysics ndash Core Dynamics ElsevierSCURTU N FUTTERER B amp EGBERS C 2010 Pulsating and travelling wave modes of natural

convection in spherical shells Phys Fluids 22 114108SOLOMATOV V S 1995 Scaling of temperature and stress-dependent viscosity convection Phys

Fluids 7 266ndash274STEMMER K HARDER H amp HANSEN U 2006 A new method to simulate convection with

strongly temperature- and pressure-dependent vicosity in a spherical shell applications to theEarthrsquos mantle Phys Earth Planet Inter 157 223ndash249

STENGEL K C OLIVER D S amp BOOKER J R 1982 Onset of convection in avariable-viscosity fluid J Fluid Mech 120 411ndash431

SUGIYAMA K CALZAVARINI E GROSSMANN S amp LOHSE D 2007 Non-OberbeckndashBoussinesqeffects in two-dimensional RayleighndashBenard convection in glycerol Europhys Lett 8034002

TACKLEY P J 1993 Effects of strongly temperature-dependent viscosity on time-dependentthree-dimensional models of mantle convection Geophys Res Lett 20 2187ndash2190

TACKLEY P J 1996 Effects of strongly variable viscosity on three-dimensional compressibleconvection in planetary mantles J Geophys Res 101 3311ndash3332

TRAVNIKOV V EGBERS C amp HOLLERBACH R 2003 The GEOFLOW-experiment on ISS (PartII) numerical simulation Adv Space Res 32 181ndash189

WHITE D B 1988 The planforms and onset of convection with a temperature-dependent viscosityJ Fluid Mech 191 247ndash286

YANAGISAWA T amp YAMAGISHI Y 2005 RayleighndashBenard convection in spherical shell withinfinite Prandtl number at high Rayleigh number J Earth Sim 4 11ndash17

YAVORSKAYA I M FOMINA N I amp BALYAEV Y N 1984 A simulation of central symmetryconvection in microgravity conditions Acta Astronaut 11 179ndash183

YOSHIKAWA H N CRUMEYROLLE O amp MUTABAZI I 2013 Dielectrophoretic force-driventhermal convection in annular geometry Phys Fluids 25 024106

ZHANG J CHILDRESS S amp LIBCHABER A 1997 Non-Boussinesq effect thermal convection withbroken symmetry Phys Fluids 9 1034ndash1042

ZHANG P LIAO X amp ZHANG K 2002 Patterns in spherical RayleighndashBenard convection a giantspiral roll and its dislocations Phys Rev E 66 055203

ZHONG S MCNAMARA A TAN E MORESI L amp GURNIS M 2008 A benchmark study onmantle convection in a 3-D spherical shell using CitcomS Geochem Geophys Geosyst 9Q10017

ZHONG S ZUBER M T MORESI L amp GURNIS M 2000 Role of temperature-dependentviscosity and surface plates in spherical shell models of mantle convection J Geophys Res105 (B5) 11 063ndash11 082

Sheet-like and plume-like thermal flow in a spherical convection experiment performed under microgravity

Introduction

Thermally driven flow in liquids of temperature-dependent viscosity

Thermo-electro-hydrodynamic driven flow in spherical shells

Different radial variations of radial gravity

Temperature-dependent viscosity

Thermal expansion in microgravity

Effects from Prandtl number radius ratio and boundary conditions

Summary

A generic example for flow visualization

Sheet-like upwellings in a self-gravitating spherical shell experiment

Experimental observation

Numerical simulation

Plume-like upwellings in a self-gravitating spherical shell experiment with enhanced viscosity ratio and at higher Rayleigh numbers



Heat transfer properties of convective patterns

Summary discussion and outlook

Acknowledgements

References

1 IntroductionConvection in spherical shells under the influence of a radial buoyancy force is of

considerable interest in geophysical flows such as mantle convection (eg Schubert ampBercovici 2009) Laboratory experiments involving this configuration are complicatedby the fact that gravity is then vertically downwards rather than radially inwardsOne alternative is to conduct the experiment in microgravity thereby switchingoff the vertically downward buoyancy force Imposing a voltage difference betweenthe inner and outer shells and using the temperature dependence of the fluidrsquosdielectric properties can further create a radial force The first such thermo-electro-hydrodynamic (TEHD) experiment was designed by Hart Glatzmaier amp Toomre(1986) using a hemispherical shell and was accomplished on the space shuttleChallenger in May 1985

A later experiment called lsquoGeoFlowrsquo not only involved a full shell instead ofa hemisphere but also was designed to be performed on the International SpaceStation (ISS) where considerably longer flight times are possible (Egbers et al 2003)Two versions of lsquoGeoFlowrsquo have now been accomplished lsquoGeoFlow Irsquo involving anessentially constant-viscosity fluid was processed from August 2008 to January 2009(Futterer et al 2010) lsquoGeoFlow IIrsquo involving a fluid with pronounced temperature-dependent viscosity was processed from March 2011 to May 2012 (Futterer et al2012) The purpose of this paper is to present an integrative comparison of the twoexperiments with one another with accompanying numerical results and with relatedresults in the literature

The arrangement of this paper is as follows With the review in sect 2 we present anextended literature study related to thermo-viscous RayleighndashBenard convection Therewe also address the differences between Cartesian and spherical geometries Thenin sect 3 we go through the physical basis for our TEHD experiment The discussionof the list of properties involved in our experiment is the basis for linking thelsquoclassicalrsquo spherical RayleighndashBenard system (driven by Archimedean buoyancy) withour TEHD driven RayleighndashBenard experiment The final objective in this sectionis the calculation of the electric Rayleigh number from the experimental data andreordering it into domains available from the literature The lsquoGeoFlowrsquo experimentseries uses a very specific flow visualization the Wollaston schlieren interferometry orWollaston prism shearing interferometry The observed patterns of convection deliverfringe images which requires additional knowledge on interpretation possibilities Thisis demonstrated with a generic example in sect 4 Then in sectsect 5 and 6 we discussthe experimentally observed flow regimes and corresponding numerical models forboth experiment series lsquoGeoFlow Irsquo and lsquoGeoFlow IIrsquo Based on further numericalsimulations the deliberation of sect 7 is on heat transfer properties related to differentconvective patterns We summarize and give conclusions in sect 8 involving a discussionon the nonlinear effects considered in our numerical models and an outlook forrecently performed experiment series lsquoGeoFlow IIbrsquo

2 Thermally driven flow in liquids of temperature-dependent viscosityIn mantle dynamics the basics of fluid flow movement and its interaction with

tectonic plates are still the object of study As extensively reviewed in Schubert ampBercovici (2009) research on those topics is focused on theoretical numerical andexperimental materials and methods and moreover on the geophysical observationsthemselves Especially laboratory experiments have the character of lsquoexploring newphysics and testing theoriesrsquo (Davaille amp Limare 2009) There authors summarizethe results from a huge series of tank experiments which are designed to understand

Ra= αg1Td3

νrefκ (21)

Pr = νref

κ(22)

γ = νref

ν(Tref+1T)= νcold

νhot (23)

Steady sluggish-lid

Time-dependent

Mobile-lid

Sluggish-lid

Stagnant-lid

Baumgardner (1985)

Zhong et al (2008)

10ndash1107 108 109106105104

1010103

fE = qEminus 12

E2nablaε +nabla(

12ρE2 partε

partρ

) (32)

E(r)= riro

ro minus ri

1r2er V0 (34)

gE = 2ε0εr

ro minus ri

1r5er (38)

Prminus1

RaE = αE1TgEd3

νrefκ

)minus5

r2i r7

references

RaEhot = 2ε0V20

r2i r7

εr(Thot)αE(Thot)1

Finally we have

RaEhot = γ(

Cubicridge

Mobile-lid

Sluggish-lid

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GAIA num

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

Ra= αg1Td3

νrefκ (21)

Pr = νref

κ(22)

γ = νref

ν(Tref+1T)= νcold

νhot (23)

Steady sluggish-lid

Time-dependent

Mobile-lid

Sluggish-lid

Stagnant-lid

Baumgardner (1985)

Zhong et al (2008)

10ndash1107 108 109106105104

1010103

fE = qEminus 12

E2nablaε +nabla(

12ρE2 partε

partρ

) (32)

E(r)= riro

ro minus ri

1r2er V0 (34)

gE = 2ε0εr

ro minus ri

1r5er (38)

Prminus1

RaE = αE1TgEd3

νrefκ

)minus5

r2i r7

references

RaEhot = 2ε0V20

r2i r7

εr(Thot)αE(Thot)1

Finally we have

RaEhot = γ(

Cubicridge

Mobile-lid

Sluggish-lid

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GAIA num

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

Steady sluggish-lid

Time-dependent

Mobile-lid

Sluggish-lid

Stagnant-lid

Baumgardner (1985)

Zhong et al (2008)

10ndash1107 108 109106105104

1010103

fE = qEminus 12

E2nablaε +nabla(

12ρE2 partε

partρ

) (32)

E(r)= riro

ro minus ri

1r2er V0 (34)

gE = 2ε0εr

ro minus ri

1r5er (38)

Prminus1

RaE = αE1TgEd3

νrefκ

)minus5

r2i r7

references

RaEhot = 2ε0V20

r2i r7

εr(Thot)αE(Thot)1

Finally we have

RaEhot = γ(

Cubicridge

Mobile-lid

Sluggish-lid

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GAIA num

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

Steady sluggish-lid

Time-dependent

Mobile-lid

Sluggish-lid

Stagnant-lid

Baumgardner (1985)

Zhong et al (2008)

10ndash1107 108 109106105104

1010103

fE = qEminus 12

E2nablaε +nabla(

12ρE2 partε

partρ

) (32)

E(r)= riro

ro minus ri

1r2er V0 (34)

gE = 2ε0εr

ro minus ri

1r5er (38)

Prminus1

RaE = αE1TgEd3

νrefκ

)minus5

r2i r7

references

RaEhot = 2ε0V20

r2i r7

εr(Thot)αE(Thot)1

Finally we have

RaEhot = γ(

Cubicridge

Mobile-lid

Sluggish-lid

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GAIA num

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

fE = qEminus 12

E2nablaε +nabla(

12ρE2 partε

partρ

) (32)

E(r)= riro

ro minus ri

1r2er V0 (34)

gE = 2ε0εr

ro minus ri

1r5er (38)

Prminus1

RaE = αE1TgEd3

νrefκ

)minus5

r2i r7

references

RaEhot = 2ε0V20

r2i r7

εr(Thot)αE(Thot)1

Finally we have

RaEhot = γ(

Cubicridge

Mobile-lid

Sluggish-lid

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GAIA num

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

fE = qEminus 12

E2nablaε +nabla(

12ρE2 partε

partρ

) (32)

E(r)= riro

ro minus ri

1r2er V0 (34)

gE = 2ε0εr

ro minus ri

1r5er (38)

Prminus1

RaE = αE1TgEd3

νrefκ

)minus5

r2i r7

references

RaEhot = 2ε0V20

r2i r7

εr(Thot)αE(Thot)1

Finally we have

RaEhot = γ(

Cubicridge

Mobile-lid

Sluggish-lid

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GAIA num

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

gE = 2ε0εr

ro minus ri

1r5er (38)

Prminus1

RaE = αE1TgEd3

νrefκ

)minus5

r2i r7

references

RaEhot = 2ε0V20

r2i r7

εr(Thot)αE(Thot)1

Finally we have

RaEhot = γ(

Cubicridge

Mobile-lid

Sluggish-lid

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GAIA num

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

RaE = αE1TgEd3

νrefκ

)minus5

r2i r7

references

RaEhot = 2ε0V20

r2i r7

εr(Thot)αE(Thot)1

Finally we have

RaEhot = γ(

Cubicridge

Mobile-lid

Sluggish-lid

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GAIA num

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

Cubicridge

Mobile-lid

Sluggish-lid

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GAIA num

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

Cubicridge

Mobile-lid

Sluggish-lid

Cubic6 plumes

Cubic7 plumes

10ndash1

107 108 109106105104 1010103

GeoFlow I expnum

GAIA num

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

number Ra and RaE

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

Arrhenius law

αE =minus 1εr

partεr

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

35 Summary

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

(times 104)

011 12 13 1410 15

C (GeoFlow II)

H(x) = 1dprime

int dprime

0T(x z) dz (41)

h(u) = 14

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

HP(x) = h( 13 |x|) (43)

HS(x) = h( 13 |x2|) (44)

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

int ro

T(ϕ θ r) dr (51)

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

10ndash1

ndash1

10ndash1

ndash1

ndash60

ndash20

ndash100

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

104 105 106 107

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

(a) (c)

(b) (d)

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

Racold = 1times 103

Racold = 1times 104

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

Racold = 1times 105

Racold = 1times 106

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

Rayleigh number

104 105

(a) (b)

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

GeoFlow I

partε

parttT

Dt2ρc (81)

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

R E F E R E N C E S

182 283ndash301

Introduction







Summary










Acknowledgements

References

Documents

J. Fluid Mech. (2013), . 735, pp. doi:10.1017/jfm.2013.507 Sheet … · 2020. 1. 24. · 650 B. Futterer et al. To rearrange all the latter results in our context, we ﬁrst refer