Plume/lithosphere interaction at Hawaii and Iceland.Read more

The Hawaiian island chain and Iceland are the two best-known examples of volcanic
edifices generated when an ascending mantle plume impinges on the base of the oceanic
lithosphere. We have investigated both cases using a combination of analytical lubrication
theory and a more realistic three-dimensional numerical convection code that includes heat
transfer and a parameterized melting model. In the Hawaiian case, the buoyant plume material
spreads beneath the Pacific plate while being advected northwestward by the plate motion.
This results in a quasi-parabolic distribution of buoyant material that supports a large
topographic swell (3000 km long, 1200 km wide, 1400 m high) and its associated gravitational
potential (geoid) anomaly. Comparison of these observables with the predictions of the numerical
and analytical models allows us to constrain several poorly known characteristics of the Hawaiian
plume, including its buoyancy flux, excess temperature, and minimum viscosity. Melting occurs both
in a primary melting zone above the plume stem and in a weaker secondary melting zone 300 to 500 km
downstream, separated by an interval where no melting occurs. On this basis we proposed that the preshield-,
shield-, and postshield stages of Hawaiian volcanism are generated by the primary melting zone, and the rejuvenated
stage by the secondary melting zone.

More recently, we investigated in detail the shape of the Hawaiian swell,
which is imperfectly predicted by models with Newtonian temperature-dependent viscosity.
Using a lubrication model with a power-law viscosity (a so-called generalized Newtonian fluid),
we showed that the height of the swell decays with distance x downstream from the hotspot as x^(-1/(3 n + 2)),
where n is the power-law index. Comparing this prediction to the observed swell topography, we find that the swell
shape is better fit by a non-Newtonian shear-thinning rheology with n = 3.5 than by a Newtonian (n = 1) rheology.
The value n = 3.5 is the one determined experimentally for olivine, the dominant mineral in Earth's uppermost mantle

The salient feature of Iceland is that the associated mantle plume rises not beneath intact old lithosphere,
as at Hawaii, but rather beneath a spreading ridge where the lithosphere is young and very thin. Important
observables for this case are the amplitude and the along-ridge extent of the topography anomaly supported
by the plume. Adapting our Newtonian models (analytical and numerical) for this case, we found that the
topography could be generated by a wide (several hundred km) and relatively cool (temperature anomaly 100 degrees) plume.

Mechanics of polycrystals and the generation of seismic anisotropy.Read more

Progressive deformation of upper mantle rocks via dislocation creep causes their constituent crystals to take on a non-random
orientation distribution (crystallographic preferred orientation or CPO)
whose observable seismological signatures include shear-wave splitting and azimuthal dependence of surface wave speeds.
Comparison of these signatures with mantle flow models thus allows mantle dynamics to be unraveled on global and regional scales.
However, existing self-consistent models of CPO evolution are computationally expensive when used in conjunction with numerical convection models.
We have recently developed a new method, 'ANPAR', which is based on an analytical parameterization of the crystallographic spin predicted by the second-order (SO)
self-consistent theory. When applied to olivine polycrystals (dunites), our parameterization fits the predictions of the SO model almost perfectly
(variance reduction > 99.5%), yet runs 50000 times faster. The ANPAR model thus shows promise for calculating CPO in large three-dimensional and/or
time-dependent flow models. Currently we are working to extend ANPAR to multiphase aggregates (e.g. olivine plus enstatite) and to lower-mantle phases (e.g. post-perovskite).

A thin stream of honey falling from a sufficient
height onto a surface creates a whirling coil-like structure,
a phenomenon known as 'liquid rope coiling'.
Using a combination of analytical, numerical, and
experimental approaches, I and colleagues from
Iran and Holland have
determined a complete regime diagram for
liquid rope coiling. It comprises four distinct
dynamical regimes (viscous, gravitational,
inertio-gravitational, and inertial) depending on
how the viscous forces that resist the bending of the
rope are balanced. The most interesting behavior occurs
in the inertio-gravitational regime, where
multiple coiling modes coexist at a fixed fall height
within a certain range. Here, the long trailing 'tail' of
the rope acts as a distributed pendulum with an
infinite sequence of natural oscillation frequencies.
If one of these is close to the frequency imposed by
the coil, then the tail enters into resonance with
the coil, manifest as a substantial increase in
the amplitude of the tail's whirling motion.

The experiments leading to the above results
were performed using fluids with very high viscosity,
up to 100000 times that of water. More recently,
we have performed experiments
using fluids with lower viscosity, in the range
500-6000 times that of water. These
experiments show that a viscous rope can
exhibit five different states: axisymmetric
stagnation flow; steady coiling; periodic
folding with steady rotation of the folding
plane; periodic collapse of the hollow
cylinder formed by a coiling rope; and
supercoiling, wherein the hollow cylinder
itself coils periodically instead of collapsing.
Up to three of these states can be
observed for given fixed values of the
viscosity, the flow rate, and the fall height.
The states observed depend on a
dimensionless parameter that measures
the ratio of the rope's radius to
the coiling radius.

We have also investigated a generalization of liquid rope coiling
in which the jet falls onto a horizontal belt moving with a constant
velocity in its own plane. When the belt is moving rapidly, the
jet is simply stretched out and leaves a straight trace on the belt.
Below a critical belt speed, however, the fluid starts to meander
back and forth on the belt. As the belt speed decreases further,
other striking patterns appear, including alternating loops, a 'W'
pattern, double coiling, etc. The resemblance of these patterns
to textile stitches led the original investigators of this
system at the University of Cambridge to call it the
'fluid mechanical sewing machine' (FMSM). Our first contribution
was a linear stability analysis of the steady dragged state,
which predicted a critical belt speed for the onset of
meandering in excellent agreement with laboratory measurements.
Subsequently, we ventured into the nonlinear regime with the help
of a numerical code for nonstationary viscous ropes
newly developed by colleagues in Paris and New York. This code allowed
us to determine a complete phase diagram for the different
patterns as functions of the belt speed and the fall height.
Most recently, we and colleagues from the
University of Cambridge derived a simple reduced model for the FMSM
patterns that has only three degrees of freedom, but which
nevertheless predicts very well the sequence of patterns
observed as a function of the belt speed in the non-inertial limit.
My current work on liquid ropes focusses on two phenomena:
(1) the generation of spiral waves of air bubbles when the
loops of rope laid down during coiling are not concentric;
and (2) the instability of a vertical rope ejected downward
from a rapidly rotating nozzle.

Elastic rods include such familiar objects as electrical cables,
mountaineering rope, and concrete reinforcement rods.
The equations governing their behavior are similar
in many ways to those describing liquid ropes,
the primary difference being that elastic ropes
are nearly inextensible in most situations. Our
first contribution to this field was to determine
a complete regime diagram for the steady coiling
of an elastic rod falling onto a surface. Elastic
rod coiling displays the same regimes as liquid
coiling, with the addition of a 'whirling shaft' regime
that corresponds to standing elastic waves.
We next studied the elastic analog of the fluid-mechanical
sewing machine, in which thin elastic threads are fed
downward onto a moving belt. The patterns observed are
mostly similar to the liquid case, but there also
a few new patterns that appear to have no liquid analog.

Most recently, and on a more playful note, we have
studied the mechanics of the lasso used in Western
'trick roping' shows. We consider the simplest rope trick,
the Flat Loop, and use numerical continuation to
classify the steadily rotating solutions in a bifurcation diagram and analyse
their stability. Matched asymptotic expansions are used to
determine the shape of the rope in the elastic boundary
layer near the sliding loop ('honda'), and we derive a macroscopic criterion for the sliding of the honda in terms of the microscopic Coulomb static friction criterion. Our predictions agree well with rapid-camera observations of a professional trick roper and with laboratory experiments using a mechanical 'robo-cowboy'.

Toricelli's curtain: morphology of laminar viscous jets under gravity. Read more

It has been known since the seventeenth century
that a jet of water issuing horizontally from a hole in
the side of a bucket describes a parabolic trajectory.
However, this is no longer true when the fluid issues
from a long tube.
Recent experiments at FAST
on laminar jets issuing from a horizontal tube
show that the initially round jet typically
evolves into a thin vertical fluid curtain
bounded by a primary jet above
and a secondary jet below. The two jets then collide further
downstream to form a 'fluid chain'. Finally, injected dye
reveals the presence of a recirculating flow with
helical streamlines around the axis of the primary jet.
We call this whole phenomenon 'Toricelli's curtain'
in honor of Toricelli's pioneering work on liquid jets.
We are currently investigating this problem using
a combination of analytical, numerical, and
experimental approaches, focusing on the morphology
of the jet as a function of the governing
dimensionless parameters.
(collaboration with M. Rabaud, FAST).

Both seismic tomography and laboratory experiments indicate that subducted lithosphere
has a complex three-dimensional morphology.
In order to understand the factors controlling the deformation of subducting lithosphere,
we have carried out three-dimensional boundary-element numerical simulations
of a dense fluid sheet with thickness h and viscosity eta_2
sinking in an ambient mantle with viscosity eta_1.
Instantaneous solutions with an infinitely deep mantle show that the dynamics
are controlled by a
dimensionless 'stiffness' S, which determines whether the slab's
sinking speed depends on the viscosity of the ambient mantle (S<1) or on
that of the sheet itself (S>10).
Using the more realistic configuration of an ambient fluid layer with a
finite depth H, we carried out a systematic investigation of the slab's
interaction with the bottom boundary as a
function of eta_2/eta_1 and H/h. These solutions delineate a rich regime diagram
of different subduction modes, including trench-retreating, slab folding, trench-advancing,
and a new 'advancing-folding' mode in which slab folding is preceded by
advancing trench motion. The solutions demonstrate that mode selection is
controlled by the dip of the leading edge of the slab at the time when
it first encounters the bottom boundary.

My current work on subduction involves developing a new
hybrid boundary-integral/thin-sheet (BITS) model for a thin
viscous sheet immersed in a second fluid. This model has
several advantages over a full boundary-integral representation:
thin-sheet theory is built into the model from the beginning;
only a single surface (the sheet's midsurface) needs to be discretized;
and the sheet may have a nonlinear rheology.
(PhD thesis of Bingrui Xu).

Publications

Journal articles

On the importance of advection in
determining the local isotopic composition of the mantle
Richter F M, Ribe N M
Earth Planet. Sci. Lett. 43, 212-222 (1979)

Observations of flexure and the geological evolution of the Pacific Ocean basin
Watts A B, Bodine J H, Ribe N M Nature 283, 532-537 (1980)

On the interpretation of frequency response functions for oceanic gravity and bathymetry
Ribe N M Geophys. J. R. Astr. Soc.70, 273-294 (1982)

The distribution of intraplate volcanism in the Pacific Ocean basin: a spectral approach
Ribe N M, Watts A B Geophys. J. R. astr.
Soc.71, 333-362 (1982)

Diapirism in the Earth's mantle: experiments on the motion of a hot sphere in a fluid with temperature dependent viscosity
Ribe N M
J. Volcanol. Geotherm. Res.16, 221-245 (1983)

On geoid heights and flexure of the lithosphere at seamounts
Watts A B, Ribe N M J. Geophys. Res. 89, 11152-11170 (1984)

On the determination of the deflection of the vertical using satellite altimetry
Watts A B, Horai K-I, Ribe N M
Marine Geodesy 8, 85-127 (1984)

The generation and composition of partial melts in the Earth's mantle
Ribe N M Earth Planet. Sci. Lett. 73, 361-376 (1985)

The deformation and compaction of partially molten zones
Ribe N M Geophys. J. R. astr. Soc. 83, 487-501 (1985)

Goethe's critique of Newton: a reconsideration
Ribe N M
Stud. Hist. Phil. Sci. 16, 315-335 (1985) Italian translation:
La critica di Goethe a Newton: una riconsiderazione, Il Minotauro XIV, 54-68 and
XV, 55-67 (1987)

Melt segregation driven by dynamic forcing
Ribe N M Geophys. Res.
Lett. 13, 1462-1465 (1986)

A stagnation-point flow model for melt extraction from a mantle plume
Ribe N M, Smooke M D J. Geophys. Res. 92, 6437-6443 (1987)

Theory of melt segregation: a review
Ribe N M J. Volcanol.
Geotherm. Res. 33, 241-253 (1987)

On the dynamics of mid-ocean ridges
Ribe N M J. Geophys. Res.93, 429-436 (1988)

Dynamical geochemistry of the Hawaiian plume Ribe N M Earth
Planet. Sci. Lett.88, 37-46 (1988)

The effect of lateral viscosity variations on surface observables
Koch D M, Ribe N M Geophys. Res. Lett. 16,
535-538 (1989)

A continuum theory for lattice preferred orientation Ribe N M
Geophys. J. 97, 199-207 (1989)

Mantle flow induced by back-arc spreading
Ribe N M Geophys.
J. 98, 85-91 (1989)

Seismic anisotropy and mantle flow
Ribe N M J. Geophys. Res. 94, 4213-4223 (1989)

A theory for plastic deformation and
textural evolution of olivine polycrystals
Ribe N M, Yu Y J. Geophys. Res. 96, 8325-8335 (1991)

The dynamics of thin shells with variable viscosity
and the origin of toroidal flow in the mantle Ribe N M Geophys. J.
Int., 110, 537-552 (1992)

On the relation between seismic anisotropy and finite
strain Ribe N M J. Geophys. Res.} , 97, 8737-8747 (1992)

Three-dimensional modeling of plume-lithosphere interaction
Ribe N M, Christensen U R J. Geophys. Res.99, 669-682 (1994)

The global distribution of
hotspots and instability of D"
Ribe N M, de Valpine D P Geophys. Res. Lett.21, 1507-1510 (1994)

The dynamics of plume-ridge interaction, 1: Ridge-centered plumes
Ribe N M, Christensen U R, Theissing J
Earth Planet. Sci. Lett. 134, 155-168 (1995)

Radioactive disequilibrium among 238U series nuclides in
recent volcanic rocks: a model for chronology and mechanism of
formation Turekian K K, Krishnaswami S, Ribe N M, Reinitz I M Geochem. Int. 33, 1-14 (1996)

The dynamics of
plume-ridge interaction, 2: Off-ridge plumes
Ribe N M J. Geophys. Res. 101, 16,195-16,204 (1996)

Cartesian optics and the mastery of nature Ribe N M
Isis 88, 42-61 (1997).

The dynamics of
plume-ridge interaction--III. The effects of ridge migration
Ribe N M, Delattre W L
Geophys. J. Int. 133, 511-518 (1998)

Spouting and planform selection in the Rayleigh-Taylor
instability of miscible viscous fluids Ribe N M
J. Fluid Mech. 377, 27-45 (1998)

The dynamical origin of
Hawaiian volcanism Ribe N M, Christensen U R
Earth Planet. Sci. Lett.171, 517-531 (1999)

Geoid height versus
topography for a plume model of the Hawaiian swell Cserepes L, Christensen U R, Ribe N M (2000)
Earth Planet. Sci. Lett. 178, 29-38.

Dynamic elevation of
the Cordillera, western United States
Lowry A R, Ribe N M, Smith R B J. Geophys. Res.105
23,371-23,390 (2000)

Stretching and bending of thin viscous sheets
Ribe N M J. Fluid Mech. 433, 135-160 (2001)

A kinematic model for recrystallization
and texture development in olivine polycrystals
Kaminski E, Ribe N M
Earth Planet. Sci. Lett. 189, 253-267 (2001)

A general theory for the dynamics of thin
viscous sheets Ribe N M J. Fluid Mech. 457, 255-283 (2002)

Time scales for the evolution of
seismic anisotropy in mantle flow Kaminski E, Ribe N M Geochem. Geophys.
Geosyst.10 (2002)

Exploratory experimentation: Goethe, Land, and color theory
Ribe N M, Steinle F Physics Today 55/7 43-49 (2002)

Periodic folding of viscous sheets Ribe N M
Phys. Rev. E 68, 036305 (2003) [discussion:
'Liquids fold according to density-viscosity ratio: New theory
sheds light on plate tectonics and pancake batter',
Nature Science Update , 15 September 2003,
www.nature.com/nsu/030908/030908-19.html]

Optics, polemics and ethics in Goethe's critique
of Newton Ribe N M Arch. int. hist. sci. 53,
257-263 (2003)

Through Thick and Thin (News and Views)
Ribe N M Nature 427, 793-795 (2004)

D-Rex, a program for calculation of seismic anisotropy in the convective
upper mantle Kaminski E, Ribe N M, Browaeys J T Geophys. J. Int. 158, 744-752 (2004)

Coiling of viscous jets Ribe N M
Proc. R. Soc. Lond. A 460, 3223-3239 (2004)

Liquid rope coiling on a solid surface
Maleki M, Habibi M, Golestanian R, Ribe N M, Bonn D
Phys. Rev. Lett. 93, 214502 (2004)

Partial melting and upwelling rates
beneath the Azores from a U-series perspective
Bourdon B, Turner S P, Ribe N M
Earth Planet. Sci. Lett. 239, 42-56. (2005)

Multiple coexisting states of liquid rope coiling
Ribe N M, Huppert H E, Hallworth M A, Habibi M, Bonn D
J. Fluid Mech. 555 275-297 (2006)

Stability of liquid rope coiling
Ribe N M, Habibi M, Bonn D
Phys. Fluids 18 084102 (2006)

Dynamics of mantle plumes: insights from U-series geochemistry
Bourdon B, Ribe N, Stracke A, Saal A E, Turner S P
Nature 444, 713-717 (2006)

Stability of a dragged viscous thread:
onset of 'stitching' in a fluid mechanical 'sewing machine'
Ribe N M, Lister J R, Chiu-Webster S
Phys. Fluids 18, 124105 (2006)

Dynamics of liquid rope coiling
Habibi M, Maleki M, Golestanian R, Ribe N, Bonn D
Phys. Rev. E 74, 066306 (2006)

Buckling instabilities of subducted lithosphere beneath the transition zone
Ribe N M, Stutzmann E, Ren Y, van der Hilst R
Earth Planet. Sci. Lett. 254} 173-179 (2007)

Coiling of elastic ropes Habibi M, Ribe N M, Bonn D
Phys. Rev. Lett. , 99, 154302 (2007) [press commentary: "Researchers
uncover physics of coiling ropes",
http://www.physorg.com/news112357305.html; "Unravelling the mysteries of
coiling ropes'', http://physicsworld.com/cws/article/news/31564, 23/10/2007]

Spontaneous generation of spiral waves by a hydrodynamic instability
Habibi M, M\oller P, Ribe N M, Bonn D
Europhys. Lett. 81, 38004 (2007)

Meandering instability of a viscous thread
Morris S W, Dawes J H P, Ribe N M, Lister J R
Phys. Rev. E 77 066218 (2008)

Deformation modes of subducted
lithosphere at the core-mantle boundary:
An experimental investigation Loubet N, Ribe N M, Gamblin Y
Geochem. Geophys. Geosyst. , 10, Q10004 (2009)

Bending mechanics and mode selection in free subduction:
a thin-sheet analysis Ribe N M Geophys. J. Int. 180, 559-576 (2010)

Buckling of liquid columns
Habibi M, Rahmani Y, Bonn D, Ribe N M
Phys. Rev. Lett. 104, 074301 (2010)

Inferring nonlinear mantle rheology from the shape of the Hawaiian swell
Asaadi N, Ribe N M, Sobouti F
Nature 473, 501-504 (2011)

Pattern formation in a thread falling onto a moving belt: an 'elastic sewing machine'
Habibi M, Najafi J, Ribe N M
Phys. Rev E 84, 016219 (2011)

All bent out of shape : Buckling instabilities of sheared fluid layers
Ribe N M J. Fluid Mech. 694, 1-4 (2012)

A numerical investigation of the
fluid mechanical sewing machine Brun P-T, Ribe N M, Audoly B Phys. Fluids 24, 043103 (2012)

Dynamics of free subduction from 3-D
boundary-element modeling Li Z, Ribe N M J. Geophys. Res.117, B06408 (2012)

New constraints on the origin of the Hawaiian swell from wavelet analysis
of the geoid to topography ratio Cadio C, Ballmer M, Panet I, Diament M, Ribe N
Earth Planet. Sci. Lett. 359-360, 40-54 (2012)

Delayed capillary breakup of falling viscous threads Javadi A, Eggers J, Bonn D, Habibi M, Ribe N M
Phys. Rev. Lett. 110, 144501 (2013)

Dynamical similarity and density (non-) proportionality in experimental tectonics
Ribe N M, Davaille A
Tectonophys. 608, 1371-1379 (2013)

Small-scale convection
in a plume-fed low-viscosity layer beneath a moving plate
Agrusta R, Arcay D, Tommasi A, Davaille A, Ribe N, Gerya T
Geophys. J. Int. 194, 591-610 (2013)

Development of texture and seismic anisotropy during the onset of subduction
Di Leo J F, Walker A M, Li Z.-H., Wookey J, Ribe N M, Kendall J-M, Tommasi A
Geochem. Geophys. Geosyst. , 15, 192-212 (2013)

Subduction-induced mantle flow, finite strain, and seismic anisotropy: Numerical modeling
Li Z, Di Leo, J F, Ribe, N M.
J. Geophys. Res. 119, 5052-5076 (2014)

Liquid supercoiling
Habibi M, Hosseini S M, Khatami M H, Ribe N M
Phys. Fluids 26, 024101 (2014)

An introduction to the dynamics of the lasso
Brun P-T, Ribe N M, Audoly B
Proc. R. Soc. London A 470, 20140512 (2014)

Liquid ropes: A geometrical model for thin viscous jet instabilities
Brun P-T, Audoly B, Ribe N M, Eaves T S, Lister J R
Phys. Rev. Lett. 114, 174501 (2015)

Analytical parametrization of self-consistent polycrystal mechanics: Fast calculation of upper mantle anisotropy
Goulding N J, Ribe N M, Castelnau O, Walker A M, Wookey J
Geophys. J. Int. 203, 334-350 (2015)

A hybrid boundary-integral/thin-sheet equation for
subduction modelling
Xu B, Ribe N M
Geophys. J. Int. 206, 1552-1562 (2016)

Liquid rope coiling: a synoptic view
Ribe N M
J. Fluid Mech. 812, R2 (2017)

Book chapters

Fluid mechanics of mantle plumes
Ribe N M, Davaille A, Christensen U in Mantle
Plumes - A Multidisciplinary Approach , eds. J. R. R. Ritter
and U. R. Christensen, pp. 1-48, Springer, Berlin (2007)

Analytical Approaches to Mantle Dynamics
Ribe N M
in Treatise
on Geophysics , vol. 7, Mantle Dynamics , 2nd ed., ed. D. Bercovici, pp. 145-196,
Elsevier Scientific (2015)

Liquid rope coiling
Ribe N M, Habibi M, Bonn D
Ann. Rev. Fluid Mech. 44, 249-266 (2012)

Mariotte Ribe N M in The Dictionary of Seventeenth-Century
French Philosophers , ed. Luc Foisneau (London, New York: Thoemmes Continuum), 823-826 (2008)

Popularizing articles

Instabilites de flambage dans les fluides visqueux :
du laboratoire au manteau terrestre
Ribe N M, Habibi M, Bonn D, Stutzmann E (2008)
Reflets de la physique , no. 11,
octobre 2008, 9-12 (2008)

The liquid rope trick Ribe N M, Habibi M, Bonn D Sci. American 310, 66-71 (2014)