Plate motions, hotspots, and plumes

Plate motions, hotspots, and plumes These notes cover a rather disparate collection of pieces of information. Hotspots are geophysically important not only as a means for the Earth to lose heat, but potentially as a kinematic marker. Hotspots are best defined as time-transgressive volcanic lineaments. This is an observational definition. Plumes, in contrast, are a physical process of upward transport of material, usually hot, rising across slower-moving mantle. Hotspots need not be created by plumes, and plumes need not create hotspots (a recent paper, for instance, argues that hotspots preferentially occur where lithospheric stress is extensional to only mildly compressional). To understand some of the problems associated with hotspots as markers, we must first understand how plates move, how such movement is reconstructed, and then examine how stationary hotspots might really be. Plumes Let us consider the possibility of a plume: a long-lived conduit to some depth arising at some anomalously hot area. The creation of a plume is most easily modeled as the rise of a sphere through a viscous medium with a cylindrical conduit attached. The sphere at the top is the plume head. If this sphere has radius r and density contrast with surrounding mantle of ρ (which is negative), then the terminal velocity of the body is u =! r2 g"# 3$ m = r2 g# m %"T 3$ m (1) where the substitution to a temperature difference uses the familiar assumption of a simple thermal expansion (this is equation 6-242 of Turcotte and Schubert, 2 nd ed.). Note that while we have formulated this as a purely thermal feature, it is possible to generate a plume by compositional differences. The viscosity η m is of the surrounding mantle. If we consider the upward flow of material in the conduit, the rate of flow by volume Q V is (for laminar flow) Q V =! " #$gr 4 8 % c = " &$ m #TgR 4 8 % c (2) where R is the radius of the conduit and η c is the viscosity of the fluid in the conduit (Eqn. 6-48 of Turcotte and Schubert). This can in turn be converted to a buoyancy flux B (the amount of excess buoyancy delivered up the conduit) by multiplying by - ρ again,

Hotspots and plate motions Physics & Chemistry of the Solid Earth, p. 2 B =! "#gr 4 "# 8 $ c =! % 2 # 2 m "T 2 gr 4 8 $ c (3) The heat flux Q H is simply the volume flux times the amount of heat the volume is carrying, so: Q H =! m c p "TQ V = c p B # (4) Because the flux up the conduit must equal the rate of rise of the plume head, we can set Q V equal to u times the crosssectional area of the conduit, πr 2. The buoyancy flux itself can be estimated by multiplying the crosssectional area of the swell by the velocity of the surface track of the hotspot times the density difference between the mantle and water (for submarine hotspots) or air (subaerial hotspots). Representative values from Hawaii (estimated from the buoyancy flux of 7400 kg s -1 producing a swell centered on Hawaii) would lead to a flux of about 12 km 3 /yr (volcanic additions to Hawaii are about 0.1 km 3 /yr) and the velocity u would be about 54 cm/yr with a radius (from (3)) of 84 km. Other plumes (especially away from ridges) would have smaller fluxes.

Hotspots and plate motions Physics & Chemistry of the Solid Earth, p. 3 Plate Motions First, consider the relative motion of two plates on the Earth s surface. At two points, the two plates will not move relative to one another. These points are where their relative axis of rotation intersects the plates. This is termed the Euler pole. The Euler pole goes through the center of the Earth (if it didn t, plates would move away or towards the Earth s center). The magnitude of the vector is the angular motion of one plate relative to the other; if we are interested in the motion of plate j relative to plate i, then the Euler pole would be ω ji and the velocity at a point r (where r is a vector from the center of the Earth) would be v ji =! ji " r (5) The cross product reaches a maximum when you are 90 away from the pole of rotation. This means that relative motion changes with distance from the pole (and changes polarity across the pole). This can be recalculated as a local velocity vector without too much difficulty (see p. 291, Stein and Wysession, 2003) for a rotation pole at latitude θ and longitude φ and a point at latitude λ longitude µ: v North = a! cos" sin ( µ # $ ) v East = a! [ sin" cos% # cos" sin%cos ( µ # $ )] (6) where a is the radius of the Earth. Usually you find relative plate motions expressed as the latitude and longitude of the Euler pole and the relative motion. Thus North America relative to the Pacific plate has an angular motion of 7.8 x 10-7 /yr about a pole at 48.7 N, 78.2 W for the NUVEL-1 plate velocity model of DeMets et al, 1990. Because of the right-hand rule usage, this works out to North America moving to the south (more or less) in the western U.S. For a specific point, say here in Colorado at 40 N, 105 W, you d get 31.17 mm/yr towards 146.76 clockwise from north (so 33.24 east of south). A handy reference is a plate motion calculator maintained by UNAVCO: http://sps.unavco.org/crustal_motion/dxdt/nnrcalc/. This is handy, but where does this info come from? In recent years, space-based geodesy has allowed us to measure plate motions directly, but this has not been a major contributor to plate motions used in most geologic studies, and of course these are merely documenting rates at the present. In the geologic past, we only have limited information, and as far as a quantitative record of motion between two plates we only have information from spreading centers. Destructive plate margins do not preserve a record of the duration or rate of subduction, nor is the true relative motion easily derived. Transform margins can provide an important constraint on the angle of relative motion, but this turns out to only be of much practical use in the ocean basins. Continental transforms tend to have too much deformation alongside them to be of much use in

Hotspots and plate motions Physics & Chemistry of the Solid Earth, p. 4 getting plate motions. Thus quantitative geologic estimates of plate motions come from the record of seafloor spreading. In point of fact, this is usually done by identifying the intersection of a given magnetic anomaly with its counterpart on the opposite side of a ridge. The finite Euler pole must then lie along a great circle bisecting the two points. As more points are added, the position of the Euler pole can be more tightly constrained. [This can also be done for plates now gone, if you work from the remaining half and assume spreading was symmetric]. Using the spreading histories produces a so-called plate circuit reconstruction. If you can walk from one plate to another across spreading centers, then you can add up the sequence of rotation poles to get the net pole between the starting and ending plates of the circuit. It so happens that this is a long circuit for motion between the Pacific and North American plates (the circuit is North America to Africa to Antarctica to the Pacific, sometimes tossing in the Africa to India to Australia to Antarctica circuit for good measure). As hotspots were recognized, it was proposed early on that these were the surface manifestations of deep-seated plumes that were fixed at some great depth (frequently the core-mantle boundary). This proposal led to the idea that hotspots could constitute a fixed reference frame. If hotspots do not move, then it is far easier to determine the relative position of plates in the past. For North America-Pacific, usually the motion of North America is taken relative to Africa, then Africa to hotspots, and then hotspots to Pacific. This can be done over a longer period of time than full plate circuit reconstructions because uncertainty in some plate boundaries no longer interferes with getting the net motion. This has been the basis of the most influential plate reconstruction for the Mesozoic interactions of North America with oceanic plates to the west that came out of David Engebretson s thesis work in the 1980s with Allan Cox. Unfortunately there is no reason to think that hotspots are fixed regardless of their origin. If they come from the core-mantle boundary, it is not clear that the plumes will stay fixed at their origin. As plumes rise, they pass through convecting mantle that presumably deflects them (note that the upward velocity of the Hawaiian plume that we calculated is comparable to the motion of the Pacific plate). Observationally, hotspots are inferred to move relative to one another at rates of about 1 cm/yr to perhaps higher values (e.g., Koppers et al., EPSL 2001). Their use in making plate reconstructions is therefore more limited than is frequently acknowledged.

Hotspots and plate motions Physics & Chemistry of the Solid Earth, p. 5 Some Hotspot Papers: Beier, C., Rushmer, T., and Turner, S.P., Heat sources for mantle plumes, G^3, 9, Q06002, 2008. Consider heats sources of CMB based plumes (they list as Afar, Easter Island, Hawaii, Louisville, and Samoa) reject concentrations of U,Th, K or transport of material across CMB, instead end up with conductive heat transport across CMB. Boschi, L., Becker, T., and Steinberger, B., Mantle plumes: Dynamic models and seismic images. Geochem. Geophys. Geosyst. (2007) vol. 8 (10) pp. 20. Test a number of tomo models for presence of plumes. Followup paper later tries to improve the statistics on these being identifiably plumes. Christiansen, R. L., G. R. Foulger, and J. R. Evans, Upper-mantle origin of the Yellowstone hotspot, Geol. Soc. Am. Bull., 114, 1245-1256, 2002. The continued crusade against Yellowstone hotspot. Foulger et al., Seismic tomography shows that upwelling beneath Iceland is confined to the upper mantle, GJI 146, 504-530, 2001. Argues that Iceland plume only goes to ~410 km (though the tomography here shows that thermal anomaly to that depth not a lot different looking than the Wolfe et al nature paper (385, 245-247, 1997). Foulger, G.R., J. H. Natland, D. C. Presnall and D. L. Anderson (ed.), Plates, Plumes and Paradigms, GSA Spec. Paper 388, 881 pp., 2005 Koppers et al., Testing the fixed hotspot hypothesis using 40 Ar/ 39 Ar age progressions along seamount trails, EPSL, 185, 237-252, 2001. (Apparently Jason Morgan didn t realize the result despite being an author). Shows that Pacific hotspots must be moving relative to one another.) Montelli, R., G. Nolet, F. A. Dahlen, G. Masters, E. R. Engdahl, nad S-H. Hung, Finitefrequency tomography reveals a variety of plumes in the mantle. Science (2004) vol. 303 (5656) pp. 338-343. Claims to image several plumes to CMB (Ascension, Azores, Canary, Easter, Samoa, Tahiti) with diameters of several hundred kilometer. [This thing is a touchstone for the plume community cited a LOT] Schutt, D. L. and Dueker, K., Temperature of the plume layer beneath the Yellowstone hotspot Geology, 36 [8], 623-626, Aug 2008. Invert surface wave data and infer mantle plume under Yellowstone has potential temperature >55-80 degrees than? something. (This should point to all their tomography under Yellowstone) Sleep NH, Mantle plumes from top to bottom, EARTH-SCIENCE REVIEWS, 77(4), 231-271, Aug. 2006. Suggests that tomographic imaging of plume tails is the key to testing this hypothesis.