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 Local regions of thin lithosphere act as catchments of hot buoyant plume material. Unless replenished, the trapped plume material cools by convection to the mantle adiabat by several tens of million years. In particular, currently hot material from the ∼130 Ma, Paraná starting plume head is unlikely to supply 85 Ma to recent volcanism on the mainland of Brazil and the Martin Vaz and Fernando hot spots. Rather, a plume tail may now underlie southern Brazil where tomographic studies detect conduit-shaped velocity anomaly through the upper mantle. If the tomographic study in fact found a plume tail, the track crossed the Amazon rift at ∼85 Ma and (since then) lateral flow along thin regions of the lithosphere from the tail fed widespread feeble volcanism including the flow line hot spots of Martin Vaz and Fernando.
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 The lithosphere acts as an upside-down drainage pattern for buoyant ponded plume material. Closed regions of thin lithosphere beneath continents may act as catchments [Artyushkov et al., 1980] (Figure 1). The trapped plume material may later partially melt and produce volcanism well after the plume head or pond tail event that originally filled the catchment. In particular, the divide between the interior of the continent and the adjacent ocean basin may be breached. When this occurs, the plume material may flow out across a passive margin along flow lines toward the nearby ridge. Alternatively, an active plume tail may fill a continental catchment that “underbottoms” (like a dam overtops), again producing a flow line hot spot. In both cases, the kinematics are analogous to valley glaciers fed by an ice cap. Physically, the channeling hot plume material is analogous to lava tubes in pahoehoe flowing over rough terrain where the fastest flowing distributaries remain hot.
 The fate of trapped plume material is of interest with regard to prolonged volcanism following a major plume event. It is also relevant to the inversion of sedimentary basins formed above thin lithosphere and their subsequent subsidence as lithosphere thermally equilibrates [Sleep, 1997]. Possible example, include the Irish Sea [Cope, 1994] and the Iceland plume, Africa basins and the east African plume [Ebinger and Sleep, 1998], and the Michigan basin which is not obviously associated with a plume [Kaminski and Jaupart, 2000].
 I concentrate on the fate of trapped plume material with respect to subsequent volcanism. I select Brazil and the adjacent ocean basin to provide real features to discuss with respect to catchments of plume material (Figures 2 and 3). The Paraná starting plume head impinged on this region at ∼130 Ma and produced widespread volcanism [e.g., O'Connor and Duncan, 1990]. The subsequent 85 Ma to recent volcanism on land and out to sea are of interest here. A key question is whether this prolonged volcanism is the aftermath of ponded Paraná plume material. Alternatively, the volcanism may be associated with another currently active plume or it may not be directly related to plumes.
VanDecar et al.  and Schimmel et al.  detected a conduit shaped feature with tomography in the upper mantle beneath southern Brazil (Figure 3). They attribute this feature to a fossil remnant of the ∼130 Ma Paraná-Tristan plume. Another independent plume currently in this area seems more likely. A buoyant region at several hundred kilometers depth would have ascended buoyantly and thermally equilibrated with its surroundings since ∼130 Ma. The South American plate has also moved significantly relative to the subasthenospheric mantle since that time. A well-developed (Rio Grande) plume tail track shows this movement across the ocean basin (Figure 2).
 I examine the geology of this region in section 2 with respect to the spatial and temporal extent of prolonged volcanism. I discuss its phenomenology with respect to plumes in section 3 with regard to the well-posed hypotheses that its source is still-hot material from the Paraná plume or material from the plume tail indicated by tomography. In section 4, I appraise hypotheses involving the persistence of trapped plume material using heat and mass transfer.
2. Hot Spot Geology of Eastern Brazil and the Adjacent Ocean Basin
 A prominent feature of the ocean floor near Brazil is that some hot spot tracks are parallel to the flow lines defined by fracture zones. The Martin Vaz (Martin Vas) and Fernando de Noronha (Figure 2) hot spots are examples, which begin on the South American continent. These tracks coexist with those having the southeast-striking directions expected from traditional absolute plate motions. This creates a problem of which direction to chose in a compilation. For example, O'Connor and Duncan  and O'Connor and le Roex  give different absolute track directions for the South Atlantic from the same data. Geodynamicists would like to know which hot spots track their underlying plumes because they expect that convection currents within the mantle advect the tail conduits and the deep sources of plumes [Steinberger and O'Connell, 1998; Steinberger, 2000].
 A complication is that the Bahia and Pernambuco seamounts on the South American plate occur nearby. They are paired with the St. Helena group on the Africa plate and hence should be about the age of the underlying oceanic crust [O'Connor and le Roex, 1992]. Only two absolute ages of 73–83 Ma and 58–66 Ma are available [Bryan and Cherkis, 1995]. These whole rock 39Ar-40Ar ages are not particularly reliable. The wide area covered by the Bahia and Pernambuco seamounts does not allow me to define a track. I do not consider these features further in this paper.
 I discuss the geology and Martin Vaz, Fernando, and the adjacent mainland with respect to plumes. There is good evidence that the Paraná starting plume impinged on South American continent at ∼130 Ma and that the younger oceanic hotpots follow flow lines. The reliable dates exist for Martin Vaz and Fernando and the onshore region of Brazil. Tomographic data constrain the possible location of a modern plume on the mainland [VanDecar et al., 1995; Schimmel et al., 2003]
2.1. Martin Vaz
 Martin Vaz is north of the Rio Grande Rise. It follows a flow line parallel to fracture zones. There is weak indication of age progression. Taken as a whole, the inland region of Brazil was volcanically active from 132 to 67 Ma [Gibson et al., 1997] (Figures 2 and 3). The coastal region was active from 80 to 54 Ma [Gibson et al., 1997]. Normal faulting accompanied this episode [Potter, 1997]. The nearshore Abrolhas platform was active from 50 to 40 Ma [Fodor et al., 1989]. There is a weak tendency of its edifices to be along flow lines. Columbia seamount was active <3 Ma [Fodor and Hanan, 2000]. Trindade (since 5 Ma) and Martin Vaz Islands have been active recently [Marques et al., 1999; Siebel et al., 2000]. Taken literally, the data give a track rate of ∼21 mm yr−1 over the 825 km between the coast and Columbia seamount and ∼91 mm yr−1 over the 275 between Columbia seamount and Martin Vaz. Fodor and Hanan , aware of this issue, postulate that Columbia was active at 10 Ma.
 Osmium isotopic data indicate a considerable contribution of lithospheric mantle to the source region of the mainland volcanism [Carlson et al., 1996]. The rocks are typically alkaline and products of small amounts of partial melting and similar to those in East Africa [e.g., Barbosa de Albuquerque Sgarbi and Gaspar, 2002]. The degree of mantle melting decreased eastward along the track since 85 Ma [Siebel et al., 2000]. These authors attribute this effect to weakening of the plume over time.
 There is good evidence that lithospheric thickness variations affected volcanic activity as would be expected for ponded plume material. Eruption at 85 Ma occurred above locally thin mobile belt lithosphere [Gibson et al., 1997]. Thompson et al.  discuss channeling of ponded plume material and pervasive pressure release melting by variations in lithospheric thickness beneath coastal southeastern Brazil. Fodor et al.  discuss these processes with respect to the Abrolhas platform.
2.2. Fernando de Novonha
 The Fernando de Novonha chain follows a flow line at latitude of 5°S (Figures 2 and 3). Rivalenti et al.  document volcanism on the island from 12–0 Ma. In the nearby coastal region of Rio Grande do Norte volcanism lasted from 30 to ∼13–15 Ma [Rivalenti et al., 2000; Fodor et al., 2002]. Xenolith studies indicate that relatively thin continental lithosphere, <100 km, exists beneath the region of ∼13–30 Ma volcanism in northeastern Brazil [Fodor et al., 2002]. Taken literally, there is a weak track progression to the east. However, sampled volcanism began on Fernando at 12 Ma soon after it shut down at ∼13 Ma on the mainland. I do not attempt to give a track rate over this 700-km distance.
 The associated interior region of Brazil, the eastern Maranhão province, had volcanism between 129 and 125 Ma [Baksi and Archibald, 1997]. These authors note that this is slightly after the time of the most intense Paraná activity and 3000 km away from the center of activity in Serra Geral and discuss the problem of tying the volcanism to a known hot spot. Volcanism occurred ∼200 Ma in the western part of the Maranhão province [Baksi and Archibald, 1997]. I associate this volcanism with the Bahama starting plume head centered beneath northeastern Florida.
3. Phenomenology and Hypotheses
 A question arises as to whether mantle plumes impinged on eastern Brazil and if so how many and when. Most obviously, the Rio Grande Rise-Walvis track starts at ∼130 Ma with the Paraná starting plume head. In a standard model, its plume tail was well out into the Atlantic Ocean before volcanism recommenced at ∼85 Ma. In recognition of this time gap in volcanism, Gibson et al.  and Thompson et al.  attribute the Martin Vaz hot spot to a starting plume head in the Brazilian interior at 85 Ma.
 Three hypotheses come to mind for the volcanism in northeastern Brazil and the Martin Vaz and Fernando tracks as flow line hot spots: (1) All the volcanism is an aftermath of the Paraná starting plume head. In that case, the plume material remained hot for the ∼120 Ma since this plume tail moved out to sea. (2) The Paraná plume had no direct effect on the area by 85 Ma. A different plume came into the area or started with a plume head at ∼85 Ma. This plume currently supplies material both to Martin Vaz and Fernando. I prefer a plume tail track entering the area at ∼85 Ma to a starting plume head at that time given the feeble nature of the volcanism compared to with Paraná starting plume head. (3) Separate Martin Vaz and Fernando plumes exist beneath the continent and feed flow line hot spots through separate breaches.
 I briefly discuss the necessary kinematics of a possible single plume now underlying Brazil. I place the present plume where VanDecar et al.  and Schimmel et al.  detected a conduit-shaped region of low seismic velocities down to the base of the upper mantle (Figure 3). By 85 Ma, the track needs to be south of the Amazon rift, a region of presumably relatively thin lithosphere for plume material to flow south and east, rather than westward toward the Paleozoic break-up margin of the rift. The lithospheric relief beneath the Guaporé shield needs to channel material toward southeast Brazil and toward the coast near Fernando. That is, the cratonal lithosphere south of the Amazon rift needs to slope to the southeast for plume material to flow in that direction and that at least two distributaries need to exist to supply two flow line hot spots. Thick lithosphere needs to exist beneath San Francisco craton along the coast so that any plume material trapped beneath thinner lithosphere in the continental interior could escape seaward through breaches to form flow line hot spots. Gibson et al. , Thompson et al. , and Fodor et al.  delineate regional aspects of the drainage system in southern Brazil.
 A volcanic track across the Amazon rift and through the continent to the north is not evident. Geomorphology provides some evidence of uplift along the track as drawn (Figure 3), although the topography and tectonics of the continent are dominated by the Andes mountains to the west, the break-up margin to the east, and the effects of the Paraná plume in the interior [Potter, 1997]. For example, faulting and folding have occurred during the Cretaceous and Tertiary within the Amazon rift [Costa et al., 2001]. Basement rocks are exposed over much of Brazil (Figure 3). Apatite fission track studies indicate exhumation at the ∼120–130 Ma time of continental break-up and Paraná plume head impingement and a second period of rapid exhumation between 60–80 Ma [Gallagher et al., 1994; Harman et al., 1998]. The Guyana shield, an uplifted region of old crust, is at the northern end of the track as drawn in Figure 3 where fission track data are unavailable. A Pre-Miocene divide extended south from the Guyana shield across the present Amazon [Potter, 1997; cf., Costa et al., 2001]. I suggest that plume material heated the base of the Guyana and Amazon basin lithosphere, producing uplift, but did not cause any mapped igneous activity. The exposure of basement over much of Brazil indicates erosion above lithosphere underplated by plume material at ∼130 Ma and subsequent to 85 Ma.
 I cannot follow the track to farther to the north. As drawn, the track extrapolates onto Caribbean plate. However, the oceanic crust that was present seaward of a passive-margin at the time of the track has since been subducted. A track to the east of the Antilles arc would extrapolate onto the African plate. There are, however, no edifices of the right age. Some volcanism is associated with strike-slip tectonics between the Demerara plateau and the Guinea margin, which continued until ∼65 Ma [Benkhelil et al., 1995]. The Guinea plateau at ∼59 Ma [Bertran et al., 1993] and the Sierra Leone rise at ∼54 Ma [Jones et al., 1991] are too young to be part of a track into southern Brazil.
4. Heat and Mass Transfer Theory
 The hypotheses discussed above involve the lateral flow of plume material toward regions of thin lithosphere and the persistence of hot plume material within closed regions of locally thin lithosphere act as catchments for plume material. The persistence of ponded plume material is clearly relevant to South America has the Paraná plume impinged on this region. Lateral flow is an attractive hypothesis for the Martin Vaz and Fernando hot spots.
 I organize my discussion of the physics of ponded plume material by discussing governing equations of heat and mass transfer and the relevant physical parameters in section 4.1. I consider the lateral spreading of plume material in section 4.2. My intent is to show that material from a starting plume head spreads out quickly after the plume impinges and its low viscosity is necessary for efficient lateral flow. I examine the persistence of ponded plume material in section 4.3. I show that conduction alone does not fully cool thick plume material over 120 m.y. Convection is a much more effective enemy of the thermal anomaly.
4.1. Physical and Mathematical Model
 I constrain the hypotheses involving the persistence of plume material and the flow of this material beneath flowing hot spots. To do this, I need to solve the combined heat flow and force balance equations. The heat flow equation is
where T is potential temperature, t is time, V is velocity, κ is thermal diffusivity, and there are no heat sources. (The radioactivity in the continental crust is too shallow to have much effect.) The vertical component of the momentum equation in three-dimensions is
where x is the horizontal coordinate in the plane of the convection cell and y are horizontal coordinate perpendicular to that plane in the direction of plate motions, z is depth, and P is the “scalar” pressure, τij is the deviatoric stress tensor, Δρ = −αρΔT is the excess density from temperature, α is the volume thermal expansion coefficient, ρ is a reference density, and ΔT is the temperature above the mantle adiabat.
 I discuss nonlinear rheology to show that my scaling results are robust. The deviatoric stress τij is related to the tensor strain rate tensor in an isotropic material by
where η is the viscosity given by equation (4), τ the square root of the second invariant of τijτij, τref is a reference shear traction (which may be set for convenience without loss of generality), and n is the power law exponent. The familiar case of linear or Newtonian viscosity is n = 1.
 Certain physical parameters in equations (1)–(3) are reasonably well constrained. I assume that these parameters are constant in my calculations. These include the thermal conductivity k = 3 W m−1 K−1, the volume specific heat is ρC = 4 MJ m−3 K−1, the thermal diffusivity is κ ≡ k/ρC = 0.75 × 10−6 m2 s−1, the density is 3400 kg m−3; and the thermal expansion coefficient is α = 3 × 10−5 K−1. The acceleration of gravity is constant at g = 9.8 m s−2. The mantle adiabat is 1300°C.
 I treat the rheology as an unknown quantity to be varied in modeling. For simplicity, I represent viscosity as a function of temperature
where η0 is the viscosity at the mantle adiabat, ΔT is the temperature above the mantle adiabat, and Tη is the temperature scale for viscosity.
Equations (1)–(4) result in a formal initial value and boundary value problem. I approach its solution in two ways. First, I do scaling and analytical models. In that case, I presume that rigid lithosphere overlies plume material and that adiabatic mantle underlies the plume material and any lithosphere not underlain by plume material. I assume simple conditions for the plume material to obtain closed-form results. I compute numerical convection models of plume material ponded within a region of thin lithosphere to illustrate the basic feature of this process.
4.2. Lateral Flow of Plume Material
 Plumes are sources of hot material not candle-like sources of heat. The lateral flow of plume material once it has impinged on the base of the lithosphere is central to this paper, that is, the 85–0 Ma spreading of material from a plume tail and the ∼130 Ma spreading of material from the Paraná starting plume head.
 My approach is to present scaling relationships and to reason from analogy from simpler regions. I illustrate the tendency to plume material to flow along channels form by the relief at the base of the lithosphere as proposed for the region beneath Brazil and the flow lines of Fernando and Martin Vaz. I show that a ponded plume head spreads rapidly at first and soon approaches its final extent. This implies a short widespread episode of volcanism as is observed in the Paraná event. It also implies the catchments for ponded starting plume material fill quickly.
4.2.1. Lubrication Theory for Lateral Flow
 To show that the results are robust, I extend the lubrication theory of Huppert  to obtain dimensional expressions for nonlinear viscosity. I replace the plume material with an equivalent layer of thickness local thickness S and density deficit with the normal asthenosphere ΔρP. The shear traction at the base of the lithosphere is then
where ∂Z/∂x is the slope of the base of the plume material. We assume that viscous forces within the plume material rather than the surrounding mantle retard its spread. Sleep et al.  discuss the applicability of this criterion and present scaling relationships for the spread of inviscid plume material retarded by normal mantle. They note that scaling relationships obtained by assuming that the plume material retards spreading represent the results of sophisticated numerical models by Albers and Christensen . With this assumption, the flow velocity is dimensionally
The flux of plume material is
The dominant parameters that are likely to vary within the Earth in equations (6) and (7) are the density contrast, the viscosity, the thickness of the plume material, and the slope of the base of the plume material.
4.2.2. Lateral Flow of Channeled Plume Material
 The lateral flow of plume material within channels along the base of the lithosphere is an attractive explanation for linear hot spots that lack good age progression and run from thick toward thin lithosphere [Morgan, 1978]. In the case of Martin Vaz and Fernando, the flow velocity in equation (6) needs to be great enough to overcome drag from the moving plate and the flux in equation (7) needs to be enough to large enough to supply material to fill the channel. The drainage pattern beneath Brazil needs to be able to supply these channels.
 As the physical properties of the Earth are not well constrained, it is productive to use analogy and scaling. The Salas y Gomez hot spot on the Nazca plate provides a good analogy to Martin Vaz and Fernando in that flow is perpendicular to the spreading direction [Kingsley et al., 2002; Simons et al., 2002]. Comparing parameters in equations (6) and (7) indicates that the South Atlantic and Nazca situations are comparable. The distance traveled by plume material, over 1000 km from Salas y Gomez to the west rift of the Easter microplate is comparable to that of Fernando and Martin Vaz from the mainland. The slope driving flow scales inversely with the paleospreading rate and inversely with the square root of plate age. Using plate ages 9 Ma and 121 Ma and (half) spreading rates of 80 and 20 mm yr−1 for the Nazca and South Atlantic plates, respectively, gives a relative ratio of slopes of 1/22:1/24. The absolute velocities in the direction of the track are also similar, ∼30 mm yr−1 for Nazca and ∼47 mm yr−1 [Gripp and Gordon, 2002]. If anything, the viscosity of the far traveled Martin Vaz and Fernando material is higher than that of the fresh Salas y Gomez material. A moderate increase in the thickness of the channel S and hence the slope would tend to offset the effects of higher viscosity in equations (6) and (7).
 The east African plume may provide an analogy to a plume beneath Brazil. Ebinger and Sleep  proposed that a single strong plume beneath east Africa supplies the Afar hot spot, the Cameroon hot spot, various North African hot spots, and the Comoros hot spot. Flow occurs through channels of thin lithosphere. Plume material flows under oceanic lithosphere significant distances beneath the Comoros and seaward of the Cameroons after leaving the continent.
 Well-defined linear features like Martin Vaz and Fernando coexist with broad features like the Abrolhas platform. That is, there is a tendency for the plume material to follow channels but some broad flow may exist. There is also a tendency of the eastern ends of the chains to be active defining a weak age progression. Both effects are fully three-dimensional. I discuss them qualitatively.
 The tendency of plume material to follow channels is basically viscous fingering [Sleep, 2002a]. Hot fluid regions flow faster than cool viscous ones. The passage of plume material thins the lithosphere increasing the thickness of the channel and hence the flow in equations (6) and (7). I attribute the Martin Vaz and Fernando tracks to the local failure of two places in the lithospheric dams on each end of the San Francisco craton. The breaching of a dam starts the top of a well-defined channel, probably with a sudden surge of plume material. The initial velocity of the surge is high and the flow is not greatly perturbed by the absolute velocity of the plate. Hot spots, like Martin Vaz and Fernando, once established along flow lines act as channels as long as they continue to be fed by plume material.
 The tendency of recent volcanism to occur near the eastern ends of the tracks is in part that it is easier to observe the growth of a new edifice near the flow front. In addition, dikes once frozen modify the stress filed in the plate making it hard for other dikes to follow. This tends to shut down volcanism on the older parts of the track.
4.2.3. Spreading of a Blob of Starting Plume Material
 Starting plume heads are an attractive explanation for large igneous provinces. In this paper, I am interested on how long the material from the Paraná plume head stayed hot enough to be the source for volcanic rocks. I begin, by considering the spreading and cooling of a plume of material ponded beneath flat lithosphere. This flow would fill any catchments beneath the lithosphere.
 Following Huppert , I treat the blob of plume material as a radially symmetric region of radius R with a maximum thickness at the center SC. The volume of the blob stays constant Q ≈ R2SC. If the blob spreads out beneath a relatively flat surface, the slope of the base of the blob is dimensionally SC/R and the thickness is dimensionally SC. Integrating the velocity in equation (6) then yields
The central thickness at this time is dimensionally
This implies that the material thins and spreads rapidly at first and slowly thereafter, especially for n > 1.
 I obtain the final thickness of the material from a Peclet number argument following Sleep et al. . That is the cooling time of the material by conduction is dimensionally SC2/κ. Equating the conductive cooling time to the spreading time for a given thickness yields that the thickness at quenching is
The time to reach this quenching thickness is dimensionally
The quenching thickness and quenching time are weakly dependent on the volume Q of the blob and on its viscosity for reasonable values of n from 1–5.
 The question arises as to whether a slightly warm blob of plume material would spread and cool slowly enough that it could produce a prolonged period of hot spot activity. Such a warm blob might conceivably form by entrainment of large volumes of normal mantle. Its density contrast with normal asthenosphere would be small and its viscosity nearly that of its surroundings. A fresh blob with low viscosity and high density contrast provides some calibration, as igneous events typically attributed to starting plume heads are typically brief, last only a few million years. For an example, I compare a warm blob with a hot blob that has 5 times the density contrast and is two orders of magnitude less viscous. The quenching time for the warm blob is 8 and 5 times that of the hot blob for n equals to 1 and 5, respectively. The quenching thicknesses are 2.8 and 2.2 that of that hot blob for n equals to 1 and 5, respectively. I conclude that a warm blob continues to spread only a few times longer than a hot one. That is, it may take a few tens of million years to spread and cool, but not over 100 m.y. Warm spreading blobs thus are not an attractive mechanism for attributing modern volcanism in South America to the Paraná starting plume head.
4.3. Persistence of Ponded Plume Material
 A conceivable mechanism to generate flow line hot spots requires that Paraná plume material stay warm beneath cratons for a long period of time, ∼130 m.y. That is, the plume material still needs to be fluid enough to flow laterally toward Martin Vaz and Fernando and hot enough to melt when it ascends to shallower depths after a catchment beneath the continent fails. It also needs to be buoyant relative to normal asthenosphere. I show that this is possible, but only in contrived situations. I thus favor a modern plume beneath Brazil.
 The conditions of plume material just after lithospheric relief traps it are not evident. Both a starting plume head and a plume tail consist of hot central material and a surrounding halo of mantle warmed by conduction and entrainment of the hot material. The hot buoyant plume material ponds above the less buoyant halo material. A catchment may fill with only halo material if the hot material is too shallow to cross a divide or it may fill with mainly hot material.
 The maximum thickness of trapped plume material is limited by the thickness of cratonal lithosphere acting as dams. Xenolith data indicate that many cratons are 200–250 km thick [e.g., Rudnick and Nyblade, 1999]. The lithosphere of rifted basins and young mobile belts is often less than 100 km thick. This implies that trapped ponded plume material may be over 100 km thick. A still greater thickness is possible beneath narrow failed ocean basins surrounded by cratons.
 Once trapped, ponded plume material loses heat through the overlying lithosphere. At a minimum, the plume material must survive conductive heat loss to stay hot. I consider this end-member case in section 4.3.1. I consider the more realistic case where plume material cools by convection in section 4.3.2 with scaling relations and in section 4.3.3 with numerical models.
4.3.1. Conductive Heat Transfer
 A naive approach is to assume that heat transfer through the lithosphere and the plume material is essentially conduction. This contrived situation places a limit on how long trapped plume material can persist beneath cratons. It also provides a basis with which to compare convective heat transfer.
 For simplicity, I let the plume material have a constant potential temperature, 200 K hotter than the normal mantle adiabat. I emplace this material beneath 100-km-thick lithosphere with a linear conductive geotherm.
Figure 4 shows the geotherms within ponded plume material as a function of time after emplacement. A 100-km thick layer approaches the mantle adiabat by 120 m.y. and a small region hotter than the mantle adiabat persists at 200 m.y. for 150-km thick layer.
 This implies that thick ponded plume material from the ∼130 Ma starting plume can supply modern hot spots if convection within the material is quite sluggish. This situation, however, is unlikely in that the material needs to be fluid to flow laterally to produce the flow line hot spots, but needs to be very viscous to remain static. Chemically buoyant plume material may still convect internally. I address convection in the following section.
4.3.2. Scaling Relationships for Convective Heat Transfer
 Techniques, collectively called parameterized convection, constrain the heat flow at the base of the lithosphere. A thin rheologically active boundary layer flows while the bulk of the lithosphere acts as a stagnant lid.
 I first consider the heat flow through relatively thin lithosphere beneath which plume material may pond. The lithosphere neither thickens nor thins with time if the conductive heat flow from below balances the convective heat flow from below. This heat flow is the critical heat flow to thin the lithosphere. It is also the heat flow if conduction through a nearly rigid, chemically buoyant lithosphere is the rate-limiting step in cooling the plume material.
 The conductive heat flow through such a lithosphere lid is
where k is thermal conductivity, ZL is the thickness of the lithosphere, and TL is the temperature difference across the lithosphere with plume material present. The time for plume material of thickness SP to cool is then
where κ is thermal diffusivity, ΔT is the excess temperature (above that of normal asthenosphere) of the plume material, and SP is the thickness of the plume material. Letting ΔT = 200 K, SP = 100 km, ZL = 100 km, and TL = 1500 K (and using the material properties in section 4.1) yields a cooling time of ∼55 m.y. This calculation indicates that plume material stays hot for a geologically significant time but not long enough for 130 Ma plume material to form modern flow line hot spots.
 The heat flow in equation (12), 45 W m−2, needs to be compared with the heat flow supplied by free convection. The heat flow at quasisteady state for a linear fluid is
where Tη is the temperature to change the viscosity by a factor of e in equation (4) and ηH is the viscosity of the adiabatic halfspace [Davaille and Jaupart, 1993a, 1993b]. More generally, the convective heat flow for a nonlinear fluid is proportional to
 I constrain the heat flow in equations (14) and (15) extrapolating from the present heat flow beneath cratons. From equations (15) and (4), the convective heat flow through plume material ΔT hotter than normal mantle is exp[ΔT/((n + 2)Tη)] higher than that through normal mantle.
 One possibility is that the chemical buoyancy of the lithosphere suppresses convection beneath cratons and that the convective heat flow in the absence of chemical buoyancy is that at the base of old oceanic lithosphere, ∼40 mW m−2 [Davaille and Jaupart, 1994; Doin et al., 1997]. This heat flow without any enhancement from the presence of plume material would cool the 100-km thick layer in equation (13) in 61 m.y. I do not discuss this case further as hot plume material does not persist for a long time.
 I consider an alternative possibility where plume material persists longer. The conductive heat flow through cratons is the convective heat flow in equations (14) and (15) for normal mantle. I select a 230-km thick craton with a convective heat flow of 17 mW m−2 at the current mantle adiabat of 1300°C to provide examples. Venerable approximations for Tη in the mantle are 43 K (1 order of magnitude per 100 K) and 100 K. For a linear fluid, the convective heat flow for plume material that is 200 K hotter than normal mantle is 4.6 and 1.9 that of normal mantle or 79 and 33 mW m−2, respectively. For n = 5, the heat flows are 33 and 23 mW m−2, respectively. The latter heat flow would cool the 100-km-thick layer of plume material discussed in this section in 108 m.y.
 It appears from this analysis that convection is not greatly enhanced by high temperature if the viscosity is nonlinear and Tη is large. The situation, however, is unlikely to apply in the Earth. I first present a direct constraint on n and Tη. I then briefly consider the effect of lithospheric relief on convection.
 First, the temperature contrast across the rheological boundary layer provides an observable constraint. It scales with Tη as
[Solomatov, 1995; Solomatov and Moresi, 2000]. Xenolith geotherms constrain it directly to <300 K [Rudnick and Nyblade, 1999]. At a minimum, temperature range cannot be so large that little temperature variation remains for a rigid lithospheric lid. For example, the combination of Tη = 100 K and n = 5 implies a temperature range of 720 K, which is probably excessive. If one restricts the boundary layer temperature range to <300 K and 1 ≤ n ≤ 5, the minimum heat flow is 26.5 mW m−2 and the cooling time is 93 m.y.
 Second, the base of the lid cannot be flat if it is to trap plume material. The lateral gradients of density associated with relief on the base of the lithosphere enhance convection. The convective heat flow formula in equation (15) applies strictly if the base of the stagnant lid is flat. The stresses for convection within a flat boundary layer then scale as
where ZB is the thickness of the rheological boundary layer, which is proportional to the rheological temperature contrast in equation (16). The stresses associated with lithospheric relief scale as
where H is the relief on the base of the lithosphere. The velocity of flow within the boundary layer determines the heat transport. It is
where the stress is obtained from either equations (17a) or (17b). The conductive and convective heat flows within the boundary layer are comparable if it is quasisteady state,
where a is an unknown dimensionless constant of order 1 and the equation applies at the limit of significant relief aH ≫ ZB. (The heat flow is approximately qv when there is only modest relief). The effect of relief in equation (20) to modestly increase heat flow is weakly dependent on n. The exponent increases from 1/3 for n = 1 to 5/7 for n = 5. The term within the bracket increases using equation (16) decreases by a factor of 3 over that range for a given Tη.
Equation (20) suffices to show that lithospheric relief modestly increases the vigor of convection. The effect hastens the cooling of trapped plume material relative to what would occur within a flat ponded layer. This result holds for both linear and nonlinear viscosity.
4.3.3. Numerical Models of Convective Heat Transfer
 I present a series of two-dimensional convection models to illustrate the basic properties of convection within ponded plume material. I use the material parameters discussed in section 4.1 and the boundary conditions discussed in appendix A.
 The rheological parameters in the Earth's mantle are not well constrained. I compute models with linear viscosity for simplicity. I constrain the predicted heat flow in equation (14) to provide 17 mW m−2, which implies a 230-km-thick steady state lithosphere. I assume various values of Tη in equation (4). From equation (14), the viscosity of the adiabatic mantle η0 scales proportional to Tη4 at a given heat flow. This precludes very high values of Tη as the plume material would be too viscous to flow laterally. I use Tη of 40, 60, and 100 K in models and η0 of 0.1975, 1, and 7.7 × 1019 Pa s, respectively. I begin with model 1 where Tη is 60 K to illustrate the procedure.
 I select generic starting conditions to represent a pronounced catchment region of thin lithosphere within a craton. To do this, I start with a stratified model. The temperature increases linearly from 0°C at the surface to 1300°C at 230-km depth except for a small perturbation at 210-km depth to start convection. The temperature below 230-km is 1300°C. After 5 m.y., I replace the center region of the model between 350 and 550 km from the edge with a linear temperature increase from 0°C to 1300°C from the surface to 15-km depth and 1300°C below that. I allow the model to convect for an additional 85 m.y. to allow the thin lithosphere to thicken and a transition region to form between the thin and the thick lithosphere. The region of thin lithosphere strongly modulates the convection into downwellings at the edges of the thin region (Figure 5).
 I then put plume material of temperature 1600°C into rectangular region between 300 and 600 km from the edge of the model and 300 to 450-km depth (Figure 5). This is unrealistic but, by the time the plume material impinges on the lithosphere, it has entrained normal mantle and has a range of temperatures. The temperature field after the plume material has ponded with the catchment after 5 m.y. is reasonable (Figure 6). I consider this to be a realistic starting condition for the aftermath of a starting plume head.
 I ignore the plume tail. In the case under consideration, the plume tail associated with the Paraná starting plume head did not loiter long beneath South America. Rather, it quickly moved into the South Atlantic Ocean by ∼120 Ma where it generated the paired Rio Grande and Walvis tracks [e.g., O'Connor and Duncan, 1990].
 One objective involves the fate of the plume material after it has ponded. Figure 6 illustrates the gross properties of the aftermath of a start plume head. A thin layer of plume material ponds beneath the thick lithosphere on each side of the catchment. This material convects vigorously. Strong downwellings exist on each side of the catchment. Fluid upwells within the catchment. The hottest plume material rises to the top of the catchment, cools, and downwells along the sides of the catchment. By 40 m.y., only a small region of warm material plume material remains at the top of the catchment.
 I increase the width of the catchment in model 2 (Figure 7), keeping the material parameters in model 1. The zone of thinned lithosphere and the rectangle of plume material extend between 550 to 950 km from the edge of the model. The catchment is wide enough that a downwelling develops within it. The plume material cools somewhat more slowly than in model 1 because the relief on the edges of the catchment has less overall effect.
 I compute model 3 (Figure 8) with Tη = 40 K and model 4 (Figure 9) with Tη = 100 K to appraise the effect of that parameter. As expected, the model with low Tη cools more rapidly than the model with high Tη. The effect, however, is modest and the excess temperature of the plume material is gone by 80 m.y. (Figure 9).
 I make some simple general inferences from the numerical models. First, most of the thermal anomaly of the plume material is gone by 40 m.y. By that time, plume material remains only at the top of catchments. The edges of the catchment strongly modulate convection. The plume material advects upward and is carried out of the catchment once cooled. Convection continues to flush the catchment even after the thermal anomaly of the plume material is gone.
 It has not escaped that the numerical models are relevant to the subsidence of cratonal basins. This topic is beyond of the scope of the paper as it involves the catchment after no hot plume material remains. I do note that my model after the plume material is gone (Figure 9) grossly resembles the starting conditions in the Michigan basin in the models of Kaminski and Jaupart .
5. Geological Implications and Conclusions
 The scaling relationships and numerical models in section 3 allow general conclusions to be made about the geology of South America and the fate of trapped plume material in general. I discuss a general negative result related to the post-85 Ma volcanism involving the Paraná plume and a local positive inference involving a modern plume conduit detected by tomography.
 My negative result is that trapped plume material cools to the mantle adiabat in several tens of million years. It is unreasonable that hot material from the Paraná starting plume head at ∼130 Ma still persists beneath South America and supplies the Martin Vaz and Fernando hot spots. Moderate amounts of hot Paraná plume material may have remained at ∼85 Ma when igneous activity recommenced. This material may have been remobilized by impingement of fresh plume material or by tectonics at that time. The options that remain for the Late Cretaceous to Cenozoic volcanism are that it is not related to plumes or that one or more plumes more recent than the Paraná one have impinged on the area.
 The nonplume option is certainly tempting in that the Paraná plume event probably left easily melted metasomatized lithospheric mantle in its wake. There is no shortage of tectonics to produce cracks for magma to vent from the mantle since 85 Ma. These include the Andes, a break-up transform to the northeast until ∼65 Ma [Benkhelil et al., 1995], and the modern Caribbean margin. The interior of the continent was effected, for example, within the Amazon rift [Costa et al., 2001]. Cracks for dikes would be most likely to form within weak areas of locally thin lithosphere. These areas are also those most likely to have partial melting within metasomatized heterogeneities and to have had metasomatism in the Paraná event.
 My positive inference involves tomographic data that indicates the presence of a modern plume conduit beneath southeastern Brazil [VanDecar et al., 1995; Schimmel et al., 2003] (Figure 3). This inference is well enough posed to compare with geology. The track of this plume is such that it could produce volcanism south of the Amazon rift from 85 Ma to the present. The plume material would need to flow laterally beneath the craton and the ocean basin to feed the mainland, the Abrolhas, the Fernando, and the Martin Vaz volcanism. Widespread exhumation after 85 Ma on the craton may indicate uplift associated with underplated plume material.
 This inference is an application of lateral flow of ponded plume material, a part of the traditional plume hypothesis [Morgan, 1978] for which sophisticated numerical models exist for the ocean basins [Albers and Christensen, 2001]. It is also testable. Seismologists can constrain the relief on the base of the lithosphere and find hot ponded plume material. At present, South American tomographic data have poor resolution at the base of the lithosphere [VanDecar et al., 1995; Schimmel et al., 2003]. Tomographers can better resolve (or even refute) the plume conduit. Once tomographers constrain the relief on the base of the South American lithosphere and the location of the plume orifice, numerical modelers can represent the flow in three-dimensions and compare it with the observed volcanism and exhumation.
Appendix A:: Numerical Calculations
 Mathematically, I need to solve equations (1) and (2) for given boundary conditions for stress and temperature and a given initial condition for temperature at every point. I use a numerical code modified after Andrews  and Sleep [2002b]. I assume rectangular box that is 500 km deep and either 900 or 1500 km wide with a uniform 5-km grid. There is one temperature condition on each boundary. The top boundary condition is natural; the temperature is 0°C. The side boundary conditions are symmetry ∂T/∂x = 0. The basal boundary condition is that the temperature of incoming material is the mantle adiabat, 1300°C. There are two stress and flow boundary conditions at each boundary. I express them in terms of a stream function where ∇ × (ψj) = V, where j is the unit vector perpendicular to the plane of the model. I set the stream function to zero at the two top rows of the numerical model to represent the rigid lithosphere. The side boundary conditions are no material flow perpendicular to the model and no shear stress. The bottom boundary condition is that the first and third derivatives of the stream function with depth are zero. This permeable boundary condition lets fluid enter and leave the domain of the model and do no work on the interior of the model [Moore et al., 1998].
 This research was in part supported by NSF grant and EAR-0000747. I thank John VanDecar and Marcelo Assumpção for helpful comments and Paul Tackley and James Conder for helpful reviews.