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Energetics of the Earth and the Missing Heat Source Mystery

Don L. Anderson

Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125

dla@gps.caltech.edu

Overview

Global heat flow estimates range from 30 to 44 TW (Table 1a; Reference List A). Estimates of the radiogenic contribution (from the decay of U, Th and K in the mantle), based on cosmochemical considerations, vary from 19 to 31 TW (Table 1b). Thus, there is either a good balance between current input and output, as was once believed (“the Chondritic Coincidence”), or there is a serious missing heat source problem, up to a deficit of 25 TW. Attempts to solve the perceived deficit problem include invoking secular cooling and deep, hidden heat-source layers (e.g., Kellogg et al., 1999).

In many studies it has been assumed that there should be a steady-state balance, or close to it, between current radioactive heat production in the mantle and current heat flow, and that very little heat is generated in the upper mantle. This view ignores mass balance considerations, other sources of energy (Reference List C), secular cooling, delays in the system, and the wide range of radioactive contents of upper mantle materials. Among the problems commonly cited with various models of mantle thermal history are very high mantle temperatures in the Archaean, survival of ancient cratonic roots, komatiitic temperatures, over-heating of the lower mantle, freezing of the core, an imbalance between the helium and heat flow budgets and a perceived missing heat source. A common perception is that there is not enough radioactivity in the upper mantle to provide a significant contribution to current heat flow. These problems can be avoided by recognizing that:

  1. heat flow is a three-dimensional problem and that heat flow is diverted into the ocean basins;
  2. continents tend to move towards cold downwelling mantle;
  3. at high mantle temperatures water is removed from both the mantle and the lithosphere, stiffening the system;
  4. the mantle is probably chemically layered, extending the cooling time, and
  5. the upper mantle cannot be entirely composed of ultradepleted MORB and barren peridotite.

It is useful to compile the sources of energy and global heat flow before addressing perceived problems. It is also useful to investigate possible shortcomings of various theoretical models. It turns out that the unknown hydrothermal contribution to cooling of old oceanic lithosphere and the temperature dependence of thermal conductivity are key issues (Hofmeister & Criss, 2003). If lattice and radiative conductivity are as high as currently calculated, heat that leaves the core is mostly conducted down the core adiabat (Gubbins, 1977), and contributes to the bulk mantle energy budget, rather than be piped directly to the surface e.g., in plumes (Stacey & Stacey, 1999). The present-day heat flow through the surface of the Earth is consistent with energy sources in the interior, including secular cooling, the gravitational contraction associated with cooling, and decline of radioactive abundances. Theoretical corrections to observed heat flow, and theoretical estimates of expected oceanic heat flow are both uncertain and model-dependent. Within the uncertainties of data and theory, there is no missing heat source paradox or need for substantial contribution to observed heat flow from the deep mantle.

Distribution of radioactive elements

Radioactive decay is only one of the sources of mantle heat flow (Reference List C). In most current models of geodynamics and geochemistry the lower mantle is assumed to have escaped accretional differentiation and to retain primordial values of radioactivity and noble gases. The crust is assumed to have been derived from only the upper mantle, making it extraordinarily depleted in radioactive and volatile elements. The heat productivity of the most U-poor mid-ocean ridge basalts are taken as upper bounds on the heat productivity of the whole mantle above the 650-km phase change. This combination of assumptions, plus neglect of non-radiogenic sources of heat, have led to the view by some that there is a missing energy source in the Earth.

Mass balance calculations and the 40Ar content of the atmosphere show that most, if not all, of the mantle must have been processed and degassed in order to explain the concentrations of incompatible and volatile elements in the outer layers of the Earth (Reference List B). The planetary accretional zone-refining process results in an outer shell that contains most of the U, Th and K, at a level about three times chondritic (if the outer shell is equated with the present mantle above 650 km), from which the proto-crust and basaltic reservoirs were formed. The residual (current) upper mantle retains radioactive abundances greater than chondritic while the bulk of the mantle, including the lower mantle, is essentially barren (Anderson, 1989). The outer shells of Earth contain both depleted and enriched sources and probably also contain the bulk of the terrestrial inventory of noble gases (Meibom & Anderson, 2003).

In a layered model the delay between heat generation and conduction through the surface, and the secular cooling of the Earth, extends the thermal evolution. As the mantle cools, it becomes less molten, degasses less readily and probably becomes more volatile-rich, since it is now a sink for CO2 and water. It thus experiences different styles of convection and cooling. These factors are not considered in currently popular models used by convective modelers, leading to apparent paradoxes.

In other recent models the entire mantle is assumed to have escaped chemical differentiation, except for crust extraction, and to convect as a unit, with material circulating freely from top to bottom. Recycled material is quickly stirred back into the whole mantle. In these models:

  1. chemical heterogeneities are embedded in a depleted matrix,
  2. the whole mantle is treated as uniformly heterogeneous, and
  3. the mantle is relatively cold and the hot thermal boundary layer above the core plays an essential role in piping heat to the surface.
Convection is assumed to be an effective homogenizer. Recent amendments to this idea assert that there must be a radioactive-rich layer deep in the mantle, but this is based on unlikely assumptions about upper mantle radioactivity. Most convection calculations ignore accretional differentiation and the effects of pressure and temperature on thermal properties such as thermal expansion and thermal conductivity. Current models assume that heat from the core and heat from the mantle are decoupled in the sense that plumes remove core heat and plate tectonics removes mantle heat (e.g., Stacey, 1992).

Although there is no missing energy when direct heat flow data are used, there is a mismatch between oceanic heat flow and theoretical predictions from the cooling plate model. In some compilations, about 12 TW is added to the measured global heat flow to match the square-root-of-age predictions. This is the same size as the perceived “missing” heat energy (Hofmeister & Criss, 2003).

Tables 1a & b summarize estimates of global heat flow and heat sources in the mantle (Reference Lists A & C). The references are subdivided by subject. See also other sections on heat flow and temperature at http://www.mantleplumes.org

Heat losses

Components and mechanisms of heat loss

Heat flows through the interior of the Earth by radiation, conduction and convection. Eventually, all heat flows through the surface boundary layer, primarily by lattice conductivity but some by dikes, volcanoes and hydrothermal activity.

The surface boundary condition has changed with time. As a planet cools it evolves from a magma ocean regime (the result of accretional heating and gravitational differentiation), to a thick buoyant surface layer (unsubductable basalt and refractory residue) with heat pipes and heat sheets (permeable plates), to plate tectonics, and finally to stagnant lid. The present continental crust, asthenosphere and olivine-rich (pyrolitic) upper mantle are most likely reprocessed residues from early differentiation. Currently, the surface boundary layer is a conduction boundary layer with an average thickness of 100-200 km. It is pierced in places by volcanoes that deliver a relatively small amount of heat to the surface via magma. The cooling of the mantle is mainly accomplished by the cooling of the surface plates.

In early Earth history a transient magma ocean allowed magmas to transfer their heat directly to the atmosphere. As buoyant material collected at the top, the partially molten interior became isolated from the surface. Magma, however, could break through and create “heat pipes” to carry magma and heat to the surface. Io and Venus may utilize this mechanism of heat transfer. The surface boundary condition in these cases can be viewed as a permeable plate. Present day plates can be penetrated by sills and dikes and are therefore partially permeable. As a planet cools further it may jump to a stagnant-lid state with a convecting interior. Mars and Moon may be in such a state.

Currently, the Earth’s interior is cooling by a combination of thermal conduction through the surface and the advection of cold material to the interior by slabs and delaminated continental crust. The heat generated in the interior of the Earth, integrated over some delay time, is transferred to the surface conduction boundary layer by a combination of solid-state convection, fluid flow, radiation and conduction. Crustal radioactivity is a major contributor to continental heat flow. Delaminated crustal blobs may contain much of the radioactivity in the mantle.

In the ocean basins the main contribution to the observed heat flow is the transient effect of the formation of the oceanic crust (Reference List F). Theoretically, this conducted heat flow should wane as the square-root of age but surprisingly, it is nearly constant. The background heat flow is nearly the same as under continents. In contrast to predictions from the plate and cooling half-space models there is little correlation of heat flow with age or depth of the ocean. There is also little evidence that hotspots or swells are associated with high heat flow (see Heatflow page; Stein & Stein, 2003). This indicates that the underlying mantle is not isothermal or homogeneous.

Thermal conductivity decreases rapidly with increasing temperature. The cold outer shell of the Earth is not simply a cooling boundary layer of uniform composition and conductivity, losing heat by conduction alone, as assumed in cooling-plate and half-space models (Reference List F). The fact that heat flow is not a function of square-root of age suggests that some process affects the near-surface thermal gradient without affecting the integrated density of the outer layers. Oceanic swells, on the other hand, apparently require a redistribution of mass and density. They do not appear to be purely thermal in origin. They are probably largely held up by the buoyancy of depleted harzburgite (the perisphere).

Heat loss from the continents

The mean heat flow from continents is about 80 mW/m2 (Reference List A). The heat flow that can be attributed to the continental crust itself is about half of this, or 32-40 mW/m2, though recent estimates of the average U, Th and K concentrations vary by almost a factor of 2. The other half thus comes from the mantle. The continental crust therefore accounts for 5.8 - 8 TW of the global heat flow.

Heat loss from the oceans

Because of sparse coverage, heat flow data must be averaged by age and by area of the seafloor. These estimates give about 62 mW/m2 for the average oceanic heat flow (Hofmeister & Criss, 2003). About half of this is a transient effect from the plate-forming process and half is the background flux from the mantle. Measured oceanic heat flow varies from about 300 to 25 mW/m2 with 45 to 55 mW/m2 being a representative range through old oceanic crust. The theoretical value for half-space cooling is ~ 101 mW/m2 (Pollack et al., 1993) but this is sensitive to values adopted for thermal conductivity of the mantle and crust. Theoretical plate cooling model values are infinity at zero age and 100 mW/m2 at 30 Ma. The theoretical value at large time depends on preselected parameters and boundary conditions. Theoretical “corrections” to measured heat flow are thus uncertain.

Near-axis hydrothermal cooling accounts for about 1 TW of the global heat flux. The extent of hydrothermal cooling due to off-axis circulation of cold water is usually taken as the difference between the predictions of the plate cooling model and the observed conducted heat flow but there is no theoretical basis for this. The plate cooling model predicts higher heat flows than observed out to ~ 50 Ma, and lower than observed thereafter. Some of the differences between the measured and theoretical heat flows are due to causes other than hydrothermal circulation, including off-axis intrusions and lateral variability in mantle potential temperature.

Hydrothermal circulation

The mean oceanic heat flux is sometimes determined by fitting the parameters of a half-space or plate cooling model to bathymetry and heat-flow data on older oceanic lithosphere. This approach substantially overestimates the heat flux for ages less than 40 My but the discrepancy persists out to 60 My. The calculated Quaternary flux exceeds the measurements by 500% (Hofmeister & Criss, 2003). The mean calculated oceanic value (Pollack et al., 1993) is about double the median observed flux of 65 mW/m2 for the oceans and 61 mW/m2 for the continents. Theoretical cooling models generally use a constant conductivity and ignore its temperature dependence. In addition, the hydrothermal contribution has been overestimated. According to Hofmeister & Criss (2003), both the theoretical heat flux values and the hydrothermal contribution should be reduced. In addition to the conducted heat, other processes which affect the heat flow as a function of age include intrusion, underplating, stress changes in the plate and serpentization, which are problematic to assess. Table 1a includes estimates of near-ridge hydrothermal circulation but otherwise tabulates the measured conducted heat flow values.

Expected background variations in heat flow

Boundary-layer and plate models attribute all variations in bathymetry and heat flow to conductive cooling as a function of time. However, mantle convection and plate tectonics could not exist, and are inconsistent, with an isothermal mantle. Lateral temperature variations of the mantle below the plate of at least 100°C are expected. For a 100-km-thick thermal boundary layer this implies heat flow variations of about 15% superposed on normal cooling curves. Variations in permeability at the top of the plate cause variations in the hydrothermal component of heat flow, and this component of heat flow must be allowed for separately. The important point here is that temporal changes in surface heat flux must be considered before concluding that an “energy crisis” exists.

Global heat flow variations can be estimated using seafloor age reconstructions [Reference list D; Loyd et al. (2007) and references therein]. Loyd et al. (2007) show that heat flow has decreased by 0.15% every million years during the Cenozoic due to a decrease in the area of ridge-proximal oceanic crust. This is an order of magnitude faster than estimates based on smooth, parameterized cooling models. This implies that heat flow experiences short-term fluctuations associated with plate tectonic cyclicity.

The cooling plate model assumes that the mantle beneath the plates is isothermal and homogeneous and has the same potential temperature as midocean ridges basalts. Tomography shows that the subplate mantle is heterogeneous.

 

Heat sources

Radioactivity

All estimates of terrestrial abundances of the heat-producing elements depend on meteorite compositions (Reference List C). Carbonaceous chondrites are the usual choice, but enstatite achondrites and meteorite mixes are also used. The Earth is unlikely to match any given class of meteorite since it condensed and accreted over a range of temperatures from a range of starting materials. The refractory elements are likely to occur in the Earth in cosmic ratios, but the volatile elements are depleted. The large metallic core indicates that the Earth, as a whole, is a reduced body, although at least the crust and the outer shells of the mantle are oxidized. Enstatite achondrites match the Earth in the amount of reduced iron (oxidation state) and in oxygen isotopic composition and have been used to estimate terrestrial abundances.

Estimates for the heating potential of the Bulk Silicate Earth (BSE = crust + mantle) range from 12.7 to 31 TW, although most authors obtain values in a much more restricted range (Table 1b). These are present-day instantaneous values. Heat conducted through the surface was generated some time ago, when the radioactive abundances were higher, so estimates of radioactive heating, based on current radioactive contents, are lower bounds on the contribution of radioactive elements to the present-day surface heat flow, assuming that the estimates of U, Th, and K are realistic. The allowable variation in U and Th contents of the mantle is a large fraction of the postulated discrepancy between production and heat flow. Production of heat can be much larger if K contents have been underestimated. It is of interest that, because of the short half-life of 40K, most of the 40Ar in the atmosphere was generated in early Earth history. More efficient degassing then may partly explain the large fraction of the terrestrial 40Ar that is in the atmosphere.

The amount of radioactivity in the crust must be subtracted out in order to obtain mantle abundances and heat productivities. Using 8 TW as the best estimate of crustal productivity gives < 23 TW as the current energy output from mantle radioactivity. Heat from the core (about 9 TW), solid Earth tides (1 to 2 TW) and thermal contraction (~ 2 TW) are non-radiogenic sources that may add 12-13 TW to the mantle heat flow, about the same as the current (non-delayed) mantle radiogenic contribution. The radiogenic contribution can be increased by about 25% if it takes 1 Gyr to reach the base of the lithosphere. On top of all this is secular cooling of the mantle. In a chemically stratified mantle, the outer layers cool much faster than the deeper layers. If cooling is confined to the upper 1,000 km a temperature drop of 50 K/Ga corresponds to a heat flow of 3 TW. Cooling rates of twice this value may be plausible (Reference List D).

There thus appears to be no need for any exotic heat sources or hidden sources of radioactivity in the mantle. This conclusion is independent of the uncertain contribution of hydrothermal circulation to the surface heat flow. There are implications, however, for the temperatures of the Archean mantle and the style of convection, and the mechanisms of heat removal (Reference List D). The present styles of mantle convection and plate tectonics are unlikely to have operated in the Archean (Hamilton, 2003).

Table 1: a) Measurements and estimates of global heat flow. Some of these are from a recent review by Hofmeister & Criss (2003) and some are from standard sources (e.g., AGU Handbooks). b) Sources of thermal energy in the Earth's interior (from Reference Lists A, C & E).

a) Heat flow

TW

 
 

Average of world-wide measurements

30-32

“Corrected” for ridge effect

43

Cooling halfspace model

41-44

 
 

Near-ridge hydrothermal

1

Temporal variations

2


b) Energetics

TW

 

Potential energy contributions

 

Conducted from core
8.6

Mantle differentiation

0.6

Thermal contraction

2.1

Earthquake induced gravitational energy

2

Radiated seismic energy

0.3

Tidal friction

1-2

Total (possible non-radiogenic and core sources)

~16

 
 

Radiogenic

 

Radiogenic (BSE)

19 - 31

Radiogenic (continental crust)

5.8-8

Delayed radiogenic
(1 to 2 Ga delay between production & arrival at surface)

 5

Total (radiogenic)

24-36

 
 

Secular cooling (Korenaga, 2003; Schubert et al., 1980)

 

 80 K/Ga

9

 100 K/Ga

14

 
 

Total input

39-66


Slab melting

It is often assumed that the melting involved in hotspot magmatism requires an anomalous source of energy, e.g., plumes or frictional heating. Heat is indeed required, and invoking a source with a lower melting point e.g., eclogite, does not remove this problem since latent heat of melting is still required.

The mantle is a much larger source of energy than the core. Melting can occur by placing material with a low melting point (e.g., basalt, eclogite, pyroxenite, piclogite) into the shallow mantle by subduction or delamination of continental lithosphere. The surrounding mantle serves as a heat source and the subducted/delaminated material as a heat sink. At thermal equilibrium the slabs, at least their upper portions, will be partially molten, except in the coldest parts of the shallow mantle. If the slab is rich in volatiles (H2O, CO2) this will reduce seismic velocity, even though the slab is cooler than the surrounding mantle. Subduction refertilizes the mantle, cools it, and causes major local thermal perturbations.

Some of the seismically slowest regions on Earth – low velocity zones (LVZ) – are at the tops of and above subducting slabs (mantle wedge, back-arc basins). These regions are cooled by subduction. The LVZ extend to depths of 200-300 km. Presumably if such convergent regions turn into diverging regions, such as when old sutures are reactivated during continental breakup, the fertile, volatile-rich regions will melt and may even provide dense sinkers which liberate volatiles as they sink (see Lithospheric Delamination page) giving deep LVZ and basaltic melts on top. A sinking carbonated eclogitic slab can have low seismic velocities!

Slabs and delaminated eclogite accumulating at say 650 km can be entrained into upwelling flows when continents diverge and can be part of the basaltic source material. Thus, slabs do not need to be carried around by continents in order to explain slab components in hotspot magmas, although some slab material will certainly be trapped between suturing Archaean cratons. The largest divergence suction is expected with thick separating cratons, whereas thin diverging lithosphere will suck up mainly shallow material. Material in the transition region can also be displaced by sinking slabs.

Summary

The heat budget of the Earth cannot be treated as an instantaneous one-dimensional heat flow problem, or one that involves a homogeneous mantle with uniform and static boundary conditions. Both the radial and lateral structure of the Earth must be considered. Continents affect mantle heat flow by diverting heat to the ocean basins (Lenardic, 1998) and drifting so as to be over cold downwellings (Reference List E). Chemical stratification (Reference List B) of the mantle slows down cooling of the Earth but the upward concentration of radioactive elements reduces the time between heat generation and surface heat flow. Nevertheless, one-dimensional and homogeneous models, or models with a downward increase in radioactive heating have dominated the attention of convection modelers (Reference List D). Paradoxes such as the “missing heat source problem” can be traced to non-realistic assumptions and initial and boundary conditions.

The major outstanding problems in the Earth's thermal budget and history involve the role of hydrothermal circulation near the top, and radiative heat transfer near the bottom, of the mantle. Convection modeling has not yet covered the parameter range which may be most pertinent from physical considerations and geophysical data. What is needed is a thermodynamically self-consistent approach which includes the temperature, pressure and volume-dependence of physical properties, realistic initial and boundary conditions, and the ability to model melting and various forms of heat transport.

The bottom line is that there appears to be no mismatch between observed heat flow and plausible sources of heating. The rate at which the mantle is losing heat appears to have been overestimated in the past, and the available energy sources in the interior have been underestimated. The uncertainties in the various estimates have not been fully appreciated (see Hofmeister and Criss, 2003, for a discussion). Lord Kelvin also underestimated the errors in his estimates of the age of the Earth, and neglected, understandably, a significant heat source.

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  • Schubert, G., Stevenson, D. & Cassen, P., 1980, Whole planet cooling and the radiogenic heat source contents of the Earth and Moon, J. Geophys. Res., 85, 2531-2538.
  • Schubert, G., Turcotte, D., Olson, P., 2001, Mantle convection in the Earth and planets, C. U. Press, 956 pp.
  • Stacey, F. D., 1992, Physics of the Earth, 2nd Ed. Brisbane, Brookfield Press.
  • Stacey, F. D. & Loper, D. E., 1984, Thermal histories of the core and mantle, Phys. Earth Planet. Inter., 36, 99-115.
  • Stevenson, D., Spohn, T. & Schubert, G., 1983, Magnetism and thermal evolution of the terrestrial planets, Icarus, 54, 466-489.
  • Tackley, P., 1998, Three dimensional simulations of mantle convection with a thermo-chemical basal boundary layer: in: M. Gurnis, M. et al., eds., The Core-Mantle Boundary Region, Washington, AGU, 334 pp.
  • Thompson, Sir W. (Lord Kelvin), 1890, On the Secular cooling of the Earth. Mathematical and Physical Papers, Vol III, Elasticity, Heat, Electro-Magnetism. London: C.J. Clay and sons, pp. 295-311.
  • Van Keken PE, Ballentine C.J., 1998, Whole-mantle versus layered mantle convection and the role of a high-viscosity lower mantle in terrestrial volatile evolution, Earth Planet. Sci. Lett., 156, 19-32.
  • Van Keken P.E., Ballentine C.J., 1999,. Dynamical models of mantle volatile evolution and the role of phase transitions and temperature-dependent rheology, J. Geophys. Res., 104, 7137-51.
E. Energy partitioning
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  • Lenardic, A., & L.-N. Moresi, 2001, Heat flux scalings for mantle convection below a conducting lid: resolving seemingly inconsistent modeling results regarding continental heat flow, Geophys. Res. Lett., 28, 1311-1314.
  • Lenardic, A., L. Guillou-Frottier, J.-C. Mareschal, C. Jaupart, L.-N. Moresi, & W.M. Kaula, 2000, What the mantle sees: the effects of continents on mantle heat flow, In, The History and Dynamics of Global Plate Motions, Ed: M. Richards, R. Gordon, & R. van der Hilst, AGU Press, 95-112.
  • Lenardic, A., 1998, On the partitioning of mantle heat loss below oceans and continents over time and its relationship to the Archean paradox, Geophys. J. Int., 134, 706-720.
F. Oceanic lithosphere
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  • McNutt, M. K. & A. V. Judge, 1990, The Superswell and mantle dynamics beneath the South Pacific, Science, 248, 969-975.
  • Parsons, B. & J. G. Sclater, 1977, An analysis of the variation of the ocean floor bathymetry and heat flow with age, J. Geophys. Res., 82, 803-827.
  • Phipps Morgan, J., W. J. Morgan, and E. Price, 1995, Hot spot melting generates both hot spot volcanism and a hot spot swell? J. Geophys. Res., 100, 8045-8062.
  • Phipps Morgan, J. & W. H. F. Smith, 1992, Flattening of the seafloor depth-age curve as a response to asthenospheric flow, Nature, 359, 524-527.
  • Phipps Morgan, J. & W. H. F. Smith, 1994, Correction: Flattening of the seafloor depth-age curve as a response to asthenospheric flow, Nature, 371, 83.
  • Rowley, D.B., 2002, Rate of plate creation and destruction: 180 Ma to Present, Geol. Soc. Am. Bull., 114, 927-933.
  • Sandwell, D. T., E. L. Winterer, J. Mammerickx, R. A. Duncan, M. A. Lynch, D. A. Levitt, and C. L. Johnson, 1995, Evidence for diffuse extension of the Pacific plate from the Pukapuka Ridges and crossgrain gravity lineations, J. Geophys. Res., 100, 15087-15099.
  • Sleep, N.H., 1994, Lithospheric thinning by midplate mantle plumes and the thermal history of hot plume material ponded at sublithospheric depths, J. Geophys. Res., 99, 9327-9343.
  • Smith, W. H. F. & J. Phipps Morgan, 1992, A dynamic origin for asymmetric subsidence and geoid anomalies in the south Atlantic Ocean? Eos, Trans. Am. Geophys. Union, 73, 582.
  • Stein, C. A. & S. Stein, 1994, Comparison of plate and asthenospheric flow models for the thermal evolution of oceanic lithosphere, Geophys. Res. Lett., 21, 709-712.
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