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A magma accretion model of oceanic lithosphere is proposed and its implications for understanding its thermal field examined. The new model (designated Variable Basal Accretion—VBA) assumes existence of lateral variations in magma accretion rates and temperatures at the boundary zone between the lithosphere and the asthenosphere. However, unlike the previous thermal models of the lithosphere, the ratio of advection to conduction heat transfer is considered a space dependent variable. The results of VBA model simulations reveal that the thickness of the young lithosphere increases with distance from the ridge axis, at rates faster than those predicted by Half-Space Cooling models. Another noteworthy feature of the new model is its ability to account for the main features in the thermal behavior of oceanic lithosphere. The improved fits to bathymetry have been achieved for the entire age range and without the need to invoke the ad-hoc hypothesis of large-scale hydrothermal circulation. Also, use of VBA model does not lead to artificial discontinuities in the temperature field of the lithosphere, as is the case with GDH (Global Depth Heat Flow) reference models. The results suggest that estimates of global heat loss need to be downsized by at least 25%.

Detailed understanding of large-scale variations in the thermal field of the oceanic lithosphere provides important constraints on deep tectonic processes. Nevertheless, thermal models of the lithosphere proposed to date have failed to provide a satisfactory account of some of the important features of large-scale variations in oceanic heat flow. For example, both the Half-Space Cooling [

There are however dissenting views on the subject matter of hydrothermal circulation on regional scales [

In the current work, we present a new model of oceanic lithosphere that can overcome the above-mentioned problems and present a satisfactory solution for the heat flow paradox, without the need to invoke the ad hoc hypothesis of large-scale hydrothermal circulation in stable ocean crust. To place the new model in context, we summarize the main characteristics and inherent limitations of currently accepted thermal models of the lithosphere. Next, the characteristics of thermal fields associated with upwelling of asthenospheric materials are outlined and its compatibility with the new model features examined. Following this, details of the new model fits to observational data on heat flow and bathymetry are presented, along with results of numerical simulations exploring the influence of model parameters. We point out, in addition, that empirical relations such as those proposed for GDH reference models are unnecessary. Implications of the new model results for understanding regional scale variations in global heat flow are discussed and the need to downsize the current estimates of global heat loss emphasized.

Thermal models of the lithosphere, with wide acceptance in the current literature, may be classified as falling into essentially two generic groups:

Half-Space Cooling (HSC) Models,

Constant Thickness Plate Models.

In the HSC model the basic assumption is that the temperature of the medium at origin time (

It was pointed out by McKenzie [

The assumptions in the Plate model that basal temperature and thickness of the lithosphere are constant rely on the argument that lateral movements of surface layer take place over large nearly isothermal cores present in mantle convection systems. While these may be true of oceanic lithosphere away from spreading centers, they can hardly be considered as representative of the thermal structure in regions close to the ridge axis, where nonisothermal conditions are likely to prevail at the base of the lithosphere. In particular, the assumption of constant basal temperature in zones overlying upwelling limbs of asthenosphere contradicts the vast body of observational evidences on temperature variations in intrusive magmatic and thermal metamorphic processes (e.g., [

In an apparent attempt to minimize problems of this type it has been proposed [

Yet, significant discrepancies continue to exist between the hybrid model values and observational heat flow data, for oceanic lithosphere with ages less than 55 Ma. The current consensus is that such differences arise from perturbing effects of supposed regional scale hydrothermal circulation in the ocean crust, believed to be unaccounted for in conventional heat flow measurements in the oceanic crust [

We consider now a new thermal model of oceanic lithosphere that can overcome some of the shortcomings of the HSC and Plate models discussed in the previous section. Following the premises of the previous models we also assume that lithosphere represents the boundary layer of mantle convection and that its temperatures are always at or below the melting temperature. In developing the new model it is assumed that the growth of this boundary layer, in regions away from the ridge axis, is determined not only by the cooling effects of surface heat loss but also by mass and energy exchange processes taking place at the bottom boundary of the lithosphere. In particular, we consider that the effects of basal magma accretion and lateral temperature variations of the asthenosphere play important roles in the formation of the lithosphere. The new model is designated hereafter as the Variable Basal Accretion model, abbreviated VBA.

The basal accretion may take place as a result of pressure and temperature variations in the ascending magma column, compositional changes occurring during up-flow and differential rates of migration of volatile components. It is well known in fluid dynamics studies [

In the present context of developing a new thermal model of the lithosphere the main interest is in examining the effect of basal accretion on the thickness of the lithosphere and on the surface heat flux. If accretion can be considered as a consequence exclusively of conductive heat loss from the upper surface of the lithosphere, an approximate description of the boundary layer growth and the ensuing heat flow variation at the surface can be provided on the basis of the well-known inverse square root relation of the time elapsed [

The growth of boundary layer is also affected by the presence of lateral temperature variations in the asthenosphere. The assumption of lateral temperature variations is compatible with the vast body of observational evidences on temperature fields in magmatic and thermal metamorphic processes (e.g., [

In the present case, we assume that the temperature variation in the asthenosphere, along a horizontal plane at the depth corresponding to the base of the stable lithosphere, is best represented by a relation of the type:

As mentioned earlier, the main consequence of basal accretion and lateral temperature variations is an increase in the rate of “migration” of the solidification isotherm to larger depths relative to those encountered for isothermal fluids [

Schematic illustration of solidification isotherms (

In the following sections we consider the mathematical basis of the new VBA model and compare the model predictions against observational heat flow and bathymetry data for oceanic regions. In addition, we also compare VBA model values of heat flow and bathymetry with those derived from the half-space cooling and Plate models.

Consider first the problem of two-dimensional heat transfers in a rectangular plate of thickness

The solution to the problem defined in (_{1}

The solutions (_{1}

Transfer of boundary temperature profile from a block of thickness L to the next one with larger thickness

The lithospheric block of larger thickness is positioned over a region of asthenosphere with relatively lower temperatures.

There is a reduction in the rate of basal accretion of magma.

There is loss of heat by the lithospheric block in the vertical direction towards the surface.

Note that the thermal effects of the first and second processes were not taken into account in previous models of the lithosphere (HSC and Plate models), a consequence of the assumption of isothermal asthenosphere in these models.

At this point a brief remark on the scaling constant

Decrease in thickness of the column of asthenospheric material as a function of the distance from the ridge axis,. The numbers on the curves are values of the parameter (

Results of numerical simulations indicate that the VBA model values of heat flow for the case

The intervals chosen in discretization of VBA model simulations are in the range of 10 m to 1 km. For gradual changes in the thickness of the lithosphere the computational accuracy of the results obtained in this piecewise approximation is not overly sensitive to the size of the interval chosen for discretization. On the other hand, the approach has the advantage that the effects of lateral temperature variations arising from compositional changes, which determine the variability in the lower boundary condition (see (

Values of parameters used in numerical simulations of the VBA model of the oceanic lithosphere.

Parameter | Values used in model simulations | |

Representative | Plausible Range | |

Thickness of Lithosphere | 95 km | 75–115 km |

Thermal Conductivity | 3.3 W | 3-4 W/m/ |

Density of Asthenosphere | 3330 kg/m^{3} | 3300–3600 kg/m^{3} |

Specific Heat | 1.171 KJ kg^{-1} | 1000–1500 KJ k |

Thermal Diffusivity | 25 ^{-6} km^{2}/yr | (11–44) |

Solidification Temperature | 927– | |

0.5 k | 0.4–0.7 k | |

0.002 k | 0.001–0.003 k | |

Intervals chosen for discretization of blocks | 10 m (Vertical) 1000 m (Lateral) | 10 to 1000 m |

Distribution of isotherms in the oceanic lithosphere, derived from the VBA model of the present work. The numbers on the curves are temperatures in degrees centigrade.

We now make a comparative analysis of the VBA model predictions with results of heat flow measurements in oceanic regions. Following earlier studies (e.g., [^{2}/s in Table

The variation of VBA model heat flow with age, determined on the basis of (

Comparison of VBA model values with experimental heat flow data for the oceanic lithosphere. The model values for basal accretion rates

As mentioned earlier, the VBA model leads to a family of solutions depending on the value of the factor

We now examine the dependence of the VBA model predictions on the plausible variations in the values of the main parameters. For this purpose, a number of numerical simulations were carried out for a range of values of the thickness of the plate (

Results of numerical simulations illustrating the dependence of VBA model response to changes in the values assumed for plate thickness (a), basal temperatures (b) and thermal diffusivity (c). The dark squares are the mean oceanic heat flow values.

Note that changes in thickness of the lithosphere (Figure

The classical solutions for transient temperature distributions in a plate with constant boundary temperatures have been discussed extensively in the literature [

A direct comparison between the transient component of VBA model solution (second term on the RHS of (

There are several important differences between the solutions (

Another important difference between the solutions (

Vertical distribution of transient temperatures in the Plate model (McKenzie, [

The fact that the transient component is absent for all times at the base of the lithosphere is an indication that the Plate model is inadequate in providing a satisfactory description of heat transfer processes at the lithosphere—asthenosphere boundary, mainly in regions close to the ridge axis. On the other hand, the vertical distribution of the transient component in the VBA model, illustrated in Figure

Vertical distribution of transient temperatures in the VBA model of the present work, for the parameter values listed in Table

Including the steady-state component in (

An interesting aspect of (

Note that the derivation of (

Fits to data for ocean floor bathymetry variations rather than that for surface heat flow is often considered as a relatively more rigorous test of thermal models of the lithosphere. The relation for bathymetry in VBA model has been developed following the isostatic compensation scheme discussed in earlier studies (e.g., [

The integration in (

In comparing the bathymetry results of VBA model with the observational datasets we make use of the same data sets [

Comparison of the fits of VBA and GDH reference models to bathymetry data. Note that VBA model curve (in blue color) of the present work provides a remarkably good fit for the entire age range of the oceanic lithosphere. The GDH model [

Apart from the above mentioned restriction, both VBA and GDH models provide equally good accounts of ocean floor bathymetry. The vertical temperature field of the lithosphere, derived from the VBA model, is similar to the example illustrated in Figure

At this point it is convenient to consider the sensitivity of VBA model response to the values of the parameters listed in Table

Results of numerical simulations illustrating the response of VBA model to change in the value assumed for the bathymetry constant

For old ocean crust (with ages

Results of numerical simulations illustrating the response of VBA model results to change in the value assumed for the basal temperatures (

In contrast to HSC and Plate modes, which are closely related regarding the assumptions made and in their implementation, the newly proposed VBA model of oceanic lithosphere assumes that both the thickness and the temperature of the magma rich basal segment vary with distance from the ridge axis. Estimates of regional heat flux in the VBA model are lower than those obtained in previous thermal models of the lithosphere, including the recently proposed Plate model with variable thermal conductivity (McKenzie et al. [

Global heat flow map derived from mixed data sets. For oceanic regions with ages less than 120 Ma heat flow values are calculated using (

The discrepancy of previous models of global heat flow from the experimental values has been hypothesized to originate from regional scale hydrothermal circulation in oceanic crust. As mentioned earlier, the validity of the hypothesis of regional scale hydrothermal circulation in oceanic crust is questionable, in view of available information on the thermal and hydrological characteristics of the ocean crust [

A problem of related interest is the Global Heat Loss. The VBA Model leads to estimates of regional heat flow lower than those derived from previous models, in agreement with recent results of higher-degree harmonic representation of global heat flow (Hamza et al., [

The main conclusions of the present work may be summarized as follows.

The new VBA model of the oceanic lithosphere allows incorporation of the thermal effects of variable heat input into its basal parts, whereas previous models (HSC and Plate) take into account only effects produced by surface heat loss.

The width of magma injection zone in the spreading center in VBA model is relatively narrower, and the transition to stable nonmagmatic configuration takes place on time-scales much shorter than those predicted by the conventional boundary layer theory.

The constant temperature Plate model envisaged by Mckenzie [

The VBA Model provides a vastly improved fit to experimental heat flow data for both the younger as well as the older segments of the oceanic crust.

The relation for ocean floor bathymetry derived from VBA model provides an equally good fit to observational data as that provided by the hybrid model curves of C. A. Stein and S. Stein [

The VBA model fit to bathymetry data is valid for the entire age range of the oceanic lithosphere. There is no need to introduce ad hoc adjustments (artificial changes in heat flow) in model fits for ocean floor bathymetry.

The fits of VBA model for sea floor heat flow and bathymetry have been achieved without introducing artificial discontinuities in the temperature field of the lithosphere.

Agreement of the VBA model with the observables exists for any reasonable choice of input parameters. The best agreement is obtained for values closest to those believed representative of the lithosphere, particularly the lowermost extent of the lithosphere.

Estimates of root-mean-square misfit between the VBA Model values and experimental heat flow data are relatively much better than those found for the previous models. Given the uncertainty in marine heat flow measurements and the quality of the fit, there appears to be no need to invoke the hypothesis of regional scale hydrothermal circulation in oceanic crust.

The VBA Model of the present work leads to estimates of regional heat flow that are significantly lower than those derived from previous thermal models of the lithosphere. The new estimates are in reasonable agreement with the results of higher-degree harmonic representations of global heat flow (Hamza et al., [

The current estimates of global heat loss need to be downsized by at least 25%, in support of recent assessments [

In addressing the problem described by (

The purpose of condition (

We assume that

The problem in

Hence the solution of problem in

The problem in

The condition (

We now admit that the solution of the problem in

the eigen values

the Eigen functions _{i}

Solving the auxiliary problem we have

The coefficients of the expansion

and we have

The integral on the RHS of (

It is obvious that the transform and its inverse in (

Multiply the original equation by the operator:

Multiplying the equation of the auxiliary problem by the operator

Subtracting (

Developing the integrals on the RHS,

To conclude we use the boundary conditions in (

In obtaining the solution of (

In terms of the dimensionless variables the equations for temperature (

The authors thank Professor Anne Hofmeister (Washington University, MO, USA) for critical comments and suggestions which have contributed to significant improvements in the manuscript. The present work was carried out as part of a research project for investigating thermal isostasy in southeast Brazil, with funding from Research Foundation of the State of Rio Janeiro—FAPERJ (Grant no. E-26/100.623/2007), under the program “Scientist of the State of Rio de Janeiro”. The second author has been a recipient of a Ph.D. scholarship granted by Coordenadoria de Aperfeiçoamento de Pessoal de Nível Superior—CAPES, Brazil. The third author contributed to the development of integral transform solution for variable basal heat input, as part of his Ph.D. thesis work on thermal models of continental lithosphere.

_{2}O-K

_{2}O-CaO-MgO-FeO-Fe

_{2}O

_{3}-Al

_{2}O

_{3}-SiO

_{2}-TiO

_{2}-H

_{2}O-CO

_{2}: representation, estimation, and high temperature extrapolation