the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Lower mantle geotherms, flux, and power from incorporating new experimental and theoretical constraints on heat transport properties in an inverse model

### Anne M. Hofmeister

An inverse method is devised to probe Earth's thermal
state without assuming its mineralogy. This constrains thermal conductivity
(*κ*) in the lower mantle (LM) by combining seismologic models of bulk
modulus (*B*) and pressure (*P*) vs. depth (*z*) with a new result, ∂ln(*κ*) $/$ ∂*P* ∼ $\mathrm{7.33}/{B}_{T}$, and available high temperature (*T*) data on
*κ* for lengths exceeding millimeters. Considering large samples accounts for
the recently revealed dependence of heat transport properties on
length scale. Applying separation of variables to seismologic $\partial B/\partial P$ vs. depth isolates changes with *T*. The resulting LM d*T* $/$ d*z* depends
on ${\partial}^{\mathrm{2}}B/\partial {P}^{\mathrm{2}}$ and $\partial B/\partial T,$ which
vary little among dense phases. Because seismic $\partial B/\partial P$ is
discontinuous and model dependent ∼ 200 km above the core,
unlike the LM, our results are extrapolated through this tiny layer (*D*^{′′}).
Flux and power are calculated from d*T* $/$ d*z* for cases of high (oxide) and low
(silicate) *κ*. Geotherm calculations are independent of *κ*,
and thus of LM mineralogy, but require specifying a reference temperature at
some depth: a wide range is considered. Limitations on deep melting are used
to ascertain which of our geotherm, flux, and power curves best represent
Earth's interior. Except for an oxide composition with miniscule ${\partial}^{\mathrm{2}}B/\partial {P}^{\mathrm{2}}$, the LM heats the core, causing it to melt. Deep
heating is attributed to cyclical stresses from *>* 1000 km daily
and monthly fluctuations of the barycenter inside the LM.

Heat moves when a temperature (*T*) difference exists, where the net flow is
from hotter to colder regions. This phenomenon is important to Earth because
it is dynamic. But as a consequence, the outcome of a laboratory experiment
is greatly influenced by the time dependence of the applied heat (e.g., Tye,
1969), which has led to overlooking the length-scale dependence of heat
transport and misunderstandings of experimental limitations and
uncertainties as well as of microscopic mechanisms (Hofmeister, 2019, 2021).
Thermal models in Earth science are particularly affected by these
shortcomings, due to wide variations in relevant length scales,
temperature, pressure, and material properties, such as transparency to
thermal radiation. An improved understanding, based on a new theory and
accurate data on mineral heat transport, is described next.

## 1.1 Recent findings on heat transport properties relevant to mantle studies

One incorrect presumption is that the physical properties representing heat
flow (thermal conductivity, *κ*, or its close relative thermal
diffusivity, *D*) are independent of the distance along the thermal gradient.
This static view is inconsistent with Fourier's heat equation, as follows.
Its simplest one-dimensional form is

where *t* = time and *z* represents the direction of heat flow. Equation (1)
holds for temperature changes being sufficiently small that the relevant
properties vary negligibly. Dimension analysis of this effectively constant
*T* condition provides

where *L* is the distance over which heat travels, *ζ* is a time constant,
and *u* is a characteristic speed. Thermal conductivity likewise depends
directly on *L* because it is proportional to *D*,

and because the multiplying parameters, density (*ρ*) and specific heat
(*c*_{P}) at constant pressure (*P*), are independent of *L*.

The required length-scale dependence has been masked by experimental
limitations, including ubiquitous use of similar sample lengths of
*>* 1 to *<* 5 mm. Many experiments are steady state, so time
and *ζ* are irrelevant. Other common techniques are periodic, where
these oscillations about quasi-equilibrium involve another, different time
constant (see, e.g., Tye, 1969; Zhao et al., 2016). The transient technique
of laser-flash analysis (LFA), which avoids heat losses from physical
contacts (see Vozár and Hohenauer, 2003) and monitors thermal evolution
across *L* with time, has confirmed that *D* linearly depends on *L* below about 1 mm for
electrical insulators, glasses (Hofmeister, 2019, chap. 7),
semi-conductors, metals, and alloys (Hofmeister, 2021). Results (Fig. 1) are
consistent with a linear response when *L* is small, as in Eq. (2).

Misunderstandings also stem from reliance on the historic kinetic theory of
gas (KTG) to depict heat transfer in solids. However, heat and matter move
together across long expanses in a gas, which is unlike a solid where these
motions are decoupled. Furthermore, KTG assumes elastic collisions, through which
temperature cannot change. Neither non sequitur is addressed by morphing molecular collisions in a gas into elastic scattering of pseudo-particles denoted as
phonons in a solid. Because gas data are collected under negligible *T*
gradients to avoid convection, assuming random fluctuations in all three
directions is reasonable and provides formulae mostly compatible with gas
data. Yet, the ratios of the transport properties are not correctly
described, while the ubiquitous emission of thermal radiation from all states
of matter remains unexplained. Accounting for inelasticity in molecular
collisions addresses both shortcomings in KTG (Hofmeister, 2019, chap. 5).

Regarding condensed matter, Fourier assumed heat flows into, across, and out of the stationary solid, whereby part of the heat is stored in the elements along the path. The process is diffusion, which is underscored by Fick constructing his formulation after Fourier's.

Fourier defined flux as heat per area per time and realized that

One dimension suffices for discussion since heat flows down the thermal gradient per the second law of thermodynamics. Equation (4) is fundamental: taking its spatial derivative, conserving energy, and simplifying using the definition of Eq. (3) leads to Eq. (1).

Experiments and theory show that light is the diffusing entity in solids
(Hofmeister et al., 2014; Criss and Hofmeister, 2017), which is real and
pure energy. Light, unlike a phonon, crosses interfaces. Attenuation of
light across the sample provides the length-scale dependence of Fig. 1.
These recent discoveries led to new formulae for the dependence of *κ*
on *P* and *T* and the absorption spectra of a material, which were verified against
reliable data on *κ* below 2 GPa (Fig. 2) and on *D* and *κ* from a
few kelvins to well above ambient *T* (Hofmeister, 2019, 2021). LFA measurements
of *D* at high *T* (Fig. 3a), combined with Eq. (3), show that *κ* above 1000 K is nearly constant for structures or chemical compositions more complex
than Al_{2}O_{3} (Fig. 3b). These advances are used in the present
paper to evaluate of thermal conductivity in the lower mantle (LM) while
accounting for ambiguities in the temperature and mineralogy for this
immense region of the Earth.

## 1.2 Reliability of available information on lower mantle heat transport

Thermal models are based on transport properties. To achieve high pressures
appropriate to the deep Earth, diamond anvil cell (DAC) experiments probe
tiny samples. Accurately determining *P* near 1 atm in devices geared for
extreme compression has not been achieved. Hence, results from DAC heat
transport experiments are benchmarked against independent measurements of
*D* or *κ* at ambient *P* (e.g., Hsieh et al., 2009). However, ambient data
are collected from *L* *>* 1 mm, which are ∼ 100× larger than sample thicknesses used in DACs. Extrapolation
from large to small *L* was not done and is non-linear (Fig. 1). Very high *P*
studies are difficult, leading to additional problems, as discussed in
detail by Hofmeister (2009, 2010b, 2019, 2021). To summarize, large thermal
gradients preclude use of Eq. (1) while requiring knowledge of the *T* dependence
of *D* (or *κ*) at *P*, which is the unknown sought. For tiny samples, heat
flow is two-dimensional but one-dimensional equations are used. At high *T*,
cooling occurs by ballistic radiation to the surroundings (e.g., to the
detector used to ascertain *T*), which is not addressed in Fourier's
description of heat diffusion (conduction). Thermal gradients changing
direction during the experiments of McWilliams et al. (2015) and
Konôpková et al. (2016) (see figure 13 in Hofmeister, 2021) were not
addressed in their analysis. Thermoreflectance methods (e.g., Hsieh et al.,
2009) assume the length scale over which heat diffuses, which dictates the
results per Eqs. (1) to (4). Low *P* experiments using ∼ millimeter
lengths and other techniques utilized in Fig. 2 lack these difficulties, as
discussed in previous work and Sect. 3, and are utilized here.

Importantly, thermal gradients inside Earth are low. Even in the
lithosphere, $\partial T/\partial z$ only reaches ∼ 20 K km^{−1}. Thermal transport properties vary little over a few degrees (Fig. 3). Because *T* varies less than 4 K over *L* *>* 5 km inside Earth, its
thermal length scale is immense, and so heat transfer therein is always
diffusive and isothermal properties are relevant. But in the laboratory, *L* is
∼ 10^{5} smaller, permitting ballistic
(boundary-to-boundary) transport to augment diffusion, as recognized in
minerals and rocks by Kanamori et al. (1968) and further documented by
Pertermann and Hofmeister (2006), Branlund and Hofmeister (2007), and
Merriman et al. (2018). Laser-flash experiments reduce and remove ballistic
effects via sample coatings and via models (e.g., Blumm et al., 1997; Hahn
et al., 1997). Our large and growing LFA database (e.g., Hofmeister, 2019) and
the associated theoretical model are essential to ascertain heat transport
at high mantle temperatures.

Seismic models provide velocities and density inside the Earth. Pressure is well constrained, since Earth's mass and moment of inertia provide independent boundary conditions (e.g., Anderson, 2007). Mineralogy is based on comparing laboratory data on minerals to radial models, such as the preliminary earth reference model (PREM) of Dziewonski and Anderson (1981) since radial changes depict average values. Comparison with laboratory studies in a forward-(fitting) approach is used but leads to equivocal results because temperature is not known independently. For the Earth, temperatures are changing, heat is moving, and seismic waves contribute energy to the rocks during their attenuation. Hence, conditions are not adiabatic, as previously assumed in forward-(fitting) models. In addition, minerals vary greatly in possible chemical compositions and structures.

Assessing the lower mantle is particularly uncertain because no rocks have
been exhumed from below 670 km. Inclusions in diamonds only indicate *P* and *T*
conditions when a single inclusion contains multiple phases because most, if
not all inclusions, predate their diamond host (Nestola et al., 2017).
The inference that lower mantle material is preserved in microdiamonds is based
on separated inclusions of (Mg,Fe)O and enstatite (Stachel et al., 2000). The
tetragonal garnet phase TAPP (now jeffbenite) once considered to form in
the lower mantle is now known to be stable above 13 GPa, i.e., in the
transition zone (Nestola et al., 2016).

Evaluating temperatures from seismic models via forward-fitting requires
knowledge of the mineralogy (e.g., Cammarono et al., 2003). A thermal model
is needed to account for Earth's heat being lost to space (i.e.,
non-adiabatic gradients). For the lower mantle, a wide range of *κ* values is possible due to the ambiguities in mineralogical models, even if
experimental uncertainties were small. An alternative approach to
fitting seismic velocities is needed to better understand this
immense region of the thermally evolving Earth.

## 1.3 Purpose and thesis

In view of limited knowledge of the LM, an analytical inverse approach is
used here to decipher its thermal state from a seismic reference model with
minimal assumptions. As in previous large-scale mineralogical or thermal
studies (e.g., MacDonald, 1959; Anderson, 2007; Murikami et al., 2009; Criss
and Hofmeister, 2016), average, radial temperatures are sought to describe
Earth's structure, which is reasonably represented as spherical shells.
Surface heat flux being remarkably similar for the continental and oceanic
crusts, despite the great contrast in their heat-generating elements (e.g.,
Veiera and Hamza, 2018), points to the radial thermal gradients dominating
Earth's thermal state and evolution. The low measured surface emissions of
∼ 60 mW m^{−2}, corresponding to ∼ 100 W km^{−3} of underlying rock, show that thermal evolution is now slow; i.e.,
conditions are quasi-steady state. Hence, angular (lateral) motions of heat
are unimportant to describing Earth as a whole.

Our mathematical analysis is based on decades of mineral physics efforts
which show that (1) pressure derivatives of diverse physical properties
vary far less than ambient values do and that, as *P* climbs, all properties
increase more weakly with *P*. (2) Physical properties at high *T* behave
similarly, as illustrated by the Dulong–Petit limit representing heat
capacity at high *T*. (3) Second-order *P* or *T* derivatives of physical properties
are small, which means that cross derivatives are small, and so separation
of variables reasonably describes many physical properties. Tabulated data
on diverse properties and phases (e.g., Anderson and Isaak, 1995; Bass,
1995; Fei, 1995; Knittle, 1995) illustrate these points.

If variables are separable, a property of interest (Υ) follows the form

where *f* and *g* are independent, dimensionless functions. Equation (5) describes
bulk and shear moduli from diverse elasticity experiments (e.g., Anderson
and Isaak, 1995; Bass, 1995). This finding is important to heat transport,
as bulk modulus is the prime descriptor of *κ*(*P*) per dimensional
analysis (e.g., Dugdale and MacDonald, 1955).

For the lower mantle, variations in velocities from available seismic
reference models differ negligibly except for the ∼ 200 km
above the core (*D*^{′′}) where variations among studies are small, despite
larger uncertainties for this region (see figures in Kennett et al., 1995).
Utilizing PREM suffices (see Section 2.1 for further discussion). In the LM,
excluding *D*^{′′}, velocity changes are slow and smooth, leading to
interpretation of invariant chemical composition. Since *T* changes far more
slowly with distance in the Earth than in experiments, an isothermal bulk
modulus represents the mantle values. The present paper assumes that changes
in mineralogy of the LM are secondary, i.e., that the main changes in its
seismic radial profile are from *P* and *T*, which permits use of Eq. (5). General
behavior of bulk moduli for dense materials from both compression and
elasticity studies supports this contention.

It is most fortunate that the derivatives are simply described:

where for dense and hard materials compatible with the LM, the constant *B*^{′} = $\partial B/\partial P$ is commonly near 4 and ${B}^{\prime \prime}={\partial}^{\mathrm{2}}B/\partial {P}^{\mathrm{2}}$ is negative and sufficiently small to require very high pressure
for its resolution (e.g., Sinogeikin and Bass, 1999; Zha et al., 2000).
Results center on ${B}^{\prime}=\mathrm{4}$ because this value corresponds to a harmonic
interatomic potential (e.g., Hofmeister, 1993) and anharmonicity links to
*T*, not *P*, changes (e.g., Wallace, 1972). Most measurements provide $\partial B/\partial T$ as a constant. Although second-order *T* derivatives exist,
these are small (if even resolvable) for hard oxide and silicate minerals,
as shown in compilations and more recent work (e.g., Aizawa et al., 2004).

## 1.4 Synopsis of our novel, analytical inverse approach, and organization of the report

Section 2 shows how to extract the LM temperature gradient ($\partial T/\partial z$) from pressure and depth derivatives of radial seismic reference models by using Eqs. (5) and (6), in an inverse approach. Fitting is not used, which describes familiar, forward modeling. The extraction uses generic values for $\partial B/\partial T$ and ${\partial}^{\mathrm{2}}B/\partial {P}^{\mathrm{2}}$, which describes dense phases, including the rock salt and perovskite type structures thought to occur in the deep mantle, to explore the possible range of $\partial T/\partial z$ and its depth dependence. Thus, our thermal model is independent of what phases with what compositions might exist in the LM. The method is new, so details are provided.

Section 3 sets upper and lower bounds on heat transport properties for the
LM based on verifiably accurate methods. We derive a simple formula for
$\partial \mathrm{ln}\left(\mathit{\kappa}\right)/\partial P$ from Fourier's heat equation, which
confirms that the result of Hofmeister (2021) is an identity. The sole
parameter in the identity (other than *B*_{T} and *P*) appears be a constant, as
suggested in Fig. 2 and independently by previous work (e.g., Chopelas and
Boehler, 1992). The resulting bounds on *κ*(*T*,*z*) for the LM, combined
with $\partial T/\partial z$ derived from PREM (Sect. 2), provide flux and
power across the LM, without assuming its mineralogy.

Section 4 constructs geotherms using a reference temperature at a shallow
level that avoids melting of a peridotite composition anywhere in the LM.
These geotherms are independent of *κ*. Then, we ascertain which of
our geotherms, fluxes, and powers are compatible with additional
constraints, such as phase equilibria and latent heat of melting.

Our inverse model, which is based on radial seismic changes and high *T* and *P*
behavior common to dense phases, indicates that the LM has a heat source
which is warming the outer core, while causing the inner core to melt.
Possible heat sources are discussed in Sect. 5, along with implications of
our results.

## 2.1 Features of PREM

Seismic reference models represent Earth's average interior; i.e., they are radial. Aspherical images of the Earth's internal structure are represented as perturbations to a reference mode (e.g., Ritzwoller and Lavely, 1995).

Reference model results are displayed as the fairly smooth functions of
velocities, density, and pressure as a function of depth (*z*) or radius (*s*), or
similarly as plots of bulk and shear moduli, the quantities of which are also
fairly smooth, being derived from *ρ* and the two velocities. Seismic
discontinuities are present as kinks, most of which are small in these
typical representations. In contrast, large jumps dominate plots of
derivatives of variables vs. depth (Fig. 4). Hence, this paper makes use of
the derivatives.

Taking derivatives accentuates differences, as this mathematical operation is the converse of integration, which averages and smooths. The pattern exhibited by velocity derivatives (not shown) is similar to moduli derivatives, whereas the density derivative (not shown) is relatively smooth, more like the pressure derivative, and so the depiction of Fig. 4 is inherent to PREM.

Smooth and continuous $\partial B/\partial P$ describes the lower mantle
but only between depths of 871 and 2741 km (Fig. 4). Importantly, other
reference models such as Ak135 differ negligibly from PREM velocities in
this restricted region (see Fig. 12 in Kennett et al., 1995). Our approach
(below) applies to continuous functions only, so PREM suffices to represent
all reference models of this volumetrically immense region. However, *D*^{′′},
where seismic models differ, cannot be quantitatively analyzed. We can only
extrapolate into this tiny shell. For brevity, “lower mantle” or “LM”
refers to its central region from 871 to 2741 km only, unless specified
otherwise. Extrapolation of results for the LM into the underlying *D*^{′′} layer
and up to 671 km, which defines the transition zone (TZ), is discussed in
Sects. 4 and 5.

## 2.2 Separation of variables

The geothermal gradient is defined by

PREM provides the input quantity $\partial P/\partial z$ (Fig. 4). PREM
values for *B* and $\partial B/\partial P$ as a function of depth are affected
by both compression and heating of the minerals inside the LM. Our goal is
to utilize available data to distinguish the effects of *P* and *T* on *B* from PREM.
This is possible through separation of variables and decades of data
acquisition.

Equation (6) provides the input for $\partial B/\partial P$. Because an
inverse approach is being used, we consider values compatible with many
dense phases, oxides, and silicates. The changes (differences) below *z* = 871 km are of interest, so the value of $\partial B/\partial P$ at 871 km serves
as a reference point. This approach links *B*^{′} and *B*^{′′}, as shown graphically in
Fig. 5. Hence, obtaining thermal gradients from PREM via Eq. (7) requires
some estimates of $\partial B/\partial T$ and ${\partial}^{\mathrm{2}}B/\partial {P}^{\mathrm{2}}$ but not of $\partial B/\partial P$. This is understood by
considering two end-member cases.

The case *B*^{′′} = 0 for the lower mantle sets an upper limit since *B*^{′′} is not
positive. If this commonly used limit (e.g., Knittle, 1995) applies to the
LM, then $\partial B/\partial P$ with depth solely results from compression
and thus is invariant (horizontal dashed–dotted line in Fig. 5). Consequently,
rising temperature causes the linear decrease in PREM $\partial B/\partial P$ with *z* immediately below 871 km under separation of variables.
For the second case, we consider *B*^{′′} = −0.015 GPa^{−1}, which reproduces
the decrease in PREM $\partial B/\partial P$ with *z* below 871 km. With this
match, changes in *B* of PREM solely result from compression; i.e., *T* is
constant for *z* slightly below 871 km. Hence, *B*^{′′} between 0 and −0.015 GPa^{−1}
depicts a LM that is both compressing and warming below 871 km, whereas
*B*^{′′} *>* −0.015 GPa^{−1} depicts a cooling and compressing LM just
below 871 km. This case is not shown because *T* in the outermost layers of the
Earth increases with depth, and so the top of the LM should behave likewise.

Once *z* reaches 1300 km, PREM $\partial B/\partial P$ curves have become
rather flat, but as *z* increases further, deeper than ∼ 2200 km, PREM
$\partial B/\partial P$ curves take on a positive slope with *z*, which is
linear just below 2741 km. The broad minimum near 2000 km suggests that a
maximum temperature may exist in the LM, where its manifestation depends on
mantle values of $\partial B/\partial T$ and ${\partial}^{\mathrm{2}}B/\partial {P}^{\mathrm{2}}$ (discussed below). The positive slope in PREM $\partial B/\partial P$ curves at great depths, assuming that Eq. (6) represents the
LM, shows that its deepest extent is shedding its heat downwards while
compressing, discussed further in Sect. 4.

The above findings are general. Here we emphasize that following the discovery of heat generation by radionuclides, it was recognized that Earth may be heating up rather than undergoing progressive cooling. This question was considered in some remarkable papers (e,g., MacDonald, 1959). Notably, Takeuechi et al. (1967) devoted an entire chapter of their book to this subject.

Importantly, temperature differences from the starting point at 871 km are
germane. Consequently, the input for $\partial B/\partial P$ in Eq. (7) is
the PREM curve less a line for the *P* response of *B*, i.e., Eq. (6).
Figure 5 shows the two end-member cases, discussed above, and one
intermediate case. These three examples show that *B*^{′} is controlled by the
choice of *B*^{′′}, in order to match the starting point at 871 km. The range of
values of 3.8 to 4.3 is compatible with mineral data (e.g., Knittle, 1995), with
4 being the harmonic value (Hofmeister, 1993).

## 2.3 Constraints on $\partial B/\partial T$and ${\partial}^{\mathrm{2}}B/\partial {P}^{\mathrm{2}}$ from elasticity and volumetric measurements of diverse phases

Regarding the thermal input parameter, $\partial B/\partial T$ equals −0.023 GPa K^{−1} from the average of four experiments on MgO (compiled by Bass,
1995) and is only slightly larger in magnitude, −0.029 GPa K^{−1}, for
MgSiO_{3} with the perovskite structure (Aizawa et al., 2004), now known
as bridgmanite. A large range of values is possible, per Aizawa et al. (2004) and work cited therein. More importantly, uncertainties are high
while similar values are observed for corundum, spinel, five compositions of
olivine, orthopyroxene, zircon, five garnets, seven other oxides, and four metals,
whereas framework silicates, diamond, and alkali halides have $\partial B/\partial T$ near −0.01 GPa K^{−1} (Bass, 1995; Anderson and Isaak,
1995).

An input value of −0.026 GPa K^{−1} is used, the average of which is within
reported experimental uncertainty of measurements of LM candidate minerals
and moreover describes dense silicates and oxides in general. Because
$\partial T/\partial z$ is inversely proportional to $\partial B/\partial T$, the effect of varying this parameter on the results is easily
ascertained. In contrast, a non-linear response is associated with *B*^{′′}, since a
difference with PREM is involved and PREM curves are non-linear with depth
(Figs. 4 and 5). Hence, possible *B*^{′′} values are the focus.

From the compilation of Bass (1995), *B*^{′′} is positive for silica glass and near
0 for MgAl_{2}O_{4} spinel, which is also disordered. Otherwise, *B*^{′′} for
oxides and silicates ranges from −0.03 to −1.6 GPa^{−1}. The largest
magnitude depicts orthopyroxene, which has unusually high *B*^{′} as well. For
bridgmanite, *B*^{′′} was not resolved even at compression to 155 GPa (Dorfman et
al., 2013), consistent with incompressibility of high-pressure, dense
phases. Aluminum being present makes no difference (Zhu et al., 2020).
However, ubiquitous use of the Birch–Murnaghan equation of state, which
involves trade-offs between *B* and *B*^{′} (see, e.g., Knittle, 1995), prevents
resolution of *B*^{′′} for these stiff structures. Polynomial fits are needed to
ascertain *B*^{′′}.

Many studies exist of MgO, but some ambiguity exists because elasticity data
are fit to a polynomial in pressure, whereas volumetric data are analyzed
using an equation of state, which is a general formulation for the *P* and *T* dependence
of volume (*V*). The second-order polynomial coefficient for *B*(*P*) is ${B}^{\prime \prime}/\mathrm{2}$. To make
sure this convention (i.e., Eq. 6) was used, original studies were
consulted. Elasticity measurements of MgO by Sinogeikin and Bass (1999)
provide *B*^{′′} = −0.04 ± 0.02 GPa^{−1}. The X-ray diffraction study of
Yoneda (1990) is consistent with the range of 0 to −0.029 GPa^{−1}. Zha et
al. (2000) reached the highest pressures and found that the null value
reasonably represents elasticity and volumetric data combined. For the dense
LM, *B*^{′′} is small. Again, *B*^{′} = 4 and negligible *B*^{′′} describes a harmonic solid
(e.g., Hofmeister, 1993). We consider 0 to −0.02 GPa^{−1}, mostly in steps
of 0.0025 GPa^{−1}, to calculate $\partial T/\partial z$ from Eq. (7).

## 2.4 Thermal gradients from 871 to 2741 km

Thermal gradients are shown in Fig. 6, with an example of a fit to *B*^{′′} = −0.005 GPa^{−1}. The sign convention used here is based on flow from a
central source (*s* = 0) moving outwards. All curves are well represented by
third-order polynomials (Table 1) with similarly high residuals. Fits can
be scaled to address variations from an input value of −0.026 GPa K^{−1}.
Constrains on $\partial T/\partial z$ are covered in Sect. 4.

The polynomial for $\partial T/\partial z$ is
${T}_{\mathrm{0}}+{T}_{\mathrm{1}}z+{T}_{\mathrm{2}}{z}^{\mathrm{2}}+{T}_{\mathrm{3}}{z}^{\mathrm{3}}$. n/a: not applicable because large $\left|{B}^{\prime \prime}\right|$ leads to unsupportable melting that occurs
near 670 km.
n.d.: not determined.
^{a} These values satisfy the criteria that the LM is not melted and that the outer core melts above 2750 K as
measured for the Fe–S system (see text).
^{b} Range at 2741 km from estimates of low and high *κ*. Power
at the core mantle boundary (CMB) is lower, by about 1 TW. Positive sign indicates heat flow from
the core into the LM, discussed below.
^{c} From the broad maximum in power which is nearly the same for
low and high *κ*.
^{d} Most likely to represent the LM; see text.

## 3.1 Dependence of thermal diffusivity on temperature

Three-parameter fits describe measurements of thermal diffusivity for diverse solids above room temperature up nearly to melting that are neither affected by physical contact losses nor by spurious radiative transfer gains:

The fitting coefficient *G* is near unity and *H* is small (Hofmeister et al.,
2014). When *H* = 0, the parameter *F** =*F*(298)^{G} on the right-hand side
equals *D* at 298 K. The general applicability of these formulae has been
established by additional measurements now encompassing over 200 substances
(Merriman et al., 2018; Hofmeister, 2019). Equation (8) represents sample
thickness *L* *>* 1 mm, i.e., bulk samples, and high temperatures and
thus is appropriate to the mantle.

Examples of *D*(*T*) for dense phases are shown in Fig. 3. Generally, *H* is quite
small, ∼ 0.0002 mm^{2} s^{−1} K^{−1}, but is essential to
represent high-temperature behavior (*T* *>* 1200 K) of structures
involving unit cell formulae more complex than Al_{2}O_{3}. For simple
materials such as MgO and alkali halides occupying the cubic B1 and B2
structures, *H* = 0 within uncertainty. However, Si with the diamond
structure has non-negligible *H* when impurities are present, whereas graphite,
which has a more complicated anisotropic structure and is generally impure,
has a substantial *H* term (Hofmeister, 2019, chap. 7). LFA data on glasses
show that large *H* is commonly associated with high Fe cation content (e.g.,
Sehlke et al., 2020). These findings point to absorption bands above
∼ 1000 cm^{−1} and into the visible region being associated
with the *HT* term. To provide *H* when LFA data on
(Mg_{x}Fe_{1−x})O are not available, we consider corundum, rather than
MgO, to represent an oxide-rich lower mantle. For a silicate LM, systematic
behavior of orthorhombic and cubic perovskites with various chemical
compositions (Hofmeister, 2010a) is considered. The average *D* from the three orientations of NdGaO_{3} is used to compute the *T* dependence of *D* for
orthorhombic perovskite, since this agrees with *D* near 298 K for unoriented
MgSiO_{3} from Osako and Ito (1991). A periodic technique was used, and
their sample was polycrystalline. Contact losses are more important than
ballistic gains, because the latter is reduced by physical scattering
between grains. Hence, *D*(*T*) for NdGaO_{3} in Fig. 3a represents a minimum
for a complex silicate phase in the LM.

Differences among dense silicates at high temperature are not large: this is
the basis of *D* = 1 mm^{2} s^{−1} being commonly used in geophysical
models. Near independence of *D* from *T* for complex solids at high *T* considerably
simplifies calculations (below).

## 3.2 Dependence of thermal conductivity on temperature

Figure 3b shows examples of *κ*(*T*) calculated from Eq. (3). For MgO and
Al_{2}O_{3}, *c*_{P} and *ρ* as a function of *T* are well constrained
even at high *T* (e.g., Ditmars et al., 1982; Fiquet et al., 1997; Chase,
1998). Heat capacity data on MgSiO_{3} perovskite (Akaogi and Ito, 1993)
are limited to near-ambient *T* due to back-conversion problems. Using a model
to extrapolate *c*_{P} is unnecessary, due to uncertainties in *D*(*T*) for
silicate perovskite. Since *D* varies more strongly with *T* compared to *c*_{P}
and *ρ*, which moreover respond in opposite directions during heating,
ambient values are used to estimate *κ*(*T*) for a silicate perovskite
mantle. To ascertain the uncertainty in this approach, analogous
computations were made for *κ*(*T*) of periclase and corundum. Accurate
and estimated trends for MgO differ little (Fig. 3b), but the
Al_{2}O_{3} estimate is significantly steeper with *T* than using exact
values in Eq. (3). Corundum *D*(*T*) is very flat and is better fit to a
polynomial than to Eq. (8), so the discrepancy is likely due to
extrapolation beyond the temperatures actually measured. Another factor is
that the *T* variations of both *D* and *κ* with *T* are weak when the
corresponding ambient values are low, as is indicated in Eq. (9) and evident
in Fig. 3. Thermal conductivity of other silicates depends similarly on *T* as
our estimate for perovskite (for examples, see Hofmeister et al., 2014, and
references therein).

Therefore, we estimate high-*T* mantle thermal conductivity in terms of
constant, limiting values. For an insulating silicate LM, *κ* is taken
as 2.7 W m^{−1} K^{−1}, whereas for a thermally conductive oxide LM,
*κ* is taken as 7 W m^{−1} K^{−1}. The generic value of *D* used in
geophysical models corresponds to ∼ 3.5 W m^{−1} K^{−1}.

## 3.3 Dependence of thermal conductivity on pressure

Many formulae have been proposed for *P* derivatives of transport properties,
based on dimensional analysis. An exact thermodynamic relationship,

was derived and confirmed using reliable available data on 20 different
homogeneous solids at pressures up to 2 GPa (Hofmeister, 2021). Equation (9)
excludes a typographic error in the earlier report. The physical properties,
other than *κ*, are part of the equation of state (EOS). Thermal expansivity
is defined by

Compressibility ($=\mathrm{1}/{B}_{T}$, the bulk modulus) is defined by

The dimensionless Anderson–Grüneisen parameter (*δ*_{T})
describes the opposing effects of *T* and *P* on *V*,

while clearly showing that the efficiency of expansion and compression for any given solid is related. Hence, the far-right-hand side (RHS) of Eq. (9) describes how temperature components of the EOS, not just pressure components, regulate changes in heat conduction during compression.

### 3.3.1 Derivation of $\partial \mathrm{ln}\left(\mathit{\kappa}\right)/\partial P$ from Fourier's equation

Due to the importance of compression to mantle heat transfer, we explain why Eq. (9) is exact. The original derivation considered diffusion of thermal radiation. A simpler approach is covered here.

Since ℑ is independent of pressure in experiments, taking the *P*
derivative of its definition (Eq. 4) suffices to relate $\partial \mathrm{ln}\left(\mathit{\kappa}\right)/\partial P$ to EOS parameters. The algebra is simple and not
specified here. However, one must recognize that a negative temperature
gradient ($\partial T/\partial z{|}_{P}$) is associated with
positive signs for ℑ and *κ* in Eq. (4). But Eq. (10) defining
thermal expansivity is a scalar quantity, being based on volume, which has
no direction. A positive sign for *α*, typical of most materials,
requires the thermal gradient and its inverse, $\partial z/\partial T{|}_{P}$, to be positive in an isotropic solid. In contrast, heat
flow has a well-defined direction from some origin and so is a vector
quantity. Maintaining consistent signs for both *κ* and *α*
during algebraic manipulations after differentiation of Fourier's equation
leads to Eq. (9).

### 3.3.2 Experimental validation

The hot wire/hot strip and Angstrom techniques accurately measure thermal
transport in metallic samples at low *P* (*<* 2 GPa) and low *T* (*<* 1000 K) since metal–metal contact losses are low and ballistic radiative
transfer gains are negligible (e.g., Andersson and Bäckström, 1986;
Jacobsson and Sundqvist, 1988). Moreover, use of standard *L* near millimeters in
piston–cylinder and multi-anvil apparatuses permits direct comparison of
results on diverse materials. Reliable data for insulators at high *P* over millimeter
length scales have been obtained near 298 K using the hot strip/hot wire or
Angstrom methods (e.g., Andersson, 1985; Osako et al., 2004). Samples are
single crystals, glasses, and disks of fine-grained soft powder (alkali
halides) that were compacted prior to study. Unlike metals, systematic
errors exist due to interface thermal resistance and ballistic transfer, but
taking a logarithmic pressure derivative minimizes these problems. Low-pressure transport property measurements and EOS results of over 20 solids,
mainly from elasticity data, are summarized in Table 3 of Hofmeister (2021).
The response of thermal conductivity to compression (Fig. 2) points to
*δ* = 7 at ambient conditions representing the average for
silicates, oxides, metals, alloys, and alkali halides.

Further verification is provided by a well-studied material with a special, negative sign of thermal expansivity. A negative sign for pressure response for thermal conductivity is expected and was indeed observed for silica glass by Andersson and Dzhavadov (1992) and Katsura (1993). Consistency with Eqs. (9) to (12) is demonstrated, since fused silica also has uncommonly positive $\partial B/\partial T$ (Spinner, 1956).

### 3.3.3 Previous EOS evidence for nearly constant *δ*_{T}

Chopelas and Boehler (1992) constrained mantle values of *δ* from 5 to
6 for metals, oxides, and alkali halides by assessing thermal expansivity at
high pressure and temperature. Anderson et al. (1992) argued for an ambient
value *δ*_{0} = 6.5 and a weak volume dependence. The focus of
these studies, along with Helffrick (2017), who further modified the *V*
dependence, is the effect of compression on *α*.

Larger *δ*_{0} = 7 was obtained from Fig. 2, which compares
measurements of ∂*κ*/∂*T* to 1/*B* for the same types of
solids. Hofmeister (2021) calculated *δ*_{0} from temperature
derivatives of bulk moduli, which are more accurate than *P* (or *V*) derivatives
of *α* for several reasons. (1) Elasticity measurements determine
$\partial B/\partial T$ as a first derivative and so are more accurate
than *x*-ray diffraction (XRD) studies which determine $\partial B/\partial T$ as well as $\partial \mathit{\alpha}/\partial P$ as second
derivatives. (2) Linear dependence of *B* on *T* exists over a wide range of
temperatures (e.g., Anderson and Isaak, 1995), which simplifies establishing
this parameter and reduces its uncertainty. (3) Because *B* is large in
magnitude whereas *α* is small, derivatives of *B* are easier to
determine accurately. Average *δ*_{0} = 7 obtained in Fig. 2
better agrees with the EOS study of Anderson et al. (1992) because they
included elasticity data in their assessment. (4) Data on thermal
expansivity at pressure from XRD methods give *α* as an average over
the temperature ranges explored, which makes this approach to *P* derivatives
of *α* very uncertain. High uncertainty in *α*, let alone its
*P* derivative, is evident in the compilation of Fei (1995).

Pressure-independent *δ* is consistent with separation of variables
describing the bulk modulus, i.e., if

Then from the RHS of Eq. (12) follows

Equation (14) equals 0 when ${\mathit{\delta}}_{\mathrm{0}}={B}^{\prime}$ and is small otherwise.
Equation (14) suggests that *δ* is a constant on the order of 4, the value of which for *B*^{′} constitutes the simple Murnaghan EOS.

Similarly exploring the middle term of Eq. (12) indicates that *δ*
also weakly depends on temperature when thermal expansivity is described by
separation of variables. Constant *δ* is thus a reasonable
first-order approximation. Exact evaluation is difficult due to the
generally small sizes of all derivatives, limitations of a polynomial
representation, and assumptions underlying the forms for the EOS.
Compressible alkali halides are very important for this endeavor but are
hydroscopic, easily deformed, and transparent in the infrared as
single crystals.

### 3.3.4 Lower and upper bounds for thermal conductivity in the LM

The pressure (or depth) dependence of *κ* in the LM is much weaker for
perovskites than oxides (Fig. 7), assuming *δ*_{0} = 7. Corundum
and perovskite have similar bulk moduli, so their different curves in Fig. 7
relate to relative efficiency of heat transfer at high *T* only.

Pure MgO would have lower *κ* than Al_{2}O_{3} with depth due to
*T* increasing beyond 1500 K (Fig. 3b) but would have higher *κ* with depth due
to *P* increasing. For this reason, corundum, which has the same *κ* as
periclase at 1500 K, is used to represent an oxide lower mantle (Fig. 7).

Calculations of geotherms, flux, and power from 871 to 2471 km are presented
here, which are extrapolated through *D*^{′′}. Comparison is made with possible
melting temperatures to eliminate cases unlikely to represent Earth's
interior. The limiting case of *B*^{′′} = 0 is unexpected but is useful for
comparisons. Importantly, our geotherms do not utilize data on thermal
conductivity and thus are essentially independent of mineralogy. In
contrast, flux and power are independent of the reference temperature, but
use *κ* and thus are affected by LM mineralogy. However, as *κ*
varies little at high *T* (Fig. 3), only the proportion of complex silicates to
simple oxides matters.

## 4.1 Calculation of temperatures across the lower mantle from PREM and a reference point

Geotherms are calculated by integrating the thermal gradients (Fig. 6) which
were obtained from PREM, from 871 km downwards, using various *B*^{′′} values (Table 1).
Results (Fig. 8) consider three values for the 871 km reference temperature
(*T*_{ref}) representing the top of the lower mantle. The minimum *T*_{ref} of 1500 K is based on temperatures for basalt extruding at the surface (e.g.,
Falloon et al., 2008), whereas the maximum *T*_{ref} of 2500 K is based on
dry melting of peridotite at 670 km (Zhang and Herzberg, 1994). The
intermediate (*T*_{ref} = 2000 K) corresponds to Takahashi's (1986)
melting curve of peridotite at ∼ 410 km, which probably
involved tiny amounts of moisture, based on sensitivity of melting to
hydration.

All geotherms (Fig. 8) are flat near 871 km, due to PREM derivatives being linear for the shallowest LM (Fig. 6). Thus, the inverse method suggests that temperatures change little from 871 km up to the shallower depth of the 671 km seismic discontinuity. Chemical composition could be changing near 670 km, as descending slabs are resolved from earthquakes down to these depths but not below (summary figures are in Hofmeister, 2020, chap. 7, and in Hofmeister et al., 2022).

Regarding the base of the LM, the geotherms smoothly decrease with depth.
Extrapolation of LM results from 2741 to 2891 km presumes similar bulk
moduli derivatives for *D*^{′′} and the LM. Although this seems unlikely, since
reaction with core material is possible, discontinuities at these depths
preclude robust analysis (Sect. 2), and modeled velocities in this region
are less certain than in the LM (e.g., Kennett et al., 1995). But, as
discussed in Sect. 2.3, as evidenced by compilations and recent work,
derivatives of *B* vary little with structure, composition, and bond type.

Melting curves of Fe–S and Fe–Ni–S systems (Chudinovskikh and Boehler, 2007;
Morard et al., 2011; Mori et al., 2017) set a minimum of ∼ 2750 K at 2891 km, whereas peridotite melting (Fiquet et al., 2010) sets a
maximum near 4100 K (Fig. 8). Only for the combination of the highest
*T*_{ref} and the limiting case of *B*^{′′} = 0 is peridotite melting reached.
Many, but not all, geotherms exceed the Fe–S eutectic. For example, if
*T*_{ref} = 2000 K, then $\left|{B}^{\prime \prime}\right|$ must be smaller than 0.0075 GPa^{−1}. Table 1 lists the range of reference temperatures consistent
with the above-mentioned phase equilibria for each *B*^{′′} value considered. Only
for the hottest possible *T*_{ref} = 2500 K can *B*^{′′} reach −0.01 GPa^{−1}. Small second derivatives are consistent with difficulty in
resolving these in experiments on dense materials, unless extreme pressures
are reached (see Zha et al., 2000) and polynomial fits are used (Sect. 2.3).

Irrespective of the temperature values, the shape of the geotherms require a
thermal maximum inside the lower mantle. For large magnitudes of *B*^{′′}, a maximum
is indicated roughly near 670 km or slightly shallower. For the smallest
$\left|{B}^{\prime \prime}\right|$, the maximum *T* in the lower mantle is reached at its
interface with the core. In all cases consistent with phase equilibria,
maximum LM temperatures are reached for *z* below 2200 km. Thus, from analyzing
PREM curves, LM temperatures climb inwards for most cases considered. A heat
source located in the LM is supported by flux and power calculations, below.

Altering our input value for $\partial B/\partial T$ from −0.026 GPa K^{−1} will either expand or contract the splayed patterns of Fig. 8,
which rest on thermal gradients of Fig. 6 obtained from Eq. (7). Comparison
of such revised curves with phase equilibria would then change the ranges of
*B*^{′′} and *T*_{ref} that avoid melting in the LM while allowing melting in the
core (Table 1, RHS), resulting in quite similar shapes. Thus, geotherms from
PREM are robust, given the subsidiary information on melting relations,
whereas the specific input values are interdependent. The shapes of the
geotherms are compatible with the process of heat diffusion, i.e., thermal
conduction, even though the calculations (depicted in Table 1 and Figs. 6
and 8) did not incorporate thermal conductivity values.

## 4.2 Calculation of flux across the lower mantle from PREM and *κ*

Flux, defined by Eq. (4), describes spherical geometry for radially changing
*T*. Temperature values are not needed to compute the amount of heat being
moved inside and across the LM, when *κ* only weakly depends on *T*. The
gradients of Fig. 6 and Table 1 lead to families of flux vs. depth curves
which depend on *B*^{′′} values for each of low and high *κ*. The sign
convention used here portrays heat from the center of a sphere moving
outwards.

All cases (Fig. 9) provide a broad peak for ℑ across the lower mantle.
For comparison, surface flux values are larger, averaging ∼ 60 mW m^{−2} for either crust (Veiera and Hamza, 2018). The height of the
peak in ℑ increases as *B*^{′′} approaches its null limit. Only for this
limiting case (*B*^{′′} = 0) is flux within *D*^{′′} large and positive, the behavior of which signifies that heat emitted from the core contributes to the flux
in *D*^{′′} and at large *z* in the LM. For the next larger value of *B*^{′′}, flux near *D*^{′′}
is positive but near zero. Thus, the vast majority of our calculations
point to a lower mantle source whose heat is being shed to its adjacent
layers. Section 4.3 provides further discussion.

Note that the curvature and a maximum in ℑ are inherent to PREM (Figs. 4 to 6; Sect. 2.3). The increase in thermal conductivity with *P* serves to
flatten these curves at great depths near the outer core and, importantly, means
that the efficiency of heat conduction increases with depth. Consequently,
heat from a source deep in the Earth is conveyed more readily inwards than
outwards.

How much heat is carried depends on the material, i.e., on the high-*T* value at ambient pressure. Our model (Fig. 7) rests on heat transport
values that are established via measurements. The thermal response at modest
temperatures (*<* 2000 K) of any given material is largely controlled
by infrared fundamentals and near-IR overtones, whose frequencies overlap
with the associated blackbody curve (Hofmeister, 2019, chap. 11). We have
not accounted for electronic transitions of Fe^{2+} augmenting heat
transfer significantly above ∼ 1000 K, as observed in various
glasses (Sehlke et al., 2020, and references therein), since the chemical
composition of the LM is not known. The purpose of this paper is to
ascertain Earth's thermal state with minimal assumptions and input
parameters. As accuracy is not possible given the scant definitive
information on LM mineralogy, such as samples, salient features not affected
by the details are pursued here.

Nonetheless, enhancements in thermal conductivity with depth would raise the
flux near 2471 km and straighten the curves, resulting in melting in the
deepest lower mantle. We suggest that *κ* cannot be significantly
larger than that considered in Fig. 7 or the boundary layer *D*^{′′} would be
significantly larger and mostly molten, which contradicts observations of
shear waves in this region (cf. behavior of velocities in the molten outer
core from PREM, Fig. 5, to those in the solid layers).

## 4.3 Calculation of power across the lower mantle from PREM and *κ*

In spherical coordinates, power is provided by

For all parameters explored, ℘ has a broad peak in the lower mantle (Fig. 10). The depth where the maximum ℘ occurs (Table 1) points to the location of a heat source. This source is in the lower mantle, unless $\left|{B}^{\prime \prime}\right|$ is larger than considered, which suggests a location near or in the TZ.

Power is supplied to the LM from the core only for the case of high *κ* with quite small $\left|{B}^{\prime \prime}\right|$, both of which are unexpected. For
this unlikely situation to exist, the core would also need to have a heat
source with a sufficiently large associated outward flux to overtake the
flux inwards from the LM source. Here, we pursue a simple explanation,
consistent with realistic input parameters, that a source in the LM supplies
heat to the core region. In detail, half of the four curves require *B*^{′′} = 0,
which is unexpected, and a third curve points to negligible power from
the core. The fourth case of an oxide (high *κ*) with low *B*^{′′} = −0.0025 GPa^{−1} is probably incompatible parameters. This inference is
consistent with large $\left|{B}^{\prime \prime}\right|$ being associated with compressible
solids like salts (e.g., Bass, 1995). Periclase, which has been viewed as a
lower mantle phase, is much more compressible than silicates with
perovskite-type structures, considered to dominate the LM (see Sect. 2.3).
If the mantle is mostly composed of dense silicates, as is the current view,
the LM is an inefficient heat transmitter and stays hot where the heat is
produced.

Heat conveyed from the lower mantle to the core could cause melting. The
latent heat of ∼ 450 kJ kg^{−1} for iron melting at high *P*
and *T* (Aitta, 2006) is close to latent heats for basalt and many other
materials. A constant rate of melting over geologic time requires 5.8 TW to
make the outer core. This value is most compatible with the case of low
*κ* (silicate) and *B*^{′′} = −0.010 GPa^{−1}. If the source is winding
down with time, as is likely, then *B*^{′′} should be more positive. The case of low
*κ* (silicate) and *B*^{′′} = −0.0075 GPa^{−1} is compatible with a wide
range of temperatures for the top of the LM and a sulfide-rich core, which
addresses the expected sulfur concentration from meteoritic models (see
tables in Lodders and Fegley, 1999). These parameters compose our best
estimate of the thermal gradient, geotherm, flux, and power, while
suggesting that heating occurs near *z* = 1900 km (Table 1).

Earth's largest zone by mass or volume is the lower mantle, which has only
been accessed remotely, via seismic data acquisitions and processing. PREM,
Ak135, and other reference models provide nearly identical velocities
(Kennett et al., 1995). The models yield smoothly varying properties and
their derivatives over most of the LM, the changes of which are taken in this
report to represent combined effects of *P* and *T* varying with depth. However, a
smooth variation in chemical composition is not precluded in utilizing
separation of variables (Sect. 2). Gradual changes in chemical composition
could be hidden in our choices for *B*^{′′} if changes in mineralogy lead to a
linear response of $\partial B/\partial P$. Specifically, mineralogy
depending linearly on density and thus on *P* would lead to an equation
equivalent to Eq. (6). Indeed, systematic dependence of velocity (and thus
of *B*) on density exists and has received much attention (see, e.g., the seminal
paper of Shankland, 1972). But because $\partial B/\partial P$ varies little
among dense phases, as underscored by commonly assuming harmonic values of 4
in analyzing data (see Sect. 2.3), geotherms calculated from PREM via the
inverse approach developed here are largely independent of mineralogical
details. Values considered for *B*^{′′} depict all three different thermal situations
that are possible below 670 km: LM temperatures can increase with depth
below the TZ, or are constant, or may decrease. Only one case was considered
for the last, due to strongly decreasing *T* being incompatible with a molten
outer core (Fig. 8).

The thermal state and gradients inward from the top of the LM indicate that
a power source exists in the lower mantle. Locations of the peak in ℘
or flux are little affected by the high-*T* values of thermal conductivity
considered in the inverse model: instead, the location of the power source
is largely controlled by *B*^{′′}. As discussed in Sect. 2.3, *B*^{′′} is not
well constrained. This ambiguity effectively lumps compositional changes
with those solely due to compression. Contrastingly, the heights of the
peaks in Figs. 9 and 10 are affected by values for *κ* and for
$\partial B/\partial T$. Neither variable depends strongly on mineralogy.
Note that the large contrast (in *κ*) explored considers vastly
different simple oxide vs. complex silicate compositions.

Notably, thermal models show that the scale length for cooling over geologic
time is ∼ 1000 km (Criss and Hofmeister, 2016), so heat
generated in the LM cannot reach the surface to any appreciable degree. A
similar scale length is obtained from Eq. (2): the Earth cools slowly
because it is large and spherical with a refractory outside. Flat geotherms
near *z* = 1000 km with a source near 2000 km are compatible with the
thermally insulating nature of rocks and our earlier cooling models. Criss
and Hofmeister (2016) used a constant *κ* that is independent of
pressure and so assumed a much more thermally insulating mantle. The flat
and low *κ* case underestimates ℑ in the lower mantle. The
consequence is a narrower thermal maxima with little heating of the core in
the calculations of Criss and Hofmeister (2016). The broad geotherms
inferred from PREM without knowing thermal conductivity are thus consistent
with the strong dependence of thermal conductivity on pressure that is
demonstrated by experiments (Fig. 2) and, importantly, is inherent to
Fourier's definition of ℑ (Sect. 3.3).

In short, results (Figs. 6, and 8 to 10) obtained by analyzing the seismologic representation of Earth's interior are consistent with studies of the equation of state, phase equilibria, and thermal transport properties. Table 1 lists the most likely thermal gradient for the parameters explored. Although other values are possible, the mostly likely gradient is unlikely to change significantly, as likelihood was deduced from melting temperatures for LM and core candidate materials.

Heat is produced in the LM, but how? Commonly considered sources are discussed next. The paper concludes with a proposal.

Heat produced in rocks by radioactive decay has been the focus of many
studies. Yet, it is well known that U, Th, and K are concentrated in the
continents, leaving very little for the mantle, if meteorites represent the
bulk Earth. Flux from the oceans suggests mantle production of <100 W km^{−3}. Such a tiny source can only heat the interior if deeply buried,
but for this case excessively high *T* results, due to the time evolution of
the heat-generating isotopes (Criss and Hofmeister, 2016).

This long-standing problem in geochemistry led to considering primordial heat as another source. This hypothesis is based on gravitational contraction producing heat and rests on Kelvin's discounted proposal for the generation of starlight. Changes in gravitational potential produce motions as per elementary physics textbooks. Conversion of gravitational potential to spin and orbits quantitatively accounts for the high kinetic energy for Earth and sister planets today, as well as high spin observed for young stars (see Hofmeister and Criss, 2012). Some accretionary heating is expected in the final stages, but this source is a small fraction of the gravitational potential, non-renewable, shallow, and winding down. Likewise, core formation is not a source of heat: rather, the planet would need to be already melted for a homogeneously accreted object to sort, since self-compression itself provides a stable density stratification.

Heat is a by-product of motions, when accompanied by deformation, non-elastic in particular, or friction. Motions are produced by forces which on large, planetary scales are gravitational in origin. On this basis, and because the previously explored sources of radiogenic and primordial heat are inadequate to describe the workings of Earth (e.g., the hypothesis of mantle convection: see Bercovici, 2007) as well as seismologic detection of a molten core, which is hot, our recent efforts have focused on forces and motions. Hofmeister et al. (2022) argue that the location of the barycenter, where the immense solar pull and orbital centrifugal forces balance, differing from that of the geocenter results in imbalanced stresses and forces that are cyclical with periods of both 1 d and 1 month, with plate tectonics being a consequence. Cyclic stresses promote failure (Schijve, 2009). Spin is important, as the force field is axially symmetric, which explains orientation of the mid-ocean ridge fracture system.

The barycenter is a point in space that the Earth spins through. Its
location relative to the geocenter is defined by masses of the Earth and
Moon plus the lunar distance, which varies over the month. The depth range
of ∼ 1450 to 2050 km, shown in Figs. 8 to 10, includes the
position of the power source indicated by *B*^{′′} more negative than −0.005 GPa^{−1}. Over the day, this point in space moves through the LM and
rarely lies in the equatorial plane, due to Earth's tilted spin axis. Thus, our results
for Earth's thermal state are compatible with cyclical stresses heating the
LM. This region is strong and plastic (or elastic) rather than brittle, like
the lithosphere, but both are underlain by fluid layers. Liquids flow under
any stress, the lack of rigidity of which adds stress to the overlying layers. The
amount of heat generated is small, ∼ 1 TW, and is conducted
away from the source in both directions. Investigating this proposal further
is beyond the scope of the present report.

No data sets were used in this article.

The contact author has declared that there are no competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Probing the Earth: experiments and mineral physics at mantle depths”. It is a result of the 17th International Symposium on Experimental Mineralogy, Petrology and Geochemistry, Potsdam, Germany, 1–3 March 2021.

I thank Robert E. Criss for critical comments.

This research has been partially supported by the National Science Foundation (grant no. EAR-2122296).

This paper was edited by Stephan Klemme and reviewed by two anonymous referees.

Aitta, A.: Iron melting curve with a tricritical point, J. Stat. Mech.-Theory E., 12, P12015, https://doi.org/10.1088/1742-5468/2006/12/P12015, 2006.

Aizawa, Y., Yoneda, A., Katsura, T., Ito, E., Saito, T., and Suzuki, I.:
Temperature derivatives of elastic moduli of MgSiO_{3} perovskite, J.
Geophys. Res., 31, L01602, https://doi.org/10.1029/2003GL018762, 2004.

Akaogi, M. and Ito, E.: Heat capacity of MgSiO_{3} perovskite, Geophys.
Res. Lett., 20, 105–108, 1993.

Anderson, D. L.: New Theory of the Earth, 2nd Edn., Cambridge University Press, Cambridge, ISBN 0-86542-335-0, 2007.

Anderson, O. L. and Isaak, D.: Elastic constants of mantle minerals at high temperatures, in: Mineral Physics and Crystallography, A Handbook of Physical Constants, edited by: Ahrens, T. J., American Geophysical Union, Washington D.C., ISBN 0-87590-852-7, 64–97, 1995.

Anderson, O. L., Isaak, D., and Oda, H.: High-temperature elastic constant data on minerals relevant to geophysics, Rev. Geophys., 30, 57–90, 1992.

Andersson, P.: Thermal conductivity under pressure and through phase transitions in solid alkali halides, I. Experimental results for KCl, KBr, KI, RbCl, RbBr and RbI, J. Phys. C Solid State, 18, 3943–3955, 1985.

Andersson, S. and Bäckström, G.: Techniques for determining thermal conductivity and heat capacity under hydrostatic pressure, Rev. Sci. Instrum., 57, 1633–1639, 1986.

Andersson, S. and Dzhavadov, L.: Thermal conductivity and heat capacity of
amorphous SiO_{2}: Pressure and volume dependence, J. Phys. Condensed
Matter., 4, 6209–6216, 1992.

Bass, J. D.: Elasticity of minerals, glasses, and melts, in: Mineral Physics and Crystallography, A Handbook of Physical Constants, edited by: Ahrens, T. J., American Geophysical Union, Washington D.C., 29–44, ISBN 0-87590-852-7, 1995.

Bercovici, D.: Mantle Dynamics Past, Present and Future: An Introduction and Overview, in: Treatise on Geophysics, Vol. 7, edited by: Schubert, G., 1–30, ISBN 9780444534569, 2007.

Blumm, J., Henderson, J. B., Nilson, O., and Fricke, J.: Laser flash measurement of the phononic thermal diffusivity of glasses in the presence of ballistic radiative transfer, High Temp.-High Pres., 34, 555–560, 1997.

Branlund, J. M. and Hofmeister, A. M.: Thermal diffusivity of quartz to 1000 ^{∘}C: Effects of impurities and the *α*-*β* phase
transition, Phys. Chem. Miner.., 34, 581–595, 2007.

Cammarano, F., Goesa, S., Vacher, P., and Giardini, D.: Inferring upper-mantle temperatures from seismic velocities, Phys. Earth Planet. In., 138, 197–222, 2003.

Chase Jr., M. W.: NIST-JANAF Thermochemical Tables, Fourth Edition, J. Phys. Chem. Ref. Data Monogr., 9, 1–1951, 1998.

Chopelas, A. and Boehler, R.: Thermal expansivity in the lower mantle, Geophys. Res. Lett., 19, 1983–1986, 1992.

Chudinovskikh, L. and Boehler, R.: Eutectic Melting in the System Fe–S to 44 GPa, Earth Planet. Sc. Lett., 257, 97–103, 2007.

Criss, R. E. and Hofmeister, A. M.: Conductive cooling of spherical bodies with emphasis on the Earth, Terra Nova, 28, 101–109, 2016.

Criss, E. M. and Hofmeister, A. M.: Isolating lattice from electronic contributions in thermal transport measurements of metals and alloys and a new model, Int. J. Modern Phys. B, 31, 75 pp., 2017.

Ditmars, D. A., Ishihara, S., Chang, S. S., Bernstein, G., and West, E. D.: Enthalpy and
Heat-Capacity Standard Reference Material: Synthetic Sapphire (*α*-A1203) from 10 to 2250 K, J. Res. Natl. Bur. Standards, 87, 159–163, 1982.

Dorfman, S. M., Meng, Y., Prakapenka, V. B., and Duffy, T. S.: Effects of
Fe-enrichment on the equation of state and stability of (Mg, Fe)SiO_{3}
perovskite, Earth Planet. Sc. Lett., 361, 249–257, 2013.

Dugdale, J. S. and MacDonald, D. K. C.: Lattice thermal conductivity, Phys. Rev., 98, 1751–1752, 1955.

Dziewonski, A. M. and Anderson, D. L.: Preliminary reference Earth model, Phys. Earth Planet. In., 25, 297–356, 1981.

Falloon, T. J., Green, D. H., Danyushevsky, L. V., and McNeill, A. W.: The composition of near-solidus partial melts of fertile peridotite at 1 and 15 GPa: implications for the petrogenesis of MORB, J. Petrol., 49, 591–613, 2008.

Fei, Y.: Thermal Expansion, in: Mineral Physics and Crystallography, A Handbook of Physical Constants, edited by: Ahrens, T. J., American Geophysical Union, Washington D.C., 29–44, ISBN 0-87590-852-7, 1995.

Fiquet, G., Richet, P., and Montagnac, G.: High-temperature thermal expansion of lime, periclase, corundum and spinel, Phys. Chem. Mineral., 27, 103–111, 1997.

Fiquet, G., Auzenda, A. L., Siebert, J., Corgne, A., Bureau, H., Ozawa, H., and Garbarino, G.: Melting of peridotite to 140 Gigapascals, Science, 329, 1516–1518, 2010.

Hahn, O., Hofmann, R., Raether, F., Mehling, H., and Fricke, J.: Transient heat transfer in coated diathermic media: a theoretical study, High Temp.-High Press., 29, 693–701, 1997.

Helffrick, G.: A finite strain approach to thermal expansivity's pressure dependence, Am. Mineral., 102, 1690–1695, 2017.

Hofmeister, A. M.: Interatomic Potentials Calculated from Equations of
State: Limitation of Finite Strain to Moderate K^{′}, Geophys. Res. Lett., 20,
635–638, 1993.

Hofmeister, A. M.: Comment on “measurement of thermal diffusivity at high pressure using a transient heating technique” in Appl. Phys. Lett. 91, 181914, 2007, Appl. Phys. Lett., 95, 096101, https://doi.org/10.1063/1.3196374, 2009.

Hofmeister, A. M.: Thermal diffusivity of perovskite-type compounds at elevated temperature, J. Appl. Phys., 107, 103532, https://doi.org/10.1063/1.3371815, 2010a.

Hofmeister, A. M.: Scale aspects of heat transport in the diamond anvil cell, in spectroscopic modeling, and in Earth's mantle, Phys. Earth Planet. In., 180, 138–147, 2010b.

Hofmeister, A. M.: Measurements, Mechanisms, and Models of Heat Transport, Amsterdam, New York, 427 pp., ISBN 978-0-12-809981-0, 2019

Hofmeister, A. M.: Heat Transport and Energetics of the Earth and Rocky Planets, Elsevier, Amsterdam, 350 pp., ISBN 978-0-12-818430-1, 2020.

Hofmeister, A. M.: Dependence of Heat Transport in Solids on Length-scale, Pressure, and Temperature: Implications for Mechanisms and Thermodynamics, Materials, 14, 449, https://doi.org/10.3390/ma14020449, 2021.

Hofmeister, A. M. and Criss, R. E.: A thermodynamic model for formation of the Solar System via 3-dimensional collapse of the dusty nebula, Planet. Space Sci., 62, 111–131, 2012.

Hofmeister, A. M., Dong, J. J., Branlund, J. M.: Thermal diffusivity of electrical insulators at high temperatures: evidence for diffusion of phonon-polaritons at infrared frequencies augmenting phonon heat conduction, J. Appl. Phys., 115, 163517, https://doi.org/10.1063/1.4873295, 2014.

Hofmeister, A. M., Criss, R. E., and Criss, E. M.: Link of planetary energetics to moon size, orbit, and planet spin: a new mechanism for plate tectonics, in: In the Footsteps of Warren B. Hamilton: New Ideas in Earth Science: Geological Society of America Special Paper 553, edited by: Foulger, G. R., Hamilton, L. C., Jurdy, D. M., Stein, C. A., Howard, K. A., and Stein, S., Geological Society of America, Boulder, CO, https://doi.org/10.1130/2021.2553(18), 2022.

Hsieh, W. P., Chen, B., Li, J., Keblinski, P., and Cahill, D. G.: Pressure tuning of the thermal conductivity of the layered muscovite crystal, Phys. Rev. B, 80, 180302, https://doi.org/10.1103/PhysRevB.80.180302, 2009.

Jacobsson, P. and Sundqvist, B.: Thermal conductivity and Lorenz function of zinc under pressure, Int. J. Thermophys., 9, 577–585, 1988.

Kanamori, H., Fujii, N., and Mizutani, H.: Thermal diffusivity of rock-forming minerals, J. Geophys. Res., 73, 595–605, 1968.

Katsura, T.: Thermal diffusivity of silica glass at pressures up to 9 GPa, Phys. Chem. Miner., 20, 201–208, 1993.

Kennett, B. L. N.: On seismological reference models and the perceived nature of heterogeneity, Phys. Earth Planet. Interiors, 159, 129–139, 2006.

Knittle, E.: Static compression measurements of equations of state, in: Mineral Physics and Crystallography: A Handbook of Physical Constants, edited by: Ahrens, T. J., American Geophysical Union, Washington D.C., 98–142, ISBN 0-87590-852-7, 1995.

Konôpková, Z., McWilliams, R. S., Gómez-Pérez, N., and Goncharov, A. F.: Direct measurement of thermal conductivity in solid iron at planetary core conditions, Nature, 534, 99–101, 2016.

Lodders, K. and Fegley Jr., B. J.: The Planetary Scientist's Companion, Oxford University Press, Oxford, ISBN 0-19-511694-1, 1998.

MacDonald, G. J. F.: Calculations on the thermal history of the Earth, J. Geophys. Res., 64, 1967–2000, 1959.

McWilliams, R. S., Konôpková, Z., and Goncharov, A. F.: A flash heating method for measuring thermal conductivity at high pressure and temperature: Application to Pt, Phys. Earth Planet. Int., 174, 24–32, 2015.

Merriman, J. M., Hofmeister, A. M., Whittington, A. G., and Roy, D. J.: Temperature-dependent thermal transport properties of carbonate minerals and rocks, Geophere, 27, 27 pp., https://doi.org/10.1130/GES01581.1, 2018.

Morard, G., Andrault, D., Guignot, N., Siebert, J., and Garbarino, G.: Melting of Fe–Ni–Si and Fe–Ni–S alloys at megabar pressures: implications for the core–mantle boundary temperature, Phys. Chem. Miner., 38, 767–776, 2011.

Mori, Y., Ozawa, H., Hirose, K., Sinmyo, R., Tateno, S., Morard, G., and Ohishi,
Y.: Melting experiments on Fe–Fe_{3}S system to 254 GPa, Earth Planet.
Sc. Lett., 464, 135–141, 2017.

Murakami, M., Ohishi, Y., Hirao, N., and Hirose, K.: Elasticity of MgO to 130 GPa: Implications for lower mantle mineralogy, Earth Planet. Sc. Lett., 277, 123–129, 2009.

Nestola, F., Burnham, A., Peruzzo, L., Tauro, L., Alvaro, M., Walter, M., and Kohn, S.: Tetragonal Almandine-Pyrope Phase, TAPP: Finally a name for it, the new mineral jeffbenite, Mineral. Mag., 80, 1219–1232, 2016.

Nestola, F., Jung, H., and Taylor, L. A.: Mineral inclusions in diamonds may be synchronous but not syngenetic, Nat. Commun., 8, 14168, https://doi.org/10.1038/ncomms14168, 2017.

Osako, M. and Ito, E.: Thermal diffusivity of MgSiO_{3} perovskite,
Geophys. Res. Lett., 18, 239–242, 1991.

Osako, M., Ito, E., and Yoneda, A.: Simultaneous measurements of thermal conductivity and thermal diffusivity for garnet and olivine under high pressure, Phys. Earth Planet., 143/144, 311–320, 2004.

Pertermann, M. and Hofmeister, A. M.: Thermal diffusivity of olivine-group minerals, Am. Mineral., 91, 1747–1760, 2006.

Ritzwoller, H. and Lavely, E. M.: Three-dimensional seismic models of the Earth's mantle, Rev. Geophys., 33, 1–66, https://doi.org/10.1029/94RG03020, 1995.

Sehlke, A., Hofmeister, A. M., and Whittington, A. G.: Thermal properties of glassy and molten planetary candidate lavas, Planet. Space Sci., 193, 105089, https://doi.org/10.1016/j.pss.2020.105089, 2020.

Shankland, T. J.: Velocity-density systematics: Derivation from Debye theory and the effect of ionic size, J. Geophys. Res., 77, 3750–3758, https://doi.org/10.1029/JB077i020p03750, 1972.

Schijve, J.: Fatigue of Structures and Materials, 2nd Edn. with Cd-Rom, Springer, Berlin/Heidelberg, Germany, ISBN 978-1-4020-6807-2, 2009.

Sinogeikin, S. V. and Bass, J. D.: Single-crystal elasticity of MgO at high pressure, Phys. Rev. B, 59, R14141, https://doi.org/10.1103/PhysRevB.59.R14141, 1999.

Spinner, S.: Elastic moduli of glasses at elevated temperature by a dynamic method, J. Am. Ceram. Soc., 39, 113–118, 10.1111/j.1151-2916.1956.tb15634.x, 2006.

Stachel, T., Harris, J. W., Brey, G. P., and Joswig, W.: Kankan diamonds (Guinea): II. Lower mantle inclusion parageneses, Contrib. Min. Petrol., 140, 16–27, 2000.

Takahashi, E.: Melting of a dry peridotite KLB-1 up to 14 GPa: Implications on the origin of peridotitic upper mantle, J. Geophys. Res., 91, 9367–9382, 1986.

Takeuechi, H., Uyeda, S., and Kanamori, H.: Debate About the Earth. Freeman, Cooper and Co, San Francisco, chap. 6, Is the Earth heating or cooling?, 253 pp., 1967.

Tye, R. P. (Ed.): Thermal Conductivity, Vol. 1–2, Academic Press, London, ISBN 10 0127054014, ISBN 13 9780127054018, 1969.

Vieira, F. and Hamza, V.: Global heat flow: new estimates using digital maps and GIS techniques, Int. J. Terr. Heat Flow Appl. Geotherm., 1, 6–13, 10.31214/ijthfa.v1i1.6, 2018.

Vozár, L. and Hohenauer, W.: Flash method of measuring the thermal diffusivity, A Review, High Temperatures-High Pressures, 35/36, 253–264, https://doi.org/10.1068/htjr119, 2003.

Wallace, D. C.: Thermodynamics of Crystals, John-Wiley and Sons Inc., New York, ISBN 13 978-0486402123, ISBN 10 0486402126, 1972.

Yoneda, A.: Pressure derivatives of elastic constants of single crystal MgO
and MgAl_{2}O_{4}, J. Phys. Earth, 38, 19–55, 1990.

Zha, C. S., Mao, H. K., and Hemley, R. S.: Elasticity of MgO and a primary pressure scale to 55 GPa, P. Natl. Acad. Sci. USA, 97, 13494–13499, 2000.

Zhang, J. and Herzberg, C.: Melting experiments on anhydrous peridotite KLB-1 from 5.0 to 22.5 GPa, J. Geophys. Res., 99, 17729–17742, 1994.

Zhao, D., Qian. X., Gu, X., Jajja, S. A., and Yang, R.: Measurement techniques for thermal conductivity and interfacial thermal conductance of bulk and thin film materials, J. Electron. Packag., 138, 040802, https://doi.org/10.1115/1.4034605, 2016.

Zhu, F., Liu, J., Lai, X., Xiao, Y., Prakapenka, V., Bi, W., Alp, E., Przemyslaw, D., Chen, B., and Li, L.:
Synthesis, elasticity, and spin state of an intermediate MgSiO_{3}-FeAlO_{3}
bridgmanite: Implications for iron in Earth's lower mantle, J.
Geophys. Res.-Sol. Ea., 125, e2020JB019964, https://doi.org/10.1029/2020JB019964, 2020.

- Abstract
- Introduction and background
- Extraction of geothermal gradients from radial seismological models in an inverse approach
- Lower mantle transport properties from theory and experiment
- Results
- Implications and conclusions
- Data availability
- Competing interests
- Disclaimer
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References

- Abstract
- Introduction and background
- Extraction of geothermal gradients from radial seismological models in an inverse approach
- Lower mantle transport properties from theory and experiment
- Results
- Implications and conclusions
- Data availability
- Competing interests
- Disclaimer
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References