Spherical harmonic analysis of earth’s conductive heat flow

Abstract

A reappraisal of the international heat flow database has been carried out and the corrected data set was employed in spherical harmonic analysis of the conductive component of global heat flow. Procedures used prior to harmonic analysis include analysis of the heat flow data and determination of representative mean values for a set of discretized area elements of the surface of the earth. Estimated heat flow values were assigned to area elements for which experimental data are not available. However, no corrections were made to account for the hypothetical effects of regional-scale convection heat transfer in areas of oceanic crust. New sets of coefficients for 12° spherical harmonic expansion were calculated on the basis of the revised and homogenized data set. Maps derived on the basis of these coefficients reveal several new features in the global heat flow distribution. The magnitudes of heat flow anomalies of the ocean ridge segments are found to have mean values of less than 150 mW/m². Also, the mean global heat flow values for the raw and binned data are found to fall in the range of 56–67 mW/m², down by nearly 25% compared to the previous estimate of 1993, but similar to earlier assessments based on raw data alone. To improve the spatial resolution of the heat flow anomalies, the spherical harmonic expansions have been extended to higher degrees. Maps derived using coefficients for 36° harmonic expansion have allowed identification of new features in regional heat flow fields of several oceanic and continental segments. For example, lateral extensions of heat flow anomalies of active spreading centers have been outlined with better resolution than was possible in earlier studies. Also, the characteristics of heat flow variations in oceanic crust away from ridge systems are found to be typical of conductive cooling of the lithosphere, there being little need to invoke the hypothesis of unconfined hydrothermal circulation on regional scales. Calculations of global conductive heat loss, compatible with the observational data set, are found to fall in the range of 29–34 TW, nearly 25% less than the 1993 estimate, which rely on one-dimensional conductive cooling models.

Appendices

Appendix 1

Harmonic coefficients and the normalization procedure

The harmonic representation of heat flow (q) in near surface layers is usually expressed as:

$$ q{\left({\theta, \phi} \right)} = {\sum\limits_{n = 0}^N}{\sum\limits_{m = 0}^n {{\left[ {A_{{nm}} \cos {\left({m\phi} \right)} + B_{{nm}} \sin {\left({m\phi} \right)}} \right]}}}{P}_{nm}^{\prime} {\left({\cos \theta} \right)} $$

(4)

where ϕ is the longitude θ = 90 − ψ, is the colatitude, P′ _nm (cos θ) is the associated Legendre function that is fully normalized and A _nm and B _nm the coefficients of the harmonic expansion. The expression for evaluation of P′ _nm is:

$$ P_{nm} = \frac{P_{nm}}{\sqrt{K_{n}^{m}}} $$

(5)

where P _nm is the associated Legendre function given by:

$$ P_{nm} (\cos \theta) = \frac{\sin^{m} \theta}{2^{n}}{\sum\limits_{t = 0}^{Int\left(\frac{n - m}{2}\right)}} \frac{(- 1)^{t} (2n - 2t)!}{t!(n - t)!(n - m -2t)!}\cos^{(n - m - 2t)} \theta $$

(6)

and

$$ K_{n}^{m} = \frac{1}{{H(2n + 1)}}\frac{{(n + m)!}}{{(n - m)!}},\left\{{\begin{array}{*{20}c} {{\rm if\,m = 0 \Rightarrow H = 0}} \\ {{\rm if\,m \ne 0 \Rightarrow H = 2}} \\ \end{array}} \right. $$

(7)

In Eq. (6) Int[(n−m)/2] refers to the largest integer that is lower than (n−m)/2.

Full normalization of associated Legendre functions (P _nm ) requires that the following equations be satisfied:

$$ {\int\limits_0^{2\pi}}{\int\limits_0^\pi {{\left[{{P}_{nm}^{\prime} (\cos \theta)\sin (m\phi)} \right]}^{2} \sin \theta \, \rm d\theta \, \rm d\phi}} = 4\pi $$

(8a)

$$ {\int\limits_0^{2\pi}}{\int\limits_0^\pi {{\left[ {{P}_{nm}^{\prime} (\cos \theta)\cos (m\phi)} \right]}^{2} \sin \theta\, \rm d\theta \,\rm d\phi}} = 4\pi $$

(8b)

The coefficients A _nm and B _nm are evaluated by fitting the harmonic expansion to the set of experimental data, which are the heat flow values (q) and their respective geographic coordinates (ϕ and θ).

Appendix 2

Relation between degree of harmonic expansion and spatial resolution

A discrete distribution of N data points over the surface of the earth at a specific latitude (θ ₀) and spaced at an angular distance of ΔΦ follow the relation:

$$ N = \frac{2\pi}{\Updelta \phi} $$

(9)

A function ${\overline{q}} (\theta_{\rm 0}, \phi)$ which represents heat flow at a specific latitude θ ₀, but continuous in ϕ between 0 and 2π may be represented as Fourier sine series:

$$ {\overline{q}} (\theta_{o}, \phi) = {\sum\limits_{m = 0}^\infty}{\left({a_{m} (\theta_{o})\cos m\phi + b_{m} (\theta_{o})\sin m\phi} \right)} $$

(10)

The coefficients a _m (θ ₀) and b _m (θ ₀) are calculated using discretized values experimental heat flow (q(ϕ)) uniformly distributed over the surface of the earth with angular spacing of Δφ. The expressions for the calculation of the coefficients are:

$$ a_{m} (\theta_{0}) = \frac{2}{N}{\sum\limits_{t = 1}^N}q_{t} (\phi)\cos \left(\frac{2\pi mt}{N}\right) $$

(11)

$$ b_{m} (\theta_{0}) = \frac{2}{N}{\sum\limits_{t = 1}^N}q_{t} (\phi)\sin \left(\frac{2\pi mt}{N} \right) $$

(12)

Obviously when ${\overline{q}} (\phi) = q(\phi)$ Eq. (10) will represent the exact value of the discretized heat flux. This condition allows determination of the “m” to be used in the expansion. The result may be obtained as follows:

• 1. Substitute Eqs. (11) and (12) in Eq. (10):

  $$ {\overline{q}} (\theta_{\rm 0}, \phi) = {\sum\limits_{m = 0}^{\infty}} {\left(\frac{2}{N}{\sum\limits_{t = 1}^N}q_{t} (\phi)\cos \left(\frac{2\pi mt}{N}\right)\cos (m\phi) +\frac{2}{N}{\sum\limits_{t = 1}^N} q_{t} (\phi)\sin \left(\frac{2\pi mt}{N}\right)\sin (m\phi) \right)} $$

  (13)

• 2. Substitute Eq. (9) in Eq. (13):

  $$ {\overline{q}} (\theta_{\rm 0}, \phi) = \frac{2}{N}{\sum\limits_{m = 0}^{\infty}}{\left({\sum\limits_{t = 1}^N}q_{t} (\phi)\cos \left(mt\Updelta \phi\right)\cos (m\phi) + {\sum\limits_{t = 1}^N} q_{t} (\phi)\sin \left({mt\Updelta \phi} \right)\sin (m\phi) \right)} $$

  (14)

• 3. The longitude “ϕ _k ” may be represented as ϕ _i  =  kΔϕ, where k is a natural number, so after substitution we have:

  $$ {\overline{q}}_{k} (\theta_{\rm 0}, k\Updelta \phi) = \frac{2}{N}{\sum\limits_{m = 0}^\infty}{\left({\sum\limits_{t = 1}^N} q_{t} (k\Updelta \phi)\cos \left({mt\Updelta \phi} \right)\cos (mk\Updelta \phi) + {\sum\limits_{t = 1}^N}q_{t} (k\Updelta \phi)\sin \left({mt\Updelta \phi} \right)\sin (mk\Updelta \phi) \right)} $$

• 4. By the principle of orthogonality, the product cos (mtΔϕ) cos (mkΔϕ) is non-zero only for t  =  k. Thus all the summation terms become zero except for the term “k”, and so we have:

  $$ {\overline{q}}_{k} (\theta_{\rm 0}, k\Delta \phi) = \frac{2}{N}{\sum\limits_{m = 0}^\infty}{\left(q_{k} (k\Updelta \phi)\cos \left({mk\Updelta \phi} \right)\cos (mk\Updelta \phi) + q_{k} (k\Updelta \phi)\sin \left({mt\Updelta \phi} \right)\sin (mk\Updelta \phi)\right)} $$
  $$ {\overline{q}}_{k} (\theta_{\rm 0}, k\Updelta \phi) = \frac{2}{N}{\sum\limits_{m = 0}^\infty}{\left(q_{k} (k\Updelta \phi)\cos^{2} \left({mk\Updelta \phi} \right) + q_{k} (k\Updelta \phi)\rm sen^{2} \left({mt\Updelta \phi} \right) \right)} $$

  which, upon simplification leads to:

  $$ {\overline{q}}_{k} (\theta_{\rm 0}, k\Updelta \phi) = \frac{2}{N}{\sum\limits_{m = 0}^\infty}q_{k} (k\Updelta \phi) $$

• 5. Applying the property of summation:

  $$ {\overline{q}}_{k} (\theta_{\rm 0}, k\Updelta \phi) = \frac{2}{N}mq_{k} (k\Updelta \phi) $$

• 6. As ${\overline{q}}_{k} (\theta_{\rm 0}, k\Updelta \phi)$ must be equal to q _k (kΔϕ) we have: $1 = \frac{2}{N}m$ which, after substitution of Eq. (9), leads to:

  $$ m = \frac{\pi}{{\Updelta \phi}} $$

Appendix 3

Table 4.

Table 4 Coefficients for harmonic expansion of degree 36