A Preliminary CMB Heat Flux map
Takumi Matsunaga1(matsunaga.t@elsi.jp), John Hernlund1(hernlund@gmail.com)
1.Earth-Life Science Institute, Tokyo Institute of Technology

Abstract

Preliminary Results
Figure4. Geotherm line and transition zone used in Hernlund(2010)

Estimating heat ﬂux at Core-Mantle Boundary(CMB) is very important forunderstanding mantle thermal convection. Thanks to the discovery of perovskite
post-perovskite transition in the deep mantle, using Clapeyron slope has apossibility that can constrain thermal gradient and geotherm above CMB.
Seismic data is very useful in estimating geotherm, but still have a lot ofuncertainties, such as eﬀect of composition heterogeneity in the lower mantle,
spin transition of iron, radioactive heat, partial melt eﬀect and thermal
conductivity in high pressure condition.
Using the maximum heat ﬂux assumed from the hypothesis 'Doublecrossing'(Hernlund,2010) and P-wave velocity map which is highly eﬀected intemperature, we can simply estimate the heat ﬂux near the CMB. This method isuseful in its simpleness, but the problem is assuming linear correlation between
P-wave velocity and temperature.

𝑇! < 𝑇"

𝑇! > 𝑇"

Entropy change

Transition Width changeProblems• Uncertainties in analytical solution
Temperature distribution:
CMB heat ﬂux:𝑧
𝑇 = 𝑇%&' − ∆𝑇𝑒𝑟𝑓𝛿 𝑥, 𝑡∆𝑇 = 𝑇%&' − 𝑇()1
Figure6.
Estimated Heat
Flux from thenumerical solution
above. The
parameters is theheight of double
cross.

2𝑘∆𝑇𝑞%&' =2𝛿 𝜋Conservation of energy:
𝜕𝑇𝜕𝑇𝜕
𝜕𝑇
𝜌𝑐!
+ 𝑣*=𝑘− 𝑇∆𝑠Γ − 𝜌𝑔𝛼𝑇𝑣* + 𝑄 + 𝜃 3𝜕𝑡𝜕𝑧𝜕𝑧 𝜕𝑧
Thermal boundary layer thickness:
𝛿 + (𝑡) =𝛿

+- ""(𝑡, ) + 4𝜅 ∫- exp 2 ∫- 𝜀̇ 𝑡 . 𝑑𝑡 .
!
!
exp 2 ∫- 𝜀̇ 𝑡 . 𝑑𝑡 .
!

𝑑𝑡 ..

4
Figure7.
Solidus line of
Pyrolitic lower
mantle.
(Nomura etal.,2014)2𝜅
𝛿 𝑡→∞ =5𝜀̇• The lower cross of the ʻDouble crossingʼ can be too thin
Figure5. Using heat ﬂux range of 0-160m𝑊𝑚!" , P-wave
velocity model(Houser et al.,2008). (Treatise,2012)Conclusions• Using the numerical solution, we can directly get the temperature distribution near theCMB, where Pv-pPv transition is there→ Assumption: D” discontinuity place is the maximum CMB heat ﬂux

Figure1. Shear wave velocity anomaly in northpaciﬁc(Y. Suzuki et al.,2021)

Figure2. Numerical solution of Geotherm doublecross(Hernlund,2010)• Just assume linear correlation of temperature anomaly and P-wave velocity anomaly,
we can have the heat ﬂux map(Figure5)
𝜕𝑙𝑛𝑉!
𝑑𝑙𝑛𝑉! ≈𝛿𝑇 12
𝜕𝑇Approach• Numerical solution method:
Using steady equation of conservation of energy(6) and ʻlever ruleʼ(7);exhibits
conservation of solute species, mass conservation(8,9)
𝜕𝜕𝑇
𝜌𝑐! 𝑇𝑣* − 𝑘= −𝑇∆𝑠Γ − 𝜌𝑔𝛼𝑇𝑣* 6𝜕𝑧𝜕𝑧
𝑋 − 𝑋!/0𝜌/0=𝜙7𝑋/0 − 𝑋!/0𝜌10
𝜕(𝜌/0 𝜙𝑣* )= Γ 8𝜕𝑧𝜕(𝜌!/0 (1 − 𝜙)𝑣* )= −Γ 9𝜕𝑧
Deﬁne Transition zone by binary loop:∆𝑇345 𝑋!/0 1 − 𝑋!/0𝑇2! =+ 𝑇, − 𝑋!/0 𝑇, − 𝑇6 10
2𝑋, 1 − 𝑋,
∆𝑇345 𝑋/0 1 − 𝑋/0𝑇7894 = −+ 𝑇, − 𝑋/0 𝑇, − 𝑇6 11
2 𝑋, 1 − 𝑋,Future Directions• Put time dependent Dynamical models- make 𝛿 time dependent (Wu et al.,2011)- ﬁx strain-rate by rheological theory• Change the parameters- thermal conductivity, CMB temperature
Figure8. Time dependent Thermal
Boundary Layer used in Wu(2011)• Fit with the seismic observations- S-wave velocity anomaly highly eﬀected by other factors:
𝜕𝑙𝑛𝑉:
𝜕𝑙𝑛𝑉:
𝜕𝑙𝑛𝑉:
𝑑𝑙𝑛𝑉: =
𝛿𝑇 +𝛿𝑋;< +
𝛿𝜑!!0 13𝜕𝑇𝜕𝑋;<𝜕𝜑!!0- get rid of P-wave anomaly by other factors: using vote map?
- test model results with velocity anomaly zones• Change the heat ﬂux range; make minimumReferences•••••Figure3. Binary phase loop (𝑇# > 𝑇$ )•Wu, B., P. Driscoll, and P. Olson (2011), A statistical boundary layer model for the mantle Dì region, J. Geophys. Res., 116, B12112,
doi:10.1029/2011JB008511.
Hernlund(2010), On the interaction of the geotherm with a post-perovskite phase transition in the deep mantle, Physics of the Earthand Planetary Interiors 180(2010) 222-234, doi:10.1016/j.pepi.2010.02.001
Yuki Suzuki, Kenji Kawai, Robert J. Geller(2021), Imaging paleoslabs and inferring the Clapeyron slope in D” beneath the northern
Paciﬁc based on high-resolution inversion of seismic waveforms for 3-D transversely isotropic structure, Physics of the Earth andPlanetary Interiors vol.321(2021), doi:10.1016/j.pepi.2021.106751
Hernlund J.W., and McNamara A.K The Core‒Mantle Boundary Region. In: Gerald Schubert (editor-in-chief) Treatise on Geophysics,
2nd edition, Vol 7. Oxford: Elsevier; 2015. p. 461-519.
Kenji Ohta(2016), Measurements of Electrical and Thermal Conductivity of Materials Deep Inside the Earth under High-Pressure
Conditions, ⾼圧⼒の科学と技術 Vol. 26, No. 2 (2016)
Ryuichi Nomura, Kei Hirose, Kentaro Uesugi, Yasuo Ohishi, Akira Tsuchiyama, Akira Miyake, Yuichiro Ueno, Low Core-Mantle Boundary
Temperature Inferred from the Solidus of Pyrolite, Science vol.343(2014), www.sciencemag.org/content/343/6170/522/suppl/DC1