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| Titel |
Damped Frank-Kamenetskii Approximation for Modelling of Terrestrial Planets with High Surface Temperatures (Venus) or Large Masses (Super-Earths) |
| VerfasserIn |
Lena Noack, Doris Breuer |
| Konferenz |
EGU General Assembly 2011
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 13 (2011) |
| Datensatznummer |
250054328
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| Zusammenfassung |
| The viscosity of a silicate mantle is strongly dependent on temperature and pressure and can
be described by the Arrhenius law [1]. Assuming realistic values of the activation energy and
volume, viscosity variations in a terrestrial mantle amount to values of about 1010Pas or
larger in the upper mantle taking into account elasto-viscous material. Many numerical codes,
however, cannot deal with such large viscosity contrasts (especially if the effect of elasticity
is ignored and viscosity contrasts increase to much higher values). In general, a smaller
viscosity contrast has the advantage of smaller computation times and less numerical
errors.
A common method to reduce the viscosity contrast is to use the Frank-Kamenetskii
approximation [2]. Applying this approximation for purely temperature-dependent viscosity,
the results are similar to those using the Arrhenius law when the convection is in the
steady-state stagnant lid regime [3; 4].
Here, we compare the Frank-Kamenetskii approximation to the Arrhenius law for both
temperature- and pressure-dependent viscosity. To this end, we have derived new
Frank-Kamenetskii parameters that also include the pressure-dependence of the viscosity and
differ from previous approximations [5; 6]. We show that using these parameters, the
depth-dependence of the approximated viscosity is comparable to the more realistic
Arrhenius viscosity. It is even possible to model a stagnant lower mantle that may form above
the core-mantle boundary in case of high activation volume or high pressure like for massive
Super-Earths [7].
It should be noted, however, that for high surface temperatures as is the case for Venus
this approximation does not represent the mantle flow as it is derived using the Arrhenius law.
With the classical Frank-Kamenetskii approximation viscosity contrast of less than ~ 105
develop, i.e. the minimal viscosity contrast necessary to be in the stagnant lid regime is not
reached. The convection regime changes to the transitional or mobile regime [3]. To
overcome this problem, we have further derived a new approximation, which we call the
Damped Frank-Kamenetskii approximation. This is a mixture between the classical law and a
second-order approximation controlled by a damping parameter. The second-order
approximation is not a linearization of the exponential viscosity term like for the classical
Frank-Kamenetskii approximation, but is a quadratic approximation with a higher accuracy.
The new method leads to a stagnant lid in all cases treated in our investigations. Note that
for planets with low surface temperatures, this new approximation can be used as
well with a damping parameter of zero that yields the standard Frank-Kamenetskii
approximation.
Our studies suggest that the classical Frank-Kamenetskii approximation with the here
derived parameters can be used to simulate the mantle dynamics in Super-Earths even
with a high pressure-dependence of the viscosity if the viscosity contrast across the
upper thermal boundary is above ~ 105 allowing a stagnant lid to form on top the
convecting mantle. If the constraint of a stagnant lid is not satisfied, e.g. for high surface
temperatures, the new Damped Frank-Kamenetskii approximation should be used
instead.
[1] Karato, S. and P. Wu (1993). Science 260, pp. 771-778. [2] Frank-Kamenetskii, D.
(1969). Diffusion and heat transfer in chemical kinetics. Plenum Press. [3] Solomatov, V.S.
and L.-N. Moresi (1996). JGR 101 E2, pp. 4737-4753. [4] Plesa, A.-C.; C. Hüttig
and D. Breuer (2010). EGU 2010, abstract EGU2010-6342. [5] Christensen, U.R.
(1984). PEPI 37, pp. 183-205. [6] Hansen, U. and D.A. Yuen (2000). EPSL 176, pp.
01-411. [7] Noack, L.; V. Stamenkovic and D. Breuer (2010). EGU 2010, abstract
EGU2010-9645. |
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