# [SVC36-P08] Long-term input rate of mafic magma inferred from a long-term heat balance model of silicic magma reservoir

Keywords:heat balance model, long-term eruption rate , long-term magma input rate, size of magma reservoir

The basic formulas of the model are following equations:

Δ

*Q*=

*Iρ*{

*cT*+

_{b}*L*(1 - φ

_{crit})} -

*E*ρc

*T*, (1)

_{eft}*Q*= 2π{

_{loss}*a*+

*b*(

*tanh*

^{-1}*e*/

*e*)}

*k*(

*T*-

_{eft}*T*). (2)

_{wrt}Here, Δ

*Q*is the difference between the power of the erupted magma (

*E*: long-term eruption rate; km

^{3}/kyr) and that of the input magma (

*I*: long term input rate; km

^{3}/kyr), and

*Q*is the power dissipation of the oblate spheroid shape magma reservoir.

_{loss}*a*is a major axis of the spheroid,

*b*is its minor axis, and

*e*is the eccentricity. ρ is the density of the rock,

*c*is the specific heat of the rock,

*L*is the latent heat of fusion,

*k*is the thermal conductivity, φ

_{crit}is the melt fraction of the eruptable magma,

*T*

_{b}is the temperature of the mafic magma,

*T*is the effective temperature of eruptible magma,

_{eft}*T*is the crustal temperature around the magma reservoir. Furthermore, assuming that Δ

_{wrt}*Q*can not be completely consumed by

*Q*, the residual power

_{loss}*Q*is:

_{ex}*Q*= Δ

_{ex}*Q*-

*Q*(3) and that of the volume increase rate (

_{loss}*R*; km

_{ex}^{3}/kyr) is:

*R*=

_{ex}*Q*/ρ

_{ex}*cT*(4).

_{eft}Next, I conducted to calculate the supply rate of mafic magma by assuming concrete cases using this model. For example, if

*a*= 3600 m was observed underneath a volcano which was geologically obtained the long-term eruption rate of 0.1 (km

^{3}/kyr), the magma input rate for maintaining the magma reservoir size (

*Q*= 0) was calculated to be 0.5 (km

_{ex}^{3}/kyr), assuming that

*T*= 100 °C, the flatness ratio of the magma reservoir was 0.7 (1-

_{wrt}*b*/

*a*) and other parameters were refferred to Koyaguchi and Kaneko (2000). This model, however, has a large dependency on

*T*. For example, if the same injection and input rate were assumed as above instance, it balanced at

_{wrt}*a*= ~ 20 km when

*T*

_{wrt}= 500 °C. With this model, although there is a large temperature dependence of the surrounding crust, it could be shown that the long-term input rate of mafic magma during the past thousands of years can be estimated by knowing the present size of the magma reservoir underneath the volcano with the known eruption rate.