# Satellite and Ground Based Thermal Observation of the 2014 Effusive Eruption at Stromboli Volcano
## 5 Discussion
### Simple Model for Waning Mass Flux  
In Figure 9, we compare our observations with the results of some model calculations for the temporal variation of the mass flux during a flank eruption at Stromboli. Model outputs are based on a very elegant and yet simple model developed by [PERSON] _et al._[8]. They used a multidisciplinary data set consisting of seismic, infrared and deformation data with which they modeled _TADR_ as a function of time. Our model embodies their geometry (see Figure 10) in which the system is divided into an upper and a lower reservoir flux, both of which contribute to _TADR_ out of the vent on the side of the volcano. The flux from the upper reservoir is _TADR\({}_{U}\)_, while the long term flux from the lower reservoir is _TADR\({}_{L}\)_ = 0.4 m\({}^{3}\)/s, which is slightly higher than the flux determined by [PERSON] _et al._[64], whose estimate is 0.3 m\({}^{3}\)/s. The flux through the tube located on the side of the main plumbing system is described by the Poiseuille flow. Figure 10: Geometry of the model for the eruptive flux from an ephemeral vent at Stromboli volcano. The model has been taken from [PERSON] _et al._[8]. Figure 9: _TADR_ in m\({}^{3}\)/s for observations along with the results of model calculations. The geometry of the model is shown in Figure 10 and the parameter varied is the length \(L\) (model 1 \(L\) = 20 m, model 2: \(L\) = 35 m, model 3, \(L\) = 50 m) of the tube feeding the flow at the vent. For more details on the model calculation, see Section 5.1. \[TADR\left(t\right)=\frac{a^{4}}{8\cdot\eta\cdot L}\cdot\Delta P\left(t\right)=TADR_{U}+TADR _{L} \tag{10}\]  
where \(a\) represents the radius of the tube feeding the flow from the ephemeral vent, \(\eta\) represents the viscosity of the magma and \(L\) represents the length of the tube that feeds the flow. The driving pressure of the flow equals the magmastatic pressure:  
\[\Delta P\left(t\right)=\rho\cdot g\cdot h\left(t\right) \tag{11}\]  
Please note that we have neglected the difference in the atmospheric pressure in this model due to the height difference between the top of the upper reservoir and the tube outlet; the magmastatic pressure is the only driving pressure. \(\rho\) is the density of the magma containing bubbles, \(g\) is the acceleration as a result of gravity while \(h(t)\) describes the magma level due to the slow emptying of the upper reservoir of size \(V_{R}(t)\) (for details, see [PERSON] _et al._[8]):  
\[V_{R}\left(t\right)=h\left(t\right)\cdot A_{0} \tag{12}\]  
where \(A_{0}\) is the base of the reservoir, which can be represented by a shallow cylindrical plumbing system immediately above the effusive vent (for details see [PERSON] _et al._, [8]). From our measurements, the initial eruptive flux \(TADR_{0}\) at the beginning of the effusive period is known. Using Equation (10), this allows us to determine the initial height h\({}_{0}\) of the upper reservoir. \[\Delta P_{0}=\frac{8\cdot\eta\cdot L\cdot TADR_{0}}{a^{4}}=\rho\cdot g\cdot h _{0}\to h_{0}=\frac{1}{\rho\cdot g}\left(\frac{8\cdot\eta\cdot L\cdot TADR_ {0}}{a^{4}}\right) \tag{13}\]  
\(h(t)\) can now b e determined from the slowly emptying upper reservoir  
\[V_{R}\left(t\right)=V_{0}-\int_{0}^{t}\left(TADR-TADR_{L}\right)dt \tag{14}\]  
with \(V_{0}\) being the initial magma volume in the reservoir \(\left(h_{0}\cdot A_{0}\right)\). Since \(TADR_{L}\) is assumed to be a constant flux, Equation (14) can be rewritten  
\[V_{R}\left(t\right)=V_{0}-TADR_{L}\cdot t+\int_{0}^{t}TADR\left(t\right)dt \tag{15}\]  
From which \(h(t)\) can be calculated  
\[h\left(t\right)=h_{0}-\frac{1}{A_{0}}\cdot\left(\int_{0}^{t}TADR\left(t \right)dt-TADR_{L}\cdot t\right). \tag{16}\]  
When we combine Equations (10), (11), and (16), we find:  
\[TADR\left(t\right)=\frac{a^{4}}{8\cdot\eta\cdot L}\cdot\left[h_{0}-\frac{1}{ A_{0}}\cdot\left(\int_{0}^{t}TADR\left(t\right)dt-TADR_{L}\cdot t\right) \right], \tag{17}\]  
which can be solved numerically for \(TADR(t)\).