# Temporal Evolution of Cooling by Natural Convection in an Enclosed Magma Chamber

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{4}) and Rayleigh (Ra = 1 × 10

^{6}) numbers yielding a low relative error of 0.11. The times for cooling the center of the chamber from 1498 to 1448 K are 40 ky (kilo years), 37 and 28 ky for rectangular, hybrid and quasi-elliptical shapes, respectively. Results show that for the cases studied, natural convection moved the magma but had no influence on the isotherms; therefore the main mechanism of cooling is conduction. When a basaltic magma intrudes a chamber with rhyolitic magma in our model, natural convection is not sufficient to effectively mix the two magmas to produce an intermediate SiO

_{2}composition.

## 1. Introduction

^{4}, fluid motion evolves through many different patterns before becoming turbulent [13]. Correctly calculating a Rayleigh number based on the temperature (Ra

_{T}) due to thermal coupling between the magma and host is complicated. A more realistic method uses the flux-based “magmatic Rayleigh number” (Ra

_{F}) [14] for natural magma convection in a chamber. In this work the Ra

_{F}is used.

^{9}–1 × 10

^{17}, or 1 × 10

^{6}–1 × 10

^{9}[15] for smaller chambers. Magmas have high Prandtl numbers of 1 × 10

^{3}–1 × 10

^{8}[16], implying that the momentum diffusion is much greater than thermal diffusion. Magmas have an even higher Lewis number of 1 × 10

^{4}–1 × 10

^{13}[17], which means that heat flow is much greater than chemical molecular diffusion. In this work, these observations about the dimensionless numbers will be very important in analyzing the simulation results.

## 2. Materials and Methods

#### 2.1. The Physical and Geological Situation

^{3}, but varying widths. Figure 1 displays the magma chamber for the most complex case as suggested by previous work in the PCB (Humay unit) [22,23]. Geological data indicate that PCB plutons are commonly 5 km thick or less [24,25]. Analysis of saline inclusions in the PCB Linga complex suggests magma crystallization pressures between 800 and 900 bar, indicating that the depth to the top of the magma chamber was between 2.9 and 3.2 km from the Earth’s surface when crystallizing [22]. In this work we locate the top of the chamber at 3 km from the surface and use a maximum thickness for the chamber of 3 km (L

_{c}= 3000 m).

_{s,0}= 298 K at the upper boundary (surface) and T

_{s,0,b}= 448 K at the lower boundary. The lower edge of the computational domain (chamber and host rock) is considered adiabatic. This could occur if the cooling below the bottom boundary of the model together with the cooling of the modeled magma chamber would lead to a small temperature gradient near the bottom boundary.

_{2}silica concentration of 50–55 wt%; thus, C

_{s}= 0.5 is used in all simulations. It is common for basaltic magma to have an SiO

_{2}concentration of 45–50 wt% and rhyolitic magma of 65–75 wt% SiO

_{2}[18]. Section 3.2.3 presents the simulation results of a chamber with two magma compositions. Initially SiO

_{2}in the lower zone of the chamber is ${C}_{f,0,b}=0.47$ and in the upper zone ${C}_{f,0,a}=0.7$, corresponding to basaltic magma intruding rhyolitic magma. This means that at the beginning of the intrusion simulation the chamber contains mostly rhyolitic magma, but with basaltic magma at L

_{intr}= 1000 m at the bottom, with all at the same temperature.

#### 2.2. Mathematical Model

^{14}by several authors [15,27]. The partial differential equations (PDEs) for mass conservation and momentum for a melt are:

_{2}in magma is:

_{2}of the host rock is considered to be constant at C

_{s}= 0.5.

_{0}and a were calculated from experimental data for basaltic magma at four temperatures: 1498, 1473, 1458 and 1448 K [4]. The resulting values used in the non-Newtonian model of Equations (9) and (10) for cooling of basaltic magma are: E

_{0}= 1.5 × 10

^{6}J/mol, a = −994.4827 J/mol K, K = 15 Pa s and n = 0.58. These parameters used in the range 1498–1448 K are constants for the non-Newtonian power law model and do not depend on either time or temperature. This means that the effects of crystals in the movement of the basaltic magma are included.

_{cl}(see Equation (22)), g is gravity and $\beta $ (1/K) is the volumetric expansion coefficient. Independent and dependent dimensionless variables are defined as:

_{2}are:

_{cl}and the magmatic Rayleigh number Ra, the method proposed by [14] for cooling lava lakes was used. This method assumes that the host regime determines the total flux of the host rock/chamber system. The flux-based Rayleigh number Ra

_{F}is related to the temperature-based version Ra

_{T}by the Nusselt number Nu, as follows:

_{F}can be related by an empirical correlation for the RB case ($Nu=0.374R{a}_{F}^{1/4}$) [14]. The earlier equations allow one to obtain the total thermal boundary layer thickness L

_{cl}, their temperature difference $\Delta {T}_{cl}$ and the magmatic Rayleigh number Ra.

^{−3}m/s) and those calculated by the correlation proposed by [17] for magma chambers heated from below (${U}_{0}$ = 0.9 × 10

^{−3}m/s).

#### 2.3. Numerical Procedure

^{−4}was used in all simulations. The convergence criteria for the φ variables for the internal iteration k, in the control volume (i,j), in eachdimensionless time step ($\Delta \tau $ = 10), are:

^{30}Pa s) and the velocities were zero (m/s). No convection is assumed in the volume occupied by the host rock nor in the solidified portion of melt below the solidus temperature (1223 K), and diffusion becomes the only transport mechanism.

**Pm**) meets the condition, ${T}_{f}\le 1448\mathrm{K}$. In order to compare reference and simulations results the following equation for the relative error (RE) was used [37]:

_{p}and y

_{p}are the abscissa and ordinate of the graphs used to obtain the RE.

#### 2.4. Mesh Study

^{−2}. The mathematical model assumed ${L}_{cl}={L}_{c}$ and $\Delta {T}_{cl}=1$. Since the driving force for natural convection is gravity acting on the Y coordinate, it is assumed that the velocity gradients would be greater in the V component of the velocity and hence the mesh convergence is studied analyzing the V profile once all variables ($\phi =U,V,T$) have converged. This is verified after $\tau =7\times {10}^{4}$. To calculate the RE, the dimensionless profiles in position V (2.8–5.2; 0.5) obtained with each mesh were compared with those of the denser mesh (mesh 1). The results shown in Table 2 indicate clear convergence when the mesh is denser. Mesh 5 with 6500 nodes distributed over the entire computational domain has a RE of 2.0933. By increasing the nodes to 9360 (mesh 4) the RE decreases to 1.1756, which is approximately 10 times smaller than results found for mesh 5. The greatest gradients occur in the chamber where there is natural convection of the magma, while in the host rock the diffusion produces smaller gradients. This leads us to formulate a mesh with a higher number of nodes in the chamber zone and a lower number of nodes in the host rock. Mesh 3 keeps the same quantity of nodes from mesh 2 in the chamber area, but decreases its number in the host rock zone. The RE is only 0.1283 even when it has about one third of the total nodes of mesh 1. The same happens with the computational time which decreases from 13,180 to 4393 s. Hence, a non-uniform mesh (mesh 3) of 196 × 58 was used in the simulations.

## 3. Results

#### 3.1. Validation

_{T}= 1 × 10

^{6}) of a highly non-Newtonian fluid with a high Prandtl number (4 × 10

^{4}) inside a square cavity surrounded by a solid was studied numerically. The calculated domain was 0.2 m in x and y directions that included a Plexiglas contour of 0.01 m thickness. Initially the fluid was static (U = V = 0) and both fluid temperatures and Plexiglas were constant at T

_{0}= 45 °C ($\theta =0$). In order to compare published results with those obtained using our algorithm, two cases were used. In the first case a constant temperature of T

_{w}= 25 °C ($\theta =-1$) outside the container was used. In the second case the same temperature T

_{w}= 25 °C ($\theta =-1$) on three boundaries was imposed, but an adiabatic condition at the bottom boundary was used. The dimensionless mathematical model introduced in Section 2.2 was used to simulate the mathematical model (using ${L}_{cl}={L}_{c}$ and $\Delta {T}_{cl}=1$) and FVM proposed in this work. In both cases, the calculated domain was discretized with five uniform meshes of 40 × 40, 180 × 180, 120 × 120, 60 × 60 and 40 × 40 nodes and three-time steps $\Delta \tau $, of 1 × 10

^{−4}, 1 × 10

^{−3}and 1 × 10

^{−2}(in dimensionless time). Simulations were carried out until reaching 1785s ($\tau $ = 1165). To calculate the apparent viscosity, the following equation was used: 10

^{−4}

^{4}is close to the basaltic magma value [39]. It is important that the Prandtl number is close to that of basaltic magma since the theory of similarity between dimensionless numbers allows prediction of the same behavior of both fluids in terms of the amount of movement and heat transfer. This is very important for the validation of magmatic flows since the high temperatures at which they occur make their study very difficult in the laboratory. The two previous characteristics of the reference problem (viscosity and Pr number) allow us to confirm that the dimensionless solution for speeds and temperatures is comparable to those that occur in basaltic magmas. In Figure 2 velocity vectors and isotherms from [38] are compared to those obtained in the present work after 1785 s using a mesh with 180 × 180 nodes.

#### 3.2. Results and Discussion

^{3}volume are simulated (where the third dimension z is considered to be of unitary magnitude). The cases studied are: rectangular (case 1), hybrid (case 2) and quasi-elliptical (case 3). The last two could have been formed by geological phenomena that have been detailed by [21]. In all cases the simulation was carried out until the temperature at point

**Pm**(12,000 m; −4500 m) was lower than 1448 K. This point is in the center of the rectangular chamber in case 1.

#### 3.2.1. Temperature Results

^{3}Pa s according to Equations (9) and (10) for the range between 1498 to 1448 K, which produce very low velocities (see Figure 6 and Figure 7) by natural convection in the chamber. Laboratory studies have shown that natural convection may not have visible effects on isotherms when a cavity is heated from the bottom and cooled from the top [41]. The results of the simulations indicate that the velocities produced by natural convection in the magma are not high enough to alter the isotherms. Conduction between the magma and host rock produces a rapid decrease in the temperature at the edges of the chamber. In general, cooling rates are minimized when the temperature difference between magma and host is small [42].

**Pm**and (b) the temperature profile at x = 12,000 m for 0.1 ky and 22 ky. Figure 5a shows that average temperatures for case 1 decrease the slowest and case 3 cools down the fastest. The average temperatures after 40, 37 and 28 ky were 1215, 1156 and 1080 K for cases 1, 2 and 3, respectively. Cooling times are related to heat flow at the edges in contact with the host rock, which depends on the shape of the chamber. These contact areas are 18, 22.3 and 28.2 km

^{2}for cases 1, 2 and 3, respectively. This in itself could explain why the cooling times in Figure 5a are different, but there is another reason. If the same boundary conditions were applied, similar heat losses should occur. However, the chambers have different shapes and this causes the edges of the chamber to come in contact with different temperatures of the host rock where the temperature increases by 25 K/km of depth. Fourier’s law indicates that the greatest heat flux at the host rock/chamber interface is found where there is the highest temperature gradient. In our case, this occurs when the host rock has lower temperatures (where the magma has a uniform initial temperature of T

_{f,}

_{0}) at a shallower depth. Case 1 has less area exposed to high gradients, while cases 2 and 3 have larger exposed areas where the temperature gradient is high. These two reasons explain the large differences in temperature between the shapes studied and highlight the effect of chamber shape on the magma’s cooling times. Small batches of magma moving from hotter to colder regions in the upper crust can achieve thermal equilibrium in 1 ky, while larger igneous systems with spatial scales of 10–100 km may have lifetimes of 100 to 10,000 ky [43].

#### 3.2.2. Velocity Results

^{−3}m/s or 31.5 km/y. Maximum velocities are observed near the lateral wall. In general, the lower local resultant velocities are in the range of 0.32–0.032 km/y. This means that in 100 years an element of the magma travels at least one complete revolution in the chamber. Case 1 is the one that retains the most heat and therefore has the most movement. Two big vortices are produced in the fluid zone in all cases. In cases 2 and 3, streamlines show the effects of the chamber shape. This is best seen at 5 ky when the fluid area is still large.

^{−4}m/s or less. Velocities within the chamber in each case are distributed in a different form. Cases 2 and 3 show positive and negative values of u and v. The velocities are displaced with time due to the solidification front for case 3. The velocity profile in case 3 has no contact with the lower part of the chamber; this produces velocities of zero near the bottom of the chamber. For this profile, case 1 displays the lowest velocities of the three studied cases. The time evolution of the average resulting velocity (V

_{res}) in the chamber is shown in Figure 7c. The V

_{res}increases rapidly during the first years of cooling. Only in case 1 is the V

_{res}value (1.1 × 10

^{−3}m/s) near the maximum calculated by Equation (11) (U

_{0}= 1.4 × 10

^{−3}m/s). The maximum V

_{res}are 5.5 × 10

^{−4}and 1.6 × 10

^{−4}m/s for cases 2 and 3, respectively. The V

_{res}decreases with: case 1 > case 2 > case 3, which is consistent with the temperature decrease for the studied chambers.

#### 3.2.3. Concentration Results

_{2}concentration when basaltic magma is intruded beneath a rhyolitic magma chamber. Both magma bodies initially had the same temperature. The basaltic magma initially had 47 wt% SiO

_{2}, the rhyolitic magma 70 wt% and the host rock 50 wt%. Using these parameters and a typical SiO

_{2}diffusion coefficient at atmospheric pressure of D = 3 × 10

^{−12}m

^{2}/s [18], no variation was observed in the concentration of SiO

_{2}for the three cases (see Figure 8a for case 3). Additional tests were made with similar results using a self-diffusivity value of D = 6.7 × 10

^{−11}m

^{2}/s obtained at 1 GPa of pressure and 1873 K for SiO

_{2}in anhydrous basaltic liquid [46]. These diffusivity values are too low to observe the SiO

_{2}transport by molecular diffusion. Advection is not capable of mixing or mingling the two contrasting magma types. It is important to remember that for the magma chambers studied here the Rayleigh, Prandtl and Lewis numbers indicate that the diffusion of momentum is much greater than that of heat, which is much greater than the molecular diffusion of SiO

_{2}. Therefore, it was expected that natural convection would not produce changes in the iso-concentration of SiO

_{2}.

^{−11}to 1 × 10

^{−6}m

^{2}/s. Results show that the minimum diffusivity necessary to observe diffusion of silica in the chamber is around 1 × 10

^{−8}m

^{2}/s (Le = 16). The result obtained using a hypothetical value for the diffusivity coefficient of D = 1 × 10

^{−8}m

^{2}/s is presented at 22 ky in Figure 8b. The advection–diffusion transport mechanisms change the concentration of SiO

_{2}while the magma is in the liquid phase. This occurs mainly in the first 10 ky of cooling. Figure 8b shows the variation in concentration level near the boundary between basalt and rhyolite. The SiO

_{2}content increases in the zone where initial basaltic magma intrudes. The rhyolite composition decreases from 0.70 to 0.60 and the basalt composition increases from 0.47 to 0.50 near the boundary interface between rhyolite and the basalt intrusion.

## 4. Conclusions

_{2}under the conditions examined in this study produced little chemical differentiation of magma.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Cross-section of a quasi-elliptical magma chamber (case 3) surrounded by a host rock approximately representing emplacement of the Humay unit in the PCB. This pluton form is a combination of the cantilever and piston mechanisms [21].

**Figure 2.**Convection lines and temperature contours after 1785 s for the cases used to validate the mathematical model and numerical procedure proposed in this work with the obtained by [38]. In (

**a**) the bottom boundary is adiabatic, while in (

**b**) all boundaries have the same non-dimensional temperature.

**Figure 3.**Time evolution of the average dimensionless fluid temperature for five meshes is compared with [38] by using relative error (RE). Results were obtained using a constant temperature of $\theta =-1$ outside the container for the first case in Section 3.1.

**Figure 4.**Temperature distributions at two different times inside the chamber and host rock for 1498 K and 1223 K correspond to magma (liquid and crystal mixture). Zones with temperatures lower than 1223 K are considered to be solid.

**Figure 5.**For the three cases studied: (

**a**) Temporal evolution of the temperature average inside the chamber and (

**b**) temperature profile (for x = 12,000 m) at two different times in the host rock and chamber.

**Figure 6.**Streamlines and v velocity distributions at two different times inside the chamber. Natural convection produces movement of magma in zones having temperatures of 1498 K to 1223 K.

**Figure 7.**Velocity profiles for the three cases studied at x = 12,000 m (at the center of the computational domain) for (

**a**) u and (

**b**) v at two different times and (

**c**) the resultant velocity average inside the whole chamber.

**Figure 8.**Silica concentration distribution (in wt fraction) within a magma chamber when basaltic magma is intruded into rhyolite magma without initially mixing and then the chamber is closed completely. Results for two diffusivities, (

**a**) using D = 3 × 10

^{−12}m

^{2}/s as a typical SiO

_{2}diffusivity for basaltic magma in rhyolite and (

**b**) using D = 1 × 10

^{−8}m

^{2}/s as an artificially high diffusivity. The original interface between the two magmas is shown as a dashed horizontal black line in Figure 8b. Streamlines show two vortices in the basaltic and rhyolitic magma that can be seen in Figure 8b.

Transport Properties | Magma [18] | Host Rock [30] | Param. | Param. | ||
---|---|---|---|---|---|---|

$\rho ({\mathrm{kg}/\mathrm{m}}^{3})$ | 2600 | 2670 | $R{a}_{F}$ | 1.62 × 10^{17} | Pr | 2.42 × 10^{5} |

${C}_{p}(\mathrm{J}/\mathrm{kg}\mathrm{K})$ | 1450 | 1000 | $\Delta {T}_{c}(\mathrm{K})$ | 75 | Le | 5.3 × 10^{4} |

$k(\mathrm{W}/\mathrm{m}\mathrm{K})$ | 0.6 | 2.65 | $Nu$ | 7426 | T_{f,0} (K) | 1498 |

$\alpha ({\mathrm{m}}^{2}/\mathrm{s})$ | 1.6 × 10^{−7} | 9.93 × 10^{−7} | ${L}_{cl}(\mathrm{m})$ | 0.4 | T_{ref} (K) | 1473 |

$\beta (1/\mathrm{K})$ * | 5 × 10^{−5} | - | $\Delta {T}_{cl}(\mathrm{K})$ | 1.01 × 10^{−2} | C_{s,}_{0} | 0.5 |

${\eta}_{ref}(\mathrm{Pa}\mathrm{s})$ | 100 | - | $Ra$ | 2.19 × 10^{13} | C_{f,}_{0,a} | 0.47 |

$D({\mathrm{m}}^{2}/\mathrm{s})$ | 3 × 10^{−12} | - | ${U}_{0}(\mathrm{m}/\mathrm{s})$ | 1.41 × 10^{−3} | C_{f,}_{0,b} | 0.7 |

**Table 2.**Different meshes studied and the relative errors (RE) between V velocity profiles in position V (2.8–5.2; 0.5), compared with the more refined mesh 1.

Mesh | Nodes X × Y | Total Nodes | N° Nodes in Chamber Zone | N° Nodes in Host Rock | RE | t (s) |
---|---|---|---|---|---|---|

1 | 300 × 90 | 27,000 | 2800 | 24,200 | 0 | 13,180 |

2 | 280 × 70 | 19,600 | 2450 | 17,150 | 0.0078 | 9836 |

3 | 196 × 58 | 11,368 | 2450 | 8918 | 0.1283 | 4393 |

4 | 156 × 60 | 9360 | 1800 | 7560 | 1.1756 | 3440 |

5 | 130 × 50 | 6500 | 1250 | 5250 | 2.0933 | 2960 |

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**MDPI and ACS Style**

Zambra, C.E.; Gonzalez-Olivares, L.; González, J.; Clausen, B.
Temporal Evolution of Cooling by Natural Convection in an Enclosed Magma Chamber. *Processes* **2022**, *10*, 108.
https://doi.org/10.3390/pr10010108

**AMA Style**

Zambra CE, Gonzalez-Olivares L, González J, Clausen B.
Temporal Evolution of Cooling by Natural Convection in an Enclosed Magma Chamber. *Processes*. 2022; 10(1):108.
https://doi.org/10.3390/pr10010108

**Chicago/Turabian Style**

Zambra, Carlos Enrique, Luciano Gonzalez-Olivares, Johan González, and Benjamin Clausen.
2022. "Temporal Evolution of Cooling by Natural Convection in an Enclosed Magma Chamber" *Processes* 10, no. 1: 108.
https://doi.org/10.3390/pr10010108