Modeling Volcanic Eruptions

Modeling Volcanic Eruptions

• The lab focuses on constructing a mathematical model of explosive volcanic eruptions.

Introduction: Magma Chamber Dynamics

• Magma resides in a chamber beneath a volcano.
• Overpressure ($\Delta P$) drives eruptions.
• Eruptions happen when $\Delta P$ exceeds the lithostatic pressure ($P_L$).
• $\Delta P$ depends on magma production rate ($Q_{in}$), magma withdrawal rate ($Q_{out}$), chamber volume ($V$), and magma compressibility ($\beta$).

Mathematical Model

• The pressure change in the magma chamber is described by the equation: $\frac{dP}{dt} = \frac{Q_{in} - Q_{out}}{\beta V}$
• Eruptions occur when $P = P_L$.
• During eruptions, $Q_{out} > 0$; otherwise, $Q_{out} = 0$.

Model Parameters

• Magma Input ($Q_{in}$): $1 \times 10^{-3}$ $km^3/yr$
• Compressibility ($\beta$): $1 \times 10^{-9}$ $Pa^{-1}$
• Chamber Volume ($V$): 10 $km^3$
• Lithostatic Pressure ($P_L$): $20 \times 10^6$ $Pa$

Eruption Dynamics

• Eruption rate ($Q_{out}$) is proportional to overpressure: $Q_{out} = k(P - P_L)$
• $k$ is a constant: $5 \times 10^{-14} \frac{km^3}{Pa \cdot yr}$

Numerical Solution

• Equation is discretized using the forward Euler method: $P_{i+1} = P_i + \Delta t \frac{Q_{in} - Q_{out}}{\beta V}$
• Timestep ($\Delta t$) used is 0.01 years.

Tasks

• Implement the numerical model in Python.
• Plot pressure ($P$) vs. time ($t$).
• Determine eruption frequency.
• Explore the effects of $Q_{in}$ and $V$ on eruption frequency through sensitivity analysis.

Python Implementation Highlights

• Key parameters such as Qin, beta, V, PL, and k are defined with their respective values and units.
• A time array t is created using numpy, ranging from 0 to tmax (100 years) with a timestep dt of 0.01 years.
• An array P is initialized to store pressure values over time, with the initial pressure set to PL.
• A for loop simulates the pressure change over time using the forward Euler method.
• Inside the loop, Qout is calculated based on whether the pressure P[i] exceeds PL.
• The pressure at each timestep P[i+1] is updated using the discretized equation.
• Matplotlib is used to plot the pressure P against time t, with labels and a title for clarity.
• The code calculates the eruption frequency by counting the number of times P exceeds PL and dividing by tmax.

Report Requirements

• Include Python code, plots, and a written summary.
• Discuss model assumptions and limitations.
• Propose future model improvements.