Deep-Earth_HeatFlow

Introduction

• A Modern Fortran (2008+) solver, Parallelising the Jacobi Method with OpenMP to model Heat-Diffusion across Earths's Core-Mantle Boundary (CMB).

Geophysics Background

Core-Mantle Boundary (CMB) Heat Transfer

• The CMB is located at a depth of approximately 3000km.

• It separates the Liquid Outer Core from the Solid Lower Mantle and controls how heat generated in the core is transferred into the matle.

• Heat Flow across the CMB plays a role in:

  • Sustaining the Geodynamo that generates Earth’s Magnetic Field,
  • Driving Mantle Convection and Plume Formation,
  • Controlling the long-term thermal evolution of the core and mantle,
  • Influencing Seismic and Chemical Structures such as Ultra-Low Velocity Zones (ULVZs).

Model for the Project

• The project models Thermal Diffusion across a Local Patch of the CMB using a simple but realistic framework.

Assumptions

• The CMB is approximated locally as a Flat, Cartesian Surface.
• Heat transport is dominated by Thermal Diffusion.
• Material properties are Homogeneous and Isotropic.
• No internal heat sources are included.
• Fluid Motion and Advection are neglected: Pure Conduction Model.

Core-Mantle Thermal Coupling

Core Side (Dirichlet Boundary)

• The Outer Core is assumed to be:

  • Well mixed by Vigorous Convection.
  • Nearly Isothermal at the CMB.

• This means we have a Fixed Temperature (Dirichlet) Boundary Condition:

  • T = T_core,

  • represents a Thermally equilibrated liquid core.

Mantle Side (Robin Boundary)

• The Lower Mantle is modeled as a Thermal Reservoir that exchanges heat with the CMB via conduction and Parameterised Convection.

• This is represented by a Robin (mixed) Boundary Condition:

  • -k(dT/dn) = h_cmb(T - T_mantle), where:
  • k is Thermal Conductivity,
  • h_cmb is an effective Mantle Heat Transfer Coefficient,
  • T_mantle is the Lowermost Mantle Temperature.

• This formation captures the idea that Mantle Dynamics regulate CMB Heat Flux rather than enforcing a fixed temperature

Governing Equation

• The temperature evolution inside the CMB patch follows the 3D Heat Diffusion Equation:

  • dT/dt = K(del . (del(T))), where:
  • T is temperature,
  • K = k/(density * c_p), Thermal Diffusivity,
  • c_p is the specific heat at constant pressure

• This equation is Discretised using the Finite Difference Method on a regular Cartesian Grid.

• The resulting System Algebraic Equations are solved via the Jacobi Iterative Method.

• The Jacobi Method is Parallelised via the OpenMP API.

Requirements

• gfortran compiler.
• OpenMP.
• Python 3.X.
• NumPy.
• Matplotlib.

Resources

• Introduction to Modern Fortran for the Earth System Sciences by Dragos B. Chirila & Gerrit Lohmann.

Running and Using the Software

• $ git clone [email protected]:MRLintern/CMB_HeatFlow.git
• $ cd CMB_HeatFlow
• $ chmod +x build.sh
• $ ./build
• $ ./build/CMB_Heat
• Now plotting the generated .bin file.
• $ cd visualisation
• $ python3 CMB_Temperature_Field.py; this will produce a plot of the 3D model. Note: this plot has been included in the sample_plot directory for viewing.

Update

• The image of the resulting model shows an anomaly between 50km to 200km in the x tangential direction.
• The anomaly consists of a Cool Patch on the base of the Outer Core surrounded by Hotter Fluid.
• This anomaly isn't a Physical Anomaly but a Numerical Anomaly from the code.
• This is to be sorted.