2D Advection Simulation Program

This project simulates 2D advection using a Gaussian distribution within a domain that mimics the atmospheric boundary layer. The simulation calculates how a cloud of material, represented by the Gaussian u(x,y), moves horizontally over time based on a velocity profile that varies logarithmically with height.

$$ \frac{\partial{u}}{\partial{t}}= - \left(v_x \frac{\partial{u}}{\partial{x}} + v_y \frac{\partial{u}}{\partial{y}}\right) $$

Features

• Customizable Gaussian Initial Conditions: Initialises a Gaussian distribution representing the material cloud.
• Boundary Conditions: Implements boundary conditions for all sides of the computational domain.
• Logarithmic Velocity Profile: Applies a height-dependent velocity profile to simulate the horizontal wind speed in the atmospheric boundary layer.

$$ v_x(z) = \frac{u_\ast}{\kappa} ln \left(\frac{z}{z0}\right) $$

• Time Evolution: Uses finite difference approximations to compute the advection over a set number of time steps.

$$ \frac{\partial{u}}{\partial{x}} \approx \left(\frac{u_{i,j} - u_{i-1,j}}{dx}\right) $$ $$ \frac{\partial{u}}{\partial{y}} \approx \left(\frac{u_{i,j} - u_{i,j-1}}{dy}\right) $$

Program Output

The program generates two output files:

1. initial.dat: Contains the initial values of the scalar field u(x, y) at the start of the simulation.

2. final.dat: Contains the final values of u(x, y) after the simulation completes.

Both files contain three columns: x, y, and u, representing the spatial coordinates and the scalar field's value at those coordinates.

Compilation and Execution

Prerequisites

• GCC Compiler: Ensure that GCC is installed on your system. The program is written in C and tested with gcc.
• OpenMP Library: Ensure that OpenMP is installed on your system and accessible by gcc.

Compilation

To compile the program, run the following command:

gcc -fopenmp -o advection2D -std=c99 advection2D.c -lm

Running the Program

After compiling, you can run the executable as follows:

./advection2D

This will generate two output files: initial.dat and final.dat, containing the initial and final values of the scalar field. You can also visualise them with the provided gnuplot scripts by running:

gnuplot plotinit.gp

gnuplot plotfinal.gp

Implementation Details

• Grid Properties: The program defines a grid of ($1000 \times 1000$) points over a domain of ($30 m \times 30 m$).

• Boundary Conditions: Material is introduced into the domain from the left boundary, while other boundaries are assigned a value of zero.

• Logarithmic Wind Profile: The horizontal wind speed is computed using a logarithmic profile, with constants:

  • $u_\ast = 0.1 m/s$
  • Roughness length $z_0 = 1.0 m$
  • Von Kármán constant $\kappa = 0.41$

• Time Stepping: The program uses the CFL condition for stable time stepping, with a CFL number of 0.9 and a total of 1000 time steps.

Notes

• The program initializes an empty computational domain and applies a boundary condition that introduces material into the domain over time.
• Ensure sufficient disk space to store the output files, especially for larger grid sizes.
• The program serves as a basic framework for further modifications, such as incorporating different velocity profiles, boundary conditions, or higher grid resolutions.