MANCompiled | Numerical Modeling of Advective Boundary Layer Dynamics

Numerical Modeling of Advective Boundary Layer Dynamics

May 22, 2026

🌪️ Project Overview

This project implements a 2D numerical solver for the Mahrt (1972) advective boundary layer model. The solver is designed to simulate cross-equatorial wind flow, focusing on the dynamics of the Somali Jet. By integrating real-world meteorological data from the ERA5 Global Catalog, we study how pressure forces, horizontal and vertical advection, eddy viscosity, and the changing Coriolis parameter across the equator interact to shape the planetary boundary layer.

💡 Research Summary Investigated boundary layer response to cross-equatorial gradients along the Somali Jet axis (60°E longitude). Developed an upwind numerical solver initialized with ERA5 data and conducted comparisons between constant and height-dependent eddy viscosity profiles.

🔬 Model Physics & Governing Equations

The Mahrt (1972) model governs the steady-state boundary layer velocity fields $( (u, v, w) )$ under the influence of pressure gradients, friction (eddy diffusion), and Coriolis acceleration:

$$ v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} = f v + K_z \frac{\partial^2 u}{\partial z^2} - \frac{1}{\rho}\frac{\partial p}{\partial x} $$

$$ v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} = -f u + K_z \frac{\partial^2 v}{\partial z^2} - \frac{1}{\rho}\frac{\partial p}{\partial y} $$

Where:

$( f = \beta y )$ is the Coriolis parameter, which varies linearly with latitude $( y )$ (beta-plane approximation).
$( K_z )$ is the vertical eddy diffusion coefficient.
$( -\frac{1}{\rho}\frac{\partial p}{\partial x} )$ and $( -\frac{1}{\rho}\frac{\partial p}{\partial y} )$ are the zonal and meridional pressure gradient forces.
The vertical velocity $( w )$ is determined from the continuity equation:

$$ \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \implies w(z) = -\int_{0}^{z} \frac{\partial v}{\partial y} dz $$

🛠️ Numerical Implementation

The solver is written in Python and uses a finite difference grid to discretize the boundary layer:

Discretization: Upwind finite difference scheme is used for horizontal and vertical advection to maintain numerical stability in strongly advective flows.
Eddy Diffusion: Standard second-order central difference scheme.
Time Integration: Explicit forward-Euler scheme used to step the equations of motion to a steady state (spin-up).
Coriolis near Equator: A reference latitude method is used to avoid singularities at the exact equator ($( f=0 )$).

Core Solver Step

def step(self, dt):
    # Calculate vertical velocity w from continuity
    self._calculate_vertical_velocity()
    
    # Calculate advection, diffusion, and Coriolis terms
    adv_u = self._advection_u()
    adv_v = self._advection_v()
    diff_u = self._diffusion_u()
    diff_v = self._diffusion_v()
    cor_u, cor_v = self._coriolis_beta_term()
    
    # Update velocity fields with pressure gradient force (dpdx, dpdy)
    u_new = self.u + dt * (adv_u + cor_u + diff_u - self.dpdx)
    v_new = self.v + dt * (adv_v + cor_v + diff_v - self.dpdy)
    
    # Enforce boundary conditions (no-slip at surface, geostrophic at top)
    u_new[0, :] = 0.0
    v_new[0, :] = 0.0
    u_new[-1, :] = u_new[-2, :]
    v_new[-1, :] = v_new[-2, :]
    
    self.u = u_new
    self.v = v_new

📊 Numerical Experiments & Findings

We performed two main simulation experiments across the domain spanning from $( -10^\circ )$ to $( +10^\circ )$ latitude and $( 0 )$ to $( 2000\text{ m} )$ altitude:

1. Experiment 1: Meridional Pressure Gradient Only

Configured with only meridional forcing ($( dp/dy \neq 0, dp/dx = 0 )$).
Results show strong cross-equatorial advection carrying southern-hemispheric momentum into the Northern Hemisphere, which delays the adaptation of the wind vector to the Northern Hemisphere Coriolis force.

2. Experiment 2: Joint Meridional and Zonal Pressure Gradients

Configured with both meridional and zonal forcing to model a realistic Somali Jet scenario.
The zonal pressure gradient drives strong zonal wind acceleration, creating a pronounced low-level wind maximum (jet core) within the boundary layer near the equator.

3. Sensitivity Analysis: Eddy Viscosity Profiles

Constant $( K_z )$: Results in classical Ekman-like spiral profiles that show uniform friction depth.
Height-Dependent $( K_z(z) )$: Simulates realistic boundary layers where turbulence decreases near the top of the boundary layer, leading to sharper shear zones and a more realistic jet core height.

⚙️ Tech Stack

Python | NumPy | xarray | Intake | Matplotlib | Jupyter Notebooks

This project was developed as part of the ICTS Monsoon School (Moist Convective Dynamics of Monsoons - II) in Bengaluru, India.