3.1 Pressure & Wind

Wind is air in motion, driven primarily by pressure differences. The pressure gradient force is the fundamental driver of atmospheric circulation.

Pressure Gradient Force

$$\vec{F}_{PGF} = -\frac{1}{\rho}\nabla p = -\frac{1}{\rho}\left(\frac{\partial p}{\partial x}\hat{i} + \frac{\partial p}{\partial y}\hat{j} + \frac{\partial p}{\partial z}\hat{k}\right)$$

Force per unit mass directed from high to low pressure

Horizontal PGF

Drives winds from high to low pressure centers

Typical: 10⁻³ to 10⁻² m/s²

Vertical PGF

Nearly balanced by gravity (hydrostatic)

$-\frac{1}{\rho}\frac{\partial p}{\partial z} \approx g$

Equation of Motion

$$\frac{D\vec{v}}{Dt} = -\frac{1}{\rho}\nabla p - 2\vec{\Omega} \times \vec{v} + \vec{g} + \vec{F}_{friction}$$

Term 1: Acceleration (rate of change of velocity)

Term 2: Pressure gradient force

Term 3: Coriolis force

Term 4: Gravitational force

Term 5: Friction

Coriolis Parameter

$$f = 2\Omega \sin\varphi$$

$\Omega = 7.292 \times 10^{-5}$ rad/s is Earth's angular velocity, $\varphi$ is latitude

Natural Coordinate Momentum Equation

$$\frac{DV}{Dt} = -\frac{1}{\rho}\frac{\partial p}{\partial s} + F_s$$

$$\frac{V^2}{r} = -\frac{1}{\rho}\frac{\partial p}{\partial n} + fV$$

Along-flow (s) and cross-flow (n) components in natural coordinates

Gradient & Cyclostrophic Wind

Gradient Wind Equation

$$\frac{V^2}{r} + fV = -\frac{1}{\rho}\frac{\partial p}{\partial n}$$

Includes centripetal acceleration for curved flow around highs and lows

Cyclostrophic Balance

$$\frac{V^2}{r} = -\frac{1}{\rho}\frac{\partial p}{\partial n}$$

Applies to small-scale vortices (tornadoes, dust devils) where $f \approx 0$

Thermal Wind Relation

$$\frac{\partial \vec{v}_g}{\partial z} = -\frac{g}{fT}\hat{k} \times \nabla_H T$$

$$\frac{\partial u_g}{\partial z} = -\frac{g}{fT}\frac{\partial T}{\partial y}, \quad \frac{\partial v_g}{\partial z} = \frac{g}{fT}\frac{\partial T}{\partial x}$$

Vertical wind shear is proportional to the horizontal temperature gradient

Interactive Simulation: Pressure Gradient & Wind

Python

Generates a 2D low-pressure field, computes pressure gradient force vectors, and overlays wind vectors deflected by the Coriolis effect (Northern Hemisphere). Edit parameters to explore different pressure patterns.

pressure_wind_sim.py74 lines

import numpy as np
import matplotlib.pyplot as plt

# Create a 2D grid
nx, ny = 60, 60
x = np.linspace(-500, 500, nx)
y = np.linspace(-500, 500, ny)
X, Y = np.meshgrid(x, y)
dx = (x[1] - x[0]) * 1000  # km to m

# Pressure field: low-pressure center at origin
r = np.sqrt(X**2 + Y**2)
p = 1013.25 - 20 * np.exp(-r**2 / (250**2))  # hPa

# Pressure gradient force components (Pa/m)
dp_dx = np.gradient(p * 100, dx, axis=1)
dp_dy = np.gradient(p * 100, dx, axis=0)
rho = 1.225
pgf_x = -dp_dx / rho
pgf_y = -dp_dy / rho

# Coriolis deflection (NH): rotate PGF 90 deg right for geostrophic approx
# Plus add PGF component for ageostrophic near-surface wind
alpha = 0.6  # blending: 0=pure PGF, 1=pure geostrophic
u_wind = alpha * pgf_y / 1e-4 + (1 - alpha) * pgf_x * 500
v_wind = -alpha * pgf_x / 1e-4 + (1 - alpha) * pgf_y * 500
speed = np.sqrt(u_wind**2 + v_wind**2)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 5.5))
fig.patch.set_facecolor('#0f172a')
for ax in (ax1, ax2):
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.set_xlabel('x (km)', color='#cbd5e1')
    ax.set_ylabel('y (km)', color='#cbd5e1')
    ax.set_aspect('equal')

# Left: pressure contour map
levels = np.arange(994, 1016, 2)
cf = ax1.contourf(X, Y, p, levels=20, cmap='coolwarm', alpha=0.85)
cs = ax1.contour(X, Y, p, levels=levels, colors='#cbd5e1', linewidths=0.7)
ax1.clabel(cs, inline=True, fontsize=7, colors='#cbd5e1')
cbar = fig.colorbar(cf, ax=ax1, shrink=0.85)
cbar.set_label('Pressure (hPa)', color='#cbd5e1')
cbar.ax.tick_params(colors='#cbd5e1')
ax1.set_title('Pressure Field & Isobars', color='#cbd5e1', fontsize=13)
ax1.annotate('L', (0, 0), fontsize=22, fontweight='bold', color='#f87171', ha='center')

# Right: wind vectors with Coriolis deflection
skip = 4
q = ax2.quiver(X[::skip, ::skip], Y[::skip, ::skip],
               u_wind[::skip, ::skip], v_wind[::skip, ::skip],
               speed[::skip, ::skip], cmap='plasma', scale=800)
ax2.contour(X, Y, p, levels=levels, colors='#475569', linewidths=0.5, alpha=0.5)
cbar2 = fig.colorbar(q, ax=ax2, shrink=0.85)
cbar2.set_label('Wind speed (arb.)', color='#cbd5e1')
cbar2.ax.tick_params(colors='#cbd5e1')
ax2.set_title('Wind Vectors (PGF + Coriolis)', color='#cbd5e1', fontsize=13)
ax2.annotate('L', (0, 0), fontsize=22, fontweight='bold', color='#f87171', ha='center')

fig.suptitle('Pressure Gradient Force & Wind Field', color='white', fontsize=15, y=1.02)
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())

print("=== Pressure Gradient & Wind Simulation ===")
print(f"Central low pressure: {p.min():.1f} hPa")
print(f"Max PGF magnitude: {np.max(np.sqrt(pgf_x**2 + pgf_y**2))*1e3:.3f} x10^-3 m/s^2")
print(f"Pressure range: {p.min():.1f} - {p.max():.1f} hPa")
print(f"Grid: {nx}x{ny}, domain: 1000x1000 km")
print("Left panel: isobars show concentric low-pressure centre")
print("Right panel: wind vectors deflected right (NH Coriolis)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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