3.2 Geostrophic Balance

In the free atmosphere, the pressure gradient force and Coriolis force are approximately in balance, producing geostrophic wind that flows parallel to isobars.

Coriolis Force

$$\vec{F}_{Cor} = -2\vec{\Omega} \times \vec{v} = -f\hat{k} \times \vec{v}$$

where f = 2Ω sin φ is the Coriolis parameter

Equator

f = 0

45°N/S

f ≈ 10⁻⁴ s⁻¹

Poles

f = ±1.46×10⁻⁴ s⁻¹

Rossby Number

$$Ro = \frac{U}{fL}$$

Ratio of inertial to Coriolis forces. When $Ro \ll 1$, geostrophic balance is a good approximation

Geostrophic Wind

$$u_g = -\frac{1}{\rho f}\frac{\partial p}{\partial y}, \quad v_g = \frac{1}{\rho f}\frac{\partial p}{\partial x}$$

Wind flows parallel to isobars with low pressure to the left (NH)

Key Features

• Wind speed proportional to pressure gradient
• Tightly packed isobars → strong winds
• NH: low pressure to left of wind direction
• SH: low pressure to right of wind direction

Geopotential Height Form

$$\Phi = gz, \quad u_g = -\frac{1}{f}\frac{\partial \Phi}{\partial y}, \quad v_g = \frac{1}{f}\frac{\partial \Phi}{\partial x}$$

On pressure surfaces, geostrophic wind is proportional to the gradient of geopotential height

Ageostrophic Wind & Acceleration

$$\vec{v}_{ag} = \vec{v} - \vec{v}_g$$

$$\frac{D\vec{v}}{Dt} = -f\hat{k} \times \vec{v}_{ag}$$

Acceleration is driven by the ageostrophic component of the wind

Thermal Wind

$$\vec{v}_T = \vec{v}_g(p_1) - \vec{v}_g(p_0) = \frac{R}{f}\ln\left(\frac{p_0}{p_1}\right)\hat{k} \times \nabla_p \bar{T}$$

Wind shear related to horizontal temperature gradient

The thermal wind blows parallel to isotherms with cold air to the left (NH). Strong temperature gradients (like the polar front) produce jet streams.

Barotropic Atmosphere

$$\nabla p \times \nabla \rho = 0$$

Density depends only on pressure; no thermal wind shear

Baroclinic Atmosphere

$$\nabla p \times \nabla \rho \neq 0$$

Density surfaces cross pressure surfaces; drives thermal wind and jet streams

Interactive Simulation: Geostrophic Wind Calculator

Python

Calculates geostrophic wind speed for different pressure gradients and latitudes, shows how the Coriolis parameter varies with latitude, and performs Rossby number analysis across spatial scales.

geostrophic_wind_calc.py72 lines

import numpy as np
import matplotlib.pyplot as plt

Omega = 7.292e-5  # Earth's angular velocity (rad/s)
rho = 1.225       # air density (kg/m^3)

lats = np.linspace(10, 80, 200)  # degrees
f = 2 * Omega * np.sin(np.radians(lats))

# Pressure gradients (hPa per 100 km -> Pa/m)
dp_dn_vals = [0.5, 1.0, 2.0, 4.0]  # hPa per 100 km

fig, axes = plt.subplots(1, 3, figsize=(15, 5))
fig.patch.set_facecolor('#0f172a')
for ax in axes:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

# Panel 1: Geostrophic wind vs latitude
ax1 = axes[0]
for dp in dp_dn_vals:
    dp_Pa_m = dp * 100 / 100000  # hPa/100km to Pa/m
    vg = dp_Pa_m / (rho * f)
    ax1.plot(lats, vg, linewidth=2, label=f'{dp} hPa/100km')
ax1.set_xlabel('Latitude (deg N)', color='#cbd5e1')
ax1.set_ylabel('Geostrophic wind (m/s)', color='#cbd5e1')
ax1.set_title('Geostrophic Wind vs Latitude', color='#cbd5e1', fontsize=12)
ax1.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
ax1.set_ylim(0, 80)

# Panel 2: Coriolis parameter f vs latitude
ax2 = axes[1]
ax2.plot(lats, f * 1e4, color='#38bdf8', linewidth=2.5)
ax2.set_xlabel('Latitude (deg N)', color='#cbd5e1')
ax2.set_ylabel('f (x10^-4 s^-1)', color='#cbd5e1')
ax2.set_title('Coriolis Parameter f', color='#cbd5e1', fontsize=12)
ax2.axhline(y=1.0, color='#f87171', linestyle='--', alpha=0.6, label='f = 10^-4')
ax2.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Panel 3: Rossby number for different scales
ax3 = axes[2]
U_vals = [1, 10, 50]  # m/s
L_vals = np.logspace(3, 7, 100)  # 1 km to 10000 km
f_45 = 2 * Omega * np.sin(np.radians(45))
for U in U_vals:
    Ro = U / (f_45 * L_vals)
    ax3.loglog(L_vals / 1000, Ro, linewidth=2, label=f'U = {U} m/s')
ax3.axhline(y=1, color='#f87171', linestyle='--', linewidth=1.5, label='Ro = 1')
ax3.axhline(y=0.1, color='#fbbf24', linestyle=':', linewidth=1, label='Ro = 0.1')
ax3.fill_between([1e0, 1e4], 0, 0.1, color='#22c55e', alpha=0.1)
ax3.set_xlabel('Length scale L (km)', color='#cbd5e1')
ax3.set_ylabel('Rossby Number', color='#cbd5e1')
ax3.set_title('Rossby Number (lat 45 deg N)', color='#cbd5e1', fontsize=12)
ax3.legend(fontsize=7, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
ax3.set_ylim(1e-4, 1e3)
ax3.text(500, 0.03, 'Geostrophic\nregime', color='#22c55e', fontsize=9, ha='center')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())

print("=== Geostrophic Wind Calculator ===")
for lat in [15, 30, 45, 60, 75]:
    fi = 2 * Omega * np.sin(np.radians(lat))
    vg = (1.0 * 100 / 100000) / (rho * fi)
    print(f"  Lat {lat:2d}N: f = {fi:.2e} s^-1, Vg = {vg:.1f} m/s (1 hPa/100km)")
print(f"\nRossby number examples (lat 45N, f={f_45:.2e}):")
for U, L, name in [(10, 1e6, 'synoptic'), (50, 1e5, 'mesoscale'), (1, 1e4, 'local')]:
    print(f"  {name}: U={U} m/s, L={L/1000:.0f} km => Ro = {U/(f_45*L):.3f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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