4.4 Jet Streams

Jet streams are narrow bands of fast-moving air at high altitudes. They are critical for weather development and are used by aircraft for fuel efficiency.

Types of Jet Streams

Polar Jet Stream

Location: 30-60°N/S at ~300 hPa (9-12 km)

Speed: 100-200 kt, highly variable

Associated with polar front and cyclones

Subtropical Jet Stream

Location: ~30°N/S at ~200 hPa (10-16 km)

Speed: 80-150 kt, more steady

Associated with Hadley cell boundary

Formation: Thermal Wind Relationship

$$|\vec{v}_T| = \frac{R}{f}\ln\left(\frac{p_0}{p_1}\right)|\nabla_p \bar{T}|$$

Strong temperature gradients produce strong jet streams

In full vector form, the thermal wind in pressure coordinates relates the wind shear to the horizontal temperature gradient:

$$\mathbf{V}_T = \mathbf{V}_g(p_1) - \mathbf{V}_g(p_0) = -\frac{R_d}{f}\int_{p_0}^{p_1} \nabla_p T \;\frac{dp}{p}$$

The geostrophic wind increases with height in proportion to the layer-mean temperature gradient

The polar jet exists because of the strong temperature contrast between tropical and polar air. Jet streams are strongest in winter when this contrast is greatest.

Subtropical Jet: Angular Momentum Conservation

The subtropical jet is maintained by the poleward transport of angular momentum from the tropics in the upper branch of the Hadley cell. Conservation of absolute angular momentum gives:

$$M = (u + \Omega a \cos\phi)\,a\cos\phi = \text{const}$$

Air rising at the equator with $u \approx 0$ and moving to $\phi = 30°$must accelerate to $u \approx 134\;\text{m/s}$ to conserve $M$ — friction and eddies reduce this to the observed 40--80 m/s

Rossby Waves

The jet stream meanders in large-scale waves called Rossby waves or planetary waves. These waves are crucial for weather pattern development.

Ridge

Northward bulge, high pressure, fair weather

Trough

Southward dip, low pressure, stormy weather

The phase speed of barotropic Rossby waves relative to the mean westerly flow is given by the Rossby dispersion relation:

$$c = \bar{u} - \frac{\beta}{k^2 + l^2}$$

where $\bar{u}$ is the mean zonal wind, $\beta = df/dy$ is the meridional gradient of the Coriolis parameter, and $k, l$ are the zonal and meridional wavenumbers

Since $\beta/(k^2+l^2) > 0$, Rossby waves always propagate westward relative to the mean flow. Long waves (small $k$) are quasi-stationary or even retrograde, while short waves move eastward with the jet.

Potential Vorticity Conservation

Rossby wave dynamics are underpinned by conservation of potential vorticity (PV). For an isentropic layer:

$$\text{PV} = \frac{\zeta_\theta + f}{\Delta p} = \text{const (following the flow)}$$

where $\zeta_\theta$ is relative vorticity on an isentropic surface and $\Delta p$ is the pressure thickness of the layer; PV maps are used to track upper-level troughs associated with jet stream dynamics

Jet Streak Dynamics

A jet streak is a localised wind maximum embedded within the jet stream. The ageostrophic wind in the entrance and exit regions of a straight jet streak creates a four-quadrant pattern of convergence and divergence. The along-stream ageostrophic wind component is:

$$v_{ag} \approx -\frac{1}{f}\frac{dV_g}{dt} = -\frac{V_g}{f}\frac{\partial V_g}{\partial s}$$

In the entrance region, air accelerates ($\partial V_g/\partial s > 0$), producing ageostrophic flow directed across the jet toward low pressure — this creates upper-level divergence in the right-entrance quadrant (Northern Hemisphere), favouring surface cyclogenesis beneath

Interactive Simulation: Thermal Wind & Jet Stream Profile

Python

Calculates the thermal wind from a meridional temperature gradient, producing a latitude-height cross-section of zonal wind. Shows the temperature field, jet stream structure with core location, and vertical wind profiles at selected latitudes.

thermal_wind_jet.py99 lines

import numpy as np
import matplotlib.pyplot as plt

# Thermal wind equation: du/dz = -(g/(f*T)) * dT/dy
g = 9.81
Omega = 7.292e-5
R_d = 287.0  # J/(kg*K)

# Latitude-height grid
lat = np.linspace(10, 80, 100)    # degrees N
z = np.linspace(0, 16000, 100)    # metres
LAT, Z = np.meshgrid(lat, z)

# Mean temperature field: decreases poleward and with height
T_surface = 300 - 40 * (LAT / 80)  # K: warm tropics, cold poles
lapse = 6.5e-3  # K/m
T = T_surface - lapse * Z
# Tropopause: higher in tropics
trop_height = 16000 - 8000 * (LAT / 80)
above_trop = Z > trop_height
T[above_trop] = T[above_trop] + lapse * (Z[above_trop] - trop_height[above_trop]) * 0.8

# Coriolis parameter
f_lat = 2 * Omega * np.sin(np.radians(LAT))
f_lat = np.clip(f_lat, 1e-5, None)  # avoid division by zero near equator

# Meridional temperature gradient (K/m) on the grid
dy = (lat[1] - lat[0]) * 111000  # degrees to metres
dT_dy = np.gradient(T, dy, axis=1)

# Thermal wind: integrate du/dz upward
dz = z[1] - z[0]
du_dz = -g / (f_lat * T) * dT_dy
u_wind = np.cumsum(du_dz * dz, axis=0)
u_wind = np.clip(u_wind, 0, None)  # westerlies only

fig, axes = plt.subplots(1, 3, figsize=(16, 5.5))
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')

# Panel 1: Temperature field (lat-height)
ax1 = axes[0]
cf1 = ax1.contourf(LAT, Z / 1000, T, levels=20, cmap='coolwarm', alpha=0.85)
cbar1 = fig.colorbar(cf1, ax=ax1, shrink=0.85)
cbar1.set_label('Temperature (K)', color='#cbd5e1')
cbar1.ax.tick_params(colors='#cbd5e1')
ax1.plot(lat, trop_height[0, :] / 1000, '--', color='#fbbf24', linewidth=2, label='Tropopause')
ax1.set_xlabel('Latitude (deg N)', color='#cbd5e1')
ax1.set_ylabel('Height (km)', color='#cbd5e1')
ax1.set_title('Temperature Field', color='#cbd5e1', fontsize=12)
ax1.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Panel 2: Zonal wind cross-section (jet stream)
ax2 = axes[1]
cf2 = ax2.contourf(LAT, Z / 1000, u_wind, levels=np.linspace(0, 60, 25), cmap='hot_r', alpha=0.9)
cbar2 = fig.colorbar(cf2, ax=ax2, shrink=0.85)
cbar2.set_label('Zonal wind u (m/s)', color='#cbd5e1')
cbar2.ax.tick_params(colors='#cbd5e1')
cs2 = ax2.contour(LAT, Z / 1000, u_wind, levels=[20, 30, 40, 50], colors='white', linewidths=0.8)
ax2.clabel(cs2, inline=True, fontsize=7, colors='white')
ax2.plot(lat, trop_height[0, :] / 1000, '--', color='#fbbf24', linewidth=1.5)
# Mark jet core
max_idx = np.unravel_index(np.argmax(u_wind), u_wind.shape)
ax2.plot(lat[max_idx[1]], z[max_idx[0]] / 1000, 'w*', markersize=15, label=f'Jet core: {u_wind.max():.0f} m/s')
ax2.set_xlabel('Latitude (deg N)', color='#cbd5e1')
ax2.set_ylabel('Height (km)', color='#cbd5e1')
ax2.set_title('Thermal Wind / Jet Stream', color='#cbd5e1', fontsize=12)
ax2.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Panel 3: Wind profiles at selected latitudes
ax3 = axes[2]
colors = ['#f87171', '#fb923c', '#fbbf24', '#34d399', '#38bdf8']
for i, lat_val in enumerate([30, 40, 50, 60, 70]):
    lat_idx = np.argmin(np.abs(lat - lat_val))
    ax3.plot(u_wind[:, lat_idx], z / 1000, color=colors[i], linewidth=2,
             label=f'{lat_val} deg N')
ax3.set_xlabel('Zonal wind (m/s)', color='#cbd5e1')
ax3.set_ylabel('Height (km)', color='#cbd5e1')
ax3.set_title('Wind Profiles by Latitude', color='#cbd5e1', fontsize=12)
ax3.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
ax3.grid(True, color='#334155', alpha=0.4)

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

print("=== Thermal Wind & Jet Stream Profile ===")
print(f"Thermal wind equation: du/dz = -(g/fT)(dT/dy)")
max_idx = np.unravel_index(np.argmax(u_wind), u_wind.shape)
print(f"Jet core location: lat={lat[max_idx[1]]:.0f}N, height={z[max_idx[0]]/1000:.1f} km")
print(f"Maximum jet speed: {u_wind.max():.1f} m/s ({u_wind.max()*1.944:.0f} kt)")
print(f"Surface temp range: {T_surface.min():.0f} - {T_surface.max():.0f} K")
for lat_val in [30, 45, 60]:
    lat_idx = np.argmin(np.abs(lat - lat_val))
    print(f"  Max wind at {lat_val}N: {u_wind[:, lat_idx].max():.1f} m/s at {z[np.argmax(u_wind[:, lat_idx])]/1000:.1f} km")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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