Module 1

Atmospheric Dynamics

Fluid mechanics on a rotating sphere: from the equations of motion to weather systems and global circulation

1.1 The Equations of Motion on a Rotating Earth

The atmosphere is a thin fluid shell on a rotating sphere. In a reference frame rotating with angular velocity \(\boldsymbol{\Omega}\), the momentum equation becomes:

\( \frac{D\mathbf{u}}{Dt} + 2\boldsymbol{\Omega} \times \mathbf{u} = -\frac{1}{\rho}\nabla p + \mathbf{g} + \nu \nabla^2 \mathbf{u} \)

Navier-Stokes in a rotating frame

where \(\mathbf{u}\) is the velocity, \(2\boldsymbol{\Omega}\times\mathbf{u}\) is the Coriolis acceleration, \(p\) is pressure, \(\rho\) is density, \(\mathbf{g}\) is effective gravity (including centrifugal), and \(\nu\) is kinematic viscosity.

In component form on the sphere, taking \(f = 2\Omega\sin\phi\) (Coriolis parameter, where\(\phi\) is latitude) and using the traditional approximation:

\( \frac{Du}{Dt} - fv = -\frac{1}{\rho}\frac{\partial p}{\partial x} + \nu\nabla^2 u \)

\( \frac{Dv}{Dt} + fu = -\frac{1}{\rho}\frac{\partial p}{\partial y} + \nu\nabla^2 v \)

\( 0 = -\frac{1}{\rho}\frac{\partial p}{\partial z} - g \quad \text{(hydrostatic)} \)

The hydrostatic approximation holds when vertical accelerations are much smaller than \(g\), valid for large-scale motions with aspect ratio \(H/L \ll 1\).

Scale Analysis & the Rossby Number

The Rossby number measures the ratio of inertial to Coriolis forces:

\( Ro = \frac{U}{fL} \)

For synoptic-scale motions: \(U \sim 10\) m/s, \(L \sim 10^6\) m,\(f \sim 10^{-4}\) s\(^{-1}\) \(\implies Ro \sim 0.1\). When \(Ro \ll 1\), the Coriolis force dominates and we obtain geostrophic balance.

Other Key Dimensionless Numbers

The dynamics of atmospheric flow are characterized by several additional dimensionless numbers:

Ekman number \(Ek = \nu/(fL^2)\): ratio of viscous to Coriolis forces. For the free atmosphere, \(Ek \sim 10^{-7}\). Near the surface (boundary layer), \(Ek \sim 1\).
Froude number \(Fr = U/(NH)\): ratio of flow speed to internal gravity wave speed. \(Fr < 1\) implies flow goes around mountains; \(Fr > 1\) implies flow goes over.
Richardson number \(Ri = N^2/(\partial u/\partial z)^2\): ratio of buoyancy to shear. \(Ri < 0.25\) implies Kelvin-Helmholtz instability.
Burger number \(Bu = (NH/(fL))^2\): ratio of Rossby deformation radius to length scale. Determines whether rotation or stratification dominates.

The Rossby deformation radius \(L_R = NH/f \approx 1000\) km sets the natural length scale for geostrophic adjustment: perturbations larger than \(L_R\) are balanced by rotation, while smaller perturbations are governed by gravity waves.

1.2 Geostrophic Balance

For \(Ro \ll 1\), neglecting friction and acceleration, the horizontal momentum equations reduce to a balance between Coriolis and pressure-gradient forces:

\( -fv_g = -\frac{1}{\rho}\frac{\partial p}{\partial x} \implies v_g = \frac{1}{f\rho}\frac{\partial p}{\partial x} \)

\( fu_g = -\frac{1}{\rho}\frac{\partial p}{\partial y} \implies u_g = -\frac{1}{f\rho}\frac{\partial p}{\partial y} \)

In vector notation:

\( \mathbf{u}_g = \frac{1}{f\rho}\,\hat{\mathbf{k}} \times \nabla p \)

Geostrophic wind: flow parallel to isobars

Key consequences: (1) wind flows parallel to isobars, not from high to low pressure; (2) in the Northern Hemisphere, low pressure is to the left of the wind direction; (3) geostrophic flow is non-divergent (\(\nabla \cdot \mathbf{u}_g = 0\) on an f-plane).

Using Geopotential Height

On constant-pressure surfaces, using geopotential \(\Phi = gz\):

\( u_g = -\frac{1}{f}\frac{\partial \Phi}{\partial y}\bigg|_p, \quad v_g = \frac{1}{f}\frac{\partial \Phi}{\partial x}\bigg|_p \)

\( \mathbf{u}_g = \frac{1}{f}\,\hat{\mathbf{k}} \times \nabla_p \Phi \)

1.3 The Thermal Wind Equation

The thermal wind describes how the geostrophic wind changes with height. Start from the geostrophic relation and differentiate with respect to \(z\), using the hydrostatic equation \(\partial p/\partial z = -\rho g\) and the ideal gas law \(p = \rho R T\):

Step 1: \( \frac{\partial u_g}{\partial z} = -\frac{1}{f\rho}\frac{\partial}{\partial y}\left(\frac{\partial p}{\partial z}\right) + \frac{1}{f\rho^2}\frac{\partial\rho}{\partial y}\frac{\partial p}{\partial z} \)

Step 2: Using hydrostatic: \( \frac{\partial p}{\partial z} = -\rho g \)

Step 3: \( \frac{\partial u_g}{\partial z} = \frac{g}{f\rho}\frac{\partial\rho}{\partial y} - \frac{g}{f}\frac{1}{\rho}\frac{\partial\rho}{\partial y} + \frac{g}{f}\frac{\partial}{\partial y}\ln\rho \)

Step 4: For ideal gas, \(\rho = p/(RT)\), so \(\partial\ln\rho/\partial y \approx -\partial\ln T/\partial y\) at constant \(p\)

The result:

\( \frac{\partial \mathbf{u}_g}{\partial z} = -\frac{g}{fT}\,\hat{\mathbf{k}} \times \nabla T \)

Thermal wind: vertical shear proportional to horizontal temperature gradient

In component form:

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

Why the Jet Stream Exists

The equator-to-pole temperature gradient (\(\partial T / \partial y < 0\) in the Northern Hemisphere) drives a westerly thermal wind (\(\partial u_g/\partial z > 0\)). The geostrophic wind increases with altitude until reaching the tropopause, producing the subtropical jet stream at \(\sim 200\) hPa with speeds of 30-70 m/s.

\( u_{\text{jet}} \approx -\frac{g}{f\overline{T}} \frac{\Delta T}{\Delta y} \cdot H_{\text{trop}} \)

With \(\Delta T \sim 40\) K, \(\Delta y \sim 3000\) km, \(H_{\text{trop}} \sim 12\) km: \(u_{\text{jet}} \sim 40\) m/s

1.4 Rossby Waves

Rossby waves (planetary waves) arise from the conservation of absolute vorticity on the \(\beta\)-plane, where \(f\) varies linearly with latitude: \(f = f_0 + \beta y\) with \(\beta = df/dy = 2\Omega\cos\phi_0/a\).

The barotropic vorticity equation on the \(\beta\)-plane:

\( \frac{\partial \zeta}{\partial t} + U\frac{\partial\zeta}{\partial x} + \beta v = 0 \)

where \(\zeta = \partial v/\partial x - \partial u/\partial y\) is relative vorticity and\(U\) is a mean zonal flow. For a wave solution\(\psi = \hat{\psi}\,e^{i(kx + ly - \omega t)}\) with \(\zeta = \nabla^2\psi = -(k^2+l^2)\psi\):

\( -i\omega(-(k^2+l^2))\hat\psi + Uik(-(k^2+l^2))\hat\psi + \beta ik\hat\psi = 0 \)

\( \omega(k^2+l^2) = Uk(k^2+l^2) - \beta k \)

The dispersion relation:

\( \boxed{\omega = Uk - \frac{\beta k}{k^2 + l^2}} \)

Phase Speed and Group Velocity

The zonal phase speed:

\( c_x = \frac{\omega}{k} = U - \frac{\beta}{k^2 + l^2} \)

Since \(\beta > 0\), the phase speed is always westward relative to the mean flow. This is the defining property of Rossby waves: their restoring mechanism is the meridional gradient of planetary vorticity (\(\beta\)-effect).

The zonal group velocity:

\( c_{gx} = \frac{\partial\omega}{\partial k} = U + \frac{\beta(k^2 - l^2)}{(k^2 + l^2)^2} \)

The group velocity can be eastward for short waves (\(k > l\)), enabling downstream energy propagation in mid-latitude storm tracks.

Stationary Rossby Waves

When \(\omega = 0\), we get stationary waves with\(k_s^2 + l^2 = \beta/U\). The stationary wavenumber:

\( K_s = \sqrt{\frac{\beta}{U}} \)

Stationary waves are forced by topography and land-sea thermal contrasts. Only waves with total wavenumber \(K < K_s\) can propagate vertically into the stratosphere (Charney-Drazin criterion). For \(U \approx 10\) m/s, only the longest planetary waves (wavenumber 1-3) propagate into the stratosphere, where they can break and drive the polar vortex.

Changes in stationary wave patterns under climate change alter precipitation and temperature patterns globally, affecting ecosystems, agriculture, and water resources.

1.5 The Hadley Cell & Angular Momentum Conservation

The Hadley cell is the thermally driven circulation between the tropics and subtropics. Air rises near the equator, flows poleward in the upper troposphere, descends in the subtropics, and returns equatorward at the surface. As air moves poleward, conservation of angular momentum determines the wind speed.

The specific angular momentum of an air parcel at latitude \(\phi\) with zonal wind \(u\):

\( M = (u + \Omega a\cos\phi)\,a\cos\phi \)

Starting at the equator (\(\phi = 0\)) with \(u = 0\):

\( M_{\text{eq}} = \Omega a^2 \)

\( \text{At latitude } \phi: \quad (u + \Omega a\cos\phi)\,a\cos\phi = \Omega a^2 \)

Solving for \(u\):

\( u(\phi) = \Omega a \left(\frac{1}{\cos\phi} - \cos\phi\right) = \Omega a \frac{\sin^2\phi}{\cos\phi} \)

Held-Hou Model: Hadley Cell Width

Held & Hou (1980) derived the extent of the Hadley cell by matching the angular-momentum-conserving upper-tropospheric flow with radiative-equilibrium temperature. The Hadley cell terminus latitude:

\( \phi_H \approx \left(\frac{5}{3}\frac{\Delta_h g H}{2\Omega^2 a^2}\right)^{1/2} \)

\(\Delta_h\) = fractional equator-pole temperature difference, \(H\) = tropopause height

With typical values (\(\Delta_h \approx 1/3\), \(H \approx 15\) km), this gives\(\phi_H \approx 30°\), consistent with the observed subtropical jet location.

1.6 Baroclinic Instability & Mid-Latitude Storms

Mid-latitude weather systems (cyclones and anticyclones) arise from baroclinic instability: the conversion of available potential energy (stored in the meridional temperature gradient) to kinetic energy. The Eady model (1949) provides a minimal framework.

Assumptions: (1) uniform \(f\), (2) uniform \(N\) (buoyancy frequency), (3) linear vertical shear \(U(z) = \Lambda z\) where \(\Lambda = |\partial u/\partial z|\), (4) bounded domain \(0 \le z \le H\).

The most unstable mode has growth rate:

\( \boxed{\sigma_{\max} = 0.31 \frac{f\Lambda}{N} = 0.31 \frac{f}{N}\left|\frac{\partial u}{\partial z}\right|} \)

Eady growth rate for baroclinic instability

Derivation Sketch

The quasi-geostrophic potential vorticity equation in the Eady model interior is:

\( \left(\frac{\partial}{\partial t} + U\frac{\partial}{\partial x}\right)\left[\nabla^2\psi + \frac{f^2}{N^2}\frac{\partial^2\psi}{\partial z^2}\right] = 0 \)

With boundary conditions from the thermodynamic equation at \(z = 0, H\):

\( \left(\frac{\partial}{\partial t} + U\frac{\partial}{\partial x}\right)\frac{\partial\psi}{\partial z} - \Lambda\frac{\partial\psi}{\partial x} = 0 \quad \text{at } z = 0, H \)

The dispersion relation for the Eady model yields unstable modes for wavenumbers below the short-wave cutoff \(k_c = 2.4 f/(NH)\), corresponding to wavelengths \(\lambda > 2\pi NH/(2.4f) \approx 3000\) km.

Physical interpretation: Growing baroclinic waves tilt westward with height. Warm air rises and moves poleward while cold air sinks and moves equatorward, converting potential energy to kinetic energy. This is why mid-latitude storms are fundamentally different from tropical cyclones (which draw energy from latent heat).

The growth rate \(\sigma \approx 0.31 f\Lambda/N\) corresponds to an e-folding time of about 2-3 days, consistent with observed cyclogenesis time scales.

Most Unstable Wavelength

The most unstable wavelength in the Eady model occurs at:

\( \lambda_{\max} = \frac{2\pi}{k_{\max}} = \frac{2\pi N H}{1.61 f} \approx 4000\;\text{km} \)

This corresponds to synoptic-scale weather systems (mid-latitude cyclones with diameter\(\sim 2000{-}4000\) km). The Charney model (which includes the \(\beta\)-effect) gives a similar result but with additional long-wave stabilization.

Under climate change, the meridional temperature gradient weakens in the lower troposphere (Arctic amplification) but strengthens in the upper troposphere. The net effect on storm intensity and tracks remains an active area of research, with implications for extreme precipitation and wind events.

The vertical structure of the most unstable Eady mode tilts westward with height by about \(\pi/4\) radians, corresponding to warm air rising and cold air sinking. This releases available potential energy at a rate proportional to the heat flux\(\overline{v'T'} \sim \sigma \Lambda H/N\).

Global Atmospheric Circulation

Three-cell model of global atmospheric circulation showing Hadley, Ferrel, and Polar cells with associated surface winds and jet streams.

Simulation: Geostrophic Wind from Pressure Field

Computing geostrophic wind vectors from a synthetic pressure field with a mid-latitude cyclone:

Python

script.py1 lines

import numpy as np\nimport matplotlib.pyplot as plt\n\n# Grid setup\nnx, ny = 50, 50\nx = np.linspace(-2000e3, 2000e3, nx)  # meters\ny = np.linspace(-2000e3, 2000e3, ny)\nX, Y = np.meshgrid(x, y)\n\n# Parameters\nf = 1e-4  # Coriolis parameter (mid-latitude)\nrho = 1.2  # air density kg/m^3\n\n# Synthetic pressure field: cyclone + background gradient\np0 = 101325  # Pa\nR_cyclone = 800e3  # radius of cyclone\nr = np.sqrt(X**2 + Y**2)\np_field = p0 - 15e2 * np.exp(-r**2 / (2*R_cyclone**2)) - 0.5e-3 * Y  # cyclone + meridional gradient\n\n# Compute geostrophic wind\ndx_m = x[1] - x[0]\ndy_m = y[1] - y[0]\ndpdy, dpdx = np.gradient(p_field, dy_m, dx_m)\nu_g = -1/(f*rho) * dpdy\nv_g = 1/(f*rho) * dpdx\nwind_speed = np.sqrt(u_g**2 + v_g**2)\n\nfig, axes = plt.subplots(1, 3, figsize=(18, 6))\nfig.patch.set_facecolor('#0a0a1a')\n\n# Panel 1: Pressure field\nax = axes[0]\nlevels = np.linspace(p_field.min()/100, p_field.max()/100, 20)\ncf = ax.contourf(X/1e3, Y/1e3, p_field/100, levels=levels, cmap='RdYlBu_r')\ncs = ax.contour(X/1e3, Y/1e3, p_field/100, levels=levels[::2], colors='white', linewidths=0.5, alpha=0.5)\ncb = plt.colorbar(cf, ax=ax, label='Pressure (hPa)')\ncb.ax.yaxis.label.set_color('white')\ncb.ax.tick_params(colors='white')\nax.set_xlabel('x (km)', color='white', fontsize=11)\nax.set_ylabel('y (km)', color='white', fontsize=11)\nax.set_title('Pressure Field (hPa)', color='#818cf8', fontsize=13, fontweight='bold')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.set_aspect('equal')\n\n# Panel 2: Geostrophic wind vectors\nax = axes[1]\ncf2 = ax.contourf(X/1e3, Y/1e3, wind_speed, levels=15, cmap='hot')\nskip = 3\nax.quiver(X[::skip,::skip]/1e3, Y[::skip,::skip]/1e3,\n          u_g[::skip,::skip], v_g[::skip,::skip],\n          color='white', alpha=0.8, scale=800, width=0.003)\ncb2 = plt.colorbar(cf2, ax=ax, label='Wind Speed (m/s)')\ncb2.ax.yaxis.label.set_color('white')\ncb2.ax.tick_params(colors='white')\nax.set_xlabel('x (km)', color='white', fontsize=11)\nax.set_ylabel('y (km)', color='white', fontsize=11)\nax.set_title('Geostrophic Wind', color='#818cf8', fontsize=13, fontweight='bold')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.set_aspect('equal')\n\n# Panel 3: Isobars + streamlines\nax = axes[2]\ncs3 = ax.contour(X/1e3, Y/1e3, p_field/100, levels=15, colors='#a5b4fc', linewidths=1)\nax.streamplot(X/1e3, Y/1e3, u_g, v_g, color=wind_speed, cmap='hot',\n              linewidth=1.5, density=1.5, arrowsize=1.5)\nax.set_xlabel('x (km)', color='white', fontsize=11)\nax.set_ylabel('y (km)', color='white', fontsize=11)\nax.set_title('Streamlines along Isobars', color='#818cf8', fontsize=13, fontweight='bold')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.set_aspect('equal')\n\nplt.tight_layout()\nplt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')\nprint('Geostrophic wind simulation complete')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Simulation: Rossby Wave Dispersion

Exploring the dispersion relation \(\omega = Uk - \beta k/(k^2 + l^2)\) and its implications for wave propagation:

Python

script.py1 lines

import numpy as np\nimport matplotlib.pyplot as plt\n\nbeta = 1.6e-11  # m^-1 s^-1 (at 45N)\nU = 10.0  # mean zonal wind m/s\n\nfig, axes = plt.subplots(2, 2, figsize=(14, 10))\nfig.patch.set_facecolor('#0a0a1a')\n\n# Panel 1: Dispersion relation omega(k) for different l\nax = axes[0, 0]\nk = np.linspace(1e-7, 8e-6, 500)\nfor l_val, color, label in [(0, '#f87171', 'l = 0'),\n                             (2e-6, '#fbbf24', 'l = 2e-6'),\n                             (4e-6, '#60a5fa', 'l = 4e-6')]:\n    omega = U*k - beta*k/(k**2 + l_val**2)\n    ax.plot(k*1e6, omega*86400, color=color, linewidth=2, label=label)\nax.axhline(y=0, color='#4f46e5', linestyle='--', alpha=0.5)\nax.set_xlabel('Zonal wavenumber k (10\u207b\u2076 m\u207b\u00b9)', color='white', fontsize=11)\nax.set_ylabel('Frequency \u03c9 (day\u207b\u00b9)', color='white', fontsize=11)\nax.set_title('Dispersion: \u03c9(k)', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\n# Panel 2: Phase speed cx vs wavelength\nax = axes[0, 1]\nwavelength = np.linspace(500e3, 15000e3, 500)  # meters\nk_arr = 2*np.pi/wavelength\nfor l_val, color, label in [(0, '#f87171', 'l = 0'),\n                             (2e-6, '#fbbf24', 'l = 2e-6'),\n                             (4e-6, '#60a5fa', 'l = 4e-6')]:\n    cx = U - beta/(k_arr**2 + l_val**2)\n    ax.plot(wavelength/1e3, cx, color=color, linewidth=2, label=label)\nax.axhline(y=0, color='#4f46e5', linestyle='--', alpha=0.5)\nax.axhline(y=U, color='#e2e8f0', linestyle=':', alpha=0.3, label='Mean flow U')\nax.set_xlabel('Wavelength (km)', color='white', fontsize=11)\nax.set_ylabel('Phase speed c_x (m/s)', color='white', fontsize=11)\nax.set_title('Phase Speed vs Wavelength', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\n# Panel 3: Hadley cell angular momentum wind\nax = axes[1, 0]\nOmega = 7.292e-5\na = 6.371e6\nphi = np.linspace(0, 50, 200) * np.pi/180\nu_AM = Omega * a * np.sin(phi)**2 / np.cos(phi)\nax.plot(phi*180/np.pi, u_AM, color='#f87171', linewidth=2.5, label='AM-conserving wind')\nax.axhline(y=0, color='#4f46e5', linestyle='--', alpha=0.5)\n# Observed subtropical jet\nax.axvline(x=30, color='#fbbf24', linestyle='--', alpha=0.7, label='Observed jet (~30\u00b0)')\nax.annotate('Subtropical jet ~30-40 m/s', xy=(30, 130), xytext=(35, 180),\n            color='#fbbf24', fontsize=11,\n            arrowprops=dict(arrowstyle='->', color='#fbbf24'))\nax.set_xlabel('Latitude (\u00b0)', color='white', fontsize=11)\nax.set_ylabel('Zonal wind u (m/s)', color='white', fontsize=11)\nax.set_title('Angular Momentum Wind', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\n# Panel 4: Baroclinic instability growth rate\nax = axes[1, 1]\nf0 = 1e-4\nN_vals = np.linspace(0.005, 0.02, 4)  # Brunt-Vaisala\nshear = np.linspace(0, 0.01, 200)  # du/dz in s^-1\n\nfor N_val, color in zip(N_vals, ['#60a5fa', '#86efac', '#fbbf24', '#f87171']):\n    sigma = 0.31 * f0 * shear / N_val\n    efolding = np.where(sigma > 0, 1/(sigma*86400), np.inf)\n    ax.plot(shear*1000, sigma*86400, color=color, linewidth=2,\n            label=f'N = {N_val:.3f} s\u207b\u00b9')\n\nax.set_xlabel('Vertical wind shear (10\u207b\u00b3 s\u207b\u00b9)', color='white', fontsize=11)\nax.set_ylabel('Growth rate (day\u207b\u00b9)', color='white', fontsize=11)\nax.set_title('Eady Baroclinic Growth Rate', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\n# Mark typical values\nax.axvline(x=3, color='#e2e8f0', linestyle=':', alpha=0.4)\nax.annotate('Typical midlat shear', xy=(3, 0.1), xytext=(5, 0.15),\n            color='#e2e8f0', fontsize=10,\n            arrowprops=dict(arrowstyle='->', color='#e2e8f0'))\n\nplt.tight_layout()\nplt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')\nprint('Rossby wave and dynamics simulation complete')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

1.7 Potential Vorticity & Its Conservation

Ertel's potential vorticity (PV) is the fundamental conserved quantity for adiabatic, frictionless flow. It combines vorticity and stratification:

\( q = \frac{1}{\rho}(\boldsymbol{\zeta}_a \cdot \nabla\theta) \)

Ertel PV: \(\boldsymbol{\zeta}_a = \nabla \times \mathbf{u} + 2\boldsymbol{\Omega}\) is absolute vorticity

For quasi-geostrophic flow on a \(\beta\)-plane, the relevant conserved quantity is the quasi-geostrophic potential vorticity (QGPV):

\( q_{QG} = \nabla^2\psi + \frac{\partial}{\partial z}\left(\frac{f_0^2}{N^2}\frac{\partial\psi}{\partial z}\right) + \beta y \)

\( \frac{Dq_{QG}}{Dt} = 0 \quad \text{(for adiabatic, inviscid flow)} \)

The conservation of PV provides the theoretical foundation for understanding atmospheric wave propagation, baroclinic instability, and the formation of weather systems. When PV conservation is violated (through diabatic heating or friction), it drives cross-isentropic circulations like the Hadley cell and storm-induced transport.

PV Thinking in Climate

PV invertibility means that given the PV distribution and boundary conditions, the entire flow field can be recovered. Upper-tropospheric PV anomalies (from stratospheric intrusions) can induce surface cyclogenesis—this is PV thinking applied to weather prediction.

In a warming climate, changes to the meridional PV gradient alter Rossby wave propagation, potentially explaining observed changes in jet stream waviness and blocking patterns that drive extreme weather. These atmospheric dynamics directly impact the ecosystems discussed in the Climate & Biodiversity course.

Atmospheric Blocking

Blocking events occur when a persistent high-pressure ridge disrupts the normal westerly flow for days to weeks. From a PV perspective, blocking involves the breaking of Rossby waves and the formation of a cut-off anticyclone. The blocking frequency depends on the PV gradient:

\( \text{Blocking} \propto \frac{|\nabla_y \overline{q}|}{\overline{U}} \)

Blocking is favored when the PV gradient is weak relative to the mean flow

Blocking events are responsible for many extreme weather events: the 2003 European heat wave, 2010 Russian heat wave, and persistent droughts. Arctic amplification may increase blocking frequency by weakening the meridional PV gradient, though this remains debated.

1.8 Stratospheric Dynamics & the Brewer-Dobson Circulation

Above the tropopause, the stratosphere is stably stratified with temperature increasing with height (due to ozone absorption of UV). The residual mean circulation, known as the Brewer-Dobson circulation, is driven by planetary wave breaking:

Air rises slowly in the tropical tropopause layer
Poleward transport by the residual circulation in the stratosphere
Descent in the polar regions (adiabatic warming)

The transformed Eulerian mean framework describes this circulation using the residual velocities \((\overline{v}^*, \overline{w}^*)\):

\( \overline{v}^* = \overline{v} - \frac{\partial}{\partial z}\left(\frac{\overline{v'\theta'}}{\partial\overline{\theta}/\partial z}\right) \)

\( \overline{w}^* = \overline{w} + \frac{1}{a\cos\phi}\frac{\partial}{\partial\phi}\left(\frac{\overline{v'\theta'}\cos\phi}{\partial\overline{\theta}/\partial z}\right) \)

The downward control principle (Haynes et al., 1991) states that the wave-driven force at a given level controls the residual circulation below it:

\( \overline{w}^* = -\frac{1}{\rho_0 a \cos\phi}\frac{\partial}{\partial\phi}\left(\frac{\cos\phi \cdot \nabla\cdot\mathbf{F}}{f}\right) \)

\(\nabla\cdot\mathbf{F}\) is the Eliassen-Palm flux divergence (wave forcing)

Climate change strengthens the Brewer-Dobson circulation (about 2% per decade in models), affecting ozone distribution, stratospheric age-of-air, and the coupling between stratospheric and tropospheric dynamics.

Sudden Stratospheric Warming

Sudden stratospheric warming (SSW) events occur when planetary waves propagate into the stratosphere and break, decelerating or reversing the polar vortex. The stratospheric temperature can rise by 30-50 K in just a few days. SSW events propagate downward, affecting surface weather for weeks afterward:

Mechanism: Rossby wave amplification and breaking in the polar stratosphere
Criterion: zonal mean wind reversal at 60\(°\)N, 10 hPa (Charlton & Polvani, 2007)
Surface impact: negative NAO pattern, cold spells in northern midlatitudes
Frequency: major events occur roughly every other winter in the Northern Hemisphere

Simulation: Thermal Wind & Jet Stream Profile

Computing the zonal wind profile from the thermal wind equation given a meridional temperature gradient:

Python

script.py1 lines

import numpy as np\nimport matplotlib.pyplot as plt\n\nfig, axes = plt.subplots(2, 2, figsize=(14, 10))\nfig.patch.set_facecolor('#0a0a1a')\n\n# Panel 1: Zonal mean temperature cross-section\nax = axes[0, 0]\nlat = np.linspace(-90, 90, 100)\nz_km = np.linspace(0, 25, 80)\nLAT, Z = np.meshgrid(lat, z_km)\n\n# Temperature model: warm tropics, cold poles, tropopause\nT_surface = 300 - 50 * (np.abs(lat)/90)**1.5\nH_trop = 16 - 8 * (np.abs(lat)/90)**2  # tropopause height\nT_field = np.zeros_like(LAT)\nfor j in range(len(lat)):\n    h_t = H_trop[j]\n    for i in range(len(z_km)):\n        if z_km[i] < h_t:\n            T_field[i, j] = T_surface[j] - 6.5 * z_km[i]\n        else:\n            T_field[i, j] = T_surface[j] - 6.5 * h_t + 2 * (z_km[i] - h_t)\n\ncf = ax.contourf(LAT, Z, T_field, levels=20, cmap='RdYlBu_r')\ncb = plt.colorbar(cf, ax=ax, label='Temperature (K)')\ncb.ax.yaxis.label.set_color('white')\ncb.ax.tick_params(colors='white')\nax.plot(lat, H_trop, 'w--', linewidth=2, alpha=0.7, label='Tropopause')\nax.set_xlabel('Latitude', color='white', fontsize=11)\nax.set_ylabel('Altitude (km)', color='white', fontsize=11)\nax.set_title('Zonal Mean Temperature', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\n\n# Panel 2: Thermal wind - zonal wind cross-section\nax = axes[0, 1]\ng = 9.81\nOmega = 7.292e-5\na_earth = 6.371e6\ndy = (lat[1]-lat[0]) * np.pi/180 * a_earth\ndz = (z_km[1]-z_km[0]) * 1000\n\nU_wind = np.zeros_like(T_field)\nfor i in range(1, len(z_km)):\n    for j in range(1, len(lat)-1):\n        phi = lat[j] * np.pi/180\n        f = 2 * Omega * np.sin(phi)\n        if abs(f) > 1e-6:\n            dTdy = (T_field[i, j+1] - T_field[i, j-1]) / (2*dy)\n            T_mean = T_field[i, j]\n            U_wind[i, j] = U_wind[i-1, j] - g/(f * T_mean) * dTdy * dz\n\ncf2 = ax.contourf(LAT, Z, U_wind, levels=np.linspace(-10, 60, 20), cmap='hot')\ncb2 = plt.colorbar(cf2, ax=ax, label='Zonal Wind (m/s)')\ncb2.ax.yaxis.label.set_color('white')\ncb2.ax.tick_params(colors='white')\nax.contour(LAT, Z, U_wind, levels=[30, 40, 50], colors='white', linewidths=0.8)\nax.plot(lat, H_trop, 'c--', linewidth=2, alpha=0.7, label='Tropopause')\nax.set_xlabel('Latitude', color='white', fontsize=11)\nax.set_ylabel('Altitude (km)', color='white', fontsize=11)\nax.set_title('Thermal Wind: Zonal Jet', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\n\n# Panel 3: Rossby wave propagation (snapshot)\nax = axes[1, 0]\nx = np.linspace(0, 40000e3, 500)\nt_vals = [0, 3, 6, 9]  # days\nk_wave = 2*np.pi / 6000e3\nl_wave = 2*np.pi / 4000e3\nbeta = 1.6e-11\nU_mean = 10.0\nomega_rossby = U_mean * k_wave - beta * k_wave / (k_wave**2 + l_wave**2)\n\nfor t_day, color, alpha in zip(t_vals, ['#60a5fa', '#86efac', '#fbbf24', '#f87171'], [0.5, 0.7, 0.85, 1.0]):\n    t_sec = t_day * 86400\n    psi = np.cos(k_wave * x - omega_rossby * t_sec)\n    ax.plot(x/1e3, psi, color=color, linewidth=2, alpha=alpha, label=f'Day {t_day}')\n\ncx = omega_rossby / k_wave\nax.annotate(f'c_x = {cx:.1f} m/s', xy=(20000, 0.8), color='white', fontsize=11)\nax.set_xlabel('Zonal Distance (km)', color='white', fontsize=11)\nax.set_ylabel('Stream function \u03c8', color='white', fontsize=11)\nax.set_title('Rossby Wave Propagation', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\n# Panel 4: Held-Hou Hadley cell width vs parameters\nax = axes[1, 1]\nOmega_val = 7.292e-5\na_val = 6.371e6\nH_vals = np.linspace(8, 20, 5) * 1e3  # tropopause heights\ndelta_h_range = np.linspace(0.1, 0.5, 100)  # fractional temp diff\n\nfor H, color in zip(H_vals, ['#60a5fa', '#86efac', '#fbbf24', '#f87171', '#c084fc']):\n    phi_H = np.sqrt(5/3 * delta_h_range * g * H / (2 * Omega_val**2 * a_val**2))\n    phi_deg = phi_H * 180 / np.pi\n    ax.plot(delta_h_range, phi_deg, color=color, linewidth=2,\n            label=f'H = {H/1e3:.0f} km')\n\nax.axhline(y=30, color='white', linestyle='--', alpha=0.4, label='Observed ~30\u00b0')\nax.set_xlabel('\u0394_h (fractional temp difference)', color='white', fontsize=11)\nax.set_ylabel('Hadley Cell Edge Latitude (\u00b0)', color='white', fontsize=11)\nax.set_title('Held-Hou Model: Cell Width', color='#818cf8', fontsize=13, fontweight='bold')\nax.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#4f46e5', labelcolor='white')\nax.set_facecolor('#0f0f2a')\nax.tick_params(colors='white')\nax.grid(True, alpha=0.2, color='#4f46e5')\n\nplt.tight_layout()\nplt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')\nprint('Thermal wind and jet simulation complete')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

References

Vallis, G. K. (2017). Atmospheric and Oceanic Fluid Dynamics (2nd ed.). Cambridge University Press.
Holton, J. R. & Hakim, G. J. (2013). An Introduction to Dynamic Meteorology (5th ed.). Academic Press.
Held, I. M. & Hou, A. Y. (1980). Nonlinear axially symmetric circulations in a nearly inviscid atmosphere. Journal of the Atmospheric Sciences, 37(3), 515-533.
Eady, E. T. (1949). Long waves and cyclone waves. Tellus, 1(3), 33-52.
Charney, J. G. (1947). The dynamics of long waves in a baroclinic westerly current. Journal of Meteorology, 4(5), 136-162.
Rossby, C.-G. (1939). Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. Journal of Marine Research, 2(1), 38-55.