4.3 Cyclones
Extratropical cyclones are large low-pressure systems that dominate mid-latitude weather. They form along fronts where contrasting air masses meet.
Norwegian Cyclone Model
Stage 1: Initial Perturbation
Small wave forms on polar front. Pressure begins falling.
Stage 2: Development
Wave amplifies, cold and warm fronts develop. Warm sector evident.
Stage 3: Mature
Strongest intensity. Cold front catching up to warm front.
Stage 4: Occlusion & Decay
Cold front overtakes warm front. System fills and weakens.
Vorticity and Cyclone Development
Cyclone intensification is governed by the vorticity equation. The tendency of relative vorticity $\zeta$ is:
$$\frac{\partial \zeta}{\partial t} = -\mathbf{V} \cdot \nabla(\zeta + f) \;-\; (\zeta + f)\,\nabla \cdot \mathbf{V} \;+\; \underbrace{\left(\frac{\partial w}{\partial x}\frac{\partial v}{\partial z} - \frac{\partial w}{\partial y}\frac{\partial u}{\partial z}\right)}_{\text{tilting}} \;+\; \text{friction}$$
The divergence term $-(\zeta+f)\nabla\cdot\mathbf{V}$ is key: upper-level divergence produces convergence at low levels, spinning up the cyclone
The QG height tendency equation predicts how geopotential height changes with time, and hence where cyclones develop and decay:
$$\left(\nabla^2 + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\frac{\partial \Phi}{\partial t} = -f_0\,\mathbf{V}_g \cdot \nabla\left(\zeta_g + f\right) \;-\; \frac{f_0^2}{\sigma}\frac{\partial}{\partial p}\left[-\mathbf{V}_g \cdot \nabla\left(-\frac{\partial \Phi}{\partial p}\right)\right]$$
Positive vorticity advection (PVA) aloft causes height falls (deepening), while warm advection causes thickness increases
QG Omega Equation & Vertical Motion
Vertical motion in cyclones is diagnosed using the QG omega equation. Rising motion ($\omega < 0$) drives clouds and precipitation:
$$\left(\nabla^2 + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\omega = \frac{f_0}{\sigma}\frac{\partial}{\partial p}\left[\mathbf{V}_g \cdot \nabla(\zeta_g + f)\right] + \frac{R}{\sigma p}\nabla^2\left[\mathbf{V}_g \cdot \nabla T\right]$$
Forcing by differential vorticity advection (term 1) and Laplacian of temperature advection (term 2)
Cyclone Structure
Warm Sector
Southerly flow, warm moist air
Cold Sector
NW flow, cold dry air
Triple Point
Junction of cold, warm, occluded fronts
Baroclinic Instability & Growth Rates
Extratropical cyclones grow by extracting available potential energy from the mean baroclinic state (the meridional temperature gradient). The Eady model gives the maximum growth rate:
$$\sigma_{\max} = 0.31 \frac{f}{N}\left|\frac{\partial u}{\partial z}\right| = 0.31 \frac{f}{N}\frac{\Lambda}{H}$$
where $f$ is the Coriolis parameter, $N$ is the Brunt-Vaisala frequency, and $\partial u/\partial z$ is the vertical wind shear (related to the temperature gradient via thermal wind)
The corresponding e-folding time is $\tau = 1/\sigma_{\max}$, typically about 1.5--2.5 days. The Petterssen-Smebye Type B development model quantifies deepening rate:
$$\frac{\partial p_s}{\partial t} \propto -f_0\left[\mathbf{V}_g \cdot \nabla(\zeta_g + f)\right]_{500} - \frac{R}{p}\left[\nabla^2(\mathbf{V}_g \cdot \nabla T)\right]_{1000\text{--}500}$$
Surface pressure deepening is driven by 500 hPa vorticity advection and lower-tropospheric warm advection; a drop of 24 hPa in 24 h classifies a system as a "bomb cyclone"
Interactive Simulation: Extratropical Cyclone Pressure Field
PythonCreates an idealized cyclone pressure field, calculates gradient wind with boundary-layer convergence, and plots isobars with wind vectors, schematic fronts, and a convergence/divergence field showing where rising motion drives cloud formation.
Click Run to execute the Python code
Code will be executed with Python 3 on the server