Atmospheric dynamics¶

working pillar 1 · Physical modeling and domain theory

In one minute

Rossby number decides the regime. \(\mathrm{Ro}\ll1\) is rotation-dominated and balanced; \(\mathrm{Ro}\sim1\) is not, and balance arguments stop applying.
Geostrophic balance — pressure gradient against Coriolis — explains why synoptic winds blow along isobars rather than across them.
Thermal wind links vertical wind shear to horizontal temperature gradient. It is why the jet stream sits above the strongest baroclinic zone.
Potential vorticity is the master variable: conserved following the flow in adiabatic, frictionless conditions, and it determines the balanced flow via inversion.
Rossby waves exist because of \(\beta\), and they propagate westward relative to the mean flow — which is why weather systems can appear to move slowly eastward.

Key results¶

Scale analysis:

\[ \mathrm{Ro} = \frac{U}{fL}, \qquad f = 2\Omega\sin\phi \approx 10^{-4}\ \mathrm{s^{-1}}\ \text{(midlatitudes)} \]

System	\(L\)	\(U\)	\(\mathrm{Ro}\)	Balanced?
Synoptic cyclone	1000 km	10 m/s	0.1	yes
Hurricane core	50 km	50 m/s	10	no
Thunderstorm	10 km	30 m/s	30	no
Ocean gyre	1000 km	0.1 m/s	0.001	strongly

Geostrophic and gradient wind:

\[ f\,\mathbf{k}\times\mathbf{u}_g = -\frac{1}{\rho}\nabla_h p, \qquad \frac{V^2}{R} + fV = \frac{1}{\rho}\left|\nabla_h p\right| \]

The curvature term makes gradient wind subgeostrophic in troughs and supergeostrophic in ridges — a 10–20% effect that matters for trajectory work.

Thermal wind:

\[ \frac{\partial \mathbf{u}_g}{\partial \ln p} = -\frac{R_d}{f}\,\mathbf{k}\times\nabla_p T \]

Cold air to the north gives westerly shear increasing with height: the midlatitude jet.

Vorticity and potential vorticity:

\[ \zeta = \mathbf{k}\cdot(\nabla\times\mathbf{u}), \qquad \frac{D}{Dt}\left(\frac{\zeta + f}{H}\right) = 0 \quad\text{(shallow water)} \]

\[ q = \frac{1}{\rho}(\zeta_a\cdot\nabla\theta) \quad\text{(Ertel PV, conserved adiabatically)} \]

The PV perspective: given the PV distribution plus boundary \(\theta\), the balanced wind and mass fields follow by inversion. A single conserved scalar contains the balanced dynamics, which is why PV maps are how dynamicists read the atmosphere.

Quasi-geostrophic omega equation — where vertical motion comes from:

\[ \left(\nabla^2 + \frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\right)\omega = \frac{f_0}{\sigma}\frac{\partial}{\partial p}\left[\mathbf{u}_g\cdot\nabla(\zeta_g+f)\right] + \frac{R}{\sigma p}\nabla^2\left[\mathbf{u}_g\cdot\nabla T\right] \]

Two forcings: differential vorticity advection, and thermal advection. Ascent (\(\omega<0\)) where positive vorticity advection increases with height, or where warm advection occurs.

Rossby waves. With \(\beta = df/dy\):

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

Always westward relative to the mean flow, and longer waves propagate westward faster — so long waves can be quasi-stationary or retrograde while short waves move east with the flow.

Mental model¶

Rotation is the organising fact. On a rotating planet, a pressure gradient does not produce flow down the gradient — it produces flow along it, with low pressure to the left in the southern hemisphere and to the right... which is exactly the kind of sign convention worth deriving rather than recalling. (Northern hemisphere: low pressure to the left of the wind.)

PV thinking is the most powerful single idea here. Instead of tracking winds and temperatures separately, track one conserved scalar. An upper-level PV anomaly "induces" a circulation beneath it, and the whole life cycle of a midlatitude cyclone becomes the interaction of an upper PV anomaly with a surface temperature anomaly.

Numerics / practice¶

Check \(\mathrm{Ro}\) before applying any balance argument. Balanced diagnostics on a convective scale are meaningless.
Use gradient rather than geostrophic wind in strongly curved flow; the error is systematic, not random.
PV on isentropic surfaces is the natural frame — that is where the conservation statement is clean.
Diabatic heating destroys PV conservation, and that is often the point: latent heating generates low-level PV, which is central to cyclone development.

Failure modes

Geostrophy applied near the equator. \(f\to0\) at the equator, so the geostrophic wind diverges. Equatorial dynamics is a different regime — Kelvin and mixed Rossby-gravity waves, not balanced flow. Roughly, geostrophic reasoning is unsafe within about 10° of the equator.

Geostrophic wind used in strongly curved flow. In a sharp trough the gradient wind is significantly subgeostrophic. Using \(u_g\) to advect a parcel through a trough overestimates its speed, and errors accumulate along a trajectory.

PV conservation assumed through diabatic processes. Latent heating, radiation and friction all generate or destroy PV. In a developing cyclone, diabatic PV generation in the warm conveyor belt is a first-order effect, not a correction.

QG applied outside its scaling. QG assumes \(\mathrm{Ro}\ll1\), small Rossby number and small relative depth changes. It fails for fronts (where along-front scales collapse), for hurricanes and for anything convective. Semigeostrophic or full primitive equations are needed.

Sign errors in hemisphere-dependent relations. \(f<0\) in the southern hemisphere flips the sense of geostrophic flow, Ekman turning and vorticity conventions. Code written and tested only on northern-hemisphere cases fails silently south of the equator.

Vorticity computed on a coarse grid. \(\zeta\) is a derivative, so it amplifies noise and is strongly resolution-dependent. Comparing vorticity fields between models at different resolutions compares smoothing as much as dynamics.

Worked example¶

Rossby number as the regime test, computed rather than recalled:

import numpy as np

Omega = 7.292e-5
for name, lat, L_km, U in (("synoptic cyclone", 45.0, 1000.0, 10.0),
                           ("hurricane core",   20.0,   50.0, 50.0),
                           ("thunderstorm",     40.0,   10.0, 30.0),
                           ("sea breeze",       35.0,   30.0,  5.0)):
    f  = 2 * Omega * np.sin(np.deg2rad(lat))
    Ro = U / (f * L_km * 1e3)
    verdict = "balanced" if Ro < 0.3 else ("marginal" if Ro < 3 else "NOT balanced")
    print(f"{name:18s} f={f:.2e}  Ro={Ro:8.3f}   {verdict}")

synoptic cyclone   f=1.03e-04  Ro=   0.097   balanced
hurricane core     f=4.99e-05  Ro=  20.048   NOT balanced
thunderstorm       f=9.37e-05  Ro=  32.002   NOT balanced
sea breeze         f=8.37e-05  Ro=   1.992   marginal

The sea breeze is the interesting case: marginal, so rotation matters but does not dominate — which is why sea-breeze fronts turn with time over a day rather than propagating straight inland.

Connections¶

Thermodynamics — static stability enters the QG equations as \(\sigma\).
NWP — which of these balances a dynamical core respects.
Boundary layer — Ekman balance and friction.
Fluids › Governing equations — the parent equations.

Sources¶

Holton & Hakim, An Introduction to Dynamic Meteorology.
Vallis, Atmospheric and Oceanic Fluid Dynamics.
Hoskins, McIntyre & Robertson (1985) — the PV perspective paper.
Archive: see the course archive.