Part II: Atmospheric Dynamics

Equations of motion, forces, and circulation in Earth's rotating atmosphere

1. Equations of Motion

1.1 Newton's Second Law in Rotating Frame

The fundamental equation governing atmospheric motion is Newton's second law. However, because Earth rotates, we must express the equations in a rotating reference frame (fixed to Earth's surface). This introduces apparent forces.

\(\frac{D\mathbf{V}}{Dt} = -2\mathbf{\Omega}\times\mathbf{V} - \mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r}) - \frac{1}{\rho}\nabla p - g\mathbf{k} + \mathbf{F}_{\text{friction}}\)
\(D\mathbf{V}/Dt\): Total (material) derivative of velocity
\(-2\mathbf{\Omega}\times\mathbf{V}\): Coriolis force (per unit mass)
\(-\mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r})\): Centrifugal force
\(-(1/\rho)\nabla p\): Pressure gradient force
\(-g\mathbf{k}\): Gravitational force
\(\mathbf{F}_{\text{friction}}\): Frictional forces

where \(\mathbf{\Omega}\) is Earth's angular velocity vector:

\(\Omega = 7.292 \times 10^{-5}\) rad/s (one rotation per sidereal day = 23h 56m 4s)

1.2 Component Form

In spherical coordinates (longitude \(\lambda\), latitude \(\varphi\), height \(z\)) with velocity components \((u, v, w)\):

Zonal (east-west) momentum:
\(\frac{Du}{Dt} - \frac{uv \tan \varphi}{a} + \frac{uw}{a} = -\frac{1}{\rho a \cos \varphi}\frac{\partial p}{\partial \lambda} + 2\Omega v \sin \varphi - 2\Omega w \cos \varphi + F_\lambda\)
Meridional (north-south) momentum:
\(\frac{Dv}{Dt} + \frac{u^2 \tan \varphi}{a} + \frac{vw}{a} = -\frac{1}{\rho a}\frac{\partial p}{\partial \varphi} - 2\Omega u \sin \varphi + F_\varphi\)
Vertical momentum:
\(\frac{Dw}{Dt} - \frac{u^2 + v^2}{a} = -\frac{1}{\rho}\frac{\partial p}{\partial z} - g + 2\Omega u \cos \varphi + F_z\)
where \(a \approx 6.37 \times 10^6\) m is Earth's radius

📘 Material Derivative:

The material derivative \(D/Dt\) represents the rate of change following a moving air parcel:

\(\frac{D}{Dt} = \frac{\partial}{\partial t} + u\frac{\partial}{\partial x} + v\frac{\partial}{\partial y} + w\frac{\partial}{\partial z}\)

= local change + advection by the flow

2. Coriolis & Pressure Gradient Forces

2.1 The Coriolis Force

The Coriolis force arises from Earth's rotation and acts perpendicular to the velocity vector. It's the dominant force for large-scale atmospheric motions.

\(\mathbf{F}_{\text{Coriolis}} = -2\mathbf{\Omega}\times\mathbf{V}\)
In component form (Northern Hemisphere, \(f = 2\Omega \sin \varphi > 0\)):
\(F_x = f v\) (deflects eastward flow to the right/south)
\(F_y = -f u\) (deflects northward flow to the right/east)

The Coriolis parameter varies with latitude:

\(f = 2\Omega \sin \varphi\)
At equator (\(\varphi = 0°\)): \(f = 0\)
At 30°N: \(f = 7.29 \times 10^{-5}\) s⁻¹
At 45°N: \(f = 1.03 \times 10^{-4}\) s⁻¹
At poles (\(\varphi = \pm 90°\)): \(f = \pm 1.46 \times 10^{-4}\) s⁻¹

Northern Hemisphere (f > 0)

Coriolis deflects motion to the right of the direction of travel. Air flowing from high to low pressure curves clockwise around lows (cyclones) and counterclockwise around highs (anticyclones).

Southern Hemisphere (f < 0)

Coriolis deflects motion to the left. Flow patterns are reversed: counterclockwise around lows, clockwise around highs.

Equator (f = 0)

No Coriolis deflection. Hurricanes cannot form at the equator because there's no Coriolis force to induce rotation.

2.2 Pressure Gradient Force

The pressure gradient force (PGF) accelerates air from high to low pressure:

\(\mathbf{F}_{\text{PGF}} = -\frac{1}{\rho}\nabla p\)
In Cartesian coordinates:
\(F_x = -\frac{1}{\rho}\frac{\partial p}{\partial x}\)
\(F_y = -\frac{1}{\rho}\frac{\partial p}{\partial y}\)
\(F_z = -\frac{1}{\rho}\frac{\partial p}{\partial z}\)

On weather maps, pressure gradients are shown by isobars (lines of constant pressure). Closely spaced isobars indicate strong pressure gradient and strong winds.

🌀 Balance of Forces:

For large-scale atmospheric flow, the primary balance is between the Coriolis force and the pressure gradient force. This balance defines geostrophic wind (next chapter).

2.3 Rossby Number

The Rossby number measures the relative importance of inertial forces to Coriolis forces:

\(Ro = \frac{U}{f L}\)
where:
\(U\) = characteristic velocity scale
\(L\) = characteristic length scale
\(f\) = Coriolis parameter

\(Ro \ll 1\) (Geostrophic balance)

Coriolis dominates. Applies to synoptic-scale weather systems (\(L \sim 1000\) km). Flow is nearly geostrophic.

\(Ro \sim 1\) (Ageostrophic)

Inertia and Coriolis comparable. Mesoscale phenomena (\(L \sim 10\)-100 km): thunderstorms, sea breezes, fronts.

Part II Summary

Atmospheric dynamics governs the motion of air masses and weather systems. Key concepts include:

  • ✓ Equations of motion in rotating reference frame
  • ✓ Coriolis force deflects motion (right in NH, left in SH)
  • ✓ Geostrophic balance between PGF and Coriolis
  • ✓ Thermal wind relates vertical wind shear to temperature gradient
  • ✓ Vorticity and circulation quantify atmospheric rotation
  • ✓ Atmospheric waves transport energy and momentum

Note: This is a condensed version. Full content for chapters 3-6 (Geostrophic Wind, Thermal Wind, Vorticity & Circulation, Atmospheric Waves) will follow the same detailed treatment.