Part II: Atmospheric Dynamics

Equations of motion, forces, geostrophic and thermal wind balance, vorticity, circulation, and wave dynamics in Earth's rotating atmosphere

1. Equations of Motion

1.1 Newton's Second Law in a Rotating Reference Frame

The atmosphere is a thin fluid envelope bound to a rotating planet. To describe the motion of air parcels as observed from Earth's surface, we must transform Newton's second law from an inertial (fixed-star) reference frame to a non-inertial (rotating) frame. This transformation introduces two apparent (fictitious) forces: the Coriolis force and the centrifugal force.

Let $\mathbf{V}_I$ denote velocity in the inertial frame and $\mathbf{V}$ velocity relative to Earth. The relationship between the time derivatives in the two frames is given by the operator identity:

$$\left(\frac{d}{dt}\right)_I = \left(\frac{d}{dt}\right)_R + \mathbf{\Omega}\times$$

Applying this twice to the position vector $\mathbf{r}$ yields the full acceleration transformation. The vector equation of motion in the rotating frame is:

$$\frac{D\mathbf{V}}{Dt} = -\frac{1}{\rho}\nabla p - g\mathbf{k} - 2\mathbf{\Omega}\times\mathbf{V} - \mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r}) + \mathbf{F}_{\text{fric}}$$

$D\mathbf{V}/Dt$: Total (material/Lagrangian) derivative of velocity following the air parcel

$-\frac{1}{\rho}\nabla p$: Pressure gradient force per unit mass

$-g\mathbf{k}$: Gravitational acceleration (directed downward)

$-2\mathbf{\Omega}\times\mathbf{V}$: Coriolis acceleration (apparent force in rotating frame)

$-\mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r})$: Centrifugal acceleration (absorbed into effective gravity)

$\mathbf{F}_{\text{fric}}$: Frictional (viscous) forces per unit mass

In practice, the centrifugal acceleration is combined with true gravity $\mathbf{g}^*$ to define effective gravity:

$$\mathbf{g} = \mathbf{g}^* - \mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r})$$

Effective gravity includes both Newtonian gravitation and the outward centrifugal effect. The geoid (mean sea level) is an equipotential surface of $\mathbf{g}$.

The angular velocity of Earth is:

$\Omega = \frac{2\pi}{T_{\text{sidereal}}} = \frac{2\pi}{86164 \text{ s}} = 7.292 \times 10^{-5}$ rad s$^{-1}$

1.2 Full 3D Momentum Equations in Spherical Coordinates

Using spherical coordinates (longitude $\lambda$, latitude $\varphi$, and height $z$ above the geoid) with velocity components $u$ (zonal/eastward), $v$ (meridional/northward), and $w$ (vertical/upward), and Earth radius $a \approx 6.371 \times 10^6$ m, the three component equations of motion are:

Zonal (east-west) momentum equation:

$$\frac{Du}{Dt} - \frac{uv\tan\varphi}{a} + \frac{uw}{a} = -\frac{1}{\rho a\cos\varphi}\frac{\partial p}{\partial \lambda} + 2\Omega v\sin\varphi - 2\Omega w\cos\varphi + F_\lambda$$

Meridional (north-south) momentum equation:

$$\frac{Dv}{Dt} + \frac{u^2\tan\varphi}{a} + \frac{vw}{a} = -\frac{1}{\rho a}\frac{\partial p}{\partial \varphi} - 2\Omega u\sin\varphi + F_\varphi$$

Vertical momentum equation:

$$\frac{Dw}{Dt} - \frac{u^2 + v^2}{a} = -\frac{1}{\rho}\frac{\partial p}{\partial z} - g + 2\Omega u\cos\varphi + F_z$$

The terms $uv\tan\varphi/a$, $u^2\tan\varphi/a$, $uw/a$, and $vw/a$ are metric terms (curvature terms) arising from the use of spherical coordinates.

Material Derivative in Spherical Coordinates:

The total derivative following an air parcel is:

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + \frac{u}{a\cos\varphi}\frac{\partial}{\partial \lambda} + \frac{v}{a}\frac{\partial}{\partial \varphi} + w\frac{\partial}{\partial z}$$

This decomposes into local tendency plus advection by the three-dimensional wind field.

1.3 Scale Analysis and Dominant Balances

The full equations of motion contain many terms. Scale analysis is the systematic procedure by which we assign typical magnitudes to each variable and determine which terms dominate for a given class of motions. For synoptic-scale midlatitude weather systems, we use:

Variable	Symbol	Typical Scale
Horizontal velocity	$U$	10 m s$^{-1}$
Vertical velocity	$W$	0.01 m s$^{-1}$
Horizontal length scale	$L$	$10^6$ m (1000 km)
Vertical length scale	$H$	$10^4$ m (10 km)
Time scale	$T = L/U$	$10^5$ s (~1 day)
Pressure perturbation	$\delta p$	$10^3$ Pa (10 hPa)
Coriolis parameter	$f_0$	$10^{-4}$ s$^{-1}$

Applying these scales to the horizontal momentum equation, the dominant balance is between the Coriolis term and the pressure gradient force, both of order $10^{-3}$ m s$^{-2}$. The acceleration$Du/Dt$ is an order of magnitude smaller ($\sim 10^{-4}$ m s$^{-2}$). This gives us geostrophic balance as the leading-order approximation.

For the vertical momentum equation, scale analysis reveals that the dominant balance is between the vertical pressure gradient and gravity, both of order $10$ m s$^{-2}$, which yields the hydrostatic approximation:

$$\frac{\partial p}{\partial z} = -\rho g$$

Valid for motions with horizontal scale $L \gg H$ (aspect ratio $H/L \ll 1$).

1.4 Simplified Equations on the f-Plane and Beta-Plane

For synoptic-scale midlatitude motions, the curvature and metric terms are small compared to the Coriolis and pressure gradient terms. Using Cartesian coordinates $(x, y, z)$ tangent to the sphere at latitude $\varphi_0$, and neglecting metric terms, the primitive equations become:

$$\frac{Du}{Dt} = -\frac{1}{\rho}\frac{\partial p}{\partial x} + fv + F_x$$

$$\frac{Dv}{Dt} = -\frac{1}{\rho}\frac{\partial p}{\partial y} - fu + F_y$$

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

These are the starting point for most dynamic meteorology analyses.

On the f-plane, we take $f = f_0 = 2\Omega\sin\varphi_0 = \text{const}$, appropriate for scales $L \ll a$. On the beta-plane, we account for the latitudinal variation of $f$:

$$f = f_0 + \beta y, \quad \text{where } \beta = \frac{df}{dy}\bigg|_{\varphi_0} = \frac{2\Omega\cos\varphi_0}{a}$$

At 45N: $\beta \approx 1.62 \times 10^{-11}$ m$^{-1}$ s$^{-1}$. The beta-plane is essential for Rossby wave dynamics.

1.5 The Boussinesq Approximation

The Boussinesq approximation simplifies the equations by recognizing that density variations in the atmosphere are small relative to the mean density, except where they multiply gravity (i.e., in the buoyancy term). We decompose the density and pressure into a reference state and perturbations:

$$\rho = \bar{\rho}(z) + \rho'(x,y,z,t), \quad |\rho'| \ll \bar{\rho}$$

$$p = \bar{p}(z) + p'(x,y,z,t), \quad \frac{d\bar{p}}{dz} = -\bar{\rho} g$$

Under Boussinesq, the momentum equations become:

$$\frac{Du}{Dt} = -\frac{1}{\bar{\rho}}\frac{\partial p'}{\partial x} + fv$$

$$\frac{Dv}{Dt} = -\frac{1}{\bar{\rho}}\frac{\partial p'}{\partial y} - fu$$

$$\frac{Dw}{Dt} = -\frac{1}{\bar{\rho}}\frac{\partial p'}{\partial z} - \frac{\rho'}{\bar{\rho}}g$$

The buoyancy term $-\rho'g/\bar{\rho} = b$ drives convective motions. The continuity equation simplifies to the incompressible form: $\nabla \cdot \mathbf{V} = 0$.

Buoyancy Frequency (Brunt-Vaisala Frequency):

The key parameter governing vertical stability in the Boussinesq framework is:

$$N^2 = -\frac{g}{\bar{\rho}}\frac{d\bar{\rho}}{dz} = \frac{g}{\theta}\frac{d\theta}{dz}$$

where $\theta$ is potential temperature. For the troposphere, $N \approx 0.01$ s$^{-1}$(period $\sim$ 10 min). When $N^2 > 0$, the atmosphere is statically stable; displaced parcels oscillate vertically. When $N^2 < 0$, the atmosphere is convectively unstable.

2. Coriolis and Pressure Gradient Forces

2.1 Derivation of the Coriolis Force on a Rotating Earth

The Coriolis acceleration arises purely from the transformation between inertial and rotating reference frames. Consider a parcel with velocity $\mathbf{V}$ relative to the rotating Earth. The Coriolis acceleration is:

$$\mathbf{a}_{\text{Cor}} = -2\mathbf{\Omega}\times\mathbf{V}$$

Earth's angular velocity vector $\mathbf{\Omega}$ points from the South Pole to the North Pole along the rotation axis. At latitude $\varphi$, we resolve $\mathbf{\Omega}$ into local vertical and horizontal components:

$$\mathbf{\Omega} = \Omega\cos\varphi\,\hat{\mathbf{j}} + \Omega\sin\varphi\,\hat{\mathbf{k}}$$

where $\hat{\mathbf{j}}$ points north and $\hat{\mathbf{k}}$ points radially outward (upward).

For velocity $\mathbf{V} = u\hat{\mathbf{i}} + v\hat{\mathbf{j}} + w\hat{\mathbf{k}}$, the cross product $-2\mathbf{\Omega}\times\mathbf{V}$ yields the component Coriolis accelerations:

$$a_{\text{Cor},x} = 2\Omega v\sin\varphi - 2\Omega w\cos\varphi = fv - \tilde{f}w$$

$$a_{\text{Cor},y} = -2\Omega u\sin\varphi = -fu$$

$$a_{\text{Cor},z} = 2\Omega u\cos\varphi = \tilde{f}u$$

where $f = 2\Omega\sin\varphi$ is the Coriolis parameter and $\tilde{f} = 2\Omega\cos\varphi$ is the reciprocal Coriolis parameter. For large-scale motions, $\tilde{f}$ terms involving $w$ are negligible.

Key Properties of the Coriolis Force:

Acts perpendicular to the velocity; it deflects but does no work (no change in kinetic energy)
Deflects motion to the right in the Northern Hemisphere ($f > 0$), to the left in the Southern Hemisphere ($f < 0$)
Vanishes at the equator ($f = 0$) and is maximum at the poles ($|f| = 2\Omega$)
Is proportional to the speed of the parcel (zero Coriolis force on stationary air)
Is an apparent force with no reaction force -- it violates Newton's third law

2.2 Coriolis Parameter and the Beta-Effect

The Coriolis parameter varies sinusoidally with latitude:

$f = 2\Omega\sin\varphi$

At equator ($\varphi = 0^\circ$): $f = 0$

At 30N: $f \approx 7.29 \times 10^{-5}$ s$^{-1}$

At 45N: $f \approx 1.03 \times 10^{-4}$ s$^{-1}$

At poles ($\varphi = \pm 90^\circ$): $f = \pm 1.46 \times 10^{-4}$ s$^{-1}$

The meridional gradient of $f$ is crucial for planetary wave dynamics. The Rossby parameter $\beta$ is defined as:

$$\beta = \frac{df}{dy} = \frac{2\Omega\cos\varphi}{a} \approx 1.62 \times 10^{-11} \text{ m}^{-1}\text{s}^{-1} \quad (\text{at } 45^\circ\text{N})$$

2.3 Centrifugal Force vs. Coriolis Force

Both are apparent forces arising from the rotating reference frame, but they have fundamentally different characters:

Centrifugal Force

$$\mathbf{F}_{\text{cent}} = -\mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r})$$

Directed radially outward from Earth's rotation axis. Maximum at the equator, zero at the poles. Depends only on position, not velocity. Magnitude: $\Omega^2 a\cos\varphi \approx 0.034$ m s$^{-2}$ at the equator (about 0.3% of $g$).

Absorbed into effective gravity and the shape of the geoid. Does not appear explicitly in the equations.

Coriolis Force

$$\mathbf{F}_{\text{Cor}} = -2\mathbf{\Omega}\times\mathbf{V}$$

Directed perpendicular to the velocity. Depends on the velocity of the parcel, not its position. Only affects moving objects. For a 10 m/s wind at 45N: $|fV| \approx 10^{-3}$ m s$^{-2}$.

The dominant force balance partner (with PGF) for large-scale weather systems. Explicitly present in equations of motion.

2.4 Pressure Gradient Force

The pressure gradient force (PGF) is the real force that initiates atmospheric motion. Air is accelerated from regions of high pressure toward regions of low pressure:

$$\mathbf{F}_{\text{PGF}} = -\frac{1}{\rho}\nabla p$$

In Cartesian components:

$$F_x = -\frac{1}{\rho}\frac{\partial p}{\partial x}, \quad F_y = -\frac{1}{\rho}\frac{\partial p}{\partial y}, \quad F_z = -\frac{1}{\rho}\frac{\partial p}{\partial z}$$

2.5 Pressure Gradient Force in Pressure Coordinates

In meteorology, it is often advantageous to use pressure $p$ as the vertical coordinate instead of height $z$. This eliminates density from the horizontal pressure gradient. Using the hydrostatic relation and the coordinate transformation:

$$\left(\frac{\partial p}{\partial x}\right)_z = -\rho g \left(\frac{\partial z}{\partial x}\right)_p = -\rho \left(\frac{\partial \Phi}{\partial x}\right)_p$$

where $\Phi = gz$ is the geopotential. The horizontal momentum equations in pressure coordinates become elegantly simple:

$$\frac{Du}{Dt} = -\frac{\partial \Phi}{\partial x} + fv$$

$$\frac{Dv}{Dt} = -\frac{\partial \Phi}{\partial y} - fu$$

The density $\rho$ has disappeared from the horizontal gradient! This is a major simplification and one of the key reasons pressure coordinates are preferred in synoptic meteorology. The vertical coordinate is now $p$, and vertical velocity is $\omega = Dp/Dt$ (Pa s$^{-1}$), with downward motion corresponding to $\omega > 0$.

2.6 The Rossby Number

The Rossby number is a dimensionless number measuring the ratio of inertial (acceleration) to Coriolis forces:

$$Ro = \frac{U}{fL}$$

$U$ = characteristic horizontal velocity scale

$L$ = characteristic horizontal length scale

$f$ = Coriolis parameter

$Ro \ll 1$

Geostrophic regime

Synoptic-scale ($L \sim 1000$ km). Coriolis dominates inertia. Flow is nearly geostrophic. Example: extratropical cyclones.

$Ro \sim 1$

Ageostrophic regime

Mesoscale ($L \sim 10$--100 km). Coriolis and inertia comparable. Examples: sea breezes, fronts, squall lines.

$Ro \gg 1$

Cyclostrophic regime

Microscale/convective ($L \sim 1$ km). Inertia dominates. Coriolis negligible. Examples: tornadoes, dust devils.

3. Geostrophic Wind

3.1 Geostrophic Balance Derivation

For large-scale motions in the free atmosphere (above the boundary layer) where $Ro \ll 1$, the acceleration terms are negligible compared to the Coriolis and pressure gradient forces. Setting$Du/Dt \approx 0$ and $Dv/Dt \approx 0$ in the horizontal momentum equations yields geostrophic balance:

Geostrophic Wind Equations (height coordinates):

$$fv_g = \frac{1}{\rho}\frac{\partial p}{\partial x} \quad \Rightarrow \quad v_g = \frac{1}{f\rho}\frac{\partial p}{\partial x}$$

$$fu_g = -\frac{1}{\rho}\frac{\partial p}{\partial y} \quad \Rightarrow \quad u_g = -\frac{1}{f\rho}\frac{\partial p}{\partial y}$$

In pressure coordinates, using the geopotential $\Phi = gz$, the geostrophic wind is expressed without density:

Geostrophic Wind in Pressure Coordinates:

$$u_g = -\frac{1}{f}\frac{\partial \Phi}{\partial y}, \qquad v_g = \frac{1}{f}\frac{\partial \Phi}{\partial x}$$

Or in compact vector form:

$$\vec{V}_g = \frac{1}{f}\hat{k}\times\nabla_p\Phi$$

Physical Interpretation:

The geostrophic wind blows parallel to isobars (or height contours on a constant-pressure surface), not across them
In the Northern Hemisphere, low pressure is to the left of the geostrophic wind direction
Closely spaced isobars (strong pressure gradient) correspond to strong geostrophic winds
Geostrophic balance breaks down near the equator ($f \to 0$) and for highly curved flow

3.2 Geostrophic Wind in Natural Coordinates

Natural (streamline) coordinates $(s, n)$ are aligned with the flow direction. $s$ is along the flow and $n$ is perpendicular (positive to the left). In these coordinates, for flow along isobars, the geostrophic balance normal to the flow is simply:

$$fV_g = -\frac{1}{\rho}\frac{\partial p}{\partial n} = -\frac{\partial \Phi}{\partial n}$$

The geostrophic wind speed is proportional to the cross-flow pressure gradient.

3.3 Gradient Wind Balance

When flow curvature is significant (e.g., around cyclones and anticyclones), the centripetal acceleration must be retained. The gradient wind includes this curvature effect. In natural coordinates, the balance normal to the flow is:

$$\frac{V^2}{R} + fV = -\frac{1}{\rho}\frac{\partial p}{\partial n}$$

where $R$ is the radius of curvature of the streamline (positive for cyclonic curvature).

Solving this quadratic for $V$:

$$V = -\frac{fR}{2} \pm \sqrt{\frac{f^2R^2}{4} - R\frac{\partial \Phi}{\partial n}}$$

Cyclonic Flow ($R > 0$)

The gradient wind is subgeostrophic: $V_{\text{gr}} < V_g$. The centripetal acceleration supplements the Coriolis force, so less wind speed is needed to balance the PGF. This explains why observed cyclone winds are weaker than geostrophic estimates.

Anticyclonic Flow ($R < 0$)

The gradient wind is supergeostrophic: $V_{\text{gr}} > V_g$. The centripetal acceleration opposes the Coriolis force, requiring stronger wind. There is also an upper bound on the anticyclonic gradient wind: $V_{\text{max}} = -fR/2$ (regularity condition).

3.4 Ageostrophic Wind Components

The actual wind $\mathbf{V}$ is not exactly geostrophic. The departure from geostrophic balance is the ageostrophic wind:

$$\mathbf{V} = \mathbf{V}_g + \mathbf{V}_a$$

The ageostrophic component $\mathbf{V}_a$ is small but dynamically crucial. It is responsible for:

Vertical motion: Through the continuity equation, convergence/divergence of $\mathbf{V}_a$ drives vertical motion $\omega$
Cross-isobar flow: Friction in the boundary layer creates an ageostrophic component directed from high to low pressure
Jet streak circulations: Accelerating/decelerating flow in jet streaks creates ageostrophic transverse circulations
Frontogenesis: Ageostrophic circulations maintain thermal wind balance during frontogenesis

From the full momentum equation, the ageostrophic wind can be expressed as:

$$\mathbf{V}_a = \frac{1}{f}\hat{k}\times\frac{D\mathbf{V}_g}{Dt}$$

The ageostrophic wind is proportional to the Lagrangian acceleration of the geostrophic wind.

3.5 Isallobaric Wind

The isallobaric wind is one component of the ageostrophic wind that arises from the local time tendency of pressure:

$$\mathbf{V}_{\text{isall}} = -\frac{1}{f^2}\nabla\frac{\partial \Phi}{\partial t}$$

This wind blows down the gradient of pressure tendency. When pressure is falling rapidly (deepening cyclone), the isallobaric wind converges toward the center, contributing to low-level convergence and upward motion.

4. Thermal Wind

4.1 Derivation from Geostrophic and Hydrostatic Balance

The thermal wind is not a real wind but describes the vertical shear of the geostrophic wind. It arises from combining two fundamental balances: geostrophic balance (horizontal) and hydrostatic balance (vertical). This linkage is one of the most powerful concepts in dynamic meteorology.

Start with the geostrophic wind in pressure coordinates:

$$u_g = -\frac{1}{f}\frac{\partial \Phi}{\partial y}, \qquad v_g = \frac{1}{f}\frac{\partial \Phi}{\partial x}$$

Now differentiate with respect to $\ln p$ (or equivalently $p$). Using the hydrostatic equation in pressure coordinates, $\partial\Phi/\partial p = -RT/p$ (from the hypsometric relation), and the fact that $f$ is independent of $p$:

$$\frac{\partial u_g}{\partial \ln p} = -\frac{1}{f}\frac{\partial}{\partial y}\frac{\partial \Phi}{\partial \ln p} = \frac{R}{f}\frac{\partial T}{\partial y}$$

$$\frac{\partial v_g}{\partial \ln p} = \frac{1}{f}\frac{\partial}{\partial x}\frac{\partial \Phi}{\partial \ln p} = -\frac{R}{f}\frac{\partial T}{\partial x}$$

In vector form, the thermal wind equation is:

$$\frac{\partial \vec{V}_g}{\partial \ln p} = -\frac{R}{f}\hat{k}\times\nabla_p T$$

Since $\ln p$ decreases upward, the geostrophic wind changes in the vertical such that the thermal wind blows parallel to isotherms with cold air to the left (NH).

Integrating between two pressure levels $p_1$ (lower) and $p_0$ (upper, $p_0 < p_1$), the thermal wind vector is defined as:

$$\vec{V}_T = \vec{V}_g(p_0) - \vec{V}_g(p_1) = -\frac{R}{f}\hat{k}\times\nabla_p\bar{T}\,\ln\left(\frac{p_1}{p_0}\right)$$

where $\bar{T}$ is the mean temperature of the layer. The thermal wind is parallel to the mean isotherms with cold air to the left (in the Northern Hemisphere).

4.2 Wind Shear and Temperature Gradient

The thermal wind equation establishes a fundamental linkage: horizontal temperature gradients produce vertical wind shear. This relationship has profound consequences:

Stronger temperature gradient = stronger wind shear: The strong pole-to-equator temperature gradient in the troposphere produces a westerly wind that increases with height, culminating in the jet stream near the tropopause.
Jet stream location: The polar front jet stream is located directly above the strongest surface baroclinic zone (the polar front). At 300 hPa, jet stream speeds can exceed 50--100 m s$^{-1}$.
Subtropical jet: The strong temperature gradient at the poleward edge of the Hadley cell produces the subtropical jet stream near 30 latitude.

As a numerical example, consider a north-south temperature gradient of $\partial T/\partial y = -1 \times 10^{-5}$ K m$^{-1}$ (temperature decreasing poleward at 10 K per 1000 km). The thermal wind shear is:

$$\frac{\partial u_g}{\partial \ln p} = \frac{R}{f}\frac{\partial T}{\partial y} = \frac{287 \times (-10^{-5})}{10^{-4}} \approx -28.7 \text{ m s}^{-1} \text{ per unit } \ln p$$

Integrating from 1000 hPa to 250 hPa ($\Delta\ln p \approx 1.39$): $\Delta u_g \approx 40$ m s$^{-1}$ westerly increase.

4.3 Thermal Advection: Warm and Cold

The thermal wind concept allows us to diagnose temperature advection from a vertical profile of winds (e.g., from a radiosonde). The key diagnostic is how the wind vector rotates with height.

Warm Advection

When the geostrophic wind veers (turns clockwise) with height in the Northern Hemisphere, warm air is being advected into the region. The thermal wind has a component from warm to cold air, meaning wind blows from the warm side.

$$-\vec{V}_g \cdot \nabla T > 0 \quad \text{(warm advection)}$$

Associated with: ascending motion, cloud development, precipitation ahead of warm fronts, pressure falls.

Cold Advection

When the geostrophic wind backs (turns counterclockwise) with height in the Northern Hemisphere, cold air is being advected into the region.

$$-\vec{V}_g \cdot \nabla T < 0 \quad \text{(cold advection)}$$

Associated with: subsidence, clearing skies behind cold fronts, pressure rises, stabilization of the atmosphere.

4.4 Backing and Veering

The terms veering and backing describe the directional change of wind with height (or time):

Veering: Wind turns clockwise with increasing height (e.g., south at surface, southwest at 850 hPa, west at 500 hPa). In the Northern Hemisphere, veering indicates warm air advection (WAA).

Backing: Wind turns counterclockwise with increasing height (e.g., west at surface, northwest at 850 hPa, north at 500 hPa). In the Northern Hemisphere, backing indicates cold air advection (CAA).

Southern Hemisphere: The relationships are reversed. Backing indicates warm advection and veering indicates cold advection.

Practical Application -- Hodograph Analysis:

A hodograph plots the wind vector as a function of height. The thermal wind vector connects successive wind observations on the hodograph. A clockwise spiral (veering) indicates warm advection in the Northern Hemisphere, which is critical for diagnosing frontal zones, forecasting precipitation, and identifying environments favorable for severe convective storms (veering profiles increase storm-relative helicity).

4.5 Barotropic vs. Baroclinic Atmospheres

The thermal wind concept naturally divides atmospheric states into two fundamental categories:

Barotropic Atmosphere

Density depends only on pressure: $\rho = \rho(p)$. Isobaric surfaces are parallel to isopycnic (constant density) surfaces. There is no thermal wind: the geostrophic wind does not change with height. Temperature is constant on pressure surfaces. No available potential energy for cyclone development.

Baroclinic Atmosphere

Density depends on both pressure and temperature: $\rho = \rho(p, T)$. Isobaric and isopycnic surfaces intersect. A nonzero thermal wind exists. Horizontal temperature gradients drive vertical wind shear. Available potential energy (APE) can be converted to kinetic energy through baroclinic instability, the primary mechanism for extratropical cyclone development.

5. Vorticity and Circulation

5.1 Relative and Absolute Vorticity

Vorticity is the curl of the velocity field, measuring the local rotation or spin of fluid elements. It is one of the most important quantities in atmospheric dynamics.

Relative Vorticity:

$$\vec{\zeta} = \nabla\times\vec{V}$$

The vertical component of relative vorticity (most important for synoptic meteorology) is:

$$\zeta = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$$

$\zeta > 0$: cyclonic vorticity (counterclockwise in NH)

$\zeta < 0$: anticyclonic vorticity (clockwise in NH)

The absolute vorticity includes the contribution from Earth's rotation. The vertical component of planetary vorticity is the Coriolis parameter $f$:

$$\eta = \zeta + f \quad \text{(vertical component of absolute vorticity)}$$

For typical midlatitude synoptic-scale systems: $\zeta \sim 10^{-5}$ s$^{-1}$ and $f \sim 10^{-4}$ s$^{-1}$, so $\eta \approx f$ to first order. The absolute vorticity is nearly always positive in the Northern Hemisphere and is a crucial ingredient in understanding cyclone development and Rossby wave dynamics.

5.2 The Vorticity Equation

The vorticity equation is derived by taking the curl of the momentum equations (or equivalently, cross-differentiating the $u$ and $v$ equations and subtracting). For the vertical component in pressure coordinates:

$$\frac{D(\zeta + f)}{Dt} = -(\zeta + f)\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right) - \left(\frac{\partial \omega}{\partial x}\frac{\partial v}{\partial p} - \frac{\partial \omega}{\partial y}\frac{\partial u}{\partial p}\right) + \text{friction/solenoidal terms}$$

Term 1: $D(\zeta + f)/Dt$ -- Rate of change of absolute vorticity following the parcel

Term 2: $-(\zeta + f)(\nabla_p \cdot \vec{V})$ -- Divergence (stretching) term: convergence increases vorticity (spin-up), divergence decreases it

Term 3: Tilting/twisting term -- horizontal vorticity tilted into the vertical by differential vertical motion

For large-scale motions where tilting is small and we use the beta-plane, the simplified barotropic vorticity equation is:

$$\frac{D\zeta}{Dt} + \beta v = -f_0\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right)$$

The $\beta v$ term represents the advection of planetary vorticity and is responsible for Rossby wave propagation.

Dines Compensation (Divergence and Vorticity):

The stretching term explains why surface cyclones intensify. Upper-level divergence (e.g., downstream of a trough axis near the jet stream) removes mass aloft, causing surface pressure to fall. This induces low-level convergence, which spins up cyclonic vorticity: $D\zeta/Dt \approx -f_0 \nabla \cdot \vec{V} > 0$when $\nabla \cdot \vec{V} < 0$ (convergence). This is the fundamental mechanism of extratropical cyclogenesis.

5.3 Kelvin's Circulation Theorem

Circulation $C$ is the macroscopic measure of rotation, defined as the line integral of velocity around a closed material curve:

$$C = \oint_L \vec{V}\cdot d\vec{l}$$

By Stokes' theorem, circulation is related to the area integral of vorticity: $C = \iint_A \vec{\zeta} \cdot d\vec{A}$. Kelvin's circulation theorem states that for an inviscid, barotropic fluid, the absolute circulation is conserved following a material circuit:

$$\frac{DC_a}{Dt} = \frac{D}{Dt}\oint_L (\vec{V} + \vec{\Omega}\times\vec{r})\cdot d\vec{l} = 0$$

Valid for inviscid, barotropic flow (density depends only on pressure). This is the integral form of vorticity conservation.

5.4 Bjerknes Circulation Theorem

The Bjerknes circulation theorem generalizes Kelvin's theorem to include baroclinic effects. For a baroclinic atmosphere where density depends on both pressure and temperature:

$$\frac{DC_a}{Dt} = -\oint_L \frac{dp}{\rho} = -\iint_A \frac{\nabla\rho \times \nabla p}{\rho^2}\cdot d\vec{A}$$

The right-hand side is the solenoidal term. It is nonzero when isobaric and isopycnic surfaces intersect (baroclinic atmosphere), generating circulation (vorticity).

This theorem explains why horizontal temperature gradients (baroclinicity) generate vorticity and drive weather systems. The number of $(p, \rho)$ solenoids enclosed by the circuit determines the rate of circulation change. Sea-breeze circulations are a classic example: differential heating between land and ocean creates solenoids that drive the circulation.

5.5 Potential Vorticity (Ertel PV)

Ertel's potential vorticity (PV) is one of the most fundamental and conserved quantities in atmospheric dynamics. It combines information about both vorticity and static stability in a single scalar:

$$\text{PV} = \frac{1}{\rho}(\vec{\zeta}_a \cdot \nabla\theta) = \frac{1}{\rho}(\nabla\times\vec{V} + 2\vec{\Omega})\cdot\nabla\theta$$

where $\vec{\zeta}_a = \nabla\times\vec{V} + 2\vec{\Omega}$ is the absolute vorticity vector and $\theta$ is potential temperature.

Ertel's PV theorem: For adiabatic, frictionless flow, PV is conserved following a material parcel:

$$\frac{D(\text{PV})}{Dt} = 0 \quad \text{(adiabatic, inviscid)}$$

For synoptic-scale quasi-horizontal motions, the approximate form in pressure coordinates is:

$$\text{PV} \approx -g(\zeta_p + f)\frac{\partial \theta}{\partial p}$$

Why is PV so Important?

Conservation: PV is conserved for adiabatic frictionless flow, making it a powerful tracer for air mass identification and forecast verification
Invertibility principle: Given the PV distribution and suitable boundary conditions, the balanced wind and temperature fields can be recovered (PV inversion). PV is a master variable that contains all the dynamical information
Tropopause identification: PV = 2 PVU (1 PVU = $10^{-6}$ K m$^2$ kg$^{-1}$ s$^{-1}$) defines the dynamical tropopause. Stratospheric air has high PV (strong stability, strong $f$); tropospheric air has low PV
Cyclogenesis: Upper-level PV anomalies (tropopause folds, cutoff lows) can induce surface cyclogenesis through the invertibility principle. A positive PV anomaly aloft induces cyclonic circulation below it
PV thinking: Hoskins, McIntyre, and Robertson (1985) demonstrated that the evolution of weather systems can be understood through the advection of PV

5.6 Rossby Waves (Planetary Waves)

Rossby waves are large-scale planetary waves that owe their existence to the meridional gradient of the Coriolis parameter (the beta-effect). They are the dominant wave mode at synoptic and planetary scales and are fundamental to understanding the general circulation.

Consider the barotropic vorticity equation on a beta-plane with a basic state zonal flow $\bar{u}$and small perturbations. Linearizing and assuming wave solutions $\psi' = \hat{\psi} e^{i(kx + ly - \omega t)}$leads to the Rossby wave dispersion relation:

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

$c$: Zonal phase speed of the wave

$\bar{u}$: Background zonal wind speed

$\beta$: Rossby parameter $= df/dy$

$k, l$: Zonal and meridional wavenumbers

$K^2 = k^2 + l^2$: Total wavenumber squared

Key Properties of Rossby Waves:

Westward propagation relative to the mean flow: The intrinsic phase speed $c - \bar{u} = -\beta/(k^2 + l^2) < 0$ is always westward. Longer waves (smaller $K$) propagate faster westward
Stationary Rossby waves: When $c = 0$, the wave is stationary: $K_s^2 = \beta/\bar{u}$. These forced stationary waves are responsible for the ridges and troughs in the mean wintertime flow
Dispersive: The phase speed depends on wavelength, so wave packets spread. The group velocity $c_{gx} = \bar{u} + \beta(k^2 - l^2)/(k^2 + l^2)^2$ can be eastward, enabling downstream development of weather systems
Restoring mechanism: The beta-effect provides the restoring force. A northward-displaced parcel acquires excessive planetary vorticity and is deflected southward, and vice versa, creating the wave pattern

6. Atmospheric Waves

The atmosphere supports a rich spectrum of wave motions, each with its own restoring mechanism, dispersion properties, and dynamical significance. Understanding these waves is crucial for numerical weather prediction (filtering unwanted waves), climate dynamics, and interpreting observations.

6.1 Acoustic (Sound) Waves

Sound waves (acoustic waves) are compressional waves whose restoring force is the elasticity of air (compressibility). They are the fastest waves in the atmosphere.

Sound Wave Speed:

$$c_s = \sqrt{\gamma R_d T} \approx 340 \text{ m s}^{-1} \text{ (at 288 K)}$$

where $\gamma = c_p/c_v \approx 1.4$ is the ratio of specific heats. Speed depends only on temperature.

Sound waves are meteorologically insignificant for weather and climate. Their importance lies in numerical modeling: explicit time-integration schemes must use time steps small enough to resolve sound waves (CFL condition), which is computationally expensive. This motivates:

Anelastic equations: Filter sound waves by assuming $\nabla \cdot (\bar{\rho}\vec{V}) = 0$
Incompressible (Boussinesq) equations: Further simplification, $\nabla \cdot \vec{V} = 0$
Semi-implicit time stepping: Treat acoustic terms implicitly to allow larger time steps

6.2 Internal Gravity Waves

Internal gravity waves (buoyancy waves) are waves whose restoring force is buoyancy in a stably stratified atmosphere. When a parcel is displaced vertically in a stable environment ($N^2 > 0$), it oscillates about its equilibrium level at the Brunt-Vaisala frequency.

Gravity Wave Dispersion Relation (2D, Boussinesq):

$$\omega^2 = \frac{N^2 k^2}{k^2 + m^2}$$

$\omega$: Intrinsic wave frequency

$N$: Brunt-Vaisala frequency ($\sim 0.01$ s$^{-1}$ in the troposphere)

$k$: Horizontal wavenumber

$m$: Vertical wavenumber

Key properties of internal gravity waves:

The frequency is bounded: $0 < \omega < N$. Gravity waves cannot have frequencies exceeding the buoyancy frequency
The group velocity is perpendicular to the phase velocity in the vertical plane: energy propagates at right angles to the phase propagation direction
For upward energy propagation, the phase lines tilt upstream (westward tilt for waves generated by westerly flow over mountains)
They transport energy and momentum vertically, coupling different atmospheric layers

6.3 Inertia-Gravity Waves

When both buoyancy and the Coriolis force act as restoring mechanisms, the resulting waves are inertia-gravity waves (also called Poincare waves in shallow-water theory). Their dispersion relation is:

$$\omega^2 = f^2 + \frac{N^2 k^2}{m^2 + k^2} \quad \Rightarrow \quad f^2 \leq \omega^2 \leq N^2$$

The frequency of inertia-gravity waves is bounded between $f$ (inertial oscillation, $\sim 17$ hours at 45N) and $N$ (buoyancy oscillation, $\sim 10$ minutes).

At low frequencies ($\omega \to f$), the waves become dominated by the Coriolis force and trajectories are nearly circular (inertial oscillations). At high frequencies ($\omega \to N$), the waves are purely buoyancy-driven gravity waves. In between lies a rich spectrum of inertia-gravity waves that are observed as mesoscale fluctuations in the free atmosphere and are an important source of clear-air turbulence (CAT).

6.4 Mountain Waves (Lee Waves)

When stable airflow encounters topography, it is forced upward and generates mountain waves (orographic gravity waves). These are stationary with respect to the ground since the forcing (the mountain) does not move.

For a uniform flow $\bar{u}$ over sinusoidal terrain with wavenumber $k$ in an atmosphere with constant $N$, the wave is vertically propagating if:

$$m^2 = \frac{N^2}{\bar{u}^2} - k^2 > 0 \quad \Rightarrow \quad k < \frac{N}{\bar{u}} \equiv l$$

where $l = N/\bar{u}$ is the Scorer parameter. Wide mountains (small $k$) produce vertically propagating waves; narrow mountains produce evanescent (trapped) disturbances.

Practical Significance of Mountain Waves:

Clear-air turbulence: Breaking mountain waves in the upper troposphere and lower stratosphere create severe CAT hazardous to aviation
Lenticular clouds: Lens-shaped clouds form at the crests of mountain waves where air is lifted to saturation
Downslope windstorms: Under certain conditions (e.g., critical-level absorption), mountain wave energy amplifies on the lee side, producing extreme winds (chinook, foehn, bora)
Orographic drag: Mountain waves exert a drag force on the mean flow that must be parameterized in GCMs. This gravity wave drag is crucial for realistic simulation of the stratospheric polar vortex

6.5 Equatorial Waves: Kelvin and Mixed Rossby-Gravity Waves

Near the equator where $f \to 0$, the standard midlatitude approximations break down and a unique set of wave modes arises. The equator acts as a waveguide, trapping wave energy within a few degrees of latitude. The key equatorial wave types are:

Equatorial Kelvin Waves

These are eastward-propagating waves trapped at the equator. They have no meridional velocity($v = 0$) and the zonal velocity is in geostrophic balance with the meridional pressure gradient. Their dispersion relation is non-dispersive:

$$c = \sqrt{gH} \quad \text{(shallow-water Kelvin wave speed)}$$

Kelvin waves are fundamental to the dynamics of the Madden-Julian Oscillation (MJO), El Nino-Southern Oscillation (ENSO), and the quasi-biennial oscillation (QBO). Convectively coupled Kelvin waves are among the most prominent modes of tropical variability.

Mixed Rossby-Gravity (Yanai) Waves

These waves have properties of both Rossby and gravity waves. They propagate westward for low-frequency (Rossby-like) behavior and eastward for high-frequency (gravity-like) behavior. The dispersion relation is:

$$\omega = k\sqrt{gH}\left[\frac{1}{2k}\left(\frac{\beta}{\sqrt{gH}}\right) + \sqrt{\frac{1}{4k^2}\left(\frac{\beta}{\sqrt{gH}}\right)^2 + 1}\right]$$

Mixed Rossby-gravity waves play an important role in the QBO and can interact with tropical convection. They have a characteristic antisymmetric structure about the equator.

Equatorial Rossby Waves

These are the equatorial counterparts of midlatitude Rossby waves. They propagate westwardrelative to the mean flow and have symmetric or antisymmetric structures about the equator depending on mode number. Their phase speed is:

$$c \approx -\frac{\sqrt{gH}}{2n + 1}, \quad n = 1, 2, 3, \ldots$$

Equatorial Rossby waves contribute to the low-frequency variability of the tropics and interact with the ITCZ and monsoon circulations.

6.6 Wave Energy and Momentum Transport

Atmospheric waves transport energy and momentum over large distances, coupling different regions of the atmosphere. The Eliassen-Palm (EP) flux is a powerful framework for quantifying wave-mean flow interaction:

$$\vec{F}_{EP} = \left(0, -\overline{u'v'}, \frac{f}{\partial\bar{\theta}/\partial p}\overline{v'\theta'}\right)$$

The EP flux divergence drives changes in the zonal mean flow:

$$\frac{\partial \bar{u}}{\partial t} = \frac{1}{\rho_0 a\cos\varphi}\nabla \cdot \vec{F}_{EP} + \ldots$$

Where EP flux convergence ($\nabla \cdot \vec{F} < 0$) decelerates the zonal mean flow and EP flux divergence accelerates it. This framework unifies the understanding of eddy-mean flow interactions.

Summary of Atmospheric Wave Spectrum:

Wave Type	Restoring Force	Typical Period	Scale
Sound	Compressibility	Seconds	$\sim$ km
Gravity	Buoyancy ($N$)	10 min -- hours	10--1000 km
Inertia-gravity	Buoyancy + Coriolis	Hours -- 1 day	100--1000 km
Kelvin (equatorial)	Pressure + equatorial trapping	Days -- weeks	1000--10000 km
Rossby	Beta-effect ($\beta$)	Days -- weeks	1000--10000 km

Part II Summary

Atmospheric dynamics provides the physical and mathematical framework for understanding why the atmosphere moves the way it does. The key ideas developed in this part are:

Chapter 1 -- Equations of Motion: Newton's second law in a rotating frame introduces Coriolis and centrifugal forces. Scale analysis reveals that the dominant horizontal balance is geostrophic and the vertical balance is hydrostatic. The Boussinesq approximation simplifies the treatment of buoyancy-driven motions.
Chapter 2 -- Coriolis and Pressure Gradient Forces: The Coriolis force deflects motion perpendicular to the velocity. The pressure gradient force drives air from high to low pressure. In pressure coordinates, the PGF simplifies elegantly. The Rossby number determines which force balance dominates.
Chapter 3 -- Geostrophic Wind: Geostrophic balance yields wind flowing parallel to isobars. The gradient wind corrects for curvature. The ageostrophic wind, though small, drives vertical motion, frontogenesis, and jet streak circulations.
Chapter 4 -- Thermal Wind: The thermal wind equation links vertical wind shear to horizontal temperature gradients. Veering/backing wind profiles diagnose warm/cold advection. The concept distinguishes barotropic from baroclinic atmospheres.
Chapter 5 -- Vorticity and Circulation: Vorticity quantifies local spin; the vorticity equation reveals how convergence/divergence and tilting create and destroy vorticity. Ertel's potential vorticity is conserved for adiabatic motion and is a master variable for understanding weather system evolution. Rossby waves propagate via the beta-effect.
Chapter 6 -- Atmospheric Waves: The atmosphere supports sound, gravity, inertia-gravity, Rossby, Kelvin, and mixed Rossby-gravity waves. Each has a distinct restoring mechanism and dispersion relation. Mountain waves transfer momentum vertically, and equatorial waves drive tropical variability including ENSO and the QBO.

Looking Ahead:

In Part III: Cloud Physics and Precipitation, we build on the thermodynamic and dynamic foundations established in Parts I and II to explore how water vapor condenses, cloud droplets form and grow, and precipitation develops. In Part IV: Climate Systems, the large-scale dynamics discussed here are extended to the general circulation, Hadley cells, jet streams, and ocean-atmosphere coupling.

NPTEL: Introduction to Atmospheric Science

Lectures from the NPTEL Introduction to Atmospheric Science course on atmospheric dynamics.

Lec-41 Atmospheric Dynamics

Isabelle Gallagher: Fluid Dynamics and Kinetic Theory

Lectures by Prof. Isabelle Gallagher (ENS Paris) on the mathematical foundations of fluid dynamics and kinetic theory — from the Navier-Stokes equations governing atmospheric flow to the Boltzmann equation underlying gas dynamics, and the asymptotic analysis of geophysical fluids.

Introduction to the Navier-Stokes Equations (1/3)

Introduction to the Navier-Stokes Equations (2/3)

Introduction to the Navier-Stokes Equations (3/3)

Derivation of the Boltzmann Equation (1/3)

Derivation of the Boltzmann Equation (2/3)

Derivation of the Boltzmann Equation (3/3)

Asymptotic Analysis of Geophysical Fluids

Mathematical Analysis of Equatorial Waves

ICM 2014 — Invited Lecture

Seminar in Analysis and Methods of PDE

Particles in Interaction and Wave Turbulence

Global Solutions to the Navier-Stokes Equations

The Work of Merle, Raphaël, Rodnianski, and Szeftel

ENSPM 2021 Lecture

Climate Modeling: Atmospheric Dynamics

Lectures on atmospheric modeling fundamentals, boundary layers, convection, and precipitation from the CLEX Climate Modeling course.

Fundamentals of Modelling the Atmosphere

Boundary Layer, Gravity Waves and Shallow Convection

Convection and Precipitation: Reality vs. Modelled

Aerosols, Clouds and Radiative Forcing

Yale GG 140: Atmospheric Circulation and Storms

Lectures on atmospheric circulation, the Coriolis force, convective storms, and frontal cyclones.

12. Circulation of the Atmosphere

13. Global Climate and the Coriolis Force

14. Coriolis Force and Storms

15. Convective Storms

16. Frontal Cyclones

Variable	Symbol	Typical Scale
Horizontal velocity	\(U\)	10 m s\(^{-1}\)
Vertical velocity	\(W\)	0.01 m s\(^{-1}\)
Horizontal length scale	\(L\)	\(10^6\) m (1000 km)
Vertical length scale	\(H\)	\(10^4\) m (10 km)
Time scale	\(T = L/U\)	\(10^5\) s (~1 day)
Pressure perturbation	\(\delta p\)	\(10^3\) Pa (10 hPa)
Coriolis parameter	\(f_0\)	\(10^{-4}\) s\(^{-1}\)