Rotation

Circular Motion

Our Earth and our galaxy both rotate. Earth spins once around its axis each day. Our galaxy’s spin carries the solar system around a complete circle over the course of about 220 million years.

We therefore observe both Earth’s atmosphere and our galaxy’s atmosphere from viewpoints that are moving in circles. Understanding the forces shaping those atmospheres becomes easier after we account for rotation of the viewpoint.

Rotating Coordinates

In a three dimensional coordinate system, we can choose the z-axis to be identical to the rotation axis and consider how things move in a plane defined by the x-axis and y-axis. If those two axes are stationary, then the trajectories of moving particles obey the usual laws of motion. However, we can also consider how the motions of those particles appear in a rotating coordinate system defined by $$x^\prime = x \cos \Omega t - y \sin \Omega t \\ y^\prime = x \sin \Omega t + y \cos \Omega t$$where $\Omega$ is an angular frequency describing the rotation rate of the $x^\prime$ and $y^\prime$ axes.

For example, $\Omega = 2 \pi \, / \, (1 \, {\rm day})$ in a coordinate system that is synchronized with Earth’s rotation. Something moving with Earth’s surface then has $x^\prime$ and $y^\prime$ coordinates that do not change.

Centrifugal Acceleration

A truly stationary particle has $x$ and $y$ coordinates that do not change. However, the $x^\prime$ and $y^\prime$ coordinates of that same particle in a rotating coordinate system execute circular motion that proceeds in a direction opposite to the rotation of the coordinate system. A familiar example is the daily motion of the Sun through our sky.

The apparent velocity of the stationary particle in a rotating coordinate system therefore appears to change with time. If the particle sits at a radial distance $r$ from the rotation axis, then its apparent speed in the rotating coordinate system is $\Omega r$. However, the direction of its apparent velocity in the rotating coordinate system changes by an angle $\Omega \, \Delta t$ over a time interval $\Delta t$. The difference in apparent velocity corresponds to a change of $\Omega^2 r \, \Delta t$ in the radial direction of the rotating coordinate system. The particle’s apparent acceleration is therefore $\Omega^2 r$ in the radial direction.

This apparent acceleration is sometimes attributed to centrifugal force, which is not an actual physical force. A stationary particle viewed from a rotating viewpoint appears to be accelerating, suggesting that a force is acting to accelerate it, but the particle is not truly accelerating.

Rotational Support

Near the solar system, the Milky Way’s atmosphere is orbiting around the galaxy at essentially the same speed as we are. Gravity is acting upon all the matter in our vicinity, accelerating it toward the galaxy’s center. However, the local gases appear to remain relatively still in a coordinate system that rotates along with us. That happens because the atmosphere’s outward centrifugal acceleration in the rotating coordinate system balances the inward acceleration that gravity would otherwise induce.

One consequence is a change in the equation of hydrostatic equilibrium. If we want to apply the principle of hydrostatic equilibrium in a rotating coordinate system, then we need to account for centrifugal acceleration away from the rotation axis. We can do that by setting $$\frac {d P} {d r_\perp} = - \, (g_\perp - \Omega^2 r_\perp) \, \rho$$where $r_\perp$ is a radial distance measured perpendicular to the rotation axis and $g_\perp$ is the component of gravitational acceleration directed toward the rotation axis.

Pressure support perpendicular to the rotation axis is therefore unnecessary to support an atmosphere with a rotation rate $$\Omega = \left( \frac {g_\perp} {r_\perp} \right)^{1/2} = \frac {v_c} {r_\perp}$$In that case, rotation alone can prevent gravity from pulling gas closer to the rotation axis.

This feature of our galaxy’s atmosphere allows some of it to be much cooler than the gravitational temperature $T_\phi$ of a non-rotating atmosphere. However, rotation supports those cool gases against gravity only in directions perpendicular to the rotation axis. Gravity can still squeeze the atmosphere in directions aligned with the rotation axis and compresses cool atmospheric gases with $T \ll T_\phi$ into a rotating disk with a thickness much smaller than its radius. Near the solar system, the cool gas disk of our own galaxy’s atmosphere therefore has a thickness only a few percent of the distance to the galaxy’s center.

Coriolis Acceleration

Earth’s atmosphere is rotating too slowly for its centrifugal acceleration to provide much opposition to the inward pull of gravity. However, another apparent force that arises in a rotating coordinate system shapes much of our weather.

Consider what happens to the trajectory of a particle moving at a speed $v_r$ directly away from the rotation axis when the particle’s trajectory is viewed in a rotating coordinate system. Its speed in the radial direction is still $v_r$, but the particle also has an apparent speed $\Omega r$ perpendicular to the radial direction. During a time interval $\Delta t$, the velocity component with speed $v_r$ appears to rotate through an angle $\Omega \, \Delta t$, producing an increment $\Omega v_r \, \Delta t$ of apparent velocity perpendicular to the radial direction. Meanwhile, the distance from the particle to the rotation axis changes by $v_r \, \Delta t$, producing an additional increment $\Omega v_r \, \Delta t$ of apparent perpendicular velocity.

The overall effect in the rotating coordinate system is a change of $2 \Omega v_r \, \Delta t$ in the particle’s apparent velocity during the time interval $\Delta t$, going in a direction perpendicular to the rotation axis. Dividing that apparent velocity change by the time interval $\Delta t$ results in an apparent acceleration of $2 \Omega v_r$. This apparent acceleration acts in a direction opposite to the system’s rotation and is attributed to a Coriolis force, named after the French scientist Gaspard-Gustave de Coriolis.

At the top of this page is an image showing a consequence of Coriolis acceleration in Earth’s atmosphere, resulting from the atmosphere’s rotation around Earth’s axis. It’s a hurricane’s eye viewed from above. The hurricane itself is a large low-pressure region that pulls in wind from both the north and the south. In the northern hemisphere, Coriolis acceleration deflects winds blowing from the north toward the west and winds blowing from the south toward the east. That happens because the north winds are moving farther from Earth’s rotation axis and the south winds are moving closer to it. Circulation of winds around a hurricane’s eye in the northern hemisphere is therefore counter-clockwise.