Coriolis Force on a Rotating Sphere

Source

Lectures: L28 — Fictitious forces in rotating frames · L29 — Centrifugal force · L30 — Coriolis force
Simulation: Coriolis on a rotating sphere (Lecture-Display, interactive)
Concept: On a rotating sphere the horizontal Coriolis parameter f = 2ω sin φ is strongest at the poles and zero at the equator.
Inquiry: Stage A (free parcels): which way does a force-free parcel bend in each hemisphere, and why not at the equator? Stage B (cyclone): add a pressure pull toward a low — which way does the inflow spiral, and why can't hurricanes form at the equator?

Pedagogy principles applied

ID	Principle	How it shows up here
H1/H2	Structural misconception	The wrong intuition ("same everywhere") lives in a prediction CHOICE, not a standing text card.
A2/C1	Every number earns its place	Only φ and the normalized f/f_max bar are shown — each read by the inquiry.
D6/D7	Scene evolves with steps	The sin φ arrow field, the equatorial dead-zone band, and the low marker are staged in as the student advances.
E1	Animation honesty	The rotation rate is exaggerated for visibility and clearly labelled as a "visual spin-up factor × Earth rate".
A3/A4	Formal tools off by default	The revision sheet opens only when the student clicks ⚛︎ The Physics.
F4	Gated observation	Next unlocks only after the student actually launches moving parcels and, for the payoff, drags the low into the equatorial no-hurricane band.

Coriolis on a Rotating Sphere

Teach me

Speed

Rotating Earth

Low-pressure centre (⇧+drag)

Parcel

|f| = 2ω sin φ (dot size)

No-hurricane band

drag to orbit · click to launch a parceldrag to orbit · click to add a parcel · shift-drag the low

Guided Inquiry

1 · The apparatus

Let’s meet our planet. A globe spins about its axis, west to east. Click anywhere on it to release a parcel of air — and once it’s released, nothing pushes the parcel sideways.

You can set its starting bearing and speed in the controls below.

2 · A parcel in the north

Let’s start in the north. A parcel is released at mid-northern latitudes, heading in whatever direction you like. Seen from the turning globe, what do you think its track over the surface does?

3 · The same release in the south

Now let’s cross the equator. The parcel is released south of it, again with nothing pushing it sideways. Which way do you think its track bends there?

4 · On the equator

So the track bends right in the north and left in the south. Now here’s a tricky one: a parcel is released on the equator, heading due east. What do you think its track does?

5 · Air pulled toward a low

Now let’s build a storm. A low-pressure centre at 25° N pulls parcels straight toward it, and surface drag is on; parcels are released at rest. Which way do you think the inflow winds?

6 · A low on the equator

One last experiment. Shift-drag the low onto the equator and release parcels at rest there. What do you think the inflow does?

Listen

Mode

▾

Set by the inquiry steps.

Launch directionE 90°

Launch speed0.35

at restfast

Parcels are released at rest — zero initial velocity.

Show |f| dots

Readouts

▾

ω_eff (dynamics)—

φ (selected)—

f / f_max —

Globe spin and Coriolis strength are exaggerated (— visual · — dynamics vs Earth) for visibility.

Coriolis parameter

φ = latitude (0 at the equator, ±π/2 at the poles); ω = spin rate. Sign convention: f > 0 in the north, f < 0 in the south. Maximum |f| at the poles, zero at the equator.

Horizontal Coriolis acceleration

ẑ is the local vertical. With this sign, f > 0 (north) deflects motion to the right; f < 0 (south) deflects to the left.

Pressure-gradient pull toward the low

Cyclone mode only — a schematic radial pull with surface drag; the balance of this and the Coriolis term produces the spiral inflow. In free-parcel mode the only horizontal force is Coriolis.

Earth's true rate (reference)

On-screen spin is slowed for viewability (not the ω in the force law). Dynamics use ω_eff = gain · ω_sim so the deflection is visible on a lecture timescale.

The Physics

The Coriolis force on a rotating sphere · what this sim shows, the physics behind it, and every control

At a glance

This sim releases parcels on a turning globe. You change where they start, how fast they go and whether a pressure low pulls on them, and see what the planet's rotation does to anything moving across its surface.

What you'll learn

Why a parcel that nothing pushes sideways can still curve.
How the bend depends on latitude and on hemisphere.
How a loop's size and period depend on speed and latitude.
How a pressure low combines with this effect, and how that depends on the low's latitude.

The globe turns about its axis. A parcel is released at latitude φ with a chosen bearing and speed.

What's on screen

Parcel (orange dot) with its track. A short arrow shows the launch bearing.
Spin axis (amber line) and the equator (pale line).
Low-pressure centre (pink), in Cyclone mode only. Shift-drag moves it.
|f| dots (yellow) and a red band near the equator, which the inquiry cards bring in when needed.
Readouts: the sim's spin rate, the latitude you last clicked, and f relative to its maximum.

See itRelease one parcel at 40° N with your current bearing and speed. Watch for: the orange dot leaving the launch point with a short arrow, and its track trailing behind.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
The globe	Click to release a parcel at that point. Drag to turn the view, scroll to zoom.	click, drag, scroll
Mode	Free parcels: the parcel is launched with your bearing and speed and nothing else acts on it. Cyclone: parcels start at rest, with a pull toward the low and surface drag. The inquiry sets the mode for you.	Free / Cyclone
Launch direction	The compass bearing a new free parcel sets off on, clockwise from north.	0°–360° opens at E 90°
Launch speed	The starting speed of a free parcel, in globe radii per second (R/s).	0–0.70 R/s opens at 0.35
Show \|f\| dots	Draws a yellow dot at each latitude, sized by the strength \|f\| of the effect there.	on / off opens on
Clear parcels	Removes every parcel and track.	button
Readouts	ω_eff, the spin rate in the sim's force law; φ, the latitude of your last click; and f/f_max, which also carries the sign.	panel
▶ Play / ⏸ Pause	Runs or freezes the globe's turn and every parcel on it.	top bar
Speed	Plays everything faster or slower on screen. The tracks themselves do not change.	0.25×–4×
↻ Reset	Clears the parcels, puts the low back at 25° N and resets the view and readouts.	top bar

Fixed: ω_eff ≈ 3.06 rad/s, about 42,000 times Earth's spin, so a loop takes seconds instead of hours. The pull toward the low and the drag are also fixed.

A turning frame adds a term

In Free mode nothing pushes the parcel once it is released. Seen from outside the globe, it would travel in a straight line at constant speed. We watch from the turning globe, so its track curves. To write Newton's second law in that frame, one extra term is needed:

The Coriolis term, for a frame turning at rate ω.

Keep only the part that acts on motion along the ground, with ẑ the local vertical:

f is the Coriolis parameter. This is the form the sim uses.

For f > 0 the term turns the velocity to the right of the motion; for f < 0, to the left.
It is perpendicular to the velocity, so it changes the direction of motion and not the speed.
It belongs to the frame. No object exerts it, it has no reaction partner, and in a non-rotating frame it is absent.

See itRelease two parcels with the same bearing and speed, one at 45° N and one at 45° S. Watch for: the two tracks bending in opposite senses.

Latitude sets the strength

Only the part of the spin along the local vertical turns motion along the ground. At latitude φ that part is ω sin φ, which gives

The other part, ω cos φ, points along the ground toward the pole and does not turn motion along the ground.

Left: the |f| dots on the globe. Right: the spin split at latitude φ into ω sin φ (local vertical) and ω cos φ (toward the pole).

f is positive in the north, negative in the south, and zero on the equator.
|f| is largest at the poles. In the sim, f = 2 · 3.06 · sin φ s⁻¹: 4.33 s⁻¹ at 45° and 5.30 s⁻¹ at 60°.
How fast the ground itself moves under the parcel does not enter.

See itLaunch two parcels due east at your current speed, one on the equator and one at 60° N, with the |f| dots on. Watch for: the equatorial track staying straight while the one at 60° N curls into a loop.

Speed sets size, f sets period

The sideways acceleration has size f v and is always perpendicular to v, so the speed stays fixed and the velocity turns at rate f:

Double v and the radius doubles, but one turn still takes 2π/f.

φ bearing v

Bearing and v move the sim's own sliders too. φ only sets the latitude for these numbers.

These formulas assume f stays constant. In the sim the parcel drifts to other latitudes as it loops, so the loop does not close exactly, and bigger loops take a little longer. At 60° N, 2π/f is 1.18 s, and the sim's loops take 1.22 s at v = 0.2 and 1.32 s at v = 0.6.

See itLaunch from 60° N at v = 0.2 R/s, then from the same point at v = 0.6 R/s. Watch for: the second loop about three times as wide as the first, closing in a similar time.

A low plus Coriolis makes a cyclone

Cyclone mode adds a pull toward the low and a drag. A parcel released at rest feels only the pull at first, because the Coriolis term needs a velocity. As it speeds up inward, it is turned.

Cyclone mode's recipe: the pull toward the low, the Coriolis term, and drag.

Rest-released inflow computed from the sim's cyclone rule, at 25° N and 25° S.

Where f > 0, every inward-moving parcel is turned to its right, so the inflow circulates anticlockwise around the low. Where f < 0 it circulates clockwise.
Drag removes energy, so the parcel spirals in instead of circling forever.

See itPut a low at 25° N and release three parcels at rest around it. Watch for: the parcels turning to their right as they fall in, so the inflow winds anticlockwise.

No spiral near the equator

The turning that organises the inflow comes from f, and f shrinks toward the equator. For a low on the equator, the pull and the drag bring parcels straight in, and neither side is preferred.

In the sim, parcels released at rest around a low on the equator, within 5° of it in latitude, turn by less than 10° on the way in and arrive in under a second.
There is no sharp edge. The winding grows steadily as the low moves poleward, and the red band is only the sim's ±5° marker, not a sharp threshold.
In the real atmosphere the same reasoning makes tropical cyclones almost never form within about 5° of the equator.

See itRelease four parcels at rest around a low on the equator, then do the same with the low at 25° N. Watch for: the equatorial parcels falling straight in, then the 25° N inflow winding into a spiral.

Where it matters

The same quantities with Earth's real ω = 7.29 × 10⁻⁵ s⁻¹.

Weather. Large-scale winds circle anticlockwise around lows in the northern hemisphere and clockwise in the southern. Coriolis is one needed ingredient of a tropical cyclone, along with warm ocean water and little wind shear.
Oceans. After a storm passes, drifting buoys trace loops like the ones in the sim, with period 2π/f ≈ 12 h / sin φ (about 17 h at 45°).
Not for sinks. Over a bathtub or a sink, a few metres and a few minutes, the effect is far too small to decide which way the water swirls.

Go deeper: geostrophic balance

Far from the equator, steady large-scale wind is close to a balance between the pressure force and the Coriolis term, f ẑ × v = −(1/ρ)∇p.
The wind then blows along the lines of constant pressure, not across them. In the northern hemisphere it keeps low pressure on its left.

Summary

In one line. A free parcel on a turning globe curves because the rotating frame adds −f ẑ × v, with f = 2ω sin φ: right in the north, left in the south, and not at all on the equator.

Key points

The term is perpendicular to the velocity: it turns the motion without changing its speed.
f = 2ω sin φ sets its strength: zero on the equator, largest at the poles, opposite in sign across it.
A free parcel loops with radius r = v/f and period 2π/f, to a good approximation.
With a pressure force added, the turning winds the inflow around a low, except where f is small.

Common mix-ups

"The Coriolis force pushes the parcel sideways." It is a term the rotating frame adds. No object exerts it.
"The effect is strongest at the equator, because the ground moves fastest there." The ground's speed does not enter. f is zero on the equator.
"It speeds the parcel up." It only turns the velocity. (In the sim the speed wobbles by a few percent, because positions are stepped in latitude and longitude.)

Things to try on your own

Release due-east parcels at 20°, 40° and 60° N at the default speed, and order the three loops by size. Check against r = v/f.
In Cyclone mode, shift-drag the low to 5°, 10° and 15° N, release parcels at rest, and compare how much the inflow winds.