Coriolis Effect

The setup

You're observing motion from a rotating reference frame (e.g., Earth's surface). An object moves freely (no forces beyond gravity). Newton's first law says it should travel in a straight line. But to an observer rotating WITH the frame, it appears to curve sideways.

The "Coriolis force" is the fictitious force that makes Newton's laws work in the rotating frame:

F_C = -2·m·ω × v

where ω is Earth's rotation vector (along axis, ~7.27 × 10⁻⁵ rad/s), v is velocity in the rotating frame.

Direction of deflection

Hemisphere	Direction of motion	Coriolis deflection
Northern	Any horizontal direction	To the RIGHT of motion
Southern	Any horizontal direction	To the LEFT of motion
Equator (horizontal)	—	Zero
Equator (vertical)	Falling object	Deflects EAST

Weather and ocean

The Coriolis effect is most dramatic on large scales — atmospheric circulation, ocean currents.

Phenomenon	Coriolis role
Hurricanes (NH)	Counterclockwise rotation around low pressure
Hurricanes (SH)	Clockwise rotation
Trade winds	Deflected from north-south to easterly (NE in NH, SE in SH)
Westerlies (mid-latitudes)	West-blowing winds; deflected southerly poleward in NH
Jet stream	West-east flow due to Coriolis on poleward-flowing air
Ocean gyres	Major ocean currents form gyres because of Coriolis
Ekman transport	Surface water deflects 90° from wind direction

Real-world Coriolis effects

Setting	Coriolis significance
Hurricanes (1000 km, 24 hr)	Dominant — without Coriolis, no rotation
Long-range artillery (60 km, 1 min)	~50-100 m correction needed
Foucault pendulum (10 m, 24 hr)	Demonstrably visible (slow rotation)
River meanders (km scale, days)	Slight asymmetric erosion
Sniper rifle shots (1 km, 1 sec)	~2-5 cm correction at long range
Bathtub draining (1 m, 1 min)	Negligible (~10⁻⁵ vs other effects)
Walking/running (10 m, 10 sec)	Imperceptible

JavaScript — Coriolis simulation

// Coriolis force on object moving in horizontal plane at given latitude
const EARTH_OMEGA = 7.292e-5;  // rad/s

function coriolisForce(mass, vx, vy, latitudeDeg) {
  // ω vector at latitude (in local horizontal frame):
  // ω_north = ω cos(latitude), ω_up = ω sin(latitude)
  const lat = latitudeDeg * Math.PI / 180;
  const omega_z = EARTH_OMEGA * Math.sin(lat);  // vertical component
  // F_C = -2m·ω × v. For horizontal motion, only ω_z matters.
  // ω_z × v_horizontal: ω_z * (-vy, vx, 0)·(-1) = (vy, -vx, 0)·ω_z (sign care)
  // F_C = -2m·ω_z × (vx, vy, 0) = -2m·ω_z * (-vy, vx, 0) = (2m·ω_z·vy, -2m·ω_z·vx, 0)
  return [
    2 * mass * omega_z * vy,
    -2 * mass * omega_z * vx
  ];
}

// Long-range artillery: 1 kg shell, 800 m/s northward at latitude 45°N
const F_C = coriolisForce(1, 0, 800, 45);
console.log(`Coriolis force: ${F_C[0].toExponential(3)} N (eastward)`);
// Acceleration ~ 5e-2 × 800 = 0.041 m/s². Over 60 s flight, deflection = ½ × 0.041 × 60² = 73 m

// Time it would take for one full rotation of Foucault pendulum
function foucaultPeriod(latitudeDeg) {
  const lat = latitudeDeg * Math.PI / 180;
  return (24 * 3600) / Math.sin(lat);  // sidereal seconds per full rotation
}

console.log(`Paris (49°N): ${(foucaultPeriod(49) / 3600).toFixed(1)} hours`); // ~32 hours
console.log(`Equator: ${foucaultPeriod(0) === Infinity ? "Never rotates" : "wrong"}`);

// Simulate trajectory of long-range projectile (north-bound at 45°N)
function ballisticPath() {
  const trajectory = [];
  let x = 0, y = 0, z = 0;
  let vx = 0, vy = 800, vz = 100;  // mostly north, slight upward
  const dt = 0.1;
  for (let t = 0; t < 60; t += dt) {
    // Forces: gravity + Coriolis (no drag for simplicity)
    const F_C = coriolisForce(1, vx, vy, 45);
    const ax = F_C[0];
    const ay = F_C[1];
    const az = -9.81;
    vx += ax * dt;
    vy += ay * dt;
    vz += az * dt;
    x += vx * dt;
    y += vy * dt;
    z += vz * dt;
    trajectory.push({ t, x, y, z });
    if (z < 0) break;
  }
  return trajectory;
}

const path = ballisticPath();
const final = path[path.length - 1];
console.log(`Hit point: ${final.x.toFixed(0)} m east, ${final.y.toFixed(0)} m north`);
// East deflection ~ 50-100 m

// Bathtub Coriolis check
function bathtubCoriolis(diameter_m, drain_time_s, latitude = 45) {
  const v_typical = diameter_m / drain_time_s;  // typical inflow velocity
  const F_C_per_kg = 2 * EARTH_OMEGA * Math.sin(latitude * Math.PI / 180) * v_typical;
  return F_C_per_kg;
}

console.log(`Bathtub Coriolis (1m, 30s): ${bathtubCoriolis(1, 30).toExponential(2)} N/kg`);
// ~ 5e-7 — way less than tiny perturbations from any other source

Where Coriolis matters

• Meteorology. Hurricane formation, jet stream dynamics, atmospheric circulation patterns. Without Coriolis, weather would be very different.
• Oceanography. Ocean gyres, surface currents, Ekman transport, deep-ocean circulation. Major ocean currents are Coriolis-shaped.
• Ballistics. Long-range artillery, ICBMs, cruise missiles — corrections for Coriolis (and Earth rotation) are essential.
• Foucault pendulum. Direct demonstration of Earth's rotation. Found in many science museums.
• Astronomy. Solar system dynamics, motion of moons around rotating planets, accretion disks.
• Engineering — large rotating machinery. High-speed centrifuges, gyroscopes, navigation systems all account for Coriolis.
• Aerospace navigation. Satellite tracking, Earth-orbit calculations, aircraft autopilots compensate for rotating reference frame.

Common mistakes

• Believing it determines bathtub draining direction. NO. Coriolis is far too weak at small scales; basin shape, initial velocity, and other perturbations dominate.
• Treating it as a real force. Coriolis is fictitious — appears only in rotating frames. In inertial frames, no Coriolis.
• Assuming uniform direction. Coriolis deflection direction depends on hemisphere AND direction of motion. NH right, SH left of motion direction.
• Forgetting it's small at small scales. Coriolis dominates at hundreds-of-km scales (hurricanes). It's negligible for ordinary motions.
• Confusing with centrifugal. Both are fictitious in rotating frames. Centrifugal — outward, depends on position. Coriolis — perpendicular to velocity, depends on motion.
• Misapplying at equator. Coriolis vanishes for horizontal motion at equator (no vertical ω component). But vertical motion at equator does have Coriolis (eastward deflection).