Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain

Your Friend the Coriolis Force

Dale Durran

February 9, 2016

1 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Gaspard Gustave de Coriolis

I 1792-1843

I Presented results on rotating systems to the Acad´emiedes Sciences in 1831

2 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Pop Quiz

Does the Coriolis force on an object moving due east vary between Fairbanks and Honolulu?

3 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Outline

Derivation

The Merry-Go-Round

The Inertial Oscillation

Swirl Around a Drain

4 / 40 Setting s to be the position vector drawn from an origin at the center of the Earth, r,

Vf = Vr + Ω × r

Setting s to be the velocity in the fixed frame Vf D D f V = r V + Ω × V Dt f Dt f f D = r V + 2Ω × V + Ω × (Ω × r) Dt r r

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Rates of Change in a Rotating Frame

For any vector s, D D f s = r s + Ω × s Dt Dt

5 / 40 Setting s to be the velocity in the fixed frame Vf D D f V = r V + Ω × V Dt f Dt f f D = r V + 2Ω × V + Ω × (Ω × r) Dt r r

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Rates of Change in a Rotating Frame

For any vector s, D D f s = r s + Ω × s Dt Dt Setting s to be the position vector drawn from an origin at the center of the Earth, r,

Vf = Vr + Ω × r

5 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Rates of Change in a Rotating Frame

For any vector s, D D f s = r s + Ω × s Dt Dt Setting s to be the position vector drawn from an origin at the center of the Earth, r,

Vf = Vr + Ω × r

Setting s to be the velocity in the fixed frame Vf D D f V = r V + Ω × V Dt f Dt f f D = r V + 2Ω × V + Ω × (Ω × r) Dt r r

5 / 40 Does the Coriolis force on an object moving due east vary between Fairbanks and Honolulu?

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Two Apparent Forces

Coriolis force 2Ω × Vr is independent of position.

Centrifugal force Ω × (Ω × r) is independent of the velocity.

6 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Two Apparent Forces

Coriolis force 2Ω × Vr is independent of position.

Centrifugal force Ω × (Ω × r) is independent of the velocity.

Does the Coriolis force on an object moving due east vary between Fairbanks and Honolulu?

6 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Outline

Derivation

The Merry-Go-Round

The Inertial Oscillation

Swirl Around a Drain

7 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Balls on the Merry-Go-Round

Classic Merry-Go-Round

MIT Merry-Go-Round

8 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Merry-Go-Round as Model for the Earth

Applying the merry-go-round model to the Earth

9 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Pop Quiz

Is the merry-go-round a good basis for our intuition about Coriolis forces on the Earth?

10 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Outline

Derivation

The Merry-Go-Round

The Inertial Oscillation

Swirl Around a Drain

11 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain The Inertial Oscillation

Cushman-Roisin: Introduction to Geophysical Fluid Dynamics

12 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain The Inertial Oscillation

13 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Pop Quiz

Can we combine the centrifugal force with gravity and forget it?

Does the Coriolis force “drive” the inertial oscillation?

14 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Coriolis Force on E-W Motion

Ahrens: Essentials of Meteorology

15 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Coriolis Force on E-W Motion

Lutgens and Tarbuck: The Atmosphere 16 / 40 On a rotating planet, what causes a parcel moving east in mid-latitudes to deflect to the south?

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain A More Correct Trajectory

On a non-rotating planet, what trajectory will a fricton-free hockey puck follow if launched due east from mid-latitudes?

17 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain A More Correct Trajectory

On a non-rotating planet, what trajectory will a fricton-free hockey puck follow if launched due east from mid-latitudes?

On a rotating planet, what causes a parcel moving east in mid-latitudes to deflect to the south?

17 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain No Inertial Oscillation on the Rotating Plane

trajectory of experimenter

point of possible catch

trajectory of puck

launch point

18 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Question Review

Is the merry-go-round a good basis for our intuition about Coriolis forces on the Earth?

19 / 40 In the rotating frame D r V + 2Ω × V = 0. Dt r r Thus, in the fixed frame D f V = Ω × (Ω × r). Dt f What does the RHS represent?

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Inertial Oscillation View in the Fixed Frame

Consider an f -plane tangent to the North Pole and define

Vr = ui + vj, Ω = Ωk

20 / 40 Thus, in the fixed frame D f V = Ω × (Ω × r). Dt f What does the RHS represent?

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Inertial Oscillation View in the Fixed Frame

Consider an f -plane tangent to the North Pole and define

Vr = ui + vj, Ω = Ωk

In the rotating frame D r V + 2Ω × V = 0. Dt r r

20 / 40 What does the RHS represent?

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Inertial Oscillation View in the Fixed Frame

Consider an f -plane tangent to the North Pole and define

Vr = ui + vj, Ω = Ωk

In the rotating frame D r V + 2Ω × V = 0. Dt r r Thus, in the fixed frame D f V = Ω × (Ω × r). Dt f

20 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Inertial Oscillation View in the Fixed Frame

Consider an f -plane tangent to the North Pole and define

Vr = ui + vj, Ω = Ωk

In the rotating frame D r V + 2Ω × V = 0. Dt r r Thus, in the fixed frame D f V = Ω × (Ω × r). Dt f What does the RHS represent?

20 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain The Earth As a Perfect Sphere

I Non-rotating earth

I True gravity points to the center of the earth.

Polar Axis Polar I True gravity is perpendicular to the True Gravity spherical surface.

Equator

21 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain The Earth Deforms

I Centrifugal force acts on the crust in a rotating Centrifugal Force earth.

I Net force, apparent Polar Axis Polar gravity, shears the crust.

True Gravity I The earth deforms. Apparent Gravity

Equator

22 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain The Oblate Spheroid

Centrifugal Force I Net force, apparent gravity, again

Polar Axis Polar perpendicular to the crust.

True Gravity

Apparent Gravity

Equator

23 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain ”Horizontal Component of True Gravity

Component of True Gravity

Centrifugal Force I True gravity has a non-zero projection onto

Polar Axis Polar the geopotential surfaces.

True Gravity

Apparent Gravity

Equator

24 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Pop Quiz

Suppose the earth were topography-free and covered with frictionless ice. A hockey puck is sitting at rest at the north pole. Donald Trump claims he is strong enough to slap the puck into a target placed at any point on the globe. You measure the speed of Donald’s fastest slap shot and discover he is a liar. You also measure the speed of Bernie Sander’s fastest shot and discover that Bernie might indeed be able to slap the puck into a target placed at any point on the earth. What criteria did you use to differentiate between the capabilities of Donald and Bernie?

25 / 40 In the non-rotating framework, the equator is uphill.

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Solution

In order to hit targets in the southern hemisphere, Donald and Bernie need to be able to slap the puck “over” the earth’s equatorial bulge.

requator = rpole + 21 km

1 mv 2 = mgz 2 i max

vi = 642 m/s

26 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Solution

In order to hit targets in the southern hemisphere, Donald and Bernie need to be able to slap the puck “over” the earth’s equatorial bulge.

requator = rpole + 21 km

1 mv 2 = mgz 2 i max

vi = 642 m/s

In the non-rotating framework, the equator is uphill.

26 / 40 Component form, origin at the North Pole

d2x + Ω2x = 0 dt2 d2y + Ω2y = 0 dt2 At the North Pole, f = 2Ω.

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Inertial Oscillation Viewed from the Fixed Frame

Recall D f V = Ω × (Ω × r). Dt f

27 / 40 At the North Pole, f = 2Ω.

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Inertial Oscillation Viewed from the Fixed Frame

Recall D f V = Ω × (Ω × r). Dt f

Component form, origin at the North Pole

d2x + Ω2x = 0 dt2 d2y + Ω2y = 0 dt2

27 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Inertial Oscillation Viewed from the Fixed Frame

Recall D f V = Ω × (Ω × r). Dt f

Component form, origin at the North Pole

d2x + Ω2x = 0 dt2 d2y + Ω2y = 0 dt2 At the North Pole, f = 2Ω.

27 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Comparion of Inertial Oscillation Trajectories

Dashed: fixed frame, Solid: rotating frame.

28 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Comparion of Inertial Oscillation Trajectories

Puck, denoted by square, initially at North Pole.

29 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Comparion of Inertial Oscillation Trajectories

30 / 40 I Coriolis and centrifugal forces:

I Object conserves linear momentum in the fixed frame. I Coriolis forces, centrifugal, and gravitational forces:

I Motion is a forced oscillation I Object does not conserve its linear momentum in the fixed frame.

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain One Source of Confusion

The Coriolis force is commonly invoked as the sole driver of motions at are actually governed by either:

31 / 40 I Coriolis forces, centrifugal, and gravitational forces:

I Motion is a forced oscillation I Object does not conserve its linear momentum in the fixed frame.

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain One Source of Confusion

The Coriolis force is commonly invoked as the sole driver of motions at are actually governed by either:

I Coriolis and centrifugal forces:

I Object conserves linear momentum in the fixed frame.

31 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain One Source of Confusion

The Coriolis force is commonly invoked as the sole driver of motions at are actually governed by either:

I Coriolis and centrifugal forces:

I Object conserves linear momentum in the fixed frame. I Coriolis forces, centrifugal, and gravitational forces:

I Motion is a forced oscillation I Object does not conserve its linear momentum in the fixed frame.

31 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Rotating Turntable with Coriolis and Centrifugal Forces

Relation between Coriolis and Centrifugal Forces

32 / 40 I Does the Coriolis force “drive” the inertial oscillation?

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Question Review

I Can we combine the centrifugal force with gravity and forget it?

33 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Question Review

I Can we combine the centrifugal force with gravity and forget it?

I Does the Coriolis force “drive” the inertial oscillation?

33 / 40 On the sphere, Du tan φ − uv = 2Ωvsinφ Dt a implies conservation of absolute angular momentum (per unit mass) about the polar axis

D [(Ωa cos φ + u)a cos φ] = 0. Dt (a = radius of the Earth).

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Coriolis and N-S Motion

A parcel moving north turns east to conserve angular (not linear) momentum.

34 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Coriolis and N-S Motion

A parcel moving north turns east to conserve angular (not linear) momentum. On the sphere, Du tan φ − uv = 2Ωvsinφ Dt a implies conservation of absolute angular momentum (per unit mass) about the polar axis

D [(Ωa cos φ + u)a cos φ] = 0. Dt (a = radius of the Earth).

34 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Coriolis and E-W Motion

Does the conservation of angular momentum uniquely determine the path of a parcel moving to the east?

35 / 40 Better preliminary question: why do we come back down in the same place when we jump straight up? Answer: The component of true gravity projecting onto the geopotential surface keeps us orbiting the pole. Answer: The component of gravity projecting onto the geopotential surface is too weak to keep the parcel orbiting around the pole at its initial distance from the polar axis. (Parcel moves southward and upward.)

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Coriolis and E-W Motion

What causes a parcel moving east to deflect to the south?

36 / 40 Answer: The component of true gravity projecting onto the geopotential surface keeps us orbiting the pole. Answer: The component of gravity projecting onto the geopotential surface is too weak to keep the parcel orbiting around the pole at its initial distance from the polar axis. (Parcel moves southward and upward.)

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Coriolis and E-W Motion

What causes a parcel moving east to deflect to the south? Better preliminary question: why do we come back down in the same place when we jump straight up?

36 / 40 Answer: The component of gravity projecting onto the geopotential surface is too weak to keep the parcel orbiting around the pole at its initial distance from the polar axis. (Parcel moves southward and upward.)

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Coriolis and E-W Motion

What causes a parcel moving east to deflect to the south? Better preliminary question: why do we come back down in the same place when we jump straight up? Answer: The component of true gravity projecting onto the geopotential surface keeps us orbiting the pole.

36 / 40 (Parcel moves southward and upward.)

Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Coriolis and E-W Motion

What causes a parcel moving east to deflect to the south? Better preliminary question: why do we come back down in the same place when we jump straight up? Answer: The component of true gravity projecting onto the geopotential surface keeps us orbiting the pole. Answer: The component of gravity projecting onto the geopotential surface is too weak to keep the parcel orbiting around the pole at its initial distance from the polar axis.

36 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Coriolis and E-W Motion

What causes a parcel moving east to deflect to the south? Better preliminary question: why do we come back down in the same place when we jump straight up? Answer: The component of true gravity projecting onto the geopotential surface keeps us orbiting the pole. Answer: The component of gravity projecting onto the geopotential surface is too weak to keep the parcel orbiting around the pole at its initial distance from the polar axis. (Parcel moves southward and upward.)

36 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Outline

Derivation

The Merry-Go-Round

The Inertial Oscillation

Swirl Around a Drain

37 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Pop Quiz

Does the Coriolis force determine the direction of the drain swirl?

38 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain On the Equator

Fun with tourists

Tourists having fun

39 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain References

Stommel, H.M. and D.W. Moore, 1989: An Introduction to the Coriolis Force. Columbia University Press, 297 pp.

Durran, D.R., 1993: Is the Coriolis Force Really Responsible for the Inertial Oscillation? BAMS, 74, 2179-2184.

Durran, D.R., and S.K. Domonkos, 1996: An Apparatus for Demonstrating the Inertial Oscillation. BAMS, 77, 557-559.

40 / 40