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.