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 Académie des 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 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Rates of Change in a Rotating Frame For any vector s, Df Dr s= s +Ω×s Dt Dt 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, Df Dr s= 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, Df Dr s= 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 Df Vf Dt = = Dr Vf + Ω × Vf Dt Dr Vr + 2Ω × Vr + Ω × (Ω × r ) Dt 5 / 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. 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 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 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 + v j , Ω = Ωk 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 + v j , Ω = Ωk In the rotating frame Dr Vr + 2Ω × Vr = 0. Dt 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 + v j , Ω = Ωk In the rotating frame Dr Vr + 2Ω × Vr = 0. Dt Thus, in the fixed frame Df Vf = Ω × (Ω × r ). Dt 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 + v j , Ω = Ωk In the rotating frame Dr Vr + 2Ω × Vr = 0. Dt Thus, in the fixed frame Df Vf = Ω × (Ω × r ). Dt What does the RHS represent? 20 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Tr ue Gr av ity Polar Axis The Earth As a Perfect Sphere I Non-rotating earth I True gravity points to the center of the earth. I True gravity is perpendicular to the spherical surface. Equator 21 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain I Centrifugal force acts on the crust in a rotating earth. I Net force, apparent gravity, shears the crust. I The earth deforms. Centrifugal Force Tr ue Gr av ity Ap pa ren tG rav ity Polar Axis The Earth Deforms Equator 22 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Centrifugal Force Tr ue Gr av ity Ap pa ren tG rav ity Polar Axis The Oblate Spheroid I Net force, apparent gravity, again perpendicular to the crust. Equator 23 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain ”Horizontal Component of True Gravity y Centrifugal Force Tr ue Gr av ity Ap pa ren tG rav ity Polar Axis Co mp on en to f Tr ue Gra vit I True gravity has a non-zero projection onto the geopotential surfaces. 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 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 2 mv = mgzmax 2 i 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 2 mv = mgzmax 2 i vi = 642 m/s In the non-rotating framework, the equator is uphill. 26 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Inertial Oscillation Viewed from the Fixed Frame Recall Df Vf = Ω × (Ω × r ). Dt 27 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Inertial Oscillation Viewed from the Fixed Frame Recall Df Vf = Ω × (Ω × r ). Dt Component form, origin at the North Pole d 2x + Ω2 x = 0 dt 2 d 2y + Ω2 y = 0 dt 2 27 / 40 Derivation The Merry-Go-Round The Inertial Oscillation Swirl Around a Drain Inertial Oscillation Viewed from the Fixed Frame Recall Df Vf = Ω × (Ω × r ). Dt Component form, origin at the North Pole d 2x + Ω2 x = 0 dt 2 d 2y + Ω2 y = 0 dt 2 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 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 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 I Object conserves linear momentum in the fixed frame. Coriolis forces, centrifugal, and gravitational forces: I I Motion is a forced oscillation 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 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 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, tan φ Du − 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 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 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 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 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
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