Introduction to Observational Physical Oceanography 12.808 Class 16, 10 November, 2009

1. Introduction to Observational Physical Oceanography 12.808 Class 16, 10 November, 2009 1:05 to 2:25 these slides are online at www.whoi.edu/science/PO/people/jprice/class/miscart/Class16-10Nov09.ppt

2. Where are we? • Stirring and mixing • Advection and diffusion last class • Section 7: Momentum balance on a rotating Earth, geostrophy, and the large scale circulation of the oceans • the Coriolis force • Autonomous Systems Lab Tour today’s class

3. Assigned reading: -Hakkinen and Rhines, Science, 2004 -Kerr, Science, 2004 To be presented on Tuesday, November 17th Volunteers?

4. the Coriolis force 1) What is the Coriolis force and why or how does it arise? The Coriolis force is an ‘inertial force’ that arises because the Earth is rotating with respect to the fixed stars and is therefore a noninertial reference frame. 2) Is it important? Oh yes, the momentum balance for large scale, extra-tropical motions of the atmosphere and oceans is: Coriolis force ~ pressure gradient. 3) A thorough understanding of the Coriolis force requires mathematical care and detail that we will not attempt here; see 'Topics in Fluid Dynamics', MIT/OCW:

5. Photo courtesy of UCSD Consider a projectile fired from the North Pole towards a target at the equator: Eastward velocity of the surface of the Earth is FASTEST at the equator. Initially, the projectile has same eastward velocity of the Earth, but as it moves southward the eastward velocity of the Earth becomes larger than that of the projectile. In relation to the Earth, the projectile moves southward AND westward!

6. towards the Coriolis force: 1) Review of inertial forces/noninertial reference frames. a) linear acceleration b) rotation at a steady speed, centrifugal force c) steady rotation of fluid-attached reference frame. 2) The Coriolis force in 3-d; the horizontal components. 3) Modes of the horizontal momentum equation (next class) a) inertial oscillations b) steady, forced (geostrophic) motion 4) What am I supposed to know from all of this? (next class)

7. The equations of motion Motion of a particle given by the sum of all forces on that particle There are many forces on a parcel of water, including pressure gradient, friction, gravity, and the Coriolis force.

8. inertial and noninertial reference frames

9. W = mg

10. a W = m(g + a)

11. W = m(g + a)

12. - a = -g

13. dynamics seen from an inertial reference frame noninertial frame inertial frame

14. dynamics observed from a noninertial reference frame noninertial frame inertial frame

15. an ‘inertial force’

16. The Earth is rotating and therefore non-inertial! Ω Z’ Y’ X’ Z X0,Y0,Z0 Y Our goal: find the inertial accelerations that we on Earth experience X

17. inertial forces due to steady, circular motion r a centrifuge at rest

18. W mW2r r W steady rotation

19. W W steady rotation

20. vector cross-product magnitude: Ia x bI = IaI IbI sin(q) the signed area of a parallelogram: direction: use the right hand rule to find that the the product vector is perpendicular to a and b for more on the vector cross product see links in slide 45

21. the centripetal acceleration associated with steady, circular motion time = t r t + dt

22. W W steady rotation

23. Inertial forces to this point: 1.) are relative to a reference frame 2.) may be accounted for exactly, provided we know the acceleration of the reference frame, 3.) an acceleration (change in direction of a velocity vector) results from steady rotation alone.

24. plumb bobs g a level surface a fluid at rest a level surface

25. plumb bobs show the local vertical g a fluid at rest set the fluid into a steady rotation; a level surface is then a parabola. W g a level surface

26. W r a level surface g

27. How is Earth similar and different from this spinning disk? Earth is a sphere, and is in a gravitational/rotational balance. Our definition of level is still a good one. F = (Req – Rpole)/Req for Earth, F = 21/6370 =0.0033 for Saturn, F = 0.1 centrif The Earth is out of round just enough so that the centrifugal force is cancelled for things that are at rest in the rotating (Earth) frame.

28. balanced motion in a rotating (parabolic) ref frame balanced at this r with V = WxR

29. balanced motion in a rotating (parabolic) ref frame balanced at this r with V = WxR What happens to the balance of forces if we move outward?

30. will not balanced if displaced to this larger r if balanced at this r = r0 If displaced to a larger r, the particle will be in an environment: 1) having a larger positive radial velocity, so the particle will have a negative (relative) azimuthal velocity, and 2) it will be accelerated back toward r = r0 because the radial slope at r > r0 will be larger than required to balance the centrifugal force of the particle.

31. In the fixed/inertial frame (looking at the spinning table from above) it appears that the particle moves outward on the table and continues to rotate with the same radial velocity, while the table spins underneath it. Video courtesy Wikipedia!

32. In the rotating/non-inertial frame, the particle moves outward on the table but also appears to veer toward the right (developing a negative radial velocity) Video courtesy Wikipedia!

33. The velocities seen in the two different reference frames are different! The difference between the two is simply the solid body rotation velocity! Video courtesy Wikipedia!

34. The velocity of the particle as seen in the non-inertial reference (rotating) frame can be expressed as follows: Velocity in fixed frame Velocity in rotating frame Solid body rotation The difference between the two velocities is simply the solid body rotation!!!

35. (for the details of this, see 'a Coriolis tutorial')

36. acceleration seen in the rotating frame Coriolis force centrifugal force cancelled by Earth's oblate shape acceleration seen in the inertial frame (and rotated)

37. Polaris acceleration seen in the rotating frame other forces acting on the parcel Coriolis force W

38. The Coriolis force has a vertical component (normal to a level surface) that is much smaller than gravity and usually ignored; the horizontal component of the Coriolis force is often very important: W Northern latitudes: the horizontal component of C is toward the equator, to the right of V. The Coriolis force tends to deflect winds and currents to the right C = -2WxV a velocity V that is east Southern latitudes, the horizontal component of C is also toward the equator, which is now to the left of V C = -2WxV

39. in the northern hemisphere the horizontal component of the Coriolis force is to the right.... ‘what’s it do right on the equator?’ in the southern hemisphere the horizontal component of the Coriolis force is to the left.... W C a velocity V that is east C

40. the 3-D Coriolis force W the vertical component of C C = -2WxV a velocity V that is east (into the board) horizontal component; in northern latitudes, the horizontal component of C is toward the equator, or to the right of V

42. in the northern hemisphere the horizontal component of the Coriolis force is to the right.... in the southern hemisphere the horizontal component of the Coriolis force is to the left.... W C so what does it do right on the equator?? a velocity V that is east C C

43. in the northern hemisphere the horizontal component of the Coriolis force is to the right.... in the southern hemisphere the horizontal component of the Coriolis force is to the left.... W C on the equator, the Coriolis force is exactly vertical; there is no horizontal component a velocity V that is east C C

44. sea level pressure, mb, at mid-latitudes 1) outside of the tropics, winds are almost parallel to the lines of constant pressure (isobars) 2) speeds are greatest where the pressure gradient is largest, i.e., where isobars are closer together 3) the wind is almost in geostrophic balance: L 480 L H H H 840 H L L from FNMOC, 'Fleet Numerical'

46. modes of the horizontal momentum eqns: C V local, time-dependent inertial oscillations the horizontal velocity vector will rotate at the rate -f = -2Wsin(latitiude), which is clockwise in the N hemisphere, c-clockwise in S hemisphere

47. inertial oscillations observed in ocean currents at 25 m depth following a hurricane ~ 2p/f data thanks to Zedler, et al., JGR 2003

48. steady, forced motion; C + F = 0 V in this steady forced balance, the velocity is normal to the force, F, and flows to the right of F (N hemisphere) C F when F is a pressure gradient, this is known as geostrophic balance

49. Coriolis force: what am I supposed to know? 1) Coriolis force, C, is an inertial force that is a consequence of Earth’s rotation. It is not 'ad hoc' or 'fictitious' or 'virtual', as is sometimes implied. 2) the 3-d vector form: C = -2W x V. 3) the horizontal components are usually the important part: Cx = fV; Cy = -fU, where U, V are east and north velocity components and f is the Coriolis parameter, f = 2 W sin(lat), W is Earth’s rotation rate. 4) the local, time-dependent mode of the horizontal momentum equations is an inertial oscillation. 5) the steady, forced mode is geostrophic (or Ekman) balance, both of which are very important.

50. end of class 17