From observations to models:

1. From observations to models: Equations of motion. Computer Modeling. Coastal Upwelling.

2. Reference • Open University: Ocean Circulation. • 1st edition, 238 pp., 1989 • 2nd edition, 286 pp., Butterworth-Heinemann, 2001/2nd revised edition, 286 pp., Butterworth-Heinemann, 2004 • Section 4.2.2: Why is there a Gulf Stream?

3. The flow pattern that results from combining (1) Ekman transport and (2) geostrophic flow in an intuitive way is: Figure 4.10 from Open University (1989) Steady-state flow pattern after geostrophic adjustment The pattern of the wind stress in (a) will result in a clockwise torque on the ocean in (b).

4. Sverdrup combined these two components mathematically.

5. Sverdrup relation At any location in the ocean the total meridional flow (that is, the net north-south flow) is determined by the rate of change of the Coriolis parameter f with latitude and the torque, or curl, of the wind stress t: where Based on Section 4.2.2 of Open University (1989)

6. Notes on Sverdrup relation • Sverdrup’s theory could not • take account of a western boundary, • explain the existence of intense western boundary currents like the Gulf Stream.

7. The asymmetrical North Atlantic gyre, intensified in the west (blue) and the more or less symmetrical wind field which overlies it [Figure 4.3 from Open University (1989)]

8. Stommel-Munk model • Stommel (1948) and Munk (1950) solved this problem by including friction in the equations of motion.

9. In the simplest case, friction is taken directly proportional to the current velocity (“Rayleigh friction”) .

10. Models of ocean circulation such as the Stommel-Munk model are being improved and refined all the time. • They are all based on the “equations of motion”. • Only in simple cases the equations of motion can be solved analytically, usually they must be solved numerically.

11. Reference • Open University, Oceanography Course Team (1989), Ocean Circulation, 238 pp. • Sections 4.2.3: The equations of motion

12. Equations of motion • Newton’s second law of motion: or applied to a fluid moving over the surface of the Earth

13. Equations of motion • In dealing with moving fluids, consider forces acting “per unit volume”: • Apply in three dimensions x, y, z at right angles to one another

14. Accelerations (rates of change of velocity with time) in x-, y- and z-directions: • Current velocities in x-, y- and z-directions: Figure 4.15 from Open University (1989)

15. What forces might lead to acceleration in the horizontal x- and/or y-directions, and therefore need to be included in the equations of motion for those directions? Based on Section 4.2.3 of Open University (1989)

16. What forces might lead to acceleration in the horizontal x- and/or y-directions, and therefore need to be included in the equations of motion for those directions? • Coriolis force • horizontal pressure gradient force • wind stress and other frictional forces

17. The Coriolis force is proportional to the sine of the latitude.

18. For a particle of mass m moving with speed u, the Coriolis force is given by: where Wis the angular velocity of the Earth about its axis and f is latitude.

19. Using the abbreviation the expression for the Coriolis force becomes: • As a force per unit volume:

20. Equation of motion for the x-direction: • Similarly for the y-direction • left-hand side: dv/dt • right-hand side: replace “x-direction” by “y-direction”

21. Equations of motion in mathematical terms: acceleration Coriolis force pressure gradient force contributions from other forces

22. Equations of motion in mathematical terms: • Fxand Fymay include • wind stress • friction • tidal forcing

23. Why does the equation for flow in the x-direction have rfvas the Coriolis term rather than rfu, and vice versa for flow in the y-direction? Based on Section 4.2.3 of Open University (1989)

24. Why does the equation for flow in the x-direction have rfv as the Coriolis term rather than rfu, and vice versa for flow in the y-direction? • Because the Coriolis force acts at right angles to the current (the Coriolis force acting in the x-direction is proportional to the velocity in the y-direction and vice versa)

25. What is the main force in the vertical or z-direction?

26. What is the main force in the vertical or z-direction? • The force due to gravity, written as rg(weight per unit volume)

27. Equation of motion for thez-direction: gravitational force acceleration pressure gradient force contributions from other forces • In the ocean, vertical accelerations are generally very small, and dw/dtmay often be neglected.

28. Rewrite the equation of motion for thez-direction assuming that dw/dt = 0, and that there are no forces acting vertically other than the weight of the water and the vertical pressure gradient force. What equation do you obtain?

29.  • This is the hydrostatic equation, which relates the pressure p at a given depth h to the weight of the overlying seawater.

30. Principle of continuity • Continuity of mass • means mass must be conserved • is effectively continuity of volume, because seawater is virtually incompressible • used in conjunction with equations of motion • provides extra constrains

31. Principle of continuity • Continuity of volume during flow Figure 4.16 from Open University (1989)

32. How does the flow pattern of the subtropical gyre exemplify the principle of continuity?

33. Mathematical equation used to express the principle of continuity: • Any change in the rate of flow in (say) the x-direction must be compensated for by a change in the rate of flow in the y- and/or z-direction(s). Based on Section 4.2.3 of Open University (1989)

34. Computer modeling • Volume occupied by water divided into “boxes” by a grid ( Discretization in space and time, next lecture) • Rate of change of flow through sides of each box calculated using equations of motion. Based on Section 4.2.2 of Open University, 2nd edition

35. Grid boxes in a three-dimensional ocean model [Figure 3-30 from Ruddiman (2001)]

36. Most modern models also include information about the transport of heat and salt, which are conservative properties. • Away from the sea-surface, they can only be changed by mixing. • Density can be computed from temperature and salinity using an equation of state. Based on Section 4.2.2 of Open University, 2nd edition

37. Process models • omitt unnecessary detail in order to reveal fundamental processes at work • Predictive models • include as many factors as possible, so as to be as realistic as possible Based on Section 4.2.2 of Open University, 2nd edition

38. Example of a predictive model Near-surface geostrophic currents on October 1, 1995 calculated by the “Parallel Ocean Program” numerical model developed at the Los Alamos National Laboratory. Stewart (2006, Figure 15.1)

39. Summary: equations of motion • The equations of motion – i.e. the mathematical equations used to investigate water movements in the ocean – are simply Newton’s Second Law, force = mass acceleration, applied in each of the three coordinate directions. They are most easily solved by considering equilibrium flow, in which there is no acceleration. When this is done for the equation of motion in a vertical direction, it becomes the hydrostatic equation. • Another important principle governing flow in the oceans is the principle of continuity, which expresses the fact that the mass (and, because water is virtually incompressible, volume) of water moving into a region per unit time must equal that leaving it per unit time. Excerpt from the Open Oniversity course, second edition

40. Today computer modelling is a valuable oceanographic tool. Process models (e.g. the simulations of the North Atlantic circulation by Sverdrup, Munk and Stommel) are not attempts to replicate the real situation in all its complexity, but experimental mathematical constructs intended to reveal the fundamental factors that determine the ocean circulation. Models may also be used in a predictive mode. Excerpt from the Open Oniversity course, second edition

41. Additional material:Coastal upwelling • Same two types of current flow resulting from wind stress as in Sverdrup’s theory of the subtropical gyre • Ekman transport at right angles to the direction of the wind in the Northern Hemisphere • Geostrophic flow in response to horizontal pressure gradients

42. Coastal upwelling • Coastal upwelling is the result of a divergence of surface water away from the coastal boundary. • In which direction must the wind blow to cause most upwelling? Based on Section 4.4 of Open University, 2nd edition

43. The bottom current must replace the water that the surface current carries away. Coast-parallel wind generates Ekman drift current away from the coast [Figure 2.27a from Ruddiman (2001)]

44. South-west African upwelling area Where does the upwelled water come from? 23°S 27°S

45. Essentials of coastal upwelling (shown for the northern hemisphere): Initial stage Open University, 2nd edition, Figure 4.36 (a) and (b)

46. Essentials of coastal upwelling (shown for the northern hemisphere): After geostrophic adjustment Open University, 2nd edition, Figure 4.36 (c) and (d)

47. Water that upwells to the surface comes from only ~100-200 m depth, but is nevertheless significantly colder than surface water.

48. Mean anomaly in the sea-surface temperature off north-west Africa for April Open University, 2nd edition, Figure 4.39

49. Upwelling water from 100-200 m depth Mixed layer “Stau” at continent Thermocline “Meteor” 1925/27 Albert Defant, Das Kaltwasserauftriebsgebiet vor der Küste Südwestafrikas, Länderkundliche Forschung, Festschrift Norbert Krebs, Stuttgart 1936.

50. Section through active upwelling area Upwelling at shelf break and coast `William Scoresby’ 1950 T. John Hart and Ronald I. Currie, The Benguela Current, Discovery Reports, Vol XXXI, Cambridge 1960.