Law of Motion (Physics):

1. Law of Motion (Physics): F = m * a Sum of all forces = time-rate of change of (mass*velocity) � if measured in an appropriate frame of reference (one that does NOT rotate) � applies to particles (Lagrangian view) Need to address both, but first,what forces act on the fluid?

2. Forces acting on geophysical fluid • Gravity • Pressure gradient force • Friction (dissipation) (viscous force) Towards center of Earth

3. Forces acting on geophysical fluid • Gravity • Pressure gradient force • Friction (dissipation) (viscous force)

4. Example of pressure gradientsdaily surface pressure map(in the atmosphere, we can simply measure the pressure at the surface)

5. Pressure gradient force in ocean? • Small deviations of sea surface drive all flows - usually much less than 1 meter height. Pressure gradient is very difficult to measure directly. Gulf Stream example 1 meter height Equivalent to pressure of 1 dbar, since water density is ~ 1000 kg/m3 1 meter height 100 kilometers width (rest of water column: 5 km deep) LOW HIGH Pressure gradient is directed from low to high. Calculate size. Pressure gradient force is directed from high to low (water pushed towards lower pressure).

6. Compute acceleration due to PGF • Take the example of the Gulf Stream and compute the velocity after 1 year of acceleration. • You’ll find it’s ridiculously large (compared with the observed 1 m/sec). Why can such a large pressure gradient be maintained without large velocities? (Earth’s rotation - Coriolis to be discussed later)

7. Friction, Viscosity, and Turbulence Turbulent: acceleration >> friction Laminar: acceleration ~ friction An unsolved problem of physics On his death-bed Physics Nobel Prize winner Werner Heisenberg is reported to have said: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." http://www.eng.auburn.edu/users/thurobs/Turb.html

8. x Shear stress  is the force F applied tangentially to a surface A (think winds over the ocean), e.g., =F/A z u For water and air it is found experimentally that the stress  is proportional to the strain u/z (change of velocity in the direction of F perpendicular to A)  ~ u/z How does this accelerate the flow?

9. Acceleration occurs if the stress at the upper surface differs from the stress at the lower surface

10. Eddy diffusivity and eddy viscosity • Molecular viscosity and diffusivity are extremely small • We know from observations that the ocean behaves as if diffusivity and viscosity are much larger than molecular • The ocean has lots of turbulent motion (like any fluid) • Turbulence acts on larger scales of motion like a viscosity - think of each random eddy or packet of waves acting like a randomly moving molecule carrying its property/mean velocity/information

11. Eddy diffusivity and viscosity Gulf Stream (top) and Kuroshio (bottom): meanders and makes rings (closed eddies) that transport properties to a new location

12. Example of surface drifter tracks: dominated to the eye by variability (they can be averaged to make a very useful mean circulation) Eddy diffusivity and viscosity

13. Law of Motion (Physics): F = m * a Sum of all forces = time-rate of change of (mass*velocity) � if measured in an appropriate frame of reference (one that does NOT rotate) � applies to particles (Lagrangian view) What about flow fields where velocity does not depend on tracking particles all the time?

14. Lagrangian View: Observations Salinity Fresh riverine water turns right at the coast. From Muenchow (1992)

15. Eulerian View Model Predictions of Winds (pink) Ocean Currents (black) Surface Salinity (color) Fresh riverine waters turn right at the coast. From John Wilkin, Rutgers U.

16. Advection of velocity --> field accelerations Particle acceleration Du/Dt Equals Local plus field acceleration: Du/Dt = u/t + (u u/x + v u/y + w u/z) local + field acceleration zero local acceleration, but particles change velocity (passing through above field)

17. Advection (Eulerian perspective) • Move “stuff” - temperature, salinity, oxygen, velocity, etc. • By moving stuff, we might change the value of the stuff at the next location. We only change the value if there is a gradient of the stuff (difference from one point to the next) • Proportional to velocity • Proportional to gradient in the same direction as the velocity, e.g., v T/y is the advection of temperature in the y-direction Low temperature or velocity gradient High temperature or velocity • strong flow • to gradient --> large • weak flow • to gradient --> small strong flow || to gradient --> zero Advection:

18. Law of Motion (Physics): F = m * a Sum of all forces = time-rate of change of (mass*velocity) � if measured in an appropriate frame of reference (one that does NOT rotate) Lets walk the plank … … Coriolis and Centrifugal accelerations … ficticious forces in a rotating reference (earth)

19. Movie time: 1. Merry-go-round (fast and slow) acceleration = coriolis+centrifugal 2. Idealized motion of merry-go-round acceleration = coriolis+centrifugal 3. Coriolis simulation without friction acceleration = coriolis 4. Coriolis simulation with friction acceleration=coriolis+friction

20. Rotating coordinates • The Earth is rotating. We measure things relative to this “rotating reference frame”. • Quantity that tells how fast something is rotating: Angular speed or angular velocity  = angle/second 360° is the whole circle, but express angle in radians (2 radians = 360°) For Earth: 2 / 1 day = 2 / 86,400 sec = 0.707 x 10-4 /sec Also can show  = v/R where v is the measured velocity and R is the radius to the axis of rotation (therefore v =  R) R

21. Rotating coordinates • Vector that expresses direction of rotation and how fast it is rotating: vector pointing in direction of thumb using right-hand rule, curling fingers in direction of rotation R

22. Centripetal and Centrifugal forces (now looking straight down on the rotating plane) Centripetal force is the actual force that keeps the ball “tethered” (here it is the string, but it can be gravitational force) Centrifugal force is the pseudo-force (apparent force) that one feels due to lack of awareness that the coordinate system is rotating or curving centrifugal acceleration = 2R

23. Effect of centrifugal force on earth and ocean Radius: Equatorial6,378.135 km Polar6,356.750 km Mean6,372.795 km (From wikipedia entry on Earth) The ocean is not 20 km deeper at the equator, rather the earth itself is deformed. We bury the centrifugal force term in the gravity term (which we call “reduced gravity”), and ignore it henceforth.

24. Coriolis effectInertial motion:motion in a straight line relative to the fixed starsCoriolis effect:apparent deflection of that inertially moving body just due to the rotation of you, the observer.Coriolis effect deflects bodies (water parcels, air parcels) to the right in the northern hemisphere and to the left in the southern hemisphere

25. Coriolis force Additional terms in momentum equations, at latitude  x-momentum equation: -sinv = -f v y-momentum equation: sinu = f u f is the “Coriolis parameter”. It depends on latitude.

26. Movie time: 1. Merry-go-round (fast and slow) acceleration = coriolis+centrifugal 2. Idealized motion of merry-go-round acceleration = coriolis+centrifugal 3. Coriolis simulation without friction acceleration = coriolis 4. Coriolis simulation with friction acceleration=coriolis+friction

27. Complete force balance with rotation Three equations: Horizontal (x) (west-east) acceleration + advection + Coriolis = pressure gradient force + viscous term Horizontal (y) (south-north) acceleration + advection + Coriolis = pressure gradient force + viscous term Vertical (z) (down-up) acceleration +advection (+ neglected very small Coriolis) = pressure gradient force + effective gravity (including centrifugal force) + viscous term

28. X: Y: Z: Momentum Equation in Cartesian co-ordinates (x,y,z) Acceleration = p-gradient + friction + gravity local acceleration field acceleration (or advection or nonlinear advection)

29. Momentum Equations in Spherical co-ordinates Radial: Latitude: Longitude: Acceleration = pressure gradient + friction + gravity