2. Conservation of Salt: Conservation of Heat: Equation of State: Conservation of Mass or Continuity: Equations that allow a quantitative look at the OCEAN

3. Conservation of Momentum (Equations of Motion) Newton’s Second Law: Conservation of momentum as they describe changes of momentum in time per unit mass

4. Pressure gradient + friction+ tides+ gravity + Coriolis Pressure gradient: Barotropic and Baroclinic Coriolis: Only in the horizontal Gravity: Only in the vertical Friction: Surface, bottom, internal Tides: Boundary condition REMEMBER, these are FORCES PER UNIT MASS Forces per unit mass that produce accelerations in the ocean:

5. Pressure gradient + friction+ tides + gravity + Coriolis Pressure gradient: Barotropic and Baroclinic Coriolis: Only in the horizontal Gravity: Only in the vertical Friction: Surface, bottom, internal Tides: Boundary condition REMEMBER, these are FORCES PER UNIT MASS

6. Net Force in ‘x’ = Net Force per unit mass in ‘x’ = Total pressure force/unit mass on every face of the fluid element is:

7. Illustrate pressure gradient force in the ocean z 12 Pressure Gradient?Pressure Gradient Pressure Gradient Force Pressure of water column at 1 (hydrostatic pressure) : Hydrostatic pressure at 2 : Pressure gradient force caused by sea level tilt: BAROTROPIC PRESSURE GRADIENT

8. Pressure gradient + friction+ tides + gravity+ Coriolis Pressure gradient: Barotropic and Baroclinic Coriolis: Only in the horizontal Gravity: Only in the vertical Friction: Surface, bottom, internal Tides: Boundary condition REMEMBER, these are FORCES PER UNIT MASS

9. Acceleration due to Earth’s Rotation Remember cross product of two vectors:and

10. Now, let us consider the velocity of a fixed particle on a rotating body at the position The body, for example the earth, rotates at a rate, To an observer from space (us): This gives an operator that relates a fixed frame in space (inertial) to a moving object on a rotating frame on Earth (non-inertial)

11. This operator is used to obtain the acceleration of a particle in a reference frame on the rotating earth with respect to a fixed frame in space 0 Acceleration of a particle on a rotating Earth with respect to an observer in space Coriolis Centripetal

12. The equations of conservation of momentum, up to now look like this: Coriolis Acceleration CvCv ChCh

13. CvCv ChCh

14. Making: f is the Coriolis parameter This can be simplified with two assumptions: 1)Weak vertical velocities in the ocean ( w << v, u ) 2)Vertical component is ~5 orders of magnitude < acceleration due to gravity

15. Eastward flow will be deflected to the south Northward flow will be deflected to the east f increases with latitude f is negative in the southern hemisphere

16. Pressure gradient + friction+ tides + gravity+ Coriolis Pressure gradient: Barotropic and Baroclinic Coriolis: Only in the horizontal Gravity: Only in the vertical Friction: Surface, bottom, internal Tides: Boundary condition

17. Pressure gradient + friction+ tides + gravity+ Coriolis Pressure gradient: Barotropic and Baroclinic Coriolis: Only in the horizontal Gravity: Only in the vertical Friction: Surface, bottom, internal Tides: Boundary condition REMEMBER, these are FORCES PER UNIT MASS

18. Centripetal acceleration and gravity g has a weak variation with latitude because of the magnitude of the centrifugal acceleration g is maximum at the poles and minimum at the equator (because of both r and lamda )

19. Variation in g with latitude is ~ 0.5%, so for practical purposes, g =9.80 m/s 2

20. Friction (wind stress) z W u Vertical Shears (vertical gradients)

21. Friction (bottom stress) z u bottom Vertical Shears (vertical gradients)

22. Friction (internal stress) z u1u1 Vertical Shears (vertical gradients) u2u2 Flux of momentum from regions of fast flow to regions of slow flow

23. Shear stress has units of kg m -1 s -1 m s -1 m -1 = kg m -1 s -2 Shear stress is proportional to the rate of shear normal to which the stress is exerted at molecular scales µ is the molecular dynamic viscosity = 10 -3 kg m -1 s -1 for water; it is a property of the fluid or force per unit area or pressure: kg m s -2 m -2 = kg m -1 s -2

24. Net force per unit mass (by molecular stresses) on u

25. If viscosity is constant, becomes: And up to now, the equations of motion look like: These are the Navier-Stokes equations Presuppose laminar flow!

26. Compare non-linear (advective) terms to molecular friction Inertial to viscous: Reynolds Number Flow is laminar when Re < 1000 Flow is transition to turbulence when 100 < Re < 10 5 to 10 6 Flow is turbulent when Re > 10 6, unless the fluid is stratified

27. Low Re High Re

28. Consider an oceanic flow where U = 0.1 m/s; L = 10 km; kinematic viscosity = 10 -6 m 2 /s Is friction negligible in the ocean?

29. Frictional stresses from turbulence are not negligible but molecular friction is negligible at scales > a few m. - Use these properties of turbulent flows in the Navier Stokes equation

30. Upon applying mean and fluctuating parts to this component of motion: -The only terms that have products of fluctuations are the advective terms - All other terms remain the same, e.g., Navier-Stokes equations x (or E) component 0 What about the advective terms?

31. 0 Reynolds stresses are the Reynolds stresses arise from advective (non-linear or inertial) terms

32. This relation (fluctuating part of turbulent flow to the mean turbulent flow) is called a turbulence closure eddy (or turbulent) viscosities The proportionality constants ( A x, A y, A z ) are the eddy (or turbulent) viscosities and are a property of the flow (vary in space and time)

33. 10 -1 10 5 m 2 /s A x, A y oscillate between 10 -1 and 10 5 m 2 /s 10 -5 10 -1 m 2 /s A z oscillates between 10 -5 and 10 -1 m 2 /s A z << A x, A y but frictional forces in vertical are typically stronger eddy viscosities are up to 10 11 times > molecular viscosities

34. Equations of motion – conservation of momentum