Conservation of Salt:

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

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

3. Forces per unit mass that produce accelerations in the ocean: Pressure gradient + Coriolis + gravity + friction + tides 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

4. Pressure gradient + gravity + Coriolis + friction + tides 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

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

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

7. Pressure gradient + Coriolis + gravity + friction + tides 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

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

9. , 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)

10. 0 Acceleration of a particle on a rotating Earth with respect to an observer in space Coriolis Centripetal 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

11. Coriolis Acceleration Ch Cv The equations of conservation of momentum, up to now look like this:

12. Ch Cv

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

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

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

16. Pressure gradient + Coriolis + gravity + friction + tides 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

17. 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)

18. Variation in g with latitude is ~ 0.5%, so for practical purposes, g =9.80 m/s2

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

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

21. Friction (internal stress) Vertical Shears (vertical gradients) z u1 u2 Flux of momentum from regions of fast flow to regions of slow flow

22. 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 Shear stress has units of kg m-1 s-1 m s-1 m-1 = kg m-1 s-2 or force per unit area or pressure: kg m s-2 m-2 = kg m-1 s-2

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

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

25. Reynolds Number Inertial to viscous: Compare non-linear (advective) terms to molecular friction Flow is laminar when Re < 1000 Flow is transition to turbulence when 100 < Re < 105 to 106 Flow is turbulent when Re > 106, unless the fluid is stratified

26. Low Re High Re

27. Consider an oceanic flow where U = 0.1 m/s; L = 10 km; kinematic viscosity = 10-6 m2/s Is friction negligible in the ocean?

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

29. x (or E) component 0 Navier-Stokes equations • 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., What about the advective terms?

30. 0 are the Reynolds stresses arise from advective (non-linear or inertial) terms

31. This relation (fluctuating part of turbulent flow to the mean turbulent flow) is called a turbulence closure The proportionality constants (Ax, Ay, Az) are the eddy (or turbulent) viscosities and are a property of the flow (vary in space and time)

32. Ax, Ay oscillate between 10-1 and 105m2/s Azoscillates between 10-5 and 10-1m2/s Az<< Ax, Ay but frictional forces in vertical are typically stronger eddy viscosities are up to 1011 times > molecular viscosities

33. Equations of motion – conservation of momentum