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

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

3. Circulación típica en un fiordox z
Circulación típica en un fiordo

4. Aceleraciones x z

5. Gradiente de presión z x
Debido a la pendiente del nivel del mar (barotrópico)
Debido al gradiente de densidad (baroclínico)

6. Fricción x z
Debida a gradientes verticales de velocidad (divergencia del flujo de momentum)

7. Coriolis x z
Debido a la rotación de la Tierra; proporcional a la velocidad

8. Balance de momentum x z

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

10. 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

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

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

13. Precipitación pluvial y Ríos
Aporte aproximado por lluvia:
2000 mm por año
area superficial: 350 km por 10 km = 3.5x109 m2
200 m3/s
Dirección Meteorológica de Chile
Aporte aproximado por ríos:
1000 m3/s
Milliman et al. (1995)
Descarga de Agua Dulce
Precipitación pluvial y Ríos

14. 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

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

16. 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)

17. 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
Acceleration of a particle on a rotating Earth with respect to an observer in space
Coriolis
Centripetal

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

19. f is the Coriolis parameter
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

20. f increases with latitude
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

21. 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

22. 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

23. 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)

24. Variation in g with latitude is ~ 0
Variation in g with latitude is ~ 0.5%, so for practical purposes, g =9.80 m/s2

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

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

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

28. 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
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

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

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

31. 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 < 105 to 106
Flow is turbulent when Re > 106, unless the fluid is stratified

32. Low Re
High Re

33. Consider an oceanic flow where U = 0
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?

34. 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 equations
The only terms that have products of fluctuations are the advection terms
All other terms remain the same, e.g.,

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

36. 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)

37. Ax, Ay oscillate between 101 and 105 m2/s
Az oscillates between 10-5 and 10-1 m2/s
Az << Ax, Ay but frictional forces in vertical are typically stronger
eddy viscosities are up to 1011 times > molecular viscosities

38. Equations of motion – conservation of momentum