1. Basic dynamics ●The equations of motion and continuity scaling
Hydrostatic relation
Boussinesq approximations
●Geostrophic balance in ocean’s interior
●Momentum flux at the interface:
Surface wind stress
(calculation and properties)
●Ekman layer
●Ocean upwelling

2. The Equation of Motion
Newton’s second law in a rotating frame. (Navier-Stokes equation)
: Acceleration relative to axis fixed to the earth.
: Pressure gradient force.
: Coriolis force, where
: Effective (apparent) gravity.
: Friction molecular kinematic viscosity.

3. If ρa is a constant,

5. Gravity: Equal Potential Surfaces
g changes about 5%
9.78m/s2 at the equator
(centrifugal acceleration 0.034m/s2,
radius 22 km longer)
9.83m/s2 at the poles)
equal potential surface
normal to the gravitational vector
constant potential energy
the largest departure of the mean sea surface from the “level” surface is about 2m (slope 10-5)
The mean ocean surface is not flat and smooth
earth is not homogeneous

8. In Cartesian Coordinates:
where

9. Accounting for the turbulence and averaging within T:

10. Given the zonal momentum equation
If we assume the turbulent perturbation of density is small
i.e.,
The mean zonal momentum equation is
Where Fx is the turbulent (eddy) dissipation
If the turbulent flow is incompressible, i.e.,

11. Eddy Dissipation Then Ax=Ay~102-105 m2/s >> Az ~10-4-10-2 m2/s
Reynolds stress tensor and eddy viscosity: ,
Then
Where the turbulent viscosity coefficients are anisotropic.
Ax=Ay~ m2/s
Az ~ m2/s
>>

12. Reynolds stress has no symmetry:
A more general definition:   if
(incompressible)

13. Continuity Equation Mass conservation law
In Cartesian coordinates, we have or
For incompressible fluid,
If we define
and
, the equation becomes

14. Scaling of the equation of motion
Consider mid-latitude (φ≈45o) open ocean
away from strong current and below sea surface.
The basic scales and constants:
L=1000 km = 106 m
H=103 m
U= 0.1 m/s
T=106 s (~ 10 days)
2Ωsin45o=2Ωcos45o≈2x7.3x10-5x0.71=10-4s-1
g≈10 m/s2
ρ≈103 kg/m3
Ax=Ay=105 m2/s
Az=10-1 m2/s
Derived scale from the continuity equation
W=UH/L=10-4 m/s  

15. Scaling the vertical component of the equation of motion 
Hydrostatic Equation
accuracy 1 part in 106

16. Boussinesq approximation
Density variations can be neglected for its effect on mass but not on weight (or buoyancy).
Assume that
where
, we have
where 
Then the equations are
(1)
(2)
(3)
where
(The term
(4)
is neglected in (1) for energy
consideration.)

18. Pressure and Depth Hydrostatic pressure:
where d is depth (instead of height)
If we choose:
ρ=1000 kg/m3 (2-4% lower than ρ of sea-water)
g=10 m/s2 (2% higher than gravity)
then p=1 decibar (db) is equivalent to 1 m of depth
(p=1 db = 0.1 bar = 106 dyn/cm2 = 105 Pa (N/m2))
True d is 1-2% less than the equivalent decibar depth.

19. Geostrophic balance in ocean’s interior

20. Scaling of the horizontal components
(accuracy, 1% ~ 1‰)
Zero order (Geostrophic) balance
Pressure gradient force = Coriolis force

21. Re-scaling the vertical momentum equation
Since the density and pressure perturbation is not negligible in the vertical momentum equation, i.e.,
, and ,
The vertical pressure gradient force becomes

22. Taking into the vertical momentum equation, we have
If we scale ,
and assume
then 
and
(accuracy ~ 1‰)

23. Geopotential
Geopotential Φ is defined as the amount of work done to move a parcel of unit mass through a vertical distance dz against gravity is
The geopotential difference between levels z1 and z2 (with pressure p1 and p2) is
(unit of Φ: Joules/kg=m2/s2).

24. Dynamic height Given , we have where
is standard geopotential distance (function of p only)
is geopotential anomaly. In general,
Φ is sometime measured by the unit “dynamic meter” (1dyn m = 10 J/kg). which is also called as “dynamic height” (D)
Units: δ~m3/kg, p~Pa, D~ dyn m
Note: Though named as a distance, dynamic height (D) is still a measure of energy per unit mass.

26. Dynamic height: an example

27. Geopotential and isobaric surfaces
Geopotential surface: constant Φ, perpendicular to gravitiy, also referred to as “level surface”
Isobaric surface: constant p. The pressure gradient force is perpendicular to the isobaric surface.
In a “stationary” state (u=v=w=0), isobaric surfaces must be level (parallel to geopotential surfaces).
In general, an isobaric surface (dashed line in the figure) is inclined to the level surface (full line).
In a “steady” state ( ),
the vertical balance of forces is
The horizontal component of
the pressure gradient force is

29. Geostrophic relation The horizontal balance of force is
where tan(i) is the slope of the isobaric surface.
tan (i) ≈ 10-5 (1m/100km) if V1=1 m/s at 45oN (Gulf Stream).
In principle, V1 can be determined by tan(i). In practice, tan(i) is hard to measure because
p should be determined with the necessary accuracy
(2) the slope of sea surface (of magnitude <10-5) can not be directly measured (probably except for recent altimetry measurements from satellite.)
(Sea surface is a isobaric surface but is not usually a level surface.)

30. Calculating geostrophic velocity using hydrographic data
The difference between the slopes (i1 and i2) at two levels (z1 and z1) can be determined from vertical profiles of density observations.
Level 1:
Level 2:
Difference:
i.e.,
because A1C1=A2C2=L and B1C1-B2C2=B1B2-C1C2
because C1C2=A1A2
Note that z is negative below sea surface.

31. Since ,
and
we have
The geostrophic equation becomes

36. “Thermal Wind” Equation
Starting from geostrophic relation
Differentiating with respect to z
Using Boussinesq approximation Or
Rule of thumb: light water on the right. .

37. ,
The geostrophic current we calculate actually the “Thermal Wind”
Analytically, or
similarly

38. Since currents in deep ocean are much weak, there may exist a level (z2) where v1 >> v2 so that we can reasonably assume v2≈0 (level of no motion (LNM)). Then
The rule for direction is the same for both p and ΔD. In practice, however, we see sections of hydrographic data (T, S, or σt). In that case,
A rule for current direction is: (In northern hemisphere)
Relative to the water below it, the current flows with the “lighter water on its right”
In a vertical section, the isopycnals (curves of constant ρ or δ) slope downward from left to right.
With respect to temperature, it is the “warmer water on its right”

40. An Example: The Gulf Stream

41. Level of No Motion (LNM)

42. For a barotropic flow, we have is geostrophic current.
 p and ρ surfaces are parallel
For a barotropic flow, we have
is geostrophic current.
Since 
Given a barotropic and hydrostatic conditions, 
and
Therefore,
And  So

43. Baroclinic Flow:
and
There is no simple relation between the isobars and isopycnals.
slope of isobar is proportional to velocity
slope of isopycnal is proportional to vertical wind shear.

44. 1½ layer flow Simplest case of baroclinic flow:
Two layer flow of density ρ1 and ρ2.
The sea surface height is η=η(x,y) (In steady state, η=0). The depth of the upper layer is at z=d(x,y). The lower layer is at rest. 
For z > d,
For z ≤ d,
If we assume 
The slope of the interface between the two layers (isopycnal) =
times the slope of the surface (isobar).
The isopycnal slope is opposite in sign to the isobaric slope.

48. Example: sea surface height and thermocline depth

49. Planetary boundary Layer near the sea surface: the effect of wind

50. Surface wind stress
Approaching sea surface, the geostrophic balance is broken, even for large scales.
The major reason is the influences of the winds blowing over the sea surface, which causes the transfer of momentum (and energy) into the ocean through turbulent processes.
The surface momentum flux into ocean is called the surface wind stress ( ), which is the tangential force (in the direction of the wind) exerting on the ocean per unit area (Unit: Newton per square meter)
The wind stress effect can be constructed as a boundary condition to the equation of motion as

51. Wind stress Calculation
Direct measurement of wind stress is difficult.
Wind stress is mostly derived from meteorological observations near the sea surface using the bulk formula with empirical parameters.
The bulk formula for wind stress has the form
Where is air density (about 1.2 kg/m3 at mid-latitudes), V (m/s), the wind speed at 10 meters above the sea surface, Cd, the empirical determined drag coefficient

52. Drag Coefficient Cd
Cd is dimensionless, ranging from to (A median value is about ). Its magnitude mainly depends on local wind stress and local stability.
• Cd Dependence on stability (air-sea temperature difference).
More important for light wind situation
For mid-latitude, the stability effect is usually small but in tropical and subtropical regions, it should be included.
• Cd Dependence on wind speed.

53. Cd dependence on wind speed in neutral condition
Large uncertainty between estimates
(especially in low
wind speed).
Lack data in high wind

54. Annual Mean surface wind stress
Unit: N/m2, from Surface Marine Data (NODC)

55. December-January-February mean wind stress
Unit: N/m2, from Surface Marine Data (NODC)

56. June-July-August mean wind stress
Unit: N/m2, from Surface Marine Data (NODC)

57. The primitive equation
(1)
(2)
(3)
(4)
Since the turbulent momentum transports are , ,
etc
We can also write the momentum equations in more general forms
At the sea surface (z=0),
turbulent transport is wind stress. ,

58. Assumption for the Ekman layer near the surface
Az=const
Steady state
Small Rossby number
Large vertical Ekman Number
Homogeneous water (ρ=const)
f-plane (f=const)
no lateral boundaries (1-d problem)
infinitely deep water below the sea surface

59. Ekman layer Near the surface, there is three-way force balance
Coriolis force+vertical dissipation+pressure gradient force=0
Take
and let
( , ageostrophic (Ekman) current, note that is not small in comparison to in this region)
then

60. The Ekman problem Boundary conditions At z=0, As z→-∞, , . , . Let
, .
, .
Let
(complex variable)
At z=0,
As z→-∞,

61. The solution Assuming f > 0, the general solution is
Using the boundary conditions, we have
Set
, where
and
note that

62. Define
and
we have
• At the sea surface (z=0), the surface current flows at 45o to the right of the wind direction
• Current decreases exponentially with depth and. At the same time, its direction changes clockwise with depth (The Ekman spiral).
• DE (≈100 m in mid-latitude) is regarded as the depth of the Ekman layer. DE is not the mixed layer depth (hm). The latter also depends on past history, surface heat flux (heat balance) and the stability of the underlying water. In reality, DE < hm because hm can be affected by strong wind burst of short period. ,
• At DE, the current magnitude is 4% of the surface current and its direction is opposite to that of the surface current. .

63. Other properties
(1) Relationship between surface wind speed W and (Vo, DE).
Wind stress magnitude (
,  ) ,
(2) Relationship between W and DE.
Ekman’s empirical formula between W and Vo.
, outside ±10o latitude
(3) There is large uncertainty in CD (1.3 to 1.5 x 10-3 ±20% for wind speed up to about 15 m/s). CD itself is actually a function of W.
(4)
has an error range of 2-5%.

64. More comments
(1) DE is not the mixed layer depth (hm). The latter also depends on past history, surface heat flux (heat balance) and the stability of the underlying water. In reality, DE < hm because hm can be affected by strong wind burst of short period.
(2) Az = const and steady state assumptions are questionable.
(3) Lack of data to test the theory. (The Ekman spiral has been observed in laboratory but yet to be found in field observations).
(4) Vertically integrated Ekman transport does not strongly depend on the specific form of Az.

65. Ekman Transport
Starting from a more general form of the Ekman equation
(without assuming AZ or even a specific form for vertical turbulent flux
Integrating from surface z=η to z=-2DE (e-2π=0.002), we have
where
and
are the zonal and meridional mass transports by the
by the Ekman current. Since
, we have 

66. Ekman transport is to the right of the direction of the surface winds

67. Ekman pumping through the layer: . Assume and let , we have
Integrating the continuity equation
through the layer: .
Assume
Where and are volume transports.
Assume and let , we have
and let
, we have
is transport into or out of the bottom of the Ekman layer to the ocean’s interior (Ekman pumping).
, upwelling
, downwelling
Water pumped into the Ekman layer by the surface wind induced upwelling is from meters, which is colder and reduces SST.

68. Upwelling/downwelling are generated by curls of wind stress

69. Coastal and equatorial upwelling
Coastal upwelling: Along the eastern coasts of the Pacific and Atlantic Oceans the Trade Winds blow nearly parallel to the coast towards the Doldrums. The Ekman transport is therefore directed offshore, forcing water up from below (usually from m depth).
Equatorial Upwelling: In the Pacific and Atlantic Oceans the Doldrums are located at 5°N, so the southern hemisphere Trade Winds are present on either side of the equator. The Ekman layer transport is directed to the south in the southern hemisphere, to the north in the northern hemisphere. This causes a surface divergence at the equator and forces water to upwell (from about  m).

70. An example of coastal upwelling
Note how all contours rise towards the surface as the coast is approached; they rise steeply in the last 200 km. On the shelf the water is colder, less saline and richer in nutrients as a result of upwelling.
Water property sections in a coastal upwelling region, indicating upward water movement within about 200 km from the coast. (This particular example comes from the Benguela Current upwelling region, off the coast of Namibia.) The coast is on the right, outside the graphs; the edge of the shelf can just be seen rising to about 200 m depth at the right of each graph.

71. Cold SST associated with the coastal and equatorial upwelling

72. Properties of Sea Water
What is the pressure at the bottom of the ocean relative to sea surface pressure? What unit of pressure is very similar to 1 meter?
What is salinity and why do we use a single chemical constituent (which one?) to determine it? What other physical property of seawater is used to determine salinity? What are the problems with both of these methods?
What properties of seawater determine its density? What is an equation of state?
What happens to the temperature of a parcel of water (or any fluid or gas) when it is compressed adiabatically? What quantity describes the effect of compression on temperature? How does this quantity differ from the measured temperature? (Is it larger or smaller at depth?)
What are the two effects of adiabatic compression on density?
What are σt and σθ? How do they different from the in situ density?  
Why do we use different reference pressure levels for potential density?
What are the significant differences between freezing pure water and freezing seawater?

73. Ocean State, Heat and Water Budget
What is the permanent thermocline? Where is it located horizontally and vertically?
What are the differences between the permanent and seasonal thermoclines?
Where are the warm pool and the major cold tongues?
What affects the distribution of surface salinity?
What are the four elements of the surface heat fluxes?
What are the major factors determining the basic features of the surface heat flux components?
Can you explain some major features of the surface heat flux distributions based on some basic atmospheric and oceanic circulation knowledge?
Why does the net longwave heat loss decrease while SST increase?
Why there are large heat loss in winter over the Gulf Stream?

74. Basic Dynamics
What are the differences between the centrifugal force and the Coriolis force? Why do we treat them differently in the primitive equation?
What is the definition of dynamic height?
In geostrophic flow, what direction is the Coriolis force in relation to the pressure gradient force? What direction is it in relation to the velocity?
Why do we use a method to get current based on temperature and salinity instead of direct current measurements for most of the ocean?
How are temperature and salinity information used to calculate currents? What are the drawbacks to this method?
What is a "level of no motion"? Why do we need a "level of known motion" for the calculation of the geostrophic current? (What can we actually compute about the velocity structure given the density distribution and an assumption of geostrophy?)
What are the barotropic and baroclinic flows? Is there a “thermal wind” in a barotropic flow?
What can you expect about the relation between the slopes of the thermocline depth and the sea surface height?