1. The Equations of Motion
Chapter 7
Some Mathematics:
The Equations of Motion
Physical oceanography
Instructor: Dr. Cheng-Chien Liu
Department of Earth Sciences
National Cheng Kung University
Last updated: 24 October 2003

2. Introduction Response of a fluid to
Internal force
External force
 basic equations of ocean dynamics
Chapter 8: viscosity
Chapter 12: vorticity
Table 7.1
Conservation laws  basic equations

3. Dominant Forces for Ocean Dynamics
Gravity Fg
Wwater  P(x)  P
Revolution and rotation  DFg  tides, tidal current, tidal mixing
Buoyancy FB
DT  Dr  FB (vertical direction)  upward or sink
Wind Fw
Wind blows  momentum transfer  turbulence  ML
Wind blows  P(x)  P  waves

4. Dominant Forces for Ocean Dynamics (cont.)
Pseudo-forces
 motion in curvilinear or rotating coordinate systems
a body moving at constant velocity seems to change direction when viewed from a rotating coordinate system  the Coriolis force
Coriolis Force
The dominant pseudo-force influencing currents
Other forces: Table 7.2
Atmospheric pressure
Seismic

5. Coordinate System Coordinate System  find location
Cartesian Coordinate System
Most commonly use
Simpler  spherical coordinates
Convention:
x is to the east, y is to the north, and z is up.
f-plane
Fcor = const (a Cartesian coordinate system)
Describing flow in small regions

6. Coordinate System (cont.)
b-plane
Fcor  latitude (a Cartesian coordinate system)
Describing flow over areas as large as ocean basins
Spherical coordinates
(r, q, f)
Describe flows that extend over large distances and in numerical calculations of basin and global scale flows

7. Types of Flow in the Ocean
Flow due to currents
General Circulation
The permanent, time-averaged circulation
Meridional Overturning Circulation
The sinking and spreading of cold water
Also known as the Thermohaline Circulation
the vertical movements of ocean water masses  Dr  DT and DS
The circulation in meridional plane driven by mixing
Wind-Driven Circulation
The circulation in the upper kilometer  wind
Gyres
Wind-driven cyclonic or anti-cyclonic currents with dimensions nearly that of ocean basins.

8. Types of Flow in the Ocean (cont.)
Flow due to currents (cont.)
Boundary Currents
Currents owing parallel to coasts
Western boundary currents  fast, narrow jets
e.g. the Gulf Stream and Kuroshio
Eastern boundary currents  weak
e.g. the California Current
Squirts or Jets
Long narrow currents
with dimensions of a few hundred kilometers
Nearly  west coasts
Mesoscale Eddies
Turbulent or spinning flows on scales of a few hundred kilometers

9. Types of Flow in the Ocean (cont.)
Oscillatory flows due to waves
Planetary Waves
The rotation of the Earth  restoring force
Including Rossby, Kelvin, Equatorial, and Yanai waves
Surface Waves (gravity waves)
The waves that eventually break on the beach
The large Dr between air and water  restoring force
Internal Waves
Subsea wave ~ surface waves
r = r (D)  restoring force
Tsunamis
Surface waves with periods near 15 minutes generated by earthquakes

10. Types of Flow in the Ocean (cont.)
Oscillatory flows due to waves (cont.)
Tidal Currents
 tidal potential
Shelf Waves
Periods  a few minutes
Confined to shallow regions near shore
The amplitude of the waves drops off exponentially with distance from shore

11. Conservation of Mass and Salt
Dm = 0 & DS = 0  net fresh water loss  minimum flushing time
Net fresh water loss = R + P – E
QL  bulk formula  large amount of ship measurements (T, q, …)  impossible
Dm = 0  Vi + R + P = Vo + E
DS = 0  ri Vi Si = ro Vo So
Measure Vi assume ri  ro
Estimate the minimum flushing time
Example
Fig 7.2: Box model  qout = q/t dt + q/x dx + qin

12. The Total Derivative (D/Dt)
D/Dt = ∂/dt + u·( )
A simple example of acceleration of flow in a small box of fluid
qout = q/t dt + q/x dx + qin
Dq/Dt = q/t + u q/x
3D case: D/Dt = /t + u/x + v/y + w/z
The simple transformation of coordinates from one following a particle to one fixed in space converts a simple linear derivative into a non-linear partial derivative

13. Conservation of Momentum: Navier-Stokes equation
Newton’s 2nd law
F = D(mv)/Dt
Dv/Dt = F/m = fm = fp+ fc+ fg + fr
Pressure gradient fp = -p/r
Coriolis force fc = -2W  v
W =  10-5 radians/s
Gravity fg = g
Friction fr
Dv/Dt = -p/r -2W  v + g + fr

14. Conservation of Momentum: Navier-Stokes equation (cont.)
Pressure term
ax = -(1/r) (p/x)
dFx = p dy dz-(p + dp) dy dz = -dp dy dz
Source:

15. Conservation of Momentum: Navier-Stokes equation (cont.)
Gravity term
g = gf - W  (W  R)
Source:

16. Conservation of Momentum: Navier-Stokes equation (cont.)
The Coriolis term

17. Conservation of Momentum: Navier-Stokes equation (cont.)
Momentum Equation in Cartesian Coordinates

18. Conservation of mass: the continuity equation
For compressible fluid
Source:

19. Conservation of mass: the continuity equation (cont.)
Oceanic flows are incompressible
Boussinesq's assumption
v << c (sound speed)
When v  c, Dv  Dr
Phase speed of waves << c
c   in incompressible flows
Vertical scale of the motion << c2/g
The increase in pressure produces only small changes in density
r  const, except the pressure term (rg)

20. Conservation of mass: the continuity equation (cont.)
For incompressible flow
The coefficient of compressibility b
b = 0 for incompressible flows

21. Solutions to the Equations of Motion
Solvable in principle
Four equations
3 momentum equations
1 continuity equation
Four unknowns
3 velocity components: u, v, w
1 pressure p
Boundary conditions
No slip condition: v//(boundary) = 0
No penetration condition: v(boundary) = 0

22. Solutions to the Equations of Motion (cont.)
Difficult to solve in practice
Exact solution
No exact solutions for the equations with friction
Very few exact solutions for the equations without friction
Analytic solution
For much simplified forms of the equations of motion
Numerical solution
Solutions for oceanic flows with realistic coasts and bathymetric features must be obtained from numerical solutions (Chapter 15)

23. Important concepts
Gravity, buoyancy, and wind are the dominant forces acting on the ocean
Earth's rotation produces a pseudo force, the Coriolis force
Conservation laws applied to flow in the ocean lead to equations of motion; conservation of salt, volume and other quantities can lead to deep insights into oceanic flow

24. Important concepts (cont.)
The transformation from equations of motion applied to fluid parcels to equations applied at a fixed point in space greatly complicates the equations of motion. The linear, first-order, ordinary differential equations describing Newtonian dynamics of a mass accelerated by a force become nonlinear, partial differential equations of fluid mechanics.
Flow in the ocean can be assumed to be incompressible except when de-scribing sound. Density can be assumed to be constant except when density is multiplied by gravity g. The assumption is called the Boussinesq approximation

25. Important concepts (cont.)
Conservation of mass leads to the continuity equation, which has an especially simple form for an incompressible fluid.