Download Chapter 7: Some Mathematics: 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
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
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
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
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
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
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.
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
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
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
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
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 nonlinear partial derivative
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 = 7.292  10-5 radians/s
Gravity fg = g
Friction fr
• Dv/Dt = -p/r -2W  v + g + fr
W = 7.292  10-5 radians/s
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:http://oceanworld.tamu.edu/resources/ocng_textbook/chapter07/chapter07_06.htm
Conservation of Momentum:
Navier-Stokes equation (cont.)
 Gravity term
• g = gf - W  (W  R)
Source:http://oceanworld.tamu.edu/resources/ocng_textbook/chapter07/chapter07_06.htm
Conservation of Momentum:
Navier-Stokes equation (cont.)
 The Coriolis term
Conservation of Momentum:
Navier-Stokes equation (cont.)
 Momentum Equation in Cartesian
Coordinates
Conservation of mass:
the continuity equation
 For compressible fluid
Source:http://oceanworld.tamu.edu/resources/ocng_textbook/chapter07/chapter07_06.htm
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)
Conservation of mass:
the continuity equation (cont.)
 For incompressible flow
• The coefficient of compressibility b
 b = 0 for incompressible flows
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
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)
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
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
Important concepts (cont.)
• Conservation of mass leads to the continuity
equation, which has an especially simple form
for an incompressible fluid.