Post on 22-Dec-2015
Some Mathematics Some Mathematics
The Equations of MotionThe Equations of MotionPhysical oceanographyInstructor Dr Cheng-Chien Liu
Department of Earth Sciences
National Cheng Kung University
Last updated 24 October 2003
Chapter 7Chapter 7
IntroductionIntroduction
Response of a fluid toResponse of a fluid tobull Internal force
bull External force
basic equations of ocean dynamicsbasic equations of ocean dynamicsbull Chapter 8 viscosity
bull Chapter 12 vorticity
Table 71Table 71bull Conservation laws basic equations
Dominant Forces for Ocean Dominant Forces for Ocean DynamicsDynamics
Gravity FGravity Fgg
bull Wwater P(x) Pbull Revolution and rotation Fg tides tidal current
tidal mixing
Buoyancy FBuoyancy FBB
bull T FB (vertical direction) upward or sink
Wind FWind Fww
bull Wind blows momentum transfer turbulence ML
bull Wind blows P(x) P waves
Dominant Forces for Ocean Dominant Forces for Ocean Dynamics (cont)Dynamics (cont)
Pseudo-forcesPseudo-forces motion in curvilinear or rotating coordinate
systemsbull a body moving at constant velocity seems to change
direction when viewed from a rotating coordinate system the Coriolis force
Coriolis ForceCoriolis Forcebull The dominant pseudo-force influencing currents
Other forces Table 72Other forces Table 72bull Atmospheric pressurebull Seismic
Coordinate SystemCoordinate System
Coordinate System Coordinate System find location find location Cartesian Coordinate SystemCartesian Coordinate System
bull Most commonly use bull Simpler spherical coordinatesbull Convention
x is to the east y is to the north and z is up
ff-plane-planebull Fcor = const (a Cartesian coordinate system)
Describing flow in small regions
Coordinate System (cont)Coordinate System (cont)
-plane-planebull Fcor latitude (a Cartesian coordinate system)
Describing flow over areas as large as ocean basins
Spherical coordinatesSpherical coordinatesbull (r )
Describe flows that extend over large distances and in numerical calculations of basin and global scale flows
Types of Flow in the OceanTypes of Flow in the Ocean
Flow due to currentsFlow due to currentsbull General Circulation
The permanent time-averaged circulation
bull Meridional Overturning Circulation The sinking and spreading of cold waterAlso known as the Thermohaline Circulation
the vertical movements of ocean water masses T and S
The circulation in meridional plane driven by mixing
bull Wind-Driven CirculationThe circulation in the upper kilometer wind
bull GyresWind-driven cyclonic or anti-cyclonic currents with dimensions nearly
that of ocean basins
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Flow due to currents (cont)Flow due to currents (cont)bull Boundary Currents
Currents owing parallel to coasts Western boundary currents fast narrow jets
eg the Gulf Stream and Kuroshio Eastern boundary currents weak
eg the California Current
bull Squirts or JetsLong narrow currents
with dimensions of a few hundred kilometers Nearly west coasts
bull Mesoscale EddiesTurbulent or spinning flows on scales of a few hundred kilometers
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to wavesOscillatory flows due to wavesbull Planetary Waves
The rotation of the Earth restoring forceIncluding Rossby Kelvin Equatorial and Yanai waves
bull Surface Waves (gravity waves)The waves that eventually break on the beachThe large between air and water restoring force
bull Internal WavesSubsea wave ~ surface waves = (D) restoring force
bull TsunamisSurface waves with periods near 15 minutes generated by earthquakes
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to waves (cont)Oscillatory flows due to waves (cont)bull Tidal Currents
tidal potential
bull Shelf WavesPeriods a few minutes Confined to shallow regions near shoreThe amplitude of the waves drops off exponentially with
distance from shore
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
IntroductionIntroduction
Response of a fluid toResponse of a fluid tobull Internal force
bull External force
basic equations of ocean dynamicsbasic equations of ocean dynamicsbull Chapter 8 viscosity
bull Chapter 12 vorticity
Table 71Table 71bull Conservation laws basic equations
Dominant Forces for Ocean Dominant Forces for Ocean DynamicsDynamics
Gravity FGravity Fgg
bull Wwater P(x) Pbull Revolution and rotation Fg tides tidal current
tidal mixing
Buoyancy FBuoyancy FBB
bull T FB (vertical direction) upward or sink
Wind FWind Fww
bull Wind blows momentum transfer turbulence ML
bull Wind blows P(x) P waves
Dominant Forces for Ocean Dominant Forces for Ocean Dynamics (cont)Dynamics (cont)
Pseudo-forcesPseudo-forces motion in curvilinear or rotating coordinate
systemsbull a body moving at constant velocity seems to change
direction when viewed from a rotating coordinate system the Coriolis force
Coriolis ForceCoriolis Forcebull The dominant pseudo-force influencing currents
Other forces Table 72Other forces Table 72bull Atmospheric pressurebull Seismic
Coordinate SystemCoordinate System
Coordinate System Coordinate System find location find location Cartesian Coordinate SystemCartesian Coordinate System
bull Most commonly use bull Simpler spherical coordinatesbull Convention
x is to the east y is to the north and z is up
ff-plane-planebull Fcor = const (a Cartesian coordinate system)
Describing flow in small regions
Coordinate System (cont)Coordinate System (cont)
-plane-planebull Fcor latitude (a Cartesian coordinate system)
Describing flow over areas as large as ocean basins
Spherical coordinatesSpherical coordinatesbull (r )
Describe flows that extend over large distances and in numerical calculations of basin and global scale flows
Types of Flow in the OceanTypes of Flow in the Ocean
Flow due to currentsFlow due to currentsbull General Circulation
The permanent time-averaged circulation
bull Meridional Overturning Circulation The sinking and spreading of cold waterAlso known as the Thermohaline Circulation
the vertical movements of ocean water masses T and S
The circulation in meridional plane driven by mixing
bull Wind-Driven CirculationThe circulation in the upper kilometer wind
bull GyresWind-driven cyclonic or anti-cyclonic currents with dimensions nearly
that of ocean basins
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Flow due to currents (cont)Flow due to currents (cont)bull Boundary Currents
Currents owing parallel to coasts Western boundary currents fast narrow jets
eg the Gulf Stream and Kuroshio Eastern boundary currents weak
eg the California Current
bull Squirts or JetsLong narrow currents
with dimensions of a few hundred kilometers Nearly west coasts
bull Mesoscale EddiesTurbulent or spinning flows on scales of a few hundred kilometers
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to wavesOscillatory flows due to wavesbull Planetary Waves
The rotation of the Earth restoring forceIncluding Rossby Kelvin Equatorial and Yanai waves
bull Surface Waves (gravity waves)The waves that eventually break on the beachThe large between air and water restoring force
bull Internal WavesSubsea wave ~ surface waves = (D) restoring force
bull TsunamisSurface waves with periods near 15 minutes generated by earthquakes
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to waves (cont)Oscillatory flows due to waves (cont)bull Tidal Currents
tidal potential
bull Shelf WavesPeriods a few minutes Confined to shallow regions near shoreThe amplitude of the waves drops off exponentially with
distance from shore
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Dominant Forces for Ocean Dominant Forces for Ocean DynamicsDynamics
Gravity FGravity Fgg
bull Wwater P(x) Pbull Revolution and rotation Fg tides tidal current
tidal mixing
Buoyancy FBuoyancy FBB
bull T FB (vertical direction) upward or sink
Wind FWind Fww
bull Wind blows momentum transfer turbulence ML
bull Wind blows P(x) P waves
Dominant Forces for Ocean Dominant Forces for Ocean Dynamics (cont)Dynamics (cont)
Pseudo-forcesPseudo-forces motion in curvilinear or rotating coordinate
systemsbull a body moving at constant velocity seems to change
direction when viewed from a rotating coordinate system the Coriolis force
Coriolis ForceCoriolis Forcebull The dominant pseudo-force influencing currents
Other forces Table 72Other forces Table 72bull Atmospheric pressurebull Seismic
Coordinate SystemCoordinate System
Coordinate System Coordinate System find location find location Cartesian Coordinate SystemCartesian Coordinate System
bull Most commonly use bull Simpler spherical coordinatesbull Convention
x is to the east y is to the north and z is up
ff-plane-planebull Fcor = const (a Cartesian coordinate system)
Describing flow in small regions
Coordinate System (cont)Coordinate System (cont)
-plane-planebull Fcor latitude (a Cartesian coordinate system)
Describing flow over areas as large as ocean basins
Spherical coordinatesSpherical coordinatesbull (r )
Describe flows that extend over large distances and in numerical calculations of basin and global scale flows
Types of Flow in the OceanTypes of Flow in the Ocean
Flow due to currentsFlow due to currentsbull General Circulation
The permanent time-averaged circulation
bull Meridional Overturning Circulation The sinking and spreading of cold waterAlso known as the Thermohaline Circulation
the vertical movements of ocean water masses T and S
The circulation in meridional plane driven by mixing
bull Wind-Driven CirculationThe circulation in the upper kilometer wind
bull GyresWind-driven cyclonic or anti-cyclonic currents with dimensions nearly
that of ocean basins
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Flow due to currents (cont)Flow due to currents (cont)bull Boundary Currents
Currents owing parallel to coasts Western boundary currents fast narrow jets
eg the Gulf Stream and Kuroshio Eastern boundary currents weak
eg the California Current
bull Squirts or JetsLong narrow currents
with dimensions of a few hundred kilometers Nearly west coasts
bull Mesoscale EddiesTurbulent or spinning flows on scales of a few hundred kilometers
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to wavesOscillatory flows due to wavesbull Planetary Waves
The rotation of the Earth restoring forceIncluding Rossby Kelvin Equatorial and Yanai waves
bull Surface Waves (gravity waves)The waves that eventually break on the beachThe large between air and water restoring force
bull Internal WavesSubsea wave ~ surface waves = (D) restoring force
bull TsunamisSurface waves with periods near 15 minutes generated by earthquakes
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to waves (cont)Oscillatory flows due to waves (cont)bull Tidal Currents
tidal potential
bull Shelf WavesPeriods a few minutes Confined to shallow regions near shoreThe amplitude of the waves drops off exponentially with
distance from shore
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Dominant Forces for Ocean Dominant Forces for Ocean Dynamics (cont)Dynamics (cont)
Pseudo-forcesPseudo-forces motion in curvilinear or rotating coordinate
systemsbull a body moving at constant velocity seems to change
direction when viewed from a rotating coordinate system the Coriolis force
Coriolis ForceCoriolis Forcebull The dominant pseudo-force influencing currents
Other forces Table 72Other forces Table 72bull Atmospheric pressurebull Seismic
Coordinate SystemCoordinate System
Coordinate System Coordinate System find location find location Cartesian Coordinate SystemCartesian Coordinate System
bull Most commonly use bull Simpler spherical coordinatesbull Convention
x is to the east y is to the north and z is up
ff-plane-planebull Fcor = const (a Cartesian coordinate system)
Describing flow in small regions
Coordinate System (cont)Coordinate System (cont)
-plane-planebull Fcor latitude (a Cartesian coordinate system)
Describing flow over areas as large as ocean basins
Spherical coordinatesSpherical coordinatesbull (r )
Describe flows that extend over large distances and in numerical calculations of basin and global scale flows
Types of Flow in the OceanTypes of Flow in the Ocean
Flow due to currentsFlow due to currentsbull General Circulation
The permanent time-averaged circulation
bull Meridional Overturning Circulation The sinking and spreading of cold waterAlso known as the Thermohaline Circulation
the vertical movements of ocean water masses T and S
The circulation in meridional plane driven by mixing
bull Wind-Driven CirculationThe circulation in the upper kilometer wind
bull GyresWind-driven cyclonic or anti-cyclonic currents with dimensions nearly
that of ocean basins
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Flow due to currents (cont)Flow due to currents (cont)bull Boundary Currents
Currents owing parallel to coasts Western boundary currents fast narrow jets
eg the Gulf Stream and Kuroshio Eastern boundary currents weak
eg the California Current
bull Squirts or JetsLong narrow currents
with dimensions of a few hundred kilometers Nearly west coasts
bull Mesoscale EddiesTurbulent or spinning flows on scales of a few hundred kilometers
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to wavesOscillatory flows due to wavesbull Planetary Waves
The rotation of the Earth restoring forceIncluding Rossby Kelvin Equatorial and Yanai waves
bull Surface Waves (gravity waves)The waves that eventually break on the beachThe large between air and water restoring force
bull Internal WavesSubsea wave ~ surface waves = (D) restoring force
bull TsunamisSurface waves with periods near 15 minutes generated by earthquakes
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to waves (cont)Oscillatory flows due to waves (cont)bull Tidal Currents
tidal potential
bull Shelf WavesPeriods a few minutes Confined to shallow regions near shoreThe amplitude of the waves drops off exponentially with
distance from shore
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Coordinate SystemCoordinate System
Coordinate System Coordinate System find location find location Cartesian Coordinate SystemCartesian Coordinate System
bull Most commonly use bull Simpler spherical coordinatesbull Convention
x is to the east y is to the north and z is up
ff-plane-planebull Fcor = const (a Cartesian coordinate system)
Describing flow in small regions
Coordinate System (cont)Coordinate System (cont)
-plane-planebull Fcor latitude (a Cartesian coordinate system)
Describing flow over areas as large as ocean basins
Spherical coordinatesSpherical coordinatesbull (r )
Describe flows that extend over large distances and in numerical calculations of basin and global scale flows
Types of Flow in the OceanTypes of Flow in the Ocean
Flow due to currentsFlow due to currentsbull General Circulation
The permanent time-averaged circulation
bull Meridional Overturning Circulation The sinking and spreading of cold waterAlso known as the Thermohaline Circulation
the vertical movements of ocean water masses T and S
The circulation in meridional plane driven by mixing
bull Wind-Driven CirculationThe circulation in the upper kilometer wind
bull GyresWind-driven cyclonic or anti-cyclonic currents with dimensions nearly
that of ocean basins
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Flow due to currents (cont)Flow due to currents (cont)bull Boundary Currents
Currents owing parallel to coasts Western boundary currents fast narrow jets
eg the Gulf Stream and Kuroshio Eastern boundary currents weak
eg the California Current
bull Squirts or JetsLong narrow currents
with dimensions of a few hundred kilometers Nearly west coasts
bull Mesoscale EddiesTurbulent or spinning flows on scales of a few hundred kilometers
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to wavesOscillatory flows due to wavesbull Planetary Waves
The rotation of the Earth restoring forceIncluding Rossby Kelvin Equatorial and Yanai waves
bull Surface Waves (gravity waves)The waves that eventually break on the beachThe large between air and water restoring force
bull Internal WavesSubsea wave ~ surface waves = (D) restoring force
bull TsunamisSurface waves with periods near 15 minutes generated by earthquakes
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to waves (cont)Oscillatory flows due to waves (cont)bull Tidal Currents
tidal potential
bull Shelf WavesPeriods a few minutes Confined to shallow regions near shoreThe amplitude of the waves drops off exponentially with
distance from shore
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Coordinate System (cont)Coordinate System (cont)
-plane-planebull Fcor latitude (a Cartesian coordinate system)
Describing flow over areas as large as ocean basins
Spherical coordinatesSpherical coordinatesbull (r )
Describe flows that extend over large distances and in numerical calculations of basin and global scale flows
Types of Flow in the OceanTypes of Flow in the Ocean
Flow due to currentsFlow due to currentsbull General Circulation
The permanent time-averaged circulation
bull Meridional Overturning Circulation The sinking and spreading of cold waterAlso known as the Thermohaline Circulation
the vertical movements of ocean water masses T and S
The circulation in meridional plane driven by mixing
bull Wind-Driven CirculationThe circulation in the upper kilometer wind
bull GyresWind-driven cyclonic or anti-cyclonic currents with dimensions nearly
that of ocean basins
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Flow due to currents (cont)Flow due to currents (cont)bull Boundary Currents
Currents owing parallel to coasts Western boundary currents fast narrow jets
eg the Gulf Stream and Kuroshio Eastern boundary currents weak
eg the California Current
bull Squirts or JetsLong narrow currents
with dimensions of a few hundred kilometers Nearly west coasts
bull Mesoscale EddiesTurbulent or spinning flows on scales of a few hundred kilometers
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to wavesOscillatory flows due to wavesbull Planetary Waves
The rotation of the Earth restoring forceIncluding Rossby Kelvin Equatorial and Yanai waves
bull Surface Waves (gravity waves)The waves that eventually break on the beachThe large between air and water restoring force
bull Internal WavesSubsea wave ~ surface waves = (D) restoring force
bull TsunamisSurface waves with periods near 15 minutes generated by earthquakes
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to waves (cont)Oscillatory flows due to waves (cont)bull Tidal Currents
tidal potential
bull Shelf WavesPeriods a few minutes Confined to shallow regions near shoreThe amplitude of the waves drops off exponentially with
distance from shore
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Types of Flow in the OceanTypes of Flow in the Ocean
Flow due to currentsFlow due to currentsbull General Circulation
The permanent time-averaged circulation
bull Meridional Overturning Circulation The sinking and spreading of cold waterAlso known as the Thermohaline Circulation
the vertical movements of ocean water masses T and S
The circulation in meridional plane driven by mixing
bull Wind-Driven CirculationThe circulation in the upper kilometer wind
bull GyresWind-driven cyclonic or anti-cyclonic currents with dimensions nearly
that of ocean basins
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Flow due to currents (cont)Flow due to currents (cont)bull Boundary Currents
Currents owing parallel to coasts Western boundary currents fast narrow jets
eg the Gulf Stream and Kuroshio Eastern boundary currents weak
eg the California Current
bull Squirts or JetsLong narrow currents
with dimensions of a few hundred kilometers Nearly west coasts
bull Mesoscale EddiesTurbulent or spinning flows on scales of a few hundred kilometers
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to wavesOscillatory flows due to wavesbull Planetary Waves
The rotation of the Earth restoring forceIncluding Rossby Kelvin Equatorial and Yanai waves
bull Surface Waves (gravity waves)The waves that eventually break on the beachThe large between air and water restoring force
bull Internal WavesSubsea wave ~ surface waves = (D) restoring force
bull TsunamisSurface waves with periods near 15 minutes generated by earthquakes
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to waves (cont)Oscillatory flows due to waves (cont)bull Tidal Currents
tidal potential
bull Shelf WavesPeriods a few minutes Confined to shallow regions near shoreThe amplitude of the waves drops off exponentially with
distance from shore
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Flow due to currents (cont)Flow due to currents (cont)bull Boundary Currents
Currents owing parallel to coasts Western boundary currents fast narrow jets
eg the Gulf Stream and Kuroshio Eastern boundary currents weak
eg the California Current
bull Squirts or JetsLong narrow currents
with dimensions of a few hundred kilometers Nearly west coasts
bull Mesoscale EddiesTurbulent or spinning flows on scales of a few hundred kilometers
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to wavesOscillatory flows due to wavesbull Planetary Waves
The rotation of the Earth restoring forceIncluding Rossby Kelvin Equatorial and Yanai waves
bull Surface Waves (gravity waves)The waves that eventually break on the beachThe large between air and water restoring force
bull Internal WavesSubsea wave ~ surface waves = (D) restoring force
bull TsunamisSurface waves with periods near 15 minutes generated by earthquakes
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to waves (cont)Oscillatory flows due to waves (cont)bull Tidal Currents
tidal potential
bull Shelf WavesPeriods a few minutes Confined to shallow regions near shoreThe amplitude of the waves drops off exponentially with
distance from shore
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to wavesOscillatory flows due to wavesbull Planetary Waves
The rotation of the Earth restoring forceIncluding Rossby Kelvin Equatorial and Yanai waves
bull Surface Waves (gravity waves)The waves that eventually break on the beachThe large between air and water restoring force
bull Internal WavesSubsea wave ~ surface waves = (D) restoring force
bull TsunamisSurface waves with periods near 15 minutes generated by earthquakes
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to waves (cont)Oscillatory flows due to waves (cont)bull Tidal Currents
tidal potential
bull Shelf WavesPeriods a few minutes Confined to shallow regions near shoreThe amplitude of the waves drops off exponentially with
distance from shore
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Types of Flow in the Ocean (cont)Types of Flow in the Ocean (cont)
Oscillatory flows due to waves (cont)Oscillatory flows due to waves (cont)bull Tidal Currents
tidal potential
bull Shelf WavesPeriods a few minutes Confined to shallow regions near shoreThe amplitude of the waves drops off exponentially with
distance from shore
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Conservation of Mass and SaltConservation of Mass and Salt
mm = 0 amp = 0 amp SS = 0 = 0 net fresh water loss net fresh water loss minimum flushing timeminimum flushing timebull Net fresh water loss = R + P ndash E
QL bulk formula large amount of ship measurements (T q hellip) impossible
bull m = 0 Vi + R + P = Vo + E
bull S = 0 i Vi Si = o Vo So
bull Measure Vi assume i o
bull Estimate the minimum flushing time
ExampleExamplebull Fig 72 Box model qout = qt t + qx x + qin
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
The Total Derivative (DDt)The Total Derivative (DDt)
DDDt Dt = = partpartdt + udt + u bull A simple example of acceleration of flow in a
small box of fluidbull qout = qt t + qx x + qin
bull DqDt = qt + u qxbull 3D case DDt = t + ux + vy + wzbull 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
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Conservation of Momentum Conservation of Momentum Navier-Stokes equationNavier-Stokes equation
Newtonrsquos 2Newtonrsquos 2ndnd law lawbull F = D(mv)Dt
bull DvDt = Fm = fm = fp+ fc+ fg + fr
Pressure gradient fp = -pCoriolis force fc = -2 v
= 7292 10-5 radianss
Gravity fg = g
Friction fr
bull DvDt = -p -2 v + g + fr
= 7292 10-5 radianss
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Pressure termPressure termbull ax = -(1) (px)
Fx = p y z-(p + p) y z = -p y z
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Gravity termGravity termbull g = gf - ( R)
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
The Coriolis termThe Coriolis term
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Conservation of Momentum Conservation of Momentum Navier-Stokes equation (cont)Navier-Stokes equation (cont)
Momentum Equation in Cartesian Momentum Equation in Cartesian CoordinatesCoordinates
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Conservation of mass Conservation of mass the continuity equationthe continuity equation
Sourcehttpoceanworldtamueduresourcesocng_textbookchapter07chapter07_06htm
For compressible fluidFor compressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
Oceanic flows are incompressibleOceanic flows are incompressiblebull Boussinesqs assumption
v ltlt c (sound speed) When v c v
Phase speed of waves ltlt c c in incompressible flows
Vertical scale of the motion ltlt c2g The increase in pressure produces only small changes in density
bull const except the pressure term (g)
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Conservation of mass Conservation of mass the continuity equation (cont)the continuity equation (cont)
For incompressible flowFor incompressible flowbull The coefficient of compressibility
= 0 for incompressible flows
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Solutions to the Equations of MotionSolutions to the Equations of Motion
Solvable in principleSolvable in principlebull Four equations
3 momentum equations1 continuity equation
bull Four unknowns3 velocity components u v w1 pressure p
bull Boundary conditionsNo slip condition v(boundary) = 0
No penetration condition v(boundary) = 0
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Solutions to the Equations of Motion Solutions to the Equations of Motion (cont)(cont)
Difficult to solve in practiceDifficult to solve in practicebull Exact solution
No exact solutions for the equations with frictionVery few exact solutions for the equations without friction
bull Analytic solutionFor much simplified forms of the equations of motion
bull Numerical solutionSolutions for oceanic flows with realistic coasts and
bathymetric features must be obtained from numerical solutions (Chapter 15)
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Important conceptsImportant concepts
bull Gravity buoyancy and wind are the dominant forces acting on the ocean
bull Earths rotation produces a pseudo force the Coriolis force
bull 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)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Important concepts (cont)Important concepts (cont)
bull 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
bull 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)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid
Important concepts (cont)Important concepts (cont)
bull Conservation of mass leads to the continuity equation which has an especially simple form for an incompressible fluid