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

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

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

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

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

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

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

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

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 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

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 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)

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 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 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)

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