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- External gravity wave (Shallow-water gravity wave)
- Internal gravity (buoyancy) wave
- Inertial-gravity wave: Gravity waves that have a large enough wavelength
to be affected by the earth’s rotation.
- Rossby Wave: Wavy motions results from the conservation of potential
vorticity.
- Kelvin wave: It is a wave in the ocean or atmosphere that balances the
Coriolis force against a topographic boundary such as a coastline, or a
waveguide such as the equator. Kelvin wave is non-dispersive.
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- Overview of Gravity waves
- Surface Gravity Waves
- “Shallow” Water
- Shallow-Water Model
- Dispersion
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- This chapter marks the beginning of more detailed study of the way the
atmosphere-ocean system tends to adjust to equilibrium.
- The adjustment processes are most easily understood in the absence of
driving forces. Suppose, for instance, that the sun is "switched
off," leaving the atmosphere and ocean with some non-equilibrium
distribution of properties.
- How will they respond to the gravitational restoring force?
- Presumably there will be an adjustment to some sort of equilibrium. If
so, what is the nature of the equilibrium?
- In this chapter, complications due to the rotation and shape of the
earth will be ignored and only small departures from the hydrostatic
equilibrium will be considered.
- The nature of the adjustment processes will be found by deduction from
the equations of motion
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- Gravity waves are waves generated in a fluid medium or at the interface
between two media (e.g., the atmosphere and the ocean) which has the
restoring force of gravity or buoyancy.
- When a fluid element is displaced on an interface or internally to a
region with a different density, gravity tries to restore the parcel
toward equilibrium resulting in an oscillation about the equilibrium
state or wave orbit.
- Gravity waves on an air-sea interface are called surface gravity waves
or surface waves while internal gravity waves are called internal waves.
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- “Shallow” in this lecture means that the depth of the fluid layer is
small compared with the horizontal scale of the perturbation, i.e., the
horizontal scale is large compared with the vertical scale.
- Shallow water gravity waves are the ‘long wave approximation” end of
gravity waves.
- Deep water gravity waves are the “short wave approximation” end of
gravity waves.
- Deep water gravity waves are not important to large-scale motions in the
oceans.
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- Normal Mode & Equivalent Depth
- Rigid Lid Approximation
- Boussinesq Approximation
- Buoyancy (Brunt-Väisälä ) Frequency
- Dispersion of internal gravity waves
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- As an introduction to the effects of stratification, the case of two
superposed shallow layers, each of uniform density, is considered.
- In reality, both the atmosphere and ocean are continuously stratified.
- This serves to introduce the concepts of barotropic and baroclinic modes.
- This also serves to introduce two widely used approximations: the rigid
lid approximation and the Boussinesq approximation.
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- The motions corresponding to these particular values of ce or
μ are called normal modes of oscillation.
- In a system consisting of n layers of different density, there are n
normal modes corresponding to the n degrees of freedom.
- A continuously stratified fluid corresponding to an infinite number of
layers, and so there is an infinite set of modes.
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- An N-layer fluid will have one barotropic mode and (N-1) baroclinic
modes of gravity waves, each of which has its own equivalent depth.
- Once the equivalent depth is known, we know the dispersion relation of
that mode of gravity wave and we know how fast/slow that gravity wave
propagates.
- For a continuously stratified fluid, it has an infinite number of modes,
but not all the modes are imporptant. We only need to identify the major
baroclinic modes and to find out their equivalent depths.
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- Rigid lid approximation: the upper surface was held fixed but could
support pressure changes related to waves of lower speed and currents of
interest.
- Ocean models used the "rigid lid" approximation to eliminate
high-speed external gravity waves and allow a longer time step.
- As a result, ocean tides and other waves having the speed of tsunamis
were filtered out.
- The rigid lid approximation was used in the 70's to filter out
gravity wave dynamics in ocean models. Since then, ocean model have
evolved to include a free-surface allowing fast-moving gravity wave
physics.
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- This approximation states that density differences are sufficiently
small to be neglected, except where they appear in terms multiplied
by g, the acceleration due to gravity (i.e., buoyancy).
- In the Boussinesq approximation, which is appropriate for an almost-
incompressible fluid, it assumed that variations of density are small,
so that in the intertial terms, and in the continuity equation, we may
substitute r by r0, a
constant. However, even weak density variations are important in
buoyancy, and so we retain variations in r in the buoyancy term in the vertical equation of
motion.
- Sound waves are impossible/neglected when the Boussinesq approximation
is used, because sound waves move via density variations.
- Boussinesq approximation is for the problems that the variations of
temperature as well as the variations of density are small. In these
cases, the variations in volume expansion due to temperature gradients
will also small. For these case, Boussinesq approximation can simplify
the problems and save computational time.
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- After the approximations, there is no h in the two continuity equation è They can be
combined to become one equation.
- The two momentum equations can also be combined into one single equation
without h.
- At the end, the continuity and momentum equations for the upper and
lower layers can be combined to solve for the dispersive relation for
the baroclinic mode of the gravity wave.
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- A fluid parcel in the presence of stable stratification (N2 >0)
will oscillate vertically if perturbed vertically from its starting
position.
- In atmospheric dynamics, oceanography, and geophysics, the Brunt-Vaisala
frequency, or buoyancy frequency, is the angular frequency at which a
vertically displaced parcel will oscillate within statically stable
environment.
- The Brunt–Väisälä frequency relates to internal gravity waves and
provides a useful description of atmospheric and oceanic stability.
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- A Kelvin wave is a type of low-frequency gravity wave in the ocean or
atmosphere that balances the Earth's Coriolis force against a
topographic boundary such as a coastline, or a waveguide such as the
equator.
- Therefore, there are two types of Kelvin waves: coastal and equatorial.
- A feature of a Kelvin wave is that it is non-dispersive, i.e., the phase
speed of the wave crests is equal to the group speed of the wave energy
for all frequencies.
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- Coastal Kelvin waves always propagate with the shoreline on the right in
the northern hemisphere and on the left in the southern hemisphere.
- In each vertical plane to the coast, the currents (shown by arrows) are
entirely within the plane and are exactly the same as those for a long
gravity wave in a non-rotating channel.
- However, the surface elevation varies exponentially with distance from
the coast in order to give a geostrophic balance.
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- The equator acts analogously to a topographic boundary for both the
Northern and Southern Hemispheres, which make the equatorial Kelvin wave
to behaves very similar to the coastally-trapped Kelvin wave.
- Surface equatorial Kelvin waves travel very fast, at about 200 m per
second. Kelvin waves in the thermocline are however much slower,
typically between 0.5 and 3.0 m per second.
- They may be detectable at the surface, as sea-level is slightly raised
above regions where the thermocline is depressed and slightly depressed
above regions where the thermocline is raised.
- The amplitude of the Kelvin wave is several tens of meters along the
thermocline, and the length of the wave is thousands of kilometres.
- Equatorial Kelvin waves can only travel eastwards.
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- Geostrophic Adjustment Process
- Rossby Radius of Deformation
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- The atmosphere is nearly always close to geostrophic and hydrostatic
balance.
- If this balance is disturbed through such processes as heating or
cooling, the atmosphere adjusts itself to get back into balance. This
process is called geostrophic adjustment.
- A key feature in the geostrophic adjustment process is that pressure and
velocity fields have to adjust to each other in order to reach a
geostrophic balance. When the balance is achieved, the flow at any level
is along the isobars.
- We can study the geostrophic adjustment by studying the adjustment in a
barotropic fluid using the shallow-water equations.
- The results can be extended to a baroclinic fluid by using the concept
of equivalent depth.
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- In atmospheric dynamics and physical oceanography, the Rossby radius of
deformation is the length scale at which rotational effects become as
important as buoyancy or gravity wave effects in the evolution of the
flow about some disturbance.
- “deformation”: It is the radius
that the direction of the flow will be “deformed” by the Coriolis force
from straight down the pressure gradient to be in parallel to the
isobars.
- The size of the radius depends on the stratification (how density or
potential temperature changes with height) and Coriolis parameter.
- The Rossby radius is considerably
larger near the equator.
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- For large scales (KHa « 1), the potential vorticity
perturbation is mainly associated with perturbations in the mass field,
and that the energy changes are in the potential and internal forms.
- For small scales (KHa » 1) potential vorticity perturbations
are associated with the velocity field, and the energy perturbation is
mainly kinetic.
- At large scales (KH-1 » a; or KHa « 1),
it is the mass field that is determined by the initial potential
vorticity, and the velocity field is merely that which is in geostrophic
equilibrium with the mass field. It is said, therefore, that the
large-scale velocity field adjusts to be in equilibrium with the large
scale mass field.
- At small scales (KH-1 « a) it is the velocity
field that is determined by the initial potential vorticity, and the
mass field is merely that which is in geostrophic equilibrium with the
velocity field. In this case it can be said that the mass field adjusts
to be in equilibrium with the velocity field.
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- If the size of the disturbance is much larger than the Rossby radius of
deformation, then the velocity field adjusts to the initial mass
(height) field.
- If the size of the disturbance is much smaller than the Rossby radius of
deformation, then the mass field adjusts to the initial velocity field.
- If the size of the disturbance is close to the Rossby radius of
deformation, then both the velocity and mass fields undergo mutual
adjustment.
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- An important feature of the response of a rotating ftuid to gravity is
that it does not adjust to a state of rest, but rather to a geostrophic
equilibrium.
- The Rossby adjustment problem explains why the atmosphere and ocean are
nearly always close to geostrophic equilibrium, for if any force tries
to upset such an equilibrium. the gravitational restoring force acts
quickly to restore a near-geostrophic equilibrium.
- For deep water in the ocean, where H is 4 or 5 km. c is about 200 m/s
and therefore the Rossby radius a = c/f ~ 2000 km.
- Near the continental shelves, such as for the North Sea where H=40m, the
Rossby radius a = c/f ~ 200 km. Since the North Sea has larger
dimensions than this, rotation has a strong effect on transient motions
such as tides and surges in that ocean region.
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- Equatorial Beta Plane
- Equatorial Wave Theory
- Equatorial Kelvin Wave
- Adjustment under Gravity near the Eq.
- Gill Type Response
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- In the Mid-latitudes, the primary energy source for synoptic-scale
disturbances is the zonal available potential energy associated with the
latitudinal temperature gradient; and latent heat release and radiative
heating are usually secondary contributors.
- In the tropics, however, the storage of available potential energy is
small due to the very small temperature gradients in the tropical
atmosphere. Latent heat release appears to be the primary energy source.
- The dynamics of tropical circulations is very complicate, and there is
no simple theoretical framework, analogous to quasi-geostrophic theory
for the mid-latitude dynamics, that can be used to provide an overall
understanding of large-scale tropical motions.
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- Equatorial waves are an important class of eastward and westward
propagating disturbances in the atmosphere and in the ocean that are
trapped about the equator (i.e., they decay away from the equatorial
region).
- Diabatic heating by organized tropical convection can excite atmospheric
equatorial waves, whereas wind stresses can excite oceanic equatorial
waves.
- Atmospheric equatorial wave propagation can cause the effects of
convective storms to be communicated over large longitudinal distances,
thus producing remote responses to localized heat sources.
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- f-plane approximation: On a rotating sphere such as the earth, f varies
with the sine of latitude; in the so-called f-plane approximation, this
variation is ignored, and a value of f appropriate for a particular
latitude is used throughout the domain.
- β-plane approximation: f is set to vary linearly in space.
- The advantage of the beta plane approximation over more accurate
formulations is that it does not contribute nonlinear terms to the
dynamical equations; such terms make the equations harder to solve.
- Equatorial β-plane approximation:
- cosφ » 1,
- sinφ » y/a.
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- f-plane approximation: On a rotating sphere such as the earth, f varies
with the sine of latitude; in the so-called f-plane approximation, this
variation is ignored, and a value of f appropriate for a particular
latitude is used throughout the domain.
- β-plane approximation: f is set to vary linearly in space.
- The advantage of the beta plane approximation over more accurate
formulations is that it does not contribute nonlinear terms to the
dynamical equations; such terms make the equations harder to solve.
- Equatorial β-plane approximation:
- cosφ » 1,
- sinφ » y/a.
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- The delayed oscillator suggested that oceanic Rossby and Kevin waves
forced by atmospheric wind stress in the central Pacific provide the
phase-transition mechanism (I.e. memory) for the ENSO cycle.
- The propagation and reflection of waves, together with local air-sea
coupling, determine the period of
the cycle.
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