1	Waves in the Atmosphere and Oceans 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.
2	Lecture 6: Adjustment under Gravity in a Non-Rotating System Overview of Gravity waves Surface Gravity Waves “Shallow” Water Shallow-Water Model Dispersion
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4	Goals of this Chapter 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
5	Gravity Waves 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.
6	Adjustment Under Gravity in a Non-Rotating System
7	Reduced Gravity
8	A Two-Layer Fluid System
9	Shallow Water Gravity Wave
10	Vertical Structure of Ocean
11	Shallow and Deep Water “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.
12	Internal Gravity (Buoyancy) Waves
13	Quasi-Geostrophic Approximation
14	Lecture 7: Adjustment under Gravity of a Density-Stratified Fluid Normal Mode & Equivalent Depth Rigid Lid Approximation Boussinesq Approximation Buoyancy (Brunt-Väisälä ) Frequency Dispersion of internal gravity waves
15	Main Purpose of This Lecture 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.
16	Two Fluids of Different Density
17	Two Fluids: Layer 1 ( )
18	Two Fluids: Layer 2 ( )
19	Adjustments of the Two-Fluid System
20	Normal Modes The motions corresponding to these particular values of c_e 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.
21	Structures of the Normal Modes
22	Structures of the Normal Modes
23	Equivalent Depth (H_e) 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.
24	Rigid Lid Approximation (for the upper layer)
25	Purpose of Rigid Lid Approximation 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.
26	Boussinesq Approximation (for the lower layer)
27	Purpose of Boussinesq Approximation 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.
28	After Using the Two Approximations 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.
29	Brunt–Väisälä Frequency (N) 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.
30	Internal Gravity Waves in Atmosphere and Oceans
31	Dispersion of Internal Gravity Waves
32	Kelvin Waves 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.
33	Costal Kelvin Waves 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.
34	Equatorial Kelvin Waves 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.
35	1997-98 El Nino
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37	Wave Propagation and Reflection
38	Lecture 8: Adjustment in a Rotating System Geostrophic Adjustment Process Rossby Radius of Deformation
39	Geostrophic Adjustments 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.
40	Geostrophic Adjustment Problem
41	An Example of Geostrophic Adjustment
42	Final Adjusted State
43	Rossby Radius of Deformation 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.
44	Rossby Radius and the Equilibrium State For large scales (K_Ha « 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 (K_Ha » 1) potential vorticity perturbations are associated with the velocity field, and the energy perturbation is mainly kinetic. At large scales (K_H^-1 » a; or K_Ha « 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 (K_H^-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.
45	Rossby Radius and the Equilibrium State 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.
46	What does the Geostrophic Adjustment Tell Us? 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.
47	Lecture 9: Tropical Dynamics Equatorial Beta Plane Equatorial Wave Theory Equatorial Kelvin Wave Adjustment under Gravity near the Eq. Gill Type Response
48	Overview 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.
49	Equatorial Waves 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.
50	Equatorial b-Plane Approximation 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.
51	Equatorial b-Plane Approximation 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.
52	Shallow-Water Equation on an Equatorial b-Plane
53	Equatorial Waves with n=0 (Mixed Rossby-Gravity Waves)
54	Mixed Rossby-Gravity Waves
55	Equatorial Waves with “n=-1” (Equatorial Kelvin Waves)
56	Equatorial Kelvin Waves
57	Equatorial Waveguide
58	Quai-Biennial Oscillation (QBO) (in Stratosphere)
59	Why QBO?
60	Steady Forced Equatorial Motions
61	Gill’s Response to Symmetric Heating
62	Delayed Oscillator: Wind Forcing 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.
63	Wave Propagation and Reflection