Thursday, November 13, 2008

Gravity Waves

gravity wave:
gravity waves are waves generated in a fluid medium or at the interface between two mediums (e.g. the atmosphere or ocean) which has the restoring force of gravity or buoyancy.

When a fluid parcel is displaced on an interface or internally to a region with a different density, gravity restores the parcel toward equilibrium resulting in an oscillation about the equilibrium state. Gravity waves on an air-sea interface are called surface gravity waves or surface waves while internal gravity waves are called internal waves. Ocean waves and tsunamis are examples of gravity waves.

gravity wave—(Also called gravitational wave.) A wave disturbance in which buoyancy (or reduced gravity) acts as the restoring force on parcels displaced from hydrostatic equilibrium.
There is a direct oscillatory conversion between potential and kinetic energy in the wave motion. Pure gravity waves are stable for fluid systems that have static stability. This static stability may be 1) concentrated in an interface or 2) continuously distributed along the axis of gravity. The following remarks apply to the two types, respectively. 1) A wave generated at an interface is similar to a surface wave, having maximum amplitude at the interface. A plane gravity wave is characteristically composed of a pair of waves, the two moving in opposite directions with equal speed relative to the fluid itself. In the case where the upper fluid has zero density, the interface is a free surface and the two gravity waves move with speeds
where U is the current speed of fluid, g the acceleration of gravity, L the wavelength, and H the depth of the fluid. For deep-water waves (or Stokesian waves or short waves), H >> L and the wave speed reduces to
For shallow-water waves (or Lagrangian waves or long waves), H << L, and
All waves of consequence on the ocean surface or interfaces are gravity waves, for the surface tension of the water becomes negligible at wavelengths of greater than a few centimeters ( see capillary wave). 2) Heterogeneous fluids, such as the atmosphere, have static stability arising from a stratification in which the environmental lapse rate is less than the process lapse rate. The atmosphere can support short internal gravity waves and long external gravity waves. The short waves (of the order of 10 km) have been associated, for example, with lee waves and billow waves. Such waves have vertical accelerations that cannot be neglected in the vertical equation of perturbation motion. The long gravity waves, moving relative to the atmosphere with speed ±(gH)½, where H is the height of the corresponding homogeneous atmosphere, have small vertical accelerations and are therefore consistent with the quasi-hydrostatic approximation. In neither type of gravity wave, however, is the horizontal divergence negligible. For meteorological purposes in which neither type is desired as a solution, for example, numerical forecasting, they may be eliminated by some restriction on the magnitude of the horizontal divergence. The above discussion is based upon the method of small perturbations. In certain special cases of water waves, for example, the Gerstner wave or the solitary wave, a theory of finite-amplitude disturbances exists. See shear-gravity wave.
Gill, A. E., 1982: Atmosphere–Ocean Dynamics, Academic Press, 95–188.

internal gravity wave—(Also called internal waves, gravity waves .) A wave that propagates in density-stratified fluid under the influence of buoyancy forces.

The dispersion relation is given by frequency
in which N is the buoyancy frequency and kh is the horizontal component of the wavenumber vector k. For all wavenumbers, internal gravity waves have frequency smaller than N. Their group velocity is perpendicular to the phase velocity such that the vertical component of the group velocity is opposite in sign to the vertical component of the phase velocity.

Gravity waves occur at interfaces between high and low density fluids. Most people are familiar with water surface waves, which act between water (as in lakes or oceans) and the air.

Where low density water overlies high density water in the ocean, internal gravity waves propagate along the boundary. They are especially common over the continental shelf regions of the world oceans and where brackish water overlies salt water at the outlet of large rivers.

There is typically little surface expression of the waves, aside from slick bands that can form over the trough of the waves.

Wavelengths vary from centimetres to kilometres with periods of seconds to hours.

Meteo 422 – Lecture 28 – Topographic gravity waves using the perturbation method

Dr. George S. Young

The derivations below generally follow those in the course text: Holton's "An Introduction to Dynamic Meteorology"

Goals: Use the perturbation method to develop the theory of those internal gravity waves driven by flow over mountains. Discover how these theoretical results relate to the different types of mountain lee waves and how they can be used to forecast downslope windstorms.

* What are topographic gravity waves?
o Topographic gravity waves are the internal gravity waves that result when flow over mountains displace air in the vertical
o They are often called mountain lee waves, mountain waves, or lee waves.
o They propagate upstream

· With the horizontal phase speed matching the wind speed

· So that they remain fixed in position relative to the terrain

* Why do we care about topographic gravity waves?
o Mountain lee waves cause some of the largest vertical velocities in the atmosphere

· Severe weather

· Flight safety

· High altitude soaring

o Mountain wave drag needs to be parameterized in NWP models

* What additional assumptions do we make when applying the perturbation method to topographic gravity waves?
o Sinusoidal ridges

· This approximation doesn't hold up if you have isolated ridges

· But section 9.4 in Holton shows a more sophisticated version of the solution that is appropriate for isolated ridges.

o We assume that the waves are fixed (standing) relative to the terrain.

· That is, the Earth-relative frequency of the waves is zero.

· This assumption is appropriate unless the weather is changing rapidly.

o Holton again assumes that the mean wind and stability don't change with height

· This assumption works poorly near fronts.

· Much of the exciting weather associated with mountain waves results from this failure.

o There are patches for all of these problems in the more sophisticated versions of the internal gravity wave theory covered in section 9.4.

* Deriving the internal gravity wave dispersion relation via the perturbation method
o We'll make use of our pervious results to skip steps in the perturbation method wherever possible

· We're using the same equations of motion as last lecture.

· And the same linearization as last lecture.

o The wave equation is simpler however because the local derivative is zero for a standing wave. Not having a local derivative in the original equations of motion simplifies the derivation somewhat (Holton shows none of it) and results in a much simpler wave equation.

Note that this is only a 2nd order PDE instead of the 4th order PDE we had for traveling gravity waves.

* Now we derive the dispersion relation from the wave equation, using the same technique we did for traveling gravity waves.
o We first assume a solution of the usual form for a wave equation (i.e. the real part of a complex exponential).

where w-hat is complex and the phase allows for variations in x, y, and t.

· Note that we'll require k to be real so that the waves are sinusoidal in the horizontal, but allow m to be complex so that the waves can decay or grow with height.

· Remember the relationship between these wavenumbers and the corresponding wavelengths

o Then we plug this assumed solution into the wave equation to get the dispersion relation.

This equation will provide us with the key to determining what weather conditions allow vertically propagating waves versus vertically trapped waves.

Either or

* Vertically propagating versus decaying waves
o Vertically propagating waves occur when m is real (i.e. the wave is sinusoidal in the vertical)

· This requires m2 to be greater than zero.

o In contrast, vertically trapped waves occur when m is imaginary (i.e. the wave is exponential in the vertical)

· This requires m2 to be less than zero.

· Boundedness (i.e. nothing goes to infinity) requires that the resulting exponential in the vertical be exponential decay rather than growth.

o Consider a prototypical mountain – sinusoidal in the horizontal and of amplitude hm.

· Note that the flow must parallel the ground at the surface (it can't go through rock!). So the vertical velocity at the surface is just the wind speed times the slope.

· This gives us a lower boundary condition.

o Imaginary m (i.e. vertically trapped waves) occur when uk>N (i.e. the frequency of the mountain relative to the flow is greater than the buoyancy frequency of the resulting waves).

· This means it takes the air less time to cross the mountain than it does to complete one buoyantly driven oscillation.

· Or, equivalently, a freely traveling wave couldn't propagate up stream as fast as the mountain is.

where μ = magnitude of m.

o Real m (i.e. vertically propagating waves) occur when uk
· This means it takes the air more time to cross the mountain than it does to complete one buoyantly driven oscillation.

· Or, equivalently, a freely traveling wave can outrun the mountain and so tilts forward with height.

* Non-linearity
o Beware of the consequences of our assumed linearity

· We've assumed that the velocity perturbations are small relative to the mean wind.

· In real mountain lee wind storms they are often nearly equal to the mean wind.

· If they become equal to the mean wind, the wave breaks (like surf).

o So the results we've derived break down as the waves become more severe.

* Useful results
o Stable stratification, wide ridges, and weak wind favor vertically propagating waves

· The Rockies produce these often because the Front Range is about 100 km wide

· In contrast, the Appalachians tend to produce vertically trapped waves because the ridges are 1 to 10 km wide.

o Vertically propagating mountain lee waves tilt upwind with height

· So expect strong vertical motions OVER the mountain for vertically propagating waves. There may however be very little vertical motion downwind of the mountain.

· In contrast vertically trapped (i.e. horizontally propagating) waves have strong vertical motions both over the lee slopes and far downwind of the mountain.

· This difference is important for aviation forecasting – especially turbulence aloft forecasts.
Also refer,M1

for derivation.
the book named..........
Dynamics of the Atmosphere: A Course in Theoretical Meteorology
By Wilford Zdunkowski, Andreas Bott
it is available in Lib. Page:426

Reference is an email from
Anish Kumar.M.Nair

No comments: