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- Perturbation Method
- Properties of Wave
- Shallow Water Gravity Waves
- Rossby Waves
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- With this method, all filed variables are separated into two parts: (a)
a basic state part and (b) a deviation from the basic state:
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- Assumptions 1: : the basic state variables must themselves satisfy the
governing equations when the perturbations are set to zero.
- Assumptions 2: the perturbation fields must be small enough so that all
terms in the governing equations that involve products of the
perturbations can be neglected.
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- If terms that are products of the perturbation variables are neglected,
the nonlinear governing equations are reduced to linear differential
equations in the perturbation variables in which the basic state
variables are specified coefficients.
- These equations can then be solved by standard methods to determine the
character and structure of the perturbations in terms of the known basic
state.
- For equations with constant coefficients the solutions are sinusoidal or
exponential in character.
- Solution of perturbation equations then determines such characteristics
as the propagation speed, vertical structure, and conditions for growth
or decay of the waves.
- The perturbation technique is especially useful in studying the
stability of a given basic state flow with respect to small superposed
perturbations.
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- Perturbations in the atmosphere can be represented in terms of a Fourier
series of sinusoidal components:
- L: distance around a latitude
circle,
- s: planetarywave number, an
integer designating the number of waves around a latitude circle
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- e i(kx+ly+nz+vt)
- k: zonal wave number
- l: meridional wave number
- n: vertical wave number
- v: frequency
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- The phase velocity of a wave is the rate at which the phase of the wave
propagates in space.
- The phase speed is given in terms of the wavelength λ and period T
(or frequency ν and wavenumber k) as:
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- In cases where several waves add together to form a single wave shape
(called the envelope), each individual wave component has its own
wavenumber and phase speed.
- For waves in which the phase speed varies with k, the various sinusoidal
components of a disturbance originating at a given location are at a
later time found in different places. Such waves are dispersive.
- For nondispersive waves, their phase speeds that are independent of the
wave number.
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- Some types of waves, such as acoustic waves, have phase speeds that are
independent of the wave number.
- In such nondispersive waves a spatially localized disturbance consisting
of a number of Fourier wave components (a wave group) will preserve its
shape as it propagates in space at the phase speed of the wave.
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- For dispersive waves, the shape of a wave group will not remain constant
as the group propagates.
- Furthermore, the group generally broadens in the course of time, that
is, the energy is dispersed.
- When waves are dispersive, the speed of the wave group is generally
different from the average phase speed of the individual Fourier
components.
- In synoptic-scale atmospheric disturbances, however, the group velocity
exceeds the phase velocity.
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- The group velocity of a wave is the velocity with which the overall
shape of the wave's amplitudes (i.e. envelope) propagates through space.
- Two horizontally propagating waves of equal amplitude but slightly
different wavelengths with wave numbers and frequencies differing by 2δk
and 2δν, respectively. The total disturbance is thus:
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- Waves in fluids result from the action of restoring forces on fluid
parcels that have been displaced from their equilibrium positions.
- The restoring forces may be due to compressibility, gravity, rotation,
or electromagnetic effects.
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- Shallow water gravity waves can exist only if the fluid has a free
surface or an internal density discontinuity.
- The restoring force is in the vertical so that it is transverse to the
direction of propagation.
- The back-and-forth oscillations of the paddle generate alternating
upward and downward perturbations in the free surface height, which
produce alternating positive and negative accelerations. These, in turn,
lead to alternating patterns of fluid convergence and divergence.
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- The wave type that is of most importance for large-scale meteorological
processes is the Rossby wave, or planetary wave.
- In an inviscid barotropic fluid of constant depth (where the divergence of the horizontal velocity
must vanish), the Rossby wave is an absolute vorticity-conserving motion
that owes its existence to the variation of the Coriolis parameter with
latitude, the so-called β-effect.
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- Considering a closed chain of fluid parcels initially aligned along a
circle of latitude with ζ = 0 at time t0, then the chain
displaced δy from the original latitude at time t1.
- Due to the conservation of absolute vorticity, we know:
- Here, β ≡ df/dy is the planetary vorticity gradient.
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- For a typical midlatitude synoptic-scale disturbance, with similar
meridional and zonal scales (l ≈ k) and zonalwavelength of order
6000 km, the Rossby wave speed relative to the zonal flow is
approximately −8 m/s.
- Because the mean zonal wind is generally westerly and greater than 8
m/s, synoptic-scale Rossby waves usually move eastward, but at a phase
speed relative to the ground that is somewhat less than the mean zonal
wind speed.
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- For longer wavelengths the westward Rossby wave phase speed may be large
enough to balance the eastward advection by the mean zonal wind so that
the resulting disturbance is stationary relative to the surface of the
earth.
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