Gravity waves derivation
ATM 562 - Fall, 2015

Introduction
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Gravity waves describe how environment responds to disturbances, such as by oscillating parcels
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Goal: derive “dispersion relation” that relates frequency (period) of response to wavelength and stability
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Simplifications: make atmosphere 2D, calm, dry adiabatic, flat, non-rotating, constant density

Starting equations continuity equation

Expand total derivatives

Perturbation method
Calm, hydrostatic, constant density environment
(“basic state”)

Start w/ potential temperature used log tricks and: for basic state

Apply perturbation method
Base state cancels... makes constants disappear...
Useful approximation for x small: speed of sound

Vertical equation
Do perturbation analysis, neglect products of perturbations
Rearrange, replace density with potential temperature

Full set of linearized equations

Obtaining the frequency equation
#1 differentiate horizontal and vertical equations of motion subtract top equation from bottom

Obtaining the frequency equation
#2
…where we differentiated w/r/t x . Since continuity implies then
Since the potential temperature equation was then
(differentiated twice w/r/t x , rearranged)

Final steps...
differentiate w/r/t t again and plug in expressions from last slide
Pendulum equation.... note only w left

Solving the pendulum equation
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We expect to find waves -- so we go looking for them!
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Waves are characterized by period P, horizontal wavelength L x and vertical wavelength L z
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Relate period and wavelength to frequency and wavenumber

A wave-like solution where...
This is a combination of cosine and sine waves owing to Euler’s relations

Differentiating #1

Differentiating #1 now differentiate again

Differentiating #2 now differentiate twice with respect to time
Do same w/r/t z , and the pieces assemble into a very simple equation

The dispersion equation
A stable environment, disturbed by an oscillating parcel, possesses waves with frequency (period) depending the stability (N) and horizontal & vertical wavelengths (k, m) that can be determined by how the environment is perturbed

Wave phase speed
Example:
Wave horizontal wavelength 20 km vertical wavelength 10 km and stability N = 0.01/s

Wave phase speed
Note as you make the environment more stable waves move faster .
Note that TWO oppositely propagating waves are produced.

Add a mean wind… intrinsic frequency flow-relative phase speed ground-relative phase speed

Wave phase tilt
Wave tilt with height depends on intrinsic forcing frequency and stability
(e.g., smaller ω … smaller cos α
…larger tilt)

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