gravity waves derivation atm 562 - fall, 2015. introduction gravity waves describe how environment...
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Gravity waves derivation
ATM 562 - Fall, 2015
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Introduction• Gravity waves describe how
environment responds to disturbances, such as by oscillating parcels
• Goal: derive “dispersion relation” that relates frequency (period) of response to wavelength and stability
• Simplifications: make atmosphere 2D, calm, dry adiabatic, flat, non-rotating, constant density
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Starting equations
continuity equation
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Expand total derivatives
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Perturbation method
Calm, hydrostatic, constant density environment(“basic state”)
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Start w/ potential temperature
used log tricks and:
for basic state
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Apply perturbation method
Useful approximation for x small:Base state cancels... makes constants disappear...
speed of sound
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Vertical equation
Do perturbation analysis,neglect products of perturbations
Rearrange, replace density with potential temperature
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Full set of linearized equations
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Obtaining the frequency
equation #1differentiate horizontal and vertical equations of motion
subtract top equation from bottom
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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)
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Final steps...
differentiate w/r/t t again and plug in expressions from last slide
Pendulum equation.... note only w left
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Solving the pendulum equation• We expect to find waves -- so we
go looking for them!
• Waves are characterized by period P, horizontal wavelength Lx and vertical wavelength Lz
• Relate period and wavelength to frequency and wavenumber
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A wave-like solution
where...
This is a combination of cosine and sinewaves owing to Euler’s relations
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Differentiating #1
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Differentiating #1
now differentiate again
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Differentiating #2now differentiate twice with respect to time
Do same w/r/t z, and the pieces assemble intoa very simple equation
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The dispersion equation
A stable environment, disturbed byan 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
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Wave phase speed
Example:Wave horizontal wavelength 20 km
vertical wavelength 10 km andstability N = 0.01/s
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Wave phase speed
Note as you make the environment more stablewaves move faster.
Note that TWO oppositely propagatingwaves are produced.
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Add a mean wind…
intrinsic frequency
flow-relative phase speed
ground-relative phase speed
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Wave phase tilt
Wave tilt with height depends onintrinsic forcing frequency and stability
(e.g., smaller ω … smaller cos α…larger tilt)
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