ATM 562 Fall 2015 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 ID: 473822
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Slide1
Gravity waves derivation
ATM 562 - Fall, 2015Slide2
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
densitySlide3
Starting equations
continuity equationSlide4
Expand total derivativesSlide5
Perturbation method
Calm, hydrostatic, constant density environment
(“basic state”)Slide6
Start w/ potential temperature
used log tricks and:
for basic stateSlide7
Apply perturbation method
Useful approximation for
x
small:
Base state cancels... makes constants disappear...
speed of soundSlide8
Vertical equation
Do perturbation analysis,
neglect products of perturbations
Rearrange, replace density with potential temperatureSlide9
Our first pendulum equation
We now know an oscillating parcel
will disturb its environment
and p’ plays a crucial, non-negligible
roleSlide10
Full set of linearized equationsSlide11
Obtaining the frequency equation #1
differentiate horizontal and vertical equations of motion
subtract top equation from bottomSlide12
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)Slide13
Final steps...
differentiate w/r/t
t
again and plug in
expressions from last slide
Pendulum equation.... note only
w
leftSlide14
Solving the pendulum equation
We expect to find waves -- so we go looking for them!
Waves are characterized by period P, horizontal wavelength L
x
and vertical wavelength L
z
Relate period and wavelength to frequency and wavenumberSlide15
A wave-like solution
where...
This is a combination of cosine and sine
waves owing to Euler’s relationsSlide16
Differentiating #1Slide17
Differentiating #1
now differentiate againSlide18
Differentiating #2
now differentiate twice with respect to time
Do same w/r/t
z
, and the pieces assemble into
a very simple equationSlide19
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 Slide20
Wave phase speed
Example:
Wave horizontal wavelength 20 km
vertical wavelength 10 km and
stability N = 0.01/sSlide21
Wave phase speed
Note as you make the environment more stable
waves move
faster
.
Note that TWO oppositely propagating
waves are produced.Slide22
Add a mean wind…
intrinsic frequency
flow-relative phase speed
ground-relative phase speedSlide23
Wave phase tilt
Wave tilt with height depends on
intrinsic forcing frequency and stability
(e.g., smaller
ω …
smaller cos
α
…larger tilt)Slide24
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