Gravity waves derivation

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

• 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

Useful approximation for x small:Base state cancels... makes constants disappear...

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 #1differentiate 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• 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

A wave-like solution

where...

This is a combination of cosine and sinewaves owing to Euler’s relations

Differentiating #1

Differentiating #1

now differentiate again

Differentiating #2now differentiate twice with respect to time

Do same w/r/t z, and the pieces assemble intoa very simple equation

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

Wave phase speed

Example:Wave horizontal wavelength 20 km

vertical wavelength 10 km andstability N = 0.01/s

Wave phase speed

Note as you make the environment more stablewaves move faster.

Note that TWO oppositely propagatingwaves are produced.

Add a mean wind…

intrinsic frequency

flow-relative phase speed

ground-relative phase speed

Wave phase tilt

Wave tilt with height depends onintrinsic forcing frequency and stability

(e.g., smaller ω … smaller cos α…larger tilt)

[end]