CLIMATE/EARTH 401�Geophysical Fluid Dynamics�Dynamical Meteorology

Richard B. Rood

rbrood@umich.edu

https://clasp.engin.umich.edu/people/richard-b-rood/

https://openclimate.org/

Analysis Barotropic Wave: Problem

• As background the next five slides are discussed in lecture 15.3.2

An equation for geopotential tendency

Start with the QG �vorticity equation

Use the definition of geostrophic wind

Plug this in, and we immediately have an equation for the time rate of change of the geopotential height

Consider a barotropic fluid

Form a perturbation equation

Barotropic height tendency equation

Divide wind into mean and perturbation, assume no mean north/south wind. The mean is taken around circle of latitude (zonal mean)

Substitute mean and perturbation, take average, form perturbation equation after linearizing.

Assume a “wave like solution”

Dispersion relation. Relates frequency and wave number to flow. Must be true for waves.

Stationary wave?

Wind must be positive, from the west, for a wave.

Think about this from a vorticity perspective

• The next four slides rewrite the GQ vorticity equation in terms of advection of vorticity.

Rewrite the QG vorticity equation

Expand material derivative

Rewrite the QG vorticity equation

Use the

continuity equation

Combine the advection terms

Rewrite the QG vorticity equation

Advection of vorticity

Consider the vorticity advection

Advection of vorticity

Advection of relative vorticity

Advection of planetary vorticity

Exercise: Explore the role of wavelength more quantitatively

• We now develop a quantitative way to compare “long waves” with “short waves”

Assume that the geopotential takes the form of a wave

Horizontal wavenumber

Mean Ф Gradient in y

Wave

Mean

Use the relationship between geostrophic wind and geopotential

Using the proposed solution on previous page. Write out u_g and v_g

Divide the geostrophic wind into mean and perturbation

Divide wind into zonal mean and perturbation, assume no mean north/south wind.

Using the proposed wave-like solution, identify the mean and perturbation winds.

​

Prove that the zonal average of the perturbation is zero.

Plug our results into the advection of relative vorticity

Using the results from previous, calculate the advection of relative vorticity.

Plug our results into the advection of planetary vorticity

Using the results from previous, calculate the advection of planetary vorticity.

Compare advection of planetary and relative vorticity

For a wave with meridional wavelength of 1000 km, U = 10 m/sec, at what zonal wavelength are the relative and planetary vorticity of the same magnitude?

Compare advection of planetary and relative vorticity

Consider the geopotential tendency

For the zonal wavelength calculated in the previous question:

• Is the wave amplitude increasing or decreasing?
• For a zonal wave with wavelength that is 50% smaller than previous, is the wave amplitude increasing or decreasing?
• For a zonal wave with wavelength that is 50% larger than previous, is the wave amplitude increasing or decreasing?
• What are the wave numbers for the longer and shorter waves?

Consider these idealized charts

Geopotential Map (Northern Hemisphere)

Φ₀ - ΔΦ

Φ₀ + ΔΦ

Φ₀

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L

L

H

Where is geostrophic approximation most accurate?

Where is the ageostrophic wind the strongest?

What is the sign of the geostrophic wind?

*

w

Geopotential Map (Northern Hemisphere)

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L

L

H

Φ₀ - ΔΦ

Φ₀ + ΔΦ

Φ₀

*

w

At points A, B, C:

• What is the sign of planetary vorticity?
• What is the sign of the relative vorticity?
• What is the sign of advection of planetary vorticity?
• What is the sign of advection of relative vorticity?

Geopotential Map (Northern Hemisphere)

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

L

L

H

At Point W,

• For the larger wave number, calculated above, is the geopotential increasing or decreasing?
• For the smaller wave number, calculated above, is the geopotential increasing or decreasing?

Φ₀ - ΔΦ

Φ₀ + ΔΦ

Φ₀

*

w

End: Analysis Barotropic Wave: Problem

Richard B. Rood

rbrood@umich.edu

https://clasp.engin.umich.edu/people/richard-b-rood/

https://openclimate.org/