1 of 12

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/

2 of 12

Scale Analysis: Horizontal Momentum Equation

3 of 12

Scale Analysis

  • Remember, we want to solve the system of equations that describes the atmosphere so that we can
    • understand how the atmosphere works
    • predict the motion and state of the atmosphere
  • Scale analysis 🡪 simplify the equations
  • Identify which processes are most important

4 of 12

Consider x and y components of the momentum equations

Remember the units—each term must have �units of acceleration

m/sec2 or L/t2

5 of 12

Let us define:

acceleration

6 of 12

Scale factors for “large-scale” mid-latitude

7 of 12

Scale analysis of the horizontal momentum equation

  • Write out the scales for the terms in the horizontal momentum equation.

8 of 12

What are the scales of the terms?�Horizontal momentum equations:

U*U/L

U*U/a

U*W/a

Uf

Wf

9 of 12

What are the scales of the terms?� For “large-scale” mid-latitude

U·U/L

U·U/a

U·W/a

ΔP/ρL

Uf

Wf

νU/H2

10-4

10-5

10-8

10-3

10-3

10-6

10-12

10 of 12

What are the scales of the terms?�For “large-scale” mid-latitude

Largest Terms

U·U/L

U·U/a

U·W/a

ΔP/ρL

Uf

Wf

νU/H2

10-4

10-5

10-8

10-3

10-3

10-6

10-12

11 of 12

Consider only the largest terms

This is a dominant balance between the

  • pressure gradient force
  • Coriolis force (the dominant sin(φ) components of� the Coriolis force)

This is the geostrophic balance.

12 of 12

End: Scale Analysis: Momentum Equation

Richard B. Rood

rbrood@umich.edu

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

https://openclimate.org/