1. Quasi-Geostrophic Motions in the Equatorial Areas
Matsuno (1966)
Todd Barron
EAS 8802
2 Oct 2007

2. Purpose of what Matsuno attempted to accomplish
Discuss the behaviors of the Rossby and gravity waves in the equatorial area (more precise than previous studies) and to answer the following questions:
Can we get 2 waves of different types in the tropics?
Is there quasi-geostrophic motion even at the equator?
Is it possible to eliminate the gravity oscillations by use of the filtering procedures?

3. Background
In tropical regions, there are 2 types of waves can be obtained: Rossby and gravity.
What separates these waves are the difference in frequencies and their relationship to pressure and velocity fields.
As we will see, waves confined near the equator exhibit both gravity and Rossby characteristics.
Rossby<<gravity in f

4. The Model
Matsunso used the “divergent barotropic model” (a layer of incompressible fluid of homogenous density w/ a free surface under hydrostatic balance).
Foundation of most of the derived equations:
Equations of motion and mass conservation or continutiy

5. The Model
Matsuno converted the basic equations into non-dimensional form:
Taking units of T and l as:
We can now transform to non- dimensional form:
Can be tinkered with to interpret some fields in other ways

6. Frequency Equation Derived to consider E-W propagating waves.
Matsuno started with the basic eqs. and assumed a factor of e so now we have:
iωt+ikx
Eliminating u and φ gives us:
with boundary conditions of: v0 when y-+∞
Boundary conditions are only satisfied when the
constant, is equal to an odd integer now written as:

7. Frequency Equation
The previous equation gives a relation b/t the frequency and the longitudinal wave number for some meridional mode.
The equation is a cubic root to ω. Therefore we have 3 roots when the wave number and frequency is given. Two roots are expected to gravity waves (one E and W propagating) and the other being a Rossby wave.

8. Frequency Equation
By solving the cubic equation with arbitrary k values, Matsuno was able to graphically show frequencies:

9. Frequency Equation What if n=0? 3 roots will be obtained from ω
Plugging n=0 into the equation and factoring yields:
Matsuno was able to classify the 3 roots as:
When he consider a continuous parameter of:

10. Frequency Equation
So, for n=0, the frequency of the E propagating gravity wave is NOT separated from that of the Rossby wave… they coincide at k=1/√2
Matsuno suggested that one of the roots should be rejected since the boundary conditions are not satisfied when we solve for φ in the following relation:

11. Frequency Equation So what does all that boil down to?
The westward propagating gravity wave and Kelvin wave do NOT exist in the lowest mode (close to Equator).
What we have is a combo of the two.

12. Equatorial Waves
Matsuno’s eigenvalues obtained can be graphically shown:
½ of a wavel is shown in the x direction.
A clear distinction is noticed b/t gravity and Rossby waves at n=1. Strong zonal flow near equator in Rossby wave solution due to geost. Balance b/t pg and wind
Similar situation for n=2. Geost. Balance holds again but we have somesort of ccw vortex at the equator w/ no counterpart in the pressure field.
Matsuno suggested it might come from the vanishing of the CorF?

13. Equatorial Waves
Here only 2 waves exist in a low mode: E and W gravity. W wave was taken from previous diagram which classified Rossby for k>1/sq2 and gravity k<1/sq2.
Pressure and diagram of Rossby in fig 7. No marked difference b/t gravity at k=.5 and Rossby at k=1

14. Summing up Matsuno’s diagrams
No marked difference b/t the Rossby and gravity waves confined near the Equator. Cannot apply mathematical filtering to equatorial motions since there are no physical reason to distinguish the two.

15. Wave Trapping
Matsuno investigated this topic more thoroughly than previously studies.
Found that propagation velocity is larger for higher latitudes, which means the wave generated near the equator will be reflected and reflected toward the equator.
Not the case for Rossby waves
Suggested more studies needed to be done concerning what way the wave is refracted or trapped.
Yoshida was not complete b/c he derived an equastion of sfc eleveations which was diffcult to e dealt w/ and he suggested the existence of trapped waves from the asymptotic behaviors of the solution.
* Velocity of the Rossby waves is smaller in high lats. Cannot expect reflections of Rossby at high lats

16. Forced Stationary Motion
Matsuno considered a stationary state resulting from some external causes.
Transformed to non-dimensional terms
*Examines what motion is caused when some external forces are working.
*Wanted to address the problem of what motions and sfc elevations will be caused when the mass sources and sinks are put alternatingly along the equator.
F, Q are the x, y componets of forces and mass sources or sinks. Given as external forces. Added the au, av terms to
w/ boundary conditions of:

17. Forced Stationary Motion
Flow field is settled as to be geostropic correstponding to the pressure field.
Anitcyclones and cylones are established
Point out strong zonal flow confined near equator caused by the impression of the mass sources and sinks.

18. Conclusion
At the lowest mode, the westward propagating wave exhibits features found in both Rossby and gravity waves.
At lower modes, Rossby and gravity waves are confined near the equator

19. Conclusions
For stationary motions, high and low pressure cells are split along the equator.
Caused by deviations of the sfc elevations being less that that in higher latitudes in magnitude.
Because of this, strong zonal flow is noticed along the equator.