Quasi-Geostrophic Motions in the Equatorial Areas

Matsuno (1966)

Todd Barron

EAS 8802

2 Oct 2007

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:

1. Can we get 2 waves of different types in the tropics?

2. Is there quasi-geostrophic motion even at the equator?

3. Is it possible to eliminate the gravity oscillations by use of the filtering procedures?

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.

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:

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:

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: v0 when y-+∞

Boundary conditions are only satisfied when the

constant, is equal to an odd integer now written as:

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.

Frequency Equation

• By solving the cubic equation with arbitrary k values, Matsuno was able to graphically show frequencies:

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:

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:

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.

Equatorial Waves• Matsuno’s eigenvalues obtained can be graphically shown:

Equatorial Waves

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.

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.

Forced Stationary Motion

• Matsuno considered a stationary state resulting from some external causes.

Transformed to non-dimensional terms

w/ boundary conditions of:

Forced Stationary Motion

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

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.