6.2 Matsuno-Gill Response

Learning Objectives

After completing this lecture, students should be able to:

understand the physical meaning of forced tropical circulations
explain why damping is introduced in the shallow-water model
distinguish between freely propagating waves and forced steady-state responses
demonstrate the steady-state response to prescribed tropical forcing
identify Kelvin-wave and Rossby-wave components of the Matsuno-Gill response
analyze how forcing amplitude, width, and latitude affect the circulation response

Concept

The free equatorial waves introduced in the previous section describe how disturbances propagate in the absence of external forcing and damping.

In the tropical atmosphere, however, large-scale circulation is often driven by persistent diabatic heating associated with convection. Examples include:

tropical rainfall systems,
the Intertropical Convergence Zone,
monsoon convection,
El Niño-related heating anomalies,
the Madden-Julian Oscillation.

Localized tropical heating can excite both eastward and westward responses. The resulting circulation pattern is often interpreted as a combination of an eastward Kelvin-wave response and a westward Rossby-wave response.

This idealized forced response is commonly known as the Matsuno-Gill response.

Model / Equation

Forced Shallow-Water Equations

To study the response to prescribed heating, we add a forcing term \(Q\) and weak damping to the nondimensional shallow-water equations.

The governing equations become

\[ \frac{\partial u}{\partial t} = yv - \frac{\partial p}{\partial x} - ru, \]

\[ \frac{\partial v}{\partial t} = -yu - \frac{\partial p}{\partial y} - rv, \]

\[ \frac{\partial p}{\partial t} = -\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} -\epsilon p + Q. \]

Here,

\(u\) is the zonal velocity,
\(v\) is the meridional velocity,
\(p\) is the pressure or geopotential anomaly,
\(Q\) is the prescribed heating,
\(r\) is the momentum damping coefficient,
\(\epsilon\) is the pressure damping coefficient.

The forcing \(Q\) acts as a source term in the pressure equation and represents prescribed tropical heating.

The damping terms,

\[ -ru, \qquad -rv, \qquad -\epsilon p, \]

remove energy from the system and prevent unlimited growth under persistent forcing.

Gaussian Heating

In the Python model, the prescribed heating is represented by a Gaussian function:

\[ Q(x,y) = Q_0 \exp \left[ -\frac{1}{2} \left( \frac{x-x_Q}{\sigma_x} \right)^2 \right] \exp \left[ -\frac{1}{2} \left( \frac{y-y_Q}{\sigma_y} \right)^2 \right], \]

where

\(Q_0\) is the forcing amplitude,
\(x_Q\) is the zonal center of the forcing,
\(y_Q\) is the meridional center of the forcing,
\(\sigma_x\) is the zonal width,
\(\sigma_y\) is the meridional width.

By modifying these parameters, we can examine how the tropical circulation response depends on the size, sign, and latitude of the heating.

Steady-State Response

A true steady-state solution is obtained by assuming that the tendency terms vanish:

\[ \frac{\partial u}{\partial t} = \frac{\partial v}{\partial t} = \frac{\partial p}{\partial t} = 0. \]

Under this assumption, the Coriolis force, pressure-gradient force, damping, and external forcing are in balance.

Rather than solving the steady-state equations directly, we solve the time-dependent shallow-water equations numerically and introduce weak damping terms.

As the integration proceeds, the tendency terms gradually become smaller,

\[ \frac{\partial u}{\partial t} \rightarrow 0, \qquad \frac{\partial v}{\partial t} \rightarrow 0, \qquad \frac{\partial p}{\partial t} \rightarrow 0, \]

and the model approaches an equilibrium state.

Therefore, the steady-state response can be obtained by integrating the time-dependent equations forward in time until the solution converges. This equilibrium solution is known as the Matsuno-Gill response and provides a simple framework for understanding tropical circulations associated with localized convection and diabatic heating.

Physical Interpretation

The Matsuno-Gill response is characterized by an asymmetric circulation pattern around the heating region.

A typical response to equatorial heating includes:

an eastward Kelvin-wave-like response,
a westward Rossby-wave-like response,
low-level convergence toward the heating region,
circulation gyres west of the heating,
pressure or geopotential anomalies associated with the forced circulation.

The east-west asymmetry arises because Kelvin and Rossby waves have different propagation characteristics.

Kelvin waves propagate eastward along the equator, while Rossby waves propagate westward and form gyre-like circulation patterns.

Connection to the Python Model

The Python model solves the forced nondimensional shallow-water equations using a forward-Euler time integration scheme.

For example, the zonal momentum equation is implemented as

dudt = Y * v - ddx(p) - r * u
u_new = u + DT * dudt

Similarly,

dvdt = -Y * u - ddy(p) - r * v

dpdt = -(ddx(u) + ddy(v)) - eps * p + Q

For the Matsuno-Gill response, the forcing and damping terms are activated:

r = 0.08
eps = 0.08
Q = forcing_term(X, Y, t)

The model is initialized from rest and integrated until the response approaches a balanced state.

Sample Code

Python scripts for this lecture are provided separately.

Template file:

Template: 6_1-shallow_water_template.py

Example command:

python 6_1-shallow_water_template.py

Exercise

Follow the steps below to simulate the Matsuno-Gill response.

Step 1. Generate the Matsuno-Gill Response

In this step, we introduce a prescribed forcing term and weak damping to simulate the steady-state response of the tropical atmosphere.

Create a new script:

cp 6_1-shallow_water_template.py 6_2-shallow_water_MG.py

Use the following configuration:

DT = 0.02
NT = 600

r = 0.08
eps = 0.08

Use zero initial conditions:

A = 0.0

p = A * env_x * env_y * np.sin(phase)
u = np.zeros_like(p)
v = np.zeros_like(p)

Activate the Gaussian forcing in the forcing_term() function:

Q0 = 2.5
xQ = 0.0
yQ = 0.0

sigxQ = 0.75
sigyQ = 0.75

Q = Q0 * np.exp(-0.5 * ((X - xQ) / sigxQ)**2) \
       * np.exp(-0.5 * ((Y - yQ) / sigyQ)**2)

To visualize the prescribed heating, overlay forcing contours on the pressure field. Add the following code after each contourf() call:

Qplot = forcing_term(X, Y, 0.0)

ax.contour(
    x, y, Qplot,
    levels=np.arange(-3.0, 3.5, 0.5),
    colors="k",
    linewidths=1.0
)

To ensure negative contours are shown as dashed lines, add the following import and setting near the top of the script:

import matplotlib as mpl
mpl.rcParams["contour.negative_linestyle"] = "dashed"

Positive heating contours will be drawn as solid black lines, while negative heating contours will be drawn as dashed black lines.

Run the model:

python 6_2-shallow_water_MG.py

Examine the generated files:

6_2-shallow_water_MG.mp4
6_2-shallow_water_MG.png

Questions:

Does the solution approach a steady state?
What circulation develops east of the heating?
What circulation develops west of the heating?
Can you identify the Kelvin-wave response?
Can you identify the Rossby-wave response?

Step 2. Elongated Heating

The Matsuno-Gill response depends on the horizontal scale of the prescribed heating. In this step, increase the horizontal extent of the forcing.

Modify the forcing width:

sigxQ = 2.0
sigyQ = 1.5

Run the model again.

You may also experiment with:

sigxQ = 4.0
sigyQ = 2.0

Questions:

How does the circulation pattern change as the heating becomes broader?
Does the Kelvin-wave response extend farther eastward?
Does the Rossby-wave response become easier to identify?
How does the horizontal scale of the response compare with the horizontal scale of the forcing?

Step 3. Negative Heating and Forcing Amplitude

Next, reverse the sign of the forcing with the default forcing width:

Q0 = -2.5

sigxQ = 0.75
sigyQ = 0.75

Run the model and compare the response with the positive-heating case.

Then experiment with different forcing amplitudes:

Q0 = 1.0

Q0 = 5.0

Questions:

How does the circulation change when the sign of the forcing is reversed?
How does increasing the forcing amplitude affect the response?
Does the spatial pattern change, or only the magnitude?
Are the Rossby and Kelvin responses still present?

Step 4. Off-Equatorial Heating

Finally, move the heating away from the equator.

First, move the heating slightly north of the equator:

yQ = 0.5

Run the model and examine the response.

Then move the heating farther north by testing:

yQ = 1.0

yQ = 1.5

yQ = 2.0

Compare how the circulation pattern changes as the heating moves farther from the equator.

Finally, move the heating south of the equator:

yQ = -1.5

Run the model again and compare the response with the Northern Hemisphere cases.

Questions:

How does the circulation change when the heating is moved away from the equator?
Is the response still symmetric about the equator?
How does the Rossby-wave response change?
How does the Kelvin-wave response change?
Which hemisphere exhibits the strongest circulation response?

Step 5. Summary

Compare the following experiments:

Experiment	Modification
Step 1	Positive equatorial heating
Step 2	Elongated heating
Step 3	Negative heating and different amplitudes
Step 4	Off-equatorial heating

Focus on the following comparisons:

sign of the forcing
forcing amplitude
forcing size
forcing latitude
Kelvin-wave response
Rossby-wave response
symmetry about the equator

These experiments demonstrate how localized tropical heating generates large-scale tropical circulations through the interaction of equatorial Kelvin and Rossby waves.

Key Points

The Matsuno-Gill response describes the steady-state circulation generated by localized tropical heating.
The forced shallow-water equations include both heating and damping terms.
Damping prevents unlimited growth and allows the model to approach equilibrium.
The steady-state response can be obtained by integrating the time-dependent equations until the tendencies become small.
Equatorial heating produces an eastward Kelvin-wave-like response and a westward Rossby-wave-like response.
The response depends on the sign, amplitude, width, and latitude of the forcing.
The Matsuno-Gill framework provides a simple model for understanding tropical circulations associated with convection and diabatic heating.

Homework

Complete the following tasks:

Generate one example of a Matsuno-Gill response simulation.
Submit the corresponding MP4 animation file.
Provide a brief discussion, approximately one paragraph, describing:

the forcing configuration,
the Kelvin-wave-like response,
the Rossby-wave-like response,
whether the solution approaches a steady state.

Optional Further Experiments

Students interested in further exploration may investigate one or more of the following:

Modify the damping parameters (\(r\) and \(\epsilon\)) and examine how the steady-state solution changes.
Experiment with multiple heating centers.
Investigate the response to time-varying heating.
Compare positive and negative heating responses.
Examine strongly off-equatorial heating.
Compare the simulated Matsuno-Gill response with observed tropical convection associated with ENSO or the Madden-Julian Oscillation (MJO).

Discussion Questions

Why does tropical heating produce an asymmetric response east and west of the forcing region?
What roles do Kelvin and Rossby waves play in the Matsuno-Gill response?
Why is damping needed to obtain a steady-state solution?
How does the location of tropical heating influence the large-scale circulation response?
What aspects of the real tropical atmosphere are not represented by this idealized shallow-water model?