Madden-Julian Oscillation

Overview

The Madden-Julian Oscillation (MJO) represents the largest subseasonal (30 to 90 day) atmospheric variability in the low latitude tropical atmosphere. MJO is an eastward traveling atmospheric pattern, moving between 4 to 8 m/s, which influences the location and strength of rainfall in the tropics[1].

(Image credit: NOAA-PSL)

MJO was discovered in 1971 by Roland A. Madden and Paul R. Julian[2] at the National Center for Atmospheric Research (NCAR).

Wheeler-Hendon Phase Diagram

Overview

The Wheeler-Hendon diagram is a phase plot and diagnostics tool[3] to quantify the strength and significance of the MJO over time. The plot represents the Real-Time Multivariate MJO (RMM) which is made up of a pair of empirical orthogonal functions (EOF). EOFs are created from the combined equatorially averaged 850-hPa and 200-hPa zonal wind and the outgoing longwave radiation (OLR).

• Outgoing longwave radiation: The longwave radiation emitted from the top of the Earth’s atmosphere into space

• Zonal wind: The zonal wind anomaly at 200-hPa and 850-hPa represents the changes in the atmospheric circulation

The pair of principal components from the EOFs form two indices: RMM1 and RMM2. The phase relationship between the two zonal winds determine the motion of the MJO.

• RMM1: Represents the MJO’s amplitude. A larger RMM1 indicates stronger MJO activity. Negative values indicate a inactive or suppressed MJO phase and positive values indicate an active or enhanced MJO circulation

• RMM2: Represents the MJO’s phase. The phase determines the geographic position of the MJO’s convection center. It is split into eight phases, each representing to a specific equatorial region

Workflow

import pandas as pd
import numpy as np
from matplotlib import pyplot as plt
import matplotlib.ticker as ticker  # force integer ticks on axis
import matplotlib.lines as lines  # add phase division lines

Example Data

For the purpose of this notebook, we will be pulling data from NOAA’s Physical Science Laboratory (PSL). PSL offers near real-time and historical RMM MJO data.

def retrieve_NOAA_data(data_type=None):
    # https://www.psl.noaa.gov/mjo/
    url = f"https://www.psl.noaa.gov/mjo/mjoindex/{data_type.lower()}.cpcolr.1x.txt"
    column_names = ["YYYY", "MM", "DD", "HH", "RMM1", "RMM2", "Phase"]
    try:
        df = pd.read_csv(url, sep=r'\s+', header=None, names=column_names)
        print(f"Retrieving data from: {url}\n")
        return df
    except Exception:
        print(f"invalid url: {url}")
        return pd.DataFrame(columns=column_names)

We will be retrieving the data related to the Real-time OLR MJO Index (ROMI). See PSL MJO for more daily MJO index values for dates from 1979.

noaa_df = retrieve_NOAA_data("ROMI")
noaa_df

Retrieving data from: https://www.psl.noaa.gov/mjo/mjoindex/romi.cpcolr.1x.txt

	YYYY	MM	DD	HH	RMM1	RMM2	Phase
0	1991	1	1	0	0.12526	-0.06945	0.14323
1	1991	1	2	0	0.18542	-0.04887	0.19175
2	1991	1	3	0	0.23960	-0.03933	0.24281
3	1991	1	4	0	0.27446	0.01990	0.27518
4	1991	1	5	0	0.27714	0.07555	0.28725
...	...	...	...	...	...	...	...
12852	2026	3	10	0	-1.40194	0.00433	1.40195
12853	2026	3	11	0	-1.41309	-0.13210	1.41925
12854	2026	3	12	0	-1.41134	-0.27556	1.43799
12855	2026	3	13	0	-1.43734	-0.42081	1.49768
12856	2026	3	14	0	-1.45038	-0.63790	1.58446

12857 rows × 7 columns

The table above shows NOAA PSL MJO time series data as:

• YYYY: Year

• MM: Month (1-12)

• DD: Day (1-31)

• HH: Hour (always 0, can be ignored)

• RMM1: Real-Time Multivariate MJO Principal Component 1

• RMM2: Real-Time Multivariate MJO Principal Component 2

• Phase: Amplitude of MJO where sqrt(RMM1^2 + RMM2^2)

For the purpose of a plot, we will be using the datetime (YYYY-MM-DD) and RMM1 and RMM2.

Plotting Phase Diagram

The function mjo_phase_plot() below has four input arguments, including three optional arguments with default values.

• data: pandas dataframe containing data (data requires the columns: ["YYYY", "MM", "DD", "RMM1", "RMM2"])

• radius: circle in the center of the plot which represents when MJO phase value is significant (defaults to 1)

• month_colors: colors for each month on plot (color for the 12 months defaults to ["tomato", "dodgerblue", "orange", "green", "magenta", "yellow", "pink", "cyan", "gold", "lime", "darkviolet", "olive"])

• rmm_axis_label: replace phase axis labels with RMM1/RMM2 labels (defaults to False, with all phase and geographical locations)

def mjo_phase_plot(data=None, radius=1, month_colors=[], rmm_axis_label=False):
    fig = plt.figure(figsize=(8, 8))
    ax = fig.add_subplot(111)

    # set x and y axis labels
    ax.set_xlim(-4, 4)
    ax.set_ylim(-4, 4)
    ax2 = ax.twinx()
    ax3 = ax.twiny()
    ax2.set_xlim(-4, 4)
    ax2.set_ylim(-4, 4)
    ax3.set_xlim(-4, 4)
    ax3.set_ylim(-4, 4)
    if not rmm_axis_label:
        ax.set_ylabel(
            "Phase 1 (Western Hemisphere, Africa) Phase 8", fontsize=20, labelpad=20
        )
        ax.set_xlabel("Phase 2 (Indian Ocean) Phase 3", fontsize=20, labelpad=20)
        ax2.set_ylabel(
            "Phase 5 (Maritime) Phase 4", fontsize=20, labelpad=40, rotation=270
        )
        ax3.set_xlabel("Phase 7 (Western Pacific) Phase 6", fontsize=20, labelpad=20)
    else:
        ax.set_xlabel("RMM2", fontsize=20, labelpad=20)
        ax.set_ylabel("RMM1", fontsize=20, labelpad=20)

    # add amplitude circle
    amplitude_circle = plt.Circle((0, 0), radius, fill=False)
    ax.add_patch(amplitude_circle)

    # add phase diagram lines
    ## Phase 1-8
    phase_line = lines.Line2D([-4, -radius], [0, 0], linestyle="--", dashes=(5, 5))
    ax.add_line(phase_line)
    ## Phase 1-2
    phase_line = lines.Line2D(
        [-radius / np.sqrt(2), -4],
        [-radius / np.sqrt(2), -4],
        linestyle="--",
        dashes=(5, 5),
    )
    ax.add_line(phase_line)
    ## Phase 2-3
    phase_line = lines.Line2D([0, 0], [-radius, -4], linestyle="--", dashes=(5, 5))

    ax.add_line(phase_line)
    ## Phase 3-4
    phase_line = lines.Line2D(
        [radius / np.sqrt(2), 4],
        [-radius / np.sqrt(2), -4],
        linestyle="--",
        dashes=(5, 5),
    )
    ax.add_line(phase_line)
    ## Phase 4-5
    phase_line = lines.Line2D([radius, 4], [0, 0], linestyle="--", dashes=(5, 5))

    ax.add_line(phase_line)
    ## Phase 5-6
    phase_line = lines.Line2D(
        [radius / np.sqrt(2), 4],
        [radius / np.sqrt(2), 4],
        linestyle="--",
        dashes=(5, 5),
    )
    ax.add_line(phase_line)
    ## Phase 6-7
    phase_line = lines.Line2D([0, 0], [radius, 4], linestyle="--", dashes=(5, 5))
    ax.add_line(phase_line)
    ## Phase 7-8
    phase_line = lines.Line2D(
        [-radius / np.sqrt(2), -4],
        [radius / np.sqrt(2), 4],
        linestyle="--",
        dashes=(5, 5),
    )
    ax.add_line(phase_line)

    # add tick marks
    full_tickmarks = np.arange(-4, 4.5, 0.5)
    ax.set_xticks(full_tickmarks)
    ax.set_yticks(full_tickmarks)
    # hide tickmarks for top/right axis)
    ax2.set_yticks(full_tickmarks)
    ax3.set_xticks(full_tickmarks)

    # plot whole numbers integar tick marks
    ax.xaxis.set_major_locator(ticker.MultipleLocator(1.0))
    ax.yaxis.set_major_locator(ticker.MultipleLocator(1.0))
    ax2.yaxis.set_major_locator(ticker.MultipleLocator(1.0))
    ax3.xaxis.set_major_locator(ticker.MultipleLocator(1.0))
    # increase length of major ticks
    ax.tick_params(which="major", length=10)
    ax2.tick_params(which="major", length=10)
    ax3.tick_params(which="major", length=10)

    # plot minor ticks every 0.25
    ax.xaxis.set_minor_locator(ticker.MultipleLocator(0.25))
    ax.yaxis.set_minor_locator(ticker.MultipleLocator(0.25))
    ax2.yaxis.set_minor_locator(ticker.MultipleLocator(0.25))
    ax3.xaxis.set_minor_locator(ticker.MultipleLocator(0.25))
    # increase length of minor ticks
    ax.tick_params(which="minor", length=5)
    ax2.tick_params(which="minor", length=5)
    ax3.tick_params(which="minor", length=5)

    # plot with title
    first_index = data.iloc[0]
    start_date = (
        f'{int(first_index["YYYY"])}-{int(first_index["MM"])}-{int(first_index["DD"])}'
    )
    last_index = data.iloc[-1]
    end_date = (
        f'{int(last_index["YYYY"])}-{int(last_index["MM"])}-{int(last_index["DD"])}'
    )

    plt.title(
        f"MJO Phase: ROMI: {start_date} to {end_date}",
        fontsize=30,
        fontweight="bold",
        pad=20,
    )

    # Setup Months Text at Bottom of Plot
    if len(month_colors) == 0:
        # default colors for all twelve months
        month_colors = [
            "tomato",
            "dodgerblue",
            "orange",
            "green",
            "magenta",
            "yellow",
            "pink",
            "cyan",
            "gold",
            "lime",
            "darkviolet",
            "olive",
        ]

    month_dict = {
        1: "Jan",
        2: "Feb",
        3: "Mar",
        4: "Apr",
        5: "May",
        6: "Jun",
        7: "Jul",
        8: "Aug",
        9: "Sep",
        10: "Oct",
        11: "Nov",
        12: "Dec",
    }

    all_months = data["MM"].unique()
    all_month_names = [
        month_name
        for month_mm, month_name in month_dict.items()
        if month_mm in all_months
    ]  # convert number to name (example: 01 to Jan)

    # dynamic spacing for month labels
    start_index = -3.3 + ((12 - len(all_month_names)) * 0.28)
    for i, month in enumerate(all_month_names):
        if i == 0:
            # First Element
            text = ax.text(start_index, -3.7, month, color=month_colors[i], fontsize=13)
        else:
            # Remaining Elements
            text = ax.annotate(
                f"  {month}",
                xycoords=text,
                xy=(1, 0),
                verticalalignment="bottom",
                fontsize=13,
                color=month_colors[i],
            )

        # filter data based on each month
        last_index = data[data["MM"] == all_months[i]].index[-1]

        # include the first day of the next month in data to preserve continuity
        next_index = []
        if last_index + 1 in data.index:
            next_index = [last_index + 1]
        filtered_df = data.loc[
            (data["MM"] == all_months[i]) | (data.index.isin(next_index))
        ]
        x_data = filtered_df["RMM2"]
        y_data = -filtered_df["RMM1"]

        # Plot data
        plt.plot(x_data, y_data, zorder=1, c=month_colors[i])
        plt.scatter(x_data, y_data, zorder=1, c=month_colors[i])

        # plot dark mark at the start and end
        if i == 0:
            plt.scatter(x_data.iloc[[0]], y_data.iloc[[0]], c="black", s=150)
        if i + 1 == len(all_month_names):
            plt.scatter(x_data.iloc[[-1]], y_data.iloc[[-1]], c="black", s=150)

    # Annotate text every 5 days
    days_text = [5, 10, 15, 20, 25, 30]
    days_filtered = data[data["DD"].isin(days_text)]
    for index, row in days_filtered.iterrows():
        plt.text(
            x=row["RMM2"],
            y=-row["RMM1"],
            s=int(row["DD"]),
            ha="center",
            zorder=2,
            fontweight="bold",
        )
    plt.show()

Plot Diagram for 2025

Collect a subset of the data, the first five months of 2025.

df_2026 = noaa_df[(noaa_df["YYYY"] == 2025) & (noaa_df.MM.between(1, 5))]

df_2026

	YYYY	MM	DD	HH	RMM1	RMM2	Phase
12419	2025	1	1	0	-0.15161	-0.83951	0.85309
12420	2025	1	2	0	-0.01538	-1.03555	1.03566
12421	2025	1	3	0	0.04397	-1.19613	1.19693
12422	2025	1	4	0	0.09447	-1.28177	1.28524
12423	2025	1	5	0	0.16386	-1.34737	1.35730
...	...	...	...	...	...	...	...
12565	2025	5	27	0	0.12277	-0.03427	0.12746
12566	2025	5	28	0	0.06407	-0.14853	0.16176
12567	2025	5	29	0	-0.01231	-0.20038	0.20076
12568	2025	5	30	0	-0.06424	-0.25173	0.25980
12569	2025	5	31	0	-0.13798	-0.25259	0.28782

151 rows × 7 columns

mjo_phase_plot(data=df_2026)

Plot Diagram Most Recent Data

Display months of the most recent year.

all_years = noaa_df["YYYY"].unique()
most_recent_year = all_years[-1]
all_months_in_recent_year = noaa_df[(noaa_df["YYYY"] == most_recent_year)][
    "MM"
].unique()
print(
    f"Most recent year {most_recent_year} has {len(all_months_in_recent_year)} months"
)

Most recent year 2026 has 3 months

df_recent = noaa_df[(noaa_df["YYYY"] == most_recent_year)]
mjo_phase_plot(data=df_recent)

Curated Resources

To learn more about Madden-Julian Oscillation:

• NOAA (Physical Sciences Laboratory): Madden-Julian Oscillation

• NWS (Climate Prediction Center): Current and Forecast Conditions

References:

---

[1]

Madden-julian oscillations (mjo). URL: https://psl.noaa.gov/mjo/.

[2]

Roland A. Madden and Paul R. Julian. Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. 1971. doi:10.1175/1520-0469(1971)028<0702:DOADOI>2.0.CO;2.

[3]

Matthew C. Wheeler and Harry H. Hendon. An all-season real-time multivariate mjo index: development of an index for monitoring and prediction. 2004. doi:10.1175/1520-0493(2004)132<1917:AARMMI>2.0.CO;2.