# Minimal Mechanisms Responsible for the Dispersive Behavior of the Madden–Julian Oscillation

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## Abstract

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## 1. Introduction

## 2. Theoretical Description

#### Dispersion Relation

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The low-frequency modes of $\omega \left(k\right)$ versus k from Equation (19): when $\alpha $ = 0.25, ${\Gamma}_{q}$ = −0.05, $\Lambda $ = 0.05, and $\u03f5$ = 0.04. (

**a**) Frequency (cycles per day) of eastward mode (blue open circles) and westward mode (red dots); (

**b**) Growth rate (day${}^{-1}$) of eastward mode (blue open circles) and westward mode (red dots).

**Figure 2.**The eastward propagating moisture mode: when $\alpha $ = 0.25, ${\Gamma}_{q}$ = −0.05, $\Lambda $ = 0.05, and $\u03f5$ = 0.04.

**Figure 3.**The role of the feedback parameter EWF on the eastward propagating moisture mode ($\Lambda $ = 0, 0.5, 1.5, and 3); other parameters are fixed ($\alpha $ = 0.25, ${\Gamma}_{q}$ = −0.05).

**Figure 4.**Role of MCF on the eastward propagating moisture mode (${\Gamma}_{q}$ = 0, 1, −0.5, and −1.5); other parameters are fixed ($\alpha $ = 0.25, $\Lambda $ = 0.05).

**Figure 5.**Role of MRT on eastward propagating moisture mode ($\tau $ = 2 h, 12 h, 36 h, and 48 h); other parameters are fixed ($\Lambda $ = 0.5, ${\Gamma}_{q}$ = −0.05).

Description | Parameter | Definition/Units | Average Value |
---|---|---|---|

Dry gravity wave speed | C | $\mathrm{m}{\mathrm{s}}^{-1}$ | 50 |

Time scale | ${T}_{0}$ | ${\left[2C\beta \right]}^{\frac{1}{2}}$ | 8.33 h |

Length scale | ${L}_{0}$ | ${\left[\frac{C}{2\beta}\right]}^{\frac{1}{2}}$ | 40,000 km |

Meridional gradient of Coriolis parameter | $\beta $ | ${\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$ | $2.28\times {10}^{-11}$ |

Moisture convergence feedback * | ${\Gamma}_{q}$ | $1-\left(\frac{H}{\overline{q}}\right)\frac{d\overline{q}}{dz}$ | −0.05 |

Convective time lag * | $\alpha $ | [$\frac{{T}_{0}}{\tau}$] | 0.25 |

Rayleigh damping * | $\u03f5$* | [$\frac{{T}_{0}}{{T}_{d}}$] | 0.04 |

Evaporation-wind feedback * | $\Lambda $ | $\rho {C}_{D}\Delta {\overline{q}}_{s}\frac{{T}_{0}}{q}$ | 0.05 |

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**MDPI and ACS Style**

Mamidi, K.; Mathew, V.
Minimal Mechanisms Responsible for the Dispersive Behavior of the Madden–Julian Oscillation. *Climate* **2023**, *11*, 236.
https://doi.org/10.3390/cli11120236

**AMA Style**

Mamidi K, Mathew V.
Minimal Mechanisms Responsible for the Dispersive Behavior of the Madden–Julian Oscillation. *Climate*. 2023; 11(12):236.
https://doi.org/10.3390/cli11120236

**Chicago/Turabian Style**

Mamidi, Kartheek, and Vincent Mathew.
2023. "Minimal Mechanisms Responsible for the Dispersive Behavior of the Madden–Julian Oscillation" *Climate* 11, no. 12: 236.
https://doi.org/10.3390/cli11120236