Phenomena and Processes: A New MJO Diagnostic Framework using Moisture Mode Theory as the Testbed

Chang, Chun-Hao; Tseng, Kai-Chih; Maloney, Eric D.

doi:10.5194/egusphere-2026-1153

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https://doi.org/10.5194/egusphere-2026-1153

Preprints

12 Mar 2026

| 12 Mar 2026

Phenomena and Processes: A New MJO Diagnostic Framework using Moisture Mode Theory as the Testbed

Chun-Hao Chang, Kai-Chih Tseng, and Eric D. Maloney

Abstract. An unified diagnostic framework is proposed to bridge theoretical, phenomenological, and process-oriented approaches for investigating the Madden-Julian Oscillation (MJO). Building upon a physical theory (moisture mode theory in this study) and linear inverse modeling, the framework links the statistical behavior of observable indices to the underlying physical processes governing column moisture evolution. Applied to the MJO in ERA5 reanalysis and 15 CMIP6 models, the framework reveals that most models simulate an MJO that propagates too slowly eastward across the basins, and decays too rapidly, especially over the Maritime Continent. By projecting model biases in column-integrated water vapor-based MJO indices onto individual terms of the moisture budget, we diagnose the physical origins of their errors. Systematic biases are primarily tied to misrepresented horizontal moisture advection and compensating errors between vertical moisture transport and convective drying, while their relative importance varies across basins. This process-resolved perspective explains the inter-model diversity in MJO simulations and provides a physically interpretable bridge between dynamical theory, model evaluation, and observational constraints—offering a transferable framework for diagnosing variability in other climate phenomena.

Received: 28 Feb 2026 – Discussion started: 12 Mar 2026

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Chun-Hao Chang, Kai-Chih Tseng, and Eric D. Maloney

Status: final response (author comments only)

RC1:
'Comment on egusphere-2026-1153', Anonymous Referee #1, 18 Mar 2026
General Comment
This manuscript proposes a robust and innovative unified framework for diagnosing MJO biases in General Circulation Models (GCMs). The integration of moisture-mode theory, Linear Inverse Modeling (LIM), and the process-resolved attribution of errors (via inner-product and unit circle analysis) is a significant methodological advance. The results clearly identify systematic GCM shortcomings, particularly the slow propagation bias, and successfully attribute these errors to physical processes like horizontal moisture advection and convective-associated terms.
The paper is generally well-written and logically structured
Major/Moderate Comments
Limitations of Linear Models: While the paper mentions limitations in the data (lack of hourly data, coarse vertical resolution), The paper should more directly address the inherent limitations of Moisture Mode Theory and LIM. Is the linear assumption of LIM adequate for capturing the full non-linear MJO life cycle? Could the authors briefly discuss or justify why the linear approximation is sufficient for diagnostic purposes in this context?

Clarification (L45): It's stated that the linear/quasi-linear nature of the theories allows for direct evaluation of metrics like growth rate and frequency. Please clarify if the framework itself diagnoses these properties from the budget terms or if it uses the traditional LIM-derived metrics and then attributes the bias.

Methodology (L70): The data uses 6 pressure levels (1000, 850, 700, 500, 250, 100 hPa). Given the sensitivity to vertical advection (e.g., L197), can the authors comment on how much the coarse resolution is believed to affect the budget terms, especially in the boundary layer?

Term Summation Clarity (L137, Eq. 7): The sum runs from i=1 to 27. Please clarify what the index i represents. If I am not mistaken they are the 27 advection terms (3 terms per timescale x 3 timescales x 3 directions) plus the one apparent moisture sink term, totaling 28 processes. A brief in-line list of the terms being summed or a reference to a table listing them would improve clarity, as the total number of terms (advection, sink, etc.) is not immediately obvious.

Finding Detail (L197): The paper states: "The vertical advection term has significantly positive contribution to total bias in the Indian Ocean, but the result does not hold in the Maritime Continent." Given the regional importance of the MC, is there a physical reason (e.g., topography, boundary layer, mean state) for this divergence that can be speculated upon?

Applicability Beyond MJO (L252): In the conclusion, the authors state the framework is a "transferable pathway for diagnosing the fundamental dynamics of diverse climate phenomena beyond the MJO." It would strengthen the claim to briefly elaborate on this. Could the authors provide an example of which other climate phenomena this framework might be applied to (e.g., equatorial waves, monsoonal intraseasonal variability) and why?

Minor Comments
Abstract (L1): "An unified diagnostic framework..." should be "A unified diagnostic framework".

Definition/Context (L40): The paper relies heavily on the Weak Temperature Gradient (WTG) approximations. A very brief sentence on why the WTG approximation is justified for the MJO's large scale and slow evolution would be helpful for non-specialist readers.

L168: The sentence "For WH region (Fig. 5a), most of the GCMs have bias vectors pointing upward, which indicates that the propagation speeds are generally slower in GCMs within this region." Remove the final two words for conciseness: "...slower in GCMs."

L197: The phrase "...than of meridional." Should be "...than that of meridional."
Citation: https://doi.org/10.5194/egusphere-2026-1153-RC1
- AC1:
  'Reply on RC1', Chun-Hao Chang, 29 Mar 2026
  Replies for the major/moderate comments:
  The reason why we use LIM in this paper is to align with moisture-mode theory. While we need to acknowledge that linear theories/modeling are unable to include (i) state-dependent features; (ii) interactions between perturbation and mean state. However, the proposed framework is not limited to linear theories/modeling. It is fine to replace the LIM with other advanced non-linear data driven models (e.g. transformer), the state-dependent growth rate/frequency can be estimated through tangent linear modeling. Furthermore, our current analyses are already state-dependent to some degree (i.e. different results across different basins), which include the non-linear characteristics of the MJO.
  
  This framework estimates the growth rate and frequency through the LIM-derived metrics. And we have to acknowledge that the growth rate/frequency estimated through LIM-derived metrics might not be fully consistent with the linear or quasi-linear theories themselves.
  
  From previous budget analysis studies (e.g. Sobel et al. 2014; Adames and Wallace 2015; Tseng et al. 2015), the vertical velocity in the boundary layer does not linearly change with respect to pressure; and there is usually local maximum of vertical moisture/MSE advection at around 900 hPa during the mature phase of MJO, so we believe that the lack of data within the boundary layer could reduce the contribution of the vertical advection term.
  
  We would spell out that the summation of 27 terms stands for the 3x3x3 moisture advection terms in the context. While Eq. 7 works only as an outline of this framework, the matching between each index i and each moisture budget term is not necessarily important here.
  
  We believe so, but our results don't support this argument. Given the time and spatial resolutions of our data are coarse, those processes (e.g. afternoon thunderstorm, topographic phase-lock of convection) might be related to this divergence aren't able to be identified here. And we think it is worthy to dig in in the future.
  
  Annular mode would be one of the perfect phenomena applicable to this framework. The dynamics of the annular mode can be described with zonal momentum budget equation, the strengthening and the meridional shift of the jet can be described with two EOF modes of zonal mean zonal wind as well. So we think this framework could be a good approach to evaluate the abilities of GCMs simulating the annular mode.
  
  For the minor comments, we will modify the manuscript accordingly.
  
  Citation: https://doi.org/10.5194/egusphere-2026-1153-AC1
RC2:
'Comment on egusphere-2026-1153', Anonymous Referee #2, 22 Apr 2026
Overview: The authors proposed a potentially novel and useful framework for diagnosing processes related to MJO simulation errors. I see the proposed framework could be very helpful to the community, but I believe the authors should further clarify the techniques so the community can actually apply them. Further discussion of the advantages and disadvantages of the new framework relative to prior work would make the paper more valuable. In addition, having a strong visualization of the diagnostic makes it more attractive, which is another area I suggest the authors improve.

Comments:
It would be helpful to clarify the use of LIM in this work. I am confused about whether the authors are only using the “concept” of LIM to develop the framework or are actually using LIM as part of the framework. If it is the latter, I did not understand exactly how LIM is used in constructing the model diagnostics.

I would like more explanation of how to calculate the terms on the right-hand side of Equation 7. Are you projecting the calculated moisture budget term onto the EOF of ERA5 moisture to find “M” for each term, then finding its time tendency?

Readability of Figs. 6-8 should be improved. It was extremely difficult to identify which term is plotted where and to find the supporting evidence for the arguments made in the text. Below are suggested changes that could help the readability of these figures:
Plot leading order terms only (instead of all 28 terms)

Avoid using light colors for the horizontal axis labels (term colors), as they were very difficult to see.

Darken the box-and-whisker plots to make them more visible. I did not even notice at first that they were on the figures.

Make the text of all labels larger. I had to zoom in a lot to read them.

The authors should further clarify the novelty of their diagnostic framework relative to prior work, including Anderson and Kuang (2012) and Adames and Ming (2018). Those studies also show the process diagnostics for the maintenance and propagation of a mode. I think those diagnostics are also phenomenological and process-oriented. Based on my understanding, the authors’ new diagnostic is similar to the one used in the aforementioned prior work but requires an additional step involving EOF projections. A potential advantage of using EOF projections is to simplify the diagnostic and remove potential noise, but at the same time, the resultant diagnostic could lose information. For example, when the MJO structure deviates from the structure captured by the first EOFs (e.g., during El Nino, Kessler 2001), the EOF-based diagnostics could become less accurate. Additional discussion to clarify the advantages and disadvantages of the authors’ new diagnostic framework relative to prior work would be helpful.

References:
Andersen, J. A., and Z. Kuang, 2012: Moist Static Energy Budget of MJO-like Disturbances in the Atmosphere of a Zonally Symmetric Aquaplanet. Climate, 25, 2782–2804, https://doi.org/10.1175/JCLI-D-11-00168.1.

Adames, Á. F., and Y. Ming, 2018: Moisture and Moist Static Energy Budgets of South Asian Monsoon Low Pressure Systems in GFDL AM4.0. Atmos. Sci., 75, 2107–2123, https://doi.org/10.1175/JAS-D-17-0309.1.

Kessler, W. S., 2001: EOF Representations of the Madden–Julian Oscillation and Its Connection with ENSO. Climate, 14, 3055–3061, https://doi.org/10.1175/1520-0442(2001)014<3055:EROTMJ>2.0.CO;2.
Citation: https://doi.org/10.5194/egusphere-2026-1153-RC2
- AC2:
  'Reply on RC2', Chun-Hao Chang, 31 May 2026
  Replies for the comments:
  We didn't build a LIM for long-term integration and analyze the corresponding processes, but instead using the concept of LIM (i.e. tangent linear model) to investigate the linear approximation of GCM output and ERA5 in our study. For example, the instantaneous moisture tendency is estimated by subtracting the q' at this timestep from the q' at the next timestep. Then we performed the inner product analysis and the unit circle analysis to examine the relation between moisture tendency and various moisture processes.
  
  We compute the moisture budget terms from ERA5 and GCMs' data (line 92-95), then project the computed moisture budget terms (with dimension of [time, lat, lon]) onto the two EOF modes (with dimension of [lat, lon]) to get equation 7. The remained indices have dimension of [time,].
  
  We have adjusted the figures to improve the readability.
  
  Andersen and Kuang 2012 defined their MJO reference time series by choosing a grid point which has the largest OLR variance. Adames and Ming 2018 defined their SMD index through calculating the domain average of a space-time filtered precipitation. Both studies calculated the regression maps for each moisture/MSE budget terms to extract the MJO/SMD-related signal. The contribution of different moisture/MSE budget terms on propagation/maintenance are examined through projecting the budget terms onto the tendency/total field with respect to a spatial domain.
  
  The major discrepancies between Andersen and Kuang 2012/Adames and Ming 2018 and our study include: (i) time-space filtering and EOF analysis; (ii) how to diagnose the propagation and maintenance.
  
  For the first discrepancy, we have to say that both time-space filtering and EOF analysis are transforming some variables from physical space to Fourier or phase space, they are actually mathematically equivalent. However, applying EOF analysis is not only for data compression or noise removal, but build the MJO phase space. By projecting the moisture budget equation onto the two EOF modes, we link a widely used and accessible performance metric (MJO indices) to a process-oriented diagnostic tool (moisture budget equation). Furthermore, our framework is not limited to single variable, but has the potential to be applied to a multi-variate system. For example, in the Trio-interaction theory (Wang et al. 2016), the prognostic variables include horizontal winds in free troposphere and boundary layer, geopotential height, and moisture. Put it under our framework, we can incorporate all of the candidate variables without a prior assumption of the underlying dynamics; while the framework in previous studies focusing more on the covarying of different processes with the tendency terms neglecting the potential multivariable interactions. Linking comprehensive processes to simple diagnostics (such as MJO indices) is very helpful on model assessment because the computation of MJO indices is simple, but it is not the case for mometum/moisture/MSE budget. As long as we can build the relation between the biases of MJO indices and the biases of different physical processes, we can diagnose the discrepancy of a model from the indices without digging into a bunch of budget terms. While we have to notice that the EOF patterns might not be as representative as the climatology drifts, this is inevitable when applying EOF analysis.
  
  For the second discrepancy, the typical approach defines the propagation/maintenance component as the projection onto the total field/tendency. Our approach, however, defines propagation/maintenance component as the tangential/radial component of dM/dt vector in the MJO phase space. This allows us to calculate the growth rate/frequency, and their relation to different moisture budget terms; while the assessment of growth rate and frequency is not that intuitive when adpating the typical way.
  
  Citation: https://doi.org/10.5194/egusphere-2026-1153-AC2

Chun-Hao Chang, Kai-Chih Tseng, and Eric D. Maloney

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Short summary

This study developed a unified Madden-Julian Oscillation (MJO) diagnostic framework that bridges the gap between two existed types of diagnostics (i.e. phenomenological diagnostics and process-oriented diagnostics). Utilizing this framework, we can attribute simulated MJO biases in general circulation models (GCMs) to specific physical processes.


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