1. Introduction
The main variables and definitions, and their units.
However, the MSE budget is not exact (Riehl and Malkus 1958; Romps 2015). It is an approximation of the total energy equation (Randall 2015; Neelin 2007; Romps 2015; see the next section). Previous work has argued that these approximations are accurate at the spatial and temporal scales of tropical deep convection (Riehl and Malkus 1958; Madden and Robitaille 1970). Within this context, the MSE budget can be considered to be a simplified form of the entropy budget (Soriano et al. 1994). Madden and Robitaille (1970), Betts (1974), Soriano et al. (1994), Romps (2015), Marquet (2016), and Yano and Ambaum (2017) provide thorough discussions on the MSE, its accuracy, and its strengths and weaknesses with an emphasis on tropical convection.
Even though the MSE budget has been widely employed to understand the thermodynamics of large-scale systems, its accuracy has been examined to a lesser degree. Neelin (2007) noted that the assumptions made to obtain Eq. (2) may not be accurate at the large scale. He recommended that these assumptions should be evaluated further. Sobel et al. (2014) indicated that residuals in Eq. (2) are indeed small, but they nevertheless opted to use a variant of the MSE budget that is more accurate than Eq. (2). Other authors have done the same (Hill et al. 2017; Smyth and Ming 2020; Adames et al. 2021) (see appendix C) or have opted to use more accurate budgets such as moist entropy (Raymond 2013; Jiang et al. 2018).
With these studies taken into account, one may ask, “Is the MSE budget shown in Eq. (2) appropriate to use in the context of large-scale motions?” The goal of this study is to answer this question by providing a discussion of the derivation of the MSE budget and examining the conditions in which the budget is accurate. The next section presents the theory behind the derivation of the MSE budget for three-dimensional flow. In section 3, we perform a scale analysis of the energy equation and present the conditions necessary to obtain the MSE budget from it. In section 4, we compare the results of the scaling with reanalysis and observations. The scaling is also analyzed on the basis of a linear regression analysis of MJO events observed during the DYNAMO field campaign. Concluding remarks are offered in section 5.
2. Derivation
In writing Eqs. (3a)–(3e) we have made several assumptions and approximations. First, since we are only considering large-scale motions, we have assumed that motions are in hydrostatic balance. This is an accurate approximation since the vertical velocities are weaker than the horizontal winds outside convective updrafts’ cores. Second, we are not including the effect of ice processes (i.e., freezing and sublimation) on Eqs. (3d) and (3e). Last, we ignore the effects of water vapor and hydrometeor content on the gas constant (Rd) and Cp. We refer the reader to Soriano et al. (1994), and Yano and Ambaum (2017) for derivations of the MSE budget that include these contributions.
It is worth pointing out that squall lines and other forms of organized convection can transport kinetic energy upgradient (LeMone et al. 1984; Moncrieff and Klinker 1997). Hence, caution should be exercised when interpreting the scaling of these terms when these types of systems are present.
By examining Eq. (12) we see that the first term on the rhs is a vertical diffusion of K, which can be either positive or negative depending on how K is vertically distributed. On the other hand, the second term (
For the MSE budget in Eq. (2) to be derived as an approximation of Eq. (15) or Eq. (18), the following conditions must be satisfied:
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Condition 1: The column integrated tendency in geopotential must be much smaller in amplitude than the MSE tendency, i.e., |∂tΦ| ≪ |∂tm|.
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Condition 2: Temporal and spatial fluctuations in K must be much smaller than those in MSE.
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Condition 3: The surface kinetic energy flux (FK) must be negligibly small.
It is important to note that while m is on the order of 105 J kg−1, its fluctuations are on the order of 103 J kg−1 (Adames et al. 2021). On the other hand, K fluctuations range from 1 J kg−1 in some equatorial waves, up to 103 J kg−1 in mature TCs. Furthermore, because the column integral of Φ is equivalent to that of RdT (appendix C), it follows that Φ fluctuations are roughly Rd/(Cp + Rd) ≈ 0.22 those of DSE. Thus, ∂tΦ could potentially be nonnegligible in Eq. (15). A more careful examination of the relative magnitude of the terms in Eq. (15) is warranted.
3. Scale analysis
We will now perform a scale analysis on Eq. (15) in order to understand when the MSE budget is an accurate approximation of Eq. (15). As in Adames (2022), we begin by creating nondimensional versions of Eqs. (3a)–(3e). This section will focus on nongeostrophic scaling in the momentum equations. See Table 2 for a list of the scales, along with their definitions, and units; note that all scales, except lowercase Greek letters, are shown in straight font. Results for geostrophic scaling are shown in appendix D.
The main scales, definition, and units.
From an examination of Eq. (15) we see that the most important factor determining the accuracy of the MSE budget is the value of Nmode, implying that the accuracy of the MSE budget hinges on having Lυq anomalies that are larger than the CpT anomalies. The Nmode is most sensitive to the ratio
The second most important factor is the magnitude of U relative to cp. In most tropical motions |U| ≤ |cph|, except in tropical cyclones, where U can be much larger.
Now let us consider a typical tropical easterly wave to examine the magnitude of the three conditions, with scaling values obtained from previous studies (Kiladis et al. 2006; Janiga and Thorncroft 2013; Rydbeck and Maloney 2015; Vargas Martes et al. 2023). The wave has a phase speed of cph ∼ 10 m s−1, a wind speed of U ∼ 3 m s−1,
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Condition 1: ∂tΦ ≪ ∂tm. Plugging in the scales for the easterly waves yields an Nmode value of 0.4. Since Rd/Cp ≈ 0.28, we have that
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Condition 2: ∂tK ≪ ∂tm. The scaling is as in condition 1, but multiplied by U/(2cph), which is roughly 0.15 for the easterly wave. Thus, we have that
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Condition 3: FK ≪ ∂tm. We will use a scaling based on FK since values of Cd over the ocean have been observed to be approximately 1.5 × 10−3 (Stull 2006). Plugging Eq. (38) into Eq. (37), we find that the scaling for condition 3 is as in condition 2 multiplied by CdUgτ/P, yielding
So all conditions are satisfied for this easterly wave example. Note that for these systems conditions 2 and 3 are easily satisfied, and only condition 1 can lead to a noticeable (albeit small) error if it is dropped from Eq. (15).
4. Insights from observations and reanalysis
a. Data and methods
The observational data used in this study were obtained from the Dynamics of the Madden–Julian Oscillation (DYNAMO) field campaign conducted from October 2011 to March 2012 (Yoneyama et al. 2013). The sounding grid during the DYNAMO period consists of two quadrangular arrays straddling the equator. We used data from the northern sounding array (NSA) located in the central equatorial Indian Ocean for 3 months, from 10 October to 31 December 2011. The NSA was defined by the following locations: Gan Island (0.69°N, 73.51°E), R/V Revelle (0°, 80.5°E), Colombo (6.91°N, 79.878°E), and Malé (4.91°N, 73.53°E). The sounding array dataset was quality controlled and bias corrected to produce DYNAMO NSA (version 3a) legacy dataset (Ciesielski et al. 2014). Precipitation is obtained from the 3B42 dataset of the Tropical Rainfall Measuring Mission (TRMM) (Huffman et al. 2007). The TRMM data are averaged over the NSA DYNAMO domain to produce a rainfall time series.
Three-dimensional fields including zonal (u) and meridional (υ) winds, geopotential (Φ), temperature (T), and specific humidity (q) are obtained from the fifth generation of the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis (ERA5; Hersbach et al. 2019). We use ERA data from January 1980 through December 2015. In addition, the MJO activity during the DYNAMO period is assessed through the OLR MJO index (OMI; Kiladis et al. 2014).
The power spectrum calculation procedure is similar to previous studies (e.g., Wheeler and Kiladis 1999; Yasunaga et al. 2019; Inoue et al. 2020). All terms are first partitioned into symmetric and antisymmetric components over the 10°S–10°N latitude belt. To prevent aliasing, the first three harmonics of the seasonal cycle are removed. Anomalies were partitioned into 96-day segments, as in Wheeler and Kiladis (1999). Then for each segment, the linear trend is removed, and the ends of the series are tapered to zero. Finally, the power spectra were computed.
b. Comparison with observations and reanalysis
In Fig. 1 we see that the column geopotential tendency (∂t⟨Φ⟩) is roughly an order of magnitude smaller than ∂t⟨m⟩, while ∂t⟨K⟩ is roughly two orders of magnitude smaller. These two terms remain relatively flat in increasing frequency. In contrast, FK is roughly three orders of magnitude smaller than ∂t⟨m⟩ at time scales longer than 10 days. It decreases in amplitude with increasing frequency, consistent with the scaling shown in Eq. (37). Interestingly, FK starts increasing with frequency at subdiurnal time scales, perhaps because of the influence of convection or boundary layer processes. Overall, Fig. 1 shows that the three conditions discussed above are satisfied at most frequencies in the DYNAMO NSA data. Furthermore, we see that ∂tLυ⟨q⟩ is on the same order as ∂t⟨m⟩, consistent with the idea that moisture must account for a large fraction of the MSE variance for the MSE budget to be accurate.
The fractional quantities of the terms relative to the MSE anomalies in ERA5 data are also shown as a function of zonal wavenumbers and temporal frequencies (Fig. 2). From examination of Fig. 2 we see that K is roughly 1/100 to 1/30 times the amplitude of ⟨m⟩, consistent with the analysis done on the DYNAMO NSA data in Fig. 1. However, we see variability in the wavenumber–frequency space that is not seen in the more limited analysis presented in Fig. 1. For example, we see a large
When we examine the relative amplitude of ⟨Φ⟩ with respect to ⟨m⟩ (Figs. 2c,d), we see that there is a region where the magnitude exceeds 0.2 near westward-propagating zonal wavenumber 1 and time scales of 2.5 days, possibly corresponding to Rossby–Haurwitz waves (Hendon and Wheeler 2008). In other regions of the spectrum, we see that ⟨Φ⟩ is nearly always less than a tenth of the magnitude of ⟨m⟩. The ratio also decreases with increasing wavenumber. Overall,
c. Accuracy of the MSE budget in the MJO
Last, we examine the amplitudes of ∂t⟨m⟩, ∂t⟨Φ⟩, and ∂tK in the evolution of the MJO events that occurred during the DYNAMO field campaign. We performed a lag regression of the sounding data on the first principal component of the OMI index as in Snide et al. (2021). The results, shown in Fig. 3, show that the MSE tendency peaks near 20 W m−2. The tendencies in ⟨Φ⟩ and ⟨K⟩ are so small that they are barely discernible in Fig. 3. A close inspection of the two terms (not shown) reveals that ∂t⟨Φ⟩ has a peak amplitude of ∼0.8 W m−2, 4% the magnitude of ∂t⟨m⟩. The term ∂t⟨K⟩ peaks at ∼0.2 W m−2, only 1% the magnitude of ∂t⟨m⟩. The term FK is even smaller than ∂t⟨K⟩, and cannot be distinguished from a horizontal line in Fig. 3. The smallness of these three terms is consistent with Eq. (37).
Let us now compare these results with those of the scale analysis. Mayta and Adames Corraliza (2023) found that the value of Nmode for the MJO over the Indian Ocean is ∼0.2. According to the scaling in Eq. (37), ∂t⟨Φ⟩ should be 6.7% the magnitude of the MSE tendency. If we assume that U ∼ cp, then ∂t⟨K⟩ should be 3.3% the magnitude of the MSE tendency.
5. Discussion and conclusions
In this study, we use a scale analysis, data from ERA5, and sounding data from the DYNAMO northern sounding array to examine the accuracy of the MSE budget. Our results show that the budget is most accurate when a nondimensional parameter Nmode has a small value, implying that the MSE budget is most accurate when moisture plays a dominant role in the MSE budget. Additionally, the horizontal winds need to be relatively weak and the surface kinetic energy flux small. These three criteria are most likely to be satisfied in tropical motions with propagation speeds and horizontal winds of 10 m s−1 or less. It is also accurate when considering climatological-mean circulations such as the Hadley cell, the ITCZ, and the storm track since Nmode ≃ 0 in these cases. The scale analysis indicates that the MSE budget is less accurate in systems with strong horizontal winds such as tropical cyclones, as Ma et al. (2015) also noted. It is also likely to be less accurate in fast-propagating systems such as inertio-gravity waves.
Spectral analysis of the conditions necessary to obtain the MSE budget in Eq. (2) from Eq. (15) suggests that the neglected terms are smaller than what the scale analysis suggests. Thus, even though the neglected terms become larger with increasing Nmode, the neglected terms rarely exceed ∼10% of the tendency. A linear regression analysis of the MJO events that occurred during the DYNAMO field campaign also shows that the conditions described in Eq. (37) are easily satisfied in the MJO. Thus, our results show that the MSE budget as defined in Eq. (2) is reasonably accurate when applied in the context of large-scale systems as long as Eq. (37) is qualitatively satisfied. Fortuitously, most of the applications of the MSE budget in the tropics have been to systems that satisfy Eq. (2), such as the MJO, equatorial Rossby waves, and monsoon low pressure systems (Maloney 2009; Andersen and Kuang 2012; Kim et al. 2014; Mayta et al. 2022; Gonzalez and Jiang 2019; Adames and Ming 2018).
When all these results are considered together, we can find a potential explanation as to why Eq. (2) appears to be widely applicable in the tropics. When the atmosphere is in WTG balance, the ratio
In spite of these results, we still recommend that readers examine the conditions in Eq. (37) prior to the application of an MSE budget, especially in faster-propagating systems or systems with strong horizontal winds.
When Nmode ≪ 1 moisture governs the thermodynamics of a tropical weather system, and the system is classified as a moisture mode (Sobel et al. 2001; Raymond et al. 2009; Adames 2022; Mayta et al. 2022).
Acknowledgments.
ÁFAC was supported by NSF CAREER Grant 2236433, and by the University of Wisconsin startup package. The authors thank Spencer Hill and two anonymous reviewers for comments that significantly improved the contents of the manuscript.
Data availability statement.
ERA5 data are available at https://www.ecmwf.int/en/forecasts/datasets/reanalysis-datasets/era5/). The DYNAMO data used in this study are available at http://johnson.atmos.colostate.edu/dynamo/products/array_averages/.
APPENDIX A
Derivation of the Column-Integrated Em Budget
APPENDIX B
DSE Budget
APPENDIX C
Equivalence Between Energy and Column Moist Enthalpy
APPENDIX D
Geostrophic Scaling
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