On the Accuracy of the Moist Static Energy Budget When Applied to Large-Scale Tropical Motions

Ángel F. Adames Corraliza aDepartment of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin

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Víctor C. Mayta aDepartment of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin

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Abstract

The moist static energy (MSE) budget is widely used to understand moist atmospheric thermodynamics. However, the budget is not exact, and the accuracy of the approximations that yield it has not been examined rigorously in the context of large-scale tropical motions (horizontal scales ≥ 1000 km). A scale analysis shows that these approximations are most accurate in systems whose latent energy anomalies are considerably larger than the geopotential and kinetic energy anomalies. This condition is satisfied in systems that exhibit phase speeds and horizontal winds on the order of 10 m s−1 or less. Results from a power spectral analysis of data from the DYNAMO field campaign and ERA5 qualitatively agree with the scaling, although they indicate that the neglected terms are smaller than what the scaling suggests. A linear regression analysis of the MJO events that occurred during DYNAMO yields results that support these findings. It is suggested that the MSE budget is accurate in the tropics because motions within these latitudes are constrained to exhibit small fluctuations in geopotential and kinetic energy as a result of weak temperature gradient (WTG) balance.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ángel F. Adames Corraliza, angel.adamescorraliza@wisc.edu

Abstract

The moist static energy (MSE) budget is widely used to understand moist atmospheric thermodynamics. However, the budget is not exact, and the accuracy of the approximations that yield it has not been examined rigorously in the context of large-scale tropical motions (horizontal scales ≥ 1000 km). A scale analysis shows that these approximations are most accurate in systems whose latent energy anomalies are considerably larger than the geopotential and kinetic energy anomalies. This condition is satisfied in systems that exhibit phase speeds and horizontal winds on the order of 10 m s−1 or less. Results from a power spectral analysis of data from the DYNAMO field campaign and ERA5 qualitatively agree with the scaling, although they indicate that the neglected terms are smaller than what the scaling suggests. A linear regression analysis of the MJO events that occurred during DYNAMO yields results that support these findings. It is suggested that the MSE budget is accurate in the tropics because motions within these latitudes are constrained to exhibit small fluctuations in geopotential and kinetic energy as a result of weak temperature gradient (WTG) balance.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ángel F. Adames Corraliza, angel.adamescorraliza@wisc.edu

1. Introduction

The moist static energy (MSE) is a measure of total energy that is approximately conserved for hydrostatic, moist adiabatic vertical motions. It is frequently defined as
mCpT+Φ+Lυq,
where Cp is the specific heat of dry air at constant pressure, T is the temperature, Φ = gz is the geopotential, Lυ is the latent energy of vaporization, and q is the specific humidity (see Table 1 for variable definitions and units). The MSE budget is most commonly written in isobaric coordinates as
DmDt=Qrωm¯p,
where Qr is the radiative heating rate, and ωm¯ is the turbulent flux of MSE by motions that are of scales much smaller than the system being examined, usually defined as scales smaller than a typical GCM grid point (∼100 km). The quasi-conservative property of MSE, its simplicity, and the fact that is discussed in textbooks (Emanuel 1994; Randall 2015) and in seminal papers (Riehl and Malkus 1958; Yanai et al. 1973; Arakawa and Schubert 1974) have favored its widespread use. The MSE budget has been used to understand the thermodynamic processes of the ITCZ (Back and Bretherton 2005; Bischoff and Schneider 2014; Byrne and Schneider 2016; Popp and Silvers 2017), the Madden–Julian oscillation (Maloney 2009; Andersen and Kuang 2012; Kim et al. 2014), tropical convection (Riehl and Malkus 1958; Yanai et al. 1973; Inoue and Back 2015, 2017), convectively coupled waves (Sumi and Masunaga 2016; Gonzalez and Jiang 2019; Nakamura and Takayabu 2022; Mayta et al. 2022), midlatitude storm tracks (Barpanda and Shaw 2017; Shaw et al. 2018), convective self-aggregation (Bretherton et al. 2005; Wing et al. 2017), and tropical cyclones (McBride 1981; Wing et al. 2019).
Table 1.

The main variables and definitions, and their units.

Table 1.

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

We will start our derivation by considering the basic equations for large-scale flow. For a moist atmosphere in isobaric coordinates, the equations are written as
DvDt=fk×vhΦωv¯p,
Φp=1ρ,
h·v=ωp,
DCpTDtωρ=Q1,
DLυqDt=Q2,
where
DDt=t+v·h+ωp
is the material derivative in isobaric coordinates, v is the horizontal wind vector, f is the planetary vorticity, ωv¯ is the vertical momentum flux by eddies much smaller than the large-scale system in question, ρ is the density, p is the pressure, and ωDp/Dt is the vertical velocity in pressure coordinates. The variables Q1 and Q2 are the apparent heat source and latent energy sink, defined as in (Yanai et al. 1973) as
Q1=Qc+Qr+D,
Q2=Qc+ωm¯p,
where
Qc=Lυ(CE)ωs¯p
is the convective heating, where C and E are the condensation and evaporation rates, respectively, and s = CpT + Φ is the dry static energy (DSE). Note that Q1 has contributions from convection (Qc), radiative heating (Qr), and dissipative heating (D; Bister and Emanuel 1998).

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.

By invoking hydrostatic balance [Eq. (3b)] we write Eq. (3d) as
CpDTDt+ωΦp=Q1,
which can be added to Eq. (3e) to obtain
DhDt+ωΦp=Qr+Dωm¯p,
where
hCpT+Lυq
is the moist enthalpy.
The MSE budget is obtained by invoking the Lagrangian derivative of geopotential:
DΦDt=Φt+v·hΦ+ωΦp
and replacing ωpΦ in Eq. (7) with the other terms in Eq. (9), yielding the following:
DmDt=Φt+v·hΦ+Qrωm¯p+D.
As noted by Neelin (2007), it is common to assume that DΦ/DtωpΦ to obtain Eq. (2), rather than using Eq. (9) (Emanuel et al. 1994; see section 4 of Betts 1974). As we will see below, this assumption is erroneous at the large scale since ∂tΦ and v⋅∇hΦ are not always negligibly small. We also show in appendix B that these terms make the DSE budget inaccurate.
We can eliminate v⋅∇hΦ from Eq. (10) by invoking the kinetic energy budget, which is obtained by taking the dot product of the terms in Eq. (3a) and v. This procedure eliminates the Coriolis force. Furthermore, as in Bister and Emanuel (1998), and Randall (2015), we make the following expansion:
v·ωv¯p=v·ωv¯pωv¯·vp.
Let us assume that the smaller-scale vertical flux of horizontal momentum is predominantly downgradient so that ωv¯μpv, where μ is the eddy transfer coefficient. By doing this, we can write Eq. (11) as
v·ωv¯p=μ2Kp2μ(Kp)2D,
where
K=12(u2+υ2)
is the kinetic energy per unit mass.

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 (D) is always negative and acts to dissipate K. The dissipated K turns into heat, hence why it shows up in Eq. (3d).

By applying Eq. (12), the resulting K budget is written as
DKDt=v·hΦ+μ2Kp2D.
Adding Eqs. (10) and (14) yields what we will refer to as the “moist nonstatic energy equation”:
DEmDt=Φt+Qrωm¯p+μ2Kp2,
where
Em=m+K
is the moist nonstatic energy, not to be confused with the mechanical energy (Randall 2015).
To simplify the turbulent fluxes in Eq. (15), many researchers choose to invoke the column-integrated MSE budget rather than its vertically resolved form. The mass-weighted vertical integral of Em is written as
Em=1g0psEmdp,
where ps = 1000 hPa is the surface pressure. The angle brackets denote a mass-weighted vertical integral over the atmospheric column. After column integration, we write the Em budget as
Emt=Φth·vEm+Qr+LυE+HFK,
where E is the surface evaporation rate, H is the surface sensible heat flux, and
FK=μ2Kp2=Cdρs|vs|3
is the surface K flux, where Cd is a bulk surface drag coefficient, and ρs is the surface density. Further details on how to obtain Eq. (18) from Eq. (15) are shown in appendix A. An alternate form of Eq. (18) is shown in appendix C

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:

  1. Condition 1: The column integrated tendency in geopotential must be much smaller in amplitude than the MSE tendency, i.e., |∂tΦ| ≪ |∂tm|.

  2. Condition 2: Temporal and spatial fluctuations in K must be much smaller than those in MSE.

  3. 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.

Table 2.

The main scales, definition, and units.

Table 2.

The independent variables scale as
(x,y)=L(x^,y^),p=Pp^,t=τt^,
while the zonal and meridional winds have the same scaling:
(u,υ)=U(u^,υ^).
It follows that the kinetic energy K scales as
K=U22K^.
From an examination of Eq. (3a) we see that we can scale all the terms except the pressure gradient force. Since we are focusing on nongeostrophic scaling, it follows that the pressure gradient force has the same scaling as the acceleration. Hence the geopotential scale is
Φ=UcphΦ^,
where
cph=Lτ
is a phase speed scale.
We will simplify the scale of T relative to that of Adames (2022) by assuming that it scales as Φ/Rd, as in the scaling of Charney (1963) and Yano and Bonazzola (2009). Thus, the temperature anomalies scale as
T=UcphRdT^.
For the thermodynamic and moisture budgets, it is convenient to scale Qc and Q2 identically, while having an independent scale for Qr:
(Qc,Q2)=Q(Q^c,Q^2),Qr=RQ^r.
Since the vertical variations in s and q differ from their horizontal and temporal fluctuations, we consider their variations independently. Following Adames (2022), the scales of the vertical gradients of DSE and latent energy are
sp=Sps^p^,Lυqp=Lpq^p^,
from which we can define their ratio as
α^=LpSp,
which is also known as the Chikira (2014) parameter. Since q is concentrated in the lower troposphere, α^ tends to be larger in this region than in the upper troposphere. Typical lower tropospheric values of α^ often exceed 2 (Janiga and Zhang 2016; Wolding et al. 2016), and decrease to zero in the upper troposphere. For the purposes of scaling, we can think of α^ as a reduction of the static stability by convection (Emanuel 1994), akin to the one minus the normalized gross moist stability used in previous studies (Neelin et al. 1987; Sobel et al. 2001; Fuchs and Raymond 2005). Within this context, one may expect α^ to be on the order of 0.8–0.9 in the convectively active regions of the tropics (Benedict et al. 2014; Inoue and Back 2017). Intriguingly, this is a similar value to the free troposphere average of α^ that Vargas Martes et al. (2023) obtained over the east Pacific ITCZ and the West African monsoon.
Up to this point, we have not defined the scale for the vertical velocity. Conventionally, this scale is obtained from mass continuity [Eq. (3c)]. However, for convectively coupled motions, it is more appropriate to use a thermodynamic scaling based on the weak temperature gradient (WTG; Sobel et al. 2001) approximation. As in Adames (2022), we make this scaling considering the radiation-driven vertical velocity (ωr) and the convective vertical velocity (ωc) separately. The scale for ωr is obtained directly from WTG balance:
Wr=RSp,
where Wr is the scale for ωr. For ωc, we invoke the WTG moisture budget [Eq. (3) in Adames and Maloney 2021] and assume that the vertical MSE advection from convection is on the same order of magnitude as the vertical latent energy advection by radiative heating:
ωcmpωrLυqp,
which we can use to obtain the scale of ωc as
WcαRSp(1α).
Last, we find a scale for q by assuming that the moisture tendency scales in proportion to the vertical MSE advection by convective motions, which yields the following scaling:
Lυqα^τRq^.
By replacing all the terms of in Eq. (15) by their corresponding scales and dividing the equation by α^R we obtain the nondimensional Em budget:
E^mt^Ucpv^·^hE^m1α^ω^E^mp^=NmodeRdCp(Φ^t^+Uμτ2cphP22K^p^2)+1α^(Q^rF^mp^),
where
E^m=q^+Nmode(T^+RdCpΦ^+Rd2CpUcphK^)
is the nondimensional moist nonstatic energy,
Nmodecph2c2α^(1α^)
is the “moisture mode number,”1 a nondimensional scale that describes the relative magnitude of the CpT tendency to the Lυq tendency, as in Adames et al. (2019) and Adames (2022), where
c=(RdCpSpP)1/2
is the scale for the phase speed of dry gravity waves.
By examining the terms in Eq. (33), we find that the Em budget [Eq. (15)] can be simplified into the conventional MSE budget when the following three conditions are met:
(1,U2cph,Uμτ2cphP2)RdCpNmode1.
The first two conditions in parenthesis represent the Eulerian Φ tendency and the Lagrangian K tendency, respectively. The third condition is related to the vertical diffusion of K and requires knowledge of μ. A scaling value for it can be obtained by equaling the column integral of μp2K to Cd|vs|3, yielding the following:
μgCdUP.

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 cph2/c2, a quantity that is tied to the robustness of the WTG approximation (i.e., how large the residual between ωps and Q1 is) (Adames 2022). For c = 50 m s−1, we find that cph ∼ 10 m s−1 or less for the geopotential tendency to be roughly an order of magnitude smaller than the MSE tendency, assuming that all the terms in parenthesis are unity or smaller. Since Adames (2022) found that systems that propagate slower than 15 m s−1 are in WTG balance, it follows that the MSE budget in Eq. (2) is accurate in systems that are in WTG balance.

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, α^0.9, P ∼ 105 Pa, and τ ∼ 105 s. We now evaluate the three conditions in Eq. (37) individually:

  1. 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
    RdCpNmode101.
  2. 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
    U2cphRdCpNmode102.
  3. 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 CdU/P, yielding
    CdUgτP(U2cph)RdCpNmode103.

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.

The amplitudes of the terms relative to the column-integrated MSE anomalies and the MSE tendency are examined via the aforementioned spectral analysis. We use the square root of the individual power spectra:
R(K,m)=P(K)/P(m),
where P is the power spectrum, and angle brackets denote a mass-weighted integral from the surface to the top of the atmosphere. We use this method rather than the ratio of the cospectrum and the power spectrum used by Yasunaga et al. (2019) and Inoue et al. (2020) because we are mainly interested in the relative amplitude between the terms, not in their covariance. However, the results are not sensitive to the method used.

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 ∂tm⟩, while ∂tK⟩ 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 ∂tm⟩ 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 ∂tLυq⟩ is on the same order as ∂tm⟩, consistent with the idea that moisture must account for a large fraction of the MSE variance for the MSE budget to be accurate.

Fig. 1.
Fig. 1.

Square root of the ratios of the power spectra (R) as obtained from Eq. (39): R(Lυtq,tm) (blue), R(tΦ,tm) (red), R(tK,tm) (orange), and R(FK,tm) (gray). The data used are from DYNAMO’s northern sounding array. Note that both axes are on a logarithmic scale.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0005.1

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 R(K,m) along the dispersion curve of Kelvin waves with equivalent depths greater than 90 m (Fig. 2a). We also see a reduction in R(K,m) with decreasing spatial scales (increasing magnitude of the zonal wavenumber).

Fig. 2.
Fig. 2.

(a),(b) The square root ratio of the power spectra R(K,m) and (c),(d) the R(Φ,m) for the (left) symmetric and (right) antisymmetric components averaged over the 10°S–10°N latitude belt. The dispersion curves correspond to equatorial waves from Matsuno (1966) (each wave is labeled) with equivalent depths of 8, 25, and 90 m. Note that the shading interval in (a) and (b) is different than in (c) and (d).

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0005.1

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, R(Φ,m) is smaller in ERA5 data than in the DYNAMO data, but still within the expected scaling values suggested by Eq. (37).

c. Accuracy of the MSE budget in the MJO

Last, we examine the amplitudes of ∂tm⟩, ∂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 ∂tm⟩. The term ∂tK⟩ peaks at ∼0.2 W m−2, only 1% the magnitude of ∂tm⟩. The term FK is even smaller than ∂tK⟩, and cannot be distinguished from a horizontal line in Fig. 3. The smallness of these three terms is consistent with Eq. (37).

Fig. 3.
Fig. 3.

Time series of (top) precipitation and (bottom) ∂tm⟩ (blue), ∂t⟨Φ⟩ (red), ∂tK (orange), and FK (black dashed lines) from DYNAMO data from the northern sounding array. In both panels, the data are lag regressed onto the first principal component of the OMI from lag day −20 to day 20.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0005.1

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 Ucp, then ∂tK⟩ 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 cph2/c2 is much smaller than unity (Adames 2022). Mayta and Adames (2023) and Mayta and Adames Corraliza (2023) have shown that cph2/c2 governs the magnitude of Nmode. Furthermore, since slowly evolving convectively coupled systems tend to be in thermal wind balance in addition to WTG balance, it follows that the kinetic energy must remain much smaller than the MSE (see Table 4 in Adames 2022). The smallness of Nmode and the kinetic energy facilitates the satisfaction of Eq. (37). Thus, we posit that the accuracy of Eq. (2) is largely a result of WTG balance being upheld throughout the tropics.

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.

1

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

To obtain Eq. (18) we invoke mass continuity [Eq. (3c)] to convert the MSE advection into the MSE flux divergence:
u·m=·(um),
where u = v + ωk. If ω = 0 at p = 0 and at ps, then column integration of Eq. (A1) eliminates the vertical MSE flux term. We can switch the order of the horizontal divergence operator and the vertical integration, leading to
·(um)=h·vm.
We must also integrate the two turbulent flux terms, which we will group together. The column integral of the eddy flux divergences is equal to the eddy fluxes evaluated at p = 0 and ps. Assuming that there are no fluxes at p = 0, the column integration simplifies to an evaluation at the surface:
1g0psp(ωm¯+v·ωv¯)dp=1g(ωm¯+v·ωv¯)s.
The first term on the rhs is the sum of the surface latent and sensible heat fluxes:
(ωm¯)sg=LυE+H,
while the term (ωv¯)s can be evaluated using the bulk aerodynamic formula (Bister and Emanuel 1998; Emanuel 2003):
vs·(ωv¯)sgvs·Cdρs|vs|vs=Cdρs|vs|3.
Using Eqs. (A1)(A5) we can obtain Eq. (18).

APPENDIX B

DSE Budget

The DSE budget is often written as
DsDt=Q1.
This equation is an approximation of a more general DSE equation that we write in its column-integrated form as
s+Kt=Φth·v(s+K)+Qr+LυPFK.
To examine whether Eq. (B1) is a good approximation of Eq. (B2), Fig. B1 is as in Fig. 1 but comparing the amplitude of FK and the tendencies in ⟨Φ⟩ and ⟨K⟩ relative to the column DSE tendency. We see that ∂tK⟩ is an order of magnitude smaller than the DSE tendency at time scales of ∼30 days, and diminishes to nearly two orders of magnitude at time scales of 2 days and shorter. The geopotential fluctuations, on the other hand, are 0.2–0.3 times the amplitude of the DSE tendency at all frequencies. The term FK is the smallest and exhibits a distribution across frequencies that is very similar to that shown in Fig. 1. Based on the results of Fig. B1, we can neglect terms associated with ∂tK and FK from Eq. (B2), but not ∂t⟨Φ⟩. An accurate approximation of Eq. (B2) is
CpTth·vs+LυP+Qr+H.
The conventional DSE budget in Eq. (B1) is not recommended to use in large-scale motions due to the significant residual that adding ∂t⟨Φ⟩ can bring to the budget.
Fig. B1.
Fig. B1.

As in Fig. 1, but showing R(CptT,ts) (purple), R(tΦ,ts) (red), R(tK,ts) (orange), and R(FK,ts) (gray).

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0005.1

APPENDIX C

Equivalence Between Energy and Column Moist Enthalpy

The column-integrated moist enthalpy h is equivalent to the sum of column-integrated internal, potential, and latent energies:
h=E,
where, following the notation of Hill et al. (2017) and Smyth and Ming (2020),
E=CυT+Φ+Lυq,
where Cυ is the specific heat of air at constant volume. We can verify this identity by following Lorenz (1955) and using integration by parts to show that the mass-weighted integral of Φ is equivalent to the vertical integral of pressure:
0pszdp=0psd(pz)+0pdz=0pdz,
where we note that the integral of d(pz) vanishes since z = 0 at the surface and p = 0 at the top of the atmosphere. By invoking the ideal gas law p = ρRdT, we obtain that
Φ=0ρRdTdz=RdT.
We can then use Eq. (C4) to verify the identity in Eq. (C1).

APPENDIX D

Geostrophic Scaling

We can scale the main variables as in the main text, but using geostrophic scaling rather than equatorial scaling. In this case, geopotential and temperature scales as
Φ=fULΦ^T=fULxRdT^.
Other variables scale in the same way. With this definition, Nmode takes the following form:
NmodeLy2Ld2α^(1α^),
where
Ld=cf
is the Rossby radius of deformation.

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