Moist Static Potential Vorticity Budget in Tropical Motion Systems

Ángel F. Adames; Rosa M. Vargas Martes; Haochang Luo; Richard B. Rood

doi:10.1175/JAS-D-21-0161.1

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Fig. 1.

Contribution of (top) dry PV advection $\tilde{M} υ^{'} β_{d}$ and (middle) moist PV advection $(1 - \tilde{M}) υ^{'} β_{m}$ to the (bottom) GPV advection $υ^{'} β_{G}$ . The contoured fields are (top) $\tilde{M} q_{d}^{'}$ , (middle) $(1 - \tilde{M}) {q^{'}}_{m}$ , and (bottom) $q_{G}^{'}$ . Values of $\tilde{M} = 0. 2$ , β_d = 2.2 × 10⁻¹¹ m⁻¹ s⁻¹, and β_m = 0.8 × 10⁻¹¹ m⁻¹ s⁻¹ were used. In the three panels the advection has been multiplied by 10¹¹ to facilitate interpretation. The contour interval for all panels is 2 × 10⁻⁶ s⁻¹.

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Fig. 2.

Schematic describing MSPV quasi-conservation. For inviscid, adiabatic processes, a vortex filament will remain enclosed between two moist isentropic surfaces as it follows the flow.

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Fig. 3.

Schematic describing how MSPV is redistributed in a synoptic-scale system that exhibits cyclonic rotation in the lower troposphere in a moist atmosphere with no planetary vorticity, so that $P_{ms} ≃ - g ζ \partial_{p} m$ . (a) Because the lower troposphere is potentially unstable, deep moist convection can develop that rises at the moist adiabatic lapse rate (approximately conserving m). (b) The convection will transport MSPV from the lower to the upper troposphere as vortex filaments rise following the surfaces of constant MSE. (c) Because the MSE gradient reverses sign in the midtroposphere, it follows that parcels must rotate anticyclonically in order to quasi-conserve MSPV.

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Fig. 4.

(a)–(e) Anomalous precipitation (shading), 700-hPa geopotential height anomalies (contours) and the horizontal wind field (arrows) regressed onto the leading PC of 2–6-day OLR variance over the eastern Pacific basin (10°S–30°N, 150°–75°W). Each panel is a lag regression with the lag days indicated in the title. Contour interval is every 0.5 m starting at 1 m. Positive z anomalies are shown as solid contours while negative anomalies are shown as dashed contours. (f)–(j) As in (a)–(e), but showing the column-integrated water vapor (CWV′) as shading and the column-averaged MSPV anomalies as contours. Contour interval is every 1.5 × 10⁻² PVU starting at 3 × 10⁻² PVU. (k)–(o) As in (a)–(e), but showing anomalous horizontal MSPV advection as shading and the MSPV tendency as contours. Contour interval is every 0.0025 PVU day⁻¹ starting at 0.005 PVU day⁻¹.

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Fig. 5.

As in Fig. 4, but showing the anomalous horizontal advection of MSPV as contours and (a),(b) the horizontal advection of dry PV by the anomalous horizontal winds [ $- 〈 (1 - α) v' \cdot β_{d} 〉$ ]; (c),(d) the horizontal advection of the moist PV by the anomalous horizontal winds [ $- 〈 α v' \cdot β_{d} 〉$ ] and (e),(f) the total advection of mean MSPV by the anomalous horizontal winds [ $- 〈 v' \cdot \nabla_{h} {\bar{P}}_{ms} 〉$ ] as shading. Linear regressions onto (a),(c),(e) PC1 and (b),(d),(f) PC2 of eastern Pacific 2–6-day filtered OLR variability. Contour interval is every 0.0025 PVU day⁻¹ starting at 0.005 PVU day⁻¹.

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Fig. 6.

Fractional contribution of the terms in Eq. (31) to the propagation of the MSPV anomalies over the Caribbean and eastern Pacific region (5°–30°N, 130°–60°W). The bars are obtained by calculating each term for PC1 and PC2 and then averaging the two PCs together.

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Fig. 7.

Wave activity convergence from the anomalous winds during the PEW life cycle averaged over 10°–20°N. The red line shows the convergence of dry wave energy $(1 - \bar{α}) C_{d}$ , the blue line shows the convergence of moist wave energy $\bar{α} C_{m}$ , the green line shows $- \nabla_{h} \cdot \bar{v} A$ , and the black line shows the sum of the three terms, C_t. The lines are constructed by calculating each term for PC1 and PC2 and then averaging the two PCs together. For increased clarity, the data were smoothed using a 10° running mean.

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Editorial Type:: Article

Article Type:: Research Article

Moist Static Potential Vorticity Budget in Tropical Motion Systems

Ángel F. Adames

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

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https://orcid.org/0000-0003-3822-5347

Rosa M. Vargas Martes

Rosa M. Vargas Martes ^aDepartment of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin

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Haochang Luo

Haochang Luo ^bDepartment of Climate and Space Science and Engineering, University of Michigan, Ann Arbor, Michigan

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Richard B. Rood

Richard B. Rood ^bDepartment of Climate and Space Science and Engineering, University of Michigan, Ann Arbor, Michigan

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Online Publication:: 03 Mar 2022

Print Publication:: 01 Mar 2022

DOI:: https://doi.org/10.1175/JAS-D-21-0161.1

Page(s):: 763–779

Received:: 10 Jun 2021

Accepted:: 08 Nov 2021

Published Online:: 03 Mar 2022

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

Analyses of simple models of moist tropical motion systems reveal that the column-mean moist static potential vorticity (MSPV) can explain their propagation and growth. The MSPV is akin to the equivalent PV except it uses moist static energy (MSE) instead of the equivalent potential temperature. Examination of an MSPV budget that is scaled for moist off-equatorial synoptic-scale systems reveals that α, the ratio between the vertical gradients of latent and dry static energies, describes the relative contribution of dry and moist advective processes to the evolution of MSPV. Horizontal advection of the moist component of MSPV, a process akin to horizontal MSE advection, governs the evolution of synoptic-scale systems in regions of high humidity. On the other hand, horizontal advection of dry PV predominates in a dry atmosphere. Derivation of a “moist static” wave activity density budget reveals that α also describes the relative importance of moist and dry processes to wave activity amplification and decay. Linear regression analysis of the MSPV budget in eastern Pacific easterly waves shows that the MSPV anomalies originate over the eastern Caribbean and propagate westward due to dry PV advection. They are amplified by the fluxes of the moist component of MSPV over the Caribbean Sea and over the eastern Pacific from 105° to 130°W, underscoring the importance of moist processes in these waves. On the other hand, dry PV convergence amplifies the waves from 90° to 100°W, likely as a result of the barotropic energy conversions that occur in this region.

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

Abstract

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

Keywords: Potential vorticity; Thermodynamics; Water vapor; Tropical variability

1. Introduction

It is well known that large-scale (≥1000 km across) tropical motion systems organize and are themselves modulated by deep convection (Kiladis et al. 2006, 2009; Feng et al. 2020). These so-called convectively coupled tropical motion systems play an important role in the hydrologic cycle of the tropics, occasionally leading to floods (Lubis and Jacobi 2015; Hunt and Fletcher 2019; Dominguez et al. 2020). They can also lead to tropical cyclone formation (Maloney and Hartmann 2000; Gall et al. 2010; Blake and Kimberlain 2013; Núñez Ocasio et al. 2020). Some of these systems exhibit slow propagation and strong vorticity anomalies. In this group we find equatorial Rossby waves, easterly waves, monsoon low pressure systems, the Madden–Julian oscillation (MJO), as well as other tropical depression–like disturbances (Yasunaga and Mapes 2012; Inoue et al. 2020).

Numerous studies into these slowly propagating systems have revealed that tropospheric water vapor plays an important role in their evolution (Raymond and Fuchs 2009; Rydbeck and Maloney 2015; Fuchs-Stone et al. 2019; Adames and Maloney 2021). There is increasing evidence that the MJO has properties of a moisture mode, a type of tropical disturbance whose thermodynamics are largely governed by water vapor [Raymond and Fuchs 2009; see Adames and Maloney (2021) for further review]. New studies also suggest that convectively coupled equatorial Rossby waves may exhibit properties of moisture modes (Gonzalez and Jiang 2019; Fuchs-Stone et al. 2019). Water vapor also explains the majority of variance in rainfall in east Pacific easterly waves (Wolding et al. 2020). It may also play an important role in monsoon low pressure systems (Diaz and Boos 2019; Clark et al. 2020; Adames 2021), and in African easterly waves (Russell et al. 2020).

In spite of the growing evidence of water vapor’s importance in slowly evolving tropical motion systems, our understanding on how moisture and convection interact with the large-scale circulation remains incomplete (Grabowski and Moncrieff 2004; Ahmed and Neelin 2018). Nonetheless, recent studies based on simple moist models of the atmosphere suggest that we may gain insight onto these interactions by examining a quantity defined as “gross” potential vorticity (GPV) (Adames and Ming 2018; Adames 2021). The GPV combines moist static energy (MSE) and PV into a single quantity that is quasi-conserved under moist adiabatic, inviscid, and hydrostatic processes. Furthermore, the processes that lead to the propagation and growth of waves in their simple models were succinctly described by the terms in the GPV equation.

More recently, Adames (2021) hypothesized that GPV may be physically related to the equivalent PV (Emanuel 1983; Martin et al. 1992) or, more specifically, a quantity that resembles equivalent PV but defined with MSE. While the former is applied to shallow-water and two-layer models of the atmosphere, the latter could be used to diagnose the processes that lead to the evolution of systems in the real atmosphere. The goal of this study is to examine this hypothesis. We will show that a quantity referred to as the moist static PV (MSPV) does indeed capture many of the features of GPV and quantifies the relative role of dry and moist processes in the evolution of off-equatorial tropical motion systems. We will also demonstrate the usefulness of the MSPV budget by applying it to eastern Pacific easterly waves (PEWs).

The remainder of this study is structured as follows. Section 2 reviews the shallow-water model of Adames and Ming (2018, hereafter AM18) and discusses the physical relevance of GPV. Section 3 defines the MSPV as the three-dimensional analog of GPV and discusses the processes that lead to its evolution. Scale analysis of the MSPV budget is shown in section 4. The MSPV budget is linearized and interpreted in section 5. An example of the potential applications of MSPV and its budget is shown in section 6. The conclusions are offered in section 7.

2. Evolution of tropical motion systems in a linear shallow-water model with prognostic moisture

a. Review of the AM18 model

Using a set of shallow-water equations on a beta plane with prognostic moisture, AM18 derived a quantity they defined as GPV. We can write the GPV as follows:

{q^{'}}_{G} = \tilde{M} ζ' + \frac{f_{0}}{{\hat{s}}_{p}} (C_{p} T' + L q'),

(1)

where ζ′ is the anomalous relative vorticity, T′ is the anomalous temperature, q′ is the specific humidity anomaly, L = 2.5 × 10⁶ J kg⁻¹ is the latent energy of vaporization, C_p = 1005 J kg⁻¹ K⁻¹ is the specific heat at constant pressure, f₀ is the planetary vorticity at a reference latitude,

{\hat{s}}_{p}

can be thought of as a shallow-water dry static stability, and

\tilde{M}

is the dimensionless moist static stability (referred to as the gross moist stability). The variable

\tilde{M}

is defined as

\tilde{M} = \frac{{\hat{s}}_{p} - L {\hat{q}}_{p}}{{\hat{s}}_{p}} = 1 - \hat{α},

(2)

where

{\hat{q}}_{p}

can be thought of as a shallow-water equivalent of a vertical moisture gradient, referred to as the gross moisture stratification (Yu et al. 1998). The hats denote that they are shallow-water mean fields, and their units are different from the equivalent fields that will be shown in the next section. The nomenclature of the shallow-water model has been slightly modified from AM18 to facilitate comparison with the subsequent sections. For the same purpose we have also defined

\hat{α} = L {\hat{q}}_{p} / {\hat{s}}_{p}

as the shallow-water climatological-mean Chikira (2014) parameter. For reference, the relevant variables discussed in this section are summarized in Table 1.

Table 1

The variables and definitions used in section 2.

It is worth noting that, to a reasonable approximation, C_pT′ can be replaced by s′ in Eq. (1), where s′ is the dry static energy (DSE) anomaly. By doing this, the gross PV can be written as

{q^{'}}_{G} ≃ \tilde{M} ζ' + f_{0} {\hat{s}}_{p}^{- 1} m'

, where m′ = s′ + Lq′ is the anomalous MSE. We can define the shallow-water moist static stability perturbation as

\tilde{M}' = m' {\hat{s}}_{p}^{- 1}

, which leads to

{q^{'}}_{G} ≃ \tilde{M} ζ^{'} + f_{0} {\tilde{M}}^{'} .

(3)

This definition of GPV will be particularly helpful in establishing the similarities between GPV and the moist static PV discussed in the following section.

One of the useful features of GPV comes from its decomposition into dry and moist components:

{q^{'}}_{G} = \tilde{M} q_{d}^{'} + (1 - \tilde{M}) q_{m}^{'},

(4a)

where

q_{d}^{'}

and

q_{m}^{'}

are the dry and moist quasigeostrophic (QG) potential vorticity anomalies, respectively, written as follows:

q_{d}^{'} = ζ^{'} + f_{0} \frac{C_{p} T^{'}}{{\hat{s}}_{p}},

(4b)

q_{m}^{'} = f_{0} (\frac{q^{'}}{{\hat{q}}_{p}} + \frac{C_{p} T^{'}}{{\hat{s}}_{p}}) .

(4c)

While

q_{d}^{'}

is simply the shallow-water equivalent of the PV in QG theory (Holton and Hakim 2012), the physical meaning of

q_{m}^{'}

warrants further discussion. Inspection of Eq. (4c) shows that

q_{m}^{'}

is reminiscent to moist enthalpy or MSE, except that q′ and C_pT′ are divided by their mean vertical gradients. AM18 showed that the budget equation for

q_{m}^{'}

is similar to that of MSE. Thus, we can interpret

q_{m}^{'}

as describing the evolution of the moist thermodynamics of a wave. A similar quantity to

q_{m}^{'}

was derived by Lapeyre and Held (2004) [see their Eq. (2)].

AM18 found that when $\tilde{M} \to 0$ (or $\hat{α} \to 1$ ), which can be thought to represent a saturated troposphere, moist enthalpy dominates the evolution of convectively coupled waves. Conversely, when $\tilde{M} \to 1$ (or $\hat{α} \to 0$ ), the atmosphere is dry and the GPV equation simplifies to the dry QG PV equation. Thus, the GPV can be thought of as a generalization of QG PV that includes the effects of moist thermodynamics.

The usefulness of using q_G′ comes from evaluation of its budget. In AM18, the GPV budget was written as

\frac{\partial q_{G}^{'}}{\partial t} = - υ^{'} β_{G},

(5)

where υ′ is the meridional wind anomaly, β_G is the GPV gradient, the weighted sum of the mean meridional dry and moist PV gradients:

β_{G} \equiv \tilde{M} β_{d} + (1 - \tilde{M}) β_{m} .

(6)

The dispersion relation of moist Rossby waves can be obtained from Eq. (5) if the moisture anomalies are neglected, in which case it can be written as follows:

ω = - \frac{β_{G} k}{\tilde{M} (k^{2} + l^{2}) + k_{d}^{2}},

(7)

where k_d is the inverse of the Rossby radius of deformation, k is the zonal wavenumber, l is the meridional wavenumber and ω is the wave’s angular frequency. Examining Eq. (7) reveals that the numerator on the right-hand side (rhs) of Eq. (7) is the rhs term of Eq. (5). Meanwhile, the denominator term on the rhs term of Eq. (7) is the left-hand side of Eq. (5) (AM18). When moisture is included and precipitation is parameterized in terms of it, the dispersion relation will contain an imaginary term that is positive when both the moist and dry PV tendency are in the same direction, a mechanism of growth AM18 termed as “moisture-vortex” instability. AM18 also found that systems that grow from moisture-vortex instability exhibit an in-phase component between q_G and the advection of mean GPV by the anomalous meridional wind.

An example of how moist and dry PV advection contribute to the evolution of a convectively coupled Rossby wave is shown in Fig. 1. For $\tilde{M} = 0. 2$ , moist PV advection contributes nearly twice as much to GPV evolution as dry PV advection does, even though β_m is less than half the magnitude of β_d in this figure. Thus, the value of $\tilde{M}$ is critical to determining the importance of moist processes in the evolution of a convectively coupled system.

Fig. 1.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0161.1

b. Wave activity evolution

The aforementioned results indicate that

{q^{'}}_{G}

can provide information about the dynamics of convectively coupled systems that cannot be obtained by examining dry PV alone. To examine this possibility, we can turn Eq. (5) into an equation for the evolution of wave activity akin to those that describe the Eliassen–Palm (EP) flux (see chapter 10 in Vallis 2017). We will multiply all terms in Eq. (5) by

{q^{'}}_{G} / β_{G}

and define the shallow wave activity density (or more accurately, its pseudo-momentum) as follows:

A \equiv \frac{{q^{'}}_{G}^{2}}{2 β_{G}},

(8)

which leads to the following equation for the evolution of wave activity:

\frac{\partial A}{\partial t} = - υ' q_{G}^{'} .

(9)

Equation (9) indicates that wave activity increases where there is a flux of anomalous GPV by the anomalous meridional winds. We can write this equation in terms of the convergence of a wave activity flux, which we break down into dry and moist components:

\frac{\partial A}{\partial t} = \tilde{M} C_{d} + (1 - \tilde{M}) C_{m},

(10a)

C_{d} \equiv - υ' q_{d}^{'},

(10b)

C_{m} \equiv - υ' q_{m}^{'} .

(10c)

Equation (10) reveals that the evolution of A is due to the weighted sum of the convergences of dry and moist wave activity (C_d and C_m, respectively). In a moist atmosphere, where

\tilde{M} \to 0

, the evolution of wave activity is determined by

{q^{'}}_{m}

. Conversely, in a dry atmosphere,

\tilde{M} \to 1

, and dry wave activity governs the evolution of wave activity.

By examining Fig. 1 we observe that $q_{m}^{'}$ is shifted westward toward meridional wind anomalies of the opposite sign. This phasing indicates that there is a net positive flux of $q_{m}^{'}$ when averaged over the horizontal domain of Fig. 1, indicating that C_m > 0. In contrast, $q_{d}^{'}$ is in quadrature with υ′. Thus these waves are growing from moist wave energy convergence alone. This result indicates that the moisture-vortex instability mechanism described in AM18 can be understood in terms of C_m.

3. Moist static potential vorticity as an analog of GPV

The results of the previous section suggest that GPV can summarize the dynamics of convectively coupled tropical motion systems in a shallow-water model. However, the GPV equation cannot be readily applied to observations, reanalysis and GCM output because of the underlying shallow-water and linearity assumptions. In this section, we seek to obtain a quantity that is reminiscent to the GPV but applicable to the stably stratified atmosphere. Table 2 summarizes the main variables that will be used from this section onward.

Table 2

The main variables and definitions used in this study from section 3 onward. Note that the PVU is defined as 10⁻³ J m² kg⁻² s⁻¹.

A candidate field variable that exhibits many similarities to GPV is the moist static Ertel PV (MSPV). In isobaric coordinates, the MSPV can be written as follows:

P_{ms} = - g ω_{a} \cdot \nabla m,

(11a)

where g = 9.81 m s⁻² is the gravitational acceleration;

ω_{a} = η i + ξ j + ζ_{a} k

(11b)

is the absolute vorticity vector, where ζ_a = ζ + f is the vertical component of the absolute vorticity;

\nabla = i \frac{\partial}{\partial x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial p}

(11c)

is the gradient operator in isobaric coordinates; and

m = C_{p} T + Φ + L q

(11d)

is the MSE. The MSPV in Eq. (11) is very similar to the equivalent PV defined in Martin et al. (1992); Moore and Lambert (1993) and McCann (1995), except that the equivalent potential temperature (θ_e) is replaced by MSE. Both MSE and θ_e are approximately conserved in moist adiabatic motions. Specifically, MSE is conserved under hydrostatic, moist adiabatic processes. In deriving the MSE, the temporal tendency and horizontal advection of pressure, as well as nonhydrostatic pressure perturbations are neglected (Madden and Robitaille 1970; Betts 1974). For large-scale tropical motions these terms are negligible, but can be large in systems with strong winds and pressure perturbations such as mature tropical cyclones. In spite of these differences, we choose to use MSE because it allows us to make a more direct comparison with the GPV defined in the previous section. Furthermore, the fact that MSE is defined as a sum of dry static energy and latent energy will allow for a straightforward decomposition of the MSPV into its dry and moist components, as was done for the GPV. Because m is used instead of θ_e, the units of MSPV differ from those of equivalent PV. We address this issue by defining the PV unit (PVU) as 10⁻³ J m² kg⁻² s⁻¹. This will cause the MSPV to have the same PVUs as equivalent PV.

It is worth noting that in isobaric coordinates k points toward the surface. It is also worth pointing out that the vertical components of both ∇ and ω_a have different units than their horizontal components. While this notation may cause some confusion, it makes Eq. (12) more compact and easily comparable to PV in height and isentropic coordinates. The same notation has been used by previous studies that analyzed PV in isobaric coordinates (Zhang and Ling 2012; Russell et al. 2020).

a. MSPV budget

The conservation equation for MSPV can be written as follows:

\frac{D P_{ms}}{D t} = G + B - F_{r},

(12a)

where

\frac{D}{D t} = \frac{\partial}{\partial t} + u \frac{\partial}{\partial x} + υ \frac{\partial}{\partial y} + ω \frac{\partial}{\partial p}

(12b)

is the material derivative in isobaric coordinates;

B = g f \frac{\partial v_{g}}{\partial p} \cdot \nabla_{h} m

(12c)

is the baroclinic (solenoidal) generation of MSPV, written as the horizontal advection of MSE by the thermal wind as in Martin et al. (1992), where v_g = u_gi + υ_gj is the geostrophic wind vector;

F_{r} = g \nabla m \cdot (\nabla \times F_{r})

(12d)

is the frictional dissipation of MSPV, where F_r is the frictional dissipation of momentum; and

G = - g ω_{a} \cdot \nabla \dot{m}

(12e)

is the diabatic generation of MSPV, where

\dot{m} \equiv \frac{D m}{D t} = Q_{r} - \frac{\partial \bar{ω' m'}}{\partial p}

(12f)

describes the sources and sinks of MSE, where Q_r is the radiative heating rate and

\partial_{p} \bar{ω' m'}

is the vertical divergence of the turbulent flux of MSE.

Equation (12a) is similar to the one shown by Martin et al. (1992), except it includes the frictional dissipation term and we replace θ_e with MSE. An alternate form of Eq. (12a), in which all the rhs terms are expressed in the form of a forcing vector, is shown in the first section of the appendix.

b. Interpretation of MSPV

By definition, the MSPV is the absolute vorticity that is normal to the MSE surfaces. The definition of MSPV differs from conventional Ertel PV in that m replaces the potential temperature (θ). However, this replacement leads to differences in the properties and physical interpretation of the two PV quantities. For inviscid, adiabatic processes, dry PV is conserved following the flow. A simplified, linear version of PV behaves as the wave equation for Rossby waves. Thus, dry PV not only serves as a tracer, it also summarizes the dynamics of inviscid, adiabatic geophysical flows.

In contrast, MSPV is not conserved under inviscid, moist adiabatic processes, as in the equivalent PV discussed by Schubert et al. (2001). Furthermore, because ∂_pm > 0 in the lower troposphere, it follows that MSPV tends to be negative in the lower troposphere when the absolute vorticity is positive (Fig. 2). In contrast, dry PV is positive in this layer for the same vorticity anomaly since ∂_pθ < 0 in the free troposphere. Furthermore, the fact that the tropical lower troposphere is potentially unstable (∂_pm > 0, Fig. 3a) can lead to the development of deep moist convection. Undiluted rising parcels within deep convection approximately conserve MSE. If we assume that the mass circulation associated with the undiluted updraft is bounded by two MSE surfaces of the same value in the upper and lower troposphere (Riehl and Malkus 1958), it follows that the updraft will be associated with a constant MSE surface that connects the lower and upper troposphere, as shown in Fig. 3b.^¹ Because MSE surfaces are impermeable to MSPV in the same way that isentropes are impermeable to Ertel PV (Haynes and McIntyre 1987, 1990), it follows that the isobaric MSPV concentration of rising parcels is constrained to be enclosed within the MSE surfaces described in Fig. 3b. Thus, in the absence of large sources of MSPV, deep tropical convection acts to redistribute MSPV in the troposphere. Furthermore, because the MSE gradient reverses sign in the mid troposphere (Fig. 3a), it follows that parcels must rotate anticyclonically as they exit the updraft in the upper troposphere. The transport of MSPV from the lower to the upper troposphere occurs at time scales that are much shorter than the time scale of synoptic-scale motion systems (Fig. 3).

Fig. 2.

Schematic describing MSPV quasi-conservation. For inviscid, adiabatic processes, a vortex filament will remain enclosed between two moist isentropic surfaces as it follows the flow.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0161.1

Fig. 3.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0161.1

Because of the vertical redistribution of MSPV by convection, it follows that the vertically averaged MSPV (

〈 P_{ms} 〉

) is the actual quasi-conserved variable, as is also the case with MSE (Andersen and Kuang 2012). We define the column-averaged MSPV as

〈 P_{m s} 〉 = \frac{1}{Δ p} \int_{p_{t}}^{p_{s}} P_{m s} d p,

(13)

where p_t = 100 hPa is the tropopause pressure, p_s = 1000 hPa is the surface pressure, and Δp = 900 hPa is the depth of the troposphere. We choose to perform column-averaging rather than column integrating so that the units of the MSPV tendency are in the form of PVU s⁻¹.

4. Scaling of the MSPV budget for off-equatorial synoptic-scale tropical motion systems

A comparison of Eq. (12) with Eq. (5) reveals that the MSPV and GPV budgets are quite different. This is a result of the assumptions made in the formulation of the AM18 model as well as the linearization. A budget that is more reminiscent of Eq. (5) can be obtained from scaling an linearization of Eq. (12a). In this section we will scale Eq. (12a) for convectively coupled synoptic-scale tropical motion systems with a vertical scale that spans the entire troposphere. While MSPV can be applied even near the equator, we will emphasize off-equatorial tropical motion systems in order to compare the MSPV budget more directly to the GPV discussed in section 2. Magnitudes for the different field variables are shown in Table 3. The scales were obtained using ERA5 data and closely follow the scale analysis done by Adames et al. (2021). Scaling of the MSPV reveals the following magnitudes:

P_{ms} = - \underset{10^{- 6}}{\underset{︸}{g η \frac{\partial m}{\partial x}}} - \underset{10^{- 6}}{\underset{︸}{g ξ \frac{\partial m}{\partial y}}} - \underset{10^{- 5}}{\underset{︸}{g ζ_{a} \frac{\partial m}{\partial p}}},

(14)

which indicates that the vertical component of MSPV governs its distribution. Thus, the MSPV can be expressed to leading order as follows:

P_{ms} ≃ - g ζ_{a} \frac{\partial m}{\partial p} .

(15)

We can use the simplified expression for MSPV in Eq. (15) to obtain the scales of the material derivative of MSPV:

\frac{D P_{ms}}{D t} = \underset{10^{- 10}}{\underset{︸}{\frac{\partial P_{ms}}{\partial t}}} + \underset{10^{- 10}}{\underset{︸}{u \frac{\partial P_{ms}}{\partial x}}} + \underset{10^{- 10}}{\underset{︸}{υ \frac{\partial P_{ms}}{\partial y}}} + \underset{10^{- 11}}{\underset{︸}{ω \frac{\partial P_{ms}}{\partial p}}},

(16)

which indicates that the evolution of MSPV is governed by the Eulerian derivative and the horizontal advection terms.

Table 3

Scales used for the different field variables.

We can scale the baroclinic term B by assuming that the geostrophic wind is of similar magnitude to the horizontal winds (∼10 m s⁻¹). Through the thermal wind equation, such a wind could be obtained by assuming a horizontal temperature gradient of 1 K (1000 km)⁻¹. The resulting scaling is written as follows:

B = \underset{10^{- 11}}{\underset{︸}{g f \frac{\partial v_{g}}{\partial p} \cdot \nabla_{h} m}},

(17)

which shows that it is much smaller than the MSPV tendency.

Last, scale analysis of the MSPV generation term yields the following magnitudes:

G = - \underset{10^{- 11}}{\underset{︸}{g η \frac{\partial \dot{m}}{\partial x}}} - \underset{10^{- 11}}{\underset{︸}{g ξ \frac{\partial \dot{m}}{\partial y}}} - \underset{10^{- 11}}{\underset{︸}{g ζ_{a} \frac{\partial \dot{m}}{\partial p}}} .

(18)

By neglecting G, F_r, and B, and dropping the vertical advection term yields the following approximate MSPV budget:

\frac{D_{h} P_{ms}}{D t} ≃ 0,

(19a)

where

\frac{D_{h}}{D t} = \frac{\partial}{\partial t} + v \cdot \nabla_{h}

(19b)

is the horizontal material derivative and v = ui + υj is the horizontal wind vector. Equation (19a) reveals that MSPV is quasi-conserved following the horizontal flow in synoptic-scale tropical motion systems.

It is important to note that the scaling presented here only applies to synoptic-scale systems. The leading-order terms are likely to be different if the analysis is performed on disturbances whose spatial and/or temporal scale differ from those assumed in this section. Furthermore, because the leading-order terms are only an order of magnitude larger than the other terms, it follows that small departures from the scaling assumptions of this section can cause the smaller terms to become nonnegligible. Nonetheless, we will use the scaled MSPV budget as a starting point to understand the evolution of moist tropical motion systems.

5. Linearized MSPV

a. Linearized MSPV and its relation to GPV

We can linearize MSPV with respect to a slowly varying background state. Anomalies of MSPV will be denoted by primes, while overbars depict the background state. By expanding P_ms into its mean and anomaly fields, and dropping the nonlinear terms yields the following:

P_{ms}^{'} = - g (ζ^{'} {\bar{m}}_{p} + {\bar{ζ}}_{a} {m^{'}}_{p}),

(20)

where

{\bar{m}}_{p} = L {\bar{q}}_{p} - {\bar{s}}_{p} m_{p}^{'} = L q_{p}^{'} - s_{p}^{'}

(21a)

is the mean and anomalous vertical MSE gradient, respectively, where

{\bar{s}}_{p} = - \frac{\partial \bar{s}}{\partial p} s_{p}^{'} = - \frac{\partial s^{'}}{\partial p}

(21b)

{\bar{q}}_{p} = \frac{\partial \bar{q}}{\partial p} q_{p}^{'} = \frac{\partial q^{'}}{\partial p}

(21c)

are the mean and anomalous static stability and moisture stratification, where s = C_pT + Φ is the dry static energy. With these definitions we can also define the dry and moist components of the MSPV:

P_{ms}^{'} = (1 - \bar{α}) {P^{'}}_{d} + \bar{α} {P^{'}}_{m},

(22a)

where the dry PV anomaly is written as

P_{d}^{'} = g (ζ^{'} {\bar{s}}_{p} + {\bar{ζ}}_{a} s_{p}^{'}),

(22b)

and the moist component of the MSPV is

P_{m}^{'} = {\bar{P}}_{d} (\frac{s_{p}^{'}}{{\bar{s}}_{p}} - \frac{q_{p}^{'}}{{\bar{q}}_{p}})

(22c)

and

\bar{α} = \frac{L {\bar{q}}_{p}}{{\bar{s}}_{p}} \cdot

(22d)

The term α can be thought of as a measure of potential instability since a layer is potentially unstable when α > 1 (see the second section of the appendix). Note that in a dry atmosphere

\bar{α} = 0

, so that MSPV approximately becomes the Ertel PV. During moist adiabatic processes, where parcels conserve m as they ascend through the troposphere α = 1. In this case

P_{d}^{'}

plays no role in the evolution of convectively coupled waves, and it is

P_{m}^{'}

that drives the evolution of the system. As Eq. (22c) indicates,

P_{m}^{'}

describes the relative change in the stratification of DSE and latent energy. If

{\bar{m}}_{p} < 0

, increasing the moisture stratification will act to further decrease m_p, and thus leading to a more negative MSPV assuming that

{\bar{ζ}}_{a} > 0

. Similarly,

P_{m}^{'}

can also be thought of as a measure of relative changes in the stability of the column since

P_{m}^{'} = - {\bar{P}}_{d} α^{'} / \bar{α}

. Similar variables to

P_{m}^{'}

have been obtained in other studies that analyze variants of MSPV (Stechmann and Majda 2006; Smith and Stechmann 2017; Wetzel et al. 2020).

Equations (22) and (22) can be thought as the full atmosphere equivalents of Eqs. (3) and (4a). We can show that

P_{ms}^{'} \sim g {\bar{s}}_{p} {q^{'}}_{G} .

(23)

The main difference between MSPV and GPV are in their dimensional units. In spite of having different dimensions, the two variables have the same physical interpretation.

b. Anomalous MSPV budget

After scaling and linearization, the equation for the evolution of the MSPV anomalies can be written as

\frac{{\bar{D}}_{h} P_{ms}^{'}}{D t} = - v' \cdot \nabla_{h} {\bar{P}}_{ms},

(24a)

where

\frac{{\bar{D}}_{h}}{D t} = \frac{\partial}{\partial t} + \bar{v} \cdot \nabla_{h} .

(25)

We can decompose

\nabla {\bar{P}}_{ms}

into the dry and moist PV gradient by noting that

{\bar{P}}_{ms}

can be rewritten as

{\bar{P}}_{ms} = (1 - \bar{α}) {\bar{P}}_{d} .

(26)

By multiplying Eq. (26) by ∇_h and after some rearrangement of the terms, we can express the horizontal MSPV gradient as

\nabla_{h} {\bar{P}}_{ms} = (1 - \bar{α}) β_{d} + \bar{α} β_{m},

(27)

where

β_{d} \equiv \nabla_{h} {\bar{P}}_{d}

(28)

β_{m} \equiv {\bar{P}}_{d} (\frac{\nabla_{h} {\bar{s}}_{p}}{{\bar{s}}_{p}} - \frac{\nabla_{h} {\bar{q}}_{p}}{{\bar{q}}_{p}})

(29)

are the dry and moist PV gradients, respectively. We use a similar notation to the moist and dry PV gradients in section 2 to suggest their physical connection, although the PV gradients in this section are vectors with different units (cf. Tables 1 and 2).

With the decomposition of

\nabla_{h} {\bar{P}}_{ms}

, we can write the anomalous MSPV budget as

\frac{{\bar{D}}_{h} P_{ms}^{'}}{D t} = - (1 - \bar{α}) v' \cdot β_{d} - \bar{α} v' \cdot β_{m},

(30)

where the horizontal advection terms are now decomposed into the advection of dry and moist PV. Note that Eq. (30) can be thought as equivalent to Eq. (5) but more general. A similar equation can be obtained without linearization, as shown in the third section of the appendix.

The column-averaged budget takes the following form:

\begin{array}{l} \frac{\partial 〈 P_{ms}^{'} 〉}{\partial t} = {〈 v \cdot \nabla_{h} P_{ms} 〉}^{'} \\ = - 〈 \bar{v} \cdot \nabla_{h} P_{ms}^{'} 〉 - 〈 (1 - \bar{α}) v^{'} \cdot β_{d} 〉 - 〈 \bar{α} v^{'} \cdot β_{m} 〉 \end{array},

(31)

where the contribution of the background flow to the advection of MSPV is now shown explicitly on the rhs of Eq. (31).

c. The moist static wave activity equation

One feature of PV is that it can describe the propagation of wave energy. In the midlatitudes, the propagation of wave energy can be described in terms of the EP flux (Andrews and McIntyre 1976) and its more generalized forms that have been derived thereafter (Plumb 1985; Takaya and Nakamura 2001). We can obtain an analogous equation but for MSPV by following Takaya and Nakamura (1997). We multiply Eq. (24a) by

{P^{'}}_{ms} / | \nabla_{h} {\bar{P}}_{ms} |

. Assuming that the horizontal MSPV gradient varies in space more slowly than

{P^{'}}_{ms}^{2}

, we can define the equivalent wave activity density (or more accurately, the equivalent pseudo-momentum) as follows:

A = \frac{1}{2} \frac{{P^{'}}_{ms}^{2}}{| \nabla_{h} {\bar{P}}_{ms} |},

(32)

and turn Eq. (24a) into an equation for wave activity evolution:

\frac{\partial A}{\partial t} = C_{t},

(33)

where C_t is the total wave activity convergence:

C_{t} \equiv - P_{ms}^{'} v^{'} \cdot \frac{\nabla_{h} {\bar{P}}_{ms}}{| \nabla_{h} {\bar{P}}_{ms} |} - \nabla_{h} \cdot \bar{v} A .

(34)

Note that for a dry atmosphere, C_t can be written as the convergence of a wave activity flux as in Takaya and Nakamura (2001), C_t = −∇ ⋅ F. It also describes the EP flux after zonal averaging. The EP flux can be extended to include the effects of water vapor for a near-saturated atmosphere (Stone and Salustri 1984; Dwyer and O’Gorman 2017). However, such an assumption would lead to large inaccuracies at the time scales that we are interested in since water vapor fluctuates significantly in the tropics while temperature does not. Thus, it would be more appropriate to derive an EP flux that is applicable for an unsaturated atmosphere. However, we were unable to obtain such an expression. As a result, we keep the notation in terms of a wave activity convergence.

As was done in the previous section, we can break down C_t into contributions from dry and moist processes:

C_{t} = (1 - \bar{α}) C_{d} + \bar{α} C_{m} - \nabla_{h} \cdot \bar{v} A,

(35)

where we define the convergence of dry and moist wave activity fluxes as follows:

C_{d} \equiv - P_{d}^{'} v^{'} \cdot \frac{\nabla_{h} {\bar{P}}_{ms}}{| \nabla_{h} {\bar{P}}_{ms} |},

(36)

C_{m} \equiv - P_{m}^{'} v^{'} \cdot \frac{\nabla_{h} {\bar{P}}_{ms}}{| \nabla_{h} {\bar{P}}_{ms} |},

(37)

noting that

- v^{'} {P^{'}}_{d} = - \nabla \cdot F_{d}

, where F_d is the dry wave activity flux, or equivalently the extended EP flux (Plumb 1985; Takaya and Nakamura 2001). With these definitions and after column-averaging the wave activity tendency equation becomes:

\frac{\partial 〈 A 〉}{\partial t} = - \nabla_{h} \cdot 〈 \bar{v} A 〉 + 〈 (1 - \bar{α}) C_{d} 〉 + 〈 \bar{α} C_{m} 〉 \cdot

(38)

Equation (38) shows the relative importance of moist and dry wave energy convergence (C_d and C_m, respectively) on the evolution of wave activity. The dry wave energy convergence is given by the flux of dry PV, as in the EP flux formulation, except that the zonal and meridional fluxes are weighted by

\partial_{x} {\bar{P}}_{ms} / | \nabla_{h} {\bar{P}}_{ms} |

and

\partial_{y} {\bar{P}}_{ms} / | \nabla_{h} {\bar{P}}_{ms} |

, respectively. The moist wave energy convergence can be thought to be similar to a weighted MSE flux. It is worth noting that the wave activity equation largely mirrors the MSPV budget in Eq. (31). Both equations contain a term that is weighted by

(1 - \bar{α})

, a term that is weighted by

\bar{α}

and a term involving the mean flow. In spite of the similarities between the two equations, a process that is dominant in Eq. (31) might not contribute significantly to Eq. (38). For example, it is possible for

- (1 - \bar{α}) v^{'} \cdot β_{d}

to dominate the evolution of MSPV while

\bar{α} C_{m}

to dominate the evolution of wave activity.

6. MSPV evolution in east Pacific easterly waves

a. Data and methods

To show an example of the usefulness of MSPV, we will investigate its variability in PEWs. We will make use of data from the fifth reanalysis from the European Centre for Medium-Range Weather Forecasts (ECMWF) (ERA5; Hersbach et al. 2019). We used instantaneous fields with a time interval of 12 h, spanning the 40-yr interval of 1979–2018. The ERA5 data used have horizontal resolution of 0.5° × 0.5° and 27 vertical levels from 1000 to 100 hPa. We make use of the following ERA5 pressure level fields: specific humidity (q), temperature (T), geopotential (Φ), vorticity (ζ), and the zonal and meridional winds (u and υ, respectively). Additionally, we make use of ERA5 column-integrated water vapor (CWV), precipitation rate (P), and outgoing longwave radiation (OLR).

Anomaly fields are obtained by removing the seasonal mean at each grid point for the months of July–September (JAS), when easterly wave activity in the Northern Hemisphere is the strongest (e.g., Mekonnen et al. 2006; Cheng et al. 2019). A Lanczos filter (Duchon 1979) is performed on the anomalous data to retain easterly wave-related variability on the order of 2–6 days (Mekonnen et al. 2006; Leroux et al. 2010; Cheng et al. 2019). From here on, primed variables (e.g., ${P^{'}}_{ms}$ ) will describe 2–6-day filtered anomalies while fields with an overline (e.g., ${\bar{P}}_{ms}$ ) denote JAS averages. While it would be more accurate to calculate the overlined fields using a 6-day low-pass filter, we use a simple climatological average since our main interest is to showcase the potential usefulness of the MSPV budget. A more rigorous calculation of the MSPV budget applied to easterly waves will be shown in a future paper.

The variability associated with PEWs is extracted by performing an EOF analysis on the filtered OLR data for the months of JAS, over a domain that ranges from 10°S–30°N and from 150° to 75°W. The first two EOFs of OLR variability in the eastern Pacific explain 3.16% and 3.01% of the filtered OLR variance, respectively. This procedure mirrors that of Rydbeck and Maloney (2014, 2015), except that we choose to perform the analysis on OLR rather than 700-hPa vorticity and our range of months is narrower. In spite of these differences, our resulting patterns are similar to those found by Rydbeck and Maloney (2014, 2015). The leading EOF patterns that correspond to PEWs are insensitive to the choice of domain or field variable, although the PEWs explain a larger fraction of the total variance if a smaller domain is chosen. They also explain a larger fraction of total variance if a spatially smoother field such as geopotential is used. However, we opt to use OLR since we are specifically interested in the moist processes associated with PEWs and how it impacts the MSPV budget.

The figures shown in this section are obtained through regression analysis of the first two principal components (PCs) of the aforementioned EOF analysis. We will refer to these as the PCs of PEW activity. Following Adames and Wallace (2014), the regression maps are obtained via the following equation:

D = N^{- 1} S P^{T},

(39)

where

D

is the linear regression pattern in dimensional units,

S

is a two-dimensional matrix representing a field variable (e.g., P_ms),

P

is the nondimensional PC, and N is the sample size. Statistical significance of the regression maps is obtained by applying a two-tailed t-test at the 95% level under the null hypothesis that the regression maps are uncorrelated with the PCs. We have defined the contour interval in such a way that all the contoured and shaded fields presented are statistically significant. In plots where vector arrows are shown, only arrows that are statistically significant are shown.

b. MSPV evolution in PEWs

Maps of anomalous geopotential height z, horizontal winds v, precipitation P, and $〈 P_{ms} 〉$ regressed upon the first PC of anomalous OLR are shown in Fig. 4a. The maps are lag regressions that show the evolution of PEWs from lag day −2 to lag day 2. Negative (positive) lags represent the days before (after) PC1 attains a maximum amplitude. At lag day −2, an anomalous anticyclone collocated with suppressed rainfall can be seen southwest of Mexico. Both the z and P anomalies associated with these features exhibit a southwest to northeast tilt, a common feature of easterly waves in this region (Serra et al. 2008; Rydbeck and Maloney 2015). The anticyclone is flanked by regions of enhanced rainfall.

Fig. 4.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0161.1

Another anomalous cyclone is seen over the Caribbean Sea near the Dominican Republic. From examination of the z anomalies, it is unclear whether this cyclone is related to the PEWs. It is clearly separated from the PEWs and exhibits a broader horizontal structure that does not tilt with longitude. A region of weak suppressed rainfall is also seen to the east of the Yucatan Peninsula in association with an anticyclone that dissipated over this region.

At lag day −1 (Fig. 4b), the anomalous anticyclone propagated slightly to the northwest. It has strengthened slightly, although the suppressed precipitation anomalies that are associated with it have weakened. An anomalous cyclone has developed to the east of the anticyclone nearly in phase with the enhanced rainfall. Suppressed rainfall anomalies have now appeared south of Guatemala, possibly in association to the suppressed rainfall anomalies that were previously east of the Yucatan Peninsula. The anomalous cyclone over the Caribbean has propagated westward, and has reduced its distance to the PEWs. It has also weakened substantially. To the east of the Caribbean cyclone, an anomalous anticyclone has appeared.

At lag day 0 (Fig. 4c), a new anticyclone is seen developing over the eastern Pacific south of Oaxaca. At this time the anomalous cyclone over the Caribbean has dissipated. Only a weak cyclonic flow is seen in association with it to the east of the Yucatan Peninsula as well as weak precipitation anomalies. However, the anticyclone to the west has amplified. At lag day 1, the weak cyclonic flow is centered over and to the South of the Yucatan Peninsula, and its features are reminiscent of a Central American Gyre (Papin et al. 2017). The regression maps at lag day 2 mirror those of lag day −1 with reversed polarity. The only notable difference is that circulation anomalies over the Caribbean are no longer evident.

The results of Figs. 4a–e suggest that westward-propagating anomalies exist both in the eastern Pacific and over the Caribbean. However, it is not clear from these maps whether the anomalies at the two locations are physically related. Examination of $〈 {P^{'}}_{ms} 〉$ (Figs. 4f–j) reveals what appears to be a single wave train of $〈 {P^{'}}_{ms} 〉$ that extends from 130° to 60°W. Anomalies in $〈 P_{ms} 〉$ develop over the eastern Caribbean and propagate westward. While the z anomalies substantially weaken as they approach the Yucatan Peninsula, the $〈 P_{ms} 〉$ anomalies do not. Instead, the $〈 P_{ms} 〉$ anomalies approach and cross central America over the course of two days. The emergence of the $〈 P_{ms} 〉$ anomalies over the eastern Pacific appears to be related to the development of the geopotential and precipitation anomalies in this region, although the latter are shifted to the southeast of the $〈 P_{ms} 〉$ anomalies.

Positive anomalies in CWV′ precede the development of the $〈 P_{ms} 〉$ anomalies and are initially associated with anomalous southerly winds (Fig. 4f). Once the $〈 P_{ms} 〉$ anomalies develop they closely follow the evolution of the CWV. The CWV anomalies exhibit a similar horizontal structure to the $〈 P_{ms} 〉$ anomalies over the Caribbean, although they are shifted to the east of them. Such a zonal shift is expected and is also seen in the moist shallow-water waves shown in Fig. 1. That the $〈 P_{ms} 〉$ anomalies resemble the CWV anomalies over the Caribbean, combined with the weak signature in geopotential height observed in that region suggests that the $〈 P_{ms} 〉$ anomalies are largely the result of anomalies of the moist component of the MSPV $({P^{'}}_{m})$ . Over the eastern Pacific, the $〈 P_{ms} 〉$ anomalies match the location of the CWV anomalies near the coast of Mexico, but not farther south. This is likely a result of the $〈 P_{ms} 〉$ anomalies being scaled by the mean ambient absolute vorticity, which becomes small near the equator.

The propagation of the MSPV anomalies is largely the result of the anomalous horizontal advection of MSPV ( ${〈 v \cdot \nabla_{h} P_{ms} 〉}^{'}$ ), as shown by the strong correspondence of the contours and shading in Figs. 4k–o. The shading is in phase with the contours at nearly all stages of the PEW life cycle, with only two exceptions. One exception is the time when the anomalies are developing over the eastern Caribbean near lag days −2 and 0. The other exception is during the decay stage of the wave over the eastern Pacific, when the contours have nearly vanished, indicating a weak to nonexistent MSPV tendency. The mismatch between the MSPV tendency and horizontal MSPV advection suggests that a nonnegligible amount of MSPV generation is occurring at this time. Nonetheless, the MSPV advection closely matches the MSPV tendency during the rest of the PEW life cycle, consistent with the scaling discussed in section 4.

Since horizontal MSPV advection is the dominant contributor to the propagation of the MSPV anomalies in PEWs, it is instructive to consider the processes that drive this advection. In Fig. 5, we show the three contributions to the anomalous horizontal MSPV advection. Dry PV advection (Figs. 5a,b) is largely in phase with the MSPV tendency throughout the domain. This phasing is robust at the times when both PC1 and PC2 become a maximum. The only time the correspondence is slightly weaker is during the initial stage of PEW development in the eastern Caribbean at the time when PC1 is a maximum, consistent with the overall weaker correspondence of total MSPV advection found in Fig. 4.

Fig. 5.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0161.1

Moist PV advection (Figs. 5a,b), on the other hand, is stronger to the south of the MSPV tendency, and does not align with the MSPV tendency, except over the far western portion of the domain at the time when PC1 is a maximum. Instead, it acts to slow down propagation to the east of 100°W. Last, advection of MSPV by the mean flow contributes to the propagation of the MSPV anomalies over the Caribbean Sea, but does not contribute significantly to MSPV propagation over the eastern Pacific.

We can summarize the results from Fig. 5 by projecting the individual advective terms in Eq. (31) to the

〈 {P^{'}}_{ms} 〉

tendency. This projection is obtained as in Clark et al. (2020) by calculating the integral of the advective term with

〈 \partial_{t} {P^{'}}_{ms} 〉

over a region with an area

A

. The result is then divided by the integral of the square of

〈 \partial_{t} {P^{'}}_{ms} 〉

Proj (X, 〈 \partial_{t} {P^{'}}_{ms} 〉) = \frac{\iint_{A} X \cdot 〈 \partial_{t} P_{ms}^{'} 〉 d A}{\iint_{A} 〈 \partial_{t} P_{ms}^{'} 〉 \cdot 〈 \partial_{t} P_{ms}^{'} 〉 d A},

(40)

where

A

is the region covering 5°–30°N, 130°–60°W, which encompasses the Caribbean Sea and eastern Pacific, and X is a term in Eq. (31). We perform this projection for both PC1 and PC2 of easterly wave activity, and average the result. The result of this calculation reveals that dry PV advection is indeed the largest contributor to MSPV propagation, explaining up ∼90% of the tendency (Fig. 6). Advection by the background winds also favors westward propagation, but its contribution is roughly 1/3 that of the dry PV advection. The overall contribution from advection of the moist component is small and acts against the westward propagation of the anomalies. A small residual in the budget is found, possibly a result of neglecting variations in the background state, or from nonlinear MSPV advection.

Fig. 6.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0161.1

To understand how the PEWs grow over the Caribbean and eastern Pacific region, Fig. 7 shows the contribution of $(1 - \bar{α}) C_{d}$ and $\bar{α} C_{m}$ to the evolution of A over the region. Because we are interested in wave activity convergence over the PEW life cycle, Fig. 7 is constructed as an average of the two PCs that represent PEW activity. Examination of Fig. 7 reveals that dry wave activity convergence $(1 - \bar{α}) C_{d}$ is dominant near 95°W. In this region the meridional shear in the climatological-mean zonal wind is a maximum. Hence the dry wave energy convergence is possibly the result of barotropic energy conversion, as suggested by previous studies (Rydbeck and Maloney 2014; Molinari and Vollaro 2000). Over the Caribbean, moist energy conversion $\bar{α} C_{m}$ explains nearly all of the wave energy convergence. Convergence of A by the mean flow is comparable to the contributions from $(1 - \bar{α}) C_{d}$ and $\bar{α} C_{m}$ , and are largest over the eastern Pacific. Together with $\bar{α} C_{m}$ , it explains nearly all of the wave energy convergence from 105° to 130°W. When the three processes are considered together, it is clear that the amplitude of the total wave activity convergence C_t is consistent with the amplitude of MSPV advection shown in Figs. 4k–o. Thus, while the wave energy convergence explains the amplification of the MSPV anomalies as they propagate across the Caribbean Sea and eastern Pacific, it does not explain their appearance over the eastern Caribbean Sea.

Fig. 7.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0161.1

7. Discussion and conclusions

Shallow-water models that include moist processes and prognostic moisture indicate that a quantity referred to as “gross potential vorticity” (GPV) can summarize the dynamics of moist waves (Adames and Ming 2018; Adames 2021). Guided by the shallow-water equations, we propose that the moist static potential vorticity (MSPV) may serve as an analog to GPV in a stratified atmosphere. The MSPV is similar to the equivalent PV, which has been used in studies of supercell thunderstorms (Rotunno and Klemp 1985), heavy precipitation bands (Clark et al. 2002), moist symmetric instability (Emanuel 1983; Martin et al. 1992) and in extratropical cyclones (Emanuel et al. 1987; Cao and Cho 1995). Furthermore, the budget equation for MSPV largely mirrors that of equivalent PV (Martin et al. 1992). Quantities similar to GPV and MSPV have also been derived in other idealized studies (Lapeyre and Held 2004; Monteiro and Sukhatme 2016). However, to the authors’ best knowledge, MSPV has not been extensively applied to the study of moist tropical motion systems.

A scale analysis of the MSPV budget was performed using scales that are typical for off-equatorial synoptic-scale tropical motion systems. The scaling indicates that horizontal advection of MSPV is the largest contributor to the MSPV tendency. This advection can be broken down into contributions from dry and moist processes, as in the shallow-water GPV. The relative contributions of these processes to the evolution of MSPV depends on α, the ratio between the vertical gradients of latent energy and dry static energy. In moist regions where the tropospheric-mean value of α is near unity, moist processes are expected to govern the evolution of MSPV, while dry processes are expected to play a minor role. In a dry troposphere (α → 0), the MSPV equation approximately becomes the traditional Ertel PV (referred to as the dry PV). The scale analysis indicates that other processes in the MSPV budget are substantially smaller than the horizontal advection. While these processes are neglected in this study, we acknowledge that they could be important when seeking a more complete understanding of moist systems.

The MSPV budget can also be used to obtain an equation for the evolution of wave activity. In a similar vein to the MSPV budget, the convergence of wave activity can be broken into dry and moist contributions, with α once again determining their relative contributions.

That α appears in the MSPV and equivalent wave activity budgets may be an indication of its physical importance to moist tropical motion systems. Previous studies have shown that α determines the sign and magnitude of diabatic processes in the moisture equation (Chikira 2014; Wolding and Maloney 2015; Wolding et al. 2016; Janiga and Zhang 2016) (see the second section of the appendix). In simple models, α is related to $1 - \tilde{M}$ , where $\tilde{M}$ is the normalized gross moist stability (section 2). Studies based on these models have shown that moisture governs the thermodynamics of tropical motion systems when $\tilde{M} \to 0$ (Adames et al. 2019; Ahmed et al. 2021). Other studies have found that a realistic value of α is needed to simulate a robust MJO (Chikira 2014). The vertical profile of α is also important in understanding the MJO’s response to climate change (Bui and Maloney 2019; Maloney et al. 2019). Our results indicate that a realistic value of α is also needed in order to properly simulate moist tropical motion systems in general. More work is needed in order to fully understand the role of α in the tropics, and its implication for MSPV and moisture evolution.

Application of the MSPV budget to PEWs shows that MSPV advection governs the evolution of MSPV, consistent with the scaling presented in section 4. It also showed that the MSPV anomalies originate over the eastern Caribbean Sea. Horizontal MSPV advection does not explain the MSPV tendency at this point, indicating that the scaling assumptions in section 4 break down at this time. We hypothesize that at this time, MSPV is generated by the baroclinic term B. Over this region, vertical wind shear in association with the Caribbean low-level jet (Amador 1998; Muñoz et al. 2008) can zonally advect the moisture anomalies that are seen at this time (Fig. 4f), generating negative MSPV. The enhanced moisture at this time is likely due to anomalous southerly flow advecting humid air from northern South America into the Caribbean.

Once developed, the MSPV anomalies continue to amplify as they propagate westward across the Caribbean. Examination of the wave activity equation suggests that amplification of the waves at this time is due to the flux of the moist component of MSPV ( $P_{m}^{'}$ ), which can be interpreted as a convergence of moist wave activity. Once the MSPV anomalies arrive at the eastern Pacific, moist wave activity conversion diminishes while dry wave energy convergence significantly amplifies. At this time, the larger geopotential height anomalies associated with PEWs is observed. We attribute the large dry wave energy convergence from 105° to 85°W as the result of barotropic energy conversions, consistent with the findings of previous studies (Molinari et al. 1997; Molinari and Vollaro 2000; Rydbeck and Maloney 2014). Once the MSPV anomalies propagate to the west of ∼100°W moist convergence becomes the dominant source of wave activity once again.

Our results suggest that the life cycle of PEWs is at least partly tied to waves that arrive to the eastern Pacific from the Caribbean, in agreement with previous studies (Molinari and Vollaro 2000; Serra et al. 2010). However, we do not reject the possibility that some PEWs are excited in the eastern Pacific, as was shown by Rydbeck et al. (2017). Our results also indicate that both dry and moist processes play a key role in these waves, and both need to be considered in order to gain a complete understanding of their behavior.

In addition to PEWs, the column MSPV budget can be used to gain new insights onto the dynamics of moist tropical motion systems, specifically in understanding the relative role of moist and dry processes in their evolution. Application of the MSPV budget to monsoon low pressure systems (Hurley and Boos 2015) could lead to new insights to the dynamics of these systems. Recent studies suggest that these systems grow from a combination of barotropic instability and processes involving moist convection, possibly involving moisture-vortex instability (AM18; Adames 2021; Diaz and Boos 2019, 2021). A combination of moist convection and dry processes may also explain the dynamics of African easterly waves (Berry and Thorncroft 2012; Janiga and Thorncroft 2013; Russell et al. 2020).

Analysis of the MSPV budget could also provide insights onto the processes that lead to tropical cyclone (TC) genesis. A recent study by Murthy and Boos (2018) showed that the radial gradient of surface latent heat fluxes was important for the formation of tropical depressions. By inspecting Eq. (12a), we can see that horizontal gradients in surface heat fluxes are included in the MSPV generation term G. Furthermore, studies by Raymond et al. (2007) and Tang (2017a,b), suggest that a horizontal gradient in the normalized gross moist stability, i.e., the normalized divergence of column MSE fluxes, is also important for TC genesis. We hypothesize that this contribution is implicit in the flux convergence of MSPV. Thus, the MSPV budget elucidates the physical processes that lead to TC genesis. However, we note that there is evidence that the MSPV in a TC evolves from being negative for weak TCs to being approximately zero in mature TCs (Peng et al. 2019), consistent with the notion of slantwise moist neutrality in these systems (Emanuel 1986). Future work should examine how MSPV evolves both during cyclogenesis and during intensification.

There are some caveats to the use of MSPV. First, MSPV is not conserved for moist adiabatic processes since it contains a baroclinic generation term (Schubert et al. 2001). Furthermore, it cannot be inverted without some additional information about the moisture field. For example, Smith and Stechmann (2017) show that a similar quantity to MSPV, referred to precipitating QGPV can be jointly inverted with a quantity M that includes additional information about the water vapor field. While PV definitions that use the virtual potential temperature may still be preferable when looking for an invertible quantity, we posit that the evolution of MSPV offers a more complete summary of the dynamics and thermodynamics of moist tropical motion systems. Nonetheless, we acknowledge that MSPV is a complex quantity whose behavior differs significantly from Ertel PV. It is unclear whether the benefit of using a variable that summarizes the dynamics of moist systems outweighs the potential hindrances that its complexity brings.

In conclusion, the column-averaged MSPV budget has the potential to further our understanding of tropical motion systems. The budget could be applied to systems that exhibit such characteristics such as African easterly waves, monsoon low pressure systems, tropical depressions, among others. It can also be used in GCM intercomparison studies to analyze how well models capture the relative contribution of dry and moist processes to the evolution of these systems. All of these have the potential to be fruitful directions for future research.

Acknowledgments.

ÁFA and HL were supported by the National Science Foundation’s Grant AGS-1841559. RVM was supported by National Science Foundation Graduate Research Fellowship Program under Grant DGE-1747503. Support was also provided by the Graduate School and the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin–Madison with funding from the Wisconsin Alumni Research Foundation. ÁFA would like to thank Kuniaki Inoue and Hannah Zanowski for conversations that helped in the interpretation of the MSPV budget.

APPENDIX

Additional Details on the Interpretation of the MSPV Budget

a. Alternate form of the MSPV budget

An interesting feature of Eq. (12a) is that all the terms on the rhs can be written in terms of the divergence of a forcing vector as in Haynes and McIntyre (1987):

\frac{\partial P_{ms}}{\partial t} = - \nabla \cdot J_{ms},

(A1)

where

J_{ms} = u P_{ms} + g (ω_{a} \dot{m} - m f \frac{\partial v_{g}}{\partial p} - m \nabla \times F_{r})

(A2)

is the MSPV forcing. The forcing J_ms consists of the MSPV flux, MSPV generation along vortex lines, MSE flux by the thermal wind, and frictional dissipation. In steady state, J_ms describes the circulation of MSPV.

b. Further discussion on α

The term α was originally described by Chikira (2014), who used it to understand the evolution of moisture in the MJO cycle. Through the application of the WTG approximation, the equation for moisture conservation can be written as

\frac{D_{h} L q}{D t} ≃ - (1 - α) Q_{c} + α Q_{r} - L \frac{\partial \bar{ω' q'}}{\partial p},

(A3)

where Q_c is the convective heating, Q_r is the radiative heating, and

\bar{ω' q'}

is the vertical eddy flux of moisture. Examination of the WTG moisture equation reveals that radiative heating becomes more effective at moistening the troposphere as α increases. Furthermore, increasing α reduces the net drying by latent heat release via compensating vertical moisture advection (Adames and Maloney 2021). When α > 1 latent heat release moistens the atmosphere instead of drying it. With all other processes held constant, a value of α near unity implies that convection is less efficient at drying the troposphere. This occurrence tends to be related with a near zero “gross moist stability” (Neelin and Held 1987; Raymond and Fuchs 2009; section 2), a condition that leads to the existence of moisture modes.

The physical relevance of α can be considered more generally by considering the criterion for potential instability. In height coordinates, a layer is potentially unstable if ∂_zθ_e < 0. In isobaric coordinates, a layer is potentially unstable if m_p > 0, where m_p = Lq_p − s_p. By rearranging the terms we can show that the moist static stability can be written as

m_{p} = s_{p} (α - 1) \cdot

(A4)

Examination of Eq. (A4) reveals that a layer is potentially unstable when α > 1 and it is stable when α < 1.

c. Scaled MSPV equation without linearization

The horizontal MSPV can be decomposed into dry and moist contributions even without linearization. Noting that

P_{ms} ≃ (1 - α) P_{d}

, and expanding the horizontal gradient in P_ms yields the following expression:

\frac{\partial P_{ms}}{\partial t} ≃ - (1 - α) v \cdot β_{d} - α v \cdot β_{m},

(A5)

where β_d and β_m are as in Eq. (30), but for the full q and s fields, not their background-mean components.

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Stechmann, S. N., and A. J. Majda, 2006:

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Moist Static Potential Vorticity Budget in Tropical Motion Systems

Abstract

Abstract

1. Introduction

2. Evolution of tropical motion systems in a linear shallow-water model with prognostic moisture

a. Review of the AM18 model

b. Wave activity evolution

3. Moist static potential vorticity as an analog of GPV

a. MSPV budget

b. Interpretation of MSPV

4. Scaling of the MSPV budget for off-equatorial synoptic-scale tropical motion systems

5. Linearized MSPV

a. Linearized MSPV and its relation to GPV

b. Anomalous MSPV budget

c. The moist static wave activity equation

6. MSPV evolution in east Pacific easterly waves

a. Data and methods

b. MSPV evolution in PEWs

7. Discussion and conclusions

Acknowledgments.

APPENDIX

Additional Details on the Interpretation of the MSPV Budget

a. Alternate form of the MSPV budget

b. Further discussion on α

c. Scaled MSPV equation without linearization

REFERENCES