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  • View in gallery

    Model configuration.

  • View in gallery

    The rate M that the momentum is transported from the upper layer to the lower layer over the scale LM by the archetypal mesoscale circulation: plotted as a function of the mesoscale pressure jump EM. The linear approximation around EM ∼ 0 is also represented by a dash line.

  • View in gallery

    The modification of the WISHE instability by mesoscale momentum transport. The growth rate (day−1: upper frame) and the phase velocity (m s−1: lower frame) are plotted against the global longitudinal wavenumber (k = 100 corresponds to a wavelength of 400 km). The solid curve shows the standard case (ub = 5 m s−1) without momentum transport. The long-dash and the short-dash curves show the cases with α̃M = 0.2 and 0.5, respectively.

  • View in gallery

    As in Fig. 3 but for α̃M = 1.5, 1.51, and 1.52 with the short-dash, the solid, and the long-dash curves, respectively. The effective unstable stratification is created by the mesoscale convective momentum transport with α̃M > 1.5.

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Impact of Mesoscale Momentum Transport on Large-Scale Tropical Dynamics: Linear Analysis of the Shallow-Water Analog

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  • 1 CRC-SHM, Monash University, Clayton, Victoria, Australia
  • | 2 National Center for Atmospheric Research,* Boulder, Colorado
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Abstract

The vertical transport of horizontal momentum by organized convection is a prominent process, yet its impact on the large-scale atmospheric circulation has not even been qualitatively assessed. In order to examine this problem in a simple framework the authors incorporate a nonlinear dynamical model of convective momentum flux into a linear model of the large-scale tropical atmosphere. This model has previously been used to investigate the WISHE (wind-induced surface-heat exchange) instability.

In order to implement the dynamically determined fluxes as a parameterization, a closure assumption is required to relate the relevant mesoscale parameters to the large-scale variables. The most straightforward method is to relate the low-level large-scale pressure (pL) to the mesoscale pressure perturbation (pM), which is linked to the mesoscale momentum flux by the dynamical model. The mesoscale momentum transport under this closure reduces the effective pressure gradient in the large-scale momentum equation and, consequently, the effective stratification. A sufficiently large pM may even cause an effectively unstable stratification (convective instability by mesoscale momentum transport), which is marginally realizable according to a scale analysis.

In general, the WISHE instability is suppressed by the mesoscale momentum flux under this closure because a larger effective stratification can provide a more efficient mass redistribution and, in turn, a larger potential energy for WISHE. This demonstrates that momentum transport by mesoscale convective systems can substantially modify the large-scale tropical dynamics through the WISHE mechanism.

Corresponding author address: Dr. Jun-Ichi Yano, CRC-SHM, Monash University, 3rd floor, Building 70, Wellington Rd., Clayton, Victoria 3168, Australia.

Email: jiy@vortex.shm.monash.edu.au

Abstract

The vertical transport of horizontal momentum by organized convection is a prominent process, yet its impact on the large-scale atmospheric circulation has not even been qualitatively assessed. In order to examine this problem in a simple framework the authors incorporate a nonlinear dynamical model of convective momentum flux into a linear model of the large-scale tropical atmosphere. This model has previously been used to investigate the WISHE (wind-induced surface-heat exchange) instability.

In order to implement the dynamically determined fluxes as a parameterization, a closure assumption is required to relate the relevant mesoscale parameters to the large-scale variables. The most straightforward method is to relate the low-level large-scale pressure (pL) to the mesoscale pressure perturbation (pM), which is linked to the mesoscale momentum flux by the dynamical model. The mesoscale momentum transport under this closure reduces the effective pressure gradient in the large-scale momentum equation and, consequently, the effective stratification. A sufficiently large pM may even cause an effectively unstable stratification (convective instability by mesoscale momentum transport), which is marginally realizable according to a scale analysis.

In general, the WISHE instability is suppressed by the mesoscale momentum flux under this closure because a larger effective stratification can provide a more efficient mass redistribution and, in turn, a larger potential energy for WISHE. This demonstrates that momentum transport by mesoscale convective systems can substantially modify the large-scale tropical dynamics through the WISHE mechanism.

Corresponding author address: Dr. Jun-Ichi Yano, CRC-SHM, Monash University, 3rd floor, Building 70, Wellington Rd., Clayton, Victoria 3168, Australia.

Email: jiy@vortex.shm.monash.edu.au

1. Introduction

Although organized convection on scales of tens to hundreds of kilometers (mesoscales) is a well-recognized atmospheric process, the impact of the accompanying momentum transport on the large scales of motion is poorly understood and has not been much studied. The purpose of this paper is to investigate, at a fundamental level, the large-scale response of the tropical atmosphere to momentum transport by the most highly organized regime of deep convection, namely, squall lines. These systems are quite common in certain regions or, in more basic terms, in certain types of environmental shear. We recognize that they have an impact on quantities other than the momentum field because they transport large quantities of mass. For example, mass transport strongly affects the mean thermodynamic and moisture profiles (Lafore et al. 1988). Note that the redistribution and generation of upper-tropospheric moisture through the cirrus and stratiform outflow in highly organized convective systems strongly affects the radiative fluxes, which is important to climate. However, we choose to focus on the aspects pertaining to the organized momentum flux for the reasons stated above.

A series of field experiments and accompanying studies reviewed, for example, in Houze and Betts (1981), Houze (1989), Rutledge (1991), and Moncrieff (1995) has shown that mesoscale convective systems are commonly organized into squall lines in both Tropics and midlatitudes. The most subtle effect of these systems on the large scale is arguably through the vertical transport of the momentum. Unlike the effect of cumulus, in which only the ensemble effects of thermodynamic processes on the large-scale dynamics have been traditionally considered, the organized flow associated with squall line convection causes a distinctive momentum transport. This dynamical aspect of mesoscale organization is a process not yet accounted for in globalmodels. Various basic aspects of the momentum transport problem are presented in Moncrieff (1997), including the various kinds of momentum transport and a dynamical basis for the concept of organized convection.

Regimes of organized convection and the accompanying momentum fluxes were derived analytically by Moncrieff (1981) using nonlinear idealized models. Motivated to reduce the squall line regime to first principles for purposes of parameterization, Moncrieff (1992, hereafter M92) derived a steady-state dynamical model that was a paradigm of the observationally based conceptual model of Houze et al. (1989). This model, which contains no explicit heating, is the simplest possible (archetypal) realization of the mass and momentum fluxes by squall lines.

The archetypal model is characterized by three quantities (cf. Fig. 1 of M92), namely, the normalized pressure change (or jump) across the convective system (EM), the inflow depth of the front-to-rear flow (h0), and the mesoscale downdraft depth (h). The conservation of Bernoulli energy, mass, and other properties, as well as continuity of pressure provide a one-to-one relationship among these three parameters. If one parameter is specified, then both the momentum flux and the vertical mass flux can be obtained in a consistent manner.

Liu and Moncrieff (1996) showed that neither the ambient stratification nor the latent heat release has much effect on the archetypal momentum fluxes. In addition, LeMone and Moncrieff (1994) evaluated these fluxes against observations and Wu and Moncrieff (1996) carried out comparison against numerical model results. These studies demonstrated that the archetypal model is a plausible paradigm for the momentum flux, not only by squall lines but also by similar kinds of flow-perpendicular systems such as cold-frontal rainbands and gust fronts of smaller scale.

With the validity of the model reasonably well established, we take the next step to incorporate the flux theory into a momentum parameterization for the large-scale equations. This warrants special attention because organized deep convection can occur on a scale comparable to the grid scale of global models. This creates the odd situation that organized convective systems are neither fully resolved nor sufficiently small to be adequately parameterized as a subgrid-scale process. For example, a supercluster (which can be thought of as a giant mesoscale convective system) that occurred during the December 1992 westerly wind burst during TOGA COARE was analyzed by Moncrieff and Klinker (1997). They identified a new uncertainty in a high-resolution, state-of-the-art weather prediction model having a grid length of about 90 km. This uncertainty sprung from a surrogate treatment of a supercluster as a single physical entity rather than an ensemble of mesoscale convective systems and convection that it really represents. This surrogate process caused errors in the convective momentum flux that directly affected the large-scale flow.

The motivation for this inaugural study of the large-scale impact of momentum flux is to seek a simple physical interpretation; therefore we adopt an idealized approach rather than implement the parameterization into a general circulation model (GCM). Specifically, we use the shallow-water model of Yano and Emanuel (1991, hereafter YE) as an idealization of the large-scale tropical atmosphere. The model has a standard bulk surface flux and bulk mass flux formulation for thermodynamics and a closure based on a convective quasi-equilibrium assumption. The modification of the wind-induced surface heat exchange (WISHE) instability for the large-scale tropical circulation by convective downdrafts was analyzed by YE using this model. WISHE was proposed by Emanuel (1987) and Neelin et al. (1987) as a theory for Madden–Julian waves, as an alternative to wave-CISK (cf. Emanuel et al. 1994). The model was also used to investigate other aspects of tropical dynamics (Yano et al. 1995, 1996). It was recently used to compare the impact of several parameterization categories (Yano et al. 1997) on the tropical atmosphere.

We extend the original shallow-water formulation of the model to a two-layer framework. This enables us to explicitly investigate the transport of horizontal momentum by mesoscale convection from the lower to the upper level (i.e., between lower and upper troposphere). Note that the momentum flux has a strong bimodal signature characterized by accelerations of opposing sign in the low and upper troposphere. We specifically enquire, how is the WISHE instability modified by mesoscale momentum transport and how does this affect the large-scale tropical dynamics? We address this question using a linear stability analysis that incorporates Moncrieff’s archetypal model of the mesoscale momentum transport.

To close the parameterization, the archetypal mesoscale momentum transport must be somehow linked to the large-scale (predicted) variables. In evaluating the archetypal model from observations, LeMone and Moncrieff (1994) chose the inflow depth h0 as a reference parameter in order to compare the archetypal momentum transport with observational data. In our study, we choose the mesoscale pressure jump EM as a simple closure. Because EM is functionally related (through the archetypal model) to the mesoscale change of pressure across the mesoscale system, it is reasonable to assume that it is continuous with the pressure change due to the large-scale wave response (δϕ). This closure will be presented in section 3.

As an illustration of a relevant large-scale setting we refer to the studies of the global distribution and frequency of organized mesoscale convective systems by Laing and Fritsch (1997). Their results illustrate that, rather than being uniformly distributed, large organized precipitating systems tend to be concentrated in specific regions and occur during specific periods, for example, during the Indian monsoon, in West Africa, and in Venezuela. We stress that these regions are characterizedby jetlike wind profiles and moderate-to-strong low-tropospheric shear. Jetlike profiles are known to be associated with squall line cloud systems. For example, Lafore and Moncrieff (1989) conducted a systematic numerical study of the effect of jetlike profiles on West African squall lines. Moreover, studies during TOGA COARE indicate that large-scale circulations such as Madden–Julian waves (Madden and Julian 1971) contain an ensemble of organized mesoscale systems, a behavior seen during westerly wind bursts in the previously cited Moncrieff and Klinker (1997) analysis (see their Figs. 1 and 2). Taking the TOGA COARE situation as an example, we assume a westerly mean wind in the linear stability analysis. The results can be simply transformed to apply to an easterly mean state.

In the following section we describe the basic formulation of the large-scale model. This is followed by a summary of the mesoscale momentum flux parameterization in section 3. Section 4 is devoted the shallow-water limit of the two-layer system. Finally, conclusions are summarized in section 5.

2. Basic formulation

a. Two-layer system

We use the idealized large-scale tropical model formulation of Yano and Emanuel (1991) extended to a two-layer system. In order to simplify the mathematical analysis we ignore planetary rotation and thereby retain only the horizontally one-dimensional Kelvin-like modes (Yano and Emanuel 1991); that is, a Rossby wave response cannot occur. We use two distinct types of parameterization. First, the mass flux formulation introduced by YE primarily represents the thermodynamic effect of cumulus convective scales. Second, we incorporate Moncrieff’s archetypal model of mesoscale momentum transport. For simplicity, neither the momentum transport by cumulus nor the thermodynamic effects of mesoscale convection are considered. Although thelatter effect is potentially important, its investigation is beyond the scope of this paper.

The model consists of two layers each of depth H/2, which corresponds to about 4 km on a density-weighted height scale (Fig. 1). The bottom of the lower layer contains the subcloud layer of depth hb = 500 m. We assume a vertically homogeneous potential temperature so that θ = θ(x, y). An equivalent potential temperature (θe) is assigned to the subcloud layer and another to the middle level of the troposphere. We identify variables in these layers by the subscripts b and m, respectively.

The vertical discretization of the model is obtained by vertically integrating the hydrostatic equation ∂(δϕ)/∂p = −δα from the top of the subcloud layer to an arbitrary level. The variables ϕ and α are the geopotential and the specific volume, respectively, and δ designates a deviation from the mean value on a constant pressure surface. In integrating the right-hand side, we relate the specific volume α to the potential temperature θ by δα = (∂α/θ*e)pδθ*e, (∂α/θ*e)p=Cp/θ0(∂T/p)θ*e, and δθ*e = γδθ, where θ*e is the saturated equivalent potential temperature, Cp is the specific heat at constant pressure, θ0 = 300 K is the reference potential temperature (a surface value), and γ = Γdm is the ratio of the dry adiabat to the moist adiabat. (See appendix A for definition of symbols.)

We define T and Tb as the temperature at an arbitrary level and the top of the subcloud layer, respectively. We then obtain
i1520-0469-55-6-1038-e2-1
Averaging of (2.1) over the model troposphere gives
i1520-0469-55-6-1038-e2-2a
where
i1520-0469-55-6-1038-e2-2b
and the subscript t designates a tropospheric mean. The potential temperature anomaly will generate a baroclinic mode of the same magnitude but of opposite sign in these layers. The mean geopotential δϕt contributes solely to the barotropic mode. The midlevel geopotential
δϕmϵkδθδϕt
is obtained from (2.1) by assuming Tt = (Tb + Tm)/2.
The potential temperature fluctuation is obtained from the thermodynamic equation
i1520-0469-55-6-1038-e2-3
where
i1520-0469-55-6-1038-e2-4
represents the Lagrangian time derivative advected bya mean tropospheric wind vt ≡ (vb + vm)/2, where v is a horizontal vector. The first term on the right-hand side of (2.3) represents adiabatic heating due to the vertical motion w, where N = 10−2 s−1 is the Brunt–Väisälä frequency and g = 9.8 m s−2 the acceleration of gravity. We assume that the parameterized heating (conv) by cumulus convection is exactly balanced by adiabatic cooling, in accordance with a standard mass flux formulations (cf. Yanai and Johnson 1993). We can therefore write
i1520-0469-55-6-1038-e2-5a
where wc is the total cumulus updraft given as a grid domain average.1 The radiative cooling is represented using a Newtonian approximation,
i1520-0469-55-6-1038-e2-5b
with R0 = 1 K day−1 and τR = 50 days.
We assume a Klemp and Durran (1983) gravity wave radiation condition at the free upper boundary. Note that, since rotational modes are omitted, the barotropic mode cannot exist without a free surface. It follows that
i1520-0469-55-6-1038-e2-6
where wt is the vertical velocity at the tropopause, k is the horizontal wavenumber, and δϕm is defined by (2.2c).
Mass continuity follows from the incompressible approximation integrated from the bottom to top of each layer:
i1520-0469-55-6-1038-e2-7a
For simplicity we have assumed that the wind speed is uniform in each layer (vb and vm, respectively). The equation of motion for each layer is given by
i1520-0469-55-6-1038-e2-8a
where the surface drag coefficient CD = 1 × 10−3 and the Lagrangian time derivatives are
i1520-0469-55-6-1038-e2-9a
Assuming that the momentum fluxes are zero at top and bottom, the redistribution of horizontal momentum (QM) by the mesoscale circulation has the same mean value, but the opposite sign, in each layer [see Moncrieff 1992, Eq. (15)]. Quantification of the large-scale effects of this redistribution of momentum is our primary aim. The detailed expression for QM is derived in section 3.
Moist thermodynamics are described by
i1520-0469-55-6-1038-e2-10a
with the surface heat flux E and the downdraft D given by bulk formulas
i1520-0469-55-6-1038-e2-11a
where Cθ = 1.2 × 10−3 is the evaporation rate and θ*eb the saturated potential temperature of the subcloud layer. The convective downdraft
wdϵpwcϵp
is assumed to be consistent with a constant “precipitation efficiency” ϵp = 0.9. The environmental subsidence is
wewwc
Note that wd and we are grid averages, like wc (see fn 1).

The precipitation efficiency ϵp measures the rate at which total water reaches the surface, by which we mean water generated by both cumulus-scale and mesoscale processes. Since the highest precipitation rate is due to the cumulus-scale processes, the precipitation efficiency is taken to be ϵp = 0.9 by weighting toward convective precipitation. As a result, 1 − ϵp can be considered as an evaporation rate of total precipitation. Roughly the same factor with the negative sign applied to the convective buoyancy gives the negative buoyancy that drives the downdrafts and gives Eq. (2.11c). It should be emphasized that mesoscale downdrafts can be also accounted for by this term. The dependence of the WISHE instability on ϵp has already been investigated by Yano and Emanuel (1991, see their Fig. 2).

b. Nondimensionalization

The M92 organized momentum transport is formulated in terms of nondimensional quantities. Velocity, length, and time are scaled by U0, H, and H/U0, respectively. We set U0 = 10 m s−1, defined as the strength of the relative inflow to a typical mesoscale organized system (see section 3). We use the same scaling as in M92 and, for convenience, omit δ from the dimensional variables. This gives δϕ = U20ϕ and δθ = (U20/ϵk)θ with a corresponding nondimensionalization for the equivalent potential temperature.

The diagnostic equations (2.11c,d) are unaltered. We obtain the nondimensional set of prognostic equations from (2.8a,b), (2.3), (2.10a,b),
i1520-0469-55-6-1038-e2-12a
and the following diagnostic equations from (2.2a,c), (2.7a,b), (2.6),
i1520-0469-55-6-1038-e2-13a
Using (2.5a,b), (2.11a,b), the source terms on the right-hand side of (2.12) are
i1520-0469-55-6-1038-e2-14a
We have introduced the following nondimensional parameters:
i1520-0469-55-6-1038-e2-15a

c. Linearization

In the following analysis, () designates base-state values and (′) the deviation from this state. For convenience, the primes are omitted when there is no likelihood of ambiguity. Unless otherwise stated, all variables are either a mean value or a deviation from the mean. We assume a homogeneous positive (westerly) wind u as a mean basic state. The westerly mean state is chosen bearing in mind a westerly wind burst environment in which organized mesoscale systems, whose momentum flux we seek to parameterize, commonly occur. Note that results for the easterly mean wind case can be immediately obtained by reversing the phase velocity and by changing the sign of the mesoscale phase velocity cM (e.g., making the results applicable to West African squall lines).

We also assume the large-scale mean vertical motion is zero (w = 0), so the cumulus updraft is exactly compensated by environmental subsidence:
wcwe
Note that although cumulus updrafts are always more concentrated and vigorous than environmental subsidence, the grid-averaged effect should balance if there is no direct interaction among neighboring grid volumes (as is the case in all GCMs). Due to the assumption that radiative cooling balances the convective heating, we have
wcR0N̂.
Furthermore, the balance between the environmental descent and the surface flux in the subcloud layer, and radiative cooling in the middle troposphere lead to
i1520-0469-55-6-1038-e2-16c
We superimpose a small amplitude disturbance on the mean state defined by (2.16a,b,c,d). Using (2.12), the linear equations for the deviations are then
i1520-0469-55-6-1038-e2-17a
i1520-0469-55-6-1038-e2-17d
where
i1520-0469-55-6-1038-e2-18a
We assume that the horizontal coordinate x and the wind component u are positive in the eastward direction. We retain the diagnostic equations (2.13). The term proportional to αθ in the right-hand side of (2.17d) represents the surface heat flux proportional to the surface wind ub, namely, the WISHE process.
It is convenient to express the momentum equations (2.17a, b) in the terms of barotropic ut and baroclinic uc components:
i1520-0469-55-6-1038-e2-19a
where
i1520-0469-55-6-1038-e2-20a
and
i1520-0469-55-6-1038-e2-21
By definition, uc > 0 means a negative value of the vertical shear (i.e., easterly shear). Because of the integral constraint on momentum generation, the mesoscale momentum tendency QM appears in the baroclinic component but not in the barotropic component.

We make additional approximations to facilitate analytic tractability. Considering the smallness of the parameters, we can legitimately approximate 1/τ̂Rαdαe → 0. We further assume that the subcloud layer is much shallower than the troposphere, namely, δ → 0. In physical terms the first set of assumptions removes certain linear damping terms, whereas the second assumption means that the subcloud layer instantaneously adjusts to a change induced by the tropospheric dynamics. These limits are identical to those in section 3b(2) of YE. The second assumption was called the “boundary-layer quasi-equilibrium” by Raymond (1995) and“subcloud-layer entropy equilibrium” by Emanuel (1995).

Hence, (2.17d) reduces to αθubλ(wdwe) = 0, which means that the wind-induced surface flux is always balanced by downdraft cooling. Downdraft cooling is further related to the cumulus updraft wc by (2.11c,d). With this approximation, the surface flux uniquely defines the cumulus updraft wc. After algebraic manipulation employing (2.13c) we obtain
i1520-0469-55-6-1038-e2-22
The cumulus updraft (wc) is therefore proportional to the surface wind (ub). The component of wc in phase with ub, represented by the first term in the right-hand side, is the basic mechanism for the WISHE instability. We can remove WISHE by setting αθ = 0. Note that we do not have to solve the midlevel moisture equation (2.17e) in this limit.

3. Mesoscale momentum flux parameterization

We use the archetypal momentum flux representation of M92, which is an expression for the total flux (i.e., mean + deviation). A decomposition of the total flux into a mean and a deviation is not made unless otherwise stated. All quantities are presented in terms of the nondimensionalization introduced in section 2b. We identify the mesoscale variables and parameters by the subscript M but otherwise adopt the notation of M92. This model is a paradigm for mesoscale convective system dynamics and provides the momentum transport on a characteristic scale LM. We use it as a grid-scale to subgrid-scale model of organized momentum flux. Following M92, the model is asymptotically linked to the large-scale (resolved scale) circulation over the distance LM. Although the theoretical considerations upon which the model is built does not provide an a priori estimate forLM, this distance is about 100 km based on observations and numerical model results.

The mesoscale momentum flux divergence is given by M92 in terms of
i1520-0469-55-6-1038-e3-1
where 〈*〉 designates the total ensemble average over LM. In a steady nonrotating system, provided the momentum flux vanishes at top and bottom of the convective layer, horizontal momentum can be redistributed in the vertical, but mean horizontal motion cannot be generated because the vertical integral momentum flux divergence is identically zero (we call this the integral constraint on the momentum flux).

a. Vertically continuous model

In implementing the archetypal model into the large-scale momentum equation, we assume that a typical mesoscale system occupies a fraction (αM) of a grid domain. Hence, in a vertically continuous system, the mesoscale momentum transport would be implemented as
i1520-0469-55-6-1038-e3-2a
where the acceleration due to the momentum flux divergence is
i1520-0469-55-6-1038-e3-2b
By substitution of (3.1) into (3.2b), we obtain
i1520-0469-55-6-1038-eq1
In this definition, x is positive in the direction of propagation of the mesoscale system in a coordinate frame moving along with the mean wind (or equivalently, along a Galilean-transformed coordinate). More generally, assuming that the x coordinate is aligned along the direction of propagation of the disturbance, we can take the sign of the phase velocity cM into account as
i1520-0469-55-6-1038-e3-3
When the inflow strength (ui), scaled by U0, departs from unity, (3.3) should, strictly speaking, be multiplied by ui. Apart from an explicit scale analysis in section 4, we allow for this in a simple way through the mesoscale fractional area αM,

b. Two-layer model

The corresponding expression of QM for the two-layer system in (2.17a, b) and (2.19b) is obtained by integrating (3.3) from the bottom to the middle of the atmospheric column to give
i1520-0469-55-6-1038-e3-4a
We have introduced the normalized mesoscale momentum flux divergence
i1520-0469-55-6-1038-e3-4b
The expression for ϒ is obtained from Eq. (16) of M92 and substituted into (3.4b). The resulting acceleration, M, is a function of the nondimensional pressure jump EM = 2αMΔpM/U20, where ΔpM is a dimensional change of pressure across LM. The obtained formula (see appendix B) is plotted in Fig. 2. The flux divergence increases and the mesoscale downdraft is stronger for smaller, negative values of EM (Fig. 2 of M92). In physical terms this corresponds to a propagating system; that is, a system that propagates relative to the mean wind at all levels.
As a closure we assume that the mesoscale pressure gradient EM/LM is correlated with the large-scale gradient ∂δϕ/∂x by a factor ϵM, giving
i1520-0469-55-6-1038-e3-5
Note that the sign of EM is defined as the change of pressure measured from front-to-rear across the system (see M92). The rationale for the closure is that the archetype is expected to be (asymptotically) continuous with the large-scale pressure field over the distance LM.When the mesoscale pressure efficiency parameter ϵM is unity, the mesoscale pressure pM perturbation is continuous with its large-scale counterpart so we expect that ϵM ∼ 1. A more formal estimate is given in section 4.
It is convenient to linearize M with respect to EM, by referring to appendix B:
i1520-0469-55-6-1038-eq2
The M92 solution has two physically distinct modes of behavior called symmetric and asymmetric. The cross-system change of pressure is identically zero in the symmetric mode, in contrast with the nonzero pressure change in the asymmetric mode. Nevertheless, the symmetric mode is nontrivial because it overturns the fluid (i.e., a nonzero mass transport and momentum flux). In other words, M ≠ 0 even if EM = 0. The asymmetric mode becomes identical to the symmetric mode for the special case of EM = 0, which also has a nondegenerate momentum flux. By substitution into (3.4a) and using (3.5),
i1520-0469-55-6-1038-eq3
We now divide QM into mean and deviation parts, namely
i1520-0469-55-6-1038-e3-6a
In physical terms, once a large-scale disturbance is created (in our framework as a Kelvin-like wave disturbance but in general terms by a variety of mechanisms), we envisage that mesoscale organized systems occur and move at a characteristic phase speed cM. For example, organized mesoscale convective systems occur within superclusters in the tropical Western Pacific (Moncrieff and Klinker 1997; Figs. 1 and 2) and squall lines within easterly waves in West Africa. Equation (3.6a) essentially represents a homogeneous feedback of the mesoscale organization to the large-scale flow. In the following linear perturbation analysis, the term (3.6b) will be incorporated into (2.17a,b) and (2.19b).

4. Shallow-water limit

The two-layer system introduced in section 2 contains two limiting cases based on the upper-level stratification Ŝ. First, when Ŝ → 0, in which case ϕm = 0 follows from (2.13e), there is no pressure disturbance and no flow in the upper layer (i.e., um = 0). This corresponds to a strong wave absorption at the top of the atmosphere, which we call the sponge-layer limit. A more useful one is Ŝ → ∞, which is the shallow-water limit used by YE.We have wt, = 0 from (2.13e), which further implies ut = 0 from (2.13c,d). The perturbation is completely baroclinic provided there is no mean vertical shear (uc = 0). We now consider this limit in detail by setting ub = uc, ub = ut.

We assume a wave solution of the form exp[ik(xct)], where c = cr + ici is a complex phase velocity consisting of the real phase velocity cr and the imaginary part ci, which gives the growth rate σ = kci. We obtain
i1520-0469-55-6-1038-e4-1a
The complex stratification
i1520-0469-55-6-1038-e4-2
is an expression for the thermodynamic modification due to a combination of the surface heat flux and moist convection (i.e., WISHE effect) defined by using (2.22). The nonzero imaginary part means that the positive temperature anomaly slightly lags the positive wind anomaly when the WISHE parameter αθ is nonzero.
We now consider the effect of the mesoscale momentum transport under the pressure closure (3.6b). In the shallow-water limit, the mesoscale pressure perturbation EM/2 is assumed to be proportional to the potential temperature by setting δϕ′ = −θ in (3.6b). By substitution, the sum of the pressure gradient and the mesoscale momentum source terms gives
i1520-0469-55-6-1038-e4-3
where the effective mesoscale fractional area α̃MϵMαM has been introduced. Equation (4.3) means that the effective pressure force changes from θ to (1 − 2α̃M/3)θ through the action of the mesoscale momentum flux divergence. This reduces the effective stratification from to (1 − 2α̃M/3) and also reduces the complex stratification Ñ* defined by (4.2). The reduction of the effective stratification stems from the fact that a stronger mesoscale pressure change from front-to-rear of the mesoscale system (i.e., EM < 0) accelerates the low-level flow in the direction of propagation of the mesoscale organization [cf. (3.6b)], aided by a stronger mesoscale downdraft (cf. Fig. 2 of M92). This counteracts the large-scale pressure force, which then decelerates the low-level flow in the direction of propagation.
By taking the determinant of (4.1a,b) and substituting (4.3) we obtain the complex dispersion equation
i1520-0469-55-6-1038-e4-4

A positive growth rate occurs when the square root is of sign opposite to the mean flow. An unstable WISHE disturbance therefore propagates upwind (relative to thesurface wind) because the stronger wind enhances the surface flux and increases the generation of available potential energy for convection. Therefore both low-level convergence and convection shifts in this direction. This upwind propagation mechanism is illustrated in Fig. 1 of Emanuel (1987) and is a characteristic signature of WISHE.

We now consider the zero surface friction limit → 0 (i.e., ĈD → 0) in order to simplify the expression. As shown by YE, both the growth rate and the phase velocity asymptotically approach constant values in the high wavenumber limit (cf. Fig. 3 below). This is basically due to the convective downdraft. The effect of the momentum transport on the WISHE instability is readily seen in this limit (k → ∞), in which case,
i1520-0469-55-6-1038-e4-5a
The growth rate σ is proportional to the WISHE parameter αθ. The reduction of the effective stratification by the pressure-driven mesoscale momentum transportresults in a decreased growth rate and reduced phase velocity of WISHE, provided 1 > 2α̃M/3. The reduction of the phase velocity is explained in terms of a slower-moving internal gravity wave associated with a weaker (effective) stratification. In order to understand the reduction of the growth rate, it needs to be appreciated that the WISHE instability is produced through a redistribution of tropospheric mass, as expressed by the thermodynamic equation (4.1b). The adjustment of the atmosphere toward convective neutrality, responding to the wind-induced subcloud-layer moisture anomaly, gives rise to such an effect [Eq. (2.22)]. Hence, the WISHE instability is amplified by a stronger effective density stratification (1 − 2α̃/3) and vice versa.2

A more fundamental change occurs if the threshold α̃M = 3/2 is exceeded; that is, if the effective stratification (1 − 2α̃M/3) of the atmosphere changes sign, leading to an unconditional thermodynamic instability. Physically, this will occur when the magnitude of the pressure-driven momentum transport exceeds the direct pressure forcing. This occurs if the mesoscale downdraft is sufficiently strong. Note that the propagating organized system commonly has a strong downdraft (see M92, Fig. 2). The degree to which this can be realized can be assessed by the following scale analysis. Recall that the effective mesoscale fractional area is defined by α̃M = ϵMαMui, with explicit reference to the inflow strength ui. The mesoscale dynamical efficiency ϵM is estimated as ϵM ≡ |ΔpM|/|δpL| ∼ L/LMpM/δpL), where L is the scale of the large-scale disturbance and δpL is the large-scale pressure anomaly. In terms of the mesoscale fractional area αM, we need separate considerations for the large-scale and the mesoscale limits.

In the large-scale limit where LLM, we expect that αM is independent of L. We further assume in this limit that ΔpMδpL and ui ∼ 1. Hence, α̃M ∼ (L/LM)αM > 1 is required to achieve an effective unstable stratification. We estimate L > LM/αM ∼ 104 km if LM ∼ 100 km and αM ∼ 10−2. This implies that the unstable stratification may be realized through the action of a planetary-scale disturbance containing an ensemble of mesoscale systems. This is a plausible interaction of organized convection with the Madden–Julian oscillation.

On the other hand, in the small-scale limit we expect that the fractional area occupied by the mesoscale convection scales as αM ∼ (LM/L)2 by assuming that a single mesoscale system occupies an entire grid domain. By substitution, we obtain α̃M ∼ (LM/L)ui ΔpMpL. Therefore, α̃M is larger at small scales in this limit. When we take the mesoscale limit LLM, we identify two scenarios for realizing α̃M > 1; that is, either ΔpM/δpL > 1 or ui > 1. Consequently, the mesoscale momentum transport can self-induce an effective unstable stratification if there is a sufficiently large pressure disturbance and/or a strong relative inflow. We refer to this instability as the effective convective instability by mesoscale momentum transport (CIMMT).

We plot the direct numerical results from (4.4) in Figs. 3 and 4 without the limit of F → 0. For simplicity, we do not consider the scale dependence of α̃M. Below the threshold α̃M < 3/2 (Fig. 3), both the growth rate and the absolute phase velocity decrease as α̃M increases. This is to be expected from (4.5a,b): the cases with α̃M = 0, 0.2, and 0.5 are shown by the solid, the long-dash, and the short-dash curves. At the threshold α̃M = 3/2 (short-dash curve in Fig. 4), the disturbance is neutral and purely advected by the zonal wind. Above the threshold α̃M > 3/2, the disturbance is again destabilized and grows faster at smaller scales, which is typical of convective instabilities. The growth rate is sensitive to α̃M. Only a slightly higher value than the threshold (α̃M = 1.51 and 1.52 by the solid and long-dash curves in Fig. 4, respectively) provides a much larger growth than the WISHE mode, ostensibly due to a fast timescale represented by the Väisälä frequency N. Note that above this threshold the disturbance propagates downwind in contrast to the WISHE instability.

The faster growth rate for the smaller scale under CIMMT is compatible with a scale analysis, which suggests that a mesoscale convective system can be self-induced by this instability. It is emphasized that the fastest growth rate in the smallest scale in this scenario does not contradict the scale separation principle. Note that Moncrieff’s archetypal model does not strictly rely on this principle per se, because it predicts the total momentum flux, not the eddy perturbations which follow from the Reynolds averaging approach. In other words, the archetypal model represents a coherent structure rather than an ensemble of quasi-random fluctuations.

5. Conclusions

The impact of mesoscale momentum transport on the dynamics of the Tropics was assessed through a linear analysis of the analogue shallow-water atmospheric model of Yano and Emanuel (1991). In order to simplify the analysis, the model contained only a Kelvin-type wave mode. Furthermore, the subcloud layer is assumed to adjust instantaneously to tropospheric changes, which closes the mass flux representation for the thermodynamic effects of ordinary cumulus convection.

The mesoscale momentum flux, as formulated by the Moncrieff (1992) archetypal dynamical model that approximates squall-line-like cloud systems, was incorporated in an idealized large-scale model of the tropical atmosphere. For the closure, we assumed that the mesoscale momentum flux is proportional to the large-scale pressure wave perturbation (consistent with the archetypal formulation). The action of mesoscale momentum transports was found to reduce both the growth rate and phase velocity of the WISHE mode.

A reduced phase speed for large-scale convectively driven disturbances in the tropical atmosphere is an attractive feature. Most linear theories, including those based on WISHE (Yano et al. 1995), produce a much faster propagation than is realistic for cloud clusters, superclusters, and Madden–Julian waves. From a linear analysis and numerical experiments, respectively, Wang and Rui (1990) and Salby et al. (1994) proposed that surface friction reduces the phase speed of the large-scale disturbances. Although our result indirectly invokes surface fluctuations, the reduction of phase speed is due primarily to convective momentum transport. This implies that the propagation of superclusters may be partly controlled by the intensity of organized mesoscale convection within them.

Furthermore, it is inferred that the large-scale effective stratification can be absolutely destabilized, provided the mesoscale momentum transport is sufficiently strong (the effective convective instability by mesoscale momentum transport, CIMMT). A scale analysis suggests that this instability may be realized not only at the mesoscale limit (if the perturbations are sufficiently intense) but also at larger scales (provided the large-scaledisturbances contain a sufficiently large population of mesoscale convective systems). Thus its impact on Madden–Julian waves warrants further study.

Based on these results, momentum transport by organized convection has an identifiable effect on large-scale tropical dynamics and the responsible mechanisms have been quantified. While more realistic models are certainly required, our conclusions point to a potential future issue for general circulation models when the horizontal resolution improves beyond a certain threshold (guaranteed by inevitable advances in computer power). Arguably, this juncture has already been reached because state-of-the-art global weather forecasting models sometimes attempt to explicitly treat the largest tropical organized cloud systems (superclusters) as identified by Moncrieff and Klinker (1997).

We caution that our idealized study should be regarded as preliminary because of the simple shallow-water analogue linear model and the omission, for example, of a more explicit description of the thermodynamic effects of mesoscale downdrafts or the convective life-cycle scheme of Yano et al. (1995, 1996, 1997).3 Either could substantially modify the large-scale thermodynamics. On the other hand, the idealized approach enabled us to isolate pertinent dynamical mechanisms. The results also suggest that the mesoscale momentum flux, which is a distinguishing feature of organized precipitating convection, should be given serious attention. In order to strengthen and quantify this conclusion, we plan to report the full two-layer linear analysis in the near future and to perform cloud-resolving numerical experiments to ascribe more realism to the results.

Acknowledgments

The major part of the work was performed when J.I.Y was a visitor at NCAR during July–September 1995. J.I.Y is supported by the Australian Government Cooperative Research Centre’s Program.

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  • ——, 1997: Momentum transport by organized convection. The Physics and Parameterization of Moist Convection. NATO Advanced Study Series C: Mathematics and Physical Sciences, Kluwer, in press.

  • ——, and E. Klinker, 1997: Mesoscale cloud systems in the Tropical Western Pacific as a process in general circulation models. Quart. J. Roy. Meteor. Soc.,123, 805–827.

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APPENDIX A

List of Symbols

  1. CD = 1 × 10−3  surface drag coefficient
  2. ĈD  surface drag coefficient [nondimensional, Eq. (2.15a)]
  3. Cθ = 1.2 × 10−3  evaporation rate by wind
  4. Ĉθ  evaporation rate by wind [nondimensional, Eq. (2.15b)]
  5. cM  phase velocity for the mesoscale convective system (along the coordinate moving with the vertical-mean wind)
  6. Db/Dt  Lagrangian time derivative for the lower layer [Eq. (2.9a)]
  7. Dm/Dt  Lagrangian time derivative for the upper layer [Eq. (2.9b)]
  8. Dt/Dt  Lagrangian time derivative advected by a mean tropospheric wind vt [Eq. (2.4)]
  9. EM  nondimensional mesoscale pressure jump
  10. F  nondimensional surface friction rate [Eq. (2.18a)]
  11.   modified nondimensional surface friction rate [Eq. (2.21)]
  12. H = 8 km  depth of troposphere (density-weighted scale in log-p coordinate). Also used as the length scale for the nondimensionalization.
  13. h  mesoscale downdraft depth in Moncrieff’s archetypal model
  14. h0  inflow depth of the front-to-rear flow in Moncrieff’s archetypal model
  15. hb = 500 m  depth of subcloud layer
  16. L  scale of the large-scale disturbance
  17. LM  horizontal extent of the mesoscale system
  18.   nondimensional stratification [Eq. (2.15d)]
  19. pL  low-level large-scale pressure
  20. pM  mesoscale pressure
  21. conv  convective heating
  22. QM  momentum redistribution by the mesoscale circulation
  23. QM  homogeneous part of the momentum redistribution
  24. QM  deviation of the momentum redistribution
  25. M  normalized momentum redistribution [Eq. (3.4b)]
  26. R  radiative cooling by long wave radiation
  27. R0  nondimensional cooling rate of the atmosphere [Eq. (2.15e)]
  28. Ŝ  upper-level stratification [nondimensional, Eq. (2.15g)]
  29. Tb = 300 K  temperature at the top of subcloud layer
  30. Tt  temperature averaged over the troposphere
  31. U0 = 10 m s−1  strength of the relative inflow to the mesoscale organized system used as a velocity scale
  32. ub  lower-layer wind disturbance (westerly)
  33. ub  lower-layer mean wind (westerly)
  34. uc  baroclinic wind disturbance [westerly, Eq. (2.20)]
  35. uc  baroclinic mean wind [westerly, Eq. (2.20)]
  36. ui  inflow strength in Moncrieff’s archetype
  37. um  upper-layer wind disturbance (westerly)
  38. um  upper-layer mean wind (westerly)
  39. ut  barotropic wind disturbance [westerly, Eq. (2.20)]
  40. ut  barotropic mean wind [westerly, Eq. (2.20)]
  41. vm  upper-layer wind vector, defined at the midlevel
  42. vt  mean tropospheric wind vector, barotropic wind
  43. w  vertical velocity at the middle troposphere
  44. wc  total cumulus updraft
  45. wd  convective downdraft
  46. we  environmental subsidence
  47. αd  nondimensional downdraft damping rate [Eq. (2.18d)]
  48. αe  nondimensional evaporative damping rate [Eq. (2.18b)]
  49. αM  fractional area occupied by the mesoscale convective systems
  50. α̃M  effective mesoscale fractional area
  51. αθ  WISHE parameter [nondimensional evaporation rate by wind, Eq. (2.18c)]
  52. γ = Γdm = 1.7  the ratio of dry to moist adiabatic lapse rates
  53. ΔpM  dimensional change of the pressure crossing over the whole domain of the mesoscale organization
  54. δ  nondimensional depth of subcloud layer [Eq. (2.15c)]
  55. δpL  large-scale pressure anomaly
  56. δα  perturbation specific volume
  57. ϵk  thermodynamic efficiency [Eq. (2.2b)]
  58. ϵM  mesoscale pressure efficiency parameter
  59. ϵp = 0.9  “precipitation efficiency” coefficient
  60. θeb  equivalent potential temperature of the subcloud layer
  61. θ*eb  saturated equivalent potential temperature of the subcloud layer
  62. θem  equivalent potential temperature at the midtroposphere
  63. λ  nondimensional cooling rate by downdraft [Eq. (2.18e)]
  64. τR = 50 day  longwave radiative relaxation time (constant for Newtonian cooling term)
  65. τ̂R  longwave radiative relaxation time [nondimensional, Eq. (2.15f)]
  66. ϕb  subcloud-layer geopotential
  67. ϕt  geopotential averaged over the troposphere
  68. ϕm  midlevel geopotential
  69. ϒ  momentum flux divergence due to the mesoscale circulation [Eq. (3.1)]

APPENDIX B

Explicit Formula for M

Explicit formula for M defined by Eq. (3.4b) is given by
i1520-0469-55-6-1038-eq4
where
i1520-0469-55-6-1038-eq5

Fig. 1.
Fig. 1.

Model configuration.

Citation: Journal of the Atmospheric Sciences 55, 6; 10.1175/1520-0469(1998)055<1038:IOMMTO>2.0.CO;2

Fig. 2.
Fig. 2.

The rate M that the momentum is transported from the upper layer to the lower layer over the scale LM by the archetypal mesoscale circulation: plotted as a function of the mesoscale pressure jump EM. The linear approximation around EM ∼ 0 is also represented by a dash line.

Citation: Journal of the Atmospheric Sciences 55, 6; 10.1175/1520-0469(1998)055<1038:IOMMTO>2.0.CO;2

Fig. 3.
Fig. 3.

The modification of the WISHE instability by mesoscale momentum transport. The growth rate (day−1: upper frame) and the phase velocity (m s−1: lower frame) are plotted against the global longitudinal wavenumber (k = 100 corresponds to a wavelength of 400 km). The solid curve shows the standard case (ub = 5 m s−1) without momentum transport. The long-dash and the short-dash curves show the cases with α̃M = 0.2 and 0.5, respectively.

Citation: Journal of the Atmospheric Sciences 55, 6; 10.1175/1520-0469(1998)055<1038:IOMMTO>2.0.CO;2

Fig. 4.
Fig. 4.

As in Fig. 3 but for α̃M = 1.5, 1.51, and 1.52 with the short-dash, the solid, and the long-dash curves, respectively. The effective unstable stratification is created by the mesoscale convective momentum transport with α̃M > 1.5.

Citation: Journal of the Atmospheric Sciences 55, 6; 10.1175/1520-0469(1998)055<1038:IOMMTO>2.0.CO;2

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

1

The “cumulus vertical velocity” may be estimated by dividing wc by the fractional area σc occupied by cumulus convection. For economy, we do not explicitly indicate the factor σc.

2

This should not be confused with the moisture stratification measured by λ, which has the opposite effect through suppressing instability through the downdraft effect (cf. Fig. 4 of YE). Also note that the “total” downdraft (wdwe) is proportional to 1/ϵp [cf. Eq. (2.11c), d].

3

This was originally called the grid column scheme in Yano et al. (1995, 1996).

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