## 1. Introduction

The Madden–Julian oscillation (MJO) is notoriously difficult for global models to simulate (Lin et al. 2006). Furthermore, there is no universally accepted theory of this phenomenon. Zhang (2005) describes the essential characteristics of the MJO in his comprehensive review. These can be summarized briefly as follows:

- (i) The MJO is a global-scale, convectively coupled, eastward-moving disturbance with roots in the tropics. The convective aspects are most obvious in the Indian and western Pacific Oceans, with a small convectively coupled region in the far eastern Pacific north of the equator in the boreal summer.
- (ii) Eastward-moving variance has a broad range of periods extending over 30–100 days. The broad convective envelope of the MJO moves eastward in the Indian and west Pacific Oceans at about 5 m s
^{−1}. Though the convective signal disappears outside of the regions noted above, a global, eastward-propagating signal exists in the zonal wind in the upper troposphere that moves much more rapidly through the nonconvectively coupled longitudes. - (iii) Easterly wind anomalies exist in the lower troposphere to the east of the active convection and in the upper troposphere to the west, while westerly anomalies occur to west of the convection in the lower troposphere and to the east in the upper troposphere. Cyclonic wind anomalies occur on the north and south flanks of the convection in the lower troposphere, whereas anticyclonic anomalies occur in the upper troposphere. All of this is consistent with the Gill (1980) model for the response of the troposphere to near-equatorial heating.
- (iv) At low levels the strongest winds in equatorial latitudes are westerlies, which tend to occur either coincident with or to the west of the most intense convection. The former pattern tends to develop more in the western Pacific, while the latter is more prevalent in the Indian Ocean.
- (v) Along the equator there is a westward tilt to the heating and vertical velocity pattern of the MJO in convectively coupled regions.
- (vi) The large-scale convective envelope of the MJO contains a myriad of synoptic-scale disturbances moving both to the east and west. These have been variously identified as equatorial Kelvin waves, mixed Rossby–gravity waves, inertio-gravity waves, easterly waves, tropical cyclones, and westerly wind bursts.
- (vii) The MJO follows latitudinal excursions in the sea surface temperature (SST) maximum, with the strongest signals north and south of the equator lagging the respective solstices by about 2 months. In between, the MJO tends to be somewhat weaker. Significant interactions exist between the MJO and the Asian monsoon.

Zhang (2005) describes the extensive history of theory and modeling aimed at understanding the MJO. In the interests of brevity this is not repeated here. Instead we focus on recent promising developments.

The treatment of convection in large-scale atmospheric models is without a doubt the most problematic aspect of simulations of the MJO (Slingo et al. 1996). There is still no reliable method for testing the performance of cumulus parameterizations. However, the evaluation of such parameterizations against cloud-resolving numerical models of convection (CRMs) has begun to change this situation. Derbyshire et al. (2004) showed, using two different CRMs, that deep convection is very sensitive to tropospheric relative humidity and that a number of widely used cumulus parameterizations did not reproduce this sensitivity. This behavior has been confirmed by other numerical models (Lucas et al. 2000; Tompkins 2001; Raymond and Zeng 2005) and in observations (Sherwood 1999; Bretherton et al. 2004; Back and Bretherton 2006). Raymond (2007) showed that tuning a cumulus parameterization to a CRM in the context of the weak temperature gradient approximation (Sobel and Bretherton 2000; Sobel et al. 2001; Mapes 2004; Raymond and Zeng 2005) results in more rapidly developing zonal wind variance on intraseasonal time scales compared to the untuned version of Raymond (2001). The weak temperature gradient approximation when applied to a single-column model approximates the effect of the tropical environment by forcing the column to maintain buoyancy equilibrium with its surroundings via an imposed vertical velocity profile that counters heating with diabatic cooling. This vertical velocity, along with the convergence and divergence it implies, also transports moisture.

Grabowski (2001, 2003) found that a large-scale model in which the cumulus parameterization is replaced by a small CRM in each grid box produces an MJO-like disturbance. The key element facilitating the production of this disturbance is the buildup of humidity in the convecting region. Such a “superparameterization” was also used by Khairoutdinov et al. (2005) in place of the standard Zhang and McFarlane (1995) parameterization in the National Center for Atmospheric Research’s Community Atmosphere Model (CAM). Comparison with the standard CAM showed a much enhanced MJO. Plots of rainfall rate versus saturation fraction (precipitable water divided by saturated precipitable water) in these two simulations show much more realistic results in the version with the superparameterization than in the standard version (H. Zhu 2008, personal communication). The standard version exhibits too much rainfall at low saturation fractions and not enough at higher saturation fractions compared to Bretherton et al. (2004).

The gross moist stability (GMS; Neelin and Held 1987) is an important concept when dealing with the dynamics of convectively coupled systems. Roughly speaking, the GMS is proportional to the net export because of lateral flows through the sides of a dynamical system of some quantity that is conserved by moist processes. No such quantity is precisely conserved, but two of the most useful are the moist static energy and the moist entropy. The GMS is normalized by some quantity proportional to the amount of convection occurring in the system of interest. Various different normalizing quantities are used as well, such as the total mass flowing through the system or the amount of moisture extracted by the system from the flow.

Theoretical support exists for the idea that moisture anomalies constitute the dynamical backbone of stationary or slowly propagating convectively coupled modes in the tropics (Fuchs and Raymond 2002, 2005, 2007; Raymond and Fuchs 2007). Free-standing “moisture modes” in a homogeneous environment were found in these results to be unstable for a wide range of spatial scales if the GMS is negative. However, moisture modes can also exist with positive GMS when there is an independent forcing mechanism, for example, advection across a moisture gradient in a rotating environment (Sobel et al. 2001) or self-enhanced surface heat fluxes and suppression of radiative losses, as in a tropical cyclone (Emanuel 1986; Raymond et al. 2007). The latter situation can be represented as an effective negative GMS (Su and Neelin 2002; Sugiyama 2009) in situations where the GMS as formally defined is positive.

The effects of negative GMS in a convectively coupled system have not been extensively considered. López-Carrillo and Raymond (2005) found from Doppler radar and sounding observations that the net moist entropy export by convective systems, and hence the GMS, could take on either positive or negative values in the equatorial western Pacific. The positive values were associated with mature mesoscale systems with a heavy stratiform component, while negative values were associated with more convective regimes and a vertical mass flux maximum located at lower levels. The saturation fraction tended to be larger for positive GMS cases.

Bretherton et al. (2005) found negative GMS in a large-domain CRM in which the convection underwent self-aggregation. Such concentration of convection into a limited region surrounded by dry air can be considered to be a case of moisture mode instability.

Negative GMS occurs when the outflow-weighted moist entropy is less than the inflow-weighted value of this variable. For a simple dipole structure with inflow at low levels and outflow at high levels, this can happen when the level of maximum vertical mass flux is lower than the minimum in the environmental moist entropy profile. This occurred in the results of López-Carrillo and Raymond (2005). However, the situation is more complex in the numerical results of Bretherton et al. (2005); in this case low moist static energy air is exported from the simulated convection in a thin detrainment layer near the top of the boundary layer. Such detainment is often seen from deep convective regions in the tropics, as documented by Zhang et al. (2004). Raymond and Blyth (1986) observed such detrainment in continental convection and attributed it to buoyancy sorting following vertical mixing of air in convection. However, irrespective of how negative GMS is achieved, it results in convection moistening its own environment rather than drying it, as occurs when the GMS is positive. Sobel and Neelin (2006) found that negative GMS associated with boundary layer flows tends to produce more concentrated intertropical convergence zones in a global model of intermediate complexity.

The purpose of this paper is to present evidence that the MJO is a moisture mode that depends on the presence of negative GMS for its existence. To this end we first explore the sensitivity of the analytical model results of Raymond and Fuchs (2007) to both positive and negative GMS, showing that moisture modes only appear in this model with negative GMS, at least in the absence of additional forcing. Next we analyze results from a slightly enhanced version of the toy equatorial beta-plane aquaplanet model of Raymond (2007) in which a strong, MJO-like disturbance develops. Finally, we examine the output of the National Oceanic and Atmospheric Administration’s operational final analysis (FNL), which incorporates the same physics as their Global Forecast System (GFS) and the associated Climate Forecast System (CFS). The CFS, like many other global models, is known to produce a rather weak MJO (J.-L. Lin 2008, personal communication). The toy model exhibits negative GMS in the equatorial western Pacific whereas the FNL does not. These results together suggest that the MJO is in essence a moisture mode and that negative GMS is required for its existence.

## 2. Analytical model results

*h*is the height of the tropopause, w is the vertical velocity,

*s*is the moist entropy, and

*s*is the dry entropy. This normalization is somewhat different from the gross moist stability of Neelin and Held (1987). Because of this, we refer to Γ

_{d}*as the*

_{M}*normalized gross moist stability*(NGMS). The NGMS is adjusted in this model by varying the vertical profile of moist entropy, since the vertical velocity exhibits to a good approximation a fixed fundamental baroclinic mode sinusoidal dependence on height for all modes.

The dispersion relation for the linearized theory in Raymond and Fuchs (2007) includes the effects of nonzero NGMS. However, plots were only presented in that paper for zero NGMS. Figure 1 shows the dispersion relations for moisture modes occurring with various negative values of the NGMS but no wind-induced heat exchange or cloud–radiation interactions. Note that instability is strongest at shorter wavelengths, decreasing somewhat at lower wavenumbers. This decrease is due primarily to the effects of CIN. Turning off CIN removes most of this decrease. Moisture modes are stable for positive NGMS. This follows from the instability mechanism for moisture modes; instability occurs when convection produces an environment that is favorable for the production of more moisture mode convection, that is, by increasing the saturation fraction. This occurs when the NGMS is negative.

Figure 2 shows the corresponding dispersion relation for convectively coupled gravity waves. This mode is only weakly sensitive to the NGMS, with positive NGMS values resulting in slightly increased instability. The instability of this mode depends primarily on the CIN term in the convective forcing, and the weak dependence on NGMS therefore makes sense.

For our purposes here, the important result from this model is that instability of the moisture mode depends on the existence of negative NGMS. Cloud–radiation interactions and wind-induced heat exchange (WISHE) are able to destabilize the moisture modes as well, but only by virtue of supplying moisture feedbacks similar to those supplied by negative NGMS.

## 3. The toy beta-plane model

### a. Governing equations

*θ*is the potential temperature and

*θ*and

_{B}*θ*are the potential temperatures at the surface and domain top. Here

_{T}*θ*is held constant and is also assumed to be a constant pressure surface, while

_{T}*θ*is in general a function of position and time. The constant pressure upper boundary condition is reflective, so a sponge layer is placed in the upper part of the domain.

_{B}The governing equations for this model are as follows:

*η*is the mass density in

*x*−

*y*−

*σ*space,

**v**is the horizontal wind in this space, and

*S*=

_{σ}*dσ*/

*dt*is the vertical velocity in

*σ*coordinates;

**u**=

*η*

**v**,

*M*is the Montgomery potential, Π is the Exner function,

*f*=

*βy*is the Coriolis parameter, and

*K*and

_{h}*K*are the horizontal and vertical eddy mixing coefficients;

_{σ}*r*is the total cloud water (vapor plus advected condensate) mixing ratio, and

_{t}*η*=

_{t}*ηr*.

_{t}*σ*-coordinate vertical velocity iswhere

*S*=

_{θ}*dθ*/

*dt*,

*S*is

_{B}*S*at the surface, and Δ

_{θ}**v**

*=*

_{B}**v**−

**v**

*, with*

_{B}**v**

*being*

_{B}**v**at the surface.

*x*, Δ

*y*, and Δ

*σ*are the

*x*,

*y*, and

*σ*cell sizes, and

*C*= 1.

_{K}*p*can be obtained from the hydrostatic equation in

*σ*coordinates:subject to the upper boundary condition of fixed pressure at the top of the domain. The Exner function is related to the pressure bywhere

*p*is a constant reference pressure and

_{R}*κ*= (

*C*−

_{p}*C*)/

_{υ}*C*, with

_{p}*C*and

_{p}*C*being the specific heats at constant pressure and volume. Finally, the Montgomery potential is obtained from the identity

_{υ}*dM*=

*d*(

*θ*Π) +

*d*Φ = Π

*dθ*+

*θd*Π +

*d*Φ, and the hydrostatic equation written in the form

*θd*Π = −

*d*Φ,subject to the condition at the surface:where

*g*is the acceleration of gravity and

*h*(

*x*,

*y*) is the height of the terrain, taken here to be zero. The geopotential is obtained by solving Eq. (9):where

*z*is the height above the zero level.

### b. Diabatic and frictional parameterizations

The toy convective and surface flux parameterizations of Raymond (2007) are used with modifications discussed below, as is the toy radiation scheme of Raymond and Torres (1998) and Raymond (2000), to determine the source terms for momentum, heat, and moisture **S*** _{υ}*,

*S*, and

_{θ}*S*. The convective scheme is a kind of convective adjustment in which the adjustment is done as a relaxation over the full vertical depth of the convective layer, as defined by the height of undilute neutral buoyancy from the planetary boundary layer. Surface fluxes are obtained using simple bulk flux formulas. For convenience, the convective and surface flux parameterizations are presented in the appendix.

_{r}The radiation scheme is quite crude but is tuned roughly to the scheme used in the National Center for Atmospheric Research’s Community Climate Model, version 2 (Kiehl et al. 1994).

Two small but significant changes are made in the cumulus parameterization relative to the version presented by Raymond (2007). In that version the deep convective adjustment rate *a* is assumed to be constant. [Note that *a* is called *η* in Raymond (2007), but this clashes with the definition of *σ*-coordinate air density in the present paper.] However, one would expect adjustment to take place more rapidly in intensely convective regimes than in regimes with weaker convection. We find that precipitation is a strong function of tropospheric humidity and therefore take the saturation fraction *H* (A12) to be a proxy for strong convective mixing. The deep convective adjustment rate *a* (A11) is thus made proportional to *H ^{s}*, where

*s*is a constant. The form (humidity)

*also appears in the source term for convective precipitation, and comparison of the current version of the cumulus parameterization with the CRM of Raymond (2007) suggests a value*

^{s}*s*= 6, which makes both convective precipitation and mixing strongly increasing functions of tropospheric humidity.

The other change is to the vertical dependence of the source term for convective precipitation *μ*(*z*). This is made small at low levels to avoid the production of strong precipitation from shallow clouds. A somewhat smoother form than previously used is defined in the current version of the parameterization by (A14).

Raymond (2007) showed that comparison of a parameterization with a CRM in the context of the weak temperature gradient approximation provides a rational means for tuning a parameterization—constants are adjusted until a reasonable fit between the CRM and the parameterization is obtained. As shown in Fig. 3, the predicted saturation fraction as a function of wind speed for the current parameterization is much less at low wind speeds than that of Raymond (2007) primarily because of the addition of a mixing coefficient that depends on saturation fraction.

### c. Numerical considerations

As in Raymond (2001, 2007), (3)–(5) are integrated using Lax–Wendroff differencing, while the surface potential temperature equation employs simple upstream differencing. The simplicity of the latter is justified because the surface potential temperature is governed primarily by local effects in the tropics, with only secondary tendencies due to advection. The eddy mixing coefficients *K _{h}* and

*K*are augmented above

_{σ}*σ*= 0.8 to produce a sponge layer at the top of the domain to absorb upward-propagating waves. Also as in Raymond (2007), cloud–radiation interactions are turned off to eliminate this source of variability. A small amount of initial variance is introduced into simulations by making random perturbations to the initial humidity field at low levels.

*η*,

**u**, and

*η*to suppress the short vertical wavelength computational mode to which cell-centered calculations are subject. This takes the formfor each variable

_{t}*χ*where the tilde indicates the smoothed variable, the subscript indicates the vertical level, and

*λ*= 0.01. Symmetric mirror conditions are used at the surface and top to insure conservation. The horizontal fourth-order smoothing used by Raymond (2007) was also eliminated here as it was found to be unnecessary with the changes in the cumulus parameterization mentioned above. Version sigma-066 of the model is used for the computations in this paper.

_{V}### d. Definition of NGMS

*σ*from 0 to 1. The variable

*s*is the moist entropy and

*r*is the total cloud water mixing ratio. We further subdivide this into two pieces: a part having to do with horizontal entropy advection,and a part associated with vertical advection, which is more closely related to the Neelin and Held (1987) definition of gross moist stability:The approximation in this equation comes from neglecting the time derivative in the mass continuity Eq. (3) and then integrating by parts in

_{t}*σ*. The denominator in (17) is equal to the latent heating associated with moisture convergence. Following Back and Bretherton (2006), we denote

*T*[

_{R}*η*

**v**· ∇

*s*] the “horizontal advection” part of the entropy divergence and

*T*[

_{R}*ηS*(∂

_{σ}*s*/∂

*σ*)] the “vertical advection” part.

It is a bit awkward to have two possibly conflicting definitions of NGMS in (1) and (17). However, the differences in these definitions are less than they appear to be at first sight. First, the linearized model of section 2 does not admit horizontal advection in the moving reference frame of the mean flow. It is thus omitted from the definition of NGMS in (1). Second, the difference between normalization in terms of mixing ratio rather than dry entropy is small. This is because *L* times the moisture removed from a parcel passing through a convective system approximately equals *T _{R}* times the specific dry entropy added to the parcel, assuming that the parcel conserves moist entropy. Thus, Γ

_{R}_{2}≈ Γ

*.*

_{M}## 4. Toy model results

### a. Moisture modes over uniform SST

We first examine the development of moisture modes in the toy model under conditions of uniform SST, zero ambient wind, no rotation, and varying spatial resolution of 250, 125, 62.5, and 31.25 km. In all cases a domain of 5000 km × 5000 km is used in the horizontal with doubly periodic boundary conditions. In the vertical, 20 levels are used, equally spaced in *σ*. The simulation is initiated with a single-column radiative–convective equilibrium profile with random humidity perturbations at low levels. All simulations were run for a minimum of 5000 ks, or about 58 days. Figure 4 shows rainfall rate, saturation fraction, and surface winds for these simulations at 2500 ks into the simulation.

The simulations show that a cellular convective structure develops for each of the model resolutions. As model grid size increases, the representation of finescale detail decreases as expected. Because of the different random initializations, the resulting convective patterns differ for the different resolutions. However, the size of convectively active patches (≈500–1000 km) and the spacing of convective cells (≈2000 km) remain the same for grid resolutions as coarse as 125 km; only for the 250-km case do these dimensions increase. This suggests that a grid size of 125 km is sufficient to capture the basic cellular structure of convection in this model, though of course finer-scale detail is unresolved.

This is to be compared with the CRM results of Bretherton et al. (2005) on a 576 km × 576 km domain. Self-aggregation occurred in this case, producing convectively active patches of order 100 km in diameter, with typically only one patch existing in the model domain. This contrasts with patches 5–10 times bigger in our case. However, it is unclear whether the convectively active regions would have grown to larger sizes in Bretherton et al.’s CRM if it had been possible to use a larger model domain. Thus, at this point we cannot say what the scales of convective patches and their surrounding subsidence regions are in nature.

### b. Reynolds SST aquaplanet simulations

In this section idealized aquaplanet simulations with 125-km grid resolution are made in which the sea surface temperature distribution is taken from the global synthesis of SST by Reynolds and Marsico (1993). In this synthesis the SST is interpolated across continents. We are thus ignoring the vast difference between land–air and sea–air interactions, which is why we characterize our results as “idealized.” However, the MJO manifests itself largely in the oceanic domain, so it should be possible to learn something about the MJO in this semirealistic context without having to delve into the complex processes that occur in the atmosphere over land. Clearly, interactions of the MJO with the Asian monsoon will not be represented correctly in these simulations.

Another idealized aspect of the simulations is the use of an equatorial beta plane to represent the tropical and near-tropical regions of the earth. The model domain is taken to be 40 000 km in the zonal direction by 12 000 km in the meridional direction with periodic east–west boundary conditions and free-slip walls bounding the simulations to the north and south. As in the last section, 20 vertical levels are used.

Simulations are run for 50 000 ks or about 579 days. The Reynolds SSTs are scaled to an earth with a circumference of 40 000 km. To minimize spurious baroclinic wall disturbances at high latitudes, the SST is bounded below by 15°C.

Two simulations are reported, with perpetual February and August SSTs, respectively. Figure 5 shows the SST distributions used in the model in both cases as well as the intraseasonal rainfall variance in each case. The latter is defined as the variance of the rainfall rate field over the time interval 20 000–45 000 ks after application of a temporal bandpass filter of 7–90 days and a longitudinal low-pass filter with smoothing length of 3000 km. There is fair agreement with the analysis of observations by Zhang (2005), though the northern summer shows stronger rainfall variance than the southern summer, and the northern summer variance in the eastern Pacific and western Atlantic is too strong in the simulations. Perhaps these differences can be explained at least in part by the lack of land in the simulations.

Figures 6 and 7 show Hovmöller diagrams for surface zonal wind and precipitation in equatorial regions south and north of the equator for perpetual February and August simulations, respectively. Low-pass filtering on longitude (3000-km smoothing length) and time (7-day smoothing time) has been performed to isolate intraseasonal disturbances. Eastward propagation of westerly wind and precipitation anomalies is clearly seen in both cases over the Indo-Pacific warm pool with broadly distributed propagation speeds ranging from about 1 to 8 m s^{−1}. Higher speeds are typically seen in the February case. The February disturbances show more power at higher frequencies than do the August disturbances. In August the disturbances disappear near the date line (20 000 km) and reappear in the east Pacific as is observed, whereas in February disturbances continue somewhat further east into the southeast Pacific convergence zone but with no separate east Pacific signal.

Figure 8 shows two snapshots of the rainfall rate, the surface wind, and the saturation fraction over the northern part of the Indo-Pacific warm pool. The first snapshot is taken during a convectively quiescent period in this region, while the second shows a convectively active period. The strongest convection in the latter case takes the form of a continuous intertropical convergence zone (ITCZ) distorted by breaking wave disturbances. The surface winds in equatorial regions are much stronger in the active case, with strong westerlies coincident with the heavy precipitation. The disturbances have the apparent character of westerly wind bursts (Lau et al. 1989; Fasullo and Webster 2000). These disturbances generally move to the east, as does the entire pattern of strong convection. It is this latter pattern, consisting in general of one or more westerly wind bursts and the associated active ITCZ that constitutes the MJO-like disturbance.

The simulated disturbance in our model satisfies most of the criteria for the MJO listed in the introduction. In particular,

- (i) The disturbance is global in scale, convectively coupled, and moves to the east at about the right speed. The associated convection is strongest over the Indo-Pacific warm pool.
- (ii) The disturbance has a first baroclinic mode structure in the vertical and Gill model structure in the horizontal, with low-level westerlies in the equatorial region.
- (iii) The overall MJO-like disturbance consists of multiple disturbances of smaller scale.
- (iv) The convection follows the latitudinal excursions of the SST.

However, the convective heating does not tilt consistently to the west with height and the surface westerlies coincide with precipitation in the Indian Ocean region as well as the west Pacific rather than being displaced to the west.

### c. NGMS in the toy model

We now turn to a diagnosis of the NGMS exhibited by the convection in the simulated MJO, focusing on the convection in the Indo-Pacific warm pool in the perpetual August simulation. We average the numerators and denominators of (18) and (19) over the region 12 000 < *x* < 18 000 km, 0 < *y* < 1500 km, which covers the heart of the region of intraseasonal variability shown in the lower panel of Fig. 5.

Figure 9 shows scatterplots of the horizontal (*T _{R}* [

*η*

**v**·

**∇**

*s*]) and vertical (

*T*[

_{R}*ηS*(∂

_{σ}*s*/∂

*σ*)]) advective parts of the entropy divergence versus the moisture convergence {−

*L*[∇(

*ηr*

_{t}**v**)]}, with each point representing a snapshot of these quantities averaged over the Indo-Pacific warm pool as indicated above. For positive moisture convergence, Fig. 9 shows that horizontal advection generally results in entropy divergence, tending to decrease the average entropy. On the other hand, vertical advection in the model tends to increase the entropy in the column. The former effect is stronger, so overall entropy divergence and the NGMS are positive. Quantitatively, we find that time averaging the moisture convergence and the pieces of the entropy divergence results in Γ

_{R}_{1}= 0.90, Γ

_{R}_{2}= −0.20, and Γ

*= 0.70 for this region.*

_{R}The positive value of Γ* _{R}* means that on the average entropy is laterally exported from the Indo-Pacific warm pool in our model. However, the horizontal advection part of the entropy divergence is not Galilean invariant; viewing this quantity in a reference frame moving with a convective disturbance could potentially yield a much smaller value for the contribution of horizontal advection to the entropy divergence, especially if the disturbance drifts with the wind in a region of low vertical shear. This is likely to happen with moisture modes, which tend to move at a speed close to that of the wind. On the other hand, the vertical advection term is Galilean invariant, so its value would not change in a moving reference frame. It is thus an intrinsic property of the convection.

We hypothesize that the MJO-like disturbances appearing over the warm pool in our model are moisture modes and are therefore dependent on the existence of negative values of Γ_{R}_{2}.

## 5. NGMS in the FNL

The analysis system used to initialize the GFS model of the National Centers for Environmental Prediction (NCEP) is called the FNL. Though the FNL and GFS are spectral models with a sigma vertical coordinate, the FNL data are made available in 1° grid form on pressure levels. The FNL is a variational analysis system in which the previous GFS model output is used in conjunction with observations to derive an initialization for the next GFS run (Parrish and Derber 1992). Over tropical oceans where conventional observations are sparse, vertical velocity profiles and other aspects of analysis data should be determined largely by GFS physical parameterizations. However, the location, structure, and intensity of large-scale convectively coupled systems such as the MJO should appear in the analysis whether or not the GFS supports such systems in free running mode. J.-L. Lin (2008, personal communication) concludes that, in common with many other global atmospheric models (Lin et al. 2006), the GFS does not satisfactorily simulate the MJO.

Figure 10 shows the moisture convergence in the FNL averaged over 15°N–0° for June through September 2005. Two MJO events are evident in the western Pacific during this period: one near day 210, the other near day 250. Eastward propagation is evident in both of these events from the Indian Ocean to the eastern Pacific.

Figure 11 shows the outgoing longwave radiation (OLR) for the same period and latitudinal average. Comparison with Fig. 10 shows that low OLR values roughly correspond with regions of strong moisture convergence in the FNL, indicating that the FNL is indeed seeing major convectively coupled disturbances in the tropical regions.

*must be modified accordingly. We define the horizontal advection part of Γ*

_{R}*asand the vertical advection part asin pressure coordinates, where the square brackets now indicate an integral over the range 100–1000 hPa.*

_{R}Figure 12 shows daily average FNL values over June–September 2005 of the horizontal and vertical advection parts of the entropy divergence versus the moisture divergence over the Indo-Pacific warm pool. In agreement with Fig. 9, the averaging domain is 108°–162° in longitude and 0°–14° in latitude.

Unlike our toy model results, both the horizontal and vertical advective parts of the moist entropy divergence are positive, with Γ_{R}_{1} = 0.25, Γ_{R}_{2} = 0.21, and Γ* _{R}* = 0.47, implying positive NGMS. These results, which differ fundamentally from those of our toy model, are almost certainly a function of GFS/CFS/FNL model physics and not a result of observations, which are sparse over tropical oceans. We conclude that the GFS/CFS model is unable to support moisture modes over the Indo-Pacific warm pool, at least without strong surface flux and cloud radiative forcing, and hypothesize that the inability of the GFS/CFS to simulate the MJO is related to this fact.

## 6. Discussion and conclusions

In this paper we explore the hypothesis that the Madden–Julian oscillation is driven by moisture mode instability. For this mode to function, the precipitation rate must be a strong positive function of the saturation fraction. In addition, the deep convection that produces the precipitation must act to further moisten the atmosphere. Generally, a negative value of the normalized gross moist stability (NGMS) is needed for such moistening to occur. However, regions of enhanced precipitation often exhibit enhanced surface entropy fluxes because of enhanced winds and gustiness and suppressed tropopause fluxes associated with the reduction in outgoing longwave radiation by high stratiform cloudiness. In these cases moistening of the column can occur with weakly positive values of the NGMS. Note that, in contrast to Raymond (2001), cloud–radiation interactions are not considered in this paper.

The moisture mode is illustrated in its simplest form by the linearized model of Raymond and Fuchs (2007). In the simplest manifestation of this model, surface and tropopause entropy flux anomalies are ignored and moisture modes amplify only if the NGMS is negative. In the linear case these modes are stationary (or move with the ambient wind) unless some other factor, such as wind-induced surface heat exchange in the presence of an ambient surface flow, exists. The growth rate of these modes tends to be relatively independent of scale except that some decrease in growth rate occurs for very long wavelengths. The results of Bretherton et al. (2005) show that the growth of moisture modes, represented in that paper by the self-aggregation of convection, is possible in cloud-resolving numerical models, even though the profile of lateral inflow and outflow of entropy in this model differs from the simple linearized case.

We show that the toy large-scale model of Raymond (2007) also exhibits self-aggregation of convection in the simplified context of a three-dimensional doubly periodic domain with uniform SST and no rotation. In this model, the NGMS consists of two parts: one associated with the tropospheric-average horizontal advection of moist entropy, the other with its average vertical advection. The former can be thought of as a ventilation term in which the moist entropy in the core of a convective region is reduced by dry advection, whereas the latter is closely related to the Neelin and Held (1987) form of the gross moist stability.

It is noteworthy that self-aggregation occurs in the present model even in the absence of cloud–radiation interactions, a process that was central to the results of Raymond (2001). The stronger response of precipitation to saturation fraction in the current version of the cumulus parameterization is responsible for this change.

The toy model produces robust MJO-like disturbances on an aquaplanet with Reynolds SST distributions for February and August. These disturbances exhibit many of the characteristics of the MJO as outlined by Zhang (2005). In these simulations the vertical advection part of the NGMS over the Indo-Pacific warm pool is on average negative, whereas the horizontal advection part is positive, at least when evaluated in the earth’s reference frame. Since the horizontal advection is highly variable and can be reduced greatly in the moving reference frame of the disturbance where the relative wind is small, moisture mode instabilities can develop. We hypothesize that the MJO produced in the model and in the real world are both fundamentally driven by this moisture mode instability.

Further evidence for this view is provided by the final analysis (FNL) of NCEP’s Global Forecast System (GFS) and the associated Climate Forecast System (CFS). The GFS/CFS produces only a weak MJO. Diagnosis of the FNL shows positive values for both parts of the NGMS over the Indo-Pacific warm pool. Since observational data are likely to be insufficient to constrain mass flux profiles in the FNL over tropical oceans, these profiles and the associated vertical entropy advection are likely to be governed by model physics. We hypothesize that the absence of an MJO in the CFS is due to the fact that the vertically averaged vertical moist entropy advection is positive in the model, thus leading to the suppression of moisture modes in the absence of strong surface and tropopause entropy flux anomalies coupled to precipitation.

It would be desirable to diagnose other global models in the above manner to determine whether the relationship between negative gross moist stability and MJO formation exists in them as well.

In summary, the theoretical and modeling work in this paper suggests that the Madden–Julian oscillation is driven by moisture mode instability. However, much remains to be revealed about the detailed workings of moisture mode dynamics in this prominent but elusive phenomenon.

The authors thank Adam Sobel and an anonymous reviewer for their useful input. This work was supported by National Science Foundation Grant 0638801.

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# APPENDIX

## Cumulus Parameterization

*θ*, total cloud water mixing ratio

_{e}*r*, and the horizontal velocity

_{t}**v**they arewhere

*F*,

_{e}*F*, and

_{r}**F**

*are the surface fluxes of the corresponding variables.*

_{υ}*χ*(

*z*) iswhere

*ρ*(

*z*) is the atmospheric density profile,

*D*is the depth of the convective layer, andEquations (A1)–(A3) are designed so that, for instance,so that the first two terms represent a conservative redistribution or adjustment that attempts to homogenize vertically the variable in question. The third term on the right of these equations represents the vertical distribution of surface fluxes. This distribution is weighted by the rate function

*a*(

*z*), the form of which is discussed below.

*i*on the left sides of the above equations equals either

*d*for deep convection or

*s*for shallow convection. The two convective regimes are assumed to exist simultaneously in varying proportions. Actual profiles of convective source terms are computed as weighted averages of the shallow and deep convective profiles, indicated respectively by subscripted

*s*and

*d*:etc.

*ϵ*is a measure of this inhibition:where Δ

*θ*is a constant,

_{e}*θ*

_{eb}is the mean equivalent potential temperature in the height range 0 <

*z*<

*b*where

*b*is the fixed PBL depth, and

*θ*

_{et}is the mean saturated equivalent potential temperature in the range

*b*<

*z*< 2

*b*. The function

*x, y*) is a throttle function; that is,

*λ*represents the maximum mixing rate and the function Λ(

_{c}*x*) = lim

_{y→0}

*x, y*).

Precipitation production and evaporation in shallow convection is assumed to take the same form as in deep convection (see below), though the precipitation rate in the shallow case tends to be very small or zero.

*a*(

*z*) equalswhere

*s*is a constant called the stiffness parameter. The stiffness parameter determines the “stiffness” of the relationship between humidity on one hand and convective mixing and precipitation rate on the other. The convective layer depth

*D*is assumed to be the height of the level of neutral buoyancy for parcels with average characteristics of the PBL. In calculating

*D*, the highest level of neutral buoyancy is taken for these parcels, thus ignoring possible inversions and stable layers at intermediate altitudes. The saturation fraction

*H*is defined

*C*(

_{R}*z*) represents the conversion of total cloud water (vapor plus cloud droplets) to precipitation, and it takes a rather complex forminvolving three terms. The third term represents the evaporation of precipitation. The quantities

*r*and

_{s}*r*are the saturation and precipitation mixing ratios and

_{p}*λ*is a constant controlling the evaporation rate.

_{e}The second term represents the formation of precipitation in regions that are saturated on the large scale, and *λ _{s}* is the rate constant controlling this process. The quantity Γ =

*dθ*/

_{e}*dz*, and Γ

*is a constant threshold value of Γ, while Γ*

_{t}*defines the range over which the throttle function turns on. The purpose of the throttle function in the second term of (A13) is to suppress the conversion of cloud water to precipitation in saturated regions where Γ < 0. Since grid-scale saturation with negative Γ is unstable, it cannot exist for long. Such regions are interpreted as being mainly unsaturated but pierced by saturated columns of convection of sufficient density to render the average cloud water content saturated. The instability in this case resides solely in the convective columns, and it is inappropriate to turn on “stratiform” processes in this case.*

_{s}The first term represents the conversion of cloud water to precipitation by convective columns in regions that may or may not be saturated on the grid scale. This process is governed by the rate constant *λ _{p}*. The process is assumed to be proportional to the relative humidity

*r*/

_{υ}*r*to the power

_{s}*s*[the stiffness parameter used in (A11)], where

*r*is the water vapor mixing ratio. It is rate limited to a value such that

_{υ}*C*Δ

_{Rp}*t*< 0.3

*r*/

_{t}*s*to avoid relaxation oscillations. The quantity

*C*is that part of

_{Rp}*C*associated with convective precipitation production and Δ

_{R}*t*is the time step.

*μ*(

*z*) takes the assumed formIt thus causes convective precipitation formation to turn on gradually with height, with the scaling being governed by

*z*. In any case, convective precipitation is not allowed above the convective top

_{p}*z*=

*D*. In addition, if the convective precipitation term as calculated above removes more than 30% of

*r*in a single time step, it is limited to this 30% magnitude, thus suppressing unphysical oscillations in the simulation.

_{t}*P*=

*ρw*is assumed to adjust instantly to conditions in the grid box and thus obeys the time-independent vertical advection equationsubject to

_{t}r_{p}*P*= 0 at the top of the domain. Thus, integrating (A15) down from the top yields the precipitation rate at the surface.

*γ*=

*L*/(

*C*);

_{p}T_{R}*L*=

*L*+

_{c}*L*is the sum of the latent heats of condensation and freezing,

_{f}*C*is the specific heat of air at constant temperature, and

_{p}*T*= 300 K is a constant reference temperature. The total cloud water mixing ratio

_{R}*r*is used instead of the vapor mixing ratio, because the difference between the two is likely to be small when averaged over grid boxes typical of large-scale models.

_{t}*S*

_{ec}and

*S*

_{er}are the convective and radiative contributions to the equivalent potential temperature source,

*S*is the convective source of total water mixing ratio, and

_{r}*γ*is defined above. The total precipitation rate is the weighted average of the deep and shallow rates:

*χ*are calculated using a simple bulk formula:where

*T*

_{ss}is the temperature of the sea surface,

*T*is the temperature of the air in the boundary layer adjacent to the surface, Δ

_{b}*T*and

*W*are a constants,

*C*is the (constant) drag coefficient, a subscripted

_{D}*b*indicates a boundary layer value, and a subscripted ss indicates a sea surface value. The throttle function turns the fluxes off when

*T*>

_{b}*T*

_{ss}+ Δ

*T*/2. Table A1 lists values of parameters used in the cumulus parameterization.

Table A1. Typical values of parameters used in the cumulus parameterization.