In this article, we explore the simplest possible configuration that allows for the interaction of a convective cluster with a mean flow. This is motivated by a desire to better understand processes influencing the propagation of organized deep convection in the tropics. In simulations of radiative convective equilibrium (RCE), a single aggregated cluster can develop from randomly distributed convective fields despite homogeneous initial conditions, boundary conditions, and forcing e.g.,. Convective self-aggregation exhibits many similarities to organized deep convection in the tropics including phenomena such as the Madden–Julian Oscillation (MJO), which is an eastward-propagating intraseasonal variability in the tropics . Some studies suggested that the MJO may itself be an expression of self-aggregation . This idea is supported by recent studies showing that MJO-like phenomena are observed in rotating RCE simulations in cloud-resolving models . Further support for this point of view comes from the observational study by , who found that the mean state of the atmosphere during an active phase of the MJO resembles the self-aggregation state in the sense that a higher degree of the convective organization is associated with more outgoing longwave radiation. This leads us to the more basic question of how convective self-aggregation responds to the imposition of a mean flow.

and proposed that the interaction between wind and the surface enthalpy flux in a mean flow may be important for the MJO propagation. They demonstrated that, in mean easterlies, winds are amplified by the convective-scale circulation to the east of convection, leading to a positive anomaly of the surface enthalpy flux. This favors the initiation of convection on the upwind side of the cluster, resulting in the upstream propagation of convection. called this the wind-induced surface heat exchange (WISHE) feedback. Self-aggregation studies also showed that in the absence of mean wind, WISHE contributes to the maintenance of aggregation as the enhanced surface enthalpy flux favors the development of deep convection on the periphery of the existing convection .

This line of thinking leads us to attempt to study a much simpler problem, which is how convective self-aggregation responds to the imposition of a background mean flow. As a first step, we focus on how asymmetries in the surface flux, in response to a mean flow, affect the propagation of a convective cluster in RCE. We impose a large-scale mean flow in simulations of RCE in the form of a shear-free wind, a setup that has not been investigated in previous simulations of RCE. We hypothesize that, on the upwind side of a convective cluster, the mean flow adds constructively to the near-surface component of the convective-scale circulation, enhancing the surface enthalpy flux, and vice versa on the downwind side. The asymmetry in the thermodynamic response to the mean wind leads to a slow upwind propagation of the deep convective system. In addition to the thermodynamic response, we also investigate the dynamic response to the mean flow, and that is how the modified surface wind field affects the surface momentum fluxes. The simulations show that the thermodynamic response to asymmetry in the mean winds is strongly coupled to changes in the momentum budget, which equilibrates the near-surface winds, due to a mean wind contribution to the surface drag in ways that damps and eventually eliminates asymmetries in the surface heat and moisture fluxes. We perform a mechanism denial experiment to suppress the dynamic response and quantify to what extent the propagation can be attributed to the thermodynamic response.

In Sect.  we describe the simulation design including a mechanism denial experiment and discuss the limitations of the setup. Sect.  shows how a convective cluster propagates in the mean flow with different mean wind speeds. In Sect.  we examine the thermodynamic response. In Sect.  we explore the surface momentum flux and discuss the mechanism denial experiment. Conclusions are given in Sect. .

We conduct numerical simulations using the University of California Los Angeles Large-Eddy Simulation (UCLA-LES) model. The UCLA-LES solves the anelastic equations with a third-order Runge–Kutta method for the temporal discretization and with centered difference in space for momentum . Full radiation is computed by using Monte Carlo spectral integration , including radiative properties of ice clouds . A two-moment microphysical parameterization for mixed-phase clouds is used to represent cloud water, rain water, cloud ice, snow, and graupel, explicitly . Subgrid-scale fluxes are modeled with a Smagorinsky model.

A 567×567×27 km3 domain size is used with horizontal grid spacing of 3 km to resolve deep convection. The 63 vertical grid levels are stretched, starting from a grid spacing of 75 m at the first model level up to 1376 m near the model top. The small vertical grid spacing near the surface allows us to better resolve the boundary layer's vertical structure. There is no rotation and no diurnal cycle. The experimental design of the UCLA-LES simulations follows . In contrast to using interactive sea surface temperature (SST) of their experiments, we prescribe an SST of 301 K.

We consider two types of simulations. In a first set of experiments we conduct numerical simulations with different background wind speeds. In an effort to isolate the thermodynamic effects of the convective circulation on the evolution of the self-aggregated convective cluster, we subject the flow to mean wind whose presence is encoded through the surface fluxes. This is equivalent to simulating a situation subject to a large-scale mean wind using a Galilean transform to avoid numerical artifacts of advection but neglecting any restoring force for the wind. Under such a transform, surface fluxes are not invariant, and the effect of the mean wind is accounted for only through the surface flux calculation, which spins down the wind. Effects of WISHE-like asymmetries in the surface fluxes will then be present in so far as they affect the flow on timescales shorter than those associated with the spin-down of the mean wind due to surface drag. In the long run when the effect of the modified surface fluxes is transferred to the atmosphere above by the momentum flux, the velocity in the atmosphere naturally reduces towards that of the surface, until the whole column is in balance again and stagnant compared to the surface. (Note that this equilibrium response is different from the equilibrium response of a nudging approach, where a background flow is maintained. For the transient response, we expect a similar behavior of both approaches.) For the mechanism denial experiment, a mean flow over the surface is maintained by including the influence of the mean wind only in the surface enthalpy equation but not in the surface momentum equation. The first set of experiments is described in Sect. , and the additional experiment in Sect .

Figure  illustrates the daily average spatial pattern of the convective cluster on the last day in the experiments. All simulations show that the quasi-circular pattern of the convective cluster lasts until the end of the simulation period, and the horizontal scale of the cluster size is comparable among all simulations, although the spatial variability of precipitable water is weak for UB4 compared to the other experiments. The standard deviation of the daily average precipitable water on the last simulation day is 14.2, 12.1, and 10.4 kgm-2 for UB0, UB2, and UB4, respectively. This standard deviation varies in time and, for example, is as low as 10.9 kgm-2 on day 6 in the control case UB0. The domain mean precipitable water on the last day increases with increasing ub, having the daily mean values of 26.5, 30.4, and 34.3 kgm-2 for UB0, UB2, and UB4. The larger domain mean precipitable water with increasing ub might be associated with our simulation setup of a propagating cluster in double periodic boundary condition which results in nine full transits through the domain in the case of UB4 . Despite this artifact, the convective cluster remains organized over the simulation period in all experiments, and we expect this difference to play a minor role in the following analysis.

We estimate the propagation speed of a convective cluster by tracking the cluster in the simulation domain. We find all grid columns where the precipitable water (PW) is greater than 62 kgm-2, and we define a convective cluster with the grid points at each output time step. The motion of the cluster is determined by tracking the PW-weighted mean center of the cluster with time. Only x-direction motion is considered, because the cluster propagates in the x direction. Changing the threshold level does not affect the estimated propagation speed. Since in the model setup the surface effectively moves with a constant speed below the atmospheric column, the absolute propagation velocity of the convective cluster to the model surface uabs is calculated as the sum of the relative velocity of the cluster to the model grid urel and the mean wind speed ub:

3uabs=urel+ub.

When urel=0ms-1, the convective cluster remains motionless in the model reference frame but is effectively moving at the speed of ub by virtue of the Galilean transformation (pure advection). If the hypothesized effect of WISHE was realized, then the convective cluster would move against the mean wind (e.g., urel<0 ms-1). Thus, we expect uabs<ub if the WISHE feedback regulates the propagation of the convective cluster.

Figure a shows uabs for each experiment. A 24 h running average is applied to the temporal evolution of uabs to present the long-term evolution more clearly. After imposing ub, the convective cluster begins to propagate. For the simulations where the momentum fluxes are allowed to feel the effect of the mean wind, uabs decreases from what pure advection would suggest to near-zero values at day 15. The decrease of uabs corresponds to our hypothesis (uabs<ub), but it is masked by the spin-down of the mean wind due to surface drag. Estimating the final value of uabs by averaging it over the last 5 d, we arrive at 0.23±0.31,0.10±0.47, and 0.29±0.76 ms-1 for UB0, UB2, and UB4, respectively. (At this point the convective cluster appears stationary to the observer.) Additional simulations with ub of 1 and 3 ms-1 show agreement in that the propagation speed decreases in the first few days and eventually the propagation speed converges to zero (not shown). Additional simulation days for UB4 (until day 30) corroborate that UB4 reaches a quasi-equilibrium state (not shown). The strong fluctuation around the mean is due to the oscillating features of aggregation . This fluctuation hinders our ability to unambiguously distinguish between a slow propagation speed and a stationary one, although its amplitude is comparable to the one with no mean wind (UB0). Since the cluster is formed by a group of individual convective cells, the shape of cluster is not firmly fixed. The cluster expands and contracts in time (though not necessarily in all directions at the same time; see the daily PW for UB2 in Fig. ), and sometimes smaller convective cells emerge outside the main cluster (see the cloud-top height for UB0 in Fig. ). Qualitatively the simulations indicate that the aggregated cluster initially moves with the wind. As the simulations with the mean winds proceed, the convective clusters develop into the wind, and as the mean wind spins down, they become stationary with respect to the surface.

In the following sections we examine if the tendency towards stationarity is a consequence of WISHE-like asymmetries by means of an upstream and downstream difference.

The temporal evolution of the propagation speed demonstrates that the spin-down of the propagation speed occurs over a week whose timescale is longer than the convective adjustment timescale, which is on the order of hours, and the convective cluster settles around 2 weeks after it begins to propagate. We focus on two simulation periods: the transient phase for the first 5 d (day 0–4) when uabs prominently decreases and compare it to the quasi-stationary stage for the last 5 d (day 15–19) when uabs is near zero. Quantities are averaged over these periods.

The surface enthalpy flux is larger on the upwind side of a convective cluster than on the downwind side through WISHE, i.e., the modulation of uabs. Convection is expected to be located over the maximum boundary layer equivalent potential temperature θe. Hence to understand how WISHE affects its distribution, we calculate the flux of θe approximately as w′θe′‾≈w′θ′‾+lvcp(p0p)Rcpcpw′q′‾. Its form is analogous to the enthalpy (or moist static energy) flux. Focusing on the budget of θe allows us to investigate whether the development of convection is associated with the positive anomaly of the surface enthalpy flux.

Figure a, b illustrate how w′θe′‾ varies from the center of the convective cluster (r=0 km) into the environment surrounding the cluster. We place the center of the convective cluster in the center of the domain at each output time step, average the physical quantities, and partition the domain diagonally into quarters, thus defining an upwind area, a downwind area, and crosswind areas. Only the upwind and downwind areas are illustrated. The distribution of w′θe′‾ for UB0 indicates that the surface enthalpy flux is strengthened, because the low-level convergence of the convective circulation intensifies the near-surface horizontal wind in the vicinity of the main convective cluster, which is also observed in other RCE studies e.g.,. As we expected, for UB2 and UB4 in the transient phase w′θe′‾ is enhanced on the upwind side and suppressed on the downwind side. These enhancement and suppression processes of w′θe′‾ become stronger with increasing ub. In the quasi-stationary stage the spatial distribution of w′θe′‾ becomes symmetric.

In the model, the surface enthalpy flux is determined by the difference between the wind speed near the surface and the velocity of the surface, which is equal to 0 ms-1, as well as the vertical differences of specific humidity and potential temperature between the surface and the first level above the surface. The vertical differences of humidity and temperature do not have significant asymmetric features, but uh shows the same transition from asymmetry to symmetry over time as seen in w′θe′‾ (Fig. ). Immediately after ub is imposed, uh is intensified on the upwind side and reduced on the downwind side as one would expect from a superposition of ub and the local circulation associated with the convective cluster. In the later stage of imposing ub, the drag has transported its signal through the near-surface layers, and uh attains a comparable magnitude of wind speed on the upwind and downwind sides. For UB4, the off-centered local minimum of uh around r= 0 km is due to the strong modeled wind u on the downwind side in the opposite direction to ub. The distribution of uh indicates that the adjustment of the near-surface wind field modifies the response of the convection to the mean wind that one would expect from thermodynamic consideration alone.

Without Coriolis force, the tendency of the horizontal wind is obtained as follows: ∂u∂t=-V⋅∇u-cpθ∂π∂x+1ρ∂ρw′u′‾∂z,∂v∂t=-V⋅∇v-cpθ∂π∂y+1ρ∂ρw′v′‾∂z, with V the vector wind: V=(u,v,w). The first term on the right-hand side represents the advection, and the second term represents the pressure gradient force with the Exner function π=(pp0)Rdcp. The third term on the right-hand side represents the contribution of friction to the wind tendency and is related to Fm (Eq. ). For UB2 and UB4, the vertical profile of the x component of the wind in the quasi-stationary stage differs from the initially prescribed shear-free profile, while remaining constant with height for UB0 and UB2_unius (Fig. a). When ub interacts with Fm, the surface drag transports its signal through the atmosphere and the horizontal wind is substantially slowed down, particularly near the surface. The convective cluster is moving with the lower-tropospheric flow well before the whole tropospheric momentum is balanced. In the long term, we expect a balance to ensue with the whole column resting compared to the surface in UB2 and UB4.

As seen in w′θe′‾ and uh, the spatial distribution of Fm shows an asymmetry with respect to the center of the convective cluster in the transient phase and a symmetry in the quasi-stationary stage (Fig. c, d). A larger Fm corresponds to a stronger drag on uh. As a result of the intensified uh, the enhanced Fm on the upwind side exerts a strong drag on uh in the transient phase and, consequently, reduces uh on the upwind side in the quasi-stationary stage. In contrast, the suppressed Fm on the downwind side generates a weak drag, allowing uh on the downwind side to become stronger in the quasi-stationary stage. This difference, or asymmetry, in the drag acts as a source of momentum that accelerates the mean wind until it balances the mean wind, thereby eliminating the asymmetry in the drag by symmetrizing uh. As a result, the symmetric uh in the quasi-stationary stage affects not only the spatial distribution of Fm but also that of w′θe′‾. The upstream and downstream difference cannot be sustained close to the surface, because the momentum exchange limits it in our simulations. This does not rule out a sustained effect in a different system where there is an active dynamical driving of a low-level flow.

To isolate the role of a sustained thermodynamic feedback, we perform an additional simulation where u* is kept constant in space and time for the calculation of Fm but remains interactive for w′θ′‾ and w′q′‾ based on the similarity functions, and we use ub=2ms-1 for the suppressed Fm experiment (Sect. ). Due to the constant value of u*, the domain-averaged w′u′‾ lingers close to zero with small fluctuations for the simulation with suppressed dynamic feedback, UB2_unius, while being negative immediately after imposing ub for UB2 (Fig. b). The suppression of the dynamic feedback enables uh to remain asymmetric, as well as to show stronger maxima in uh for UB2_unius than for UB2 (Fig. a, b) and a persistent asymmetry of the surface enthalpy flux (Fig. a, b). The long-lasting asymmetric feature does not considerably decrease the propagation speed, resulting in the final value for uabs of 1.88±0.16 ms-1 for UB2_unius, hence propagating with a velocity only slightly slower than the mean wind speed of 2 ms-1. A Hovmöller diagram of the cloud-top height confirms the estimated propagation speed, showing that the convective cluster indeed moves against ub with a very small value of urel (Fig. ). The propagation speed is only about 5 % smaller than ub of 2 ms-1, suggesting that this small difference between uabs and ub can be associated with the thermodynamic feedback alone.

As the surface momentum flux is uncoupled from the near-surface wind field, the displacement of the convective cluster with time can be considered to be a result of the pure thermodynamic process. Assuming that the change of the lateral transport of the moisture flux is negligible, the spatial distribution of PW due to the pure thermodynamic process at a certain time PWthermo(t1) is obtained by adding the surface moisture flux anomaly ρw′q′̃ integrated over a time period [t0,t1] to the initial PW at t0:

PWthermo(t1)=PW(t0)+∫t0t1ρw′q′̃dt.

This simple thermodynamic argument gives us a displacement of PWthermo(46 h) from PW(0 h) of approximately 10 km (Fig. b), which corresponds to urel=-0.06 ms-1 and therefore uabs=1.94 ms-1. The estimated displacement of the precipitable water within the given time step due to the moisture flux anomaly agrees well with the estimated propagation speed of 1.88±0.16 ms-1 for UB2_unius (Fig. a) and confirms that the thermodynamic contribution to the propagation speed of a convective cluster is small in our simulations.

This study uses a highly simplified framework to understand how the imposition of a mean flow may influence the propagation of organized deep convection. For the simulations, we applied an RCE framework with a horizontal grid spacing of 3 km, with no rotation and with a prescribed SST of 301K. We hypothesize that the convective cluster propagates against the mean flow through the WISHE feedback, providing a favorable environment to develop convection on the upwind side of the cluster (Fig. a). Our idealized simulations with the mean flow show that organized deep convection initially propagates much slower than what pure advection suggests and eventually becomes stationary towards the end of the simulation period regardless of the imposed wind speed. The near-surface wind field in response to the mean flow modifies the surface enthalpy flux and the surface momentum flux. In return, the surface momentum flux acting as a drag decreases the near-surface wind on the upwind side of the convective cluster and increases it on the downwind side. Because of the surface drag acting on the mean background wind, the mean momentum near the surface is depleted, and on a timescale of a week the surface-relative winds and the surface-relative motion of the convective cluster vanish (Fig. b). In the simulation with the dynamic feedback removed and the WISHE-induced asymmetry in surface fluxes preserved, the effect on the propagation of convective clusters is small.

While the problem we study is probably too simple to meaningfully inform our understanding of much more complex and larger-scale processes like the MJO, it does highlight how a consideration of surface thermodynamic fluxes alone has only a small influence on the propagation of the convective cluster as well as how considering these fluxes in isolation of the associated fluxes of momentum distorts our understanding of the response to the asymmetry imposed by the mean winds. The periodic boundary conditions are limitations of our study, as they cause the effect of anomalously small fluxes to affect the inflow of the region with anomalously large fluxes in ways that damp the effect of the large fluxes. To the extent that WISHE is important for the propagation of convective self-aggregated systems, it would favor large-scale, or solitary systems, so that the moistening that leads the disturbed phase does more than simply offset the drying that lags.

A Galilean transformation can have the advantage of avoiding numerical artifacts of advection. The benefit of the approach, however, ends up being true only to a limited extent, as the convective system starts to propagate through the model grid in our study. Nevertheless, the simulations show that the convective system maintains its thermodynamic structure until the end of the simulation period when ub≤4 ms-1. For future studies, we recommend considering the momentum flux response to a large-scale motion by including a physical mechanism for maintaining a mean flow.

The simplicity of our framework and the difficulties encountered in the setup of the simulations prevent direct inferences from our study for real-world propagating deep convection, let alone the MJO. Compared to typical wind speeds in the tropics, the prescribed large-scale wind speed of up to 4 ms-1 in this study is on the low end of the range. Also, feedbacks between the degree of organization and stronger wind speeds remain an open question. Nonetheless, the basic questions it highlights – such as the role of surface momentum fluxes in WISHE-like mechanisms – are likely to be fruitful avenues to explore when pursuing understanding of more complex phenomena.