Convective self–aggregation in a mean ﬂow

. Convective self-aggregation is an atmospheric phenomenon found in numerical simulations in a radiative convective equilibrium framework of which conﬁguration captures the main characteristics of the real-world convection in the deep tropics. As tropical deep convection is typically embedded in a large-scale ﬂow, we impose a background mean wind ﬂow on convection-permitting simulations through the surface ﬂux calculation. The simulations show that with imposing mean ﬂow, the organized convective system propagates in the direction of the ﬂow but slows down compared to what pure advection would 5 suggest, and eventually becomes stationary relative to the surface after 15 simulation days. The termination of the propagation arises from momentum ﬂux, which acts as a drag on the near-surface horizontal wind. In contrast, the thermodynamic response through the wind-induced surface heat exchange feedback is a relatively small effect, which slightly retards (by about 5 % ) the convection relative to the mean wind.


Introduction
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., Tompkins and Craig, 1998;Bretherton et al., 2005;Coppin and Bony, 2015;Hohenegger and Stevens, 2016). Convective self-aggregation exhibits many similarities to organized deep convection in the tropics including phenomena such as the Madden-Julian Oscillation (MJO), 15 which is an eastward-propagating intraseasonal variability in the tropics Julian, 1971, 1972). Some studies suggested that the MJO may itself be an expression of self-aggregation (Raymond and Fuchs, 2009;Dias et al., 2017). This idea is supported by recent studies showing that MJO-like phenomena are observed in rotating RCE simulations in cloud-resolving models (Arnold and Randall, 2015;Khairoutdinov and Emanuel, 2018). Further support for this point of view comes from the observational study by Tobin et al. (2013), who found that the mean state of the atmosphere during an active phase of the MJO 20 resembles the self-aggregation state in the sense that a higher degree of the convective organization is associated with more outgoing longwave radiation. 1 10 https://doi.org/10.5194/acp-2020-875 Preprint. Discussion started: 18 September 2020 c Author(s) 2020. CC BY 4.0 License. Emanuel (1987) and Neelin et al. (1987) 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 25 the initiation of convection on the upwind side of the cluster, resulting in the upstream propagation of convection. Emanuel (1987) 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 Wing and Emanuel, 2014;Coppin and Bony, 2015).

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Motivated by the potential link between self-aggregation and the MJO, we investigate how convective self-aggregation is influenced by a background mean flow. To do so, we impose a large-scale mean flow in simulations of RCE in the form of mean barotropic 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. This asymmetric response of the surface enthalpy 35 flux to the mean flow is further hypothesized to lead an upwind propagation of the deep convective system. In this study, we test this train of thought. The simulations suggest that the equilibration of the near-surface winds, due to a mean wind contribution to the surface drag, ends up playing a dominant role in the interaction of a large-scale convective cluster with the mean wind, which motivates our analysis of the momentum budget of the self-aggregated state.
2 Simulation setup 40 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 (Stevens et al., 2005). Full radiation is computed by using Monte Carlo spectral integration (Pincus and Stevens, 2009), including radiative properties of ice clouds (Fu and Liou, 1993). A twomoment microphysical parameterization for mixed-phase clouds is used to represent cloud water, rain water, cloud ice, snow, 45 and graupel, explicitly (Seifert and Beheng, 2006a, b). Sub-grid scale fluxes are modeled with a Smagorinsky model.
A 576 × 576 × 27 km 3 domain size is used with horizontal grid spacing of 3 km to resolve deep convection. The vertical grid levels are stretched, starting from a grid spacing of 75 m at the first model level. 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 Hohenegger and Stevens (2016). In contrast to using interactive 50 sea surface temperature (SST) of their experiments, we prescribe an SST of 301 K.

Surface fluxes
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 (Matheou 55 et al., 2011) 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 time-scales shorter than those associated with the spin-down of the mean wind due to surface drag.
The surface fluxes, including the momentum flux (F m ) at the surface and the surface enthalpy flux (SEF), are defined as: with ρ being the air density at the surface, c p the isobaric specific heat and l v the specific enthalpy of vaporization. The covariances ρw u and ρw v represent the x-and y-component of momentum fluxes in kinematic units, respectively. The terms w θ and w q represent the near-surface turbulent fluxes of potential temperature and specific humidity, respectively. The turbulent fluxes are calculated from the turbulence scales of velocity u * , temperature θ * and humidity q * as w u The scale values are computed from profiles of horizontal velocity, temperature and humidity in the boundary layer based on similarity functions (Ψ m , Ψ h ) proposed by Dyer and Hicks (1970), Businger (1973), and Dyer (1974). In the model, u * is proportional to the near-surface horizontal wind u h which is defined as the wind at the first level above the surface, which is at 37.5 m in our case. We modify u h by adding a mean flow u b to it: Physically, this Galilean transform works as if we move the surface with a velocity of −u b , so that it is analogous to putting the atmospheric system on a conveyor belt.

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The aggregated state in simulations of RCE reveals hysteresis; it hardly returns to the random occurrence of convection once an aggregated state is established (Khairoutdinov and Emanuel, 2010;Muller and Held, 2012). We start from an aggregated state in order to separate the effect of a mean wind on the evolution of self-aggregation from its initiation. For this purpose, we run a simulation without a mean wind for 26 days until the convection is fully aggregated. The time scale of self-aggregation in our simulations is comparable to other self-aggregation studies in a square domain (Wing and Emanuel, 2014;Holloway et al., 75 2017; Arnold and Putman, 2018). We then restart the simulations from the aggregated state, but with a mean wind imposed.
The specification of the surface fluxes are described above. Each experiment with u b ranging from 0 to 4 m s −1 is simulated for additional 20 days. Organized convection disaggregates when u b is stronger than 4 m s −1 . Since disaggregation of organized convection is not the focus of this study, the experiments for u b of 0, 2 and 4 m s −1 are discussed and will be denoted by UB0, UB2 and UB4, respectively.

Mechanism denial experiment
UB0, UB2, and UB4 indicate that the dynamic feedback significantly modulates the propagation of the convective system as the surface momentum flux F m interacts with the near-surface wind u h through the velocity scale u * (Sect. 2.1). To identify the role of the dynamic feedback, we perform an additional simulation where we suppress the influence of F m on u h . As F m is a function of u * , we disable this feedback by setting u * to a constant value for the computation of F m . We prescribe u * as 85 a constant value of 0.09 m s −1 obtained by averaging u * over the simulation domain and the last 20 simulation days in UB0.
For the mechanism denial experiment, u * is temporally and spatially constant to disable the dynamic feedback, but remains variable for computation of w θ and w q in order to retain the WISHE feedback. By restarting a simulation with an uncoupled F m and without u b from day 22 when a single convective cluster is surrounded by the dry areas, we find that the organization of convection is not affected by the uncoupled F m (not shown). In the same way as the experiments with coupled F m , u b of 90 2 m s −1 is imposed on the mechanism denial experiment after day 26. The experiment with uncoupled F m will be denoted by UB2_unius.

Propagation speed of the organized convective cluster
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 kg m −2 , and define a convective cluster with the grid points 95 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 u abs is calculated as the sum of the relative velocity of the cluster to the model grid u rel and the mean wind speed u b : When u rel = 0 m s −1 , the convective cluster remains motionless in the model reference frame but is effectively moving at the speed of u b by virtue of the Galilean transformation (pure advection). In the case of WISHE, the convective cluster moves against the mean wind (e.g., u rel < 0 m s −1 ). Thus, we expect u abs < u b if the WISHE feedback regulates the propagation of the convective cluster. 0.29 ± 0.76 m s −1 for UB0, UB2 and UB4, respectively. The strong fluctuation around the mean is due to the oscillating features of aggregation Windmiller and Hohenegger, 2019;Patrizio and Randall, 2019). 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. 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 attain 115 a velocity that exactly compensates for the mean flow, so that they become stationary with respect to the surface.
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 u abs . Convection is expected to locate over the maximum boundary layer equivalent potential temperature θ e . Hence to understand how WISHE affects its distribution we calculate the flux of θ e approximately as w θ e ≈ w θ + 120 lv cp p0 p Rc p cp w q . Its form is analogous to the enthalpy (or moist static energy) flux. Focusing on the budget of θ e allows us to investigate whether the development of convection is associated with the positive anomaly of the surface enthalpy flux. We focus on two simulation periods: the transient phase for the first five days (day 0-4) when u abs prominently decreases and the quasi-stationary stage for the last five days (day 15-19) when u abs is near-zero. Quantities are averaged over these periods. However, the spatial distribution of w θ e does not remain asymmetric with respect to the convective center, but rather becomes symmetric in the quasi-stationary stage.
In the model, the surface enthalpy flux is determined by the difference between the wind speed near the surface and the 135 velocity of the surface, which is equal to 0 m s −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 (Fig. 3), but u h shows the same transition from asymmetry to symmetry over time as seen in w θ e (Fig. 3 top). Immediately after u b is imposed, u h is intensified on the upwind side and reduced on the downwind side as one would expect from a superposition of u b and the local circulation associated with the convective cluster. In the later 140 stage of imposing u b , u h attains a comparable magnitude of wind speed on the upwind and downwind sides. For UB4, the off-centered local minimum of u h around r = 0 km is due to the strong modeled wind u on the downwind side in the opposite direction to u b . The distribution of u h 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.

Dynamic process
145 Without Coriolis force, the tendency of the horizontal wind is obtained as follows: with V being the three components of the 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 = p p0 R d cp . The third term on the righthand side represents the contribution of friction to the wind tendency and is related to F m (Eq. 1). The vertical profile of the x-component of the wind in the quasi-stationary stage differs from the initially prescribed shear-free profile for UB2 and UB4, 150 while remaining constant with height for UB0 and UB2_unius (Fig. 4 left). When u b interacts with F m , the horizontal wind is substantially slowed down, particularly near the surface (Fig. 3 top).

Transient phase
Quasi-stationary stage As seen in w θ e and u h , the spatial distribution of F m 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. 2 bottom). A larger F m corresponds to a stronger drag on u h . As a result of the intensified u h , the enhanced F m on the upwind side exerts a strong drag on u h in the transient phase, 155 and consequently, reduces u h on the upwind side in the quasi-stationary stage. In contrast, the suppressed F m on the downwind side generates a weak drag, allowing u h 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 acerbates the mean wind until it balances the mean wind, thereby eliminating the asymmetry in the drag by symmetrizing u h . As a result, the symmetric u h in the quasi-stationary stage affects not only the spatial distribution of F m but also that of w θ e . To analyze the role of F m for the propagation of the convective cluster, we perform an additional simulation where u * is kept constant in space and time for the calculation of F m but remains interactive for w θ and w q based on the similarity functions with imposing u b of 2 m s −1 (Sect. 2.2). Due to the constant value of u * , the domain-averaged w u lingers close to zero with small fluctuations for UB2_unius, while being negative immediately after imposing u b for UB2 (Fig. 1 bottom). The showing that the convective cluster indeed moves against u b with a very small value of u rel (Fig. 5). The propagation speed is only about 5 % smaller than u b of 2 m s −1 , suggesting that this small difference between u abs and u b can be associated with 170 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 PW thermo (t 1 ) is obtained by adding the surface moisture flux anomaly ρ w q integrated over a time period [t 0 , t 1 ] to the initial 175 PW at t 0 : This simple thermodynamic argument gives us a displacement of PW thermo (46 h) from PW(0 h) of approximately 10 km, which corresponds to u rel = −0.06 m s −1 and therefore u abs = 1.94 m s −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 m s −1 for UB2_unius (Fig. 1 top) and confirms that the thermodynamic contribution to the propagation speed 180 of a convective cluster is small.

Conclusions
This study analyzes how organized deep convection propagates in an imposed mean flow, and which processes modulate the propagation speed of the convective cluster. 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 301 K. We hypothesize that the convective cluster propagates against 185 the mean flow through the WISHE feedback, providing a favorable environment to develop convection on the upwind side of the cluster (Fig. 6 left). Our idealized simulations with the mean flow exhibit that organized deep convection initially propagate 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 190 the upwind side of the convective cluster, and increases it on the downwind side (Fig. 6 left). 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 few days the surface relative winds and the surface-relative motion of the convective cluster vanishes. Even in the simulation with the dynamic feedback removed and the WISHE-induced asymmetry in surface fluxes preserved, the effect on the convection propagation of convective clusters is small.

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The periodic boundary conditions are limitations of our study in this regard, 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 latter. To the extent that WISHE is important for the propagation of convective self-aggregated systems, the experimental setup favors wave-like anomalies, rather than solitary.
Because of the analogy of radiative convective equilibrium to tropical climate, the implication for less idealized setups and 200 tropical phenomena such as the MJO is worth further investigations. Compared to typical wind speeds in the tropics, the prescribed large-scale wind speed of up to 4 m s −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. Despite these more complex interactions, the importance of surface momentum fluxes on WISHE suggests a potentially important role of dynamic feedbacks for the propagation of convection and the modification of thermodynamic feedbacks in less idealized setups.