Diurnal Cycle, Large-Scale Wind Speed, and ... - AMS Journals

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Factors Controlling Rain on Small Tropical Islands: Diurnal Cycle, Large-Scale Wind Speed, and Topography SHUGUANG WANG Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York

ADAM H. SOBEL Department of Applied Physics and Applied Mathematics, Columbia University, New York, and Lamont-Doherty Earth Observatory, Columbia University, Palisades, New York (Manuscript received 30 November 2016, in final form 24 July 2017) ABSTRACT A set of idealized cloud-permitting simulations is performed to explore the influence of small islands on precipitating convection as a function of large-scale wind speed. The islands are situated in a long narrow ocean domain that is in radiative–convective equilibrium (RCE) as a whole, constraining the domain-average precipitation. The island occupies a small part of the domain, so that significant precipitation variations over the island can occur, compensated by smaller variations over the larger surrounding oceanic area. While the prevailing wind speeds vary over flat islands, three distinct flow regimes occur. Rainfall is greatly enhanced, and a local symmetric circulation is formed in the time mean around the island, when the prevailing large-scale wind speed is small. The rainfall enhancement over the island is much reduced when the wind speed is increased to a moderate value. This difference is characterized by a change in the mechanisms by which convection is forced. A thermally forced sea breeze due to surface heating dominates when the largescale wind is weak. Mechanically forced convection, on the other hand, is favored when the large-scale wind is moderately strong, and horizontal advection of temperature reduces the land–sea thermal contrast that drives the sea breeze. Further increases of the prevailing wind speed lead to strong asymmetry between the windward and leeward sides of the island, owing to gravity waves that result from the land–sea contrast in surface roughness as well as upward deflection of the horizontal flow by elevated diurnal heating. Small-amplitude topography (up to 800-m elevation is considered) has a quantitative impact but does not qualitatively alter the flow regimes or their dependence on wind speed.

1. Introduction In much of Earth’s tropics, landmasses are sparse and scattered in the form of islands. Even relatively small islands can strongly influence the spatiotemporal patterns of deep convection (Sobel et al. 2011; Robinson et al. 2008; Smith et al. 2012; Li and Carbone 2015). Some authors have speculated that island formation over geological time has influenced important largescale climate patterns such as the Walker circulation (e.g., Molnar and Cronin 2015). More generally, because islands present constant and known perturbations to the atmosphere’s lower boundary, they present an opportunity for understanding the regulation of deep convection in the tropics. Corresponding author: Shuguang Wang, [email protected]

One region of particular interest is the Maritime Continent (MC), where thousands of islands with complex topography strongly influence the character of the deep convection that occurs over them. The islands induce strong diurnal cycles in precipitation, as well as spatial structures in time-mean rainfall that closely follow mountains and coastlines (e.g., Biasutti et al. 2012). These influences make the islands rainier than the surrounding ocean in the time mean. At intraseasonal time scales, however, there is more variability in rainfall over the ocean than over the large islands of the MC (Sobel et al. 2008, 2010). Weather over the islands is nonetheless closely related to the passing of the Madden–Julian oscillation (MJO), although with a different relationship than over the ocean. Over the islands, precipitation exhibits intraseasonal variations that are not in phase with the larger-scale MJO phases over the surrounding oceans,

DOI: 10.1175/JAS-D-16-0344.1 Ó 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

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instead leading the latter by approximately a week (Ichikawa and Yasunari 2006, 2008; Wu and Hsu 2009; Rauniyar and Walsh 2011, 2013; Peatman et al. 2014; Vincent and Lane 2016; Zhang and Ling 2017). Understanding both the time mean and intraseasonal variability over islands likely requires understanding the diurnal cycle. Convective rain over tropical landmasses— including larger islands in many cases—has a strong diurnal cycle, peaking in the late afternoon (e.g., Yang and Slingo 2001; Yang and Smith 2006; Kikuchi and Wang 2008; Hassim et al. 2016; Vincent and Lane 2016). The recent study by Peatman et al. (2014) found that 80% of the MJO precipitation signal in the MC is accounted for by changes in the amplitude of the diurnal cycle. This might suggest that numerical models that poorly capture the diurnal cycle of precipitation (as most global models do) might be unable to forecast the influence of the MJO over the MC properly. However, physical understanding of the diurnal cycle of rain over islands of varying size, topographic relief, and large-scale environment remains limited. The influences of large-scale wind, land–sea contrast in both thermal and mechanical properties, and topography are of particular interest here. Both topography and surface roughness exert mechanical forcings as horizontal wind impinges on an island from the nearby ocean, while the island also exerts a thermal forcing due to surface solar heating. We would expect the mechanical forcing to be more important as wind large-scale low-level speed increases and to have no inherent diurnal cycle (other than whatever diurnal cycle the large-scale wind has). The elevated heating imposed on the atmosphere by insolation-driven sensible heat flux from a mountain’s surface, on the other hand, should be inherently diurnal. Consistent with these qualitative ideas, a global survey of small tropical islands (Sobel et al. 2011) showed that diurnal cycles over small islands are stronger in the low-wind regime of the MC than in the high-wind regime of the Caribbean. Large-eddy simulations of convection over heated terrain by Kirshbaum (2011) indicated that deep convection can be suppressed by even relatively light large-scale prevailing winds. Analyses of observations from the Caribbean island of Dominica have shown that low-level wind controls observed shallow convection over that Caribbean island, with a transition from thermal to mechanical forcing as wind speed increases (Smith et al. 2012; Nugent et al. 2014). Though tropical, Dominica’s large-scale environment is less favorable to deep convection than that in the MC (for example), with lower sea surface temperature (SST) and a drier free troposphere. Several observational studies in the MC have, however, also demonstrated that diurnal cycles of convection may be related to the low-level

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wind there (e.g., Ichikawa and Yasunari 2006, 2008; Virts et al. 2013) in various MJO phases. These studies are informative but do not constitute a comprehensive characterization of the response of island rainfall to island size, topography, large-scale wind, and thermodynamic environment. Theory offers only limited guidance. The dynamics of dry flow over topography has been thoroughly studied in the past (e.g., Smith 1989; Roe 2005; Lin 2007). In these studies, the flow’s regime dependence on the several key parameters—such as wind speed, topography height, and aspect ratio—are often succinctly described by the Froude number and the aspect ratio of the topography. A limited number of numerical modeling studies have suggested that a similar characterization is also useful for understanding the transient behavior of moist orographic flow regimes (e.g., Jiang and Smith 2003; Smith and Barstad 2004; Colle 2004; Chen and Lin 2005; Galewsky and Sobel 2005; Galewsky 2008), with appropriate redefinition of the Froude number to account for the presence of moisture. Most of this work, however, treats the environment as essentially stable, without considering explicitly the role of triggered free convection. Yet surely such convection occurs over tropical islands during the course of the diurnal cycle. We expect in general that the dynamics of deep convective cells might lead to different precipitation patterns compared to purely mechanically forced flow (e.g., Miglietta and Rotunno 2009, 2010). Certainly, we know that deep convection occurs over tropical islands even in the absence of any large-scale wind, so mechanical forcing is clearly not relevant in that limit. The transition between that case and those with larger wind speeds, where mechanical forcing becomes more relevant, is of particular interest here. In this study we examine precipitation due to presence of a small island in idealized cloud-resolving simulations. Our focus is on the roles of large-scale wind speed, land– sea contrast, topography, and the diurnal cycle, with a goal of characterizing how these parameters control the precipitation in a deep convective environment. The rest of the paper is organized as follows. Section 2 shows observational results. Section 3 contains a description of the numerical experiments, followed by discussions of the numerical results in section 4. Conclusions are presented in section 5.

2. TRMM observations of precipitation over three islands We first present some results, derived from the precipitation radar (PR) aboard the Tropical Rainfall Measuring Mission (TRMM) satellite, on the relationship

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between low-level wind speed and rain over several islands. Specifically, we composite precipitation observed in individual TRMM PR swaths with respect to 850-hPa wind speeds over a couple of small islands in the Maritime Continent. The resolution of the native satellite footprints (approximately 5-km pixel size) is sufficiently fine to resolve precipitation over terrain with horizontal scales of tens to hundreds of kilometers, and the multiyear record we have assembled at this resolution (1998–2007) allows us to construct a climatology that makes these fine details apparent, as demonstrated in Biasutti et al. (2012). The TRMM precipitation 2A25, version 6, dataset for precipitation and wind from the ECMWF-Interim reanalysis (Dee et al. 2011) are used to derive composites over several tropical islands: Manus, Papua New Guinea, in the tropical western Pacific, as well as Java and Sumatra, Indonesia. For convenience, the TRMM PR data in the rectangular area 2.28–28S, 146.58–147.28E are used over Manus. Because satellite swaths often do not cover the whole areas of larger islands, only part of the larger islands are considered: 88–6.58S, 1088–1148E for Java and 38S–08, 1008–1048E for Sumatra. The bins for wind speeds are ,1, 1–2, 2–4, 4–6, and .6 m s21 for Java and Sumatra and ,2, 2–4, 4–6, and .6 m s21 for Manus. The number of bins for Manus is one fewer than for the others because there are considerably fewer data points with wind speeds ,1 m s21 at Manus. Figure 1 shows the relationship between rain and lowlevel wind speeds. Over all three islands, rainfall is greatest when 850-hPa wind speed is near zero and decreases as winds increase until rain reaches a minimum for wind speeds ;5 m s21. As wind speed increases further, rain increases either slightly (over Manus) or significantly (Sumatra). The nonmonotonicity of the functional relationship between rain and wind speed is consistent over all the three islands regardless of the differences in their sizes and locations, suggesting that it is a feature worthy of further study.

3. Design of idealized numerical experiments We study precipitation over islands surrounded by the ocean in simulations of radiative–convective equilibrium (RCE). We use the Advanced Research Weather Research and Forecasting (WRF) Model, version 3.3 (Skamarock et al. 2008). Boundary layer turbulence and vertical subgrid-scale eddy diffusion are treated with the Yonsei University (YSU) scheme (Hong and Pan 1996; Hong et al. 2006). This is a first-order closure scheme but also includes nonlocal countergradient transport (Troen and Mahrt 1986). In this scheme, boundary layer height is determined by the local Richardson number, temperature, and wind speed. Horizontal subgrid-scale eddy

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FIG. 1. Climatological-mean precipitation as a function of 850-hPa wind speed over three islands: (a) Manus, (b) Java, and (c) Sumatra. Winds from the ERA-Interim data and precipitation data composited from individual satellite swaths from the TRMM 2A25, version 6, data (1997–2006) with 5-km resolution are used to derive the composite.

mixing is parameterized using the 2D Smagorinsky firstorder closure scheme performed in physical space. The bulk microphysics scheme is the Morrison scheme (Morrison et al. 2009). This scheme has prognostic equations for both mixing ratio and number concentration of six species of hydrometeors: water vapor, cloud water, cloud ice, rain, snow, and graupel. The surface moisture and heat fluxes are parameterized following Monin–Obukhov similarity theory. We explore the limit of infinite Bowen ratio over the land: moisture availability and the latent heat flux are both zero over the island. The idealized numerical experiments are performed on the equatorial f plane (zero Coriolis force). As a result, we expect most synoptic variability to occur in the

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form of convectively coupled gravity waves. A number of dynamically active flow features will emerge over the island—for example, land and sea breezes, cold pools and gravity currents, and gravity waves (e.g., Qian 2008; Qian et al. 2010, 2013; Mapes et al. 2003; Love et al. 2011; Robinson et al. 2011, 2013; Moron et al. 2015). Each of these may potentially play important roles in organizing convection near the coastal regions. We will analyze these mesoscale flow features in our simulations. Our computational domain is a long narrow strip of ocean with an idealized island in the center. The computational domain is 3000 km long in the zonal (x) direction. To partially accommodate the three-dimensional development of convection, the computational domain also spans 60 km in the meridional (y) direction. The computational domain is doubly periodic. The horizontal grid spacing is 3 km. A total of 60 vertical levels are used with stretched grid spacing. The nominal top of the numerical domain is at ;30-km altitude, and both diffusion and implicit damping are used in the topmost 10 km to prevent unrealistic momentum transport from excessive gravity wave activity in some simulations. The presence of an island in an otherwise uniform ocean surface introduces several inhomogeneities in the surface conditions. First, because of the small effective heat capacity of the island surface, surface temperature over land varies greatly due to varying insolation over the course of a day, with day–night differences often on the order of 10 K. Second, the aerodynamical surface roughness length over land is significantly greater than that over ocean. In our experiments, it is set to a typical value of 0.1 m over the island. It is diagnosed over sea through the Charnock’s relation with a typical value on the order of 1024 m. Third, island topography, if present, introduces forced lift at the surface. Fourth, the Bowen ratio is different over land than ocean, with relatively greater sensible heat flux, in general. We will examine the roles of the first three differences in our experiments with a fixed Bowen ratio of infinity, meaning zero latent heat flux over land. The parameter space of Bowen ratio will be explored in the future. Radiative cooling is represented by a simple Newtonian relaxation scheme (Wang and Sobel 2011; Wang et al. 2014; Anber et al. 2015): 8 21 > < 21:2 K day QR 5 195 K 2 T > : 5 days

for

T . 201K . otherwise

The troposphere is cooled at constant rate of 1.2 K day21, which is close to the observed climatological value in Earth’s tropical troposphere. The

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stratospheric temperature is near constant at 195 K. Cronin et al. (2015) concluded that cloud–radiation interaction is not essential for the precipitation enhancement over an island. Hence, our simplified treatment not only reduces the complexity of the problem but also has additional benefit of substantially reducing computational cost. To simplify the treatment of diurnal cycles and avoid the need to compute a surface energy budget, we specify surface temperature at the lower boundary. Specifically, temperature at the island surface is varied sinusoidally from 238 to 338C over a day, while SST elsewhere is fixed to 288C. As a result, the diurnal-mean surface temperature over the island equals that of the ocean. Other studies have opted to use surface heat flux as an independent parameter at the surface (e.g., Robinson et al. 2008). Our choice of prescribed surface temperature eliminates this free parameter and allows surface fluxes to vary. Our working hypothesis is that wind speed has a strong effect on the characteristics of convection over small islands. We prescribe the prevailing wind speed by relaxing the horizontal wind V toward a given wind profile Vt as [V] 2 Vt ›V 1⋯52 , ›t t where the square brackets indicate the horizontal average over the entire domain, t is the relaxation time scale (taken as 1 h), and Vt 5 (U, 0). This relaxation of the horizontal wind does not interfere with local mesoscale circulations since the forcing is uniform over the entire domain. We will examine precipitation over the island for four different values of the prevailing wind speed: 0, 5, 10, and 15 m s21. In the rest of this article, we use the convention that the prevailing wind blows from west to east (or from left to right in the figures shown later). The mean zonal wind is maintained by relaxing the domain-averaged zonal wind to these prescribed values at a 1-h time scale. Numerical experiments with these different prevailing wind speeds will be referred to as U0, U5, U10, and U15, respectively. Variations in vertical wind shear in the free troposphere are not considered in this study. We will focus on convection over a flat island of 138-km width first. Sensitivity to island size and dimensionality of the computation domain will be discussed later. Table 1 summarizes the parameter values of the moist WRF numerical experiments. All the numerical experiments are initialized with temperature, pressure, and humidity archived from prior small-domain RCE experiments, and integrated for 4 months. The first 20 days are discarded, and 3-hourly data from the remaining 100 days (containing

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TABLE 1. List of moist WRF numerical experiments. Numerical experiments 21

Wind speed (m s ) Topography (m) Island width in 2D (km)

Island width in 3D (km)

Parameter values tested 0, 5, 10, 15 0, 400, 800 For U 5 0 m s21: 5, 11, 21, 31, 41, 61, 81, 121, 181, 241, 301 For U 5 5 m s21: 11, 21, 41, 61, 81, 121, 181, 241, 301 For U 5 10 m s21: 21, 41, 61, 81, 121, 181, 241, 301 For U 5 15 m s21: 21, 41, 61, 121, 181, 241, 301 For U 5 0 m s21: 6, 12, 30, 42, 66, 96, 126, 138, 192, 318 For U 5 5 m s21: 12, 30, 66, 96, 138, 192, 252, 510 For U 5 10 m s21: 12, 30, 42, 66, 96, 138, 192, 252 For U 5 15 m s21: 12, 30, 66, 126, 138, 192, 252

100 samples of the diurnal cycle) are used to compute mean values.

4. Results a. Precipitation Figure 2 shows a Hovmöller diagram of daily precipitation over a 30-day period for U 5 0, 5, 10, and 15 m s21 over the flat island. Sporadic precipitation episodes are seen throughout the domain in the simulations at lower U (0 and 5 m s21). Larger persistently wet and dry regions are also evident over or near the island. For larger U (10 and 15 m s21), convection is organized into slowly westward-propagating features with many fast eastward-propagating waves embedded. The estimated phase speed of these features relative to ground is ;27.5 m s21 for U10 and ;22.5 m s21 for U15, respectively. This gives flow-relative phase speed of ;17 m s21 for both cases, which is close to the speed of the large-scale convective envelopes found in some previous work (e.g., Grabowski and Moncrieff 2001). The domain-averaged time-mean precipitation in all these simulations is approximately 3.9–4.5 mm day21, consistent with the energetic constraint imposed by the nearly constant radiative cooling in these radiative– convective equilibrium runs. Without the island, the time-mean precipitation in statistical equilibrium would be distributed evenly in space because no other factors would be present to break the symmetry imposed by the translationally invariant boundary conditions and forcings, especially given the absence of cloud–radiative feedback (e.g., Wing and Cronin 2016). However, the presence of the

FIG. 2. Hovmöller diagrams of daily surface rain rate over the idealized island as a function of horizontal distance x (km) and time t (days).

island leads to great spatial inhomogeneity. For the U 5 0 cases, P is much enhanced over the island (Fig. 3a) and has a minimum a few hundred kilometers away from the island. The time-mean P is nearly symmetric about the island center. As the prevailing wind U is increased to 5 m s21 (Fig. 3b), the local maximum over the island is

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further increased to 10 and 15 m s21 (Figs. 3c and 3d, respectively), strong asymmetry in P is evident over the island, with a local maximum on the windward side and a minimum on the leeward side. This is a signature of gravity waves, generated as the large-scale wind encounters either the elevated heat source associated with the convection or the increased roughness length over the flat island, as discussed in detail below. The spatial distribution of precipitation is also progressively smoother with greater U. The mean rainfall P over the island with respect to the mean prevailing wind U is summarized in Fig. 4 (red curve), which shows a clearly nonmonotonic relationship. The increase of P as U increases over the island indicates a fundamental shift in the dynamical regime: rainfall is greatly enhanced when the prevailing largescale wind speed is small (0 m s21), but there is little rainfall enhancement when the wind speed is 5 m s21. This nonmonotonic relationship between rain and wind speed is qualitatively consistent with observations, as shown above. Because of the imposed diurnal variation of surface temperature over the island, P also shows a diurnal cycle, though one whose amplitude is a strong function of imposed wind speed. Figure 5 shows composites of P over the island. For U 5 0 m s21, a strong diurnal cycle is prominent over the island with a peak in the late afternoon, ;1500–1800 local time. Both the windward and leeward sides show nearly identical diurnal variations in P. For large U (515 m s21; Fig. 5d), the diurnal cycle becomes muted over the island. The U 5 5 and 10 m s21 cases are more complex, and it is helpful to discuss the leeward and windward sides separately. In general, the windward side of the island shows a weak maximum in the morning, while the leeward side has a peak 63 h around local noon. For U 5 5 m s21, the leeward side dominates over the windward side, and the diurnal cycle is dominated by the variations over the leeward side. For U 5 10 m s21, both the leeward and windward sides show their own diurnal cycles with nearly equal amplitude.

b. Dynamic regimes: Sea breeze and gravity waves FIG. 3. Time-averaged rainfall from 120-day integrations of three-dimensional RCE over the tropical ocean with a small island, represented by the small red box, in the middle of the computational domain, as a function of horizontal distance x (km), for U 5 (a) 0, (b) 5, (c) 10, and (d) 15 m s21. Positive U indicates winds blowing from west/left to east/right.

weakened while the local minimum becomes more pronounced on the leeward side of the island. The reduction of precipitation over the island is related to the weakening of the sea breeze, as discussed later. As U is

In this section we examine the dynamical processes in the different regimes that occur for weak and strong large-scale wind speed. We will show that the low-wind regime is characterized by deep ascent over the island and a shallow sea breeze, while the high-wind regime is dominated by gravity waves. Figure 6 displays the timeaveraged vertical motion w as a function of horizontal position x and vertical position z, centered over the island. For the U0 cases, broad ascent is observed over the island with two distinct maxima, one at upper levels and

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FIG. 4. Time-mean precipitation (solid) over the island as a function of prevailing wind speed for three different topographic heights (0, 400, and 800 m). Time-mean domain-averaged precipitation (dashed) for all the cases is shown as a reference, which varies little in these RCE integrations.

another at 900 hPa. This shallow circulation is a signature of the sea breeze as seen in diurnal variations of w (Fig. 7). The U0 cases have maximum upper-level ascent, presumably associated with deep convection, lagging the surface temperature maximum by ;6 h (Fig. 7a), while the low-level shallow ascent peaks slightly earlier, at ;1500 local time. Broad descent is found in the evening and early morning. As with precipitation, circulation is nearly symmetric with respect to the island center in the low-wind cases. Composite diurnal cycles of perturbation zonal wind u, vertical velocity w, and horizontal vorticity h (defined as h 5 ›u/›z 2 ›w/›x) show the solenoidal circulations associated with the land–sea breeze (Fig. 8) in the lower troposphere. In the U0 case, vortical flow characterized by ascent over the gradually heated island and weaker descent offshore begins to develop at 9 h nearly symmetrically at both ends of the flat island where horizontal temperature gradients are strongest. As the island surface warms continually afterward, the vortical motion intensifies and moves inland as a result of nonlinear horizontal advection by the gravity wave current. By ;15 h, the two local vortical circulation branches almost meet at the center of the island. The estimated speed is ;2 m s21 or less given that the gravity current front is located at x 5 50 km at 9 h and x 5 0 (near the center of the island) at 15 h. Broad ascent associated with deep

FIG. 5. Composite of precipitation (mm day21) diurnal cycles over the entire island for four cases: U 5 (a) 0, (b) 5, (c) 10, and (d) 15 m s21 as a function of horizontal distance and time. Two diurnal cycles are shown for clarity. White arrows indicate background wind speed in (b)–(d).

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FIG. 6. Time-averaged vertical motion (m s21) within 450-km radius from the center of the island. Horizontal bars indicate the location of the island.

convection is established in the later afternoon from 15 to 18 h. The diurnal evolution of the land–sea breeze circulation at the coast area in U0 differs markedly from linear theories of sea breeze (Rotunno 1983; Crook and Tucker 2005; Qian et al. 2009, 2012; Jiang 2012; Kirshbaum 2013) in several aspects. First, the sea breeze

FIG. 7. Diurnal composites of vertical motion (m s21) on the (left) windward and (right) lee sides of the island as a function of time (horizontal axis) and height (km; vertical axis).

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FIG. 8. Diurnal composites of vorticity (s21) and wind vectors for the two flat island cases—(left) U0 and (right) U5–from (top) 9 to (bottom) 21 h, as functions of horizontal distance x and height. Vertical velocity is scaled by 100 for visualization. Vertical bars indicate the edges of the island.

in the afternoon maximum is significantly stronger than the land breeze in the early morning. The temporal asymmetry is related to turbulent mixing of heat within a boundary layer of diurnally varying depth. The heated surface during the day significantly raises the boundary layer top, while the cooled surface over the night leads to a shallow stable boundary layer. This leads to a shallow land breeze during the night and a deep sea breeze during the day. The linear theory assumes temporally symmetric variations in heating through the

diurnal cycle (night is the exact opposite of day) and, consequently, does not predict such a diurnally asymmetric circulation. Second, slantwise gravity wave packets radiating from the center of the island in linear theories are not visible in our simulations. However, the timing of the ascent and its proximity to the surface resembles the growth–decay solution found in Kirshbaum (2013). Indeed, a closer look at the time evolution of boundary layers indicates that a neutral or weakly unstable boundary layer develops during the daytime

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underneath the stable lower troposphere because of heating at the surface, and the top of the boundary layer reaches its maximum altitude in the early afternoon. Kirshbaum (2013) shows that the vertical stratification profile provides a favorable environment for this growth– decay solution and inhibits radiating gravity waves. In contrast to the U0 case, the U5 case develops strong asymmetry at both ends of the island throughout the day. Progression of the upstream vortical motion associated with the sea breeze is overall similar to that in the U0 case but is significantly weaker. The negative horizontal vorticity patch spreads horizontally, covering the island by midafternoon (15 h). After that, descent develops over the windward side of the island associated with the sea breeze. The leeward vortical motion associated with the sea breeze begins to develop offshore in the morning as the U0 case with stronger amplitude, and it stays offshore over the whole day, limiting its influence on rain over the island. For the stronger wind cases (U . 10 m s21), the circulation changes completely. In these cases, horizontal vorticity is not a useful indicator of circulation strength as it spreads over the island. Large-scale vertical motion is no longer single signed (Fig. 7); instead, ascending and descending motions alternate with height, and they differ from the windward to the leeward sides. The diurnal evolution of the solenoidal circulation may be diagnosed by considering the equation for horizontal vorticity: ›h ›B 5 ADV(h) 2 1 ⋯, ›t ›x

(1)

where the first term is the nonlinear advection of h and the second term is the horizontal gradient of buoyancy B [defined as B 5 (g/Q)u, where g is gravity, and Q 5 300 K]. Other terms, such as diffusion, have been neglected. The retained terms are computed using variables averaged in the y direction and composited over a whole diurnal cycle. Figure 9 shows the contributions of the two terms on the right-hand side to the vorticity tendency averaged over the upwind side and downwind side of the island from the surface to 1 km for diurnalcycle composite. The horizontal buoyancy gradient is the dominant contribution, while nonlinear advection of vorticity is not important. The negative vorticity on the downwind side in U5 is attributed to the weak horizontal gradient of buoyancy (blue dashed). Hence the weakening of solenoidal sea breeze with increasing wind speed can be attributed to a reduced horizontal buoyancy gradient due to horizontal temperature advection. Essentially, the imposed large-scale wind blows the land–sea thermal contrast away.

FIG. 9. Diurnal variation of horizontal buoyancy gradient (s 22 ) and nonlinear advection (s 22 ) in Eq. (1) from the U0 (red) and U5 (blue) cases at the upwind (solid) and downwind (dashed) sides of the island (solid), as a function of local time of day.

To further show how the horizontal buoyancy gradients—essentially sea-breeze fronts—control horizontal vorticity, we take the time derivative of the above equation, and keep only the horizontal gradient term (Kirshbaum 2013) as ›Qt › ›2 h › ›B g ›u › ›u 5 2 u 1 w , 1 ; 2 ›t2 ›t ›x Q ›x ›x ›z ›x ›x where Qt is the total diabatic heating and notation for other variables follows standard convention. Here, the heating terms considered are microphysical and boundary layer turbulent mixing. The three terms are horizontal gradient of diabatic heating, horizontal advection of u, and vertical advection of u. Assuming a periodic solution in time, then we have

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g ›Qt › ›u › ›u u 2 w . 2 h; Q ›x ›x ›x ›x ›z

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(2)

Figure 10 shows time series of the three terms using diurnal composites of the state variables (so eddy contributions as deviations from the composite diurnal cycle are further neglected) for the U0, U5, U10, and U15 cases, averaged in the area of negative vorticity (the upwind half of the island from surface to ;1 km; Fig. 8). In U0, the first term dominates over the other terms; while in the other cases, the horizontal advection of u becomes substantially positive in the early afternoon, largely canceling vorticity generation due to horizontal gradient of heating. In all the cases, the vertical advection remains negligible in the near-neutrally stratified environment. This indicates that wind blowing relatively cool air toward the center of the island is the primary cause of the reduced solenoidal circulation in the nonzero-wind cases. The diurnal variations of vertical motion on the windward and leeward sides differ significantly owing to the influence of the prevailing wind U (Figs. 6d and 7d). The alternating regions of ascent and descent are signatures of stationary gravity waves. Two types of largeamplitude gravity wave structures may be identified: 1) those on both edges of the island, which have opposite phases but the same horizontal wavelength, around ;6 km, similar to the linear gravity wave solution in Qian et al. (2009, 2012), but barely resolved here because of limited resolution and 2) mountain waves with half-wavelength approximately the width of the island. Over the course of the day, these gravity waves show weak diurnal variations. Steady hydrostatic topographic waves have the simple dispersion relationship: m 5 N/U, where U is the mean wind speed, m is the vertical wavenumber, and N is the static stability [e.g., section 9.4 in Holton (2004)]. This indicates that vertical wavenumber is inversely proportional to zonal wind speed and independent of the dimensions of topography. Given the typical values of the other parameters (U 5 10 m s21, N ; 1022 s21), the vertical wavelength is estimated to be ;6 km for U10 and 9 km for U15, both in reasonable agreement with the results of the numerical experiments (Figs. 6c and 6d). The cases with intermediate wind speed (5 m s21) are transitional between the low- and high-wind regimes. The U5 case shows a shallow sea-breeze circulation, but only in the leeward side, while the windward side shows relatively weak circulations. The strong diurnal cycle of the leeside circulation is clearly related to the strong diurnal cycle in precipitation (Fig. 5b). Many observational and numerical studies have shown that precipitation over tropical oceans is

FIG. 10. Diurnal evolution of the three contributing terms (K s21 m21) to horizontal vorticity in Eq. (2): (a) horizontal gradient of diabatic heating, ›Qt /›x; (b) horizontal gradient of horizontal advection of potential temperature, 2(›/›x)(u›u/›x); and (c) horizontal gradient of vertical advection of potential temperature, 2(›/›x)(w›u/›z). All these terms are vertically averaged in the left half of the island between the surface and the tenth model level (;1 km).

closely related to free-tropospheric humidity (e.g., Sherwood 1999; Derbyshire et al. 2004; Bretherton et al. 2004; Wang and Sobel 2012; Raymond and Flores 2016). We have examined column-integrated water vapor in these numerical simulations, however, and found that it varies by less than 2 mm over a diurnal cycle. This suggests that variations in moisture do not play a significant role in our results, and we do not expect moisture–convection feedback to be crucial for understanding convection over the islands in these simulations. This is not inconsistent with our understanding that this feedback is important for other tropical convective phenomena, only that other factors are more important here because moisture variations are weak.

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The diurnal evolution of both the thermally forced solenoidal sea-breeze circulation and the mechanically forced gravity waves in these experiments suggests that the transition between these different dynamical phenomena as wind speed varies is the cause for the nonmonotonic variation of precipitation under different prevailing winds. The thermally forced sea breeze maximizes in the zero-wind case, and its decrease in strength with wind speed may be associated with the precipitation decrease as wind speed increases from 0 to 5 m s21, with its effect continuing to dominate over the relatively weak gravity waves in that lower-wind-speed range. On the other hand, forced gravity waves and their increasing amplitude in the high-wind regimes are associated with the increase in precipitation as the largescale wind speed increases from 5 to 15 m s21, while the sea breeze plays a much smaller role in that range.

c. Effect of surface friction versus diurnal cycle The diurnal cycle in surface temperature, the significantly larger roughness length over the land (0.1 m) and contrast in Bowen ratio are the primary sources of spatial inhomogeneity differentiating the surface of a flat island from the surrounding ocean. Any diurnal variations in rain can only be attributed to the diurnal cycle in surface temperature, since the other external factors do not vary in time. These diurnal variations can rectify so that the diurnal cycle influences the time-mean precipitation (Cronin et al. 2015). At the same time, the roughness heterogeneity perturbs surface winds, generates convergence near the surface over the island, and produces gravity waves. These dynamical effects of the island’s enhanced surface roughness can also contribute to variations in time-mean precipitation with wind speed. Hence, it is not clear which of the spatial inhomogeneities is primarily responsible for the variations in time-mean surface precipitation distribution with wind speed. It may be argued that Bowen ratio and diurnal cycles of surface temperature are not independent. Wet soil in general has smaller Bowen ratio and larger heat capacity, hence smaller diurnal variations in surface temperature. In this section, we will focus on two independent effects: surface roughness and diurnal cycle in surface temperature. To assess their relative importance, we perform two sets of mechanism denial experiments: one without a diurnal cycle, and the other without surface drag. In the experiments without a diurnal cycle, the surface temperature over the island is set to the same value as over the ocean (301.15 K). For the experiments testing the influence of surface drag, the surface stress tensor is set to zero so that the nonzero roughness does not influence the momentum flux (the free-slip boundary conditions, with du/dz 5 0). One might argue it is simpler to use the same

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surface roughness length everywhere. However, this would cause great reduction in surface heat fluxes over the island (because the frictional velocity is a function of roughness length) and diminish the diurnal cycle of heating. Disabling surface drag eliminates the surface roughness inhomogeneity without this unintended thermal effect. The two sets of experiments are performed for the U0, U5, and U15 cases. For the U0 cases, switching off the diurnal cycle completely shuts off all rain over the island owing to lack of land–sea breeze and surface evaporation over the island (blue curve in Figs. 11a and 11d), while surface roughness has very little influence (red curves in Figs. 11a and 11d). For U5 (Figs. 11b and 11e), the no-diurnal-cycle experiment produces more rain upstream of the island, reduced rain over the island (Fig. 10e), and much reduced rain downstream (100–400 km). Reduction in rain over the windward side with a diurnal cycle is likely due to inland progression of the sea breeze as shown in Fig. 8. In the no surface drag experiment, the diurnal cycle is able to produce a similar rainfall distribution over the island as that with surface drag, but much more rain downstream (100–400 km). The diurnal cycle and surface drag appear to have opposing effects on precipitation: the former reduces rain over the windward and increases it over the leeward side, while the latter does the opposite. For U15 (Figs. 11c and 11f), switching off either effect greatly reduces rain. The two experiments, both producing local rain maxima at the windward side and rain shadows at the leeward, indicates that gravity waves—generated either from the surface roughness induced convergence or from convective heating— are responsible for precipitation responses, and the two effects work cooperatively to produce more precipitation than either individual effect alone would, since both generate gravity waves that are nearly in phase with each other. The contrasting behavior in the U5 and U15 case further indicates that the sea breeze is important in the low-wind regime, while gravity waves are crucial in the high-wind regime.

d. Islands of different sizes Two sets of experiments are performed to examine the effects of island size. In the first set of experiments, horizontal resolution and number of grid points are kept the same, while the length of the island is increased from 6 to 300 km for U 5 0, 5, 10, and 15 m s21. The second set of experiments is performed with a smaller horizontal grid spacing, dx 5 1 km, and reduced dimensionality (no y dimension) to save computing time (Table 1). Figure 12 shows that the precipitation decreases from U 5 0 to 5 m s21 and increases as prevailing wind is increased for island larger than 90 km and smaller than

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FIG. 11. (top) Time-mean precipitation as a function of horizontal distance x over the island and surrounding ocean. (bottom) Diurnal cycles of rain over the island as a function of local time from the runs with either no diurnal cycle or no surface drag.

300 km in both the 3- and 1-km runs. However, it is more complicated for smaller and larger islands. Precipitation for smaller island is the largest for U 5 0 m s21, while cases with different U do not show consistent differences with different

resolutions and model dimensions. For U 5 0 m s21, a local maximum is found for islands 30–40 km wide. This local maximum is similar to those found by Robinson et al. (2008) and Cronin et al. (2015). For U 5 5 m s21, the

FIG. 12. Rain over flat islands of different sizes for prevailing wind speeds of 0, 5, 10, and 15 m s21. (a) 3D experiments with horizontal grid spacing dx 5 3 km. (b) 2D experiments with dx 5 1 km.

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local maximum increases to 60 km in the dx 5 3-km runs but not in the dx 5 1-km runs. The local precipitation maximum shifts to larger island with further increases in prevailing wind: x 5 80–90 km for U 5 10 m s21 and x 5 120 km for U 5 15 m s21. The results from 3D dx 5 3-km and 2D dx 5 1-km runs are broadly consistent.

e. Effect of topography We consider the effect of topography in this section. The topography of the island is specified in the domain as a function of x: 8 h i4 > < Hm 1 1 cos p x 2 x0 , if jx 2 x j , a 0 a H 5 16 , > : 0, else where Hm, a, and x0 are the peak height, half-width, and center of the island, respectively. Here we take a 5 69 km, the same as in the case of the flat island examined in previous sections. This form also yields a smooth transition from island to ocean. Figure 4 shows that the presence of topography leads to a general increase of P over the island; for example, P increases from ;8.5 mm day21 for H 5 0 m to ;15 mm day21 for H 5 400 m. Unlike the U5 case with a flat island (Fig. 3b), however, topography leads to the reemergence of a local maximum in rain over the island. In other cases, topography enhances the local maximum in rain without changing its spatial structure. The U15 cases are similar to the U10 cases: strong asymmetric responses over the island and increased rainfall due to topography. In all nonzero U cases, the greater precipitation in the windward side may be expected because the mechanically forced lift, U =H, is stronger for steeper topography. The precipitation is convective in nature, however; the orographic lifting alters the statistics of precipitation by triggering deep convective cells, rather than by purely mechanical lifting of an otherwise stable flow (e.g., Miglietta and Rotunno 2009, 2010; Kirshbaum and Smith 2009). Topography changes the response of precipitation to the island quantitatively, not qualitatively, but its impact on the circulation is substantial. Vertical motion is focused in the middle of the island in the Hm 5 400-m cases (Figs. 13a and 13b) with topography, while its maxima are located at the two ends (coasts) of the flat island (Fig. 6a) for Hm 5 0 m. Inspection of the low-level circulation indicates that a diurnally varying mountain solenoidal circulation is present and that its strength increases with topographic height (not shown). On the other hand, topographic gravity waves with larger amplitude can be clearly identified in the rest of the cases, with ascent in the windward and descent in the leeward slopes, and vertical wavelength increasing with U.

FIG. 13. Time-averaged vertical motion as a function of horizontal distance x and height z for the Hm 5 400 m. Horizontal bars indicate the island.

f. Nonlinear dry solutions of diurnal cycles over small islands To shed further light on why the prevailing wind speed has such a strong controlling influence on island rainfall in our simulations, we discuss a dry initial value problem in this section. We integrate the model

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FIG. 14. Time-mean vertical velocity as a function of height from nonlinear dry solutions for (a) U0, (b) U5, (c) U10, and (d) U15.

without microphysics, moisture, or radiation for 1 day. This short integration time is used since no meaningful thermodynamic equilibrium is expected without radiative or convective physics. The initial conditions are a horizontally uniform atmosphere with a moist neutral sounding derived from the time-averaged vertical profiles of temperature in the moist runs. We vary the size of the island and the initial zonal wind speed in these dry integrations. The model is integrated for one full diurnal cycle. The surface boundary conditions are the same as in the moist run, including the sinusoidal variation in surface temperature. Vertical turbulent diffusion is sufficient to mix air heated from the surface of the island in the boundary layer. Nonlinear dry solutions from experiments in which surface drag and diurnal cycle are disabled are also computed. Figure 14 shows vertical velocity averaged over a full diurnal cycle at the same island as shown in Fig. 6. For U0, vertical velocity is confined within the lowest 1 km and the upper-level ascent found in the moist solution is absent. The sea breeze in the lower troposphere has a similar progression (not shown) as for the moist solution (left column of Fig. 8). This solution is again similar to the growth–decay solution in Kirshbaum (2013; e.g., their Fig. 4) with a stable layer capping a neutral layer in the lowest 1 km. Low-level vortical motion associated with the sea breeze is also similar to that in the

moist solution (right column of Fig. 8) except that the amplitude is stronger in the dry solutions. The U5 case (Fig. 14b) shows descent in the lowest 1.5 km, presumably associated with the gravity wave response. Increasing the prevailing wind to 10 and 15 m s21 leads to ascent in the lowest 1 km, with magnitude proportional to U. In the U15 case, descent also develops around 4 km as part of the propagating gravity wave. Similarity in w between the control runs and the nodrag runs (red curve) indicates that the effect of the diurnal cycle determines the vertical velocity in the U0 and U5 cases, while surface drag opposes but cannot fully counter its effect. However, in the U10 and U15 cases, w is similar in the control runs (gray) and the nodiurnal-cycle runs (red), indicating that surface drag appears to be more important in these high-wind cases. Nonlinear dry solutions are also obtained for a range of island sizes. Figure 15 shows the dependence of the maximum horizontal winds toward the island center within a full diurnal cycle on the island size. The horizontal wind increases monotonically and then asymptotically approaches a limit for larger islands. This nearly replicates the result of Cronin et al. (2015, their Fig. 13a). The maximum vertical velocity decreases monotonically with the island size. Our nonlinear dry solutions with varying island size confirm their results. Hence the optimum size for precipitation remains unexplained.

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wind speed as the control parameter of primary interest. Our main conclusions are summarized as follows:

FIG. 15. Maximum inland (a) velocity (wind toward the island center) and (b) vertical velocity as a function of island size, from the nonlinear dry solutions.

Considering an infinitesimally small island, we expect that its presence should have no measurable effect on rain given larger internal variability in the moist atmosphere, but its influence should increase with size (at least within some size range) as the sea breeze and mechanically forced circulations develop, with the relative importance of each depending on wind speed. On the other hand, our dry solutions show that largescale ascent decreases with island size. We speculate that the two effects together produce the size optimum. A fuller understanding of the dependence of precipitation on island size remains a topic for future research.

5. Conclusions We have investigated precipitation over small tropical islands using a cloud-system-resolving model in a long narrow domain in radiative–convective equilibrium. The 1–3-km grid spacing allows us to study precipitation and dynamics arising from prescribed diurnally varying surface temperature over a small island in a narrowocean-channel domain with fixed SST, with prevailing

1) Prevailing wind speed strongly controls rain over the island. Island precipitation is a nonmonotonic function of wind speed. It is a maximum for zero prevailing wind, decreases as wind speed increases to ;5 m s21, and increases again with further increases of wind speed. 2) The thermally forced solenoidal circulation associated with the sea breeze is strong with zero prevailing wind and weakens with increasing wind speed as horizontal advection of temperature reduces the land–sea temperature contrast, while mechanically forced gravity waves associated with surface inhomogeneity strengthen with wind speed. We conclude that the change in the circulation regimes with wind speed leads to the nonmonotonic behavior of precipitation. 3) For flat islands, the greater surface roughness (compared to that of the surrounding ocean) plays an important role in the high-wind regime, while it is much less important in the low-wind regime. This suggests that mechanical land–sea contrast is an important factor even without topography. 4) The nonmonotonic behavior of island precipitation with respect to island size—a relationship distinct from and independent of the nonmonotonic dependence on wind speed—is overall robust. The optimal island size (that which maximizes diurnal-mean island rainfall) is approximately 30–40 km in the zero-wind regime, similar to what has been found in previous studies. This optimal size increases, and the maximum as a function of size broadens, with wind speed. 5) Small-amplitude topography increases island precipitation for all wind speeds investigated here. The nonmonotonic behavior of rain with respect to wind speed is robust in the presence of topography. However, topographic waves and mountain solenoid circulations emerge as a result of topography and both enhance vertical circulation at low levels. A variety of dynamical processes may cause variations of large-scale wind in the tropics. For example, strong westerlies—say, greater than 10 m s21—can be found during the active phase of the MJO and easterlies associated with Kelvin wave component of the MJO. It is well known that intraseasonal rain anomalies maximize with the low-level zonal wind. Rain over the islands of the Maritime Continent, however, does not follow this rule. Instead, it is high during the inactive phases despite lower humidity (Peatman et al. 2014). While a proper understanding of the response of island rainfall to the

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MJO is a topic for further investigation, our results suggest that the interplay between thermally forced and mechanically forced mesoscale dynamics and convection may be relevant to this problem. Acknowledgments. We acknowledge support from the National Science Foundation under Grants AGS-1062206, AGS-1305788, and AGS-1543932 and the Office of Naval Research under PISTON Grant (N00014-16-1-3073). We are grateful to Drs. Michela Biasutti, Casey Burleyson, and Sandra Yuter for making the TRMM 2A25 data available. We thank Dr. Timothy Cronin for his critical and insightful review and one anonymous reviewer for his/her helpful comments. We thank Drs. Ronald Smith, Todd Lane, Richard Rotunno, Larissa Back, and Gilles Bellon for sharing their insights. We acknowledge high-performance computing support from Yellowstone (ark:/85065/ d7wd3xhc) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. REFERENCES Anber, U., S. Wang, and A. Sobel, 2015: Effect of surface fluxes versus radiative heating on tropical deep convection. J. Atmos. Sci., 72, 3378–3388, doi:10.1175/JAS-D-14-0253.1. Biasutti, M., S. E. Yuter, C. D. Burleyson, and A. H. Sobel, 2012: Very high resolution rainfall patterns measured by TRMM precipitation radar: Seasonal and diurnal cycles. Climate Dyn., 39, 239–258, doi:10.1007/s00382-011-1146-6. Bretherton, C. S., M. E. Peters, and L. E. Back, 2004: Relationships between water vapor path and precipitation over the tropical oceans. J. Climate, 17, 1517–1528, doi:10.1175/1520-0442(2004)017,1517: RBWVPA.2.0.CO;2. Chen, S.-H., and Y.-L. Lin, 2005: Effects of moist Froude number and CAPE on a conditionally unstable flow over a mesoscale mountain ridge. J. Atmos. Sci., 62, 331–350, doi:10.1175/ JAS-3380.1. Colle, B. A., 2004: Sensitivity of orographic precipitation to changing ambient conditions and terrain geometries: An idealized modeling perspective. J. Atmos. Sci., 61, 588–606, doi:10.1175/1520-0469(2004)061,0588:SOOPTC.2.0.CO;2. Cronin, T. W., K. A. Emanuel, and P. Molnar, 2015: Island precipitation enhancement and the diurnal cycle in radiativeconvective equilibrium. Quart. J. Roy. Meteor. Soc., 141, 1017–1034, doi:10.1002/qj.2443. Crook, N. A., and D. F. Tucker, 2005: Flow over heated terrain. Part I: Linear theory and idealized numerical simulations. Mon. Wea. Rev., 133, 2552–2564, doi:10.1175/MWR2964.1. Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system. Quart. J. Roy. Meteor. Soc., 137, 553–597. Derbyshire, S. H., I. Beau, P. Bechtold, J.-Y. Grandpeix, J.-M. Piriou, J.-L. Redelsperger, and P. M. M. Soares, 2004: Sensitivity of moist convection to environmental humidity. Quart. J. Roy. Meteor. Soc., 130, 3055–3079, doi:10.1256/qj.03.130. Galewsky, J., 2008: Orographic clouds in terrain-blocked flows: An idealized modeling study. J. Atmos. Sci., 65, 3460–3478, doi:10.1175/2008JAS2435.1.

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