1. Introduction



The concept of mass flux provides a concise description of moist convection and a practical way to parameterize the convection-induced vertical transport in large-scale models. Using large-eddy simulations (LESs), Siebesma and Cuijpers (1995) and Wang and Stevens (2000) showed that the convective fluxes computed by Eq. (2) can account for 80%–90% of the total fluxes of conservative thermodynamic variables, such as liquid water potential temperature θl and total water mixing ratio qt, in the cloud layer of shallow cumuli, indicating that Eq. (2) provides a good estimate of the vertical transport of heat and moisture associated with shallow cumulus clouds. The mass-flux approach was originally developed for parameterizing deep cumulus convection and has been widely accepted by the community. Recently, Liu et al. (2015) examined the mass-flux representation of vertical transport of water vapor using the explicitly resolved fluxes by the cloud-resolving model (CRM) simulations of a midlatitude continental squall-line case and a tropical mesoscale convective complex case. However, although CRMs may appropriately simulate coherent convective features of deep moist convection, they cannot resolve large turbulent eddy circulations that are also important to the vertical transport. To the author’s knowledge, no evaluation has been done at the LES scale to examine to what extent the convective fluxes computed by Eq. (2) can represent the total vertical transport of heat, moisture, and hydrometeors in the cloud layer of deep convection. In recent studies of mass-flux parameterization of deep moist convection, Arakawa and Wu (2013) and Wu and Arakawa (2014) developed a generalized mass-flux framework for cumulus parameterization. Their proposed “unified parameterization” framework eliminates the assumption of σ ≪ 1 used by the traditional mass-flux parameterization to allow for a smooth transition to an explicit simulation of cloud-scale processes as the resolution increases. But the accuracy of using convective fluxes computed by Eq. (2) to represent the total vertical transport of heat and moisture in the cumulus layer is not discussed in their papers.
Therefore, the motivations of this study are to advance our understanding of vertical transport processes in the cloud layer of moist convection and to distinguish the differences between vertical transport of horizontal momentum and vertical transport of conservative thermodynamic variables. To do so, LESs of a number of well-documented and studied cloud cases are performed. The generated numerical data allow us to quantify the individual contributions of explicitly simulated atmospheric flow from turbulent eddies to coherent convective features to the vertical transport of heat, moisture, and momentum in the cloud layer, and to evaluate to what extent the convective fluxes of conservative thermodynamic variables and horizontal momentum components computed by Eq. (2) can represent the total fluxes in the cloud layer of both deep and shallow moist convection. Specifically, this paper attempts to answer the following questions: 1) How much of total fluxes of conservative thermodynamic variables in the cloud layer of both deep and shallow convection can be represented by the convective fluxes computed by the mass-flux formula? 2) Can the mass-flux approach be extended to represent the vertical momentum transport in the cloud layer? 3) How does the partition of total fluxes of dynamic and thermodynamic variables based on the mass-flux top-hat profile differ from the flux decomposition in terms of eddy scales? The paper is organized as follows. Section 2 describes the LES setup and cloud cases simulated in this study. The simulation and analysis results are presented in section 3 followed by a summary of this study.
2. LES setup and cloud cases
Six well-documented and studied cloud cases are investigated in this study. These include four shallow cloud cases and two deep convective cases. The shallow cloud cases are the Barbados Oceanographic and Meteorological Experiment (BOMEX) case (Siebesma and Cuijpers 1995), the Rain in Cumulus Over the Ocean (RICO) case (VanZanten et al. 2011), the Atlantic Stratocumulus Transition Experiment (ASTEX) case (Bretherton et al. 1999), and the Second Dynamics and Chemistry of the Marine Stratocumulus field study (DYCOMS II), first research flight (RF01) case (Stevens et al. 2005; Zhu et al. 2005). These cases provide a good representation of maritime boundary layer (MBL) clouds over the vast ocean under different climatological conditions from typical trade wind shallow cumulus and shallow clouds in transition regions to stratocumulus capped by a strong sharp inversion. Note that the parameterization of stratocumulus in large-scale models is not based on the mass-flux approach. The reason to include the stratocumulus case DYCOMS-II-RF01 in this study is to compare the vertical transport process in stratiform clouds with that in the convective clouds. The initial profiles, surface conditions, and large-scale forcing of these four cases can be found in the corresponding references provided in the paper. The simulations of all four cases are forced by the prescribed surface fluxes. The BOMEX and RICO cases are run for 6 h, and the ASTEX and DYCOMS-II-RF01 cases are run for 9 h. Since all four simulations reach the quasi-steady steady after the initial spinup period due to the prescribed surface forcing used in the simulations, the model outputs from the last 3 h are used for analysis.
As stated previously, small-scale turbulence may play an important role in momentum transport; thus, to understand the momentum transport processes in the cloud layer, high-resolution LESs are needed to resolve a range of scales from small-scale turbulence to coherent convective features. Combining this consideration and our computational ability, the model horizontal grid spacing used for all four cases is set to 25 m with a horizontal grid mesh of 640 × 640 covering an area of 16 × 16 km2. The vertical grid spacing is set to 20 m for the three shallow cumulus cases and 5 m in the cloud layer for the stratocumulus case, DYCOMS-II-RF01. The model top varies from case to case, but it is set to the height well above the cloud layer.
The two deep convective cases are selected from the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE; Webster and Lukas 1992) and the GARP Atlantic Tropical Experiment (GATE; Mason 1975). The domain size of 16 × 16 km2 used for simulating shallow cloud cases is too small to realistically simulate deep convective cloud cases, but the available computational resources do not allow for a substantial increase in the model grid mesh. As a compromise, the horizontal grid spacing is increased to 100 m but the grid mesh of 640 × 640 is still kept the same so that the model domain covers an area of 64 × 64 km2. The vertical grid spacing is also increased, 50 m below 2 km and then gradually increasing to 150 m until the model top at 23.3 km. Such an increase of model grid spacing may be acceptable for deep convection because convective elements are much stronger than those of shallow convection, and thus presumably the relative importance of turbulent eddies may be reduced. From the turbulent energy spectrum perspective, large turbulent eddies are energy-containing eddies. A key requirement of an LES is that the model grid spacing should fall in the inertial subrange, so that the model explicitly simulates large energy-containing eddies while parameterizing small eddies that are more isotropic and less flow dependent. It is, thus, unclear if the simulated results will be substantially affected because of the reduced model resolution. However, according to Bryan et al. (2003), 100-m horizontal resolution seems to be sufficient for a realistic simulation of deep moist convection. In a recent LES study of GATE convection by Khairoutdinov et al. (2009), a 100-m grid spacing is also used but with a much larger grid mesh of 2048 × 2048 × 256. The statistics of some key variables from the GATE simulation performed in this study appears to be consistent with those from Khairoutdinov et al.’s (2009) “Giga-LES” (see Fig. 2), suggesting that the smaller horizontal grid mesh (640 × 640) used in this study is acceptable. The simulation of the TOGA COARE case starts at 1800 UTC 18 December 1992, and the GATE case is selected from phase III starting at 0000 UTC 1 September 1974. Both simulations are executed for 24 h in which the surface fluxes are determined interactively based on the prescribed sea surface temperature (SST). Figure 1 shows the initial vertical profiles, large-scale forcing, and SST used for the simulations of the two deep cases. Note that different forcing strategies are used for the GATE and TOGA COARE simulations. In the GATE simulation, the total tendencies of temperature and moisture are imposed, whereas the horizontal advective tendencies and large-scale vertical velocity are imposed in the TOGA COARE simulation. However, to make an appropriate comparison of large-scale forcings between the two simulations, the total tendencies for the TOGA COARE case are plotted in Fig. 1. The large-scale forcing for the GATE and TOGA COARE simulations were derived from field experiments and are the same as those used in previous numerical simulations (e.g., Xu and Randall 1996; Grabowski et al. 1996; Wu et al. 1998; Redelsperger et al. 2000; Mechem and Oberthaler 2013). As shown in Figs. 1e–h, the large-scale forcing of the TOGA COARE case is stronger than that of the GATE case. The selection of deep convective cases with relatively weak and strong large-scale forcing is based on the consideration that the goal of this study is to evaluate the extent that the basic assumption of mass-flux approach is valid for convection under different forcing conditions over the vast tropical and subtropical ocean. Nonetheless, the peak total tendencies of the GATE case (Figs. 1e and 1g) are comparable to the mean GATE large-scale forcing used by Khairoutdinov et al. (2009). Table 1 summarizes the basic model configurations of the six cloud cases simulated in this study. Nudging is not activated in all six cases. This is based on the consideration that the focus of this study is on investigating vertical transport processes associated with moist convection, and thus a fully free development of eddy circulations with different scales is most important. All LESs are performed by the System for Atmospheric Modeling (SAM). For detailed information of model dynamics and physics, please refer to Khairoutdinov and Randall (2003). Three-dimensional data are collected at a 2-min interval. In this study, the domain size for shallow and deep moist convection is 16 × 16 km2 and 64 × 64 km2, respectively. For all cases investigated in this study, the mean wind speed below 10 km is less than 10 m s−1. Thus, the eddy residence time for shallow and deep convection is 26.7 and 106.7 min, respectively, assuming the advection speed is 10 m s−1. This indicates that 2-min output can sample an eddy in the domain sufficient times to obtain reliable statistics. The comparison of several key variables (e.g., wind components, thermodynamic variables, and hydrometeors) shows that the statistics calculated from the 2-min output is nearly identical to the SAM online statistics that collects samples at each time step. Since the two-dimensional fast Fourier transform (2D-FFT) performed in this study (to be shown later) uses the 2-min model output, for consistency all analyses presented in this paper are done using the 2-min model output.
Initial vertical profiles and large-scale/surface forcing of GATE and TOGA COARE. (a) Initial potential temperature θ. (b) Initial water vapor mixing ratio q. (c) Initial x-direction and y-direction wind components. (d) Sea surface temperature. Total temperature tendencies for (e) GATE and (f) TOGA COARE. Total moisture tendencies for (g) GATE and (h) TOGA COARE.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
Time evolution of domain-mean simulated cloud updraft mass flux of (a) GATE and (b) TOGA COARE, (c) domain-mean surface rainfall, and (d) domain-mean surface buoyancy fluxes of GATE and TOGA COARE. White vertical lines in (a) and (b) denote the time periods analyzed in this study. Black and red curves in (c) and (d) are scaled by the black and red x axes, respectively. (e)–(h) Domain-mean vertical profiles of vertical velocity (m s−1) in the updraft core, mass flux (×103 kg m−2 s−1) of the updraft core, cloud fraction (%), and precipitation flux (mm day−1), respectively, averaged over the deep convective episode indicated by the vertical white lines in (a) and (b) for GATE and TOGA COARE. An updraft core is defined to have vertical velocity >1 m s−1 for 500 m and more. The shades indicate the standard deviations of the domain-mean vertical profiles over the episode of interest.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
Basic model configuration of six cloud cases.
3. Analysis results
Unlike the four shallow cloud cases, which are forced by the constant surface fluxes, the surface fluxes in the simulations of the two deep convection cases are determined interactively based on the prescribed SST; thus, a quasi-steady state cannot be reached. Rather, the simulated convection shows a substantial temporal variation. Figures 2a–d show the time evolution of simulated cloudy updraft mass flux, surface rainfall, and surface buoyancy fluxes
To further assess the fidelity of the simulations of the two deep convective cases, Figs. 2e–h show the vertical profiles of several key variables describing the main characteristics of moist convection averaged over the convective episodes of GATE and TOGA COARE defined in Figs. 2a and 2b, where an updraft core is defined to have vertical velocity greater than 1 m s−1 for 500 m or more. This definition of core is the same as that used by Khairoutdinov et al. (2009). The updraft core vertical velocity, updraft core mass flux, cloud fraction, and precipitation flux of the GATE case have similar magnitude and vertical structure to those of Khairoutdinov et al. (2009). There are two major differences between the two simulations. One is that the convection in Khairoutdinov et al.’s (2009) simulation is a little deeper than the GATE simulation performed in this study. The other is that cloud fraction above 5 km in our simulation shows a double-layered structure, which is not shown in Khairoutdinov et al.’s (2009) simulation. These differences may be attributed to the different large-scale and surface forcing used in the simulations and should not affect the vertical transport analyses and main conclusions presented in this paper. While the vertical profiles of these variables from the TOGA COARE simulation share the main characteristics of the GATE simulation, they do show some interesting differences. One of them is the cloud fraction, which remains small at most of the altitudes in the TOGA COARE simulation except for the cloud top near 15 km. The impact of this difference in cloud fraction on vertical transport will be discussed in detail shortly. The statistics of these key variables from our TOGA COARE simulation is qualitatively consistent with that of Mechem and Oberthaler (2013), who simulated 20 days of TOGA COARE but used 200-m horizontal resolution and a grid mesh of 512 × 128 × 165, which is much smaller than the grid mesh of 640 × 640 × 186 used in this study. The good convection simulations summarized in Figs. 2e and 2h suggest that the numerical data generated in this study are appropriate for investigating the vertical transport processes of deep moist convection.
Using Eq. (1), the LES-resolved kinematic vertical fluxes of θl, qt, and horizontal momentum components (i.e.,
Decomposition of the vertical fluxes of liquid water potential temperature (K m s−1) using Eq. (1) based on the coherent cloud updraft defined by qc > 10−3 g kg−1 and w > 0.01 m s−1 of the six cloud cases. Blue, black, red, and green curves denote the total fluxes resolved by LES, convective fluxes associated with the coherent features computed by Eq. (2), flux components induced by the perturbations inside the coherent features, and environment in which the coherent features are embedded, respectively. The dashed line indicates the zero line. The profiles are averaged over the last 3 h for shallow cloud cases and the convective episode of deep cloud cases, GATE and TOGA COARE, indicated respectively, in Figs. 2a and 2b.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
As in Fig. 3, but for the vertical fluxes of the total water mixing ratio (g kg−1 m s−1).
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
The decomposition of momentum fluxes (Figs. 5 and 6) shows a different story from that of
As in Fig. 3, but for the vertical fluxes of the x-direction wind component (m2 s−2).
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
As in Fig. 3, but for the vertical fluxes of the y-direction wind component (m2 s−2).
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
The different performance of the mass-flux formula in representing the total vertical fluxes of conservative thermodynamic variables and horizontal momentum components should reflect the fundamental difference in governing vertical transport of conservative scalar and nonconservative momentum induced by eddy circulations. Physically, all convective clouds are developed from thermal plumes or cells originated at the surface that have similar thermodynamic properties. During adiabatic rising, the convective plumes or cells conserve their reversible thermodynamic properties, such as liquid water potential temperature and total water mixing ratio, to a good approximation, implying that there should be a good correlation between updraft and conserved thermodynamic properties. This suggests that most of the vertical transport of conservative thermodynamic variables should be carried out by a few strong convective updrafts. However, convective plumes or cells do not necessarily possess similar properties of horizontal momentum components since thermal plumes or cells are generated mainly due to inhomogeneous heating but not by dynamic forcing. Therefore, inside a plume updraft, momentum components may have a large variation leading to both positive and negative momentum perturbations (i.e., +u′, −u′, +υ′, and −υ′), so that the cancellation of positive and negative w′u′ and w′υ′ within the plume updraft may result in small net momentum fluxes. Moreover, as pointed out by Wu and Arakawa (2014), horizontal momentum components are not conservative variables due to the pressure gradient force. Hence, as convective updrafts rise, the change in momentum components can lead to further changes in momentum fluxes from height to height. From the mechanism of an eddy-generation perspective, in addition to buoyancy production, eddy circulations can also be generated by dynamic instabilities, such as the Kelvin–Helmholtz instability due to wind shear, which tends to produce small-scale eddies. The updrafts/downdrafts of small eddies are effective in vertical momentum transport, as they can generate large momentum fluxes, w′u′ and w′υ′. The small shear-driven eddies overlapped onto the relative large convectively driven plumes or cells provide an explanation of the difference between the vertical transport of conservative thermodynamic variables and the vertical transport of nonconservative horizontal momentum seen in the simulations.
The difference in mass-flux decomposition of vertical fluxes of conservative thermodynamic variables in the two deep convection cases shown in Figs. 3e, 3f, 4e, and 4f is interesting. Analyses show that the substantial underestimation of the LES-resolved fluxes of θl and qt by the convective fluxes computed by Eq. (2) in the GATE case is mainly due to the large fraction of cloud updraft defined by qc > 0.001 g kg−1 and w > 0.01 m s−1 used for flux decomposition. As an illustration, Figs. 7a–f show the time series of simulated vertical fluxes of total water mixing ratio qt, cloud updraft fraction defined by different thresholds of vertical velocity (qc > 0.001 g kg−1 and w > 0.01 m s−1 and qc > 0.001 g kg−1 and w > 0.7 m s−1), and the ratios of convective fluxes to the total resolved fluxes of qt corresponding to the different cloud updraft thresholds at 5-km altitude of the GATE case compared with those obtained from the TOGA COARE case. The time evolution of the convective updraft fraction defined by qc > 0.001 g kg−1 and w > 0.01 m s−1 in the two cases shows a similar characteristic; that is, a peak fraction occurs at an early time during the simulations, which is most likely associated with the model spinup. After the peak, the convective updraft fraction decreases and then reaches a quasi-steady state. In the TOGA COARE case, the convective updraft fraction after the spinup is consistently smaller than 5%, whereas the convective updraft fraction after the spinup in the GATE case is around 10%. The convective fluxes associated with the cloud updraft computed by Eq. (2) are able to account for most of the LES-resolved fluxes after the spinup period (about 70%–100%) in the TOGA COARE case, but they substantially underestimate the LES-resolved fluxes in the GATE case (less than 50%). We have done sensitivity experiments on the flux decomposition using different thresholds of vertical velocity to define the coherent convective features. The results show that the ratio of the convective fluxes computed by Eq. (2) to the LES-resolved fluxes consistently increases as the vertical velocity threshold increases. When the vertical velocity threshold increases to 0.7 m s−1, the convective updraft fraction in the GATE case reduces to the value similar to that of the TOGA COARE case (Figs. 7c and 7d). As a response, the ratio of the convective fluxes to the total resolved fluxes in the GATE case also increases to the value similar to that of the TOGA COARE case.
Time series of domain-mean vertical fluxes of total water mixing ratio (g kg−1 m s−1) at 5-km altitude for (a) GATE and (b) TOGA COARE. (c),(d) Cloud updraft fractions (%) at 5-km altitude for (c) GATE and (d) TOGA COARE. Blue and red curves indicate the cloud updraft fraction defined by qc > 10−3 g kg−1 and w > 0.01 m s−1 and qc > 10−3 g kg−1 and w > 0.7 m s−1, respectively. (e),(f) Ratio of convective fluxes to total resolved fluxes of total water mixing ratio at 5-km altitude for (e) GATE and (f) TOGA COARE. Blue and red curves indicate the ratio determined by the two differently defined cloud updrafts. (g)–(j) Decomposed vertical fluxes of GATE averaged over the convective episode indicated in Fig. 2. The decomposition is based on the cloud updraft defined by qc > 10−3 g kg−1 and w > 0.7 m s−1. The meaning of blue, black, red, and green curves is as in Figs. 3–6.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
To further confirm the importance of the fraction of coherent convective elements to the mass-flux decomposition, Figs. 7g–j show the decomposition of the LES-resolved vertical fluxes of GATE using Eq. (1) based on the coherent cloud updraft defined by qc > 10−3 g kg−1 and w > 0.7 m s−1. Compared with Figs. 3–6, the representation of the LES-resolved fluxes of θl and qt by the corresponding convective fluxes is significantly improved (Figs. 7g and 7h). This result supports the basic dynamic view of the mass-flux approach, that the organized structure consisting of strong updrafts and the associated downdrafts are mainly responsible for the vertical transport of conservative thermodynamic variables. When cloud updraft fraction is small, a few strong updrafts and the associated downdrafts can be well described by the mass-flux top-hat profile, and the contribution of the internal variation within the updrafts–downdrafts to the total transport [i.e., the first two terms in Eq. (1)] is negligible. However, as the convective updraft fraction becomes larger, the more complicated updraft–downdraft structure will be difficult to represent by the top-hat profile, and thus a higher chance for the mass-flux approach to fail. In this case, the large convective updraft fraction increases the internal variations within the updrafts and this is what we see in Figs. 3e and 4e—that the internal variation within the convective updraft contributes significantly to the total vertical fluxes of θl and qt. It should be pointed out that the threshold of 0.01 m s−1 may be too small for defining the convective updrafts. Such defined updrafts under certain circumstances may include both convective and stratiform clouds that may have very different thermodynamic properties. For example, in a study of GATE deep convection, LeMone and Zipser (1980) used a threshold of 0.5 m s−1 to define the deep convective updraft. In the GATE case simulated in this study, a value of 0.7 m s−1 significantly improves the performance of the mass-flux approach. However, a much smaller threshold (0.01 m s−1) of vertical velocity seems to work well for the TOGA COARE deep convective case and all shallow convective cases, suggesting that finding an appropriate vertical velocity threshold for defining the convective updraft in mass-flux parameterization is not scientifically trivial. Recently, Arakawa and Wu (2013) and Wu and Arakawa (2014) developed a unified mass-flux parameterization framework in which convective updraft fraction can be determined internally within the framework. Their study virtually provides a way to solve this problem. If the method for determining convective updraft fraction could appropriately account for the strong convective updrafts responsible for vertical transport, then the unified mass-flux system would significantly improve moist convection parameterization in large-scale models. Another possible method to solve this problem is to use multiple updrafts instead of a single updraft to encompass the complicated dynamic structure of moist convection. As shown by Arakawa and Wu (2013) and Liu et al. (2015), the mass-flux representation of vertical transport of heat and moisture can be improved by using multiple updrafts.
As a comparison, Fig. 8 shows the cloud updraft fraction defined by qc > 0.001 g kg−1 and w > 0.01 m s−1 of the six cloud cases averaged over the same periods as those in Fig. 3. For all the cumulus convection cases (BOMEX, RICO, ASTEX, and TOGA COARE) except for the GATE case, the cloud updraft fraction is consistently smaller than 5%. In all these cases, the mass-flux convective fluxes represent well the total fluxes of θl and qt in the cloud layer. However, in the GATE case, both the cloud updraft fraction and cloud buoyant core (defined by qc > 0.001 g kg−1 and
Cloud updraft fractions defined by qc > 10−3 g kg−1 and w > 0.01 m s−1averaged over the last 3 h for shallow cloud cases and over the convective episode for deep cloud cases indicated in Fig. 2. The cloud buoyant core fraction defined by
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
However, as indicated by Figs. 7i and 7j, no improvement is seen in the representation of total vertical momentum transport by the convective momentum fluxes due to the increased threshold of vertical velocity. This confirms that the poor representation of the total momentum fluxes in the cloud layer by the mass-flux approach is due to the fundamental difference in the physical processes that govern the vertical transport of conservative thermodynamic variables and nonconservative momentum components stated previously.
Although the mass-flux decomposition of fluxes using Eq. (1) can clearly reveal the difference between momentum and heat/moisture transport in the cloud layer (Figs. 3–7), the mass-flux top-hat profile used for decomposition oversimplifies the vertical transport induced by overturning circulations with different scales from small turbulent eddies to coherent convective cells. To better understand the transport processes in terms of scales, the LES-resolved vertical fluxes are also decomposed using 2D-FFT. A detailed description of 2D-FFT is provided in the appendix. The computed spectral energy from model output provides a way to examine the fidelity of the LESs performed in this study. Figure 9a shows an example of the normalized azimuthally integrated spectral energy of u, υ, w, θl, and qt at 290-m altitude averaged over the last 3 h from the simulation of the BOMEX case, where the definition of normalized azimuthally integrated spectral energy can be found in the appendix. In the inertial subrange, all spectra closely follow the Kolmogorov −
(a) Normalized power spectra (%) of different variables at 290-m altitude of BOMEX averaged over the last three simulation hours. (b)–(e) Phase spectra averaged over the cloud layer and over the last three simulation hours for shallow cloud cases and the convective episode for deep cloud cases indicated in Fig. 2. Here, the cloud layer is defined by qc > 10−3 g kg−1 for the shallow cloud cases. For deep cloud cases, the cloud layer is taken as 500–11 650 m for GATE and 500–15 550 m for TOGA COARE. Note that the phase spectrum of
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
To better understand the differences between momentum transport and heat/moisture transport processes in the cloud layer, the phase relationship between vertical velocity and various variables is first examined. Figures 9b–e shows the azimuthal-mean phase spectra of
Figure 10 shows the normalized azimuthally integrated cospectra
(a),(c),(e),(g),(i),(k) Normalized cospectra (%) averaged over the same cloud layer and the time period as that defined by Fig. 9. (b),(d),(f),(h),(j),(l) Accumulated normalized cospectra (%) over three total wavenumber index ranges: 1–40, 41–80, and ≥81.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
The cospectra of momentum fluxes are not only substantially different from those of
The RICO case shares the basic characteristics of the BOMEX case except that eddies with a total wavenumber index around 50 (scale of 320 m) have a negative contribution to the LES-resolved
The cospectra of deep convection cases do show a certain similarity to those of shallow cumulus cases in that eddies with high wavenumber indices contribute significantly to the momentum fluxes in both directions, although an absolute comparison in terms of scales is impossible since the model resolution used for simulating shallow and deep cloud cases are different. The main differences are in the low wavenumber indices. In the TOGA COARE cases, low wavenumber circulations have a large negative contribution to
Figures 11 and 12 show the vertical profiles of the LES-resolved
Decomposition of the vertical fluxes of liquid water potential temperature (K m s−1) by 2D-FFT of the six cloud cases. Blue, black, red, and green curves denote the total fluxes resolved by LES flux components accumulated over total wavenumber index 1–40, 41–80, and ≥81, respectively. The dashed line indicates the zero line. The profiles are averaged over the same period as that in Fig. 3. The shaded area indicates the cloud layer.
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
As in Fig. 11, but for the vertical fluxes of the total water mixing ratio (g kg−1 m s−1).
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
The vertical profiles of momentum fluxes decomposed by 2D-FFT (Figs. 13 and 14) show much more complicated characteristics than those of
As in Fig. 11, but for the vertical fluxes of the x-direction wind component (m2 s−2).
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
As in Fig. 11, but for the vertical fluxes of the y-direction wind component (m2 s−2).
Citation: Journal of the Atmospheric Sciences 72, 12; 10.1175/JAS-D-14-0332.1
4. Conclusions and discussion
How to appropriately represent the vertical transport processes associated with moist convection in large-scale models is a long standing problem in numerical weather forecasting and climate simulations. While the mass-flux approach provides a physical base for cumulus parameterization, under what conditions the convective fluxes formulated based on the mass-flux top-hat profile can represent the vertical transport of heat and moisture in the cloud layer and whether the same approach can be extended to the parameterization of momentum transport have not yet been thoroughly addressed. These two questions are investigated in this study using LESs of six well-documented cloud cases, including deep and shallow convective clouds as well as stratiform clouds. Two methods are used to decompose the LES-resolved vertical fluxes: the decomposition within the mass-flux framework in terms of coherent convective features defined by a top-hat profile and the decomposition by 2D-FFT in terms of wavenumbers. The main conclusions of this study are summarized as follows.
The convective fluxes computed based on the coherent cloud updraft can account for most of the vertical fluxes of conservative thermodynamic variables in the cloud layer for both shallow convective and stratiform cloud cases, consistent with previous studies. However, the simulations of the two deep convection cases, GATE and TOGA COARE, show that a good representation of the total vertical transport of conservative thermodynamic variables in the cumulus layer using the mass-flux approach requires an appropriate definition of the convective updraft since the convective fluxes computed based on the mass-flux formula depend strongly on the threshold of vertical velocity used for defining the convective updraft. While a too small threshold can smooth out the strong convective updrafts responsible for vertical transport by including too many weak updrafts, a too large threshold can eliminate some of the convective updrafts important for vertical transport. From the mass-flux decomposition perspective [i.e., Eq. (1)], the former causes a large internal variation within the updrafts, whereas the latter leads to a large internal variation within the environmental downdrafts. In both cases, the inappropriately defined convective updraft can cause a significant underestimation of the total vertical transport in the cumulus layer by the mass-flux formula through unrealistically increasing the contribution of internal variation within the convective updrafts to the total transport in the former and increasing the environmental contribution in the latter, but both contributions from the internal variations within the convective updraft and environment are neglected in the mass-flux parameterization. This problem may be solved by the unified mass-flux framework recently proposed by Arakawa and Wu (2013) and Wu and Arakawa (2014) in which the convective updraft fraction can be predicted internally within the framework. If their method could appropriately account for the change in convective updraft fraction under different forcing conditions, then the mass-flux parameterization would be significantly improved. In a recent study, Liu et al. (2015) showed that the representation of convective transport can be significantly improved using multiple updrafts. This could provide another possible way to improve the mass-flux representation of convection-induced vertical transport.
All cloud cases investigated in this study show that the mass-flux approach provides a poor representation of vertical momentum transport in the cloud layer. The decomposition by 2D-FFT and other analyses suggest four reasons that might be responsible for the different behaviors between the vertical transport of conservative thermodynamic variables and nonconservative momentum components. First, compared with the conservative thermodynamic variables that possess relatively simple and sometimes well-defined vertical structure, the complicated momentum distribution in the cloud layer cannot be well described by the simple top-hat profile. Second, small-scale eddies are more efficient in carrying momentum than in carrying conservative thermodynamic variables in the cloud layer. This is because shear-driven small-scale eddies can generate highly correlated perturbations of vertical velocity and horizontal winds, resulting in large w′u′ and w′υ′ but not necessarily large
Acknowledgments
This work is supported by the National Science Foundation under Grant AGS-0847332 and the BP-sponsored Gulf of Mexico Research Initiative. The author is very grateful to the three anonymous reviewers for their constructive comments, as their helpful suggestions led to improvements in this paper. All data used in this study can be accessed online (http://vortex.ihrc.fiu.edu/download/momentum_transport/).
APPENDIX
2D-FFT


























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