A quasi balance with respect to parcel buoyancy at cloud base between destabilizing processes and convection is imposed as a constraint on convective cloud-base mass flux in a modified version of the Kain–Fritsch cumulus parameterization. Supporting evidence is presented for this treatment, showing a cloud-base quasi balance (CBQ) on a time scale of approximately 1–3 h in explicit simulations of deep convection over the U.S. Great Plains and over the tropical Pacific Ocean with the Naval Research Laboratory’s Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS).1 With the exception of the smaller of two convective events in the Great Plains simulation, a CBQ is still observed upon restriction of the data analysis to instances where the available buoyant energy (ABE) exceeds a threshold value of 1000 J kg−1. This observation is consistent with the view that feedbacks between convection and cloud-base parcel buoyancy can control the rate of convection on shorter time scales than those associated with the elimination of buoyant energy and supports the addition of a CBQ constraint to the Kain–Fritsch mass-flux closure.
Tests of the modified Kain–Fritsch scheme in single-column-model simulations based on the explicit three-dimensional simulations show a significant improvement in the representation of the main convective episodes, with a greater amount of convective rainfall. The performance of the scheme in COAMPS precipitation forecast experiments over the continental United States is also investigated. Improvements are obtained with the modified scheme in skill scores for middle to high rainfall rates.
A key issue in many cumulus parameterization schemes centers on the magnitude of the mass flux of air that rises through the cloud-base level. In the schemes currently used in the forecast systems developed at the Naval Research Laboratory (NRL), two very different treatments of convective cloud-base mass flux are employed. These schemes are the version of the Emanuel scheme (Peng and Hogan 2002; Peng et al. 2004) used in the Navy Operational Global Atmospheric Prediction System (NOGAPS) (Hogan et al. 1991) and the Kain–Fritsch scheme (Kain and Fritsch 1990, 1993) used in the Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS) (Hodur 1997; Chen et al. 2003).
The Emanuel scheme (Emanuel 1991; Emanuel and Zivkovic-Rothman 1999) parameterizes cloud-base mass flux based on the boundary layer quasi-equilibrium (BLQ) hypothesis (Raymond 1995) developed for the western Pacific warm pool (and possibly other tropical oceanic regions). The hypothesis states that most of the time the boundary layer exists in a state of balance between the tendency of surface fluxes and convective downdrafts to increase and decrease, respectively, the equivalent potential temperature of the boundary layer. Although originally developed for tropical oceanic environments, the formulation of this quasi equilibrium adopted in the Emanuel convective scheme has been employed with success in weather forecasts globally (Peng et al. 2004). In such diverse application, and particularly with the implementation described by Peng et al. (2004), the updraft source layer and associated cloud base can lie well above the boundary layer; hence, the term subcloud-layer quasi equilibrium (SLQ) employed by Emanuel and Zivkovic-Rothman (1999) may be preferable.
In contrast to the subcloud-layer control of the amount of cloud-base mass flux in the Emanuel scheme, the Kain–Fritsch scheme assumes that the mass flux is governed by the amount of available buoyant energy (ABE), as originally formulated by Fritsch and Chappell (1980). The basis for this treatment is the observation of Fritsch et al. (1976) that although typically it took many hours for the generation of potential buoyant energy in a Great Plains severe storm environment, once the stored buoyant energy became available for convection, it was consumed over a comparatively short time scale.
Although both of these treatments of convective cloud-base mass flux have been employed with some degree of success in NRL forecast models, persistent model deficiencies have continued to draw attention to the area of convective parameterization. The present study was motivated, for example, by the frequent underprediction of the number of heavy rainfall events with the version of the Kain–Fritsch scheme in COAMPS (Nachamkin et al. 2005). After an extended period of testing, it was discovered that this underprediction is considerably reduced by a modified treatment of convective cloud-base mass flux that imposes a quasi balance with respect to parcel buoyancy at the cloud-base level. In section 3a below, the existence of such a cloud-base quasi balance (CBQ) on time scales of 1–3 h is demonstrated in cloud-resolving scale simulations of U.S. Great Plains and tropical Pacific convection. The CBQ treatment adopted here, described in section 2b, follows fairly closely the SLQ treatment in the Emanuel convective scheme, which is reviewed in section 2a. The modified scheme includes changes to the updraft source-layer selection (appendix A) and convective trigger perturbations (appendix B). Results presented in section 3b from column model simulations help provide insight into the behavior of the scheme, and a study of the performance of the scheme in COAMPS precipitation forecasts over the continental United States is presented in section 4. The paper concludes with a summary and discussion.
2. Cloud-base quasi balance as a constraint on convection
The concept of CBQ applied here to convective parameterization bears a resemblance to SLQ in its focus on a supposed quasi balance between processes acting to generate and remove updraft parcel buoyancy. However, whereas SLQ focuses on a half-day time-scale quasi equilibrium in the properties of the subcloud layer, CBQ focuses on a shorter time-scale quasi balance with respect to processes impacting parcel buoyancy at cloud base. Specifically, CBQ describes a 1–3-h time-scale quasi balance at cloud base between tendencies to increase and decrease the updraft parcel virtual temperature perturbation. Whereas SLQ applies to tropical oceanic environments, CBQ may have a wider distribution. From the perspective adopted here, where no convection exists there is no cloud base to speak of; hence CBQ does not apply. It follows that the time required for the moist static energy of the subcloud layer to be restored following a convective episode is not an issue with CBQ, a feature that helps to explain the differences in time scale and expected geographical coverage between SLQ and CBQ.
The CBQ constraint described in section 2b is formulated to a large degree based on the SLQ treatment in the Emanuel convective scheme. In the Emanuel scheme, SLQ is applied in the computation of convective cloud-base mass flux Mb by parameterizing its change ΔMb at each time step according to
where α, ΔTk, and D are fixed parameters and ΔTLCL is the virtual temperature difference (typically a negative quantity) between the source-layer parcel and the environment at the lifting condensation level (LCL). The “temperature deficit” ΔTk is expected to be a positive quantity dependent on the turbulence kinetic energy at the LCL (Emanuel and Zivkovic-Rothman 1999). The parameter values currently used in NOGAPS are as follows: α = 0.05 kg m−2 K−1 s−1, ΔTk = 0.9 K, and D = 0.1. This closure formulation adjusts Mb in a manner that tends to maintain near-neutral updraft parcel buoyancy at cloud base. In particular, convective mass flux is generated as long as ΔTLCL is not less than −0.9 K.
Although the above treatment has performed well in the NOGAPS Emanuel scheme (Peng et al. 2004), questions can be raised concerning its suitability for numerical weather prediction (NWP) under certain conditions. Raymond (1995) notes, for example, that the downdraft outflows that ultimately act to maintain the quasi equilibrium can over short time scales act to initiate and organize convection. The half-day time scale ascribed to the hypothesized quasi equilibrium indeed raises doubts as to its general applicability to NWP, as does its geographical limitation to tropical oceanic regions.
One additional aspect of the treatment of SLQ in the Emanuel scheme that may negatively impact its application to NWP is its disregard for the rate of buoyant energy consumption. In its current formulation, the profile adjustment over a given time step can exceed the amount required to completely eliminate the CAPE. This aspect of the Emanuel scheme stands in contrast with the Kain–Fritsch convective scheme (Kain and Fritsch 1990, 1993), in which a sufficient amount of cloud-base mass flux is computed to reduce the ABE to a small fraction of its original value within an assigned convective time scale, typically between 1800 and 3600 s. This treatment helps to avoid overadjustment of the profile with respect to CAPE. Nonetheless, the parameterized downdraft in the Kain–Fritsch scheme can produce a very considerable amount of cooling and drying of the subcloud layer (Kain et al. 2003), substantially in excess of the amount expected from the perspective of CBQ.
Although the Emanuel treatment for cloud-base mass flux is based on an assumed SLQ on a time scale of ∼12 h, the mass fluxes can in fact adjust to perturbations on much shorter time scales. For the parameter values given above, a time step of 240 s, and constant or slowly varying values of ΔTLCL + ΔTk, the computation of Mb converges to within about 20% of the limit in 1 h (starting with Mb = 0). This feature means that the parameterized convection responds fairly rapidly to changes in cloud-base buoyancy. It is reasonable to question whether the success of the Emanuel scheme for global NWP in NOGAPS may stem to some degree from a compatibility of its mass-flux scheme in this way with the concept of CBQ.
b. Modified Kain–Fritsch convective scheme
The current Kain–Fritsch parameterization in COAMPS (Chen et al. 2003) does not differ significantly from the original scheme introduced in the early 1990s (Kain and Fritsch 1990, 1993). In this version (provided by J. Kain), a step is included that helps to speed up the scheme by filtering out nonconvective grid columns. To distinguish this code from other versions, it is referred to here as the COAMPS Kain–Fritsch scheme.
The COAMPS Kain–Fritsch scheme was modified in the present work with the implementation of a CBQ constraint on cloud-base mass flux. With few exceptions, the modified scheme is otherwise unchanged. The scheme includes a modified source-layer selection (appendix A), as well as changes to the convective trigger perturbations (appendix B).
The CBQ constraint is implemented through a two-part modification. First, a very short adjustment/update time scale, τu = 2 Δt, where Δt is the model time step, is adopted in place of the corresponding 1800–3600-s time scale in the Kain–Fritsch scheme. Referred to simply as the update time scale here, this quantity serves in both schemes as the time interval for updating the convective tendencies. In the Kain–Fritsch scheme, this time scale also controls the time over which instability can be removed by convection, and is often referred to as the convective time scale. Although this time scale is applied in much the same manner in the present treatment, here τu is generally prevented from controlling to nearly the same degree the rate of stabilization because of the added constraint (described below) placed on the cloud-base mass flux. The shortened update time scale is necessary to enable the scheme to respond in a quasi-compensatory manner during the course of a convective life cycle to destabilizing processes.
The second part of the CBQ constraint implementation is an additional constraint placed on the cloud-base mass flux Mb. For this purpose, a limiting value of cloud-base mass flux, Mmax, is computed based largely on the treatment in the Emanuel scheme. This limiting value is computed at time intervals of 2 Δt by adding to Mmax the increment ΔMmax given by
The constant parameters α and D are assigned the same values as in the NOGAPS Emanuel scheme. The perturbation threshold Tp is taken to be 0.9 K, which corresponds to the minimum value allowed here for the temperature deficit ΔTk [defined in (B1)]. The quantity ΔTLCL is defined as in (1). The maximum in (2)–(4) is taken over the selected cloud source layer and all “potential source layers” that have their base within the associated cloud (see appendix A). As detailed in the appendix, the base of potential source layers must be within 400 hPa of the first model level above the surface, as in the COAMPS Kain–Fritsch scheme. By computing the maximum of ΔTLCL + ΔTk in (2)–(4), the scheme attempts to account in some manner for the potential impact of air entrained above cloud base. Because Mmax is not solely dependent on parcel buoyancy at cloud base, in order to help ensure that a CBQ is maintained, Mb is limited such that ΔTLCL + ΔTk for the selected source layer is not adjusted below zero. The quantity Mmax is reset to zero each time the scheme does not predict a cloud at a given location.
The expression for ΔMmax in (2) is much the same as for ΔMb in the Emanuel scheme (1). The linear dependence of the generation term on the virtual temperature difference maximum [max(ΔTLCL + ΔTk )] is reasonable, since from the perspective of mass conservation, updraft parcel acceleration should generate a proportional increase in mass flux due to entrainment. This assessment is admittedly a simplified view and neglects, for example, the contribution of the perturbation pressure gradient force. In the modified treatment for cases where max (ΔTLCL + ΔTk) < Tp [(3) and (4)], the linear dependence on max(ΔTLCL + ΔTk) is replaced by one that varies between linear [where max(ΔTLCL + ΔTk ) = 0.9 K] and cubic [as max(ΔTLCL + ΔTk) approaches zero]. The lesser rate of mass-flux generation than predicted by the linear relation for such cases may be justified as an allowance for the limited amount of thermally perturbed air.
In the Kain–Fritsch scheme, Mb is computed in an iterative manner, in which different values are successively tested until the amount that reduces the ABE to 1/10 of the initial value is determined. In this process, Mb is not allowed to exceed a value ML consistent with more air entering the cloud at any level than is initially present in the model grid cell at that level
The present scheme employs a similar iteration, but with the conditions
3. Cloud-resolving scale and column model simulations
a. CBQ in COAMPS cloud-resolving scale simulations of deep convection
Supporting evidence for the addition of a CBQ constraint on mass flux in the Kain–Fritsch scheme is investigated here using cloud-resolving scale data from COAMPS triple-nested simulations carried out using a 12-h data assimilation cycle. The primary focus is a rerun with 1-km horizontal grid dimension on the inner mesh of a 1 August 1995 simulation of convection over the U.S. Great Plains from a study by Waliser et al. (2002). The original 3-km grid-dimension simulation was also employed in work by Peng et al. (2004), for testing of the Emanuel convective scheme. Results from a similar rerun with a 1-km grid dimension of a tropical Pacific simulation described by Ridout (2002) are also presented.
The data assimilation cycle for the Great Plains simulation was initiated 24 h prior to the initial forecast time, and the forecast length for this rerun was just 12 h due to the increased computational and storage requirements. The simulation domain lies within the southern Great Plains (SGP) Cloud and Radiation Testbed (CART) of the Atmospheric Radiation Measurement (ARM) Program (Stokes and Schwartz 1994). The COAMPS model version is the same as for the initial 3-km grid simulation, though the contribution of horizontal eddy mixing is included here, using the scheme of Mellor and Yamada (1982). The inner 1-km mesh grid has dimensions 127 km × 127 km, which is smaller than the 216 km × 216 km inner grid used for the initial simulation.
The Great Plains simulation produced a number of deep convective cells on the 1-km mesh grid, primarily in two convective episodes spaced approximately 8 h apart. The 1-km “grid mean” rainfall rates are plotted in Fig. 1, which shows peak values in the range of 60–90 mm day−1. For these and subsequent results, the data actually correspond to a 112 km × 112 km subdomain of the entire 1-km mesh grid in order to help ensure adequate removal from the boundaries. The rates plotted in Fig. 1 are consistent with the peak 60 mm day−1 rainfall rate analyzed for the considerably larger ARM CART domain during this time period (e.g., Waliser et al. 2002). The forecast was initiated at 0000 UTC, so the simulated convection occurs at nighttime and the early morning hours, with a stabilization of the boundary layer during the course of the simulation due to low-level cooling that decreases with height above the surface. The ARM CART domain-average observations show a similar stabilization, though not quite as pronounced. Such differences may be due to local conditions not representative of the larger-scale CART domain. In this regard, examination of the 1-km grid-mean heating rates due to horizontal advection (Fig. 2) reveals the periodic passage of cold air into the 1-km mesh grid at low levels, apparently representing outflows from deep convection occurring elsewhere.
To investigate the extent to which the simulated Great Plains convection reflects a CBQ, cloud-base levels of convective clouds were estimated from the simulation data in a manner similar to that described by Peng et al. (2004). In that study, grid points were assumed to correspond to a convective cloud base if they represented the first level in the column below 600 hPa for which the cloud water mixing ratio exceeded 0.1 g kg−1, if in addition the vertical velocity exceeded a threshold of 0.5 m s−1. For the present work, in order to more accurately identify the cloud-base level, the vertical velocity threshold is set to zero, but the column is excluded from consideration if the vertical velocity does not exceed 3 m s−1 at some level between the diagnosed cloud-base level and the 200-hPa level.
For the diagnosed cloud-base points, cloud-base virtual temperature perturbations were computed with respect to the corresponding area-mean values of encompassing regions defined by a somewhat arbitrarily selected length scale. For this purpose, the 112 km × 112 km subdomain of the 1-km mesh grid described above was divided into sixteen 28 km × 28 km subregions, or “boxes.” The 28-km length scale is close to the 27-km grid dimension adopted for the COAMPS forecast experiments described in section 4, so may be particularly appropriate here. Some tests of resolution sensitivity with respect to CBQ were carried out, as discussed below. For each box, the mean cloud-base virtual temperature perturbation was computed. The 1-km grid-mean cloud-base virtual temperature perturbation was obtained by averaging the mean values from boxes for which cloud-base points were identified. The values thus obtained (Fig. 3) are almost all within 0.5 K of zero, though during the interval between the two main convective episodes, some larger-magnitude negative values are shown.
The results in Fig. 3 are consistent with, but do not necessarily prove, the existence of a cloud-base quasi balance associated with the simulated deep convection. The term quasi balance implies some degree of cancellation between opposing tendencies. To investigate such cancellation, for each of the 28 km × 28 km boxes for which convective clouds are diagnosed at a given time, a representative source level for the clouds is identified. The source level is the level for which the corresponding LCL (for a parcel with box-mean values) matches the box-mean cloud-base level. A simple interpolation with respect to the model vertical grid levels is used for this computation.
The mean results obtained for cloud-base level and source level are plotted in Fig. 4. To provide insight into the factors impacting cloud-base buoyancy in the simulation, also shown are the 1-km grid-mean virtual potential temperature tendency due to convection (Fig. 4a), the total virtual potential temperature tendency (Fig. 4b), and the virtual potential temperature tendency due to advection on the scale of the 28-km boxes (Fig. 4c). The virtual potential temperature tendency due to convection (Fig. 4a) is computed by combining the changes in temperature and moisture from the microphysics scheme with the changes due to convective mixing. The convective mixing tendencies for temperature and water vapor are set equal to the difference between the total advective tendencies and the 28-km-scale advective tendencies computed as in Ridout (2002). The treatment of the microphysical contribution seems reasonable here, though in general it can be expected to include the effect of large-scale condensation. The convective tendencies (Fig. 4a) are small near cloud base compared to higher up, and for the second convective episode one finds a significant degree of cancellation in heating rates near cloud base (Fig. 4b). The advective forcing profile (Fig. 4c) shows periods of destabilization of source-layer parcels, generally corresponding to the two main convective episodes. Further analysis (Fig. 5) suggests that the destabilization shown here is in fact the major contributor to parcel buoyancy at cloud base in this case.
To more clearly determine the existence of a quasi balance of tendencies impacting cloud-base buoyancy, the source levels determined as for Fig. 4 were used to compute tendencies in virtual temperature difference between lifted parcel and associated environment at the LCL. These tendencies were computed separately for each 28 km × 28 km box for which convection was occurring at a given time, and averaged. The results obtained were smoothed by applying a 28-min running mean filter and were plotted in Fig. 5a as they varied with forecast time. The figure shows the contribution to the tendency in the virtual temperature difference between parcel and environment at cloud-base level due separately to convection, and to other processes (the forcing tendency), as well as the net tendency. In general, the plot shows that convection tends to counter the tendency for other processes to increase parcel buoyancy at cloud base. The extent to which the opposing tendencies balance varies, but particularly during the stronger convective episode, the term quasi balance seems to describe the results. During the portion of the first convective episode most closely associated with the corresponding rainfall peak, the forcing tendency exhibits large oscillations on time scales of about 1 h, in the range often associated with individual convective cells. The net tendency has corresponding large oscillations, though on a time scale of approximately 2–3 h there does appear to be a fair degree of cancellation, and hence CBQ appears to describe this case as well. A similar plot is shown in Fig. 5b, where the forcing tendency is taken to be simply the contribution due to advection on the scale of the 28-km boxes. The results are nearly indistinguishable from Fig. 5a, showing that the primary destabilizing, or restoring, process in the simulation is a dynamical process on a scale of 28 km or greater.
The evidence for CBQ in the COAMPS Great Plains simulation is of potential relevance to other cases of dynamically forced convection in the presence of a stable boundary layer. In this regard, it should be noted that the observed nocturnal rainfall maximum over the Great Plains ARM CART for this day is typical of summertime precipitation over the Great Plains (Dai et al. 1999). An important question is the extent to which the current results extend to convection in the presence of a convective boundary layer. To address this issue, results are presented from a rerun of a COAMPS tropical western Pacific warm pool simulation described by Ridout (2002). The present simulation was carried out with a 1-km inner mesh grid following the procedure described above for the Great Plains simulation. A number of scattered convective cells develop on the inner mesh during the 12-h simulation, with maximum surface rainfall occurring during the final 4–5 h. In Fig. 6a, plots are presented corresponding to the ones in Fig. 5a, showing the contributions of both the total forcing and convection to the virtual temperature perturbation at cloud base, as well as the sum of these two quantities. The results are very similar to those for the Great Plains simulation with respect to the observed quasi balance in tendencies during the main convective episode. In Fig. 6b, plots corresponding to those in Fig. 5b are presented. The clear departure from the quasi balance in Fig. 6a illustrates that in contrast with the Great Plains case, a significant portion of the cloud-base destabilization is attributable to processes other than advection. The importance of surface fluxes and vertical turbulent mixing in this regard is evidenced by the quasi balance in Fig. 6c, which shows that combined with advection, these processes account for the major portion of the cloud-base parcel destabilization in Fig. 6a.
Tests of the sensitivity of the present CBQ analysis to a doubling of the length scale to 56 km were carried out. To a significant extent, CBQ is observed in the Great Plains case at this resolution on a time scale of 2–3 h (not shown), much as in Fig. 5a. For the tropical Pacific case (Fig. 7), the length scale doubling has the effect of rendering the total forcing with respect to cloud-base parcel buoyancy associated with the major convective episode largely unresolved; hence CBQ is not evident at this length scale.
Although the results in Figs. 5 and 6 show a CBQ in the COAMPS cloud-resolving scale simulations, it is not clear the degree to which the quasi balance controls the rate of convection. It is conceivable, for example, that the time scale associated with the CBQ corresponds to the time scale required for elimination of ABE. A means adopted here to address this question is to carry out the analysis used to create Figs. 5 and 6, while excluding data from boxes where the ABE is small. Evidence for CBQ in the filtered dataset should lend support to the view of CBQ as a potential factor in convective control on shorter time scales than those associated with the elimination of ABE. A key point is that the threshold value of ABE for inclusion in the analysis must be large enough to sustain significant convection. With this consideration in mind, results are presented here for a threshold ABE value of 1000 J kg−1. To compute the ABE corresponding to a given box, the parcels that are lifted are the grid cells on the COAMPS 1-km mesh grid that are selected as cloud-base points for the CBQ analysis described above. Parcel buoyancy is based on the box-mean environmental profile. The maximum value of ABE computed for a given box is taken as the ABE for that box. The results obtained for the Great Plains and tropical Pacific simulations are plotted in Figs. 8a and 8b, respectively. One finds that the CBQ observed in Fig. 5a for the second, more intense, convective episode is well defined in the filtered dataset. However, the quasi balance for the first convective episode in Fig. 5a is no longer present, with a shift toward net stabilization. For the tropical Pacific simulation, one finds some differences compared with Fig. 6a, though the CBQ observed for the major convective episode beginning at about tau = 6 h is present here as well.
b. Column model simulation experiments
The above analysis suggests that the present COAMPS cloud-resolving scale simulations should be useful for tests of the modified Kain–Fritsch scheme with CBQ constraint described in section 2b. For such tests, a series of column model experiments were run using forcing data from the COAMPS simulations following Ridout (2002). For each experiment, a total of 16 simulations were carried out, one corresponding to each of the 28 km × 28 km subdomains of the 1-km grid that was used for the analysis in the preceding subsection. The experiments include both single-column model (SCM) and semiprognostic model (SPM) simulations. For these simulations, time-averaged forcing data were ingested into the model every 240 s. These data include advective tendencies, surface fluxes, and radiative heating profiles. In SCM mode, the temperature and moisture profiles evolve as determined by model physics tendencies. In SPM mode, temperature and moisture profiles evolve with time corresponding to prescribed values, in this case based on the appropriate subdomain of the 1-km COAMPS grid.
The convective rainfall predicted using the COAMPS Kain–Fritsch scheme in the SCM simulations is shown in Fig. 9, together with the corresponding rainfall simulated on the COAMPS 1-km mesh grid. Convective rainfall is plotted from the SCM simulation, as it provides a clearer picture of the convective parameterization performance for these predominantly convective cases than total rainfall. One finds that for both cases, the parameterized rainfall is considerably less, overall, than the 1-km mesh grid rainfall.
By decreasing the convective time scale in the Kain–Fritsch scheme, the convective rainfall is increased, as seen in Fig. 10 for a convective time scale of 60 s (2 Δt). The temporal development of the rainfall is perhaps in some sense better represented here, though there is a considerable amount of noise. Some sensitivity tests were carried out using higher values of convective time scale, ranging up to 1800 s. The frequency of the noise in the simulated rainfall tends to decrease with increasing convective time scale, as one expects, though the size of the peaks tends to decrease as well. The noisiness here is consistent, it might be argued, with the observation of Kain and Fritsch (1998) that experiments with a decreased convective time scale in their scheme tended to degrade the realism of the simulated rainfall.
The simulated convective rainfall obtained by adding the modified source-layer selection and convective trigger perturbation values to the scheme of Fig. 10 is plotted in Fig. 11. The rainfall appears somewhat noisier, and with a 10%–20% increase compared with the Fig. 10 scheme. Despite the noisiness, this version appears to show some slight improvement over the scheme used for Fig. 10 in the COAMPS precipitation forecast experiments of section 4.
The beneficial impact of adding the CBQ mass-flux constraint to the scheme of Fig. 11 is clearly evident in Fig. 12. Despite the notable improvements, the amount of parameterized rainfall is often significantly less than the COAMPS 1-km grid rainfall. The actual extent of any deficiency is difficult to determine. The cases studied are predominantly convective, and this fact is reflected to some degree in the SCM simulations. Parameterized convection accounts for 64% of the total SCM rainfall for the Great Plains simulation (Fig. 12a) and 81% of the total for the tropical Pacific simulation (Fig. 12b). Convection accounts for a similar fraction of the total rainfall for the schemes used for Figs. 10 and 11. For the COAMPS Kain–Fritsch scheme, the corresponding percentages are 44% for the Great Plains case and 66% for the tropical Pacific case. What constitutes realistic percentages in this regard is uncertain. The rainfall on the COAMPS 1-km mesh grid is generally concentrated near the grid columns corresponding to the cloud-base points identified by the method described in section 3a. Nonetheless, 25% of the rainfall occurs more than 8 km from such grid columns in the Great Plains simulation. In the tropical Pacific simulation, the corresponding percentage is 37%. These results may suggest a significant amount of anvil rainfall or rainfall during the decaying stage of convective clouds in the simulations, which can be expected to pose problems for the parameterizations.
Comparison of Figs. 11 and 12 illustrates the effectiveness of the CBQ mass-flux constraint at controlling the rate of convection. It is helpful also to examine the corresponding SPM simulations. In SCM mode, the atmospheric profile can rapidly stabilize in response to excessive convective mass flux. This feedback, which can to some degree mask the impact of the CBQ mass-flux constraint, is not allowed in SPM mode. In the SPM simulations, one finds (Fig. 13) that the scheme without the CBQ mass-flux constraint (solid line) produces excessive amounts of precipitation. The comparatively reasonable rainfall produced by the scheme of Fig. 12 (dotted line) directly reflects the large degree of control exerted by the added constraint.
Consistent with the SCM convective rainfall comparisons, the convective mass fluxes computed in the COAMPS Kain–Fritsch scheme are substantially less than those computed in the scheme of Fig. 12. This point is illustrated in Fig. 14 for the Great Plains case. The corresponding mass fluxes diagnosed from the COAMPS simulation data using a threshold vertical velocity of 1 m s−1 to define the updrafts are also plotted. The local 28-km box-mean vertical velocity was subtracted from the 1-km mesh grid data for this purpose to filter out the contribution of larger-scale uplift. In general, the mass-flux profiles from the Fig. 12 scheme more closely resemble those from the explicit COAMPS simulation. Significant improvements are also obtained with the modified scheme in the tropical Pacific case (not shown).
4. Precipitation forecast skill in COAMPS
To evaluate the impact of the Kain–Fritsch scheme modifications on precipitation forecast skill, a series of 24-h COAMPS forecasts were carried out over the continental United States using a 12-h data assimilation update cycle. The results presented here correspond to the 27-km mesh inner grid, for which a time step of 80 s was used. The tests were run for the month of June 2003, which was fairly wet, with a series of frontal systems moving across the eastern two-thirds of the country. Also, Tropical Storm Bill brought a considerable amount of precipitation as it moved into Louisiana and tracked across southern Mississippi.
Standard measures of skill were computed for the COAMPS precipitation forecasts using the National Centers for Environmental Prediction (NCEP) stage IV multisensor 24-hourly precipitation analyses. The equitable threat score (ETH) (Schaefer 1990) is used here to measure the skill of forecasting precipitation amounts in excess of a given threshold R. Higher values of ETH signify greater skill relative to chance at predicting an event, with unity representing a perfect score. The bias scores (B) are also computed. For a given threshold R, B is the ratio of the number of forecasts of rainfall in excess of R to the number of cases in which the observed rainfall exceeds R. Hence B is a positive number, and a perfect score is unity.
To facilitate the presentation of results, the COAMPS Kain–Fritsch scheme is denoted as CKF, and the schemes used in Figs. 10, 11 and 12 are denoted as F10, F11, and F12, respectively. These conventions are listed in Table 1 for reference. Two additional schemes, identical to F12 in most respects, are also included in Table 1. One scheme, denoted as F12a, differs from F12 in that it uses (2) for all values of max(ΔTLCL + ΔTk). Scheme F12b differs from F12 in that there is no ABE constraint in the iterative solution for the convective mass flux.
Results from the CKF and F12 schemes are discussed first, as this comparison provides an overall look at the effect of the present modifications. In Table 2, the June 2003 equitable threat and bias scores obtained with these schemes are presented. One finds that there are appreciable improvements in both ETH and B, particularly in the middle to high rainfall rate range, though some caution (see below) must be attached to the significance of these results. The higher bias scores for F12 are consistent with the increased SCM convective rainfall in Fig. 12 with respect to Fig. 9 during the main convective events.
For the highest thresholds, the number of events is relatively small, so the significance of the score differences in Table 2 is particularly questionable. One means to address this issue, permitting an analysis of statistical significance, is to compute the average of score values from individual days, rather than scores based on pooling all of the forecast data together. With the exception of the bias score for the 75-mm threshold, the relative skill comparisons computed in this manner (Table 3) are much the same as portrayed in Table 2. A Student’s t test was performed to determine to what extent the mean performance differences in Table 3, scaled with the daily variability of the means, can be considered statistically significant at the 95% confidence level. One finds that the middle-range bias score differences (thresholds of 15, 25, and 35 mm) are in this respect statistically significant. In addition, the 50% improvement in the equitable threat score at 25 mm is statistically significant at the 90% confidence level. For other thresholds, the daily scores exhibit too much variability to ascribe meaningful statistical significance with the limited number of forecasts represented here.
As an example of the impact of the parameterization modifications, the rainfall amounts predicted in the 24-h, 0012 UTC 5 June 2003 forecasts with the CKF and F12 schemes are plotted in Fig. 15, along with the NCEP stage IV data. The central portion of the United States received a significant amount of rainfall that day, particularly over the southern Great Plains. The plots show a sizable improvement with the F12 scheme, which did much better in representing the extensive rainfall over Texas, in particular, which was hit by severe weather, including both isolated thunderstorms and mesoscale convective systems. The equitable threat score for the F12 forecast for the 25-mm rainfall threshold is 0.20, an improvement of 100% over the corresponding forecast with the CKF scheme.
The performance of the complete version of the modified Kain–Fritsch scheme (F12) has been examined. It is of interest to examine the impact of individual changes as well. The June 2003 equitable threat and bias scores for the schemes used in the SCM simulations for F10 and F11 are thus included in Table 2. Comparing these results with those for F12 and CKF, one finds that in general, the parameterization changes in F10 and F11 provide intermediate results between those obtained with F12 and CKF. The F11 results are generally slighly better than the F10 results. The F10 scheme, the reader is reminded, includes just the change in the convective time scale. The smaller value used in this run generally has a positive effect on the threat scores. There is some improvement in the bias scores in the middle rainfall rate range, but the high bias for low rainfall rates with CKF is made worse with this scheme.
The impact of the shortened convective time scale (F10), and the additional changes included in F11 (convective trigger perturbations and source-layer modifications) are further illustrated in Fig. 16, which shows the rainfall amounts predicted in the 24-h, 0012 UTC 5 June 2003 forecasts with the F10 and F11 schemes. Both schemes do a better job than the CKF scheme (Fig. 15a) in representing the southern extension of the rainfall into Texas. Comparisons with the F12 results (Fig. 15b) are mixed. The skill scores for this day confirm that the F10 and F11 schemes did the best for the 0.25- and 5-mm thresholds, whereas the F12 scheme generally outperformed F10 and F11 for middle and high rainfall rate thresholds.
The final results to be discussed concern the two schemes F12a and F12b. Both schemes are identical to F12, but with simple modifications. F12a differs from F12 in its use of (2) to compute Mmax for all values of max(ΔTLCL + ΔTk). F12b differs from F12 in the elimination of the ABE constraint in the iterative computation of Mb. The skill scores obtained with these schemes are shown in Table 2. The results for F12a are generally similar to those for F12, though the underprediction of heavy rainfall tends to be greater. The scores for F12b are also similar to those for F12, but somewhat worse for high rainfall rates. This finding suggests that incorporation of a constraint to prevent overadjustment with respect to CAPE in the Emanuel convective scheme may prove beneficial for heavy rainfall events.
5. Summary and discussion
Evidence is presented in this study from COAMPS Great Plains and tropical Pacific cloud-resolving scale simulations of a 1–3-h time-scale cloud-base quasi balance (CBQ) between tendencies to increase and decrease the updraft parcel virtual temperature perturbation. The simulation of a CBQ for two very different regimes, dynamically forced nocturnal convection over land and tropical maritime convection in the presence of a convective boundary layer, suggests that the phenomenon may have a wide occurrence. The primary data analysis is performed as described in section 3a, using an averaging length scale of 28 km. Tests show that a coarser resolution (56 km) analysis can in some cases render the cloud-base parcel buoyancy forcing, and associated CBQ, largely unresolved. It is shown that the observed CBQ is still present in two of three simulated convective episodes when the data analyzed is restricted to instances where the ABE exceeds a threshold of 1000 J kg−1. This evidence supports the view that CBQ can act on time scales shorter than associated with the elimination of ABE, and hence may prove a helpful addition to the mass-flux closure in the Kain–Fritsch convective scheme.
A modified version of the Kain–Fritsch scheme is described in which a CBQ constraint on convective cloud-base mass flux is implemented. The constraint has two key features. First, in the modified scheme, clouds are typically updated numerous times during a convective life cycle in order that convection can respond in a quasi-compensatory manner to processes tending to destabilize updraft parcels at the cloud-base level. The second part of the implementation is the application of a modified constraint on the cloud-base mass flux. This constraint is based largely on the closure treatment in the Emanuel convective scheme (Emanuel and Zivkovic-Rothman 1999) but includes a condition that the updraft cloud-base parcel virtual temperature perturbation (including the parameterized subgrid-scale contribution) remain nonnegative. In addition to the imposed CBQ constraint, the modified scheme adopts changes in the source-layer selection and the convective trigger perturbations. The scheme tends to increase the amount of convection in SCM simulations, yielding improvements in tests based on the COAMPS cloud-resolving scale simulations. Improvements are also obtained with the modified scheme in COAMPS precipitation forecasts over the continental United States for June 2003.
Despite significant improvements in precipitation forecasts with the modified Kain–Fritsch convective scheme in section 4, the problem of too much light rainfall remains, as well as too little rainfall in the moderate to heavy rainfall rate range. In this regard, it is noted that the COAMPS forecasts were carried out using a 12-h data assimilation cycle. In operational use, a 6-h cycle is used, which can be expected to provide somewhat better results overall. Column model tests in section 3b of this study suggest that some of the deficiency at higher rainfall rates may stem from issues related to the parameterization of rainfall from the decaying stage of convective clouds. Additionally, it should be noted that Kain (2004) recently described several modifications to the Kain–Fritsch scheme that may provide further improvements in conjunction with the present changes. Of particular relevance in regard to the weaknesses cited here is a modification to enforce a minimum entrainment rate of environmental air into the updraft, which has been found to both decrease the areal coverage of light rainfall and increase the amount of rainfall at higher rainfall rates (Kain 2004).
The authors would like to take this opportunity to thank Ying Lin for information regarding the NCEP precipitation analyses. The help of Jason Nachamkin in providing data used in preliminary experiments for this study is also gratefully acknowledged. We would also like to thank Jack Kain and an anonymous reviewer for their very helpful comments and suggestions. This research was sponsored by the Office of Naval Research, through the Naval Research Laboratory, Program Elements 0601153N and 0602435N. The work was supported in part by a grant of computing time from the Department of Defense’s High Performance Computing Program.
The source-layer selection procedure differs from the original in a manner that adopts a middle ground between the Kain–Fritsch “bottom-up” by design stabilization approach, and the source-level selection described by Peng et al. (2004). To implement the modified scheme, a search is first made for the potential source layer layermax with the maximum associated virtual temperature difference ΔTLCL + ΔTk with respect to the environment at the LCL. The term potential source layer refers to layers that satisfy certain basic requirements of the COAMPS Kain–Fritsch scheme. In particular, potential source layers comprise the minimum number of model levels such that the depth of the layer is at least 60 hPa. In addition, the base of each such layer must be within 400 hPa of the first model level above the surface, and the layer must satisfy the convective triggering criteria (appendix B). Based on a preliminary analysis of the present Great Plains simulation, the search for layermax is restricted to those layers for which the grid-scale vertical velocity at the LCL exceeds a certain threshold [ > 0; see (B.2)], unless no such layers exist, in which case all potential source layers are considered.
Once layermax has been identified, the scheme selects the lowest potential source layer either for which the associated virtual temperature difference at the LCL exceeds a fraction fsc of the value associated with layermax, or that overlaps such a layer. Currently, fsc = 0.8, a value chosen in an attempt to balance the importance of parcel buoyancy at the LCL with the sensitivity of deep convection to source-layer moist static energy, which is often higher closer to the surface. If the cloud produced with the selected source layer does not satisfy the Kain–Fritsch minimum cloud-depth requirement (3 km), the search proceeds to the next higher-up potential source layer. The search continues as necessary, but fails (no parameterized convection) if a satisfactory layer is not found at or below the level of layermax.
In the selection procedure outlined here, layers for which the convective trigger requirements are met can be passed over in favor of layers higher up with greater associated cloud-base buoyancy. An exception to this treatment is adopted for cases where a potential source layer is undergoing uplift that would bring it to its LCL within a short time period τsat, where currently τsat = 360 s. Such a layer is selected in preference to layers higher up to help prevent unrealistic large-scale saturation of the layer before the succeeding call to the convective scheme. Preliminary tests in tropical cyclone forecasts show a greater proportion of convective rainfall when this exception is adopted.
Convective Trigger Perturbations
The convective trigger perturbations differ with respect to the COAMPS Kain–Fritsch scheme. The temperature perturbation ΔT is taken to be the same as in the unmodified treatment, except that it is not allowed to fall below 0.9 K. This quantity is also used for the temperature deficit ΔTk associated with a particular source layer [(2) and (3)]. Thus
where c1 = 4.64 K m−1/3 s1/3, and is given by
The quantity is computed at each time step following
where w is the instantaneous grid-scale vertical velocity at the LCL. The quantity wT in (B2) is equal to the minimum of ZLCL × 10−5 s−1 and 0.02 m s−1, where ZLCL is the height of the LCL. For convection to occur, the quantity ΔTLCL is required to be no less than −ΔT.
The cloud-scale vertical velocity perturbation at the LCL, wLCL, is taken to be half the value computed in the unmodified scheme. Thus
where wsgn is equal to 1.0 if is greater than zero, and –1.0 otherwise. The quantity Tυ is the environmental virtual temperature at the cloud-base level. For convection to occur wLCL is required to be nonnegative. This quantity is also taken as the updraft vertical velocity at the highest model level that is not above the LCL in the Kain–Fritsch scheme.
Results from the cloud-resolving scale simulations provide some rationale for the trigger perturbation changes. To diagnose values of ΔT and wLCL from the simulation data, mean values are computed over the 28 km × 28 km boxes for which cloud-base points are identified. For a given box, the updraft temperature perturbation is computed as follows. First, for the diagnosed cloud-base points in the box, the mean difference is computed between the 1-km grid-cell temperature and the corresponding box-mean temperature at that level. The value of ΔT for the box is obtained by subtracting from this mean temperature difference the corresponding temperature difference between the lifted source-layer parcel for that box (defined as for Fig. 4) and the box-mean temperature at the associated LCL. The vertical velocity perturbation is computed as the mean difference between the cloud-base point vertical velocities and the associated box-mean values. The values for ΔT and wLCL diagnosed from the Great Plains simulation data are plotted in Fig. B1, together with the values computed as in the Kain–Fritsch scheme. For a given box, the Kain–Fritsch values are computed based on box-mean profiles and represent an average for the various cloud-base levels in that box. One finds that during the convective episodes, the parameterized temperature perturbations (Fig. B1a) are often close to the diagnosed values, but there is a tendency to be too low, especially after the precipitation rate reaches its maximum. During the convective episodes, the parameterized vertical velocity perturbations (Fig. B1b) are generally larger in magnitude than the corresponding diagnosed values. The treatment adopted here seems a reasonable starting point based on this comparison.
Corresponding author address: Dr. James A. Ridout, Naval Research Laboratory, 7 Grace Hopper Ave., Stop 2, Monterey, CA 93943-5502. Email: firstname.lastname@example.org
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