Mass Flux Model Research Articles

Characteristics of the Meiyu and Baiu Frontal Precipitation Zone in the CMIP3 20th Century Simulation and 21st Century Projection

Features of the Meiyu frontal rain zone (MFZ, in 110–125°E) and Baiu frontal rain zone (BFZ, in 125–140°E) in the 20th Century simulation (20C3M) and 21st Century projection (Special Report on Emission Scenarios (SRES) A1B), obtained by 22 climate models contributed to the World Climate Research Programme's Coupled Model Intercomparison Project phase 3 (CMIP3) are studied. Characteristics of the MFZ and BFZ in the 20th century climate simulation are compared with two sets of observed precipitation data. The latitude of the zone and the precipitation in the zone (“precipitation in FZ”) reproduced by the models are examined, using the 20-year (1980–1999) averaged values for May, June and July. The “precipitation in FZ” in each month obtained from the multi-model ensemble average (MEA) coincides approximately with the observation data. However, MEA reproduces MFZ and BFZ to the north of their observed latitude in May. MEA reasonably reproduces the latitude BFZ for June and July, and the latitude of MFZ only for June. The standard deviation (STD) of the “precipitation in FZ” does not change widely from zone to zone, and from month to month. However, STD of the latitude of zones varies widely. The STD of latitude of MFZ and BFZ became larger in July. The STD of the latitude of MFZ is larger than that of BFZ. The low and medium horizontal resolution models with moist convective adjustment scheme tend to reproduce MFZ and BFZ in the northern latitude, while the models with mass flux scheme and models with Arakawa-Schubert scheme tend to reproduce MFZ and BFZ in the southern latitude. Features of the MFZ and BFZ in the 21st century climate projection by the models are examined using the 20-year (2080–2099) averaged values for May, June and July. The characteristics of the models in regard to MFZ and BFZ in 20th century simulation are commonly found in the 21st century projection. The MFZ and BFZ shift slightly northward, and the “precipitation in FZ” decreased slightly from the 20th century to 21st century. However, these changes are very small, as compared with the respective STD. The change of MFZ and BFZ under the climate changes in the 21st century can not be definitely concluded by models of CMIP3.

Nature versus Nurture in Shallow Convection

Abstract Tracers are used in a large-eddy simulation of shallow convection to show that stochastic entrainment (and not cloud-base properties) determines the fate of convecting parcels. The tracers are used to diagnose the correlations between a parcel’s state above the cloud base and both the parcel’s state at the cloud base and its entrainment history. The correlation with the cloud-base state goes to zero a few hundred meters above the cloud base. On the other hand, correlations between a parcel’s state and its net entrainment are large. Evidence is found that the entrainment events may be described as a stochastic Poisson process. A parcel model is constructed with stochastic entrainment that is able to replicate the mean and standard deviation of cloud properties. Turning off cloud-base variability has little effect on the results, which suggests that stochastic mass-flux models may be initialized with a single set of properties. The success of the stochastic parcel model suggests that it holds promise as the framework for a convective parameterization.

Comparison of different mass transport calculation methods for wind erosion quantification purposes

AbstractQuantitative estimation of the material transported by the wind under field conditions is essential for the study and control of wind erosion. A critical step of this calculation is the integration of the curve that relates the variation of the amount of the material carried by the wind with height. Several mathematical procedures have been proposed for this calculation, but results are scarce and controversial. One objective of this study was to assess the efficiency of three mathematical models (a rational, an exponential, and a simplified Gaussian function) for the calculation of the mass transport, as compared to the linear spline interpolation. Another objective of this study was to compare the mass transport calculated from field measurements obtained from a minimum of three discrete sampling heights with measurements of nine sampling heights. With this purpose, wind erosion was measured under low surface roughness conditions on an Entic Haplustoll during 25 events. The rational function was found to be mathematically limited for the estimation of wind eroded sediment mass flux. The simplified Gaussian model did not fit to the vertical mass flux profile data. Linear spline interpolation generally produced higher mass transport estimates than the exponential equation, and it proved to be a very flexible and robust method. Using different sampling arrangements and different mass flux models can produce differences of more than 45% in mass transport estimates, even under similar field conditions. Under the conditions of this study, at least three points between the soil surface and 1·5 m high, including one point as closest as possible to the surface, should be sampled in order to obtain accurate mass transport estimates. Additionally, the linear spline interpolation and the non‐linear regression using an exponential model, proved to be mathematically reliable methods for calculating the mass transport. Copyright © 2010 John Wiley & Sons, Ltd.

Assumed Probability Density Functions for Shallow and Deep Convection

The assumed joint probability density function (PDF) between vertical velocity and conserved temperature and total water scalars has been suggested to be a relatively computationally inexpensive and unified subgrid‐scale (SGS) parameterization for boundary layer clouds and turbulent moments. This paper analyzes the performance of five families of PDFs using large‐eddy simulations of deep convection, shallow convection, and a transition from stratocumulus to trade wind cumulus. Three of the PDF families are based on the double Gaussian form and the remaining two are the single Gaussian and a Double Delta Function (analogous to a mass flux model). The assumed PDF method is tested for grid sizes as small as 0.4 km to as large as 204.8 km. In addition, studies are performed for PDF sensitivity to errors in the input moments and for how well the PDFs diagnose some higher‐order moments. In general, the double Gaussian PDFs more accurately represent SGS cloud structure and turbulence moments in the boundary layer compared to the single Gaussian and Double Delta Function PDFs for the range of grid sizes tested. This is especially true for small SGS cloud fractions. While the most complex PDF, Lewellen‐Yoh, better represents shallow convective cloud properties (cloud fraction and liquid water mixing ratio) compared to the less complex Analytic Double Gaussian 1 PDF, there appears to be no advantage in implementing Lewellen‐Yoh for deep convection. However, the Analytic Double Gaussian 1 PDF better represents the liquid water flux, is less sensitive to errors in the input moments, and diagnoses higher order moments more accurately. Between the Lewellen‐Yoh and Analytic Double Gaussian 1 PDFs, it appears that neither family is distinctly better at representing cloudy layers. However, due to the reduced computational cost and fairly robust results, it appears that the Analytic Double Gaussian 1 PDF could be an ideal family for SGS cloud and turbulence representation in coarse‐grid CRMs, mesoscale models, and GCMs if the required input moments can be predicted or diagnosed accurately.

A Higher-Order Closure Model with an Explicit PBL Top

Abstract In 2001, the authors presented a higher-order mass-flux model called “assumed distributions with higher-order closure” (ADHOC 1), which represents the large eddies of the planetary boundary layer (PBL) in terms of an assumed joint distribution of the vertical velocity and scalars. In a subsequent version (ADHOC 2) the authors incorporated vertical momentum fluxes and second moments involving pressure perturbations into the framework. These versions of ADHOC, as well as all other higher-order closure models, are not suitable for use in large-scale models because of the high vertical and temporal resolution that is required. This high resolution is needed mainly because higher-order closure (HOC) models must resolve discontinuities at the PBL top, which can occur anywhere on a model’s Eulerian vertical grid. This paper reports the development of ADHOC 3, in which the computational cost of the model is reduced by introducing the PBL depth as an explicit prognostic variable. ADHOC 3 uses a stretched vertical coordinate that is attached to the PBL top. The discontinuous jumps at the PBL top are “hidden” in the layer edge that represents the PBL top. This new HOC model can use much coarser vertical resolution and a longer time step and is thus suitable for use in large-scale models. To predict the PBL depth, an entrainment parameterization is needed. In the development of the model, the authors have been led to a new view of the old problem of entrainment parameterization. The relatively detailed information available in the HOC model is used to parameterize the entrainment rate. The present approach thus borrows ideas from mixed-layer modeling to create a new, more economical type of HOC model that is better suited for use as a parameterization in large-scale models.

A simple equilibrium model for shallow-cumulus-topped mixed layers

A new equilibrium model for shallow-cumulus-topped mixed layers is presented. A variant on the w∗ closure for the shallow-cumulus mass flux is applied that retains the convective area-fraction in its formu- lation. As opposed to being constant, the fraction is explicitly modeled using a statistical closure as a function of the saturation deficit and humidity variance at cloud base. As a consequence, important new interactions are introduced between the convective transport, humidity, and depth of the mixed layer. This mechanism, which we call the mass-flux humidity feedback, helps determine the character of the equilibrium state such that the mixed-layer top is maintained close to the cloud-base height. Due to the strong sensitivity of the mass flux to the area fraction, the latter thus acts as a regulator or valve mechanism on moist convective transport. As a consequence, the mixed-layer model is able to explain the robustness of many aspects of the shallow-cumulus boundary layer that is typically found in observations and large-eddy simulations (LESs). The model is evaluated for a single-LES case as well as for global climatology obtained from a 40-year reanalysis of meteorological data by the European Centre for Medium-range Weather Forecasts (ECMWF). LES characteristics of convective mass flux, cloud fraction, humidity variance, cloud-base height, and surface fluxes of heat and humidity are reproduced. The solution on reanalysis fields reproduces the spatial structure of mixed-layer temperature and humidity and their associated surface fluxes in the subtropical Atlantic and Pacific trade wind regions. Furthermore, the spatial structure of the convective area-fraction matches that of synoptic surface observations of frequency of occurrence of shallow cumulus. Particularly striking is the smooth onset of the convective area-fraction and mass flux along the trade-wind trajectory that is reproduced, from zero to typical trade-wind values. The cumulus onset represents the necessity for shallow-cumulus mass flux to occur in order to close the mixed-layer budgets of heat, moisture, and mass, as a response to the changing magnitude of large-scale subsidence and free tropospheric humidity along the trajectory. Finally, the mass flux model is implemented in an intermediate-complexity tropical climate model to study its behavior when fully interactive with the larger-scale flow. A climate run then shows that the model is stable, due to the mass-flux humidity feedback acting to keep the shallow-cumulus boundary layer close to its equilibrium state for long, climatological timescales.

A closure model for turbulent convection. Application to the excitation of p modes

AbstractOscillations of stellar p modes excited by turbulent convection are investigated. In the uppermost part of the solar convection zone, radiative cooling is responsible for the formation of turbulent plumes, hence the medium is characterized with downdrafts and updrafts. The motivation is to take the asymmetry of up- and downflows created by turbulent plumes into account through an adapted closure model. We built a generalized two-scale mass-flux model (GTFM) that considers both the skew introduced by the presence of two flows and the effect of turbulence onto each flow. In order to apply the GTFM to the solar case, we introduce the plume dynamics as modelled by Rieutord & Zahn (1995) and construct a closure model with plumes (CMP). The CMP enables to express third- and fourth-order velocity correlation products in terms of the second-order ones. When comparing with 3D numerical simulation results, the CMP improves the agreement for the fourth order moments in comparison with the quasi-normal approximation (QNA) or the classical mass-flux model (MFM). This excitation model reproduces the maximum of the power supplied to solar p modes, when compared with GOLF observations.

Parameterization of Pressure Perturbations in a PBL Mass-Flux Model

Abstract In a companion paper, the authors presented a boundary layer parameterization that was based on the mass-flux concept and included an internally consistent representation of the vertical flux of horizontal momentum. In the present paper, the authors show how the framework of that model can be used to determine the perturbation pressure field, by solving the anelastic pressure equation. The pressure covariances needed to close the parameterization can then be diagnosed. Tests show very encouraging agreement of the pressure statistics with results obtained from large-eddy simulations.

Using Idealized Coherent Structures to Parameterize Momentum Fluxes in a PBL Mass-Flux Model

Abstract In 2001, the authors presented a higher-order mass-flux model called assumed distributions with higher-order closure (ADHOC), which represents the large eddies of the planetary boundary layer (PBL) in terms of an assumed joint distribution of the vertical velocity and scalars such as potential temperature or water vapor mixing ratio. ADHOC is intended for application as a PBL parameterization. It uses the equations of higher-order closure to predict selected moments of the assumed distribution, and diagnoses the parameters of the distribution from the predicted moments. Once the parameters of the distribution are known, all moments of interest can be computed. The first version of ADHOC was incomplete in that the horizontal momentum equations, the vertical fluxes of horizontal momentum, the contributions to the turbulence kinetic energy from the horizontal wind, and the various pressure terms involving covariances between pressure and other variables were not incorporated into the assumed distribution framework. Instead, these were parameterized using standard methods. This paper describes an updated version of ADHOC. The new version includes representations of the horizontal winds and momentum fluxes that are consistent with the mass-flux framework of the model. The assumed joint probability distribution is replaced by an assumed joint spatial distribution based on an idealized coherent structure, such as a plume or roll. The horizontal velocity can then be determined using the continuity equation, and the momentum fluxes and variances are computed directly by spatial integration. These expressions contain unknowns that involve the parameters of the assumed coherent structures. Methods are presented to determine these parameters, which include the radius of convective updrafts and downdrafts and the wavelength, tilt, and orientation angle of the convective rolls. The parameterization is tested by comparison with statistics computed from large-eddy simulations. In a companion paper, the results of this paper are built on to determine the perturbation pressure terms needed by the model.

Prediction of methanol and water fluxes through a direct methanol fuel cell polymer electrolyte membrane

Development of a direct methanol fuel cell (DMFC) mass flux model, using conventional transport theory, is presented and used to predict the fluid phase superficial velocity, methanol and water molar fluxes, and the chemical species (methanol and water) dimensionless concentration profiles in the polymer electrolyte membrane, Nafion ® 117, of a DMFC. Implementation of these equations is illustrated to generate the numerical data as functions of the variables such as the pressure difference across the membrane, methanol concentration at the cell anode, temperature, and position in the membrane.

Using Probability Density Functions to Derive Consistent Closure Relationships among Higher-Order Moments

Parameterizations of turbulence often predict several lower-order moments and make closure assumptions for higher-order moments. In principle, the low- and high-order moments share the same probability density function (PDF). One closure assumption, then, is the shape of this family of PDFs. When the higher-order moments involve both velocity and thermodynamic scalars, often the PDF shape has been assumed to be a double or triple delta function. This is equivalent to assuming a mass-flux model with no subplume variability. However, PDF families other than delta functions can be assumed. This is because the assumed PDF methodology is fairly general. This paper proposes closures for several third- and fourth-order moments. To derive the closures, the moments are assumed to be consistent with a particular PDF family, namely, a mixture of two trivariate Gaussians. (This PDF is also called a double Gaussian or binormal PDF by some authors.) Separately from the PDF assumption, the paper also proposes a simplified relationship between scalar and velocity skewnesses. This PDF family and skewness relationship are simple enough to yield simple, analytic closure formulas relating the moments. If certain conditions hold, this set of moments is specifically realizable. By this it is meant that the set of moments corresponds to a real Gaussian-mixture PDF, one that is normalized and nonnegative everywhere. This paper compares the new closure formulas with both large eddy simulations (LESs) and closures based on double and triple delta PDFs. This paper does not implement the closures in a single-column model and test them interactively. Rather, the comparisons are diagnostic; that is, low-order moments are extracted from the LES and treated as givens that are input into the closures. This isolates errors in the closures from errors in a single-column model. The test cases are three atmospheric boundary layers: a trade wind cumulus layer, a stratocumulus layer, and a clear convective case. The new closures have shortcomings, but nevertheless are superior to the double or triple delta closures in most of the cases tested.

Interfacial mass transfer around a vapor bubble during nucleate boiling

Interfacial mass transfer from vapor bubbles affects markedly the heat transfer efficiency of nucleate boiling. The position of the interfacial zone that exhibits zero net mass flux, namely, the “zero-flux zone”, represents an essential parameter in detailed modeling works on nucleate boiling. Assuming a linear temperature profile in the superheated liquid adjacent to the heating wall, our previous work (Li et al. [10]) demonstrated the zero-flux angle as a function of wall superheat, solid-liquid- vapor contact angle, and bubble growth rate. To make a more realistic framework, we refined in this paper the proposed mass flux model by taking into account the role of thermocapillary flow that is induced by the temperature gradient around the vapor bubble, and that of non-condensable gas presented in the boiling liquid. The Hertz-Kundsen-Schrage equation describes the interfacial mass flux distribution along the vapor bubble surface. Owing to the “flattened” temperature distribution produced by thermocapillary flow, which significantly reduces the interfacial area to evaporation, the zero-flux zone shifts to the bubble base with most of the cap regime to condense vapor at the interface and to produce the thermal jet. This occurrence also weakens the dependence of bubble growth rate and of the contact angle on the location of zero-flux zone, and yields early occurrence of the “non-condensation limit” at which the entire bubble surface is subjected to evaporation. Sensitivity analysis demonstrated the significance of process parameters on the evaluation of zero-flux angle using the HKS equation.

A NEW ALGEBRAIC MASS FLUX MODEL FOR SIMULATING TURBULENT MIXING IN SWIRLING FLOW

A new algebraic turbulent mass flux model (AFM), which properly accounts for swirl–turbulence interactions, is proposed and formulated in the present article. A simplified explicit form of this new AFM is derived. By incorporating this model with the new algebraic Reynolds stress model proposed previously, a new algebraic stress/flux model (ASM/AFM) is obtained. The new ASM/AFM is applied to the simulation of swirling turbulent flow and mixing in a combustor with helium/air jet and swirling air stream. The calculated gas axial and tangential velocities, gas axial and tangential fluctuating velocities, and helium concentration are in agreement with the measured test data. Comparing with the k–ϵ model for turbulent mass transport, the helium concentration predicted by the new ASM/AFM is much improved.

A critical mass flux model for the flammability of thermoplastics

The combustion of thermoplastics is modelled using a critical mass flux hypothesis as the ignition and extinction criteria. Polymer degradation is modelled as a single-step first-order Arrhenius reaction term. A simple model for mass transport of polymer through the sample during pyrolysis is included. The degradation products are assumed to move out of the polymer instantaneously. The model consists of a nonlinear integral-differential advection-diffusion equation for the temperature in the thermoplastic, coupled to an ordinary differential equation, for the regression rate. Results are presented which quantify the effect that the thickness of the test sample has on the mass-loss rate, or equivalently heat-release rate, curve. From these we conclude that thermally thick samples are characterized by a region of steady burning which is independent of the initial sample thickness. The test method that we have in mind is the cone calorimeter.

Toward a Unified Parameterization of the Boundary Layer and Moist Convection. Part II: Lateral Mass Exchanges and Subplume-Scale Fluxes

The dissipation parameterizations developed for higher-order closure are used to parameterize lateral entrainment and detrainment in a mass-flux model. In addition, a subplume-scale turbulence scheme is included to represent fluxes not captured in the conventional mass-flux framework. These new parameterizations are tested by simulating trade wind cumulus from the Barbados Oceanographic and Meteorological Experiment (BOMEX).

Toward a Unified Parameterization of the Boundary Layer and Moist Convection. Part I: A New Type of Mass-Flux Model

Higher-order closure (HOC) models have been proposed for parameterization of the turbulent planetary boundary layer (PBL). HOC models must include closures for higher-order moments (e.g., fourth moments in thirdorder closure models), for pressure terms, and for dissipation terms. Mass-flux closure (MFC) models have been proposed for parameterization of cumulus convection and, more recently, the convective PBL. MFC models include closures for lateral mass exchanges and for pressure terms (which are usually ignored). The authors developed a new kind of model that combines HOC and MFC, which they hope will be useful for the parameterization of both the PBL and cumulus convection, in a unified framework. Such a model is particularly well suited to regimes in which the PBL turbulence and the cumulus convection are not well separated, for example, the broken stratocumulus and shallow cumulus regimes. The model makes use of an assumed joint probability distribution for the variables of interest, and the equations typically used in HOC models can be derived by integrating over the distribution. Accordingly, the model is called Assumed-Distribution Higher-Order Closure (ADHOC). The prognostic variables of ADHOC are the mean state, the second and third moments of the vertical velocity, and the vertical fluxes of other quantities of interest. All of the parameters of the distribution can be determined from the predicted moments; thereafter the joint distribution is effectively known, and so any and all moments can be constructed as needed. In this way, the usual closure problem of ‘‘higher moments’’ is avoided. The pressure-term parameterizations previously developed for HOC models are used to predict the convective fluxes and the moments of the vertical velocity. In companion papers, parameterizations of lateral mass exchanges and subplume-scale fluxes are presented, and then ADHOC is applied to several observationally based tropical, subtropical, and dry convective boundary layers.

Simulations of Trade Wind Cumuli under a Strong Inversion

The fifth intercomparison of the Global Water and Energy Experiment Cloud System Studies Working Group 1 is used as a vehicle for better understanding the dynamics of trade wind cumuli capped by a strong inversion. The basis of the intercomparison is 10 simulations by 7 groups. These simulations are supplemented by many further sensitivity studies, including some with very refined grid meshes. The simulations help illustrate the turbulent dynamics of trade cumuli in such a regime. In many respects the dynamics are similar to those found in many previous simulations of trade cumuli capped by weaker inversions. The principal differences are the extent to which the cloud layer is quasi-steady in the current simulations, evidence of weak countergradient momentum transport within the cloud layer, and the development and influence of an incipient stratiform cloud layer at the top of the cloud layer. Although many elements of the turbulent structure (including the wind profiles, the evolution of cloud-base height, the statistics of the subcloud layer, and the nature of mixing in the lower and middle parts of the cloud layer) are robustly predicted, the representation of the stratiform cloud amount by the different simulations is remarkably sensitive to a number of factors. Chief among these are differences between numerical algorithms. These sensitivities persist even among simulations on relatively refined grid meshes. Part of this sensitivity is attributed to a physically realistic positive radiative feedback, whereby a propensity toward higher cloud fractions in any given simulation is amplified by longwave radiative cooling. The simulations also provide new insight into the dynamics of the transition layer at cloud base. In accord with observations, the simulations predict that this layer is most identifiable in terms of moisture variances and gradients. The simulations help illustrate the highly variable (in both height and thickness) nature of the transition layer, and we speculate that this variability helps regulate convection. Lastly the simulations are used to help evaluate simple models of trade wind boundary layers. In accord with previous studies, mass-flux models well represent the dynamics of the cloud layer, while mixing-length models well represent the subcloud layer. The development of the stratiform cloud layer is not, however, captured by the mass-flux models. The simulations indicate that future theoretical research needs to focus on interface rules, whereby the cloud layer is coupled to the subcloud layer below and the free atmosphere above. Future observational studies of this regime would be of most benefit if they could provide robust cloud statistics as a function of mean environmental conditions.

A Minimal Three-Dimensional Tropical Cyclone Model

A minimal 3D numerical model designed for basic studies of tropical cyclone behavior is described. The model is formulated in s coordinates on an f or b plane and has three vertical levels, one characterizing a shallow boundary layer and the other two representing the upper and lower troposphere, respectively. It has three options for treating cumulus convection on the subgrid scale and a simple scheme for the explicit release of latent heat on the grid scale. The subgrid-scale schemes are based on the mass-flux models suggested by Arakawa and Ooyama in the late 1960s, but modified to include the effects of precipitation-cooled downdrafts. They differ from one another in the closure that determines the cloud-base mass flux. One closure is based on the assumption of boundary layer quasi-equilibrium proposed by Raymond and Emanuel. It is shown that a realistic hurricane-like vortex develops from a moderate strength initial vortex, even when the initial environment is slightly stable to deep convection. This is true for all three cumulus schemes as well as in the case where only the explicit release of latent heat is included. In all cases there is a period of gestation during which the boundary layer moisture in the inner core region increases on account of surface moisture fluxes, followed by a period of rapid deepening. Precipitation from the convection scheme dominates the explicit precipitation in the early stages of development, but this situation is reversed as the vortex matures. These findings are similar to those of Baik et al., who used the Betts‐Miller parameterization scheme in an axisymmetric model with 11 levels in the vertical. The most striking difference between the model results using different convection schemes is the length of the gestation period, whereas the maximum intensity attained is similar for the three schemes. The calculations suggest the hypothesis that the period of rapid development in tropical cyclones is accompanied by a change in the character of deep convection in the inner core region from buoyantly driven, predominantly upright convection to slantwise forced moist ascent.

Recovery of Surface Waters in the Northeastern U.S. from Decreases in Atmospheric Deposition of Sulfur

A simple mass flux model was developed to simulate the response of SO 4 2− concentrations in surface waters to past and anticipated future changes in atmospheric deposition of SO 4 2− . Values of bulk (or wet) SO 4 2− deposition and dry deposition of S determined from measured air concentrations and a deposition velocity were insufficient to balance watershed SO 4 2− export at the Hubbard Brook Experimental Forest, NH and for a regional survey of watersheds in the northeastern U.S. We propose two explanations for the unmeasured S source: 1) a significant underestimation of dry S deposition, and/or 2) internal watershed S sources, such as weathering and/or mineralization of soil organic S. Model simulations based on these two mechanisms agreed closely with measured stream SO 4 2− concentrations at Hubbard Brook. Close agreement between measured and model predicted results precluded identification of which of the two mechanisms controlled long-term trends in stream SO 4 2− . Model simulations indicated that soil adsorption reactions significantly delayed the response of stream water to declines in SO 4 2− inputs since 1970, but could not explain the discrepancy in watershed S budgets. Extrapolation of model predictions into the future demonstrates that uncertainty in the source of the S imbalance in watersheds has important implications for assessments of the recovery of surface water acid neutralizing capacity in response to anticipated future reductions in SO2 emissions.

Analysis of Astex-Stratocumulus Observational Data Using a Mass-Flux Approach

A mass-flux approach is applied to observational data obtained in a convective boundary layer topped with stratocumulus clouds. The observational data were obtained from aircraft measurements during the Atlantic Stratocumulus Transition Experiment (ASTEX). A conditional sampling method is used to calculate average updraft and downdraft values. The vertical fluxes calculated with the mass-flux approach are found to be proportional to the real (measured) fluxes, with a proportionality factor being about 0.6. This value is predicted by theory for two variables having a joint Gaussian distribution function; proportionality factor = 2π-1 ≈ 0.637. The horizontal fractional entrainment and detrainment rates calculated from the data (e ≈ 1–2 × 10-2 m-1) are an order of magnitude higher than the rates obtained by large eddy simulations for cumulus convection (e ≈ 2–3 × 10-3 m-1) and two orders of magnitude higher than those used in modelling cumulus convection with a mass-flux scheme in an operational weather forecast model (e ≈ 3 × 10-4 m-1). A numerical mass-flux model for the thermodynamics was developed and showed that results are in good agreement when compared with measured profiles of the liquid water content.