Applications of second-moment turbulent closure hypotheses to geophysical fluid problems have developed rapidly since 1973, when genuine predictive skill in coping with the effects of stratification was demonstrated. The purpose here is to synthesize and organize material that has appeared in a number of articles and add new useful material so that a complete (and improved) description of a turbulence model from conception to application is condensed in a single article. It is hoped that this will be a useful reference to users of the model for application to either atmospheric or oceanic boundary layers.

http://fap.if.usp.br/~hbarbosa/uploads/Teaching/Modclim2010a/MellorYamada1982.pdf

Development of a turbulence closure model for geophysical fluid problems

The two main principles underlying the use of isentropic maps of potential vorticity to represent dynamical processes in the atmosphere are reviewed, including the extension of those principles to take the lower boundary condition into account. the first is the familiar Lagrangian conservation principle, for potential vorticity (PV) and potential temperature, which holds approximately when advective processes dominate frictional and diabatic ones. the second is the principle of ‘invertibility’ of the PV distribution, which holds whether or not diabatic and frictional processes are important. the invertibility principle states that if the total mass under each isentropic surface is specified, then a knowledge of the global distribution of PV on each isentropic surface and of potential temperature at the lower boundary (which within certain limitations can be considered to be part of the PV distribution) is sufficient to deduce, diagnostically, all the other dynamical fields, such as winds, temperatures, geopotential heights, static stabilities, and vertical velocities, under a suitable balance condition. the statement that vertical velocities can be deduced is related to the well-known omega equation principle, and depends on having sufficient information about diabatic and frictional processes. Quasi-geostrophic, semigeostrophic, and ‘nonlinear normal mode initialization’ realizations of the balance condition are discussed. an important constraint on the mass-weighted integral of PV over a material volume and on its possible diabatic and frictional change is noted.

Some basic examples are given, both from operational weather analyses and from idealized theoretical models, to illustrate the insights that can be gained from this approach and to indicate its relation to classical synoptic and air-mass concepts. Included are discussions of (a) the structure, origin and persistence of cutoff cyclones and blocking anticyclones, (b) the physical mechanisms of Rossby wave propagation, baroclinic instability, and barotropic instability, and (c) the spatially and temporally nonuniform way in which such waves and instabilities may become strongly nonlinear, as in an occluding cyclone or in the formation of an upper air shear line. Connections with principles derived from synoptic experience are indicated, such as the ‘PVA rule’ concerning positive vorticity advection on upper air charts, and the role of disturbances of upper air origin, in combination with low-level warm advection, in triggering latent heat release to produce explosive cyclonic development. In all cases it is found that time sequences of isentropic potential vorticity and surface potential temperature charts—which succinctly summarize the combined effects of vorticity advection, thermal advection, and vertical motion without requiring explicit knowledge of the vertical motion field—lead to a very clear and complete picture of the dynamics. This picture is remarkably simple in many cases of real meteorological interest. It involves, in principle, no sacrifices in quantitative accuracy beyond what is inherent in the concept of balance, as used for instance in the initialization of numerical weather forecasts.

/pdf/on-the-use-and-significance-of-isentropic-potential-wx69wazwtl.pdf

On the use and significance of isentropic potential vorticity maps

A new two-moment cloud microphysics scheme predicting the mixing ratios and number concentrations of five species (i.e., cloud droplets, cloud ice, snow, rain, and graupel) has been implemented into the Weather Research and Forecasting model (WRF). This scheme is used to investigate the formation and evolution of trailing stratiform precipitation in an idealized two-dimensional squall line. Results are compared to those using a one-moment version of the scheme that predicts only the mixing ratios of the species, and diagnoses the number concentrations from the specified size distribution intercept parameter and predicted mixing ratio. The overall structure of the storm is similar using either the one- or two-moment schemes, although there are notable differences. The two-moment (2-M) scheme produces a widespread region of trailing stratiform precipitation within several hours of the storm formation. In contrast, there is negligible trailing stratiform precipitation using the one-moment (1-M) scheme. The primary reason for this difference are reduced rain evaporation rates in 2-M compared to 1-M in the trailing stratiform region, leading directly to greater rain mixing ratios and surface rainfall rates. Second, increased rain evaporation rates in 2-M compared to 1-M in the convective region at midlevels result in weaker convective updraft cells and increased midlevel detrainment and flux of positively buoyant air from the convective into the stratiform region. This flux is in turn associated with a stronger mesoscale updraft in the stratiform region and enhanced ice growth rates. The reduced (increased) rates of rain evaporation in the stratiform (convective) regions in 2-M are associated with differences in the predicted rain size distribution intercept parameter (which was specified as a constant in 1-M) between the two regions. This variability is consistent with surface disdrometer measurements in previous studies that show a rapid decrease of the rain intercept parameter during the transition from convective to stratiform rainfall.

http://kiwi.atmos.colostate.edu/pubs/i1520-0493-137-3-991.pdf

Impact of Cloud Microphysics on the Development of Trailing Stratiform Precipitation in a Simulated Squall Line: Comparison of One- and Two-Moment Schemes

The largest convective clouds are mesoscale convective systems, which account for a large portion of Earth's cloud cover and precipitation, and the patterns of wind and weather associated with mesoscale convective systems are important local phenomena that often must be forecast on short timescales. They often produce floods. Mesoscale convective systems are generally much larger than the individual cumulonimbus and lines of cumulonimbus discussed in Chapter 8 . They develop circulations on the mesoscale, which are larger in scale than the updrafts and downdrafts of individual cumulonimbus clouds. The mesoscale circulations produce large regions of stratiform (nimbostratus) precipitation of the type discussed in Chapter 6 . Often the stratiform precipitation regions trail a squall line consisting of convective cells, and a mesoscale convective vortex tends to form in the stratiform region. The heating profile in the stratiform region is positive at upper levels and negative at lower levels due to evaporation and melting of the precipitation particles. The dynamics of mesoscale circulations involve a joint adjustment to the wind shear and thermodynamic stratification of the large scale environment. Gravity-wave dynamics also contribute to the maintenance of mesoscale convective systems. This chapter reviews both the observed structure of mesoscale systems and their unique dynamics.

/pdf/mesoscale-convective-systems-144jge62r4.pdf

Mesoscale convective systems

[1] Convectively coupled equatorial waves (CCEWs) control a substantial fraction of tropical rainfall variability. Their horizontal structures and dispersion characteristics correspond to Matsuno's (1966) solutions of the shallow water equations on an equatorial beta plane, namely, Kelvin, equatorial Rossby, mixed Rossby-gravity, and inertio-gravity waves. Because of moist processes, the tilted vertical structures of CCEWs are complex, and their scales do not correspond to that expected from the linear theory of dry waves. The dynamical structures and cloud morphology of CCEWs display a large degree of self-similarity over a surprisingly wide range of scales, with shallow convection at their leading edge, followed by deep convection and then stratiform precipitation, mirroring that of individual mesoscale convective complexes. CCEWs have broad impacts within the tropics, and their simulation in general circulation models is still problematic, although progress has been made using simpler models. A complete understanding of CCEWs remains a challenge in tropical meteorology.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2008RG000266

Convectively-coupled equatorial waves

A two-dimensional, anelastic cloud model was used in attempts to numerically replicate the observed structure of a midlatitude squall line. Initial conditions were adapted from observations of the 22 May 1976 Oklahoma line. Model simulations were made with and without considering the ice phase of water. These model storms have, within the constraints of the model's geometry, replicated the basic multicellular character and general line-normal airflow typical of these lines. In addition, the structure and intensity of the subcloud cold air pool and the propagation speeds developed by the storms appear to be reasonable. Further, it was found that the initial conditions chosen resulted in model storms which were not only long-lasting but also essentially repetitive, indicating that a state of quasi-equilibrium had been attained. The storms did not decay because the environmental conditions ahead of the storms were favorable and essentially unchanged during the course of the simulations. The inclusio...

Numerical Simulation of a Midlatitude Squall Line in Two Dimensions

On 22 May 1976 a well-organized squall line with a width of more than 100 km passed through the mesonetwork of the National Severe Storms Laboratory in central Oklahoma. This squall line was evolving from its mature stage to the decaying stage when it passed the network. A total of 81 rawinsonde observations were made at nine stations within the network. These, as well as other observations, were analyzed to depict the kinematic and thermodynamic structure of the squall line. Composite and objective analysis techniques were used to arrive at the time-averaged structure on a vertical plane transverse to the squall line. The resulting air flow relative to the traveling squall line is consistent with the distribution of the thermodynamic variables and radar reflectivity. A familiar upshear tilt of the updraft is clearly indicated; the upper-level divergence maximum is located ∼70 km to the rear of the low-level convergence maximum. An interesting feature revealed by the analysis is that the horizont...

The Structure of a Midlatitude Squall Line: A Case Study

A strictly two-dimensional cloud model was used to gauge the effect of vertical wind shear on the mature phase behavior of model-simulated multicellular storms, extending the previous work of the authors. We specifically examined the propagation speed, quasi-equilibrium behavior, storm scale and updraft orientation of the model storms as a function of shear intensity. We also considered the precipitation efficiencies of our. model storms and applied density current and Rotunno–Klemp–Weisman theories to our results. Our previous work revealed that model storms could achieve a mature phase consisting of repetitive multicellular development when certain numerical obstacles were overcome. This was referred to as a “quasi-equilibrium state.” We found herein that this state was also reached by model storms even when subjected to a very wide range of low-level wind shear intensities, although the temporal behavior during this stage was clearly dependent on the shear. We also found a very systematic rela...

Effect of vertical wind shear on numerically simulated multicell storm structure

When the western coast of India lies in the path of the low-level west-southwest wind crossing the Arabian Sea during the summer monsoon season, deep convection frequently develops over the ocean off the coast. In such a situation, the maximum rainfall occurs near the coast, not over the Western Ghats. In order to study the physics underlying orographic-convective precipitation over this area, a two-dimensional compressible moist cloud model is applied. The model is written in terrain-following coordinates and includes the Coriolis force and a planetary boundary layer parameterization. The initial fields of thermodynamic variables are specified using observed data gathered upstream of the offshore precipitating systems over the Arabian Sea. Two wind profiles are considered: vertically uniform and nonuniform flows. The latter profile represents a monsoonal westerly jet at low levels and easterlies in the layer above 5 km. Three cases are considered for each wind profile by including or omitting mo...

Numerical study of orographic-convective precipitation over the eastern Arabian Sea and the Ghat Mountains during the summer monsoon

The level 3 turbulence closure model proposed by Mellor and Yamada (1974) is modified 1) to incorporate the formulations for the turbulence third-order moments and pressure terms proposed by Zeman and Lumley (1976) and 2) to introduce turbulence length scales which depend upon the stratification of the atmosphere. The vertical heat and moisture fluxes and the temperature-humidity covariance are determined from differential equations. The model includes two other differential equations, one for the turbulence kinetic energy and the other for virtual potential temperature variance. All other turbulence variables are determined from algebraic equations. The model is used to simulate the daytime evolution of the planetary boundary layer observed on day 33 of the Wangara boundary-layer experiment. The calculated vertical profiles of the mean wind, temperature and humidity are found to be in good agreement with the observations. The calculated vertical distributions of turbulence variables, including k...

Yoshi Ogura

Papers

Numerical Simulation of a Midlatitude Squall Line in Two Dimensions

The Structure of a Midlatitude Squall Line: A Case Study

Effect of vertical wind shear on numerically simulated multicell storm structure

Numerical study of orographic-convective precipitation over the eastern Arabian Sea and the Ghat Mountains during the summer monsoon

Modeling the Evolution of the Convective Planetary Boundary Layer