A new invariant called the generalized Ertel Rossby invariant （GER） was developed in this study. The new invariant is given by the dot product of the generalized vorticity and the generalized velocity. The generalized vortieity is the absolute vorticity minus the cross product of the gradient of Lagrangian--time integrated temperature and the gradient of entropy. The generalized velocity is the absolute velocity minus the sum of the gradient of Lagrangian-time integrated kinetic potential and the Lagrangian time integrated temperature multiplied by the gradient of entropy. In addition to the traditional potential vorticity, the GER invariant may provide another useful tool to study the atmospheric dynamic processes for weather phenomena ranging from large scales to small scales.
The precipitation efficiency and its relationship to physical factors are examined by analyzing a two-dimensional cloud-resolving model simulation during TOGA COARE in this study. The basic physical factors include convective avail- able potential energy, water-vapor convergence, vertical wind shear, cloud ratio, sea surface temperature, air temperature, and precipitable water. Precipitation efficiencies do not show a close relationship to air temperature nor to sea surface tem- perature nor to precipitable water. The precipitation efficiency increases as the water-vapor convergence rate increases and vertical wind shear weakens, whereas it decreases as the convective available potential energy dissipates and anvil clouds develop.
The non-uniqueness of solution and compatibility between the coupled boundary conditions in computing velocity potential and streamfunction from horizontal velocity in a limited domain of arbitrary shape are revisited theoretically with rigorous mathematic treatments.Classic integral formulas and their variants are used to formulate solutions for the coupled problems.In the absence of data holes,the total solution is the sum of two integral solutions.One is the internally induced solution produced purely and uniquely by the domain internal divergence and vorticity,and its two components（velocity potential and streamfunction） can be constructed by applying Green＇s function for Poisson equation in unbounded domain to the divergence and vorticity inside the domain.The other is the externally induced solution produced purely but non-uniquely by the domain external divergence and vorticity,and the non-uniqueness is caused by the harmonic nature of the solution and the unknown divergence and vorticity distributions outside the domain.By setting either the velocity potential（or streamfunction） component to zero,the other component of the externally induced solution can be expressed by the imaginary（or real） part of the Cauchy integral constructed using the coupled boundary conditions and solvability conditions that exclude the internally induced solution.The streamfunction（or velocity potential） for the externally induced solution can also be expressed by the boundary integral of a double-layer（or singlelayer） density function.In the presence of data holes,the total solution includes a data-hole-induced solution in addition to the above internally and externally induced solutions.
Built on the integral formulas in Part I,numerical methods are developed for computing velocity potential and streamfunction in a limited domain.When there is no inner boundary（around a data hole） inside the domain,the total solution is the sum of the internally and externally induced parts.For the internally induced part,three numerical schemes（grid-staggering,local-nesting and piecewise continuous integration） are designed to deal with the singularity of the Green＇s function encountered in numerical calculations.For the externally induced part,by setting the velocity potential（or streamfunction） component to zero,the other component of the solution can be computed in two ways：（1） Solve for the density function from its boundary integral equation and then construct the solution from the boundary integral of the density function.（2） Use the Cauchy integral to construct the solution directly.The boundary integral can be discretized on a uniform grid along the boundary.By using local-nesting（or piecewise continuous integration）,the scheme is refined to enhance the discretization accuracy of the boundary integral around each corner point（or along the entire boundary）.When the domain is not free of data holes,the total solution contains a data-hole-induced part,and the Cauchy integral method is extended to construct the externally induced solution with irregular external and internal boundaries.An automated algorithm is designed to facilitate the integrations along the irregular external and internal boundaries.Numerical experiments are performed to evaluate the accuracy and efficiency of each scheme relative to others.
In this study, we aimed to elucidate the critical role of moisture transport affecting monsoon activity in two contrasting summers over the Arabian Sea during the years 1994, a relatively wet year, and 2002, a relatively dry year. A comprehensive diagnostic evaluation and comparisons of the moisture fields were con- ducted; we focused on the precipitation and evaporation as well as the moisture transport and its divergence or convergence in the atmosphere. Monthly mean reanalysis data were obtained from the National Centers for Environmental Prediction （NCEP-I and -II）. A detailed evaluation of the moisture budgets over Pak- istan during these two years was made by calculating the latent energy flux at the surface （E - P） from the divergence of the total moisture transport. Our results confirm the moisture supply over the Arabian Sea to be the major source of rainfall in Pakistan and neighboring regions. In 1994, Pakistan received more rainfall compared to 2002 during the summer monsoon. Moisture flow deepens and strengthens over Arabian Sea during the peak summer monsoon months of July and August. Our analysis shows that vertically integrated moisture transport flux have a significant role in supplying moisture to the convective centers over Pakistan and neighboring regions from the divergent regions of the Arabian Sea and the Bay of Bengal. Moreover, in 1994, a deeper vertically integrated moisture convergence progression occurred over Pakistan compared to that in 2002. Perhaps that deeper convergence resulted in a more intense moisture depression over Pakistan and also caused more rainfall in 1994 during the summer monsoon. Finally, from the water budget analysis, it has been surmised that the water budget was larger in 1994 than in 2002 during the summer monsoon.
