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JOSEPH P. GERRITY JR.

Abstract

Work published in 1968 by Crowley is adapted to study the truncation error associated with the semimomentum scheme for approximating advection. The results indicate that higher order approximations based on the semimomentum scheme are competitive with the best results obtained by Crowley with conservation schemes and comparable to fine-mesh calculations.

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JOSEPH P. GERRITY JR.

Abstract

A system of equations governing the behavior of a physical model of the atmospheric, planetary boundary layer was formulated for solution on a digital computer. The physical-numerical model was designed to permit the investigation of the significance of certain boundary-layer processes for the development of horizontally extensive areas of low cloudiness.

Numerical solutions of the equations were computed for three synoptic cases. In each case, the forecast period was 12 hr. The initial state of the atmosphere was analyzed from synoptic surface and upper-air observations in the eastern United States. The computations were made for a finite difference grid with 1200 grid points, using a 15-min. time step. The vertical coordinate was defined by 12 grid points over each of 100 grid points in the horizontal plane. The average spacing of the horizontal grid points was 160 km. The separation of the vertical grid points expanded from 50 m. near the ground to 450 m. at the uppermost level.

The model boundary layer was subdivided into a 50-m. deep, surface contact layer and a 1950-m. deep, transition layer. Stability-dependent, constant-flux profile formulas were applied within the surface contact layer. These were used in conjunction with semi-empirical formulas to derive boundary conditions applicable at the base of the transition layer. Observed data were used to prescribe the horizontal pressure gradient force at the upper boundary of the transition layer. Within the transition layer the horizontal wind was computed by means of a diagnostic equation implying a balance of the Coriolis, pressure gradient, and eddy viscous forces. The eddy viscosity coefficient was held equal to its value at the top of the surface contact layer. The pressure gradient force was assumed to be a linear function of height; its variation was computed from the predicted temperature field. The eddy conductivity and diffusivity coefficients were assumed to be equal. They were computed as functions of the stability.

The results obtained in one of the synoptic case studies is presented in some detail. Certain statistics are presented for all three cases studied. It is concluded that, despite certain deficiencies, the model seems capable of improving the accuracy of low-cloud predictions for data-dense regions. It is also suggested that the model may have diagnostic and predictive utility for other applications which require a knowledge of the structure of the atmosphere within the planetary boundary layer.

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JOSEPH P. GERRITY JR.

Abstract

No Abstract Available.

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Joseph P. Gerrity Jr.

Abstract

Gandin and Murphy introduced an “equitable skill score” for use in evaluating categorical forecasts. For forecasts involving more than two categories, the elements of the scoring matrix are not defined uniquely. In this note, a specific formula for the general multiple-category scoring matrix is presented and proven to satisfy the necessary conditions for “equitability.” It is shown that, while it is not the only possible scoring matrix satisfying these necessary conditions, it is compatible with a logical condensation of the general K-category problem into a set of K−1 two-category problems. Each of the two-category problems is associated with one of the K−1 partitions defining the categories of the original problem.

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JOSEPH P. GERRITY JR.

Abstract

It is shown by elementary analysis that the two-step Lax-Wendroff method for integrating the advection equation is subject to nonlinear computational instability. The source of the instability lies in the possibility of lattice separation in the field of the advecting coefficient.

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JOSEPH P. GERRITY JR. and RONALD D. McPHERSON

Abstract

A limited area, fine-mesh, primitive equation barotropic model has been integrated using data observed at 500 mb. The lateral boundary conditions used in the model required that no change occur on the boundary during the 24-hr forecast. The predictions compare favorably with those obtained with the barotropic and baroclinic models in operational use at the National Meteorological Center.

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Joseph P. Gerrity Jr. and Ronald D. McPherson

Abstract

A new scheme is presented for the numerical integration of the quasi-static system of hydrodynamical equations. The main advantage of the proposed method is its efficiency for the economical production of short-range, high-resolution meteorological forecasts. The method combines an implicit formulation of the linear non-advective processes and a staggered spatial-temporal arrangement of the dependent variables upon the calculation lattice. Results of experimental integrations using a free-surface barotropic model are presented. These results confirm the theoretical order-of-magnitude reduction in computation time by the new method as compared with the conventional explicit method.

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JOSEPH P. GERRITY JR., RONALD D. McPHERSON, and PAUL D. POLGER

Abstract

Second- and fourth-order accurate finite-difference approximations of the equations governing a free surface autobarotropic fluid are compared with each other and with a second-order approximation on a one-half mesh. It is concluded that once the mesh size has been reduced sufficiently to adequately resolve the scales of interest then further reduction in mesh size would be inefficient in comparison with the use of more accurate finite-difference approximations.

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Joseph P. Gerrity Jr., Thomas L. Black, and Russell E. Treadon

Abstract

A new method is presented for obtaining the numerical solution of the production-dissipation component of the turbulent kinetic energy equation that arises in the Mellor-Yamada level 2.5 turbulent closure model. The development of this new method was motivated by the occasional appearance of large temporal oscillations in the solution provided by a previously used method. Analysis of the equation revealed that the solution should tend toward a stationary asymptotic value, which is the equilibrium value of turbulent kinetic energy for the level 2, Mellor-Yamada model. Failure to identify the correct asymptotic value in the formalism underlying the numerical solution of the equation allows the solution to overshoot the equilibrium. Repeated overshooting gives rise to an oscillation in the solution from one time step to the next. The new method prevents this from happening.

Idealized cases are used to demonstrate the performance of the new method. It has been incorporated into the eta coordinate, numerical weather prediction model being used by the National Meteorological Center.

Although the new method corrects the particular deficiency of the previous method, the integration of the equation for turbulent kinetic energy remains subject to oscillatory solutions associated with rapid variations of the Richardson number. An example of this is provided.

Additionally, even with the new method, it is sometimes necessary to revert to the level 2 model when the numerical integration of the full system of equations yields a value of turbulent kinetic energy that falls below a value associated with a singularity of the level 2.5 model. In future work, modifications of the Mellor-Yamada turbulent closure system that avoid this limitation will be investigated.

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ANDRÉ J. ROBERT, FREDERICK G. SHUMAN, and JOSEPH P. GERRITY JR.

Abstract

A rather general theory of nonlinear computational stability is reported. Instability is caused by both spatial and temporal high frequencies that, if not present initially, will appear from nonlinear interactions. It appears that through simple remedies relative stability, if not perfect stability, can be achieved.

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