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# Various Vertical Coordinate Systems Used for Numerical Weather Prediction

Akira Kasahara

## Abstract

For numerical weather prediction with primitive equations (the Eulerian hydrodynamic equations modified by the assumption of hydrostatic equilibrium), various coordinate systems are used to represent the vertical structure of the atmosphere. In this paper, we review the essential features of prediction equations, satisfying the conservation of mass and total energy, in various vertical coordinate systems. We formulate the equations of horizontal motion, hydrostatic balance, mass continuity, and thermodynamics using a generalized vertical coordinate in which any variable that gives a single-valued monotonic relationship with a geometric height can be used as a vertical coordinate. Conditions to conserve total energy in a generalized vertical coordinate are investigated.

Various prediction schemes using pressure, height, and potential temperature as a vertical coordinate are derived from the set of basic equations in the generalized coordinate system. These three coordinate systems are unique in that the features of prediction equations in each system are all distinct. We place special emphasis on handling the earth's orography as the lower boundary condition. As an extension of the original idea of Phillips applied to the pressure-coordinate system, we propose transformed height and isentropic systems. In those systems, both the top of the model atmosphere and the earth's surface are always coordinate surfaces. It is hoped that these new schemes, as in the case of the Phillips' sigma-system, will enable us to handle the effect of the earth's orography in the prediction models without lengthy coding logic.

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# THE NUMERICAL PREDICTION OF HURRICANE MOVEMENT WITH THE BAROTROPIC MODEL

Akira Kasahara

## Abstract

In order to increase the accuracy of prediction of hurricane movement beyond that which has been obtained with methods now in use, a numerical procedure based upon the barotropic model is presented here.

A vortex field is separated from the total flow to obtain the residual-steering flow field. By solving the equation for the steering field, which does not include any parameters related to the vortex, the prediction of the steering flow is executed in the ordinary manner. By solving the other equation, which includes the interaction terms between the hurricane and the steering flow, the movement of the vortex pattern is predicted. To make the latter problem more tractable, a velocity formula for the movement of the vortex center is derived on the basis of a few reasonable assumptions concerning the structure of the vortex and of the steering field. The variables in the formula are expressed only in terms of quantities determined by the steering field and a single characteristic parameter related to the scale of the hurricane. By this means, a forecast of the hurricane movement can be programmed as a subroutine which is executed in the course of the prediction of the steering flow.

Five cases of predicting the 24-hour and 48-hour movements of hurricane “Diane” and “Connie” (August 1955) at the 500-mb level are presented here. These results have been obtained by the use of a high-speed computer, from initial maps which were be-analyzed using all available data.

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# The Dynamical Influence of Orography on the Large-Scale Motion of the Atmosphere

Akira Kasahara

## Abstract

In an attempt to understand the dynamical influence of the earth's orography upon the large-scale motion of the atmosphere, the system of “shallow water” equations on the rotating earth is integrated numerically. The model consists of an incompressible, homogeneous, hydrostatic and inviscid fluid. The “beta-plane” approximation is used to simplify the model. The fluid is confined in a channel bounded on the north and south by two parallel “walls” extending in the cast-west direction. Periodicity is the boundary condition applied at the east and west boundaries to simulate the cyclic continuity of the zone with longitude. A circular obstacle of parabolic shape is placed at the bottom in the middle of the channel. The steady-state solutions in the absence of the obstacle are used as the initial conditions of the problem. Five different cases are investigated in detail. All computations were performed for an interval of 20 days (some cases were run longer) with a time step of 6 minutes.

The following main results were obtained: 1) Westerly flows past the obstacle produced a train of long waves on the lee side, which can be identified as “planetary” waves. On the other hand, easterly flows are little disturbed by the obstacle and long waves do not appear; 2) The number of waves produced in the westerly cases agrees with the number expected from the steady-state Rossby-Haurwitz wave formula for various intensities of zonal flow past the obstacle.

The results of the present calculations agree qualitatively with the data obtained in the early 1950's by Fultz, Long and Frenzen in laboratory experiments on the flow past a barrier in a rotating hemispherical shell. Finally, a theoretical consideration is given to explain the characteristic differences between westerly and easterly flows past the obstacle as observed in the numerical experiments.

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# ON CERTAIN FINITE-DIFFERENCE METHODS FOR FLUID DYNAMICS

AKIRA KASAHARA

## Abstract

Two finite-difference methods for geophysical fluid problems are described, and stability conditions of these schemes are discussed. These two schemes are formulated based upon a similar procedure given by Lax and Wendroff in order to obtain a second-order accuracy in finite-difference equations. However, the two schemes show remarkable differences in their computational stability. One scheme is stable, as one might expect, under the usual stability conditions of Courant-Friedrichs-Lewy and Lax-Wendroff. However, the other scheme is conditionally stable only if the flow is supereritical (supersonic in the case of gas dynamics) and unconditionally unstable if the flow is suberitical (subsonic).

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Akira Kasahara
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# Further Studies on a Spectral Model of the Global Barotropic Primitive Equations with Hough Harmonic Expansions

Akira Kasahara

## Abstract

This paper describes further improvement in a new spectral model of the global barotropic primitive equations (Kasahara, 1977) which utilizes Hough harmonies as basis functions. A review is presented on a method of constructing Hough harmonics (normal modes of Laplace's tidal equations) with new results of the eigensolutions for the logitudinal wavenumber zero case.

Applying this complete set of orthonormal Hough harmonics, we formulate a spectral model of the nonlinear, barotropic primitive equations (shallow-water equations) over a sphere which eliminates separate treatment of the zonally averaged component equations in the previously proposed model by the author. An example of the model calculation with Haurwitz wavenumber 6 initial conditions is presented.

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# A COMPARISON BETWEEN GEOSTROPHIC AND NONGEOSTROPHIC NUMERICAL FORECASTS OF HURRICANE MOVEMENT WITH THE BAROTROPIC STEERING MODEL

Akira Kasahara

## Abstract

The steering flow of a hurricane is obtained by eliminating the vortex pattern from the total flow field. The evolution of the steering flow is predicted by solving the barotropic nondivergent vorticity equation. Based upon the steering-flow prediction, a forecast of the hurricane movement is obtained with the use of an equation which provides for interaction between the hurricane and the steering flow.

In the geostrophic model, the geostrophic-wind assumption is used in solving the vorticity equation. In the nongeostrophic model, the stream function governed by the ‘balance equation’ is adopted. With the above two prediction models, 45 pairs of predictions of the 24-hr and 48-hr movement of hurricanes Diane and Connie (August 1955) and Betsy (August 1956) at the 500- and 700-mb levels were prepared by the use of an electronic computer.

A detailed comparison between performances of the two prediction models is presented. Generally speaking, there is a remarkable similarity between the nongeostrophic and geostrophic forecasts of hurricane movement. The accuracy of the 700-mb hurricane forecasts appears to be comparable with that obtained from the 500-mb forecasts. However, for both prediction models, the ‘resultant’ forecast, which is the vector mean of the 700- and 500-mb predicted displacements, seems to provide a significant improvement over the accuracy of a single-level barotropic forecast. These analyses suggest that further improvements are possible by advancing the prediction model from the barotropic model to the equivalent barotropic or the baroclinic model.

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# THE NUMERICAL PREDICTION OF HURRICANE MOVEMENT WITH A TWO-LEVEL BAROCLINIC MODEL

Akira Kasahara

## Abstract

A steering method of predicting hurricane movement is formulated based upon a two-level baroclinic model. The upper steering-flow field is constructed from the pressure-weighted mean of the 200- and 500-mb steering-height fields, and the lower steering field is constructed from the pressure-weighted mean of the 700- and 1000-mb steering-height fields. Here, the steering flow of a hurricane is defined as the residual field after eliminating the vortex pattern from the total-flow field.

The evolutions of the upper and lower steering flows are predicted simultaneously by solving the two-level steering-flow vorticity equations. Based upon those steering-flow forecasts, the movement of a hurricane is predicted with the use of an equation which is derived from a solution of two vortex vorticity equations. A side condition is imposed that the upper and lower vortex patterns should move with the same velocity in the corresponding steering flows.

Ten cases of predicting the movement of hurricane “Betsy” (August 1956) up to 48 hr are presented. A preliminary comparison of the forecasts with those obtained from the barotropic model is also made.

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# A NUMERICAL EXPERIMENT ON THE DEVELOPMENT OF A TROPICAL CYCLONE

Akira Kasahara

## Abstract

A model of a tropical cyclone is constructed which is based upon conservation of momentum, mass,water vapor and heat in the hydrostatic system. The horizontal and vertical eddy-exchange processes for momentum, moisture and heat are included in the equations in order to incorporate the planetary frictional (Ekman) layer into the model. The effects of the surface boundary (Prandtl) layer are simulated by the boundary conditions for the equations, which permit the evaluation of surface stress, the sensible heat transport and the evaporation of water vapor from the earth surface. The energy sources of the model are the latent heat of condensation released during the ascent of moist air and the sensible heat transported from the ocean surface.

The formulation of the finite-difference equations for the axially-symmetric case is presented, together with an examination of the computational stability. By means of a high-speed computer, two independent computations with and without the supply of latent heat were made from the same initial wind and temperature fields.

A comparison of the two cases reveals an important effect of latent heat of condensation upon the development of the tangential motion as well as its warm-core radial circulation. It is shown that the formation of cellar convective bands in the system is a manifestation of the gravitational instability which does not occur without the latent-heat supply.

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# Simulation Experiments for Meteorological Observing Systems for GARP

Akira Kasahara

The future observing system of the global atmosphere which has been contemplated for the Global Atmospheric Research Program (GARP) is a combination of various observing subsystems including satellites, constant-level balloons, automatic ocean buoys, etc., as well as conventional upper-air and surface networks. All observing subsystems are neither perfect nor ideal. Numerical experiments (with global circulation models) are needed to evaluate the subsystems in terms of the accuracy, density, and frequency of observations. They are called Observing Systems Simulation Experiments (OSSE).

The purpose of this report is to describe the present activities in OSSE. Most useful up-to-date information was obtained from the oral presentation of papers at the International Symposium on Four-Dimensional Data Assimilation, Princeton, N. J., 19–22 April 1971.

Since a global circulation model is used as an integrator and analyzer of observed data in three-dimensional space and time taken from various observing subsystems, the question of error growth of the prediction model in relation to the accuracy of observation is discussed as the predictability experiment.

Another important property of the model atmosphere is the ability to adjust the model's variables for forced prescription of incomplete observed data. This adjustment property is used to assimilate observed data in four dimensions. The method of direct substitution is used to determine the basic data requirements for observation of wind, temperature, and surface pressure.

Various methods of four-dimensional data assimilation are reviewed for the purpose of optimum design for OSSE. Finally, questions concerning the reference level information are reviewed and results of some numerical experiments conducted at the National Center for Atmospheric Research are presented.

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