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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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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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Akira Kasahara

Abstract

Recently, questions were raised concerning the ellipticity condition of the traditional balance equation, related to the application of nonlinear normal mode initialization for primitive equation prediction models. We investigate, in this paper, the occurrence of non-elliptic regions, using the FWE level IIIb analyses of the European Centre for Medium Range Weather Forecasts. We found that non-elliptic regions are ubiquitous in the tropics. For the traditional balance equation, non-elliptic regions are not realizable in the sense that balanced flows are not physically possible. To reconcile this dilemma, we postulate that the traditional balance equation lacks an additional term due, for example, to the effects of subgrid-scale motions. We evaluated this additional term as the residual by computing each term in the balance equation. This additional term modifies the ellipticity criterion of the balance equation. We call the modified ellipticity condition the realizability condition. This study concludes that the non-elliptic regions found in the tropics satisfy the realizability condition when this additional term in the balance equation is considered. This suggests that the apparent dilemma may be resolved by taking into account presently lacking physical processes in achieving the dynamical balance between the mass and wind fields, if the FGGE level IIIb analyses are supposed to be correct.

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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

Abstract

Spherical harmonics have been used to analyze global meteorological data, and because they are the solutions of a linearized nondivergent vorticity equation, it is appropriate to use them as orthogonal basis functions for analysis and prediction. However, for ultralong waves—geostrophic motions of the second type—horizontal divergence plays as essential a role as the vertical component of vorticity. Hence, it will be advantageous to use the solutions of linearized primitive equations over a sphere as basis functions. This will also permit identification of the characteristics of wave motions for the initialization of primitive equation models. Such solutions have been investigated in the past in conjunction with atmospheric tidal theories and the basic mathematical tools already available piecewise in the literature.

This paper reviews the mathematical development behind the construction of the eigensolutions (referred to as normal modes) of linearized primitive equations over a sphere. The basic state has no motion and the temperature is a function of height only. The solutions of both the vertical and horizontal structure equations are discussed.

The horizontal parts of such normal modes are called Hough harmonics Θ s l exp (isλ), where s is zonal wavenumber, λ longitude and l meridional mode index. Hough vector functions Θ s l consist of three components—zonal velocity Û, meridional velocity V̂ and geopotential height Ẑ, all of which are functions of latitude. There are three modes with distinct frequencies: eastward and westward propagating gravity waves, and westward propagating rotational waves of the Rossby/Haurwitz type. Hough harmonics are orthogonal and are conveniently used to decompose wind and mass fields simultaneously. Some examples are presented of global data decomposition in terms of Hough harmonics for studying ultralong waves in the atmosphere.

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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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Toshihisa Itano
and
Akira Kasahara

Abstract

The linear stability of a zonal flow confined in a domain within horizontal top and bottom boundaries is examined under full consideration of the Coriolis force. The basic zonal flow is assumed to be in thermal wind balance with the density field and to be sheared in both vertical and horizontal directions under statically and inertially stable conditions. By imposing top and bottom boundary conditions in this framework, the number of wave modes increases to four, instead of two in an unbounded domain, as already reported in studies on internal gravity waves. The four modes are classified into two pairs of high- and low-frequency modes: the high modes are superinertial and the low modes are subinertial. The discriminant of symmetric instability is nevertheless determined by the sign of the potential vorticity of the basic zonal flow, as in the case of an unbounded domain. The solutions satisfying the top and bottom boundary conditions are interpreted as the superposition of incident and reflected waves, revealing that the neutral solutions consist of two neutral plane waves with oppositely directed vertical group velocities. This may explain why the properties of wave behavior, such as the instability criteria, remain the same in both the bounded and unbounded domains, although the manifestation of wave activity, such as the order of dispersion relation, is quite different in the two cases. Furthermore, the slope of the constant momentum surface, the slope of the isopycnic surface including the nontraditional effect of the Coriolis force, and the ratio between the frequencies of gravity and inertial waves form an essential set of parameters for symmetric motion. The combination of these dimensionless quantities determines the fundamental nature of symmetric motions, such as stability, regardless of boundary conditions with and without the horizontal component of the planetary vorticity.

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David Williamson
and
Akira Kasahara

Abstract

Numerical experiments designed to investigate trade-offs among meteorological variables and between space and time in observing systems are conducted using the six-layer global circulation model of the National Center for Atmospheric Research (NCAR). The error growth characteristics of the NCAR model are first discussed in view of their effect on periodically updating historical data.

The updating experiments are divided into two groups. In the first group, “observed” temperature data with and without errors are periodically inserted into the model to recover the wind field. The root mean square (rms) error of the wind field is reduced by updating temperature and it approaches an asymptotic level which depends on the magnitude of the random errors in the “observed” temperature field. In the second group, “observed” wind data with and without errors are periodically updated to recover the temperature field. The rms error of the temperature field is reduced by updating winds. The asymptotic level depends on the magnitude of errors in the “observed” wind field. The results of wind updating were found to be sensitive to a slight change in the prediction model.

The scale and latitude dependence of the adaptation of meteorological variables forced by updating is also investigated. The wind is shown to adjust to temperature updating better at higher latitudes and for larger scales. The temperature adjusts to wind updating better for smaller scales, but not necessarily at lower latitudes.

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Akira Kasahara
and
Takashi Sasamori

Abstract

We discuss momentum balance and energetics in the stratosphere (18–36 km) based on the simulated mean January results described in Part I. The latitudinal distributions of component terms in the momentum budget equation clearly demonstrate the two-cell structure of the mean meridional circulation in the stratosphere as opposed to the three-cell structure in the troposphere. In contrast to the troposphere, the eddy and mean transports of momentum are equally important in the stratosphere in all latitudes with some tendency for the eddy transport to counteract the mean transport. The latitudinal distributions of component terms in the energy budget equations suggest the following mechanism for maintaining the zonal and eddy kinetic energies. The vertical flux of wave energy through the lower boundary of the stratosphere provides a major source of the eddy kinetic energy in the stratosphere. On the other hand, the zonal kinetic energy in the stratosphere appears to be maintained by energy conversion from the eddy kinetic energy against the energy loss by frictional dissipation. The present results of zonal and eddy kinetic energy budgets are compared with those based on real data of the Northern Hemisphere by various investigators and those of a numerical simulation experiment by Manabe and Hunt. The zonal internal energy budget shows that the zonal internal energy in the stratosphere is maintained by the supply of sensible heat from the troposphere and energy conversion from the eddy kinetic energy against the loss of energy by radiative cooling and the downward transport of internal energy by mean vertical motion. In addition, we find that interpretation of the energetics of the stratosphere depends upon how various energy terms are combined and energy conversion terms formulated. This special consideration is needed because the vertical transports of energy at the interface between the troposphere and the stratosphere play important roles in the energetics of the stratosphere.

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Tomio Asai
and
Akira Kasahara

Abstract

An attempt is made to investigate theoretically the controlling influence of compensating downward motions on the development of cumulus clouds and the size of the cloudless areas associated with them. The model consists of two circular concentric air columns, the inside column corresponding to the updraft (cloud) region and the outside concentric annular column to the downward motion region. The combined cell is surrounded by the atmospheric at rest. The governing equations of both the updraft and the compensating downward motion are derived from the conservation equations of momentum, heat, moisture and mass.

The differential equations are solved numerically to compute the vertical velocity, temperature, specific humidity and liquid water content in and out of the cloud as functions of height and time. Two experiments were performed with and without the effect of compensating downward motion. The main conclusions are the following: Without the effect of the compensating motion, the structure of the solitary updraft tends to a steady state. However, with the compensating motion, no tall cloud is maintained (unless there is a steady source of moisture at the cloud base), since the compensating downward motion acts as a “break.” Also, it was found that the most active cloud system develops when the ratio of the cloud area over the entire area (including the cloudless area associated with the updraft) is of the order of several per cent.

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