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Akira Kasahara
and
Yosuke Shigehisa

Abstract

In the formulation of nonlinear normal mode initialization (NNMI), we apply the separation of variables to a primitive equation model which is linearized about a basic state at rest. This results in the vertical and horizontal structure equations from which normal mode functions are constructed as their solutions. This paper is concerned with the property of the vertical normal mode functions. They vertical structure functions constructed in the framework of a vertically staggered discretized atmospheric model may not, in general, be orthogonal. There is an obvious advantage in using an orthogonal set of expansion functions. We propose in this paper the construction of orthogonal vertical structure functions for the NNMI with an atmospheric prediction model for which non-orthogonal vertical structure functions have been adopted in the past. We discuss the difference between the new formulation which yields orthogonal vertical structure functions and the old formulation which does not, in general, yield orthogonality. We also present the results of a sensitivity test with a global model on 55 day forecasts starting from two sets of initial conditions in which two versions of vertical normal mode functions are used. It is shown that the differences between the two 5-day forecasts are negligibly small.

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Akira Kasahara
and
Kamal Puri

Abstract

To represent atmospheric data spectrally in three indices (zonal wavenumber, and meridional and vertical modal indices), we propose to use three-dimensional normal mode functions (NMF's) to express the wind and mass fields simultaneously. The NMF's are constructed from the eigensolutions of a global primitive equation model and they are orthogonal functions. The vertical parts are obtained from the solutions of the vertical structure equation with the equivalent height as the eigenvalue. The vertical modal index is associated with a different value of the equivalent height. The horizontal parts of NMF's are Hough harmonics with zonal wavenumber and meridional modal index as two-dimensional scalings. The expansion of global data in terms of NMF's permits the partition of energy into two distinct kinds of motions-gravity-inertia modes and rotational modes of Rossby/Haurwitz type. Both kinds of motion are also partitioned into different vertical modes. Results of the spectral distribution of atmospheric energy, obtained by expanding in the NMF's hemispherical data of the National Meteorological Center, are presented. Information obtained will be useful to select proper horizontal and vertical computational resolutions for representation of atmospheric data.

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Akira Kasahara
and
Jian-Hua Qian

Abstract

Anticipating use of a very high resolution global atmospheric model for numerical weather prediction in the future without a traditional hydrostatic assumption, this article describes a unified method to obtain the normal modes of a nonhydrostatic, compressible, and baroclinic global atmospheric model.

A system of linearized equations is set up with respect to an atmosphere at rest. An eigenvalue–eigenfunction problem is formulated, consisting of horizontal and vertical structure equations with suitable boundary conditions. The wave frequency and the separation parameter, referred to as “equivalent height,” appear in both the horizontal and vertical equations. Hence, these two equations must be solved as a coupled problem.

Numerical results are presented for an isothermal atmosphere. Since the solutions of the horizontal structure equation can only be obtained numerically, the coupled problem is solved by an iteration method. In the primitive-equation (hydrostatic) models, there are two kinds of normal modes: The first kind consists of eastward and westward propagating gravity–inertia oscillations, and the second kind consists of westward propagating rotational (Rossby–Haurwitz type) oscillations. In the nonhydrostatic model, there is an additional kind of eastward and westward propagating acoustic–inertia oscillations. The horizontal structures of the third kind are distinguished from those of the first kind by large differences in the equivalent height. The second kind is hardly affected by nonhydrostatic effects. In addition, there are so-called external inertia–gravity mode oscillations (Lamb waves), which propagate horizontally with almost constant speed of sound. Also, there are external rotational mode oscillations that correspond to equivalent barotropic planetary waves. Those two classes of oscillations are identical to those in the hydrostatic version of model. Nonhydrostatic effects on the first kind of oscillations become significant for smaller horizontal and deeper vertical scales of motion.

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Akira A. Kasahara
and
George W. Platzman

Abstract

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Akira Kasahara
and
Warren M. Washington

Abstract

This paper describes the method of incorporating into the NCAR global circulation model the dynamic effect of mountains, the prediction of cloudiness for radiation calculations, and the calculation of ground surface temperature using a heat balance equation. Other aspects of the physics of the model and the finite-difference schemes are very similar to those discussed by the authors in 1967 and 1970. For the simulation of seasonal climate we specify two parameters: the sun's declination and the distribution of ocean surface temperatures. Since the prediction of cloudiness is parameterized in terms of the relative humidity and the vertical motion fields, solar and atmospheric radiation processes interact closely with the dynamics of the atmosphere through variations in the fields of cloudiness, temperature and water vapor. Coupling between radiation and dynamics helps to maintain stronger baroclinic activity in middle latitudes. Although a hydrologic cycle is included in the model atmosphere and the ground surface temperature is computed, a hydrologic cycle in the ground is not taken into account. Instead, it is assumed that the latent heat transport from the ground to the atmosphere and the soil heat transport below the surface are both functions of the sensible heat transport between the ground and the atmosphere.

Experiments are conducted to simulate January climate with and without the earth's orography. In both experiments the domain of continents, the January mean ocean surface temperatures, and the sun's declination for mid-January are unchanged during the time integrations. The model has a spherical horizontal mesh spacing of 5° in both longitude and latitude and six vertical layers at 3-km height increments. The time step is 6 min and both cares are integrated up to 80 days starting from an isothermal atmosphere at rest. The results of the 41–70 days of the time integration are analyzed for various diagnostic studies. Synoptic comparisons of the two experiments are made for selective meteorological variables to discuss the relative importance of the thermal and orographic influences upon the large-scale motions of the atmosphere. Detailed studies are made on the balance of momentum, water vapor, heat and energy. The present experiments indicate that the six-layer and 5° mesh model can simulate successfully a January climate and that the earth's orography plays a minor role over the thermal effect of continentality in determining the major features in the transport mechanism of momentum, water vapor, heat and energy in terms of the zonal mean state. However, for the regional aspects of general circulation the effects of orography are significant.

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Akira Kasahara
and
H. L. Tanaka

Abstract

As an alternative to the finite difference method, we explore the use of the spectral method with normalmodes as the basis functions for discretizing dependent variables in the vertical direction in order to obtainnumerical solutions to time dependent atmospheric equations. The normal modes are free solutions to the timedependent perturbation equations linearized around the atmosphere at rest. To demonstrate the feasibility ofnormal mode representation in the spectral vertical discretization, the vertical normal mode expansion is appliedto the quasi-geostrophic potential vorticity equation to investigate the traditional baroclinic instability of Charneyand Green types on a zonal flow with a constant vertical shear. The convergence of the numerical solutions isexamined in detail in relation to the spectral resolution of expansion functions.We then extend the method of vertical normal mode expansion to solve the problem of baroclinic instabilityon the sphere. Two aspects are different from the earlier example. One is use of the primitive equations insteadof the quasi-geostrophic system and the other is application of normal mode expansions in the horizontal, aswell as vertical direction. First, we derive the evolution equations for the spectral coefficients of truncated seriesin three-dimensional normal mode functions by application of the Galerkin procedure to the global primitiveequations linearized around a basic zonal flow with vertical and meridional shear. Then, an eigenvalue-eigenfunction problem is solved to investigate the stability of perturbation motions superimposed on the 30' jetexamined earlier by Simmons, Hoskins and Frederiksen. From these two examples, it is concluded that thenormal mode spectral method is a viable numerical technique for discretizing model variables in the vertical.

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Akira Kasahara
and
Desiraju B. Rao

Abstract

Stability properties of a two-layer model of homogeneous and incompressible fluid subject to gravity and rotation are investigated by the small-perturbation method. The upper and lower fluids correspond, respectively, to warm and cold air. The interface between the warm and cold layers intersects the ground and forms the surface front. The model has been used to investigate the development of frontal cyclones. Linearized equations of warm and cold layers are solved as an eigenvalue problem to find solutions growing exponentially with time using a finite-difference technique.

For prescribed values of the density ratio ε of warm and cold layers, the north-south extent D of the frontal interface, the Coriolis parameter f, the external gravity wave speed C 0, and the basic state cold air velocity ū1, we vary the values of the wavenumber k of perturbations and the basic state warm air velocity ū2, through the use of the Rossby number [Ro≡½(ū2&minusū1)k/f] and the Richardson number [Ri≡C 0(1−∈)/(ū2&minusū1)].

Orlanski has investigated the stability of a similar model in the parameter domain of Ri≲5 and Ro≲3. Eliasen has studied the stability of a frontal model in the domain of 3 Ri 6 and Ro≲0.5. In the present work, we cover the domain of 1.25≲Ri<12 and Ro≲2 which includes the region that is not investigated by either Eliasen or Orlanski.

The stability characteristics for Ri≲2 are fairly complex. For Ri≳3, there are two modes of instability, One appears in the region of Ro≲0.4 which corresponds to the type of instability found by Eliasen and it is a quasi-geostrophic baroclinic instability. The other appears for Ro≲0.9 which has an unbounded growth rate as the wavelength decreases. Orlanski pointed out the presence of instability in this parameter domain, but he has not investigated in detail the characteristics of the instability.

The kinematics for Ri=5.0 and Ro=1.2 reveals that the unstable motion is highly nongeostrophic and has a small latitudinal width. Since this instability has an unbounded growth rate as the wavelength decreases, the determination of the preferred scale of this unstable motion is very much dependent on the mechanism of momentum dissipation which is not considered in this study.

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Akira Kasahara
and
Arthur P. Mizzi

Abstract

In order to asses the uncertainty of daily synoptic analyses for the atmospheric state, the intercomparison of three First GARP Global Experiment (FGGE) level IIIb datasets is conducted. The original analyses and reanalyses produced by the European Centre for Medium Range Weather Forecasts (ECMWF) are compared with the reanalyses produced by the National Meteorological Center (NMC) system, operational in early 1987. Daily values of vorticity ζ, divergence δ, temperature T, static stability Γ, mixing ratio q, vertical motion ω, and diagnosed diabatic heating rate are compared for the period of 26 January–11 February 1979. The spatial mean and variance, temporal mean and variance, two-dimensional wavenumber power spectrum, anomaly correlation, and normalized square difference are used for comparison. Also, equivalent blackbody temperatures from the TIROS-N are used as a proxy to the vertical motion and diagnosed diabatic heating rates in the tropics.

Data are interpolated onto the σ coordinates of the NCAR Community Climate Model (CCM 1 ) with 12 vertical levels. Global data are expanded in spherical harmonics with two resolutions [triangular 13 (T 13) and T42] in order to investigate how data agreement changes depending on the horizontal length scale. Other questions to be investigated are: What aspects of the analyses have improved in the FGGE reanalyses produced at the ECMWF and NMC? What aspects of the analyses are still unsatisfactory? What can be done to further improve the analyses? Since data agreement tends to be weak in the tropics, attention is focused in the tropical belt of 30°N–30°S. One enlightening finding is that both ζ and δ of the NMC reanalyses agree more closely with the ECMWF reanalyses than their earlier analyses. This result may indicate that both the data quality and the analysis techniques have improved. More good news is found in that the agreement of ζ at T13 is excellent, while there is only slight disagreement at T42, indicating that FGGE has succeeded in describing the quasi-rotational state of the atmosphere, even in the tropics. The bad news is that interanalysis agreement of δ and q>/ is poor. Similarly, the analyses of ω show the least agreement, indicating the need for further improvement in describing the diabatically driven irrotational circulations of the tropics.

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WARREN M. WASHINGTON
and
AKIRA KASAHARA

Abstract

Results of a January simulation experiment with the two-layer version of the NCAR global circulation model are discussed. The model includes a hydrological cycle, horizontal and vertical turbulent transports of momentum, heat, and water vapor from the lower boundary and within the atmosphere, and calculations of solar and terrestrial radiation. Although the water vapor field interacts with the radiation calculations, the cloud distribution is a function of latitude and season. In this version of the model, the earth's orography is omitted as well as an explicit calculation of the surface temperature.

This version of the model has a spherical horizontal mesh spacing of 5° in both longitude and latitude and two vertical layers at 6-km height, increments. The details of the finite-difference scheme for the model are presented.

The initial conditions for this experiment are based on an isothermal atmosphere at rest. The zonal mean cloudiness, the mean sea level temperature distribution, and the sun's declination are specified for January. The early stage of the numerical integration is characterized by a Hadley-type direct circulation due to the thermal contrasts between the continents and oceans. Within 2 weeks, the Hadley circulation breaks down due to baroclinic instability. This results in the typical three-cell meridional circulation. The comparison between computed and observed January climatology is discussed together with the presentation of momentum, moisture, and energy budgets. The main result from these budget calculations is that the Hadley cell is of dominant importance in the transport of various quantities within the Tropics and that baroclinic eddies are important in midlatitudes.

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AKIRA KASAHARA
and
WARREN M. WASHINGTON

Abstract

This paper describes a model of the general circulation of the earth's atmosphere which has been developed and experimented with, since 1964, at the National Center for Atmospheric Research (NCAR), Boulder, Colo. A distinguishing feature of the NCAR model is that the vertical coordinate is height rather than pressure, though hydrostatic equilibrium is maintained in the system. In fact, the dynamical framework of the model is very similar to the one proposed by L. F. Richardson in 1922.

Various physical processes in the atmosphere, such as energy transfer due to solar and terrestrial radiation, small-scale turbulence and convection, etc., are incorporated in the model. An explicit prediction of the moisture field is avoided. Instead, it is assumed that the atmosphere is completely saturated by water vapor. Thus, the release of latent heat of condensation can be computed. In addition to a description of the model, the equations for the zonal mean and eddy energy are presented. Finally, a baroclinic stability analysis of the model is made in order to gain an insight into the finite-difference formulation of the present model. Long term (over 100 days) numerical integrations are being performed successfully with a two-layer version of the present model. Details of finite-difference schemes and the results of numerical calculations will be described in a separate article.

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