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1.
Summary Most finite-difference numerical weather prediction models employ vertical discretizations that are staggered, and are low-order (usually second-order) approximations for the important terms such as the derivation of the geopotential from the hydrostatic equation, and the calculation of the vertically integrated divergence. In a sigma-coordinate model the latter is used for computing both the surface pressure change and the vertical velocity. All of the above-mentioned variables can diminish the accuracy of the forecast if they are not calculated accurately, and can have an impact on related quantities such as precipitation.In this study various discretization schemes in the vertical are compared both in theory and in practice. Four different vertical grids are tested: one unstaggered and three staggered (including the widely-used Lorenz grid). The comparison is carried out by assessing the accuracy of the grids using vertical numerics that range from second-order up to sixth-order.The theoretical part of the study examines how faithfully each vertical grid reproduces the vertical modes of the governing equations linearized with a basic state atmosphere. The performance of the grids is evaluated for 2nd, 4th and 6th-order numerical schemes based on Lagrange polynomials, and for a 6th-ordercompact scheme.Our interpretation of the results of the theoretical study is as follows. The most important result is that the order of accuracy employed in the numerics seems to be more significant than the choice of vertical grid. There are differences between the grids at second-order, but these differences effectively vanish as the order of accuracy increases. The sixth-order schemes all produce very accurate results with the grids performing equally well, and with the compact scheme significantly outperforming the Lagrange scheme. A second major result is that for the number of levels typically used in current operational forecast models, second-order schemes (which are used almost universally) all appear to be relatively poor, for other than the lowest modes.The theoretical claims were confirmed in practice using a large number (100) of forecasts with the Australian Bureau of Meteorology Research Centre's operational model. By comparing test model forecasts using the four grids and the different orders of numerics with very high resolution control model forecasts, the results of the theoretical study seem to be corroborated.With 8 Figures 相似文献
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3.
M. Shabbar 《Boundary-Layer Meteorology》1974,6(3-4):413-421
A theory is presented both for spectral energy transfer and for the transfer of spectral components of pseudo-potential enstrophy in a homogeneous quasi-geostrophic turbulent field which is rendered anisotropic by the distortion caused by a random collection of vortices superimposed on the principal motions. The fluid is, thus, subjected to an almost irrotational distortion. The random vortices cause straining effects on turbulent velocity and temperature fluctuations and modify the energy spectrum in the spectral ranges of interest. The strain imposed by the distortion is assumed to be homogeneous. For three-dimensional quasi-geostrophic turbulence that conserves pseudo-potential enstrophy as well as energy, this theory predicts –8/3 and –4 power inertial-range energy spectra.The predictions favourably corroborate the observed spectrum of energy in the atmosphere in the region of hemispheric wave-numbers 10–16 with a –8/3 slope and at higher wave-numbers with –4 slope on a log-log energy-wave-number diagram. The transfer rates of pseudo-potential enstrophy in the range 10n16 and of energy in the rangen>16 are identically zero, while the transfer of energy in the first range is from higher to lower wave-numbers and that of the pseudo-potential enstrophy in the second range is from lower to higher wave-numbers.As compared with the earlier two-dimensional turbulence theory of Kraichnan and the quasigeostrophic turbulence theory of Charney, the present theory predicts more realistic shapes of the energy spectra of atmospheric motions at scales shorter than the baroclinic excitation scales. 相似文献
4.
Prof. Z. I. Janjić 《Meteorology and Atmospheric Physics》1995,55(1-2):1-16
Summary Two-time-level multiply-upstream semi-Lagrangian schemes were examined in the case of the self-advecting, one-dimensional nonlinear momentum conservation equation. The shock formation process was analyzed. It is pointed out that the shocks cannot be created in the truncated systems satisfying the Pudykiewicz, Benoit and Staniforth criterion.The numerical integrations were restricted to 12 h. It was shown that, at least in the sub-CFL range, increased complexity of the scheme can compensate reduced horizontal resolution. A considerable sensitivity of the schemes with respect to the time step was detected. In the super-CFL mode, several windows on various time scales were found within which the Pudykiewicz, Benoit and Staniforth criterion was satisfied. The time step of 1.44 times the maximum time step allowed by the CFL criterion was used in the semi-Lagrangian runs.The super-CFL, semi-Lagrangian solutions were diverging progressively from the sub-CFL ones as the forecasts advanced. This was also reflected in the energy spectra.Unacceptably large energy losses were encountered in the super-CFL, semi-Lagrangian runs. Most of these losses could be explained by the reduced mean wind speed, i.e., the amplitude of the zero wavenumber wave. At the same time, the energy content in the shorter waves increased. In a more complex model, such a situation would resemble a loss of zonal, and an increase of transient eddy kinetic energy.A trajectory error measure was defined as the maximum absolute value of the distance between the actual arriving point of the particle originating at the estimated departure point, and the grid point assumed to be the arrival point in the semi-Lagrangian procedure. In contrast to the sub-CFL regime, this measure could reach a considerable fraction of the grid distance in the computations with the super-CFL time steps.In the physical system considered, the trajectories are determined only by the velocities at the departure points. With the semi-Lagrangian schemes the distances traveled by the particles are estimated on the basis of the velocities at the points downstream with respect to the departure points. Thus, unless the solution is smooth (in space and time) on the scales of the extrapolation distances/times, the upstream extrapolation does not promise the convergence of the solution.With 16 Figures 相似文献
5.
钟青 《Acta Meteorologica Sinica》1999,(2)
In this paper,two formulation theorems of time-difference fidelity schemes for generalquadratic and cubic physical conservation laws are respectively constructed and proved,with earliermajor conserving time-discretized schemes given as special cases.These two theorems can providenew mathematical basis for solving basic formulation problems of more types of conservative time-discrete fidelity schemes,and even for formulating conservative temporal-spatial discrete fidelityschemes by combining existing instantly conserving space-discretized schemes.Besides.the twotheorems can also solve two large categories of problems about linear and nonlinear computationalinstability.The traditional global spectral-vertical finite-difference semi-implicit model for baroclinicprimitive equations is currently used in many countries in the world for operational weatherforecast and numerical simulations of general circulation.The present work,however,based onTheorem 2 formulated in this paper,develops and realizes a high-order total energy conservingsemi-implicit time-difference fidelity scheme for global spectral-vertical finite-difference model ofbaroclinic primitive equations.Prior to this,such a basic formulation problem remains unsolved forlong,whether in terms of theory or practice.The total energy conserving semi-implicit schemeformulated here is applicable to real data long-term numerical integration.The experiment of thirteen FGGE data 30-day numerical integration indicates that the newtype of total energy conserving semi-implicit fidelity scheme can surely modify the systematicdeviation of energy and mass conserving of the traditional scheme.It should be particularly notedthat,under the experiment conditions of the present work,the systematic errors induced by theviolation of physical laws of conservation in the time-discretized process regarding the traditionalscheme designs(called type Z errors for short)can contribute up to one-third of the totalsystematic root-mean-square(RMS)error at the end of second week of the integration and exceedone half of the total amount four weeks afterwards.In contrast,by realizing a total energyconserving semi-implicit fidelity scheme and thereby eliminating corresponding type Z errors,roughly an average of one-fourth of the RMS errors in the traditional forecast cases can be reducedat the end of second week of the integration,and averagely more than one-third reduced at integraltime of four weeks afterwards.In addition,experiment results also reveal that,in a sense,theeffects of type Z errors are no less great than that of the real topographic forcing of the model.The prospects of the new type of total energy conserving fidelity schemes are very encouraging. 相似文献
6.
7.
THE FORMULATION OF FIDELITY SCHEMES OF PHYSICAL CONSERVATION LAWS AND IMPROVEMENTS ON A TRADITIONAL SPECTRAL MODEL OF BAROCLINIC PRIMITIVE EQUATIONS FOR NUMERICAL PREDICTION* 
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下载免费PDF全文 Zhong Qing 《Acta Meteorologica Sinica》1999,13(2):226-248
In this paper,two formulation theorems of time-difference fidelity schemes for general quadratic and cubic physical conservation laws are respectively constructed and proved,with earlier major conserving time-discretized schemes given as special cases.These two theorems can provide new mathematical basis for solving basic formulation problems of more types of conservative time-discrete fidelity schemes,and even for formulating conservative temporal-spatial discrete fidelity schemes by combining existing instantly conserving space-discretized schemes.Besides.the two theorems can also solve two large categories of problems about linear and nonlinear computational instability.The traditional global spectral-vertical finite-difference semi-implicit model for baroclinic primitive equations is currently used in many countries in the world for operational weather forecast and numerical simulations of general circulation.The present work,however,based on Theorem 2 formulated in this paper,develops and realizes a high-order total energy conserving semi-implicit time-difference fidelity scheme for global spectral-vertical finite-difference model of baroclinic primitive equations.Prior to this,such a basic formulation problem remains unsolved for long,whether in terms of theory or practice.The total energy conserving semi-implicit scheme formulated here is applicable to real data long-term numerical integration.The experiment of thirteen FGGE data 30-day numerical integration indicates that the new type of total energy conserving semi-implicit fidelity scheme can surely modify the systematic deviation of energy and mass conserving of the traditional scheme.It should be particularly noted that,under the experiment conditions of the present work,the systematic errors induced by the violation of physical laws of conservation in the time-discretized process regarding the traditional scheme designs(called type Z errors for short) can contribute up to one-third of the total systematic root-mean-square(RMS) error at the end of second week of the integration and exceed one half of the total amount four weeks afterwards.In contrast,by realizing a total energy conserving semi-implicit fidelity scheme and thereby eliminating corresponding type Z errors,roughly an average of one-fourth of the RMS errors in the traditional forecast cases can be reduced at the end of second week of the integration,and averagely more than one-third reduced at integral time of four weeks afterwards.In addition,experiment results also reveal that,in a sense,the effects of type Z errors are no less great than that of the real topographic forcing of the model.The prospects of the new type of total energy conserving fidelity schemes are very encouraging. 相似文献
8.
In this paper,a numerical simulation of the geostrophic adjustment process with C-grid network is illustrated.A difference scheme which has the energy and potential vorticity conserving relation consistent with the differential equations is given,and the effect of some time difference schemes on dispersion of the gravity-inertia wave is discussed.An improved forward-backward time integration scheme is proposed for keeping the computational stability.The effect of various boundary conditions for a finite region model On the gravity-inertia wave is shown by some calculated results. 相似文献
9.
《Dynamics of Atmospheres and Oceans》1988,12(1):47-70
Two local implementations of no-slip boundary conditions are investigated for both the vorticity—streamfunction and momentum—pressure formulations of the time-dependent planar incompressible Navier-Stokes equations, as applied to barotropic ocean circulation modelling. The objective is to determine the extent to which the local accuracy and numerical consistency of these conditions affects the global solution. The effects of a non-local implementation of no-slip conditions for the vorticity—streamfunction equations are also studied. In all cases, boundary condition effects are measured by comparing time-averaged dynamics of turbulent solutions of numerical models based on the two formulations.In the model interior, the energy and enstrophy conserving Arakawa Jacobian is used for the vorticity—streamfunction equations while an extension of the energy and potential enstrophy conserving Arakawa and Lamb finite difference scheme is used for the momentum-pressure equations.Numerical experiments performed with a non-linear model similar to Bryan's barotropic ocean reveal no significant differences between the time-averaged solutions obtained with either of the two formulations, with each using either of the two local boundary conditions. A simple one-dimensional analogue of the vorticity—streamfunction equations is solved algebraically to explain the experimental results. A similar analogue suggests that an apparent inconsistency in the no-slip boundary conditions within the Cox stratified, primitive equation, ocean circulation model should not affect the accuracy or convergence of the global solution. 相似文献
10.
Martin Claussen 《Boundary-Layer Meteorology》1985,33(2):151-172
The spectral equations of turbulent kinetic energy and temperature variance have been solved by using Onsager's energy cascade model and by extending Onsager's model to closure of terms that embody the interaction of turbulent and mean flow.The spectral model yields the following results: In a stably stratified shear flow, the peak wave numbers of the spectra of energy and temperature variance shift toward larger wave numbers as stability increases. In an unstably stratified flow, the peak wave numbers of energy spectra move toward smaller wave numbers as instability increases, whereas the opposite trend is observed for the peak wave numbers of temperature variance spectra. Hence, the peak wave numbers of temperature spectra show a discontinuity at the transition from stable to unstable stratification. At near neutral stratification, both spectra reveal a bimodal structure.The universal functions of the Monin-Obukhov similarity theory are predicted to behave as
m
~
H
~ (- Z/L)-1/3 in an extremely unstable stratification and as
m
~
H
~ z/L in an extremely stable stratification. For a stably stratified flow, a constant turbulent Prandtl number is expected. 相似文献
11.
有效的正压原始方程拟能守恒保真（拟）谱模式 总被引：4，自引：0，他引：4
本工作遵循保真计算原理与方法,对正压原始方程气象传统全球（拟）谱模式方案进行改造,构造了正压原始方程拟能完全守恒（拟）谱模式新型保真计算方案,解决了正压原始方程的（非线性）计算稳定性问题和拟能守恒整体性质保持问题,改进了相应正压原始方程气象传统全球（拟）谱模式方案的计算效能。新型保真方案的数值实验表明,计算实践中,新方案在解决拟能守恒问题的同时,可解决（非线性）计算稳定问题,并在一定条件下可解决非线性计算收敛性问题。进一步的比较数值实验还表明,计算实践中,新型保真计算方案在提高相应气象传统方案的计算精度、 相似文献
12.
We discuss the nonlinear transfers possible in a quasigeostrophic fluid with a basic stratification taken from oceanic data. The energy and enstrophy conservation laws imply a cascade of energy to larger total scale (including both the horizontal scale, defined as wavelength/2π, and the deformation radius of the vertical mode). The triplet interactions among components with various horizontal scales and vertical structures, represented by the vertical mode numbers, are considered in detail for exchanges involving the barotropic and first three baroclinic modes. The initial transfer rates from one component into the other two are estimated and the most rapid transfers described as a function of the initial scale and mode number. These results suggest that barotropic motions will cascade to larger-scale barotropic motions, first baroclinic small-scale motions will transfer to first baroclinic larger scales, and first baroclinic large-scale motions will cascade to barotropic and first baroclinic motions at the deformation scale. Second and third mode motions prefer to transfer energy into small-scale (second or third mode deformation radius) first and third baroclinic mode motions.We also show the relationship of these triplet interactions to Rossby wave instabilities and resonant triads. For the latter motions, the weakness of the nonlinearity adds additional constraints which impty that the motions will tend to become zonal. 相似文献
13.
Stefano Pierini 《Dynamics of Atmospheres and Oceans》1985,9(3):273-280
In the framework of the equivalent barotropic dynamics, we show that the kinetic energy and enstrophy a small scale perturbation superimposed on a large scale solitary wave do not grow in time as long as the perturbation scale remains much less than that of the wave. This property is proved to hold for the equivalent modons of Flierl et al. which therefore turn out to be resistant to small scale perturbations. These conclusions appear to explain the numerical results of McWilliams et al. on non divergent modons. 相似文献
14.
为了模拟球面平流传输过程,本文基于球面阴阳重叠网格设计了一种两时间层半拉格朗日平流方案.该方案在球面坐标下采用新型的LE水平跳点网格,同时针对阴阳网格重叠区,采用了不同插值方法进行比较分析,且进行了相关的理想数值试验对方案设计效果进行评估.数值试验表明方案设计是成功的,阴阳网格重叠区平流对插值方案比较敏感;半拉格朗日方案能较好地模拟球面刚体平流和变形涡旋的结构、位置及演变过程,并具有较好的数值稳定性和较高的数值精度. 相似文献
15.
Dr. I. M. Navon 《Meteorology and Atmospheric Physics》1987,37(3):143-152
Summary Bayliss and Isaacson (1975) method of modifying any given difference scheme so as to ensure total conservation of the appropriate physical invariants is shown to be equivalent to the constraint restoration method of Miele et al. (1968, 1969) subject to the requirement of least-square change of the state-vector coordinates.Both methods are applied to enforce conservation of total energy and potential enstrophy in global shallow-water equations models. Some algorithmic differences between the methods are discussed as well as some implications of a posteriori enforcement of conservation of integral invariants on the performance of meteorological numerical weather prediction (NWP) models and the internal energy distribution.With 2 Figures 相似文献
16.
Construction of high-order difference schemes based on Taylor series expansion has long been a hot topic in computational mathematics, while its application in comprehensive weather models is still very rare. Here, the properties of high-order finite difference schemes are studied based on idealized numerical testing, for the purpose of their application in the Global/Regional Assimilation and Prediction System(GRAPES) model. It is found that the pros and cons due to grid staggering choices diminish with higher-order schemes based on linearized analysis of the one-dimensional gravity wave equation. The improvement of higher-order difference schemes is still obvious for the mesh with smooth varied grid distance. The results of discontinuous square wave testing also exhibits the superiority of high-order schemes. For a model grid with severe non-uniformity and non-orthogonality, the advantage of high-order difference schemes is inapparent, as shown by the results of two-dimensional idealized advection tests under a terrain-following coordinate. In addition, the increase in computational expense caused by high-order schemes can be avoided by the precondition technique used in the GRAPES model. In general, a high-order finite difference scheme is a preferable choice for the tropical regional GRAPES model with a quasi-uniform and quasi-orthogonal grid mesh. 相似文献
17.
Klaus Hasselmann 《Boundary-Layer Meteorology》1974,6(1-2):107-127
Summary The effect of white capping on the spectral energy balance of surface waves is investigated by expressing the white-cap interactions in terms of an equivalent ensemble of random pressure pulses. It is shown first that the source function for any non-expansible interaction process which is weak-in-the-mean is quasi-linear. In the case of white capping, the damping coefficient is then shown to be proportional to the square of the frequency, provided the wave scales are large compared with the white-cap dimensions. The remaining free factor is determined indirectly from consideration of the spectral energy balance. The proposed white-capping dissipation function is consistent with the structure of the energy balance derived from JONSWAP, and the existence of a
–5 spectrum governed by a non-local energy balance between the atmospheric input, the nonlinear energy transfer and dissipation. However, closure of the energy balance involves hypotheses regarding the structure of the atmospheric input function which need to be tested by further measurements. The proposed set of source functions may nevertheless be useful for numerical wave-prediction. According to the model, nearly all the momentum transferred across the air-sea interface enters the wave field. For fetchlimited and fully developed spectra in a stationary, uniform wind field, the drag coefficient remains approximately constant. However, for more general wind conditions, this will not be the case and the wave spectrum should be included in an accurate parameterisation of the air-sea momentum transfer.Contribution from the Sonderforschungsbereich Meeresforschung Hamburg of the Deutsche Forschungsgemeinschaft. 相似文献
18.
C. Freytag 《Boundary-Layer Meteorology》1978,14(2):183-198
In previous experimental investigations of the hypothesis of a lognormal distribution of kinetic-energy dissipation proposed by Kolmogorov and Obukhov in 1962, approximations for the dissipation of kinetic energy were used.By using special measuring equipment, these approximations could be avoided and dissipation could be measured directly, permitting computation of structure functions, probability densities and dissipation spectra. As a result, the predictions of the lognormal hypothesis (LNH) have been confirmed. A great part of the deviations reported elsewhere could be explained as resulting from the use of approximations. The universal parameter of the LNH could be evaluated in different ways, yielding: = 0.5 (from structure functions of u) and = 0.47 (from dissipation spectra). 相似文献
19.
In this study, a long-range energy alternative planning (LEAP) model was built to evaluate the relative priority of three kinds of policies expected to be implemented for the energy-intensive manufacturing sectors (EIMS) in China to achieve CO2 mitigation and energy conservation targets. These policies encourage (1) the use of more electricity instead of coal; (2) the continuous improvement of energy efficiency; and (3) a shift to other less energy-demanding sectors. The results indicate that the policy of shifting economic activity from the EIMS to other sectors is most helpful for China to achieve its targets of mitigating CO2 emissions and conserving energy. Encouraging the EIMS to use more electricity can help China to achieve a higher proportion of non-fossil-fuel based energy in its overall primary energy consumption. No single policy will allow China to achieve all the targets, emphasizing the need for an integrated policy design that combines all types of policies.
Key policy insights
The policy of encouraging a shift to less energy intensive industries should receive the highest priority in aiming to peak China's energy-related CO2 emissions as early as possible, and lower overall CO2 emissions, coal consumption and primary energy consumption in the long run.
Encouraging a shift to electricity should go hand-in-hand with greater energy efficiency, otherwise such a policy cannot help China significantly reduce energy-related CO2 emissions.
Encouraging the EIMS to use more electricity should receive the highest priority in helping China achieve a higher proportion of non-fossil-fuel based energy in its overall primary energy consumption.
20.
In this study, the global energy cycle of five state-of-the-art climate models is evaluated in the wave number domain for all seasons. The energy cycle estimates are based on 30?years of 6-hourly data obtained at pressure levels of all models. The models energetics are compared to those obtained from three reanalysis datasets (ERA-40, JRA-25 and NCEP-R2). The results show that the distributions of the energetics integrands and the shape of the various wave number spectra are reasonably well simulated. Many important features can be found in most models, namely both the upscale and downscale energy cascade for the wave?Cwave interactions of kinetic energy, the downscale energy cascade for the wave?Cwave interactions of available potential energy and the downscale energy transfer for the zonal?Cwave interactions of kinetic energy. However, the magnitude in the integrands distributions is generally excessive, yielding too much energy and an overactive energy cycle in the models. Accordingly, this energy excess is also reflected in the various spectra, specially but not exclusively, at the synoptic scale wave numbers for the energy conversion/transfer rates. The well known cold pole bias and the too strong tropospheric jets, along with their dislocation in some cases, still persist in the climate models. These are some of the deficiencies in the models directly implicated in the energy cycle. Apparently, simply increasing the horizontal and vertical resolutions is not enough to eliminate these deficiencies due to somewhat opposite effects achieved by refining both spatial resolutions. Therefore, more accurate physics parameterisations as well as improved numerical schemes and resolution dependence of parameterisations seem to be essential for a significant improvement in the models energetics. Moreover, efforts should be made to improve the physical processes controlling the generation of zonal available potential energy and dissipation of eddy kinetic energy, in which the synoptic scale should be fundamental, as inferred from the excessive energy conversion/transfer rates in the models spectra. 相似文献