Instability in Lagrangian stochastic trajectory models,and a method for its cure

A number of authors have reported the problem of unrealistic velocities (“rogue trajectories”) when computing the paths of particles in a turbulent flow using modern Lagrangian stochastic (LS) models, and have resorted to ad hoc interventions. We suggest that this problem stems from two causes: (1) unstable modes that are intrinsic to the dynamical system constituted by the generalized Langevin equations, and whose actual triggering (expression) is conditional on the fields of the mean velocity and Reynolds stress tensor and is liable to occur in complex, disturbed flows (which, if computational, will also be imperfect and discontinuous); and, (2) the “stiffness” of the generalized Langevin equations, which implies that the simple stochastic generalization of the Euler scheme usually used to integrate these equations is not sufficient to keep round-off errors under control. These two causes are connected, with the first cause (dynamical instability) exacerbating the second (numerical instability); removing the first cause does not necessarily correct the second, and vice versa. To overcome this problem, we introduce a fractional-step integration scheme that splits the velocity increment into contributions that are linear (U _i ) and nonlinear (U _i U _j ) in the Lagrangian velocity fluctuation vector U, the nonlinear contribution being further split into its diagonal and off-diagonal parts. The linear contribution and the diagonal part of the nonlinear contribution to the solution are computed exactly (analytically) over a finite timestep Δt, allowing any dynamical instabilities in the system to be diagnosed and removed, and circumventing the numerical instability that can potentially result in integrating stiff equations using the commonly applied explicit Euler scheme. We contrast results using this and the primitive Euler integration scheme for computed trajectories in a drastically inhomogeneous urban canopy flow.     

On the solution of the stationary,baroclinic Ekman-layer equations with a finite boundary-layer height

The stationary, Ekman-layer equations have been solved in closed form for two expressions of the eddy viscosity as a function of height, z: v _τ=cu*z(1?z/h)and v _τ=cu*z(1?z/h) ², where u* is the friction velocity, h the boundary-layer height and c a constant. The main difference between both solutions is that the quadratic K-profile leads to a velocity discontinuity at the top of the boundary layer, while the solution for the cubic profile approaches the geostrophic wind at z=h smoothly. We discuss the characteristics of the solutions in terms of a dimensionless parameter C=fh/cu*, where f is the Coriolis parameter. The dependence on C can be interpreted in terms of a varying boundary-layer height or in terms of stability. The results for C ~ 1 are related to a neutral boundary layer. They agree well with results of a second-order model. The limit C → 0 is investigated in detail. We find that the stress profile becomes linear. The velocity profile shows different characteristics depending on whether we consider a shallow or a very unstable boundary layer. The results agree with observations. Finally we consider the influence of baroclinicity on the wind and stress profiles.     

Boundary-layer flow and turbulence near porous obstacles

We derive a set of governing equations for flow through porous obstacles by employing a two-step averaging processes. The Navier-Stokes equations under the Boussinesq approximation that describe the air space of the porous obstacle are subjected to high-wavenumber a veraging, which leads to a set of high-frequency (wake) turbulence equations. We then use conventional Reynolds-averaging methods to obtain statistically steady mean and turbulence equations that include interactions between wake and shear turbulence. Our method provides a theoretical basis for the cascade of turbulent kinetic energy. We use this approach to analyze the constants and parameters of simpleK-theory and higher-order closure models. We also discuss qualitatively the theory of the turbulence energy generation process and the significance of interactions between different turbulent mechanisms.     

强迫耗散非线性发展方程准完全平方守恒格式的构造

从描述大气和海洋运动的强迫耗散非线性发展方程出发，对强迫耗散非线性大气和海洋方程组显式差分格式的计算稳定性进行了分析，构造了一类强迫耗散性发展方程的显式准完全平方守恒差分格式，理论分析和数值试验证明，这类显式准完全平方守恒差分格式是计算稳定的．值得推广应用。     

强迫耗散非线性发展方程显式差分格式的计算稳定性

基于计算准稳定的概念来分析强迫耗散非线性方程显式差分格式的计算稳定性，给出强迫耗散非线性大气方程组显示差分格式计算准稳定的判据，为设计强迫耗散非线性大气方程组计算稳定的显式差分格式提供了新的思路和理论依据。     

An invariant theory of the linearized shallow water equations with rotation and its application to a sphere and a plane

The system of linearized shallow water equations is formulated in this paper on any rotating and smooth surface M in terms of differential geometry. The system decouples into two separate equations: a scalar one for the height deviation and a vector one for the velocity field. For low and high frequencies these equations yield asymptotic equations whose solutions are the generalizations of the Poincare and Rossby waves to smooth surface. The application of these equations to the β-plane yields both new and previously known equations for the height deviation and for the velocity components. The application of the equations to the rotating spherical Earth shows that the meridional amplitudes of Poincare and Rossby waves are both described by the prolate angular spheroidal wave functions. The asymptotic and the power series expansions of the eigenvalues of these functions yield new approximations for the dispersion relations of these waves on a sphere. The new dispersion relations are very accurate in the physically relevant range of the single nondimensional model parameter – the square of the nondimensional gravity waves’ phase speed. The invariant formulation can also be applied to other surfaces that are of geophysical interest such as an oblate ellipsoid of revolution.     

THE FORMULATION OF FIDELITY SCHEMES OF PHYSICAL CONSERVATION LAWS AND IMPROVEMENTS ON A TRADITIONAL SPECTRAL MODEL OF BAROCLINIC PRIMITIVE EQUATIONS FOR NUMERICAL PREDICTION*

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.     

THE FORMULATION OF FIDELITY SCHEMES OF PHYSICAL CONSERVATION LAWS AND IMPROVEMENTS ON A TRADITIONAL SPECTRAL MODEL OF BAROCLINIC PRIMITIVE EQUATIONS FOR NUMERICAL PREDICTION

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.     

Local balance equations for atmospheric exergies and anergies

Summary The basic differential system of equations for mass specific thermodynamic potentials is extended to a system that describes the thermodynamics of finite differences. This system is characterized by mass specificexergies andanergies of the original thermodynamic potentials which are split up into exergies and anergies. Equations for the differentials of exergies and anergies replace the wellknown differential relationships of thermodynamics and are the basis of a more general energetics of the atmosphere, an energetics with an entropy flavor (Dutton, 1992). Here, a special state of reference is not prescribed. It turns out that mass specific exergy of internal energy combined with specific kinetic energy, and mass specific anergy of internal energy combined with potential energy leads to local balance equations for these energies. They represent the balance equation of total energy split up into two separate equations. The physical meaning of the two equations is clearly understood. In particular, irreversible processes and entropy production play a dominant role in all energy equations so derived. Finally, integration over the entire atmosphere leads to generalized global energy equations. The production (destruction) of kinetic energy depends on the rate of change with time of exergy of internal energy, and vice versa on the rate of change with time of anergy via a rate of change of potential energy. J. A. Dutton's (1973) relationship between global entropy difference and the sum of global kinetic energy and static entropic energy is recovered and static entropic energy is identified with global exergy of internal energy. In case of ideal gases, local exergy of enthalpy can be split up into a temperature potential and a pressure potential. For both potentials local balance equations are derived.     

Development of cloud detection methods using CFH,GTS1, and RS80 radiosondes

The accuracies of three instruments in measuring atmospheric column humidity were assessed during an upper troposphere and lower stratosphere observation campaign conducted from 7 to 13 August 2009 in ...     

正压原始方程的谱和初值问题的解I：＝常数

本文运用Ｌａｐｌａｃｅ变换法和围道积分法分析线性化的正压原始方程组的谱问题。通过严格的数学推演从形式上给出了方程组的初值问题的解，从而原则上决定了连续谱、离散谱的出现条件。与已有的研究结果相比较，本文最大的特点是保证了了解的完备性，因而相应的展开定理便是自然的了。     

Kinetic equations and stationary energy spectra of weakly nonlinear internal gravity waves

An ensemble of random-phase internal gravity waves is considered in the dynamical framework of the Euler–Boussinesq equations. For flows with zero mean potential vorticity, a kinetic equation for the mean spectral energy density of the waves is obtained under hypothesis of Gaussian statistics with zero correlation length. Stationary scaling solutions of this equation are found for almost vertically propagating waves. The resulting spectra are anisotropic in vertical and horizontal wave numbers. For flows with small but non-zero mean potential vorticity, under the same statistical hypothesis applied to the wave part of the flow, it is shown that the vortex part and the wave part decouple. The vortex part obeys a limiting slow dynamics equation exhibiting vertical collapse and layering which may contaminate the wave-part spectra. Relation of these results to the in situ atmospheric measurements and previous work on oceanic gravity waves is discussed.     

天气雷达测定回波顶高的几种误差

研究天气雷达测定回波顶高的几种误差表明:大气异常折射会引起回波顶高明显误差。但在标准大气条件下,由于几类误差相互抵消,因此近距离测定强对流云回波顶高精度较高。     

A Comparison of the Radar Ray Path Equations and Approximations for Use in Radar Data Assimilation

The radar ray path equations are used to determine the physical location of each radar measurement. These equations are necessary for mapping radar data to computational grids for diagnosis, display and numerical weather prediction (NWP). They are also used to determine the forward operators for assimilation of radar data into forecast models. In this paper, a stepwise ray tracing method is developed. The influence of the atmospheric refractive index on the ray path equations at different locations related to an intense cold front is examined against the ray path derived from the new tracing method. It is shown that the radar ray path is not very sensitive to sharp vertical gradients of refractive index caused by the strong temperature inversion and large moisture gradient in this case. In the paper, the errors caused by using the simplified straight ray path equations are also examined. It is found that there will be significant errors in the physical location of radar measurements if the earth’s curvature is not considered, especially at lower elevation angles. A reduced form of the equation for beam height calculation is derived using Taylor series expansion. It is computationally more efficient and also avoids the need to use double precision variables to mitigate the small difference between two large terms in the original form. The accuracy of this reduced form is found to be sufficient for modeling use.     

Steady flow between two reservoirs of fluid with different densities and levels through a rectangular channel section of varying depth and width

The steady hydrostatic flow through a channel of rectangular cross section connecting reservoirs of infinite width and depth and containing inviscid fluids of different densities and levels is studied. The main goal is the determination of the discharges of the lighter and denser fluids in terms of the external conditions (reservoir levels, fluid densities and variation of width and depth along a channel). It is shown that the key parameter is δ, which is the ratio of relative reservoir level difference, γ, to relative density difference, ε. If δ<0 then the denser fluid plunges under the stationary lighter layer. If δ>δ_* (1<δ_*<1.5) then the lighter fluid runs up on a wedge of stationary heavier fluid. Here δ_* depends on the geometry of the constriction. The solutions describing these regimes are stated. If 0<δ<δ_* then both layers are in motion. A qualitative analysis of the solution for arbitrary bottom shape and channel width and arbitrary ε is presented and the problem is reduced to a system of two equations which can be easily solved numerically for any particular channel profile. We give detailed analyses for the following two cases: 1) the narrowest width of the channel is on the side of the heavier fluid and the top of the sill is on the side of lighter fluid; 2) the minima in channel depth and width coincide. In the second case the discharges for one class of geometries in the Boussinesq approximation are calculated and discussed.     

再论发展方程差分格式的构造和应用

【摘　要】本文把一大类大气、海洋方程归结为一种发展方程,具体构造了若干定时间步长的显式完全平方守恒差分格式。并证明在一定条件下,这类格式也具有能量守恒、“广义能量”守恒和“平均尺度”守恒的特性,它表明这类格式具有较好的计算稳定性和省时性。文中还探讨了显式平方守恒格式与隐式平方守恒格式之间的密切联系。最后给出了令人满意的用四波的R-H波作数值检验的结果。     

On the validity of layered models of ocean dynamics and thermodynamics with reduced vertical resolution

With the purpose of studying the upper part of the ocean, the shallow water equations (in a `reduced gravity' setting) have been extended in the last decades by allowing for horizontal and temporal variations of the buoyancy field ϑ, while keeping it as well as the velocity field u as depth-independent. In spite of the widespread use of this `slab' model, there has been neither a discussion on the range of validity of the system nor an explanation of points such as the existence of peculiar zero-frequency normal modes, the nature of the instability of a uniform u flow, and the lack of explicit vertical shear associated with horizontal density gradients. These questions are addressed here through the development of a subinertial model with more vertical resolution, i.e., one where the buoyancy ϑ varies linearly with depth. This model describes satisfactorily the problem of baroclinic instability with a free boundary, even for short perturbations and large interface slopes. An enhancement of the instability is found when the planetary β effect is compensated with the topographic one, due to the slope of the free boundary, allowing for a `resonance' of the equivalent barotropic and first baroclinic modes. Other low-frequency models, for which buoyancy stratification does not play a dynamical role, are invalid for short perturbations and have spurious terms in their energy-like integral of motion.     

Barotropic and baroclinic instability in rotating stratified fluids

The linear normal mode instabilities of a parallel shear flow which varies both vertically (z) and meridionally (y) in a quasigeostrophic, rotating, stratified fluid are considered. The β effect (variation of Coriolis parameter with y) is included. Both two-layer and continuous fluids are treated. Attention is concentrated on the types of instability possible for a given shear flow. It is found that the instability can be described adequately by three nondimensional parameters: Λ, the ratio of the horizontal length scale of the shear to the internal deformation radius: δ, which is either the ratio of layer depths in the two-layer fluid or the fractional depth of variation of the stratification in the continuous fluid; and β, suitably nondimensionalized.Asymptotic analyses, confirmed by direct numerical solutions, are performed for conditions in which various parameters become large or small. The β effect is essentially quantitative, whereas Λ and δ define the type of instability as barotropic (if the kinetic energy of the mean flow feeds the growing perturbations), baroclinic (if the available potential energy of the mean flow feeds the perturbations) or mixed (a combination of the two).The case of large Λ (the most relevant for oceanographic applications) is treated in detail. It is shown that y-independent problems have only limited relevance. For a fixed deformation radius and the y scale of the mean flow increasing without limit, the asymptote is not the case of no y variation in the mean flow.     

Consistent balanced models in bounded and periodic domains

An analysis is made of a family of balanced models which are intermediate in physical content and complexity between the primitive equations and quasigeostrophy. The family is based on the linear balance equations and balance equations, and is characterized by truncations of the full vorticity and divergence equations that retain a global energy invariant. Consistent initial boundary value problems for this family of models are derived, and various aspects of numerical balanced model design are discussed.     

Analysis of quantitative precipitation forecasts using the Dynamic State Index

The German Weather Service (DWD) has two non-hydrostatic operational weather prediction models with different spatial resolution and precipitation parametrisations. The coarser COSMO-EU model has a spatial resolution of 7 km, whereas the higher-resolution COSMO-DE model has a gridspace of 2.8 km and explicitly resolves deep convection. To improve the numerical weather prediction (NWP) models it is necessary to understand precipitation processes. A central goal is the statistical evaluation of precipitation forecasts with dynamic parameters. Here, the Dynamic State Index (DSI) is used as a dynamic threshold parameter. The DSI theoretically describes the change of atmospheric flow fields as deviations from a stationary adiabatic solution of the primitive equations (Névir, 2004). For seasonal area means the DSI shows a remarkably high correlation with the precipitation forecasts provided by the COSMO-DE model. This is especially the case for the summer of 2007. The same analysis has been performed with the COSMO-EU forecast data and the results were compared with those from the COSMO-DE model. Moreover, an independent precipitation analysis, with a resolution corresponding to 7 km and 2.8 km, has been compared with respect to modelled precipitation and the DSI. In addition, correlations between the DSI and modelled as well as observed precipitation as a function of the forecast time for the different grid resolutions are also presented. The results show, that after 12 h, the correlation of the persistence forecast with the DSI reaches two thirds of the initial value. Thus, the DSI offers itself as a new dynamic forecast tool for precipitation events.