Divergence Equation Research Articles

Convergence rate estimates for difference schemes of one class of divergent equations

We derive some rate of convergence estimates for difference schemes of a quasilinear equation which diverge for x~-0 9 The exact solution of the original problem is assumed to go to infinity with rate O(I/x~. It is shown that if T~<I~/2--I/2--~[ , then the difference solution converges to the exact solution of the differential problem in the difference norm of type 2 [i] with rate O(h I12+h~i2). We use the standard notation for difference functions and their difference relations [2]. Consider the following problem:

On a Theory of the Evolution of Surface Cold Fronts

The governing vorticity and divergence equations in the surface layer are derived and the roles of the different terms and feedback mechanisms are investigated in semigeostrophic and nongeostrophic cold-frontal systems. A planetary boundary layer model is used to perform sensitivity tests to determine that in a cold front the ageostrophic feedback mechanism as defined by Orlanski and Ross tends to act as a positive feedback mechanism, enhancing vorticity and convergence growth. Therefore, it cannot explain the phase shift between convergence and vorticity as simulated by Orlanski and Ross. An alternative plausible, though tentative, explanation in terms of a gravity wave is offered. It is shown that when the geostrophic deformation increases, nonlinear terms in the divergence equation may become important and further destabilize the system.

The Error Associated with Use of Various Forms of the Divergence Equation to Diagnose Geopotential and Temperature

This study utilizes a model-generated data set to evaluate the errors associated with the diagnosis of geopotential height using three approximate forms of the divergence equation (DE). The DE was solved on the original sigma-coordinate surfaces of the model to exclude any error produced in the process of interpolating the data from observation points to the coordinate surface of the DE. An inviscid, nonlinear, divergent DE produced rms geopotential height errors as large as 3 m and rms temperature errors of over 26 C in the lower planetary boundary layer (PBL), with minimal errors above 800 mb. Elimination of the isobaric divergence in the velocity field caused additional height errors of 5-6 m and temperature errors of 0.4-1.2 C in the free troposphere. Use of a further-degraded DE without nonlinear terms caused height errors of 10-15 m in the upper troposphere with modestly increased temperature errors. Error fields at all levels had synoptic-scale features, with some mesoalpha structure. No spatial noise was apparent. Use of a temporal spectral filter to eliminate high-frequency modes from the model-generated data did not significantly influence the error associated with use of various forms of the DE.

Transient front and initial conditions in Maxwell’s theory

Through investigating the transient front, initial conditions are obtained from the time domain electromagnetic problem involving objects with edges or vertices. These conditions are required so that Maxwell’s divergence equations are satisfied away from a radiating surface. Via a simple example, it is demonstrated that when such idealized geometries are considered, the time dependence of the source in such a problem cannot be chosen arbitrarily under Maxwell’s theory.

A divergent system of non-stationary equations of motion of viscoelastic media in euler coordinates

A system of divergent equations of non-stationary motions of viscoelastic media is presented. It is shown that for continuous flows it is equivalent to the well-known system of equations in /1/. Divergent equations are preferable, for instance, from the viewpoint of their utilization in calculational algorithms. On the basis of the divergent forms obtained, relationships at discontinuities are analysed.

Selected Comments on the Use of the Divergence Equation to Obtain Temperature and Geopotentials from an Observed Wind

Abstract Recent suggestions to use the full divergence equation and nearly continuous (in time) wind observations to retrieve the geopotential and temperature of large-scale and mesoscale (but still hydrostatic) phenomena are examined. By extrapolating from results obtained in small-scale meteorology the following is suggested: 1) boundary conditions for the normal derivative of temperature (or the geopotential) can be obtained from the wind data itself and 2) objective criteria can be devised to indirectly test the quality of the results and also to decide whether Neumann (i.e., a case in which the normal derivative is specified) or Dirichlet (i.e., the function itself is specified) type boundary conditions should be enforced.

On the dynamics of mid-latitude synoptic systems with intense cumulus convection

A scale analysis appropriate for mid-latitude explosive cyclones is presented, including the effects of cumulus clouds. First-order equations are derived for the thermodynamic and vorticity fields. It is shown that the quasi-geostrophic form of the divergence equation is a good approximation, while the geostrophic wind approximation is no longer valid in the presence of intense convection. It is also found that diabatic heating by clouds is responsible for the enhancement of the large-scale vertical motion which induces a divergent wind on the same order as the rotational part. Therefore, cumulus convection affects the large-scale flow through an induced secondary circulation, which manifests its effect in the thermodynamic and vorticity equations as the advection by the divergent wind component. In addition, cumulus clouds also affect the large-scale vorticity by imparting excess cloud vorticity to the environment. Interactions between cloud and large scales are also discussed in terms of the omega and potential vorticity equations. It is shown that the dominant forcing terms, i.e. diabatic heating and vortex stretching due to cumulus clouds, tend to balance in the large-scale potential vorticity equation, and that the influence of cumulus convection appears through the smaller forcings due to cloud life-cycles and two non-geostrophic effects. A three-level model is employed to investigate the effects of cumulus convection on baroclinic instability, with Ekman-CISK as the closure condition. It is found that the various forcing mechanisms produce quite different effects on the evolution and structure of the baroclinically unstable waves. DOI: 10.1111/j.1600-0870.1986.tb00420.x

Iterational methods of solving equations of gas dynamics

Iterational methods of solving non-linear sets of difference equations, which are written in a divergent and non-divergent form and approximate the equations of gas dynamics for a heat-conducting gas, are described. For divergent equations, bearing in mind the physical viscosity, all the methods are applicable in the case of plane geometry, and for non-divergent equations some of them can be used after obvious modification in the case of any uniform geometry. The effectiveness of these methods is illustrated by the example of the solution of model problems.

Existence and structure of past asymptotically simple solutions of Einstein's field equations with positive cosmological constant

The initial value problem for Einstein's field equations with positive cosmological constant is analysed where data are prescribed at past conformal infinity. It is found that the data on past conformal infinity are given, up to arbitrary conformal rescalings, by a freely specifyble Riemannian metric and a trace-free, symmetric tensorfield of valence two, which satisfies a divergence equation. For each initial data set exists a unique (semi-global) past asymptotically simple solution of Einstein's equations. The case is discussed where in such a space-time exists a Killing vector field with a time-like trajectory which approaches a point p on conformal infinity. It is shown that in a neighbourhood of the trajectory near p the space-time is conformally flat.

Macroscopic conservation laws in the kinetic theory of a monatomic gas and their consequences

A complete set of divergent forms of macroscopic equations, which hold for arbitrary motions of a monatoroic gas is obtained using the Boltzmann kinetic equation. The physical content of the new divergent equations is revealed using the treatment of a gas cloud as an example, and exact relationships are indicated which determine the laws of motion of a cloud of gas in free space.

Calculation of Geopotential and Temperature Fields from an Array of Nearly Continuous Wind Observations

The accuracy is studied of temperature estimates derived from the divergence equation when wind observations of various spatial and temporal resolutions and accuracies are available. The basic data set used is the high resolution model data set used by Kuo and Anthes (1984a) in observing systems simulation experiments (OSSE) designed to estimate the errors in heat and moisture budgets (Kuo and Anthes, 1984b) calculated from the AVE-SESAME-1979 spatial observational network. This model data set is modified in ways to simulate wind observations that appear feasible from an operational regional network of wind profilers.

On conservation laws in one-dimensional magnetohydrodynamics in the case of infinite conductance

The complete system of divergence equations of one-dimensional magnetohydrodynamics in the case of infinite conductance is constructed. It is shown that the linearly independent conservation laws, not explicitly dependent on the coordinates, are all classical conservation laws. The same result, due to Rozhdestvenskii, is familiar for the equations of gas dynamics.

Bend Flow Calculational Methods Compared

The two widely used methods of finite‐difference viscous‐flow calculation, the vorticity transport and primitive variable approaches, are applied to the analysis of flow through a plane short‐radius bend flanked by sections of straight conduit. Additionally, the vorticity transport method is employed in a direct (using combined rectangular and polar coordinates) and an indirect calculation (using a nonorthogonal coordinate transform). The solution for all three methods are found to be in essential agreement at a Reynolds number of 72. Because of the sharp curvature, the primitive variable method required pressure adjustment using a Poisson equation; velocities were adjusted by the divergence equation and advanced in time by the momentum equations. For the short‐radius bend, the matching procedures involved in the primitive variable approach were found to be crucial in developing smooth field transitions from one coordinate system to another. The adjustment of the coefficients in the transform vorticity transport method for the same regions (but not along the junction lines) were easily made and the comparisons with results from the other methods were excellent.

Balanced models in isentropic coordinates and the shallow water equations

It is shown that there is an appropriate family of three balanced models in isentropic coordinates and the shallow water equations. “Appropriate” means the equations include exact hydrostatic, continuity and heat equations and form a complete system; some family members are second-order accurate in a midlatitude Rossby number expansion; the models conserve at least one integral invariant analogous to the primitive equation energy and enstrophy invariants; and the terms retained in the vorticity and divergence equations are the minimum required for the accuracy of the model in an asymptotic scale analysis. We compare vorticity and divergence in physical and isentropic coordinates to illustrate an aspect of the difference in physical content between balanced models defined in the different coordinates. We show that the early balanced models of Bolin and Charney are similar to members of our family in isentropic coordinates, but fail some of our criteria for appropriateness. Finally, we discuss why we think the above criteria are the right ones for balanced model families, and compare and contrast similar criteria for other intermediate models . DOI: 10.1111/j.1600-0870.1984.tb00236.x

Approximate particular solutions for the pole tide in a global ocean

*Summary. Approximate particular solutions are presented for Laplace’s tidal equations for the pole tide forcing on an ocean-covered earth. The system of equations only needs to satisfy cyclic continuity and is reduced to separate equations for the meridional velocity and the deviation of the tide from equilibrium. These two equations are characterized by a small parameter multiplying the derivatives and so approximate particular solutions are obtained by equating the undifferentiated terms with the forcing terms. With this approximation, linear friction does not introduce a phase shift in the solutions. There is a near geostrophic balance between the horizontal velocities and the small deviation of the tide from equilibrium in the momentum equations, while to the first approximation the horizontal velocity divergence in the continuity equation yields the equilibrium tide. The horizontal velocities are expressed in terms of a stream function and velocity potential, which are shown to satisfy the vorticity and divergence equations. These results show that for the long-period pole tide forcing (a velocity potential forcing in the divergence equation), both a stream function and velocity potential are necessary to describe the motion.

On the Occurrence of Atmospheric States that are Non-Elliptic for the Balance Equation

The origins of atmospheric states that are non-elliptic for the height constrained balance equation are examined from observational perspectives. Such states are commonly present in the deep tropics in Objectively analyzed data sets. In order to analyze the source of this phenomenon, we compute terms of the divergence equation (from which the balance equation derives) for disturbed periods of the GATE experiment. Meaningful residuals cannot be obtained because they are obscured by observational uncertainty of the geopotential gradients that are calculated from the hydrostatic equation using temperature observations. The geopotential fields recomputed from the divergence equation using observed wind data appear to be much better determined, but they still produce fields that are non-elliptic for the height constrained nondivergent balance equation. For the convective GATE cases, it appears that the essential balance of the divergence equation is between divergent accelerations, deforming accelerations, the divergent pressure form field and friction, while centripetal accelerations (including the Coriolis is effect) are secondary. Thus, the underlying assumption of solenoidal flow in the balance equation is fundamentally wrong in regions of tropical convection. This appears to be the physical source of the poorly posed balance equation in many non-elliptic cases.

Variational approach to the closure problem of turbulence theory

A new method is proposed to solve the closure problem of turbulence theory and to drive the Kolmogorov law in an Eulerian framework. Instead of using complex Fourier components of velocity field as modal parameters, a complete set of independent real parameters and dynamic equations are worked out to describe the dynamic states of a turbulence. Classical statistical mechanics is used to study the statistical behavior of the turbulence. An approximate stationary solution of the Liouville equation is obtained by a perturbation method based on a Langevin–Fokker–Planck (LFP) model. The dynamic damping coefficient η of the LFP model is treated as an optimum control parameter to minimize the error of the perturbation solution; this leads to a convergent integral equation for η to replace the divergent response equation of Kraichnan’s direct-interaction (DI) approximation, thereby solving the closure problem without appealing to a Lagrangian formulation. The Kolmogorov constant Ko is evaluated numerically, obtaining Ko=1.2, which is compatible with the experimental data given by Gibson and Schwartz.

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.

Tropical Cyclone Motion: Environmental Interaction Plus a Beta Effect

Abstract The dynamics of tropical cyclone motion are investigated by solving for instantaneous motion tendencies using the divergent barotropic vorticity equation on a beta plane. Two methods of solution are presented a direct analytic solution for a constant basic current, and a simple numerical solution for more general conditions. These solutions indicate that cyclone motion can be accurately prescribed by a nonlinear combination of two processes. 1) an interaction between the cyclone and its basic current (the well known steering concept), and 2) an interaction with the Earth's vorticity field which causes a westward deviation from the pure steering flow. The nonlinear manner in which these two processes combine with the effect of asymmetries in the steering current raise some interesting questions on the way in which cyclones of different characteristics interact with their environment, and has implications for tropical cyclone forecasting and the manner in which forecasting techniques are derived.

12-hour forecasting of horizontal wind components

The horizontal wind components were predicted for 12 hours, using the vorticity and divergence equations. The experiment was carried out in between the latitudes 30 deg. and 50 deg. N. The predicted values were compared with their corresponding ones verifying the suggested method.