Nondivergent Flow Research Articles

A study of numerical errors in the integration of barotropic flow on a spherical grid

The barotropic vorticity equation for horizontal nondivergent flow is integrated numerically over the northern hemisphere on a spherical grid and compared with the known analytic solution for harmonic initial conditions. At latitudes ≤45°N, points spaced each 5° lat. and 5° long. are used, the longitudinal mesh increment being increased to 10°, 15°, 20°, 30°, and 60° long. at higher latitudes. The pole is treated separately in surface spherical coordinates. In these solutions the truncation and relaxation errors have been isolated without the usual errors of initial data and boundary conditions. Variations of the over-relaxation coefficient α. from 0.28 to 0.44 and of the residue tolerance ∈ by a factor of 100 produced approximately 8 per cent variation in the magnitude of the maximum tendency error at t = 0. In a series of integrations to 48 hr, the value of the time increment Δt was varied from 1/4 hr to 3 hr, producing a variation in the magnitude of the maximum error of approximately 5 per cent. By far the largest part of the total numerical error is due to the space truncation of the finite-difference grid. This error in the 5° grid produces a phase lag of the predicted solution relative to the true solution of approximately 1° long. per day for the case of six symmetrical hemispheric waves moving eastward at 20° long. per day. A parallel integration with a 10° grid of similar geometry increased this truncation phase lag to approximately 4° long. per day.

LAGRANGIAN DYNAMICS AND THE PARCEL METHOD IN ATMOSPHERIC MODELS

Abstract The Lagrangian “parcel method” of stability analysis is systematically presented and rendered rigorous in its application to nondivergent flow on a plane with constant Coriolis parameter. The method is applied in detail to two familiar models. (1) Zonal geostrophic current with linear horizontal shear. Here the parcel method is valid only for uniform initial disturbance (“infinite wave-length”), in which case the usual stability criterion applies, −ζg <f. Another particular solution has been given by Lord Kelvin. (2) Anticyclonic vortex with concentric circular contours. Here, the parcel-method stability criterion is −ζg < f/2 (as suggested by the balance equation) and is valid for any initial disturbance with balanced velocity field. So far, no non-symmetric solution to the balance equation for this pressure field has been discovered.

The use of the deformation theorem in forecasting non-divergent flow

На основании уравнений вихря, баланса и деформации была выведена система двух прогностических уравнений для для предсказания геопотенциала и функции тока в горизонтальном бездивергентном течении. Приводится способ возможного численного решения названной быше системы.

A remark on some of the properties of horizontal non-divergent flow

В работе рассмотрены условия совнадения изобарической поверхности с уровнем нулевой дивергенции.

A Heuristic Theory of Large-Scale Turbulence and Long-Period Velocity Variations in Barotropic Flow

By combining an equation for the conservation of zonal momentum with the eddy vorticity equation for nondivergent barotropic flow through a channel, it has been found possible to express the time variations of the longitude-averaged zonal wind in terms of the mean zonal wind itself and certain statistics of the eddy velocity field. In large-scale turbulence-where the characteristic dimension of the eddies is comparable with the entire “width” of the flow— these eddy statistics take on the properties of autocorrelation functions; thus, they depend primarily on the overall scale of the eddies and intensity of turbulence, rather than on details of the flow pattern. Changes in the scale and intensity of large-scale turbulence are related to changes in the mean zonal flow through the principles of conservation of total kinetic energy and variance of vorticity. Under a general hypothesis regarding the “stability” of the eddy statistics, the equations for the zonally-averaged motion comprise a complete system of nonlinear equations, which can be integrated by finite-sum and-difference methods. Accordingly, these equations constitute a mathematical theory of large-scale “free” turbulence, and provide the basis for a numerical method of predicting the zonally-averaged motion.&#x0D; With the assumption that the eddy scale is invariant, the equation governing the mean zonal wind takes a form related to that of the classical wave equation. Numerical integrations of this equation verify that extremes of mean zonal wind (e.g., pronounced jets) maintain their identity over long periods of time, and migrate either northward or southward (or in both directions at once) in a very regular way—a phenomenon that has been observed by Riehl bt al. (1950). The results also show that a strong jet in the mean westerlies tends to split into two weaker jets, which then move in opposite directions to produce the separated double jet structure typical of widespread blocking. The latter has been described as a characteristic denouement from strong zonal flow by Cressman (1950). The theoretical speed of migration of extremes in the mean zonal wind is found to agree closely with the observed speed in a case where published data are available.&#x0D; Since these features of long-period velocity variations—the most striking and regular discovered to date—have a characteristic time scale of a week or two, numerical prediction methods based on the present theory show some promise of aiding in the preparation of extendedrange forecasts.

NOTE ON HEMISPHERIC NUMERICAL INTEGRATION OF THE BAROTROPIC MODEL

The computational procedures employed for the numerical integration of the barotropic model for two-dimensional non-divergent flow on a finite-difference grid covering all longitudes of the northern hemisphere south to approximately 10°N are described.The results of a pilot forecast are presented, and the implications of this work for both research and operational applications are discussed.

On the Use of Space-Smoothing in Physical Weather Forecasting

In section 1–2 a certain space-smoothing operation is defined and its usefulness in solving elliptic equations is demonstrated in the case of a Poisson equation. It leads to solutions in a closed form which possess the numerical simplicity of the ordinary iteration methods, but is converging more rapidly. The reverse operation of unsmoothing is also defined as far as it can be done, and it is mentioned that the combined processes of smoothing and unsmoothing are convenient tools for obtaining a spectral analysis of horizontal scalar fields, and also to remove systematical errors which are made when derivatives are taken as finite differences.&#x0D; In sections 3–6 the application of smoothing is shown in the barotropic forecasting problem. At first a general theorem is proved concerning trajectories of two-dimensional non-divergent flow. It states that if the streamfunction ψ1 of such a flow can be decomposed into two components ψ2, α of which α is individually conserved in the ψ1-motion, then the displacements of the fluid particles up to any time can be found by at first displacing in the stationary flow αt=0 = const and then adding from the resulting positions the displacements in the flow with the streamfunction ψ2. The theorem is first applied to barotropic flow. In this case the first stationary field to displace in is the deviation between the actual and smoothed flow, while the second field to displace in is the smoothed flow.&#x0D; The space-smoothing is next applied to an equation expressing the individual conservation of a quantity s in a two-dimensional non-divergent flow. The Reynolds term belonging to the smoothed equation is studied and found to depend essentially upon the deformation properties of the velocity field. The role of deformation for the net spectral flow of energy in the s-field is studied. The smoothing is in particular applied to the vorticity equation to show how this possibly can be utilized in the integration problem.&#x0D; In sections 7–11 the baroclinic case is considered. In sections 7–8 is shown the fundamental role of deformation for the interchange of potential and kinetic energy. It is found that in the advective model there is direct proportionality between the change in total kinetic energy and total thermal wind energy, and also a direct proportionality between the change in kinetic energy for the vertically mean motion and the thermal wind energy.&#x0D; In section 9 is discussed the possible importance of non-linear interference for the understanding of the creation and local distribution of disturbances in the atmosphere.&#x0D; The integration problem is discussed in sections 10–12. At first an extension of the barotropic displacement rule is given for the vertically mean motion. The trajectory problem for levels other than the mean level is touched in section 12, and a simple non-advective model discussed shortly in section 11.

HARMONIC WAVE SOLUTIONS OF THE NON-LINEAR VORTICITY EQUATION FOR A ROTATING VISCOUS FLUID

Abstract With the assumption of non-divergent horizontal flow, harmonic wave solutions of the non-linear hydrodynamic equations for a rotating, viscous fluid are obtained both for the plane and for the sphere. These solutions yield waves which may be damped, amplified, or remain unchanged, depending on the longitudinal and latitudinal extents of the waves, and also on the vertical profile of the stream function. For plane flow, the vertical profile of the velocity components is shown to be a sinusoidal function, or a hyperbolic sine and cosine of z, proportional to or independent of z, the elevation. For spherical flow, the vertical profile is shown to be proportional to a Bessel function of order n + ½, and n, an integer, is the degree of a surface spherical harmonic. Harmonic waves are amplified, neutral or damped, depending respectively on whether the argument of the Bessel function is imaginary, zero or real. No steady wave is possible in spherical viscous flow, although quasi-stationary waves, which ...

On the Changes in the Spectral Distribution of Kinetic Energy for Twodimensional, Nondivergent Flow

On the Changes in the Spectral Distribution of Kinetic Energy for Twodimensional, Nondivergent Flow

On the Changes in the Spectral Distribution of Kinetic Energy for Twodimensional, Nondivergent Flow

Total kinetic energy as well as total vorticity squared are integral quantitics which cannot change in the course of time in a twodimensional flow of a homogeneous, nondivergent, and inviscid fluid when the fluid is isolated from the surroundings. The case is considered where the fluid is defined over the total region of thc surface of a sphere. The nature of the changes in time of the spectral distribution of kinetic energy is discussed on the basis of the two conservation requirements mentioned above. It is found that only fractions of the initial energy can flow into sniallcr scales and that a greater fraction simultaneously has to flow to components with larger scales. The upper limits to the flow of kinetic energy into components with scales less than a given one are found. The conservation theorems are also used to discuss the stability of a certain stationary flow for a twodimensional motion which is not necessarily spherical. It is shown how important it is for the proof of stability that not only the kinetic energy of the disturbance is supposed to be small but also its vorticities.In chapter II molecular viscosity is taken into account for the spherical flow. Finally some conclusive remarks are offered regarding the fundamental difference between twoand threedimentional flow.

THE DISTRIBUTION OF KINETIC ENERGY IN CERTAIN STEADY BAROTROPIC CURRENTS

Abstract The spreading of an energy concentration superimposed on a uniform zonal current at a particular latitude, may be characterized by the complete disappearance of the perturbation at infinity. Under this condition the vorticity equation for a nondivergent barotropic atmosphere can be linearized exactly for steady-state motion and reduces to the well-known Helmholtz wave equation. Its solution then is quite general, and is not subjected to any restrictive condition of small perturbations. Two cases of “dispersion” were studied, the problem of the deflection of a zonal current and the problem of a zonal jet stream. In the first case the northward velocity component was prescribed as a boundary condition, in the second case the eastward velocity component. Due to the assumption of nondivergent flow, both components cannot be arbitrarily prescribed since they must satisfy the continuity requirement. Prescribing the distribution of either velocity component along a fixed meridian by a function with a pr...

THE FORMATION OF THE SEMIPERMANENT CENTERS OF ACTION IN RELATION TO THE HORIZONTAL SOLENOIDAL FIELD

Abstract The perturbation of the middle cell of the general circulation in a baroclinic atmosphere is investigated under the assumption of horizontal, divergence-free, isothermal motion. With the introduction of the horizontal solenoidal field the velocity of the trough does not differ very much from that obtained by Rossby, but the existence of a critical wave length, beyond which the perturbation will become unstable, explains to some extent the flow pattern of upper-air motions during weak-circulation weather and also provides a means of explaining how the large-scale mixing process of air masses between different latitudes takes place. It is shown that the assumption of non-divergent flow does not greatly affect the investigation of the disturbance of the horizontal motion. The inclusion of the divergence term in the perturbation equation adds to the frequency equation a term which may affect the magnitude of the critical wave length and the velocity of the disturbance by roughly ten per cent. The app...

THE MOTION OF HARMONIC WAVES IN THE ATMOSPHERE

Abstract With the assumption of nondivergent horizontal flow, harmonic wave solutions of the complete vorticity equation are obtained both for the plane and for the sphere. These solutions yield, for the velocity of propagation of the waves, formulae which are identical in form with those obtained by Haurwitz from an approximate linearized form of the vorticity equation. However, it is shown that these formulae must be interpreted quantitatively somewhat differently than heretofore. Qualitatively, the interpretation is unchanged. The motion of the waves toward the east decreases with increasing wave length, and increases with increasing westerly current.