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Abstract
The nonlinear aspects of the vorticity equation for two-dimensional planetary circulations of the earth's atmosphere may be studied by expansion of the solution in spherical surface harmonics. Some of the main mathematical problems are discussed here that arise in an examination of the “spectral” form of the vorticity equation which results from such an expansion. The truncation of the spectral equations is discussed, and a proof is given of the invariance of mean square velocity and vorticity for truncated spectra. Some comments are made on low-order systems, which are to be followed by a detailed investigation of three-component systems in the second part of this study.
Abstract
The nonlinear aspects of the vorticity equation for two-dimensional planetary circulations of the earth's atmosphere may be studied by expansion of the solution in spherical surface harmonics. Some of the main mathematical problems are discussed here that arise in an examination of the “spectral” form of the vorticity equation which results from such an expansion. The truncation of the spectral equations is discussed, and a proof is given of the invariance of mean square velocity and vorticity for truncated spectra. Some comments are made on low-order systems, which are to be followed by a detailed investigation of three-component systems in the second part of this study.
Abstract
If one regards a discrete function as a vector, the “best” approximation to the product of two discrete functions (defined for the same set of values of the argument) is not necessarily the ordinary scalar product. The “best” approximation is shown to be an approximation-in-the-mean to the product of the trigonometric interpolation polynomials (cardinal functions) which correspond to the given discrete functions. This approximation arises naturally when the product is taken in the spectral domain. However, it can be approached by the ordinary scalar product provided the input functions are smoothed. The smoothing operator is linear and easily computed; it results in the suppression of all harmonics of wave length less than four times the mesh length.
Abstract
If one regards a discrete function as a vector, the “best” approximation to the product of two discrete functions (defined for the same set of values of the argument) is not necessarily the ordinary scalar product. The “best” approximation is shown to be an approximation-in-the-mean to the product of the trigonometric interpolation polynomials (cardinal functions) which correspond to the given discrete functions. This approximation arises naturally when the product is taken in the spectral domain. However, it can be approached by the ordinary scalar product provided the input functions are smoothed. The smoothing operator is linear and easily computed; it results in the suppression of all harmonics of wave length less than four times the mesh length.
Abstract
Nonlinear aspects of the vorticity equation for nondivergent, planetary circulations of the atmosphere may be explored by representation of the solution in spherical surface harmonics. A systematic analytic investigation is presented here of the truncated spectral vorticity equation thus obtained. The principal results pertain to systems with three spherical-harmonic components (which may have as many as six degrees of freedom): the spectral equations for all three-component systems are solvable by quadratures; and in those cases in which the components exchange energy, the quadratures lead either to elliptic or to circular functions of the time. The solutions, and in particular the energy exchange therefore are periodic in all three-component systems.
Extensions to nontrivial multicomponent systems are made in a few cases for which complete solutions are possible. The simplest configuration is one with an arbitrary zonal flow and a single tesseral component. No energy exchange occurs; but the phase of the tesseral component is propagated with a speed determined by the distributions of angular velocity and vorticity gradient in the zonal flow. With an arbitrary zonal flow and two tesseral components of the same rank (equal wave number), there is a periodic exchange of energy governed by solutions expressible in terms of elliptic functions. The corresponding linearized equations may be regarded as providing a means of approximating conventional perturbation analysis of stability of an arbitrary (inviscid) zonal flow, and are found to yield an explicit necessary and sufficient condition for stability.
Abstract
Nonlinear aspects of the vorticity equation for nondivergent, planetary circulations of the atmosphere may be explored by representation of the solution in spherical surface harmonics. A systematic analytic investigation is presented here of the truncated spectral vorticity equation thus obtained. The principal results pertain to systems with three spherical-harmonic components (which may have as many as six degrees of freedom): the spectral equations for all three-component systems are solvable by quadratures; and in those cases in which the components exchange energy, the quadratures lead either to elliptic or to circular functions of the time. The solutions, and in particular the energy exchange therefore are periodic in all three-component systems.
Extensions to nontrivial multicomponent systems are made in a few cases for which complete solutions are possible. The simplest configuration is one with an arbitrary zonal flow and a single tesseral component. No energy exchange occurs; but the phase of the tesseral component is propagated with a speed determined by the distributions of angular velocity and vorticity gradient in the zonal flow. With an arbitrary zonal flow and two tesseral components of the same rank (equal wave number), there is a periodic exchange of energy governed by solutions expressible in terms of elliptic functions. The corresponding linearized equations may be regarded as providing a means of approximating conventional perturbation analysis of stability of an arbitrary (inviscid) zonal flow, and are found to yield an explicit necessary and sufficient condition for stability.
The first numerical weather prediction was made on the ENIAC computer in 1950. This lecture gives some of the historical background of that event and a partially narrative account of it.
The first numerical weather prediction was made on the ENIAC computer in 1950. This lecture gives some of the historical background of that event and a partially narrative account of it.
In 1922 Lewis F. Richardson published a comprehensive numerical method of weather prediction. He used height rather than pressure as vertical coordinate but recognized that a diagnostic equation for the vertical velocity is a necessary corollary to the quasi-static approximation. His vertical-velocity equation is the principal, substantive contribution of the book to dynamic meteorology.
A comparison of Richardson's model with one now in operational use at the U. S. National Meteorological Center shows that, if only the essential attributes of these models are considered, there is virtually no fundamental difference between them. Even the vertical and horizontal resolutions of the models are similar.
Richardson made a forecast at two grid points in central Europe and obtained catastrophic results, in particular a surface pressure change of 145 mb in 6 hours. This failure resulted partly, as Richardson believed, from inadequacies of upper wind data. Underlying this was a more fundamental difficulty which he did not seem to recognize clearly at the time he wrote his book: the impossibility of using observed winds to calculate pressure change from the pressure-tendency equation, a principle stated many years earlier by Margules. However, he did point in the direction in which a remedy was later found: suppression or smoothing of the initial field of horizontal velocity divergence.
The 6-hr time interval used by Richardson violates the condition for computational stability, a constraint then unknown. It is sometimes said that this is one of the reasons his calculation failed, but that interpretation is misleading because the stability criterion becomes relevant only after several time steps have been made. Since Richardson did not go beyond a calculation of initial tendencies—in other words, he took only one time step—violation of the stability criterion had no effect on the result.
Richardson's book surely must be recorded as a major scientific achievement. Nevertheless, it appears to have had little influence in the decades that followed, and indeed, the modern development of numerical weather prediction, which began about twenty-five years later, did not evolve primarily from Richardson's work. Shaw said it would be misleading to regard the book as “a soliloquy on the scientific stage,” but in fact that is what it proved to be. The intriguing problem of explaining this strange irony is one that leads beyond the obvious facts that when Richardson wrote, computers were nonexistent and upper-air data insufficient.
In 1922 Lewis F. Richardson published a comprehensive numerical method of weather prediction. He used height rather than pressure as vertical coordinate but recognized that a diagnostic equation for the vertical velocity is a necessary corollary to the quasi-static approximation. His vertical-velocity equation is the principal, substantive contribution of the book to dynamic meteorology.
A comparison of Richardson's model with one now in operational use at the U. S. National Meteorological Center shows that, if only the essential attributes of these models are considered, there is virtually no fundamental difference between them. Even the vertical and horizontal resolutions of the models are similar.
Richardson made a forecast at two grid points in central Europe and obtained catastrophic results, in particular a surface pressure change of 145 mb in 6 hours. This failure resulted partly, as Richardson believed, from inadequacies of upper wind data. Underlying this was a more fundamental difficulty which he did not seem to recognize clearly at the time he wrote his book: the impossibility of using observed winds to calculate pressure change from the pressure-tendency equation, a principle stated many years earlier by Margules. However, he did point in the direction in which a remedy was later found: suppression or smoothing of the initial field of horizontal velocity divergence.
The 6-hr time interval used by Richardson violates the condition for computational stability, a constraint then unknown. It is sometimes said that this is one of the reasons his calculation failed, but that interpretation is misleading because the stability criterion becomes relevant only after several time steps have been made. Since Richardson did not go beyond a calculation of initial tendencies—in other words, he took only one time step—violation of the stability criterion had no effect on the result.
Richardson's book surely must be recorded as a major scientific achievement. Nevertheless, it appears to have had little influence in the decades that followed, and indeed, the modern development of numerical weather prediction, which began about twenty-five years later, did not evolve primarily from Richardson's work. Shaw said it would be misleading to regard the book as “a soliloquy on the scientific stage,” but in fact that is what it proved to be. The intriguing problem of explaining this strange irony is one that leads beyond the obvious facts that when Richardson wrote, computers were nonexistent and upper-air data insufficient.
Abstract
The transfer of energy between mean flow and disturbance for a barotropic non-divergent fluid is investigated by solution of the vorticity equation to obtain initial tendencies corresponding to assigned flow-patterns. The initial rate and direction of energy transfer are computed for different types of mean flow and disturbance. These calculations suggest that the form, as well as the wavelength, of the disturbance is a controlling factor in determining the initial energy-transfer, and that conclusive inferences cannot be made merely from the character of the mean flow. The distribution of initial changes in mean flow is calculated in several cases; it is found that, for both initially amplified and initially damped disturbances, momentum may be transferred from both sides into a central zone of maximum mean flow.
Abstract
The transfer of energy between mean flow and disturbance for a barotropic non-divergent fluid is investigated by solution of the vorticity equation to obtain initial tendencies corresponding to assigned flow-patterns. The initial rate and direction of energy transfer are computed for different types of mean flow and disturbance. These calculations suggest that the form, as well as the wavelength, of the disturbance is a controlling factor in determining the initial energy-transfer, and that conclusive inferences cannot be made merely from the character of the mean flow. The distribution of initial changes in mean flow is calculated in several cases; it is found that, for both initially amplified and initially damped disturbances, momentum may be transferred from both sides into a central zone of maximum mean flow.
The diurnal tide in surface pressure was explained theoretically in 1967, with the help of a framework that had been built up in the preceding six years. A significant contribution to this framework was made by Bernhard Haurwitz, who in 1965 published spherical-harmonic and Hough-function analyses of the diurnal surface-pressure oscillation. The lecture also examines origins of knowledge about negative equivalent depths.
The diurnal tide in surface pressure was explained theoretically in 1967, with the help of a framework that had been built up in the preceding six years. A significant contribution to this framework was made by Bernhard Haurwitz, who in 1965 published spherical-harmonic and Hough-function analyses of the diurnal surface-pressure oscillation. The lecture also examines origins of knowledge about negative equivalent depths.
Abstract
The use of central differences on a rectangular net, in solving the primitive or vorticity equations, produces solutions on each of two lattices. By exploring this lattice structure, a formal equivalence is established between the central-difference vorticity and primitive equations. A demonstration is given also that exponential instability previously found to result from certain types of boundary conditions is suppressed by applying these conditions in such a way as to avoid coupling the lattices.
Abstract
The use of central differences on a rectangular net, in solving the primitive or vorticity equations, produces solutions on each of two lattices. By exploring this lattice structure, a formal equivalence is established between the central-difference vorticity and primitive equations. A demonstration is given also that exponential instability previously found to result from certain types of boundary conditions is suppressed by applying these conditions in such a way as to avoid coupling the lattices.
Abstract
A method is designed to calculate normal modes of natural basins. The purpose is to determine the period and configuration of free oscillations in a way that provides for the full two-dimensionality of the problem, and thus avoids limitations such as are inherent in the traditional channel approximation. The method—called resonance iteration—is amenable to detailed numerical analysis.
As a test, the lowest positive and negative modes in a rotating square basin of uniform depth are calculated for a range of rotation speeds, with results that agree well with existing evaluations of this case. Similar agreement was found in an application to Lake Erie, a basin that conforms to the channel approximation. The method was then applied to two basins where it was expected to give results differing from those obtained by earlier methods: Lake Superior and the Gulf of Mexico. The fundamental gravitational mode of Lake Superior was found to have a period of 7.84 hr, which is 9% greater than the value known from the channel approximation, and is in virtually exact agreement with a recent spectral analysis. Phases of this mode also agree with observation.
The Gulf of Mexico is of particular interest because of the still-unresolved role of normal modes in the tidal regime of that basin. With the basin completely closed, the method of resonance iteration gave a period of 7.48 hr for the slowest gravitational mode and produced a single, positive amphidromic system that imparts to this mode approximately the character of a longitudinal oscillation on the nearly west-east axis of Mexico Basin. With the Gulf open through the Yucatan Channel and the Straits of Florida, the structure of this mode, is not altered qualitatively and the period is lowered to 6.68 hr. The most significant effect of these “ports” is that they elicit an additional gravitational oscillation—the so-called Helmholtz mode—which has a period much longer than that of the slowest seiche-type oscillation, and nodal points only at the ports. This co-oscillating mode is found to have a period of 21.2 hr. The proximity of its period to that of the diurnal tide points to a revival of the traditional conception that the tidal regime in the Gulf of Mexico is affected appreciably by resonance.
Abstract
A method is designed to calculate normal modes of natural basins. The purpose is to determine the period and configuration of free oscillations in a way that provides for the full two-dimensionality of the problem, and thus avoids limitations such as are inherent in the traditional channel approximation. The method—called resonance iteration—is amenable to detailed numerical analysis.
As a test, the lowest positive and negative modes in a rotating square basin of uniform depth are calculated for a range of rotation speeds, with results that agree well with existing evaluations of this case. Similar agreement was found in an application to Lake Erie, a basin that conforms to the channel approximation. The method was then applied to two basins where it was expected to give results differing from those obtained by earlier methods: Lake Superior and the Gulf of Mexico. The fundamental gravitational mode of Lake Superior was found to have a period of 7.84 hr, which is 9% greater than the value known from the channel approximation, and is in virtually exact agreement with a recent spectral analysis. Phases of this mode also agree with observation.
The Gulf of Mexico is of particular interest because of the still-unresolved role of normal modes in the tidal regime of that basin. With the basin completely closed, the method of resonance iteration gave a period of 7.48 hr for the slowest gravitational mode and produced a single, positive amphidromic system that imparts to this mode approximately the character of a longitudinal oscillation on the nearly west-east axis of Mexico Basin. With the Gulf open through the Yucatan Channel and the Straits of Florida, the structure of this mode, is not altered qualitatively and the period is lowered to 6.68 hr. The most significant effect of these “ports” is that they elicit an additional gravitational oscillation—the so-called Helmholtz mode—which has a period much longer than that of the slowest seiche-type oscillation, and nodal points only at the ports. This co-oscillating mode is found to have a period of 21.2 hr. The proximity of its period to that of the diurnal tide points to a revival of the traditional conception that the tidal regime in the Gulf of Mexico is affected appreciably by resonance.
Abstract
In preceding parts of this study a set of normal modes was constructed as a basis for synthesizing diurnal and semidiurnal solutions of Laplace's tidal equations. The present part describes a procedure by which such solutions can be computed as eigenfunction expansions. Since the calculated normal modes are nondissipative, it is necessary to incorporate dissipation into the synthesis procedure. This is done by a variational treatment of the tidal equations.
Abstract
In preceding parts of this study a set of normal modes was constructed as a basis for synthesizing diurnal and semidiurnal solutions of Laplace's tidal equations. The present part describes a procedure by which such solutions can be computed as eigenfunction expansions. Since the calculated normal modes are nondissipative, it is necessary to incorporate dissipation into the synthesis procedure. This is done by a variational treatment of the tidal equations.
