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THE PREDICTION OF SURGES IN THE SOUTHERN BASIN OF LAKE MICHIGAN

Part I. The Dynamical Basis for Prediction

GEORGE W. PLATZMAN

Abstract

A summary of numerical computations is presented, in a form designed to aid in operational prediction of surges in the Southern Basin of Lake Michigan. The computations are based upon a dynamical model in which the surge is generated by pressure gradient and wind stress in a squall line which moves across the Basin with constant speed and direction. For each of 25 combinations of squall-line propagation speed and direction, the arrival time of the surge is determined, and the amplitude estimated, at various locations along the shore. At some locations there are a well-defined peak of surge amplitude and corresponding critical values of squall-line propagation speed and direction, associated with resonant coupling between the squall line and Lagrangian body waves. Energy computations indicate the presence of another resonant peak associated with Stokesian edge waves.

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GEORGE W. PLATZMAN

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.

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George W. Platzman

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.

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George W. Platzman

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.

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George W. Platzman

Abstract

A finite-element model recently designed for calculation of oceanic normal modes is amended here with a prescription for the proper treatment of multiple connectivity, the main effects of which are likely to be located in the Southern Ocean. For this purpose line bases, as well as point bases, are needed in the finite-element representations of volume-flux and energy-flux streamfunctions. The dynamical conditions that should be met by the circulation of velocity and energy on each boundary are found to arise as natural boundary conditions for the finite-element equations.

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George W. Platzman

Abstract

Normal modes are calculated for a homogeneous ocean occupying a connected domain consisting of the North Atlantic, South Atlantic, and Indian Oceans. Coastal configuration and bathymetry are resolved on a grid of 675 six-degree Mercator squares. The calculation is based upon the Lanczos process and is more efficient than resonance iteration. Twenty-six gravity modes were found with periods greater than 8 h, the slowest being a fundamental mode of about 67 h. The North Atlantic co-oscillates with the South Atlantic at a period of about 42 h, and has strong resonances at 23, 21, 14.4, 12.8, 8.6 and 8.3 h. Eleven topographically-induced modes of rotational type were found with periods less than 100 h; the fastest of these is a 44 h mode in the Weddell Sea. In the 6° model the fastest rotational mode of the North Atlantic is a 55 h topographic wave most prominent near the Grand Banks of Newfoundland.

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George W. Platzman

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.

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George W. Platzman

Abstract

Diurnal and semidiurnal tides of second and third degree are synthesized from 60 normal modes with period in the range 8 to 96 h. Diurnal tides, especially those of second degree, can be represented by remarkably few modes. The principal lunar diurnal constituent, for example, consists almost entirely of a single forced mode excited mainly in the Pacific and Indian Oceans. Semidiurnals are spectrally more heterogeneous, and more resonant, than diurnals, but some specific features can be attributed to individual modes. Several of the most energetic modes in the principal lunar semidiurnal constituent are prominent in the Atlantic Ocean. Together with the fact that diurnally excited modes are relatively weak in that region, this presumably accounts for the observed tendency for the total tide to be predominantly semidiurnal in the Atlantic but mixed diurnal and semidiurnal in many parts of the Pacific and Indian Oceans.

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George W. Platzman

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.

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George W. Platzman

Abstract

The linearized primitive equations for a barotropic world ocean are discretized by means of first-order, piecewise-linear finite elements. Surface elevation and Stokes/Helmholtz velocity potentials are adopted as dependent variables. On any segment of the ocean boundary, specification of elevation and simple radiation are allowed as alternative conditions. The discretized mass and momentum equations are designed to make the finite-element solution satisfy global energy balance exactly. They also permit an arbitrary choice of axes at each node of the grid and thereby avoid the “pole” problem. The model is tested by applying it to a rotating rectangular basin and to a domain consisting of the Atlantic and Indian Oceans.

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