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G. J. Haltiner

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G. J. HALTINER

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G. J. Haltiner

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The partial differential equation for heat diffusion is numerically integrated by the Runge-Kutta method. Solutions are obtained for the diurnal temperature variation with a bounded coefficient of eddy diffusivity which varies periodically with time and exponentially with height. The surface wave is represented by the sum of a diurnal and a semidiurnal harmonic wave. The results may be interpreted to apply over a fairly broad range of diffusivity values and height. With appropriate choices of the various parameters, reasonably good agreement is obtained between theoretical and observational values of amplitude reduction and phase lag as functions of height and time.

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G. J. Haltiner

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G. J. Haltiner and J. M. McCollough

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A global, multi-level, baroclinic primitive equation model is used as a vehicle to test several initialization procedures for suppressing inertial-gravity noise, including static balancing on pressure and on sigma surfaces, iterative dynamic balancing with Temperaton's averaging scheme, and finally the use of a time filter. It is tentatively concluded that static balancing, followed by dynamic balancing equivalent to a 12 h forecast, and the use of a time filter will suppress gravity noise adequately for prediction purposes and may also permit nonsynoptic data to be assimilated about every 4 h without serious trauma.

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G. J. Haltiner and Yeh-chum Wang

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A model for numerical prediction of the 1000-mb surface is developed which includes a term expressing the interchange of sensible heat between the air and the underlying surface as well as the effect of terrain-induced vertical motion. In spite of the crudeness of the non-adiabatic representation, the model shows a definite improvement over a similar adiabatic model when the two are compared in a series of prognoses. Moreover, when monthly mean isotherms may be used to represent the temperature of the underlying surface, the non-adiabatic term may be combined with the orographic term and the earth's vorticity so that there is no work added to the prognostic routine.

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G. J. Haltiner and R. T. Williams

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Recent developments in numerical weather prediction during the past several years are briefly summarized for the nonspecialist.

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G. ARNASON, G. J. HALTINER, and LT. M. J. FRAWLEY

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Two iterative methods are described for obtaining horizontal winds from the pressure-height field by means of higher-order geostrophic approximations for the purpose of improving upon the geostrophic wind. The convergence properties of the iterative methods are discussed; and in a simple theoretical case, one of the methods is found to diverge with strong cyclonic motion. Both iterative methods were applied to analyzed 500-mb. height charts and over most of the map converged in a few scans to wind values somewhere between the geostrophic wind and the wind obtained from the balance equation. However in a few locations continued iteration led to increasing differences between successively computed winds: i.e., the methods appeared to diverge. In fact, wind values in adjacent areas gradually tended to be corrupted. This lack of convergence, occurring mainly in areas of negative vorticity and additionally in the case of method II in areas of strong cyclonic vorticity, was associated with the development of excessive horizontal wind divergence, which after three or four iterations sometimes exceeded the relative vorticity. Stream functions were computed by relaxing the relative vorticity of the winds obtained by methods I and II, generally after one iteration. These were compared to the stream function obtained by solving the balance equation and no significant differences were noted. Barotropic forecasts prepared from the stream functions derived from the two methods are essentially the same as forecasts with the stream function obtained from the balance equation.

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G. J. Haltiner, H. D. Hamilton, and G. ’Arnason

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The effects of the state of the sea on the safety and economy of a ship's route are of considerable concern to the U. S. Navy and other organizations engaged in shipping. This article deals specifically with the problem of determining a ship's minimal-time route between two ports of call as reflecting an important aspect of desirable routing. This is a minimum value problem and the governing differential equations are derived by use of the Calculus of Variations. The basic theory is essentially the same for a ship on the sea as for an aircraft in horizontal flight, but in application the two problems differ in the manner in which the environment impedes the forward speed of the vehicle.

Direct solution of the governing differential equations by numerical methods appears feasible with the aid of an electronic computer and the results of a test case are presented. For this purpose it was convenient to replace a series of empirical equations relating ship's speed to wave height and direction by a single analytical expression.

Inspection of the basic differential equations and the empirical relation between ship's speed and the state of the sea indicates that in the case of moderate wave heights, the dependence on wave direction may be omitted as a good approximation. This is borne out in the test case which also shows a considerable saving in computing time because of the resulting simplification of the differential equation.

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G. J. Haltiner, L. C. Clarke, and D. G. Lawniczak Jr.

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Vertical velocities are computed at four levels over a Northern Hemisphere grid in addition to a lower boundary value which is applied at terrain pressure and includes effects of frictionally- and terrain-induced vertical velocities. The latter, as computed here, have somewhat smaller maximum magnitudes and less irregularity than the frictionally-induced vertical velocity.

The calculations show that the terrain and friction effects markedly influence the ω-fields in the lower troposphere but largely disappear by 500 mb for the rather heavily smoothed mountains used in these experiments.

Computations with several static stability parameters, namely a constant value, a value varying with pressure only, and a point-variable value, exhibit the greatest differences in the maximum vertical velocities, as much as 50 per cent, at 300 mb. At lower altitudes the differences are only about 10 per cent.

Similarly when computations were made utilizing the term fη versus fM 2 in the coefficient of ∂2ω/∂p 2 in the ω-equation, differences in ω up to 50 per cent occurred, but mostly the differences were only 10 to 25 per cent.

The computed vertical velocities during a severe west coast storm appear to show good correlation with the weather.

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