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William H. Raymond

Abstract

In this study the individual components of the horizontal moist wind are derived and expressed as a function of the horizontal pressure gradients. The drag on the large-scale horizontal wind by mesoscale diabatic activity is shown to be consistent with homogeneous Lagrangian solutions that asymptotically approach a lognormal distribution. The kinematic moisture flux components are also shown to contain homogeneous solutions that behave asymptotically as a lognormal distribution. These findings are considered relevant to the large number of observed occurrences of the lognormal distribution in cloud and rainfall statistics.

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William H. Raymond

Abstract

Applications of low-pass filters or implicit diffusion are found to be extremely beneficial in a simple model problem that contains an oven-ordered advection scheme based on a symmetric stencil. In contrast, the effects are negative when either is applied to an odd-ordered approximation of advection that uses an asymmetric stencil. Also, the discriminating power of filters is shown to be helpful in identifying how small spatial-scale information influence a numerical weather prediction forecast.

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William H. Raymond

Abstract

Using first principles, it is shown that the vertical variation of the mixing ratio can be approximated by a power law if the relative humidity is also expressed in terms of a power law. Nevertheless, variability in the relative humidity is a major source of error in estimating moisture profiles. This error arises because of a lack of knowledge about a parcel’s source region and its recent transport history. To understand the physics of relative humidity better, its temperature and pressure dependences are examined. Using this knowledge, a small correction or modification to the moisture power law is formulated by assuming that the relative humidity is known at some location far above the base of the profile. Although not always obtainable, relative humidity is available during numerical weather prediction’s data assimilation. The method works well at estimating profiles associated with large-scale moisture patterns and in cases where inversions and isolated dry or moist layers are not pervasive. Additionally, from the knowledge of the temperature and pressure dependency of the relative humidity, it is easy to construct a simple analytic method in terms of relative humidity to identify adiabatic changes induced by turbulent mixing schemes used in numerical weather forecast models. This technique is especially useful in parameterizing shallow nonprecipitating clouds when the depth of the boundary layer is known, or when using nonlocal turbulent mixing parameterizations.

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William H. Raymond

Abstract

This study introduces a moisture advection formulation that contains relative humidity. In the sigma coordinate system, rewriting the mixing ratio conservation equation in terms of relative humidity leads to an equation that explicitly contains temperature and pressure. Consequently, the governing equation, containing relative humidity, is significantly different from that used for a fluid tracer. It is shown analytically that the homogeneous part of the governing equation for relative humidity is contained in the class of differential equations that yield solutions with lognormal distributions. These findings are relevant to the large number of observations of cloud and rainfall distributions that statistically have a lognormal component. (The relationship between clouds and the probability distribution function for relative humidity is examined in the appendix.) The equation describing the temporal and spatial distribution of relative humidity can be solved using any reliable numerical approximation. All solutions are positive definite.

The formulation containing relative humidity provides an alternative method to test the sensitivity of the forecast to the moisture equation. In this study only the gridscale advection process is calculated using the relative humidity equation. Otherwise, all turbulence, physics, and numerical filtering involving moisture are performed using the conservative mixing ratio quantity. Forecast comparisons between mixing ratio and relative humidity methods show that the major differences occur in regions with large gradients. In some instances the numerical approximation of the mixing ratio conservation equation tends to produce slightly tighter gradients in regions with sharp changes. The tightness of the water vapor gradients directly influences the rainfall amounts. Consequently, rainfall maxima are slightly reduced with the relative humidity approach. Because these changes are very localized in the limited area forecasts used here, the differences in global verification statistics are small but favor the relative humidity approach. Verification statistics include dewpoint temperature, total precipitable water, and precipitation. For precipitation, forecast results are presented using 40-km and 80-km horizontal grid configurations. All other statistics use only the 80-km horizontal grid spacing.

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William H. Raymond

Abstract

Comparisons are made between hydrostatically computed temperatures and ambient temperatures associated with nine different data sources, including analyses, forecasts and conventional observations. Five-day averages and the day-to-day variations in the root-mean-square temperature differences are presented. Several different numerical and interpolation procedures are examined. Error correction and a constrained optimum procedure that minimizes ambient minus calculated hydrostatic temperature differences are introduced.

Systematic differences between ambient and hydrostatic temperatures are found to be associated with the synoptic situation. When compared with ambient temperatures, hydrostatic temperatures at 500 mb tend to he too warm at or in front of a trough and too cold behind the trough. In the vertical direction, for the eight-level configuration tested, the average hydrostatic temperatures are too cold at low levels (850, 700 mb) and too warm at upper levels, (300, 250 mb).

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William H. Raymond

Abstract

Equations describing the temporal and spatial behavior of the kinematic moisture and heat flux are introduced in this study. In these nonlinear equations, the contribution by diabatic processes to the large-scale flux is composed of two parts. One part is associated with a Rayleigh damping term, while the other arises from temporal and spatial changes in the pressure gradient term.

The influence of diabatic processes on the large-scale moisture fluxes depends greatly on the degree of balance between forcing and damping terms in the governing equations. The existence of a near balance requires a reduction in the large-scale horizontal geostrophic wind speed. From a scale analysis of the moisture flux equations it is argued that reductions in the large-scale, horizontal wind speed, observed within major cumulus cloud systems, help conserve large-scale moisture fluxes. The deviation of the wind from geostrophic conditions is easily estimated. This wind modification induct secondary vertical circulations that contribute to the convergence, creating or supporting long-lived mesoscale flows. In the tropics the wind modification has an antitriptic relationship.

These diagnostic findings suggest possible modifications to the wind field in the application of cumulus parameterization, and may be important in diabatic initialization of numerical weather prediction models.

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William H. Raymond

Abstract

In this study high-order, high-pass implicit filters are introduced. They represent symmetric filters in an implicit formulation. In this investigation their use within a finite region is examined. The effects of the boundary are investigated and comparisons with spectral methods are made to evaluate the response of the high-pass filter and to test the desirability of Neumann and reflective boundary conditions. These high-order, high-pass implicit filters are shown to be a useful tool to help evaluate and interpret meteorological fields.

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William H. Raymond

Abstract

High-order implict tangent filters are developed. The implicit tangent filters possess a highly selective amplitude response function and they can be applied relatively close to a boundary. Comparisons are made between the implict and the traditional or explict filters. Numerical simulations are performed to test the response in a limited-area model. An algorithm to maximize computing efficiency is presented. All tests indicate the desirability and utility of the implicit tangent filter.

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Arthur Garder and William H. Raymond

Abstract

The Galerkin method is used to solve numerically the nonlinear initial-boundary value problem describing the vertical temperature profile for a thermally coupled soil-atmosphere boundary layer in a simple physical setting. The basic physical processes modeled, including conduction and free convection, are simple, yet the solution to the model problem illustrates the Galerkin procedure. The numerical simulation requires special mathematical techniques for handling interior boundary or coupling conditions. The physical interpretation of the internal conditions includes consideration of time-dependent insolational heating and the thermal interaction between the atmosphere and the soil. Comparisons between the exact solutions and the Galerkin solutions for several linear problems show agreement for all cases studied.

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William H. Raymond and Arthur Garder

Abstract

Low- and high-pass traditional recursive and implicit filters are reviewed. Some similarities and differences between these two forms are illustrated. The use of recursive filters in signal processing is contrasted with the needs in meteorology. The standard techniques used in building a recursive filter with specified characteristics are described. The desirability of high-order calculations is demonstrated. Some numerical results are presented to illustrate the differences in filter selectivity in the presence of topography. To make the implicit filters competitive with the traditional recursive formalism, efficient numerical matrix inversion procedures are employed in the application of both limited area and cyclic boundary conditions.

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