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J. R. Philip

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J. R. Philip

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Windward diffusion (WD) is customarily neglected in convection-diffusion studies, though some work suggests upwind diffusive flux may be significant. Analyses of internal boundary layers produced by a step change in surface concentration of a scalar with and without WD confirm that the effect of WD is indeed negligible. The analysis with WD is, however, needed if an estimate of upwind diffusive flux is required.

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J. R. Philip

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J. R. Philip

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A one-dimensional model for transfer processes within the air layers occupied by vegetation is presented. This involves a height-dependent apparent diffusivity and vertical profiles of (i) mean concentration in the air and (ii) mean concentration at foliage surfaces. The model enables an examination of the partition and dissipation of the net radiant energy absorbed by vegetation.

In the present study diffusive resistances to H2O transfer within the plant and the soil are neglected. However, numerical examples indicate that, even in this case, the downward transfer of sensible heat tends to make the distribution of the transpiration load within vegetation more even than that of absorbed net radiation. The source distribution for sensible heat is generally of quite different shape to that for latent heat, and the “foliage Bowen ratio” systematically increases with height within the canopy.

The observations of Denmead (1964) suggest that smoothing of the transpiration load is greater than the model calculations suggest. It seems likely that diffusive resistances within the vegetation play a significant role in the smoothing process.

Source strength distributions for various entities within vegetation way differ markedly, so that attempts to infer information about diffusive resistances within vegetation from concentration profiles above the canopy alone are not soundly based.

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J. R. Philip

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The character of the Earth's atmospheric heat engine depends, inter alia, on the relatively tight linkage between surface fluxes of energy and of H20. On Mars, on the other hand, H2O-based latent heat fluxes are only a trivial fraction of total surface energy fluxes and the dominant component of the working fluid is CO2. These considerations are made quantitative through evaluation of Λ, the equivalent temperature excess at the surface for a particular component of the working fluid. The very different values (and latitudinal distribution) of Λ on the two planets signalize vividly their different meteorology. Preliminary study of the climatology of Λ on Earth brings out, in particular, the tightness of the H20-energy linkage in the tropics.

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J. R. Philip

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Forbidding supersaturation above an evaporating water (or sublimating ice) surface fixes an upper bound on β, the Bowen ratio; a similar bound holds for evaporation from unsaturated surfaces. The bound arises from the nonlinearity of the temperature dependence of saturated water vapor density. The resulting inequalities are compared with other estimates of β and related quantities. The case of supersaturation, with fog formation, lies outside the scope of this study.

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J. R. Philip

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Kirkaldy (1958) has raised objections to the usual quasi-stationary analysis (QS) of evaporation and condensation processes. Fuchs’ (1959) attempt to study the validity of QS is marred by a number of errors (which are identified in the present work). This paper presents a critical examination of QS, largely based on mathematical methods capable of establishing the accuracy of QS in cases where the exact solution is unavailable.

Exact similarity solutions for n-dimensional condensation (zero initial radius) are presented. These, and an exact solution for diffusion about a spherical body of constant radius, provide points of departure for the development of two simple and powerful approximate methods, a perturbation method and a continuity-preserving quasi-stationary analysis (CPQS). The perturbation method provides an apparently highly accurate means of using the relevant known exact solutions; but it is less versatile than CPQS. Under a wide range of circumstances, CPQS, which satisfies the diffusion equation in the Pohlhausen sense, correctly indicates the order of magnitude of the errors of QS. Where both methods apply, they give mutually consistent results.

The study indicates, that, despite the fundamental objections to QS, it is sufficiently accurate for many meteorological purposes. The success of QS arises from the particular circumstances of the three-dimensionality of the processes and of the very small ratio of vapor density differences to the density of the condensed phase. Further study is needed of the psychrometric aspects of certain problems, and of processes where conditions at the droplet surface are radius-dependent.

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J. R. Philip

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This paper is the first of a series in which the theory of local advection (the exchange due to horizontal heterogeneity) of energy and moisture will be developed and applied to a number of problems of practical and theoretical interest. The paper provides an introduction to the practical implications and physical basis of local advection, but it is mainly devoted to developing methods of analysis to be applied in later papers. The treatment aims to provide simple and rapid numerical procedures for the solution of advection problems.

Methods are given for solving the two-dimensional atmospheric-diffusion equation subject to “concentration,” “flux,” and “radiation” types of boundary conditions. Appendices give discussions of the properties of the functions entering the solutions and provide simple means for their computation. Extensive tables of the relevant functions are given for the case m = 1/7, n = 6/7 (m, n being the exponents in the power-law approximations to the vertical profiles of mean wind speed and eddy diffusivity, respectively).

The rudiments of a quantitative theory of advective inversion are developed, expressions being obtained for the equation of the inversion surface, and for the maximum height and downwind extent of the inversion.

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J. R. Philip

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Recent developments in the study of moisture and heat fields in the soil are applied to the problem of evaporation from bare soil surfaces. Most of the analysis is confined to steady conditions, but it definitely suggests that there are three phases to the desiccation of a given soil profile:

1. So long as the soil is sufficiently moist, the evaporation rate, E, is indistinguishable from that from a saturated surface, E3.

2. At intermediate moisture contents, E is independent of E3 and depends only on the soil-moisture distribution.

3. When the surface layers of the soil are sufficiently dry, E is sensitive to the heat flux in the soil, and a negative correlation between E and E3 may follow.

The first and second of these correspond to the well-known constant and falling rate phases of the “isothermal” drying of initially saturated soils and other porous media, which thus receive quantitative physical explanation. There is experimental evidence of the existence of the third phase also.

Other results of the article include :

1. Development of a (steady state) mode1 of the energy balance at imperfectly evaporating surfaces (E < E3). This leads to a simplified but suggestive approach to the microclimate of bare soils.

2. Analysis of the spatial distribution of evaporation sites within the soil.

3. Analysis of the modification of the energy balance which results from the location of evaporation sites within the soil rather than at the surface.

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J. R. Philip and J. W. Rottman

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Satellite images reveal tracks of enhanced solar reflectivity in low-level stratus clouds over the ocean that are known to be produced by the aerosols emitted from diesel-powered ships. The question arises: Can we track a ship from such images? A primitive model of convection–diffusion of particles from the moving ship to the condensation level, and of formation of the detectable ship track, is developed. The analysis takes account of various influences, such as ship and wind velocity vectors. It yields, inter alia, the dimensions of the ship track and its disposition relative to the ship. Plausible parameter values give results in the observed range. Tracking is achieved for the simplified problem, but a paucity of meteorological and other data causes practical difficulties.

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