Height Coordinates Research Articles

On the scale of baroclinic instability in deep, compressible atmospheres

A simple model of linearized, inviscid baroclinic instability in an adiabatic, hydrostatic, compressible atmosphere of arbitrary (though finite) depth, based on the well-known Eady model, is used to investigate the variation of growth rates and favoured horizontal length scales as functions of δ, the ratio of the model depth D to the density scale height Hs. Both geometric height coordinates (with w = 0 horizontal boundary conditions) and log-pressure (with ω = 0 boundary conditions) are considered. For δ > 3 and a given zonal velocity scale, growth rates are significantly reduced relative to comparable instabilities in an incompressible fluid (δ = 0), and may be suppressed altogether in a laterally-bounded channel for large enough δ at a given value of static stability. Where instability does occur, the length scales favoured are longer than for an incompressible fluid, and are generally comparable to a deformation radius based on D (rather than Hs). The relationship between these results and those obtained in comparable recent studies (which have found scales comparable to a deformation radius based on Hs to be important) is examined. Some implications for the role of baroclinic instability in the atmospheres of the major planets are also discussed.

On the scale of baroclinic instability in deep, compressible atmospheres

AbstractA simple model of linearized, inviscid baroclinic instability in an adiabatic, hydrostatic, compressible atmosphere of arbitrary (though finite) depth, based on the well‐known Eady model, is used to investigate the variation of growth rates and favoured horizontal length scales as functions of δ, the ratio of the model depth D to the density scale height Hs. Both geometric height coordinates (with w = 0 horizontal boundary conditions) and log‐pressure (with ω = 0 boundary conditions) are considered. For δ > 3 and a given zonal velocity scale, growth rates are significantly reduced relative to comparable instabilities in an incompressible fluid (δ = 0), and may be suppressed altogether in a laterally‐bounded channel for large enough δ at a given value of static stability. Where instability does occur, the length scales favoured are longer than for an incompressible fluid, and are generally comparable to a deformation radius based on D (rather than Hs). The relationship between these results and those obtained in comparable recent studies (which have found scales comparable to a deformation radius based on Hs to be important) is examined. Some implications for the role of baroclinic instability in the atmospheres of the major planets are also discussed.

Vertical Normal Modes of a Mesoscale Model Using a Scaled Height Coordinate

Vertical modes were derived for a version of the Colorado State Regional Atmospheric Mesoscale Modeling System. We studied the impact of three options for dealing with the upper boundary of the model. The standard model formulation holds pressure constant at a fixed altitude near the model top, and produces a fastest mode with a speed of about 90 m s−1. An alternative formulation which allows for an external mode, could require recomputation of vertical modes for every surface elevation on the horizontal grid unless the modes are derived in a particular way. These results have beating on the feasibility of applying vertical mode initialization to models with scaled height coordinates.

Radar Stereomapping Techniques and Application to SIR-B Images of Mt. Shasta

One of the goals of the SIR-B experiment was the definition of optimum radar incidence and intersection angles for radargrammetric stereoscopy by comparing the results from three separate data reduction methods. To this end three overlapping images of the prime radargrammetric site (Mt. Shasta in northern California) were obtained, fewer than anticipated. This paper describes the mathematical basis for radar stereomapping, along with preliminary results from one of the methods (using a so-called "analytical" stereoplotter) for the Mt. Shasta site. Height coordinate errors range from 60 to 170 m depending on the density (number per ground area) of ground control points used.

Reproducibility of repeated mountings of a noninvasive CT/MRI stereoadapter.

The reproducibility of a noninvasive computed tomography (CT)/magnetic resonance imaging stereoadapter was tested on 12 healthy volunteers. The adapter was easy to mount and detach without discomfort. Two repeated mountings of the adapter showed a good reproducibility in relation to the scalp. When extrapolated to a hypothetical target in the right thalamus, the mean differences between the two mountings were 0.07 mm in the lateral, 0.16 mm in the anteroposterior and 0.17 mm in the height coordinates. This high degree of reproducibility seems to permit a safe CT-guided functional stereotaxis, where the preoperative CT scanning can take place on one day and the surgery may be performed on any other suitable day.

Steady convection in pressure coordinates

Conservative properties of steady flow in non-hydrostatic pressure coordinates are used to formulate theoretical models of deep convection. the pressure coordinate analysis extends the range of analytic solutions previously obtained in height coordinates, because the effect of the variation of density with height is included and is more useful for comparison with observations and the implementation of dynamical cloud models in convective parametrization schemes. Previous models are made more realistic by including a non-constant thermodynamic lapse rate. In the case of steering-level convection, this is shown to be a simple reinterpretation of the constant lapse rate model, while for propagating convection there are more fundamental implications; downstream propagating solutions cannot exist if the net work done by the pressure field is zero, while upstream propagation, although dynamically possible, is not observed.

Steady convection in pressure coordinates

AbstractConservative properties of steady flow in non‐hydrostatic pressure coordinates are used to formulate theoretical models of deep convection. the pressure coordinate analysis extends the range of analytic solutions previously obtained in height coordinates, because the effect of the variation of density with height is included and is more useful for comparison with observations and the implementation of dynamical cloud models in convective parametrization schemes. Previous models are made more realistic by including a non‐constant thermodynamic lapse rate. In the case of steering‐level convection, this is shown to be a simple reinterpretation of the constant lapse rate model, while for propagating convection there are more fundamental implications; downstream propagating solutions cannot exist if the net work done by the pressure field is zero, while upstream propagation, although dynamically possible, is not observed.

Dimensional models for the perception of rectangles.

Krantz and Tversky (1975), Takane (1981), Wender (1971), and others have suggested that it is impossible to explain similarity judgments for rectangles by a simple dimensional model of the Minkowski distance type, because the psychologically compelling dimensions are not independent and interact. For reasons never made explicit, the relevant dimensions were assumed to be area and shape, rather than width and height. Reanalyses show, however, that the latter dimensions, appropriately scaled, eliminate interaction effects. To test these conclusions empirically, an experiment was carried out with two sets of rectangles. Each set was a complete 4×4 design, one in width × height coordinates, and one in area × shape coordinates. For each stimulus set, 21 subjects judged the dissimilarities of all pairs of rectangles twice. All subjects were reliable. Apart from one extreme subject in each group, all data lead to very similar MDS solutions. These solutions correspond closely to the predictions, that is, they can be well approximated with physical width and height coordinates rescaled such that the units decrease increasingly as one moves away from the origin. No interaction effects are found, but some indications for a more complicated composition rule are observed.

On the Nongeostrophic Baroclinic Instability Problem

Abstract The conventional balance-equation treatment is found to be undesirable for the purpose of filtering out internal gravity waves when the Richardson number is of order unity. A generalized formulation for such purpose is established. The nature of nonseparability in the ageostrophic baroclinic instability problem for a channel model is closely examined. It is found that there exist auxiliary boundary conditions which have to be explicitly satisfied in order to uniquely pose the eigenvalue problem. These conditions become greatly simplified if the proper filtering procedure is applied. The eigenfunction then becomes separable in its dependence on the height and cross-channel coordinates under fairly general conditions. An algorithm is then formulated to solve the highly ageostrophic baroclinic instability problem in a channel model. The effects of the meridional walls are conclusively established by closely examining the three-dimensional structure of the unstable mode. Quasi-symmetric instability m...

Torsion of structural cores on deformable foundations

An approximate method is presented for assessing the influence of foundation deformations on the warping stresses of a shear core subject to torsional loading. The core is assumed supported on elastic foundations such that the vertical displacements are proportional to the imposed forces. Solutions have been derived for two general forms of loading, a concentrated torque at any level, and a distributed twisting moment whose intensity is defined as any polynomial in the height coordinate. A numerical example is given to illustrate the reduction in warping stresses which can occur as a result of relative foundation deformations.

Automatic N (h, t) Profiles of the Isonosphere With a Digital Ionosonde

A method is described to accomplish automatic data selection and profile inversion to obtain ionospheric electron‐density profiles from digitized radio soundings. The profile inversion is based upon the well‐established formulation of Paul [1960] by which the optimum radio‐frequency sounding intervals can be specified from an approximate knowledge of the profile. The expected virtual height coordinates (h') at these frequencies (ƒ) are likewise estimated, and procedures are then used to select h'(ƒ) observations nearest the predicted coordinates from a subsequent digital ionogram. From these the next profile is obtained. The process adaptively follows the changing shape and detail of the profile. The procedure requires an average of 20 sec per profile on a standard data‐processing computer and can be adapted, with benefit to on‐line real‐time use in our ‘dynasonde,’ digital ionosonde.

Doppler radar investigation of Hawaiian rain

Doppler radar measurements in a number of Hawaiian rain echoes indicated a variable amount of convection, with vertical air velocities ranging from unmeasurably small values in some cases to values of 7 m/sec in the more convective cases. The echoes exhibiting weak vertical motions usually had a stratiform appearance when plotted on coordinates of height versus time: contours of signal intensity and Doppler velocity tended to be approximately horizontal. The intensity decreased with altitude in these cases, and vertical gradients as strong as 20 db/km were not uncommon. In the echoes exhibiting strong vertical air motions, signal intensities, echo depth, and surface rainfall rate tended to be higher. Radar reflectivity factors as high as 10 5 mm 6 /m 3 were observed. Although height-time patterns revealed less order than in the echoes with weak updraft velocities, there was a tendency for the leading (upslope) portion of a convective echo to be more active than the trailing portion. As echoes progressed further upslope they tended to become more stratiform in character. The fact that moderate rain could at times fall from shallow, stratiform clouds, taken in conjunction with the strong vertical reflectivity gradients observed in these cases, indicates a high efficiency of the precipitation process in Hawaiian orographic clouds. The observations in convective echoes clearly demonstrate that convective motions enhance the process. DOI: 10.1111/j.2153-3490.1967.tb01498.x

Doppler radar investigation of Hawaiian rain<sup>1</sup>

Doppler radar measurements in a number of Hawaiian rain echoes indicated a variable amount of convection, with vertical air velocities ranging from unmeasurably small values in some cases to values of 7 m/sec in the more convective cases. The echoes exhibiting weak vertical motions usually had a stratiform appearance when plotted on coordinates of height versus time: contours of signal intensity and Doppler velocity tended to be approximately horizontal. The intensity decreased with altitude in these cases, and vertical gradients as strong as 20 db/km were not uncommon.In the echoes exhibiting strong vertical air motions, signal intensities, echo depth, and surface rainfall rate tended to be higher. Radar reflectivity factors as high as 105 mm6/m3 were observed. Although height-time patterns revealed less order than in the echoes with weak updraft velocities, there was a tendency for the leading (upslope) portion of a convective echo to be more active than the trailing portion. As echoes progressed further upslope they tended to become more stratiform in character.The fact that moderate rain could at times fall from shallow, stratiform clouds, taken in conjunction with the strong vertical reflectivity gradients observed in these cases, indicates a high efficiency of the precipitation process in Hawaiian orographic clouds. The observations in convective echoes clearly demonstrate that convective motions enhance the process.

Optical Aberration Coefficients IX Theory of Reversible Optical Systems

This paper develops the theory of the aberrations of symmetrical reversible optical systems (sometimes called “symmetric” or “holosymmetric”). It is shown that if such a system is not working at unit magnification (m2 ≠ 1), then not all of its aberrations can be removed simultaneously. In particular it is shown that if s is the magnification associated with the pupil planes, f the focal length, σ1, σ2, ⋯,σ5 the primary (Seidel) coefficients, then (1−s2)(1−sm)σ1+[3(1−sm)2−(s−m)2]σ2+(1−m2)(1−sm)(3σ3+σ4)+(1−m2)2σ5=−12(1−m2)/f2.The variables entering into the aberrations here are the ideal image height and polar coordinates in the plane of the exit pupil. Analogous relations hold between coefficients of higher order, and some of them are derived explicitly. The number of relations in any order is counted. The usual results concerning systems working at unit magnification are derived, and the special case m = 0 (object at infinity) is discussed separately. The third- and fifth-order relations are transcribed (in the case m = 0) into the language of the “algebraic” coefficients which have been investigated in previous papers of this series and elsewhere. Finally, the problem of the imaging of plane objects is examined in some detail both when m ≠ 0 and when m = 0; and it is shown that when m2 ≠ 1 a sharp image can be formed only on an ellipsoid or hyperboloid of revolution unless m = 0. In the latter case one is of course left with a large amount of distortion when the image is plane.

LOGARITHMIC TABLES FOR COMPUTING THE SURFACE AREA OF THE BODY ACCORDING TO DUBOIS' FORMULA

In 1916, Dubois and Dubois 1 published a formula for the computation of the surface area of the body and presented a chart from which one could read the approximate surface area from the coordinates of height and weight if given in metric units. This chart permitted one to read the surface area to within approximately 150 square centimeters. Boothby and Sandiford 2 and Feldman and Umanski 3 publish nomograms for computing the surface area. In my experience, these have given large errors for the larger surface areas. These errors would be of small significance in ordinary clinical work, but they introduce an element of uncertainty in research work of any degree of refinement. Owing to the use of surface area in the computation of normal vital capacity and in research in the field of basal metabolism, a means of ascertaining surface area quickly and accurately from either English or metric units of