Richardson Criterion Research Articles

Interpretation of Steroid Receptors in Breast Cancer. A Case With Discordant Estrogen Receptor Results Using ER1D5 and H222 Antibodies

We report a case of an invasive duct breast cancer, grade 1 by Bloom and Richardson criteria, and metastatic to axillary nodes and perinodal soft tissue. Immunostaining for estrogen receptor (ER) with ERlD5 resulted in intense positive tumor cell staining with both frozen and paraffin sections. Matching studies using a different anti-estrogen receptor reagent (H222), uniformly produced negative results in 3 different assay systems (EIA-ER, ER-ICA, ER-PAR). This case reveals a discrepancy between reactivity of ERlD5 and H222 that may indicate the presence of outlaw ER variants. It also illustrates the interpretation problems that can arise with new antibodies that have not been fully validated, particularly if the results are used in therapeutic decision making. (The J Histotechnol 18:331, 1995)

Interpretation of Steroid Receptors in Breast Cancer. A Case With Discordant Estrogen Receptor Results Using ER1D5 and H222 Antibodies

AbstractWe report a case of an invasive duct breast cancer, grade 1 by Bloom and Richardson criteria, and metastatic to axillary nodes and perinodal soft tissue. Immunostaining for estrogen receptor (ER) with ERlD5 resulted in intense positive tumor cell staining with both frozen and paraffin sections. Matching studies using a different anti-estrogen receptor reagent (H222), uniformly produced negative results in 3 different assay systems (EIA-ER, ER-ICA, ER-PAR). This case reveals a discrepancy between reactivity of ERlD5 and H222 that may indicate the presence of outlaw ER variants. It also illustrates the interpretation problems that can arise with new antibodies that have not been fully validated, particularly if the results are used in therapeutic decision making. (The J Histotechnol 18:331, 1995)

A Richardson criterion for compressible swirling flows in a toroidal magnetic field

Stability conditions for an adiabatic compressible swirling flow in the presence of a toroidal magnetic field is investigated. In the absence of all the dissipative mechanisms, the flow is found to be stable to all small axisymmetric perturbations if the local Richardson number is equal to or greater than 1/4 throughout the flow domain.

Richardson criteria for stratified vortex motions under gravity

Three sufficient conditions for stability are established for a general class of rotating flows with the velocity and the density distribution varying in both the axial and radial directions. Two are the classical Richardson criteria in the axial and radial directions. The third, measured by a newly defined Richardson number, acts as a constraint on the other two. The newly defined Richardson number is a ratio between the interaction of the density variations with the force fields and the interaction of the velocity gradients in two directions. The former interaction is a result of the pressure restraint condition and is measured by a new Brunt–Väisälä frequency. The latter interaction determines whether the velocity gradients in the second direction strengthen or weaken the resultant shear effect. Because of the generality of the flow profiles being considered, the criteria established in this investigation are valid for a wide range of problems in oceanographic and atmospheric studies.

On the Stability of Vortex Motions in Compressible Stratified Fluids

Necessary conditions for the stability of a general class of compressible vortex flows are obtained using the method of generalized progressing wave expansion. The vortex motion and the density are assumed to vary in both the radial and axial directions. These conditions, representing a generalized state of statically stable distribution for the steady flow, require that the flow be stable in the centrifugal force field created by the rotation of the fluids, in the gravitational force field arisen from the density variation, and in the pressure field constraining the variation of density and forces in both the radial and axial directions. These results also suggest the likely behavior of Richardson criteria for stability of the flow yet to be derived.

Eigenvalue bounds in magnetoatmospheric shear flow

A rigorous approach by E.M. Barston (1977) to stability of Lagrangian systems is used to establish both rectangle and semicircle theorems for plane parallel flow along a horizontal but otherwise arbitrary magnetic field, permeating a perfectly electrically conducting incompressible fluid under gravity. The radius of the semicircle is reduced by magnetic effects and stable stratification. A Richardson criterion for stability against constant shear flow is also derived. The analogous problem for a compressible fluid is also discussed, and for a certain class of disturbances a 'semi-dumbbell' theorem is established which is considerably stronger than the semicircle theorem. Possible astrophysical applications are discussed.

Integral formulas with distribution kernels for irreducible projections in L2 of a nilmanifold

Let N be a simply connected nilpotent Lie group and Γ a discrete uniform subgroup. The authors consider irreducible representations σ in the spectrum of the quasi-regular representation N × L 2( Γ/ N) → L 2( Γ→) which are induced from normal maximal subordinate subgroups M ⊆ N. The primary projection P σ and all irreducible projections P ⩽ P σ are given by convolutions involving right Γ-invariant distributions D on Γ→, Pf(Γn) = D ∗ f(Γn) = <D, n · f> all f ϵ C ∞(Γ/N) , where n · f( ζ) = f( ζ · n). Extending earlier work of Auslander and Brezin, and L. Richardson, the authors give explicit character formulas for the distributions, interpreting them as sums of characters on the torus T κ = ( Γ ∩ M) · [ M, M]⧹ M. By examining these structural formulas, they obtain fairly sharp estimates on the order of the distributions: if σ is associated with an orbit O ⊆ n ∗ and if V ⊆ n ∗ is the largest subspace which saturates θ in the sense that f ϵ O ⇒ f + V ⊆ O . As a corollary they obtain Richardson's criterion for a projection to map C 0( Γ→) into itself. The authors also resolve a conjecture of Brezin, proving a Zero-One law which says, among other things, that if the primary projection P σ maps C r ( Γ→) into C 0( Γ→), so do all irreducible projections P ⩽ P σ . This proof is based on a classical lemma on the extent to which integral points on a polynomial graph in R n lie in the coset ring of Z n (the finitely additive Boolean algebra generated by cosets of subgroups in Z n ). This lemma may be useful in other investigations of nilmanifolds.

Transport Processes in the Upper Atmosphere

Studies of transport mechanisms in the upper atmosphere (diffusion, turbulence, large-scale circulation) are reviewed. A normally turbulent state of the lower thermosphere is inferred from direct observations of tracers and from thermal and composition structure. Viscous dissipation in the thermosphere and energy dissipation in turbulence working against buoyancy are discussed in relation to a critical Richardson number for initiation of turbulence (0.25, as in the stratosphere). But the usefulness of the Richardson criterion is limited when static stability is low, as is the case when strong shear-generated turbulence prevails.

The ‘Richardson’ criterion for compressible swirling flows

The stability of a compressible non-dissipative swirling flow to adiabatic infinitesimal disturbances of arbitrary orientation is considered. The resulting sufficient condition for stability is the general form of the effective Richardson criterion for swirling flows, first obtained, for axisymmetric modes only, by Howard. In addition, upper bounds to the growth rate of unstable modes are obtained and some extensions of the semicircle theorem to azimuthal disturbances are stated.

Wave-Induced Instabilities in an Atmosphere Near Saturation

Abstract In this paper the stability and the propagation characteristics of internal gravity waves that propagate in an atmosphere near saturation and, over some height range, create the appropriate thermodynamic conditions for condensation to occur for a fraction of the wave cycle, are investigated. It is shown that if the atmosphere, over some height range, is close enough to saturation, a linear stability analysis is possible and results in a modified Richardson criterion, based on a new Brunt-Vaisala frequency nav smaller than the corresponding nu with condensation effects neglected. Since nav2 can become negative, even though the atmosphere is originally statically stable, the ability of gravity waves to trigger convective processes in a moist atmosphere is demonstrated. Finally, a numerical example is presented to illustrate the alterations of the characteristics of propagation.

Generalizations of the Richardson criterion for the onset of atmospheric turbulence

The energetics of air-parcel interchange are examined for the case in which the atmospheric potential temperature θ and background wind U are continuous but otherwise arbitrarily varying in space, and in which the direction of U is free from constraint. Interchange along a single s-axis, inclined at angle α to the vertical, is shown to be energetically feasible if where g is the gravitational acceleration. This condition, which is in general far less stringent than that imposed by the standard Richardson criterion, is suggested as a condition portending the onset of anisotropic turbulence. An isotropic interchange throughout a sphere is shown to be energetically feasible if where the z co-ordinate of the Cartesian x, y, z system is directed upward. This condition, which is in general somewhat less stringent than that imposed by the elementary Richardson criterion, may well portend the occurrence of isotropic turbulence. An outer time scale for the turbulence is estimated in each case, and its relevance to the occurrence of turbulence is discussed for the case when the background parameters of the system change with time.

Generalizations of the Richardson criterion for the onset of atmospheric turbulence

AbstractThe energetics of air‐parcel interchange are examined for the case in which the atmospheric potential temperature θ and background wind U are continuous but otherwise arbitrarily varying in space, and in which the direction of U is free from constraint. Interchange along a single s‐axis, inclined at angle α to the vertical, is shown to be energetically feasible if where g is the gravitational acceleration. This condition, which is in general far less stringent than that imposed by the standard Richardson criterion, is suggested as a condition portending the onset of anisotropic turbulence. An isotropic interchange throughout a sphere is shown to be energetically feasible if where the z co‐ordinate of the Cartesian x, y, z system is directed upward. This condition, which is in general somewhat less stringent than that imposed by the elementary Richardson criterion, may well portend the occurrence of isotropic turbulence. An outer time scale for the turbulence is estimated in each case, and its relevance to the occurrence of turbulence is discussed for the case when the background parameters of the system change with time.

The winter anomaly and related transport phenomena

The Richardson criterion ( R i ), which is a measure of the presence or absence of turbulence, has been calculated from measurements taken during and after a ‘winter anomaly’. The results indicate that during the anomaly, the atmosphere is fairly stable ( R i > 1) from 60 km up to about 78 km but between 78 and 90 km, in the same altitude range where the electron density is seen to be most enhanced, the atmosphere is turbulent as defined by the condition R i < 1. This contrasts with the normal day, where the condition for the existence of turbulence extends over the entire altitude range from 68 to 86 km.

Estrogen and progesterone receptor expression in breast carcinoma

Background: Breast carcinoma is the most common malignancy diagnosed among women worldwide and second leading cause of cancer mortality. One of the hallmarks of the disease is expression of estrogen receptor and progesterone receptor that ultimately drives prognosis and treatment modalities of the patient. The objective of this study was to determine the Estrogen and Progesterone receptor status in relation to histological grade of tumor. Materials and Methods: This was an observational study, carried out in the department of Pathology, BP Koirala Memorial cancer Hospital, Bharatpur, from January 2010 to December 2010. A total of 136 patients with histological proven diagnosis of breast carcinoma was included in this study. These cases were graded according to the modified Bloom and Richardson criteria into three histological grades. We used immunohistochemistry to evaluate the expression of Estrogen and Progesterone in relation to histological grade of tumor. Results: Out of 136 cases, there were 131 (96%) cases of infiltrating ductal carcinoma with mean age of 48 years. Majority of cases were grade II (59%) followed by grade III (21%) and grade I (20%). Estrogen Receptor and Progesterone Receptor expression were seen in 28% and 19% respectively. In grade I, 16 (59%) and 10 (37%) cases out of 27 were Estrogen Receptor and Progesterone Receptor positive respectively. In grade II, 21 (26%) and 15(19%) out of 80 cases were Estrogen Receptor and Progesterone Receptor positive respectively. In grade III, 1(3%) and 1 (3%) cases were positive for Estrogen Receptor and Progesterone Receptor respectively. Conclusions: Expression of estrogen and progesterone is comparable to west with Estrogen and Progesterone showing inverse association with histological grades of tumor. Keywords: Breast carcinoma; Estrogen receptor; Progesterone receptor DOI: http://dx.doi.org/10.3126/jpn.v1i2.5401 JPN 2011; 1(2): 100-103

Heat Convection from the Earth's Surface

EVIDENCE has been presented1 which suggests quite strongly that a regime of effectively free convection of heat begins to operate in the surface layers of the atmosphere at a negative Richardson number as low as 0.02 or 0.03. It is improbable that this can take the form of regular cellular convection of the type studied by Rayleigh and recently elaborated by Malkus2, for this would require in a layer of depth h that the symbols having their usual significance. If we replace x and v by their turbulent counterparts and average through the depth of the layer, then where α is less than but comparable with unity. This transforms the Rayleigh criterion into a Richardson criterion When there is some wind and wind shear, therefore, cellular convection should not set in until −Ri exceeds a value of order 1–10, whereas the free-convection regime in fact effectively sets in at a very much smaller value.

自由大氣中の擾亂の増減に封するRichardsonの判定條件について

自由大氣中の擾亂の増減に封するRichardsonの判定條件について

The criterion of turbulence in a fluid of variable density, with particular reference to conditions in the atmosphere

AbstractThe energy equation of eddying motion in a fluid of variable density is derived by methods based on a treatment by Ertel, with particular reference to conditions in the atmosphere. The general form of the criterion for the growth or decay of turbulence is then deduced. The limitations inherent in the Richardson criterion of turbulence are examined and it is shown that the latter neglects terms which are potentially large. These represent respectively, the rate of dissipation of eddying energy into heat by molecular viscosity, the rate of working of the fluctuating gradients of static pressure on the eddying motion, and the eddy diffusion of eddying kinetic energy. Richardson's criterion also involves the assumption of a special functional form for the vertical heat flux due to turbulence, which may not be generally valid. If it is, however, the critical value of the Richardson number above which turbulence is damped out is probably less than unity, in contradiction to Richardson's analysis, but in agreement with a theoretical analysis by Schlichting, which has been shown to be in satisfactory agreement with laboratory experiments.

On the application of the richardson criterion to large ‐ scale turbulence in the free atmosphere

A brief report on an investigation into the relationship between the wind shear and the static stability of the air in the layer immediately below the tropopause was presented to the Meteorological Research Committee in October 1942 (Swinbank, M.R.P. 51, unpublished). In the course of the ensuing discussion, members of the M.R.C. suggested that the investigation should be extended to cover the entire troposphere. This has been done particularly with a view to finding empirical relationships between stability and wind shear and comparing these with the results obtained by applying the Richardson criterion to actual observations. A further demand for investigating this problem arose in connection with an investigation of the turbulence conditions that obtain near the tropopause. In the present investigation use has been made of the reports from the British radio-sonde stations for the period December 1942 to March 1943. The accuracy of the D/F method of determining the wind in the free atmosphere depends to a large extent upon the geometry of each case, and a tolerance figure is allotted to each observation assessing the accuracy. Only reports of the highest degree of accuracy (i.e. vector errors less than 5 km/hour) have been used in this investigation. In all, 1,556 simultaneous observations of lapse rate and wind shear have been investigated. The observations have been grouped according to levels, and the results are illustrated in 12 diagrams in which lapse rate is the abscissa and wind shear the ordinate. The diagrams contain also the limiting curve between decrease and increase in turbulence as computed from Richardson's criterion. Since wind observations are made at intervals of 50 millibars, the diagrams represent the observed relationship between the mean wind shear and the mean lapse rate through layers of 50 mb. depth. The limiting curve in these diagrams would then represent the limit between the conditions that determine decrease and increase in the large-scale turbulence of the atmosphere. The diagrams show that, with very few exceptions, for any degree of stability the observed wind shear is less than that required by Richardson's theory to cause the turbulence to increase. If the assumptions underlying the theory were correct and applicable to large-scale turbulence, the diagrams would indicate that large-scale turbulence almost invariably tends to decrease. Since this result appears improbable, it is shown that agreement between theory and observation can be obtained by dropping one of the assumptions made by Richardson; namely, that the coefficient.

On the application of the richardson criterion to large‐scale turbulence in the free atmosphere

AbstractA brief report on an investigation into the relationship between the wind shear and the static stability of the air in the layer immediately below the tropopause was presented to the Meteorological Research Committee in October 1942 (Swinbank, M.R.P. 51, unpublished). In the course of the ensuing discussion, members of the M.R.C. suggested that the investigation should be extended to cover the entire troposphere. This has been done particularly with a view to finding empirical relationships between stability and wind shear and comparing these with the results obtained by applying the Richardson criterion to actual observations. A further demand for investigating this problem arose in connection with an investigation of the turbulence conditions that obtain near the tropopause.In the present investigation use has been made of the reports from the British radio‐sonde stations for the period December 1942 to March 1943. The accuracy of the D/F method of determining the wind in the free atmosphere depends to a large extent upon the geometry of each case, and a tolerance figure is allotted to each observation assessing the accuracy. Only reports of the highest degree of accuracy (i.e. vector errors less than 5 km/hour) have been used in this investigation. In all, 1,556 simultaneous observations of lapse rate and wind shear have been investigated.The observations have been grouped according to levels, and the results are illustrated in 12 diagrams in which lapse rate is the abscissa and wind shear the ordinate. The diagrams contain also the limiting curve between decrease and increase in turbulence as computed from Richardson's criterion. Since wind observations are made at intervals of 50 millibars, the diagrams represent the observed relationship between the mean wind shear and the mean lapse rate through layers of 50 mb. depth. The limiting curve in these diagrams would then represent the limit between the conditions that determine decrease and increase in the large‐scale turbulence of the atmosphere.The diagrams show that, with very few exceptions, for any degree of stability the observed wind shear is less than that required by Richardson's theory to cause the turbulence to increase.If the assumptions underlying the theory were correct and applicable to large‐scale turbulence, the diagrams would indicate that large‐scale turbulence almost invariably tends to decrease. Since this result appears improbable, it is shown that agreement between theory and observation can be obtained by dropping one of the assumptions made by Richardson; namely, that the coefficient.