Renormalization group analysis of turbulence

The objective is to understand and extend a recent theory of turbulence based on dynamic renormalization group (RNG) techniques. The application of RNG methods to hydrodynamic turbulence was explored most extensively by Yakhot and Orszag (1986). An eddy viscosity was calculated which was consistent with the Kolmogorov inertial range by systematic elimination of the small scales in the flow. Further, assumed smallness of the nonlinear terms in the redefined equations for the large scales results in predictions for important flow constants such as the Kolmogorov constant. It is emphasized that no adjustable parameters are needed. The parameterization of the small scales in a self-consistent manner has important implications for sub-grid modeling

Local and Nonlocal Dispersive Turbulence

We consider the evolution of a family of 2D dispersive turbulence models. The members of this family involve the nonlinear advection of a dynamically active scalar field, the locality of the streamfunction-scalar relation is denoted by $\alpha$, with smaller $\alpha$ implying increased locality. The dispersive nature arises via a linear term whose strength is characterized by a parameter $\epsilon$. Setting $0 < \epsilon \le 1$, we investigate the interplay of advection and dispersion for differing degrees of locality. Specifically, we study the forward (inverse) transfer of enstrophy (energy) under large-scale (small-scale) random forcing. Straightforward arguments suggest that for small $\alpha$ the scalar field should consist of progressively larger eddies, while for large $\alpha$ the scalar field is expected to have a filamentary structure resulting from a stretch and fold mechanism. Confirming this, we proceed to forced/dissipative dispersive numerical experiments under weakly non-local to local conditions. For $\epsilon \sim 1$, there is quantitative agreement between non-dispersive estimates and observed slopes in the inverse energy transfer regime. On the other hand, forward enstrophy transfer regime always yields slopes that are significantly steeper than the corresponding non-dispersive estimate. Additional simulations show the scaling in the inverse regime to be sensitive to the strength of the dispersive term : specifically, as $\epsilon$ decreases, the inertial-range shortens and we also observe that the slope of the power-law decreases. On the other hand, for the same range of $\epsilon$ values, the forward regime scaling is fairly universal.Comment: 19 pages, 8 figures. Significantly revised with additional result

On the Two-point Correlation of Potential Vorticity in Rotating and Stratified Turbulence

A framework is developed to describe the two-point statistics of potential vorticity in rotating and stratified turbulence as described by the Boussinesq equations. The Karman-Howarth equation for the dynamics of the two-point correlation function of potential vorticity reveals the possibility of inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and Reynolds number parameters. For the case of large Rossby and Froude numbers, and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3 prefactor is derived for the third-order mixed correlation between potential vorticity and velocity, a result that is analogous to the Kolmogorov 4/5-law for the third-order velocity structure function in turbulence theory.Comment: 10 pages, to appear in Journal of Fluid Mechanics (2006

Two-dimensional Moist Stratified Turbulence and the Emergence of Vertically Sheared Horizontal Flows

Moist stratified turbulence is studied in a two-dimensional Boussinesq system influenced by condensation and evaporation. The problem is set in a periodic domain and employs simple evaporation and condensation schemes, wherein both the processes push parcels towards saturation. Numerical simulations demonstrate the emergence of a moist turbulent state consisting of ordered structures with a clear power-law type spectral scaling from initially spatially uncorrelated conditions. An asymptotic analysis in the limit of rapid condensation and strong stratification shows that, for initial conditions with enough water substance to saturate the domain, the equations support a straightforward state of moist balance characterized by a hydrostatic, saturated, vertically sheared horizontal flow (VSHF). For such initial conditions, by means of long time numerical simulations, the emergence of moist balance is verified. Specifically, starting from uncorrelated data, subsequent to the development of a moist turbulent state, the system experiences a rather abrupt transition to a regime which is close to saturation and dominated by a strong VSHF. On the other hand, initial conditions which do not have enough water substance to saturate the domain, do not attain moist balance. Rather, the system remains in a turbulent state and oscillates about moist balance. Even though balance is not achieved with these general initial conditions, the time scale of oscillation about moist balance is much larger than the imposed time scale of condensation and evaporation, thus indicating a distinct dominant slow component in the moist stratified two-dimensional turbulent system.Comment: 23 pages. 9 figure

Growth or decline in the Church of England during the decade of Evangelism: did the Churchmanship of the Bishop matter?

The Decade of Evangelism occupied the attention of the Church of England throughout the 1990s. The present study employs the statistics routinely published by the Church of England in order to assess two matters: the extent to which these statistics suggest that the 43 individual dioceses finished the decade in a stronger or weaker position than they had entered it and the extent to which, according to these statistics, the performance of dioceses led by bishops shaped in the Evangelical tradition differed from the performance of dioceses led by bishops shaped in the Catholic tradition. The data demonstrated that the majority of dioceses were performing less effectively at the end of the decade than at the beginning, in terms of a range of membership statistics, and that the rate of decline varied considerably from one diocese to another. The only exception to the trend was provided by the diocese of London, which experienced some growth. The data also demonstrated that little depended on the churchmanship of the diocesan bishop in shaping diocesan outcomes on the performance indicators employed in the study

Progress report on the geology of 1:50k sheet 64W (Newtonmore)

This report describes the results of solid geology fieldwork in 1:50 000 sheet 64W (Newtonmore) resulting from the 2002 summer mapping season. A full revision of the solid geology at 1:10 000 scale was completed in the north-western part of the sheet (Sheet NN69SE) while rapid mapping/reconnaissance of the solid geology has been carried out in the remaining 85% of the sheet area The superficial geology of the sheet has been completely revised and will be described in a separate report. The north-western part of the sheet contains the transition from the deeper water graded sandy and silty turbidite deposits of the Corrieyairack Subgroup upwards (and south-eastwards) into the shallow water sand-dominated deposits of the Strathtummel Subgroup. East of the A9 trunk road, the Gaick region is confirmed as a single lithostratigraphical package in the Strathtummel Subgroup recording shallow water depositional conditions, greatly thickened by D2 recumbent folding. Axial surfaces of these folds dip gently east overall with gently east plunging to subhorizontal fold axes. Axial traces are generally N-S trending. The main regional (biotite) schistosity is axial planar to these folds and locally, can be seen clearly deforming an earlier bedding near-parallel biotite fabric. The available evidence for stratigraphical younging is limited to a few well-washed river sections but shows that regional facing is always to the south in S2 across the Gaick region. No large-scale F1 folds are recognised with the exception of those at Crubenmore on the A9. Minor undulations of the main regional fabric mean that the sheet dip varies between gently north to gently east across open upright north-east-plunging folds, in marked contrast to the conspicuous pattern of reclined, north-west verging D3 folds deforming the main regional (S2) schistosity in Glen Truim and farther north-west. There appear to be no other major fold sets across this part of the Gaick region The Drummochter Dome thus takes the form of a stack of recumbent D2 folds, modified by steep zones to the north-west (Geal CharnOssian Steep Belt) and south-east (Tummel Steep belt and correlatives). The pattern of early recumbent folds and later steep belts is similar to that seen in the higher structural levels south of the Boundary Slide which include the Tay Nappe