Vladimir M. Gryanik

Third-Order Transport and Nonlocal Turbulence Closures for Convective Boundary Layers*

The turbulence closure problem for convective boundary layers is considered with the chief aim to advance the understanding and modeling of nonlocal transport due to large-scale semiorganized structures. The key role here is played by third-order moments (fluxes of fluxes). The problem is treated by the example of the vertical turbulent flux of potential temperature. An overview is given of various schemes ranging from comparatively simple countergradient-transport formulations to sophisticated turbulence closures based on budget equations for the second-order moments. As an alternative to conventional ‘‘turbulent diffusion parameterization’’ for the flux of flux of potential temperature, a ‘‘turbulent advection plus diffusion parameterization’’ is developed and diagnostically tested against data from a large eddy simulation. Employing this parameterization, the budget equation for the potential temperature flux provides a nonlocal turbulence closure formulation for the flux in question. The solution to this equation in terms of the Green function is nothing but an integral turbulence closure. In particular cases it reduces to closure schemes proposed earlier, for example, the Deardorff countergradient correction closure, the Wyngaard and Weil transport-asymmetry closure employing the second derivative of transported scalar, and the Berkowicz and Prahm integral closure for passive scalars. Moreover, the proposed Green-function solution provides a mathematically rigorous procedure for the Wyngaard decomposition of turbulence statistics into the bottom-up and top-down components. The Green-function decomposition exhibits nonlinear vertical profiles of the bottom-up and top-down components of the potential temperature flux in sharp contrast to universally adopted linear profiles. For modeling applications, the proposed closure should be equipped with recommendations as to how to specify the temperature and vertical velocity variances and the vertical velocity skewness.

A Turbulence Closure for the Convective Boundary Layer Based on a Two-Scale Mass-Flux Approach

The closure problem for the convective turbulence of the shear-free and low to moderate wind atmospheric boundary layer is considered. Non-Gaussian parameterizations are developed for fourth-order moments based on a two-scale mass-flux approach. With this approach the ballistic stirring of fluid by coherent structures is taken into account and the differences in the horizontal scales and spacing of the velocity and temperature fields are recognized. The fractional coverage of positive temperature variations is introduced, as well as the fractional coverage of positive vertical velocity fluctuations. The parameterizations are compared to those of the traditional mass-flux scheme and of the classical eddy-damped quasi-normal approach, and the principal similarities and dissimilarities are outlined. The results of testing the parameterizations against aircraft measurements at moderate wind and against large eddy simulation data of free convective conditions show good agreement between model predictions and data.

A Refinement of the Millionshchikov Quasi-Normality Hypothesis for Convective Boundary Layer Turbulence

Abstract The Millionshchikov hypothesis of quasi-normal distribution of fourth-order moments fails for convective conditions where the probability density functions of temperature and vertical velocity fluctuations are skewed. This is shown for aircraft and large-eddy simulation (LES) data, and new closures for fourth-order moments that take the skewness into account are suggested. These new closures are in very good agreement with the data.

Barotropic Beta-Plane Turbulence in a Regime with Strong Zonal Jets Revisited

Abstract The problem of quantification of barotropic beta-plane turbulence driven by small-scale stochastic forcing into regimes dominated by quasi-periodic zonal jets is revisited. It is shown that the large-scale relative vorticity in such regimes is organized into a sequence of zonal bands. Its zonal mean profile varies approximately linearly within the bands. Its mean negative slope β∗ is less than the meridional gradient of the Coriolis parameter β, and depends on the external parameters (friction, forcing, and β). The neighboring bands are connected through the vorticity fronts where the zonal mean meridional gradient is large and positive. The frontal-band vorticity structure defines piecewise parabolic profiles of asymmetric eastward and westward jets, and strong peaks in the low-k interval of turbulent zonal energy spectra, which store most of the zonal energy. The slope of their envelope depends on the structure of the frontal zones and is always steeper than −4. The presence of peaks invalidate...

Vortex intensification and collapse of the Lissajous-elliptic ring: single-and multi-filament Biot-Savart simulations and visiometrics

The collapsing ‘Lissajous-elliptic’ (LE) vortex ring is examined via quantifications of Single- and multi-filament Biot-Savart numerical simulations. In the single-filament simulations, parametric studies show simple relationships between the collapse boundary and the impulse and energy invariants. Collapse becomes non-monotonic in time, for a sufficiently small initial core ‘radius’. Self-similar, singular-like behaviour of the off-filament strain-rate growth has been observed in a small interval, just prior to core overlapping. The computation of the strain-rate eigenvalues and vortex stretching in a diagnostics box surrounding the collapse region yields patterns observed previously in continuum simulations. New diagnostics are presented, including line densities of the energy and the linear and angular momentum, all of which approach zero in the collapse region of the ring. These diagnostics may provide critical parameters for initiating surgery in a topology-changing algorithm. Our multi-filament simulations exhibit layer-like vortex regions and a ‘torus’-shaped vortex stretching pattern observed previously in continuum periodic-domain simulations of vortex reconnection. Quantifications in a cross-section of the collapse region indicate that the circulation tends to concentrate in the head or frontside of the convecting dipolar structure. This is also the location of the incipient ‘bridge’ which is evolving from the weak filaments that have been convected from the initially outer-vortex regions. The formation of this smaller scale vortex structure exhibits the largest vorticity amplification in the variable-core model simulations.

A stability‐dependent parametrization of transfer coefficients for momentum and heat over polar sea ice to be used in climate models

The interaction between sea ice and atmosphere depends strongly on the near-surface transfer coefficients for momentum and heat. A parametrization of these coefficients is developed on the basis of an existing parametrization of drag coefficients for neutral stratification that accounts for form drag caused by the edges of ice floes and melt ponds. This scheme is extended to better account for the dependence of surface wind on limiting cases of high and low ice concentration and to include near-surface stability effects over open water and ice on form drag. The stability correction is formulated on the basis of stability functions from Monin-Obukhov similarity theory and also using the Louis concept with stability functions depending on the bulk Richardson numbers. Furthermore, a parametrization is proposed that includes the effect of edge-related turbulence also on heat transfer coefficients. The parametrizations are available in different levels of complexity. The lowest level only needs sea ice concentration and surface temperature as input, while the more complex level needs additional sea ice characteristics. An important property of our parametrization is that form drag caused by ice edges depends on the stability over both ice and water which is in contrast to the skin drag over ice. Results of the parametrization show that stability has a large impact on form drag and, thereby, determines the value of sea ice concentration for which the transfer coefficients reach their maxima. Depending on the stratification, these maxima can occur anywhere between ice concentrations of 20 and 80%.

The theory of three-dimensional hetons and vortex-dominated spreading in localized turbulent convection in a fast rotating stratified fluid

The problem of lateral heat/buoyancy transport in localized turbulent convection dominated by rotation in continuously stratified fluids of finite depth is considered. We investigate the specific mechanism of the vortex-dominated lateral spreading of anomalous buoyancy created in localized convective regions owing to outward propagation of intense heton-like vortices (pairs of vortices of equal potential vorticity (PV) strength with opposite signs located at different depths), each carrying a portion of buoyancy anomaly. Assuming that the quasi-geostrophic form of the PV evolution equation can be used to analyse the spreading phenomenon at fast rotation, we develop an analytical theory for the dynamics of a population of three-dimensional hetons. We analyse in detail the structure and dynamics of a single three-dimensional heton, and the mutual interaction between two hetons and show that the vortices can be in confinement, splitting or reconnection regimes of motion depending on the initial distance between them and the ratio of the mixing-layer depth to the depth of fluid (local to bulk Rossby radii). Numerical experiments are made for ring-like populations of randomly distributed three-dimensional hetons. We found two basic types of evolution of the populations which are homogenizing confinement (all vortices are predominantly inside the localized region having highly correlated wavelike dynamics) and vortex-dominated spreading (vortices propagate out of the region of generation as individual hetons or heton clusters). For the vortex-dominated spreading, the mean radius of heton populations and its variance grow linearly with time. The law of spreading is quantified in terms of both internal (specific for vortex dynamics) and external (specific for convection) parameters. The spreading rate is proportional to the mean speed of propagation of individual hetons or heton clusters and therefore depends essentially on the strength of hetons and the ratio of local to bulk Rossby radii. A theoretical explanation for the spreading law is given in terms of the asymptotic dynamics of a single heton and within the frames of the kinetic equation derived for the distribution function of hetons in collisionless approximation. This spreading law gives an upper ‘advective’ bound for the superdiffusion of heat/buoyancy. A linear law of spreading implies that diffusion parameterizations of lateral buoyancy flux in non-eddy-resolving models are questionable, at least when the spreading is dominated by heton dynamics. We suggest a scaling for the ‘advective’ parameterization of the buoyancy flux, and quantify the exchange coefficient in terms of the mean propagation speed of hetons. Finally, we discuss the perspectives of the heton theories in other problems of geophysical fluid dynamics.

The theory of quasi-geostrophic von Kármán vortex streets in two-layer fluids on a beta-plane

In this study self-organized periodic coherent vortex structures arising in geophysical turbulent flows at low Rossby number are investigated by developing a conceptual model based on an analytical theory of von Karman vortex streets affected by stratification and differential rotation. In the framework of a quasi-geostrophic (QG) two-layer beta-plane model vortex streets with three different types of vertical structures (barotropic, upper layer and hetonic) are analysed using the point vortex approximation. The streets are found to be exact solutions of the potential vorticity equation and to be characterized by four non-dimensional parameters. Von Karman streets are semi-localized solutions which form a bridge between vortex pairs (limit of symmetric dilute streets) and two parallel vortex sheets (limit of dense streets). On the beta-plane QG von Karman streets can only move to the east, i.e. with a speed outside the range of speeds of Rossby waves, so that a dynamical asymmetry in the zonal direction is introduced. A complete classification on a diagram of states shows that critical bounds exist in the parameter space, prescribing for example a maximum distance between vortex rows beyond which no QG vortex streets can be found. Typically a fast and a slow vortex street with different flow structures are found in the region of existence. As a function of distance between vortex rows baroclinic QG vortex streets show a characteristic non-monotonic speed behaviour at scales of the order of the baroclinic Rossby radius. A wide region of possible existence of QG von Karman streets is found in atmospheric, oceanic and planetary conditions as well as in rotating tank experiments. The theory can be applied to describe the coherent part of turbulent baroclinic intermittent zonal jet-like and frontal flows and provides a scaling for such flows.

Mesoscale modelling of the Arctic atmospheric boundary layer and its interaction with sea ice

This chapter summarises mesoscale modelling studies, which were carried out during the ACSYS decade until 2005. They were aiming at the parameterisation and improved understanding of processes in the Arctic boundary layer over the open ocean and marginal sea ice zones and over the Greenland ice sheet. It is shown that progress has been achieved with the parameterization of fluxes in strong convective situations such as cold-air outbreaks and convection over leads. A first step was made towards the parameterization of the lead-induced turbulence for high-resolution, but non-eddy resolving models. Progress has also been made with the parameterization of the near-surface atmospheric fluxes of energy and momentum modified by sea ice pressure ridges and by ice floe edges. Other studies brought new insight into the complex processes influencing sea ice transport and atmospheric stability over sea ice. Improved understanding was obtained on the cloud effects on the snow/ice surface temperature and further on the near-surface turbulent fluxes. Finally, open questions are addressed, which remained after the ACSYS decade for future programmes having been started in the years after 2005.

Near‐singular collapse and local intensification of a ‘‘Lissajous‐elliptic’’ vortex ring: Nonmonotonic behavior and zero‐approaching local energy densities

This Letter observes collapse and intensification of the two parameter (b,c) ‘‘Lissajous‐elliptic’’ vortex ring. Laboratory and direct numerical studies of this ring are proposed to elucidate near‐singular and intermittent fluid phenomena at very high Reynolds number. Quantifications of single filament Biot–Savart numerical simulations with various core ‘‘radii’’ show that collapse may be nonmonotonic in time. In the collapsing region, the largest positive strain‐rate eigenvalue, α, is off the filament and exhibits self‐similar, singular‐like behavior. A signature of collapse is found in the local approach to zero of the filament energy density in the collapsing regions.