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Peter Bartello

Abstract

A set of one-dimensional turbulence models can be constructed by applying severe D-1 directional Fourier truncation to the D-dimensional fluid equations, allowing for numerical calculation over an extremely wide range of scales. At low resolution a reduced 2D model displayed both inviscid energy-enstrophy equipartition and a spectrum consistent with the enstrophy cascade phenomenology in the decay problem.

In the present note, these results are extended by using reduced models to examine both the inverse (D = 2) and direct (D = 3) energy cascades. In addition, higher numerical resolution (up to 4096 grid points) is employed to demonstrate unequivocal adherence to the 2D phenomenologies. Small-scale intermittency in the form of spatially intermittent vorticity gradients is also observed. Simulations based on the reduced 3D model were less successful. Although the truncated inviscid equilibrium agreed with an energy equipratition spectrum, forced-viscous simulations failed to show a clear Kolmogorov range. It is argued that the technique, although not justified for 3D problems, produces an interesting 1D turbulence model when applied to 2D flow.

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Peter Bartello

Abstract

Rotating stratified turbulence is examined both numerically and analytically, guided by energy and potential enstrophy conservation as well as resonant interaction theory, in order to investigate the cascade properties of rotational and wave modes at Froude numbers of order one or below, over a range of Rossby numbers. As Ro → 0, rotational modes are only weakly coupled to wave modes, and there are only weak rotational wave energy exchanges when initial conditions are random. A catalytic interaction involving two waves and a rotational mode, leaving the rotational mode unchanged, then provides the mechanism for geostrophic adjustment via a downscale cascade of wave energy. When simulations are initially balanced, gravity modes act to damp large-scale rotational modes through a transfer into intermediate-scale gravity modes and a subsequent downscale wave cascade involving the catalytic interaction. At larger Ro transfer from rotational to wave modes is important at any Froude number, and geostrophic adjustment may not take place. The consequences in terms of the proposed inverse cascade of rotational energy from convective scales are discussed.

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William Sacher and Peter Bartello

Abstract

In the current study, the authors are concerned with the comparison of the average performance of stochastic versions of the ensemble Kalman filter with and without covariance inflation, as well as the double ensemble Kalman filter. The theoretical results obtained in Part I of this study are confronted with idealized simulations performed with a perfect barotropic quasigeostrophic model. Results obtained are very consistent with the analytic expressions found in Part I. It is also shown that both the double ensemble Kalman filter and covariance inflation techniques can avoid filter divergence. Nevertheless, covariance inflation gives efficient results in terms of accuracy and reliability for a much lower computational cost than the double ensemble Kalman filter and for smaller ensemble sizes.

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William Sacher and Peter Bartello

Abstract

This paper discusses the quality of the analysis given by the ensemble Kalman filter in a perfect model context when ensemble sizes are limited. The overall goal is to improve the theoretical understanding of the problem of systematic errors in the analysis variance due to the limited size of the ensemble, as well as the potential of the so-called double-ensemble Kalman filter, covariance inflation, and randomly perturbed analysis techniques to produce a stable analysis—that is to say, one not subject to filter divergence. This is achieved by expressing the error of the ensemble mean and the analysis error covariance matrix in terms of the sampling noise in the background error covariance matrix (owing to the finite ensemble estimation) and by comparing these errors for all methods. Theoretical predictions are confirmed with a simple scalar test case. In light of the analytical results obtained, the expression of the optimal covariance inflation factor is proposed in terms of the limited ensemble size and the Kalman gain.

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Peter Bartello and Stephen J. Thomas

Abstract

The purpose of this paper is to ascertain the cost-effectiveness of semi-Lagrangian advection schemes for a wide variety of geophysical flows at all scales. The approach used is first to determine the minimum computational overhead associated with these schemes and then to examine temporal variability in the Lagrangian and Eulerian frames by employing simple turbulent cascade phenomenologies. The goal is to evaluate whether the Lagrangian variability is sufficiently slower than that of the Eulerian frame to overcome the computational overhead. It is found that the most efficient semi-Lagrangian schemes require a factor of 5–10 times more floating point operations per grid point per time step than the classic second-order leapfrog scheme.

In the enstrophy cascade of 2D or quasigeostrophic turbulence, evolution of flow quantities is considerably slower in the Lagrangian frame and semi-Lagrangian advection schemes can be very cost-effective. In an energy cascade such as the Kolmogorov range of 3D turbulence or the inverse cascade of QG or 2D turbulence, the Lagrangian evolution remains slower than the Eulerian evolution. However, the difference is very much less than in the enstrophy cascade. Since the computational overhead of semi-Lagrangian schemes is considerable, they are at best marginally cost-effective at current resolutions for these flows, which prevail in the atmosphere at scales below 300–400 km. In the presence of stationary forcing fields in the Eulerian frame, the time step must respect the advective timescale even in the Lagrangian frame, at length scales where the forcing is significant. Here semi-Lagrangian schemes are not recommended.

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Peter Bartello, Olivier Métais, and Marcel Lesieur

Abstract

It is well established that at low Rossby and Froude numbers modes possessing potential vorticity behave differently from gavity-inertial wave modes. Wave energy cascades relatively more efficiently downscale to the dissipation, resulting in a geostrophic adjustment. For this reason, it has been suggested that wave energy be subjected to relatively stronger dissipation via “divergence damping.” This study reports separate measurements of the effective eddy damping acting on wave and rotational modes in simulations of nonhydrostatic Boussinesq flow. The method employs an arbitrary cutoff wavenumber, kc, within the simulation’s range of resolved scales, in order to calculate explicitly the effect of the smaller-scale motion on wavenumbers below kc. It is found that the rotational-mode eddy viscosity resembles that found in studies of 2D turbulence, with a significant negative range, while it is positive at all wavenumbers for the wave modes.

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Olivier Asselin, Peter Bartello, and David N. Straub

Abstract

The near-tropopause energy spectrum closely follows a −5/3 power law at mesoscales. Most theories addressing the mesoscale spectrum assume unbalanced dynamics but ignore the tropopause (near which the bulk of the data were collected). Conversely, it has also been proposed that the mesoscale spectrum results from tropopause-induced alterations of geostrophic turbulence. This paper seeks to reconcile these a priori mutually exclusive theories by presenting simulations that permit both unbalanced motion and tropopause-induced effects. The model integrates the nonhydrostatic Boussinesq equations in the presence of a rapidly varying background stratification profile (an idealized tropopause). Decaying turbulence simulations were performed over a wide range of Rossby numbers. In the limit of weak flow (U ≲ 1 m s−1), the essential features of the Boussinesq simulations are well captured by a quasigeostrophic version of the model: secondary roll-ups of filaments and shallow spectral slopes are observed near the tropopause but not elsewhere. However, these tropopause-induced effects rapidly disappear with increasing flow strength. For flow strengths more typical of the tropopause (U ~ 10 m s−1), the spectrum develops a shallow, near −5/3 tail associated with fast-time-scale, unbalanced motion. In contrast to weak flows, this spectral shallowing is evident at any altitude and regardless of the presence of a tropopause. Diagnostics of the fast component of motion reveal significant inertia–gravity wave activity at large horizontal scales (where the balanced flow dominates). However, no evidence points to such activity in the shallow range. That is, the mesoscale of the model is dominated by unbalanced turbulence, not waves. Implications and limitations of these findings are discussed.

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Martin Charron, P. L. Houtekamer, and Peter Bartello

Abstract

The ensemble Kalman filter (EnKF) developed at the Meteorological Research Branch of Canada is used in the context of synthetic radial wind data assimilation at the mesoscale. A dry Boussinesq model with periodic boundary conditions is employed to provide a control run, as well as two ensembles of first guesses. Synthetic data, which are interpolated from the control run, are assimilated and simulate Doppler radar wind measurements.

Nine “radars” with a range of 120 km are placed evenly on the horizontal 1000 km × 1000 km domain. These radars measure the radial wind with assumed Gaussian error statistics at each grid point within their range provided that there is sufficient upward motion (a proxy for precipitation). These data of radial winds are assimilated every 30 min and the assimilation period extends over 4 days.

Results show that the EnKF technique with 2 × 50 members performed well in terms of reducing the analysis error for horizontal winds and temperature (even though temperature is not an observed variable) over a period of 4 days. However the analyzed vertical velocity shows an initial degradation. During the first 2 days of the assimilation period, the analysis error of the vertical velocity is greater when assimilating radar observations than when scoring forecasts initialized at t = 0 without assimilating any data. The type of assimilated data as well as the localization of the impact of the observations is thought to be the cause of this degradation of the analyzed vertical velocity. External gravity modes are present in the increments when localization is performed. This degradation can be eliminated by filtering the external gravity modes of the analysis increments.

A similar set of experiments is realized in which the model dissipation coefficient is reduced by a factor of 10. This shows the level of sensitivity of the results to the kinetic energy power spectrum, and that the quality of the analyzed vertical wind is worse when dissipation is small.

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Sisi Chen, M. K. Yau, and Peter Bartello

Abstract

This paper aims to investigate and quantify the turbulence effect on droplet collision efficiency and explore the broadening mechanism of the droplet size distribution (DSD) in cumulus clouds. The sophisticated model employed in this study individually traces droplet motions affected by gravity, droplet disturbance flows, and turbulence in a Lagrangian frame. Direct numerical simulation (DNS) techniques are implemented to resolve the small-scale turbulence. Collision statistics for cloud droplets of radii between 5 and 25 μm at five different turbulence dissipation rates (20–500 cm2 s−3) are computed and compared with pure-gravity cases. The results show that the turbulence enhancement of collision efficiency highly depends on the r ratio (defined as the radius ratio of collected and collector droplets r/R) but is less sensitive to the size of the collector droplet investigated in this study. Particularly, the enhancement is strongest among comparable-sized collisions, indicating that turbulence can significantly broaden the narrow DSD resulting from condensational growth. Finally, DNS experiments of droplet growth by collision–coalescence in turbulence are performed for the first time in the literature to further illustrate this hypothesis and to monitor the appearance of drizzle in the early rain-formation stage. By comparing the resulting DSDs at different turbulence intensities, it is found that broadening is most pronounced when turbulence is strongest and similar-sized collisions account for 21%–24% of total collisions in turbulent cases compared with only 9% in the gravitational case.

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Charmaine N. Franklin, Paul A. Vaillancourt, M. K. Yau, and Peter Bartello

Abstract

Direct numerical simulations of an evolving turbulent flow field have been performed to explore how turbulence affects the motion and collisions of cloud droplets. Large numbers of droplets are tracked through the flow field and their positions, velocities, and collision rates have been found to depend on the eddy dissipation rate of turbulent kinetic energy. The radial distribution function, which is a measure of the preferential concentration of droplets, increases with eddy dissipation rate. When droplets are clustered there is an increased probability of finding two droplets closely separated; thus, there is an increase in the collision kernel. For the flow fields explored in this study, the clustering effect accounts for an increase in the collision kernel of 8%–42%, as compared to the gravitational collision kernel. The spherical collision kernel is also a function of the radial relative velocities among droplets and these velocities increase from 1.008 to 1.488 times the corresponding gravitational value. For an eddy dissipation rate of about 100 cm2 s−3, the turbulent collision kernel is 1.06 times the magnitude of the gravitational value, while for an eddy dissipation rate of 1500 cm2 s−3, this increases to 2.08 times. Therefore, these results demonstrate that turbulence could play an important role in the broadening and evolution of the droplet size distribution and the onset of precipitation.

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