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M. K. MacVean

Abstract

An investigation of cloud-top entrainment instability (CTEI) has been carried out using a fine-resolution two-dimensional numerical model. Initial conditions having specified values of R = cpΔθe/LΔqt were used. Here, Δθe, and Δqt, are the jumps in equivalent potential temperature and total water mixing ratio across cloud top. In order to isolate the effects of entrainment across cloud top, cloud microphysics and surface fluxes were excluded from all the integrations. Radiative processes were generally also excluded, although a number of runs with longwave radiative cooling were performed. Integrations were carried out for specified values of R, using various subgrid models, including several constant values of eddy viscosity. Because the crucial process underlying CTEI is small-scale mixing, which must be parameterized in this model, only those results that are not critically dependent on the precise form of the subgrid model are likely to have any general validity. Fortunately, significant conclusions can still be drawn from the study. At values of R greater than the critical value of about 0.7 recently derived by MacVean and Mason, the cloud layer breaks up and evaporates completely within 1–2 h. On the other hand, for values of R greater than and close to the critical value of about 0.23 derived by earlier authors, no tendency for rapid dissipation of the cloud is observed. The results from the integrations that included longwave cooling at cloud top suggest that the inclusion of this process does not fundamentally modify these conclusions. Furthermore, analysis suggests that the entrainment rate in the simulations is likely to be realistic. It is concluded that CTEI may be an important mechanism governing the rapid dissipation of stratocumulus, although only at much larger values of R than earlier theoretical work had suggested. This conclusion is shown to be consistent with most of the limited, available observational data. These simulations provide strong support for the CTEI criterion proposed by MacVean and Mason. This support is, however, not unequivocal because the model is two-dimensional and, of necessity, employs a subgrid parameterization and because of the continuously increasing potential source of turbulent kinetic energy from buoyancy reversal above R = 0.23.

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M. K. Macvean

Abstract

Results from an integration of a hemispherics spectral model, starting from a Northern Hemisphere climatological zonal mean state plus a small amplitude white noise perturbation are presented. Although there is no external forcing in this model, upper tropospheric, midlatitude long-wave fields of amplitude comparable to those observed in the atmosphere are produced. Two distinct generation mechanisms which are important in consecutive phases of the integration are investigated. The first type of long-wave growth is due to the interaction of the synoptic-scale systems, which vary in amplitude with longitude, with the “mean” flow in which they are embedded. The second long-wave development, which takes the form of an intensifying jet stream extending over 60° of longitude, can be interpreted in terms of linear instability of the zonal-mean flow involving one normal mode for each of zonal wavenumbers 1, 2 and 3. These modes are slowly growing and slow-moving, their superposition yields a growing disturbance which is quasi-stationary, localized in space and coherent in time. It is suggested that such modes might be preferentially excited in the presence of realistic, large-scale orographic or diabatic forcing.

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M. K. MacVean and P. J. Mason

Abstract

In a recent paper, Kuo and Schubert demonstrated the lack of observational support for the relevance of the criterion for cloud-top entrainment instability proposed by Randall and by Deardorff. Here we derive a new criterion, based on a model of the instability as resulting from the energy released close to cloud top, by Mixing between saturated boundary-layer air and unsaturated air from above the capping inversion. The condition is derived by considering the net conversion from potential to kinetic energy in a system consisting of two layers of fluid straddling cloud-top, when a small amount of mixing occurs between these layers. This contrasts with previous analyses, which only considered the change in buoyancy of the cloud layer when unsaturated air is mixed into it. In its most general form, this new criterion depends on the ratio of the depths of the layers involved in the mixing. It is argued that, for a self-sustaining instability, there must be a net release of kinetic energy on the same depth and time scales as the entrainment process itself. There are two plausible ways in which this requirement may be satisfied. Either one takes the depths of the layers involved in the mixing to each be comparable to the vertical scale of the entrainment process, which is typically of order tens of meters or less, or alternatively, one must allow for the efficiency with which energy released by mixing through a much deeper lower layer becomes available to initiate further entrainment. In both cases the same criterion for instability results. This criterion is much more restrictive than that proposed by Randall and by Deardorff; furthermore, the observational data is then consistent with the predictions of the current theory.

Further analysis provides estimates of the turbulent fluxes associated with cloud-top entrainment instability. This analysis effectively constitutes an energetically consistent turbulence closure for models of boundary layers with cloud. The implications for such numerical models are discussed. Comparisons are also made with other possible criteria for cloud-top entrainment instability which have recently been suggested.

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M. K. MacVean and I. N. James

Abstract

Lifecycles of baroclinic waves obtained by integrating the primitive and quasi-geostrophic equations on the sphere are compared. Two basic states are considered. The first is based on Northern Hemisphere winter climatology, while the other has a maximum baroclinicity at midlevels, but no temperature gradients on the upper and lower boundaries. The earlier part of the lifecycles are broadly similar for both equation sets, although there are significant quantitative differences. In the later stages, the momentum fluxes of the decaying waves can differ greatly. Moreover, we are unable to identify any systematic pattern in these differences. We conclude that the use of quasi-geostrophic results to parameterize the modification of the mean flow by baroclinic wave activity is likely to be subject to large errors.

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B. P. Leonard, A. P. Lock, and M. K. MacVean

Abstract

The multidimensional advection schemes described in this study are based on a strictly conservative flux-based control-volume formulation. They use an explicit forward-in-time update over a single time step, but there are no “stability” restrictions on the time step. Genuinely multidimensional forward-in-time advection schemes require an estimate of transverse contributions to each face-normal flux for stability. Traditional operator-splitting techniques based on sequential one-dimensional updates introduce such transverse cross-coupling automatically; however, they have serious shortcomings. For example, conservative-form operator splitting is indeed globally conservative but introduces a serious “splitting error”; in particular, a constant is not preserved in general solenoidal velocity fields. By contrast, advective-form operator splitting is constancy preserving but not conservative. However, by using advective-form estimates for the transverse contributions together with an overall conservative-form update, strictly conservative constancy-preserving schemes can be constructed. The new methods have the unrestricted-time-step advantages of semi-Lagrangian schemes, but with the important additional attribute of strict conservation due to their flux-based formulation. Shape-preserving techniques developed for small time steps can be incorporated. For large time steps, results are not strictly shape preserving but, in practice, deviations appear to be very slight so that overall behavior is essentially shape preserving. Since only one-dimensional flux calculations are required at each step of the computation, the algorithms described here should be highly compatible with existing advection codes based on conventional operator-splitting methods. Capabilities of the new schemes are demonstrated using the well-known scalar advection test problem devised by Smolarkiewicz.

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A. R. Brown, M. K. MacVean, and P. J. Mason

Abstract

Large eddy simulations sometimes use monotone advection schemes. Such schemes are dissipative, and the effective subgrid model then becomes the combined effect of the intended model and of the numerical dissipation. The impacts on simulation reliability are examined for the cases of dry convective and neutral planetary boundary layers. In general it is found that the results in the well-resolved flow interior are insensitive to the details of the advection scheme. However, unsatisfactory results may be obtained if numerical dissipation dominates where the flow becomes less well resolved as the surface is approached.

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C.-H. Moeng, W. R. Cotton, C. Bretherton, A. Chlond, M. Khairoutdinov, S. Krueger, W. S. Lewellen, M. K. MacVean, J. R. M. Pasquier, H. A. Rand, A. P. Siebesma, B. Stevens, and R. I. Sykes

This paper reports an intercomparison study of a stratocumulus-topped planetary boundary layer (PBL) generated from ten 3D large eddy simulation (LES) codes and four 2D cloud-resolving models (CRMs). These models vary in the numerics, the parameterizations of the subgrid-scale (SGS) turbulence and condensation processes, and the calculation of longwave radiative cooling. Cloud-top radiative cooling is often the major source of buoyant production of turbulent kinetic energy in the stratocumulus-topped PBL. An idealized nocturnal stratocumulus case was selected for this study. It featured a statistically horizontally homogeneous and nearly solid cloud deck with no drizzle, no solar radiation, little wind shear, and little surface heating.

Results of the two-hour simulations showed that the overall cloud structure, including cloud-top height, cloud fraction, and the vertical distributions of many turbulence statistics, compared well among all LESs despite the code variations. However, the entrainment rate was found to differ significantly among the simulations. Among the model uncertainties due to numerics, SGS turbulence, SGS condensation, and radiation, none could be identified to explain such differences. Therefore, a follow-up study will focus on simulating the entrainment process. The liquid water mixing ratio profiles also varied significantly among the simulations; these profiles are sensitive to the algorithm used for computing the saturation mixing ratio.

Despite the obvious differences in eddy structure in two- and three-dimensional simulations, the cloud structure predicted by the 2D CRMs was similar to that obtained by the 3D LESs, even though the momentum fluxes, the vertical and horizontal velocity variances, and the turbulence kinetic energy profiles predicted by the 2D CRMs all differ significantly from those of the LESs.

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