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Y. Cheng, V. M. Canuto, and A. M. Howard
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Y. Cheng, V. M. Canuto, and A. M. Howard
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V. M. Canuto, Y. Cheng, and A. M. Howard

Abstract

It has been known for three decades that in the case of buoyancy-driven flows the widely used second-order closure (SOC) level-2.5 turbulence models exhibit divergences that render them unphysical in certain domains. This occurs when the dimensionless temperature gradient Gh (defined below) approaches a critical value Gh(cr) of the order of 10; thus far, the divergences have been treated with ad hoc limitations of the type
i1520-0469-62-5-1645-eq1
where τ is the eddy turnover time scale, g is the gravitational acceleration, α is the coefficient of thermal expansion, T is the mean potential temperature, and z is the height. It must be noted that large eddy simulation (LES) data show no such limitation. The divergent results have the following implications. In most of the ∂T/∂z < 0 portion of a convective planetary boundary layer (PBL), a variety of data show that τ increases with z, −∂T/∂z decreases with z, and Gh decreases with z. As one approaches the surface layer from above, at some z cr (∼0.2H, H is the PBL height), Gh approaches Gh(cr) and the model results diverge. Below z cr, existing models assume the displayed equation above. Physically, this amounts to artificially making the eddy lifetime shorter than what it really is. Since short-lived eddies are small eddies, one is essentially changing large eddies into small eddies. Since large eddies are the main contributors to bulk properties such as heat, momentum flux, etc., the artificial transformation of large eddies into small eddies is equivalent to underestimating the efficiency of turbulence as a mixing process.

In this paper the physical origin of the divergences is investigated. First, it is shown that it is due to the local nature of the level-2.5 models. Second, it is shown that once an appropriate nonlocal model is employed, all the divergences cancel out, yielding a finite result. An immediate implication of this result is the need for a reliable model for the third-order moments (TOMs) that represent nonlocality. The TOMs must not only compare well with LES data, but in addition, they must yield nondivergent second-order moments.

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Y. Cheng, V. M. Canuto, and A. M. Howard

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The standard approach to studying the planetary boundary layer (PBL) via turbulence models begins with the first-moment equations for temperature, moisture, and mean velocity. These equations entail second-order moments that are solutions of dynamic equations, which in turn entail third-order moments, and so on. How and where to terminate (close) the moments equations has not been a generally agreed upon procedure and a variety of models differ precisely in the way they terminate the sequence. This can be viewed as a bottom-up approach. In this paper, a top-down procedure is suggested, worked out, and justified, in which a new closure model is proposed for the fourth-order moments (FOMs). The key reason for this consideration is the availability of new aircraft data that provide for the first time the z profile of several FOMs. The new FOM expressions have nonzero cumulants that the model relates to the z integrals of the third-order moments (TOMs), giving rise to a nonlocal model for the FOMs. The new FOM model is based on an analysis of the TOM equations with the aid of large-eddy simulation (LES) data, and is verified by comparison with the aircraft data. Use of the new FOMs in the equations for the TOMs yields a new TOM model, in which the TOMs are damped more realistically than in previous models. Surprisingly, the new FOMs with nonzero cumulants simplify, rather than complicate, the TOM model as compared with the quasi-normal (QN) approximation, since the resulting analytic expressions for the TOMs are considerably simpler than those of previous models and are free of algebraic singularities. The new TOMs are employed in a second-order moment (SOM) model, a numerical simulation of a convective PBL is run, and the resulting mean potential temperature T, the SOMs, and the TOMs are compared with several LES data. As a final consistency check, T, SOMs, and TOMs are substituted from the PBL run back into the FOMs, which are again compared with the aircraft data.

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Y. Cheng, V. M. Canuto, and A. M. Howard

Abstract

Second-order turbulence models of the Mellor and Yamada type have been widely used to simulate the planetary boundary layer (PBL). It is, however, known that these models have several deficiencies. For example, assuming the production of the turbulent kinetic energy equals its dissipation, they all predict a critical Richardson number that is about four times smaller than the large eddy simulation (LES) data in stably stratified flows and are unable to distinguish the vertical and lateral components of the turbulent kinetic energy in neutral PBLs, and they predict a boundary layer height lower than expected.

In the present model, three new ingredients are employed: 1) an updated expression for the pressure–velocity correlation, 2) an updated expression for the pressure–temperature correlation, and 3) recent renormalization group (RNG) expressions for the different turbulence timescales, which yield

1) a critical Richardson number of order unity in the stably stratified PBL (at level 2 of the model),

2) different vertical and lateral components of the turbulent kinetic energy in the neutral PBL obtained without the use of the wall functions,

3) a greater PBL height,

4) closer comparisons with the Kansas data in the context of the Monin–Obukhov PBL similarity theory, in both stable and unstable PBLs, and

5) more realistic comparisons with the LES and laboratory data.

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V. M. Canuto, A. Howard, Y. Cheng, and M. S. Dubovikov

Abstract

Ocean mixing processes have traditionally been formulated using one-point turbulence closure models, specifically the Mellor and Yamada (MY) models, which were pioneered in geophysics using 1980 state-of-the-art turbulence modeling. These models have been widely applied over the years, but the underlying core physical assumptions have hardly improved since the 1980s; yet, in the meantime, turbulence modeling has made sufficient progress to allow four improvements to be made.

1) The value of Ricr. MY-type models yield a low value for the critical Richardson number, Ricr = 0.2 (the result of linear stability is Ricr = 1/4). On the other hand, nonlinear stability analysis, laboratory measurements, direct numerical simulation, large eddy simulation, and mixed layer studies indicate that Ricr ∼ 1. The authors show that by improving the closure for the pressure correlations, the result Ricr ∼ 1 naturally follows.

2) Nonlocal, third-order moments (TOMs). The downgradient approximation used in all models thus far seriously underestimates the TOMs. A new expression that includes both stratification and shear is presented here for the first time. It is obtained by solving the dynamic equations for the third-order moments.

3) Rotation. The MY-type models with rotation assume that the latter does not affect turbulence, specifically, neither the pressure correlations nor the rate of dissipation of turbulent kinetic energy. Recent studies show that both quantities are affected.

4) Mixing below the mixed layer. Thus far, the momentum and heat diffusivities below the mixed layer have been treated as adjustable parameters. A new model that allows use of the same turbulence model throughout the ocean depth is proposed.

A new model is presented that includes 1), 2), and 4). Rotation will be dealt with in a subsequent paper. The new model is fully algebraic and easy to use in an ocean code. The new model is used in an OGCM, and the predicted global temperature and salinity profiles are compared with those of the KPP model and Levitus data.

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V. M. Canuto, A. Howard, Y. Cheng, and M. S. Dubovikov

Abstract

A Reynolds stress–based model is used to derive algebraic expressions for the vertical diffusivities K α(α = m, h, s) for momentum, heat, and salt. The diffusivities are expressed as
KαRρN,Tϵ
in terms of the density ratio R ρ = α sS/∂z(α TT/∂z)−1, the Brunt–Väisälä frequency N 2 = −g ρ−10ρ/∂z, the Richardson number RiT = N 22 (Σ is the shear), and the dissipation rate of kinetic energy ϵ. The model is valid both in the mixed layer (ML) and below it. Here R ρ and N are computed everywhere using the large-scale fields from an ocean general circulation model while RiT is contributed by resolved and unresolved shear. In the ML, the wind-generated large-scale shear dominates and can be computed within an OGCM. Below the ML, the wind is no longer felt and small-scale shear dominates. In this region, the model provides a new relation RiT = cf(R ρ) with c ≈ 1 in lieu of Munk's suggestion RiTc. Thus, below the ML, the K α become functions of R ρ, N, and ϵ.

The dissipation ϵ representing the physical processes responsible for the mixing, which are different in different parts of the ocean, must also be expressed in terms of the large-scale fields. In the ML, the main source of stirring is the wind but below the ML there is more than one possible source of stirring. For regions away from topography, one can compute ϵ using a model for internal waves. On the other hand, near topography, one must employ different expressions for ϵ. In agreement with the data, the resulting diffusivities are location dependent rather than universal values.

Using North Atlantic Tracer Release Experiment (NATRE) data, the authors test the new diffusivities with and without an OGCM. The measured diffusivities are well reproduced. Also, a set of global T and S profiles is computed using this model and the KPP model. The profiles are compared with Levitus data. In the North Atlantic, at 24°N, the meridional overturning is close to the measured values of 17 ± 4 Sv and 16 ± 5 Sv (Sv ≡ 106 m3 s−1). The polar heat transport for the North Atlantic Ocean, the Indo–Pacific Ocean, and the global ocean are generally lowered by double diffusion. The freshwater budget is computed and compared with available data.

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Yu Cheng, Lisa M. Beal, Ben P. Kirtman, and Dian Putrasahan

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We investigate the interannual variability of Agulhas leakage in an ocean-eddy-resolving coupled simulation and characterize its influence on regional climate. Many observational leakage estimates are based on the study of Agulhas rings, whereas recent model studies suggest that rings and eddies carry less than half of leakage transport. While leakage variability is dominated by eddies at seasonal time scales, the noneddy leakage transport is likely to be constrained by large-scale forcing at longer time scales. To investigate this, leakage transport is quantified using an offline Lagrangian particle tracking approach. We decompose the velocity field into eddying and large-scale fields and then recreate a number of total velocity fields by modifying the eddying component to assess the dependence of leakage variability on the eddies. We find that the resulting leakage time series show strong coherence at periods longer than 1000 days and that 50% of the variance at interannual time scales is linked to the smoothed, large-scale field. As shown previously in ocean models, we find Agulhas leakage variability to be related to a meridional shift and/or strengthening of the westerlies. High leakage periods are associated with east–west contrasting patterns of sea surface temperature, surface heat fluxes, and convective rainfall, with positive anomalies over the retroflection region and negative anomalies within the Indian Ocean to the east. High leakage periods are also related to reduced inland convective rainfall over southeastern Africa in austral summer.

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B. S. Felzer, Carol R. Ember, R. Cheng, and M. Jiang

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Our broad research goal is to understand how human societies adapt to natural hazards, such as droughts and floods, and how their social and cultural structures are shaped by these events. Here we develop meteorological data of extreme dry, wet, cold, and warm indices relative to 96 largely nonindustrial societies in the worldwide Standard Cross-Cultural Sample to explore how well the meteorological data can be used to hindcast ethnographically reported drought and flood events and the global patterns of extremes. We find that the drought indices that are best at hindcasting ethnographically reported droughts [precipitation minus evaporation (P − E) measures] also tend to overpredict the number of droughts, and therefore we propose a combination of these two indices plus the PDSI as an optimal approach. Some wet precipitation indices (R10S and R20S) are more effective at hindcasting ethnographically reported floods than others. We also calculate the predictability of those extreme indices and use factor analysis to reduce the number of variables so as to discern global patterns. This work highlights the ability to use extreme meteorological indices to fill in gaps in ethnographic records; in the future, this may help us to determine relationships between extreme events and societal response over longer time scales than are otherwise available.

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Cheng-Shang Lee, Roger Edson, and William M. Gray

Abstract

This paper discusses the meteorological conditions associated with tropical cyclone formation in the north Indian Ocean during the 1979 FGGE year. Seven developing systems are composited together using FUGE Ill-b analyses to show the common circulation features surrounding developing cloud clusters. Three systems are further discussed to show different environmental influences on the low-level buildup of circulation during formation. The characteristics of these three disturbances’ 200 mb outflow patterns and a general discussion of north Indian Ocean tropical cyclone activity are also given.

Results show that tropical cyclone formation generally follows the initial increase of strong low-level winds on one side (either equatorial or polar) of a precyclone disturbance. This early buildup of wind appears to result from environmentally forced asymmetric wind surge action. Some of this increase appears to result from inward advection of velocity, but part of the increase seems to occur in situ. These initial strong azimuthal wind asymmetries are gradually reduced as the winds spread more evenly around the disturbance. A basic cyclone development process is the evolution of the low tropospheric flow from initial asymmetrical now (shear vorticity) to a more symmetrical circulation (curvature vorticity).

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