scholarly journals Monin–Obukhov Similarity and Local-Free-Convection Scaling in the Atmospheric Boundary Layer Using Matched Asymptotic Expansions

2018 ◽  
Vol 75 (10) ◽  
pp. 3691-3701 ◽  
Author(s):  
Chenning Tong ◽  
Mengjie Ding
Keyword(s):  
Surface Layer ◽  
Free Convection ◽  
Asymptotic Expansions ◽  
Atmospheric Surface Layer ◽  
Similarity Theory ◽  
Potential Temperature ◽  
Matched Asymptotic Expansions ◽  
Obukhov Length ◽  
Local Free Convection ◽  
Valid Solution

The Monin–Obukhov similarity theory (MOST) is the foundation for understanding the atmospheric surface layer. It hypothesizes that nondimensional surface-layer statistics are functions of [Formula: see text] only, where z and L are the distance from the ground and the Obukhov length, respectively. In particular, it predicts that in the convective surface layer, local free convection (LFC) occurs at heights [Formula: see text] and [Formula: see text], where [Formula: see text] is the inversion height. However, as a hypothesis, MOST is based on phenomenology. In this work we derive MOST and the LFC scaling from the equations for the velocity and potential temperature variances using the method of matched asymptotic expansions. Our analysis shows that the dominance of the buoyancy and shear production in the outer ([Formula: see text]) and inner ([Formula: see text]) layers, respectively, results in a nonuniformly valid solution and a singular perturbation problem and that [Formula: see text] is the thickness of the inner layer. The inner solutions are found to be functions of [Formula: see text] only, providing a proof of MOST for the vertical velocity and potential temperature variances. Matching between the inner and outer solutions results in the LFC scaling. We then obtain the corrections to the LFC scaling near the edges of the LFC region ([Formula: see text] and [Formula: see text]). The nondimensional coefficients in the expansions are determined using measurements. The resulting composite expansions provide unified expressions for the variance profiles in the convective atmospheric surface layer and show very good agreement with the data. This work provides strong analytical support for MOST.

Multi-point Monin–Obukhov similarity in the convective atmospheric surface layer using matched asymptotic expansions

2019 ◽  
Vol 864 ◽  
pp. 640-669 ◽  
Author(s):  
Chenning Tong ◽  
Mengjie Ding
Keyword(s):  
Surface Layer ◽  
Asymptotic Expansions ◽  
Large Scale ◽  
Dynamic Range ◽  
Fourier Transforms ◽  
Potential Temperature ◽  
Strong Support ◽  
Matched Asymptotic Expansions ◽  
Length Scale ◽  
Horizontal Length

The multi-point Monin–Obukhov similarity (MMO) was recently proposed (Tong & Nguyen, J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348) to address the issue of incomplete similarity in the framework of the original Monin–Obukhov similarity theory (MOST). MMO hypothesizes the following: (1) The surface-layer turbulence, defined to consist of eddies that are entirely inside the surface layer, has complete similarity, which however can only be represented by multi-point statistics, requiring a horizontal characteristic length scale (absent in MOST). (2) The Obukhov length $L$ is also the characteristic horizontal length scale; therefore, all surface-layer multi-point statistics, non-dimensionalized using the surface-layer parameters, depend only on the height and separations between the points, non-dimensionalized using $L$. However, similar to MOST, MMO was also proposed as a hypothesis based on phenomenology. In this work we derive MMO analytically for the case of the horizontal Fourier transforms of the velocity and potential temperature fluctuations, which are equivalent to the two-point horizontal differences of these variables, using the spectral forms of the Navier–Stokes and the potential temperature equations. We show that, for the large-scale motions (wavenumber $k<1/z$) in a convective surface layer, the solution is uniformly valid with respect to $z$ (i.e. as $z$ decreases from $z>-L$ to $z<-L$), where $z$ is the height from the surface. However, for $z<-L$ the solution is not uniformly valid with respective to $k$ as it increases from $k<-1/L$ to $k>-1/L$, resulting in a singular perturbation problem, which we analyse using the method of matched asymptotic expansions. We show that (1) $-L$ is the characteristic horizontal length scale, and (2) the Fourier transforms satisfy MMO with the non-dimensional wavenumber $-kL$ as the independent similarity variable. Two scaling ranges, the convective range and the dynamic range, discovered for $z\ll -L$ in Tong & Nguyen (2015) are obtained. We derive the leading-order spectral scaling exponents for the two scaling ranges and the corrections to the scaling ranges for finite ratios of the length scales. The analysis also reveals the dominant dynamics in each scaling range. The analytical derivations of the characteristic horizontal length scale ($L$) and the validity of MMO for the case of two-point horizontal separations provide strong support to MMO for general multi-point velocity and temperature differences.

Nearly equilibrium dissociating boundary-layer flows by the method of matched asymptotic expansions

1966 ◽  
Vol 26 (4) ◽  
pp. 793-806 ◽  
Author(s):  
George R. Inger
Keyword(s):  
Boundary Layer ◽  
Asymptotic Expansions ◽  
High Speed ◽  
Numerical Solutions ◽  
The Body ◽  
Matched Asymptotic Expansions ◽  
Form Analysis ◽  
Perturbation Problem ◽  
Valid Solution ◽  
Non Equilibrium

The approach to equilibrium in a non-equilibrium-dissociating boundary-layer flow along a catalytic or non-catalytic surface is treated from the standpoint of a singular perturbation problem, using the method of matched asymptotic expansions. Based on a linearized reaction rate model for a diatomic gas which facilitates closed-form analysis, a uniformly valid solution for the near equilibrium behaviour is obtained as the composite of appropriate outer and inner solutions. It is shown that, under near equilibrium conditions, the primary non-equilibrium effects are buried in a thin sublayer near the body surface that is described by the inner solution. Applications of the theory are made to the calculation of heat transfer and atom concentrations for blunt body stagnation point and high-speed flat-plate flows; the results are in qualitative agreement with the near equilibrium behaviour predicted by numerical solutions.

scholarly journals Atmospheric surface layer characteristics of turbulence above the Pantanal wetland regarding the similarity theory

2008 ◽  
Vol 148 (6-7) ◽  
pp. 883-892 ◽  
Author(s):  
E.P. Marques Filho ◽  
L.D.A. Sá ◽  
H.A. Karam ◽  
R.C.S. Alvalá ◽  
A. Souza ◽  
...  
Keyword(s):  
Surface Layer ◽  
Atmospheric Surface Layer ◽  
Similarity Theory ◽  
Pantanal Wetland

scholarly journals Characteristics of the Heat Flux in the Unstable Atmospheric Surface Layer

2016 ◽  
Vol 73 (11) ◽  
pp. 4519-4529 ◽  
Author(s):  
Maithili Sharan ◽  
Piyush Srivastava
Keyword(s):  
Heat Flux ◽  
Surface Layer ◽  
Temperature Gradient ◽  
Power Law ◽  
Atmospheric Surface Layer ◽  
Potential Temperature ◽  
Stability Parameter ◽  
Layer Potential ◽  
Power Law Function ◽  
The Stability

Abstract The behavior of the heat flux H with respect to the stability parameter (=z/L, where z is the height above the ground, and L is the Obukhov length) in the unstable atmospheric surface layer is analyzed within the framework of Monin–Obukhov similarity (MOS) theory. Using MOS equations, H is expressed as a function of and vertical surface-layer potential temperature gradient . A mathematical analysis is carried out to analyze the theoretical nature of heat flux with the stability parameter by considering the vertical potential temperature gradient as (i) a constant and (ii) a power-law function of heat flux. For a given value of H, two values of associated with different stability regimes are found to occur in both the conditions, suggesting the nonuniqueness of MOS equations. Turbulent data over three different sites—(i) Ranchi, India; (ii) the Met Office’s Cardington, United Kingdom, monitoring facility; and (iii) 1999 Cooperative Atmosphere–Surface Exchange Study (CASES-99; United States—are analyzed to compare the observed nature of H with that predicted by MOS. The analysis of observational data over these three sites reveals that the observed variation of H with is consistent with that obtained theoretically from MOS equations when considering the vertical temperature gradient as a power-law function of heat flux having the exponent larger than 2/3. The existence of two different values of the stability parameter for a given value of heat flux suggests that the application of heat flux as a boundary condition involves some intricacies, and it should be applied with caution in convective conditions.

scholarly journals An Extremum Solution of the Monin–Obukhov Similarity Equations

2010 ◽  
Vol 67 (2) ◽  
pp. 485-499 ◽  
Author(s):  
Jingfeng Wang ◽  
Rafael L. Bras
Keyword(s):  
Surface Layer ◽  
Atmospheric Surface Layer ◽  
Turbulent Transport ◽  
Asymptotic Properties ◽  
Field Measurements ◽  
Surface Heat ◽  
Heat Fluxes ◽  
Obukhov Length ◽  
Surface Heat Fluxes ◽  
Momentum And Heat Fluxes

Abstract An extremum hypothesis of turbulent transport in the atmospheric surface layer is postulated. The hypothesis has led to a unique solution of Monin–Obukhov similarity equations in terms of simple expressions linking shear stress (momentum flux) and heat flux to mean wind shear and temperature gradient. The extremum solution is consistent with the well-known asymptotic properties of the surface layer. Validation of the extremum solution has been made by comparison to field measurements of momentum and heat fluxes. Furthermore, a modeling test of predicting surface heat fluxes using the results of this work is presented. A critical reexamination of the interpretation of the Obukhov length is given.

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