A REVIEW ON PERTURBATION TECHNIQUES APPLIED TO THE STUDY OF THE STRATIFIED ATMOSPHERIC BOUNDARY LAYER
Cláudio C. Pellegrini1
ABSTRACT: Perturbation theory is employed to study the stratified atmospheric boundary layer over flat surfaces as static stability varies. The study has been divided into three papers. This paper concentrates in the mathematical aspects of the problem. No new analytical solutions are presented, as this study is an initial effort to clarify some dynamics aspects of the problem and its link to sta-bility parameters. The main objective is to lay foundation for future studies on cases where no ana-lytical solution is available. Nevertheless, some new links between the governing equations in first order of approximation and the dynamics of the atmospheric boundary layer are revealed.
Key words: Perturbation techniques, atmospheric boundary layer, micrometeorology.
INTRODUCTION
Problems that can be easily tackled through perturbation analysis are the ones that are similar to simpler problems for which an exactly solution exists. The word similar in this case means that the complex problem differs from the simpler only by a small dimensionless parameter, ε, so that in the limit when the resulting system is exactly solvable. The solution for the complex problem is generally written as an asymptotic expansion on the independent variable and on and the accu-racy of the result is expected to improve as gets smaller.
0
ε =
ε ε
Perturbation theory includes a number of techniques. The intermediate variable technique (IVT) has its roots on matched asymptotic expansion (MAE) analysis, one of the most successful methods. When MAE are used to solve boundary layer problems, a special variable (called interme-diate) is used to match the inner to the outer solutions. This is achieved through a co-ordinate stretching (or rescaling) of the kind , where is the independent variable, a is (generally but not necessarily) an integer and x is the stretched co-ordinate. The co-ordinate is substituted by into the equations under study, a is allowed to vary and all resulting possibilities of matching are investigated. The value of a obtained indicates where the boundary layer is and, substituted into
, is used to accomplish the matching.
/ a x =x ε x x x / a x =x ε
The IVT is a slight modification of the procedure just described and is indeed implicit on the search for boundary layers mentioned. In IVT, no outer and inner asymptotic solutions are sought; the intermediate variable is used to define the layers where different terms dominate the original equation on first order of approximation. The exponent a is abandoned and in is al-lowed to vary continuously in [
ε x =x ε/
]
0,1 . The method shows the relative importance of terms in the
1
Universidade Federal de São João del-Rei, Depto. Ciências Térmicas e dos Fluidos, Praça. Frei Orlando 170, São João del-Rei, MG, 36.307-904, [email protected]
original equation as one goes from the boundaries of the problem to the far field. The approximate equations found may be solved but, generally, no matching is proposed between them. The reason is that IVT is intended to be simple, focusing more on the physical aspects of the problem than on the solution itself. As Hinch (1995) points out, ‘Obtaining good numerical values for the solution is not
the only quest of a perturbation approximation. One can hope that the analysis will reveal some physical insights through the simplified physics of the limiting problem.’ What makes IVT different
from MAE is the fact that it does not concentrate only at the inner and outer domains of the prob-lem, but also at its overlap domains.
The present study has been divided into three papers. It employs the IVT to study the stratified atmospheric boundary layer (ABL) over flat surfaces as static stability varies. This paper concen-trates in the mathematical aspects of the problem and in particular in reviewing the IVT, generally unknown in the meteorological field. No new analytical solutions are presented, as this study is an initial effort to clarify some aspects of the ABL dynamics and their link to the static stability. The main intention is to lay foundation for future studies on cases where no analytical solution is avail-able at all. The analysis results in a division of the ABL according to the dominating terms in the governing equations and some new links between the first order approximate equations and the dy-namics of the ABL are pointed in the second and third papers (Pellegrini, 2006 a, b). In future pa-pers, we shall consider non-stationary and topographic variations effects. We shall also analyze the prognostic equations for the turbulent fluxes and probe into mathematical details of the IVT, as the influence of the scaling parameters on the problem.
REVIEW OF LITERATURE AND ORDER SYMBOLS
Recent meteorological literature contains some examples of the use of perturbation techniques in the analysis of turbulent boundary layers. The atmospheric flow over small topographic features, in particular, is several times addressed in the literature. Towsend (1972) and Knight (1977) study the turbulent flow over wavy surfaces. Jackson and Hunt (1975) propose a two-layer structure for the ABL over a gentle hill. Sykes (1980) uses asymptotic expansions to study the influence of small terrain elevations on the main flow and on turbulence. Hunt et al. (1988a) include different upwind profiles in the analysis and Hunt et al. (1988b) consider stably stratified atmosphere. In both papers a three layer structure is proposed. More recently, Baldauf and Fiedler (2003) use MAE to obtain a parameterization of the effective roughness length over flat terrain in neutral atmosphere. The use of IVT in meteorological problems is rare and was found only in Pellegrini and Bodstein (2005).
Comparisons of magnitude order made in the meteorological literature generally use symbols in a different way from perturbation theory literature. Therefore, a brief description of the tradi-tional nomenclature is presented here, and then the way symbols are used in this study are then
de-fined. Attention is restricted to functions and , where and is a scalar parameter. The independent variable x vary over some specified domain D and lies on the inter-val ( , ) u x ε v( , )xε x=( , , , )x y z t ε ε [0, 1] I = ε for ε >1 0.
According to Kevorkian and Cole (1996), for two functions u( , )x ε and v( , )xε the statement
( )
( , ) ( , )
u xε =O v xε (1)
means that there exists a function k( , )x ε >0 such that u /v ≤ x )k( ,ε for all . In words, this means that the ratio of the functions being compared is bounded above by . Thus, for exam-ple, means that
I ε ∈ ( , ) k x ε * ( )
u′ =O u u′ u* ≤k( , )xε . If k is a constant, the relation is said to be uniformly
valid. The symbol is used to indicate that the relation between two functions is bounded above
and below. Thus, if s
O
( )
( , ) s ( , )
u xε =O v xε , then, u /v ≤k1(x, )ε and v /u ≤k2(x, )ε for all ,
implying that and . For example, means that
I ε ∈
( )
u =O v v =O u( ) θ =Os( )θ0 θ θ0 ≤k1( , )x ε and
0 k2( , )
θ θ ≤ xε . Uniform validity is again defined when for k1 and k2 constants. The statement
( )
( , ) ( , )
u xε =o v x ε (2)
implies that u /v ≤δ, for δ arbitrarily small as ε →0. The meaning is that u is arbitrarily small compared v as ε →0.
Here we adopt the symbols O and but give them more restrictive definitions. This is com-mon practice in the meteorological and engineering fields, although it is rarely clearly stated. There-fore, O and hereafter imply uniform validity and and close to one. This definition cannot be formalized saying that and are because the symbol is being defined. We therefore assume that close to one means 1/ for and for O. Hinch (1995) avoids this problem using the symbol ord , informally defined as meaning ‘strictly of the order of
s O s O k1 k2 1 k k2 Os(1) Os 5≤ki ≤5,i =1,2 Os k ≤5 ’ The present definition implies that the two terms are of the same order of magnitude in the way the sen-tence is usually understood in meteorological and engineering texts.
The symbol is often used in the literature in the place of O or . Here, we avoid this use as much as possible and, when necessary, ∼ is used in place of our definition of . To express typical values of a function, = is used, as in ms
∼ Os
s
O
* 0.05
u = -1. We also avoid completely using ∼ to state that the left and right-hand side of an equation involves terms of the same order, because in perturbation texts this symbol is used to define an asymptotic expansion of some variable. For equa-tions, we use the symbol = meaning that the equality holds in some appropriate asymptotic limit.
The symbol o is used here in the same sense it has in the definition above, because it does ex-press the idea we want to communicate, i.e., that u is much smaller than v when ε →0, as in
(u v )/ z o(1
ε∂ ′ ′ ∂ = ). However, to compare functions that do not depend on ε, as the turbulent and the
average parts of a variable for example, o is employed in a more restricted sense. In these cases,
( )
( ) ( )
u x =o v x implies that u /v ≤δ, for . As in perturbation theory literature the symbol is customarily substituted by , here our definition of o is also replaced by when necessary.
1/ 5 δ < o ′ 2 2 u ⎤ ∂ ⎢ ⎥ ⎥ ∂ ⎦ GOVERNING EQUATIONS
The equations for the conservation of mass, momentum and energy for the ABL, for 2-D flows can be written as (Arya, 2001):
0 u w x z ∂ +∂ = ∂ ∂ and 0 u w x z (3) ′ ∂ +∂ = ∂ ∂ 2 2 1 (4) 0 1 u u u p u u u w u u w t x z ρ x x z ν x z ⎡ ′ ′ ′ ′ ⎡ ⎤ ∂ + ∂ + ∂ = − ∂ −⎢∂ +∂ ⎥+ ∂ + ⎢ ∂ ∂ ∂ ∂ ⎢ ∂⎣ ∂ ⎥⎦ ⎣∂ 2 2 1 1 2 2 0 0 1 w w w p w u w w w w u w g t x z z x z x z θ ν ρ θ ⎡ ⎤ ′ ′ ′ ′ ⎡ ⎤ ∂ + ∂ + ∂ = − ∂ −⎢∂ +∂ ⎥+ ⎢∂ +∂ ⎥+ ⎢ ⎥ ∂ ∂ ∂ ∂ ⎢ ∂⎣ ∂ ⎥⎦ ⎣∂ ∂ ⎦ (5) 2 2 2 h u w u w t x z x z x z θ θ θ θ θ α θ 2 θ ⎡ ⎤ ′ ′ ′ ′ ⎡ ⎤ ∂ + ∂ + ∂ = −⎢∂ +∂ ⎥+ ⎢∂ +∂ ⎥ ⎢ ⎥ ∂ ∂ ∂ ⎢ ∂⎣ ∂ ⎥⎦ ⎣∂ ∂ ⎦ (6)
Here, all variables are time mean averages with the bars omitted for simplicity and with their usual meaning. At the turbulent mass conservation equation, u and w represent turbulent quanti-ties (no bars omitted in this case). Pressure, density and potential temperature have been divided into basic state and deviation components, the basic state being assumed to be an ideal gas in hydrostatic equilibrium. In the energy equation,
′ ′
0 0 0
( , , )p ρ θ ( , , )p ρ θ1 1 1
h k c
α = ρ p is the thermal molecular diffusivity. The Coriolis force, the diabatic heat and the radiation divergence have been neglected, as usual in ABL studies.
The non-dimensional variables are defined as
1 1 2 0 * * 0 ; ; g; ; ; ; ; ; ; x z x g c g tU x z u w p u w X Z U W P U W L L L U W U u w θ θ τ θ θ ρ ′ ′ ′ ′ ′ = = = = = = Θ = = = Θ = * ′ (7)
The characteristic lengths, and , are expected to depend on the static stability of the ABL. The geostrophic velocity, , is used as a scaling velocity to u, assuming that geostrophic balanced is achieved outside the ABL. The value is used as the scaling factor for the pressure because is the dynamic pressure in eqns. (4) and (5). Friction velocities used to scale u and w
are defined as and
x L Lz g U 2 0 gU ρ 1 p ′ ′ ( )1/2 * s
u = u w′ ′ w* =
(
θ′ ′w gzi θ)
1/ 3s , where ( ) refers to values at the surface andis the ABL depth. For the friction temperature, different expressions are adopted for the unstable
s i
and stable ABL: θ*uns =(w′ ′θ )s w* and θ*st =(w′ ′θ )s u*. Velocity need not be precisely defined
for the moment. In eqn. (7), , for and O , for .
c
W
( ) 1
O φ = φ=X Z, , , , ,τU W U W′, ′,Θ′ s( )φ =1 φ = P1,Θ
Applying eqns. (7) to eqns. (3)—(6) gives the non-dimensional set of equations needed. Non-dimensional conservation of mass reads ∂U ∂X +
(
L W L Ux c z g)
∂W ∂ =Z 0 and(
x * z *)
0U′ X L w L u W′ Z
∂ ∂ + ∂ ∂ = . If L W L Ux c z g 1 or 1 L W L Ux c z g , it follows that ∂ ∂ =w z 0,
implying w =0 throughout the ABL because of the condition at the surface. The same applies to
*
x z
L w L u*. This indicates that and /
)
. The first part of thisrela-tion is also recognized to hold for meso-scale flows by Pielke (2002). Non-dimensional conserva-tion of mass then reads
(
)
/ / x z s g c L L =O U W L Lx/ z =O u ws(
* * 0 U W X Z ∂ +∂ = ∂ ∂ and 0 U W X Z ′ ′ ∂ +∂ = ∂ ∂ (8)The non-dimensional x-momentum equation is
2 2 1 2 2 * R2 L 2 L U U U P U U U W U U U W X Z X X Z X Z ε ε ε τ ε 2 ⎡ ⎤ ′ ′ ′ ′ ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ∂ ⎢ ∂ ∂ ⎥ ⎢ ⎥ + + = − − + + + ⎢ ⎥ ∂ ∂ ∂ ∂ ⎢ ∂⎣ ∂ ⎥⎦ ⎣ ∂ ∂ ⎦, (9)
where Re=νU Lg x is the Reynolds number and was used to simplify the result.
The small parameters where defined as , and .
(
/ x z s g c L L =O U W/)
x * u U*/ g ε = ε =L Lz/L ε =R 1/ ReThe non-dimensional z-momentum equation is
2 2 1 2 2 2 2 bs * 2 2 Ri L W U WX W WZ PZ L W UX W WZ R L XW ZW ε ε ε ε ε τ ⎡ ⎤ ′ ′ ′ ′ ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎡ + + ⎤ = − − ⎢ + ⎥+ ⎢ + ⎥+ ⎢ ⎥ ⎢ ⎥ ⎢ ∂ ∂ ∂ ⎥ ∂ ⎢ ∂ ∂ ⎥ ∂ ∂ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ , (10)
where Ribs =g Lθ1 z θ0Ug2 is a deviation form the basic state bulk Richardson number and, therefore,
means statically unstable flow and statically stable (opposite to traditional bulk ). The non-dimensional energy equation is
bs Ri >0 Ribs <0 Ri 2 2 2 * * P2 L 2 L U W U W X Z θ X Z X Z ε ε ε ε τ ε 2 ⎡ ⎤ ′ ′ ′ ′ ⎡ ⎤ ∂Θ+ ∂Θ+ ∂Θ = − ⎢∂ Θ +∂ Θ ⎥+ ⎢ ∂ Θ +∂ Θ⎥ ⎢ ⎥ ∂ ∂ ∂ ⎢ ∂⎣ ∂ ⎥⎦ ⎣ ∂ ∂ ⎦ (11) where ε*θ =θ*/θ0 and ε =P 1/ Re Pr. COORDINATE STRETCHING
The stretching co-ordinate transformation is , where ε is the stretching parameter and Z is the stretched vertical coordinate assumed to be . Substituting this equation eqn. into eqns. (8), (9), (10) and (11) and rearranging yields
/
Z =Z ε
(1)
s
1 0 U W X ε Z ∂ ∂ + = ∂ ∂ and 1 0 U W X ε Z ′ ′ ∂ ∂ + = ∂ ∂ (12) 2 2 2 1 * 2 2 2 2 2 1 R L L U U U P U U U W U U U W X Z X X Z X ε ε ε ε ε τ ε ε ε ε 2 Z ⎡ ⎤ ′ ′ ′ ′ ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ∂ ⎢ ∂ ∂ ⎥ ⎢ ⎥ + + = − − + + + ⎢ ⎥ ∂ ∂ ∂ ∂ ⎢⎣ ∂ ∂ ⎥⎦ ⎣ ∂ ∂ ⎦, (13) 2 2 2 2 1 2 * 2 2 bs 2 2 2 1 1 Ri L R L W U WX W W P W UX W W L XW W Z Z Z Z ε ε ε ε ε ε τ ε ε ε ε ε ⎡ ⎤ ′ ′ ′ ′ ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎡ + + ⎤ = − − ⎢ + ⎥+ ⎢ + ⎥+ ⎢ ⎥ ⎢ ⎥ ⎢ ∂ ∂ ∂ ⎥ ∂ ⎢ ∂ ∂ ⎥ ∂ ∂ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (14) 2 2 * * 2 2 2 2 2 2 1 P L L U W U W X Z X Z X θ ε ε ε ε ε ε τ ε ε ε ε Z ⎡ ⎤ ′ ′ ′ ′ ⎡ ⎤ ∂Θ ∂Θ ∂Θ ∂ Θ ∂ Θ ⎢ ∂ Θ ∂ Θ⎥ ⎢ ⎥ + + = − + + + ⎢ ⎥ ∂ ∂ ∂ ⎢⎣ ∂ ∂ ⎥⎦ ⎣ ∂ ∂ ⎦ (15) CONCLUSION
Equations (12)—(15) depend on more than one small parameter; therefore, it is necessary to establish a relation between them to proceed with the IVT analysis. The study of all possible values of is somewhat different from what is found in traditional perturbation theory because all small parameters must be considered simultaneously. We propose a way to make this calculation system-atic in a companion paper (Pellegrini, 2006a). In the paper we also analyze eqns. (12) and (13) in detail, showing some interesting links between they first order approximate forms and the dynamics of the flow.
ε
ACKNOWLEDGEMENTS: The author acknowledges the financial support obtained from FAPEMIG under project TEC-921/04 during the execution of this study. He also acknowledges the support given for the presentation of this work in the XIV CBMet, held at Florianópolis, SC, Brazil.
REFERENCES
Arya, S. P.: 2001, Introduction to micrometeorology, 2nd ed., Academic Press, 420 pp. Hinch, E. J.: 1995, Perturbation Methods, Cambridge Univ. Press, Cambridge, 160 pp.,
Hunt, J. C. R., Leibovich, S. and Richards, K.J.: 1988a, ‘The turbulent shear flow over low hills’,
Quart. J. Roy. Meteor. Soc. 114, 1435-1470.
Hunt, J. C. R., Richards, K. J., Brighton, P. W. M.: 1988b; ‘Stably stratified shear flow over low hills’, Quart. J. Roy. Meteor. Soc. 114, 859-886.
Jackson, P. S. and Hunt, J. C. R.: 1975, ‘Turbulent wind flow over a low hill’, Quart. J. Roy.
Me-teor. Soc. 106, 929-955.
Kevorkian, J., Cole, J. D.: 1996, Multiple scales and singular perturbation methods, Springer-Verlag, New York, 620 pp.
Knight, D.: 1977, Turbulent flow over a wavy boundary, Bound.-Layer Meteor. 11, 205-222.
Sykes, R. I.: 1980, ‘An asymptotic theory of incompressible turbulent boundary layer flow over a small hump’, J. Fluid Mech., 101, 647-670.
Towsend, A. A.: 1972, “Flow in a deep turbulent boundary layer over a surface distorted by water waves’, J. Fluid Mech., 55, 719-735.
Pellegrini, C.C. and Bodstein, G.C.R.: 2005, ‘A modified logarithmic law for neutrally stratified flow over low-sloped hills’, J. App. Meteor., 44 (6), 900—916.
Pellegrini, C.C.: 2006a, submitted, ‘A study of the stratified atmospheric boundary layer through perturbation techniques – part I: conservation of mass and x-momentum equations’, proceedings of the XIV CBMet, Florianópolis, SC, Brasil.
Pellegrini, C.C.: 2006b, submitted, ‘A study of the stratified atmospheric boundary layer through perturbation techniques – part II: conservation of energy and z-momentum equations’, proceed-ings of the XIV CBMet, Florianópolis, SC, Brasil.
