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Non‐iterative surface flux parametrization for theunstable surface layerK. Abdella a & Dawit Assefa ba Department of Mathematics , Trent University , Peterborough, ON, K9J 7B8 E-mail:b Department of Mathematics , Trent University , Peterborough, ON, K9J 7B8Published online: 21 Nov 2010.

To cite this article: K. Abdella & Dawit Assefa (2005) Non‐iterative surface flux parametrization for the unstable surfacelayer, Atmosphere-Ocean, 43:3, 249-257

To link to this article: http://dx.doi.org/10.3137/ao.430305

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Non-iterative Surface Flux Parametrization for the Unstable Surface Layer

K. Abdella* and Dawit Assefa

Department of Mathematics, Trent University Peterborough ON K9J 7B8

[Original manusript received 28 October 2004; in revised form 20 April 2005]

ABSTRACT This study presents a semi-analytic non-iterative solution for the Monin-Obukhov similarity equa-tions under unstable surface conditions. The solution is represented in terms of the non-dimensional Monin-Obukhov stability parameter z/L. This parameter is given as a function of the bulk Richardson number and other surface parameters including the heat and momentum roughness lengths which are generally assumed to be dif-ferent in this formulation. The proposed formulations give results that are both quantitatively and qualitativelyconsistent with the fully iterated numerical solution for a wide range of surface parameters.

RESUMÉ [Traduit par la rédaction] Cette étude présente une solution semi-analytique non itérative aux équationsde similitude de Monin-Obukhov dans des conditions de surface instables. La solution est exprimée en fonctionde la stabilité adimensionnelle z/L de Monin-Obukhov. Ce paramètre est donné comme une fonction du nombrede Richardson apparent et d’autres paramètres de surface, y compris les longueurs de rugosité de la chaleur etde la quantité de mouvement, qui sont généralement supposées différentes dans cette formulation. Les formula-tions proposées donnent des résultats qui s’accordent tant quantitativement que qualitativement avec la solutionnumérique pleinement itérée pour une vaste gamme de paramètres de surface.

ATMOSPHERE-OCEAN 43 (3) 2005, 249–257© Canadian Meteorological and Oceanographic Society

*Corresponding author’s e-mail: Kabdella@trentu.ca

1 IntroductionThe accurate determination of the surface fluxes of momen-tum, heat and moisture is of critical importance in predictingthe atmospheric structure and its evolution in time since thesefluxes represent the forcing and the lower boundary condi-tions in atmospheric boundary layer modelling. These fluxesare commonly described by the Monin-Obukhov (1954) sim-ilarity theory for a wide range of surface conditions.

The Monin-Obukhov similarity theory consists of severalhighly non-linear coupled sets of equations represented interms of the non-dimensional Obukhov stability parameter

where z is a reference height above the surface, and

L is the Obukhov stability length scale defined as

(1)

where u* is the friction velocity, θ̄0 is a reference mean tem-perature above the surface, k is the von Karman constant, g isacceleration due to gravity and is the heat flux.

Solving these equations numerically using iterative proce-dures is highly undesirable in numerical simulations withlarge atmospheric models since this can be computationallyexpensive and time consuming in terms of achieving numeri-

′ ′w θ

ζ = z

L

cal convergence. Therefore, accurate non-iterative solutionsare preferable in order to minimize these computational costsand avoid difficulties related to numerical convergence. Inlight of this, several non-iterative solutions have been pro-posed during the last few decades.

Assuming the roughness length for momentum (zo) and theroughness length for heat (zh) to be identical, Louis (1979)proposed one of the earliest non-iterative approximations ofthe Monin-Obukhov similarity equations in terms of anexpression for the stability parameter ζ as an explicit functionof the bulk Richardson number Rib. With the same assump-tion of equal momentum and heat roughness length, Byun(1990) derived an analytical solution for ζ as a function of thegradient Richardson number. However, these approximationswere inadequate since both experimental and theoretical evi-dence demonstrated that the momentum and heat roughnesslengths could differ by several orders of magnitude (seeGarratt (1978)) and neglecting the difference could result ininaccurate surface flux predictions (Braud et al., 1993).

More recently, alternative formulations have been devel-oped without the restrictive assumption of equal momentumand heat roughness lengths. Mascart et al. (1995) used a high-er order polynomial fit to obtain the explicit approximation in

Lu

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′ ′* ,3

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terms of Rib, while Launianen (1995) adopted a series expan-sion to arrive at an alternative expression. Their expressionsgave an improved estimate of the transfer coefficients formomentum and heat. However, they were typically found to be applicable only to a limited range of the roughness ratio,

(see Van Den Hurk and Holtslag (1997) for a review).

Guilloteau (1998) obtained an expression for ζ as an analyticsolution of a cubic equation. However, this scheme tends tounderestimate both the momentum and the heat transfer coeffi-cients.

Abdella and McFarlane (1996) presented another surface fluxparametrization derived using a fitting function based on theasymptotic behaviour of ζ at large and small Rib. This formula-tion has been used by the Canadian Centre for ClimateModelling and Analysis (CCCma) (McFarlane et al., 1992).However, the chosen form of the fitting function leads to inac-curate results for some surface conditions. In particular, forrough surfaces when the ratio of the roughness length ofmomentum to that of heat is above unity, their formulation leadsto an overestimation of the fluxes by up to 25%. In contrast, forsmooth surfaces with roughness length ratios which are lessthan unity, their formulation leads to an underestimation of thefluxes. Moreover, for moderate to high Rib values and for someranges of the roughness length ratio this formulation is inade-quate. A more accurate solution, based on a semi-analyticapproach, was recently proposed by De Bruin et al. (2000).However, the model parameters needed in their scheme are notanalytic functions of height. Instead, the parameters areassumed to be constants which are determined by matching theexact and the approximate formulation at one selected value ofζ. As pointed out in De Bruin et al. (2000), the constant para-meter assumption is inconvenient and limits the generality andreduces the accuracy of their scheme. Van den Hurk andHoltslag (1997) compared most of the approximate analyticmethods. They reported that, due to the inaccuracies of the exist-ing analytic solutions for a wide range of unstable conditions,the full iterative solution is still used in many operational large-scale models (see also Beljaars and Viterbo (1998) and DeBruin et al. (2000)).

In this paper, we present a semi-analytic non-iterative solu-tion under unstable surface conditions in terms of ζ given asa function of Rib, zo, zh. The derivation of this formulationleads to a bi-quadratic polynomial equation whose analyticsolution is readily available using Ferari’s method. In contrastto the work of De Bruin et al. (2000) the model parameterswhich are approximated using their asymptotic behaviour areanalytic functions of height and surface layer variables. Forvarious, carefully selected, typical surface conditions the newscheme yields results that are both qualitatively and quantita-tively consistent with the fully iterated solution.

In the next section a brief summary of surface-flux theoryis given. In Section 3 the derivation of the new solution is pre-sented followed by some results and discussion in Section 4.Finally, concluding remarks are presented in Section 5.

z

zo

h

2 Surface flux parametrizationsThe surface fluxes of momentum τ, sensible heat H and mois-ture QE can be expressed in terms of the friction velocity u*,the temperature scale θ* and the humidity scale q* respective-ly as

(2)

where ρ is the density of air, Cp is the specific heat at constantpressure and λ is the latent heat of vaporization. Assuminghorizontally homogeneous and stationary conditions, theMonin-Obukhov similarity theory states that the scaling para-meters and the gradient of the mean field variables can bewritten as follows, in terms of the non-dimensional Monin-Obukhov parameter ζ which is negative for unstable condi-tions, positive for stable conditions and zero for neutralconditions:

(3)

where u is the magnitude of the mean wind, θ is the potentialtemperature, q is the specific humidity, and φ is the corre-sponding stability function which is typically determinedempirically through observation. The commonly accepted

forms of these functions in the unstable case (i.e., for < 0)are:

(4)

where turbulent Prandtl number and γm and

γh are constants.Integrating Eq. (3) from the corresponding roughness

height to z and using Eq. (4) for the φ functions we obtain thefollowing well known flux profile relationship:

(5)

(6)

where θs = θ(zh) and the integrated stability functions aredefined as

(7)

prth

m=

=

φφ ζ 0

z

L

250 / K. Abdella and Dawit Assefa

τ ρ ρ θ ρ θ λρ= = ′ ′ = − = −u H C w C u Q u qp p E* * * * *, ,2

∂∂

= ( ) ∂∂

= ( ) ∂∂

= ( )u

z

kz

u z

kz q

z

kz

qm h q*

,*

,*

φ ζ θθ

φ ζ φ ζand

φ ζ γ ζ φ ζ φ γ ζm m h q t hpr( ) = −( ) ( ) = = −( )− −1 1

14

12, and

uku z

zz

zL

z

Lom m

o*

ln

= ( )

−

−

ψ ψ

θθ θ

ψ ψ*

ln

=( ) −( )

−

−

k z

prz

zzL

z

L

s

th

h hh

ψφ

ψφ

m

m

h

hz

L

zL

zdz

z

L

zL

zdz

=

−

=

−

∫ ∫

1 1,∫ ∫

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and their integrated expressions are given by:

(8)

(9)

where and . Similar

expressions can be written for q*. For a given set of mean and surface parameters, Eqs (1), (5)

and (6) define a non-linear set of coupled equations for thevariables L, u* and θ*. Due to the non-linearity of these equa-tions, full analytic solutions are available only in special cases(Byun, 1990). Moreover, because numerical methods basedon iterative procedures can be computationally costly, accu-rate non-iterative solutions need to be developed. In the nextsection, a new non-iterative solution is derived for the unsta-ble surface layer.

3 New parametrizationsA new non-iterative parametrization for ζ in terms of Rib andother surface layer parameters will be derived in this section.The newly parametrized ζ can then be substituted into Eqs (5)and (6) to obtain the scaling parameters u* and θ* whichyields the surface fluxes according to Eq. (2).

In order to derive the new parametrization for ζ, we firstnote that, Eqs (1), (5) and (6) can be combined to give theequation:

(10)

where

(11)

(12)

and

(13)

yz

Lh= −

1

12γx

z

Lm= −

1

14γ

The expressions for fh and fm can be simplified by applyingCauchy’s theorem for mean value, which states that if twofunctions g1(x) and g2(x) are differentiable in the interval[a,b] then there is at least one value of x, say x′ with a < x′ <b, for which

(14)

where at x = x′ and at x = x′

(Franklin, 1960).

Choosing g1 = ψh(z), g2 = ln (z), a = zh, and b = z, Eq. (12)gives

(15)

for some zt such that zh < zt < z and the derivative ψ′h is withrespect to z. However, using Eq. (7) this reduces to

(16)

Similarly, choosing g1 = ψm (z), g2 = ln (z), a = zo, and b = z,Eq. (13) gives

(17)

for some zu such that zo < zu < z.Using Eqs (4), (10), (16) and (17) we obtain

(18)

which can be written as,

(19)

′ ′( ) = ∂∂

g xg

x22′ ′( ) = ∂

∂g x

g

x11

Non-iterative Surface Flux Parametrization for the Unstable Surface Layer / 251

ψ

π

mz

L

x x

x

= +

+ +

− ( ) +

21

2

1

2

22

2ln ln

arctan

ψhz

L

y

= +

2

1

2ln

L ALf

fh

m

= 0 2

Lu z

gA

prz

z

zz

zt

h

o

s00

2

2=( )( )

=

= ( ) −θ

θθ θ θ

∆∆,

ln

ln

,

f

zL

z

Lz zh

h hh

h= −

−

( ) − ( )1ψ ψ

ln ln

f

zL

z

Lz zm

m mo

o= −

−

( ) − ( )1ψ ψ

ln ln

g b g a

g b g a

g x

g x1 1

2 2

1

2

( ) − ( )( ) − ( )

=′ ′( )′ ′( )

f

zL

z

Lz z

z

L

z

h

h hh

h

ht

t

= −

−

( ) − ( ) = −′

1 11

ψ ψ ψ

ln ln

f

z

Lz

z

z

Lh

ht

t

t

ht= −

−

=

1

1

1

φ

φ .

f

zL

z

Lz z

z

L

z

z

L

m

m mo

o

mu

u

mu

= −

−

( ) − ( )

= −′

=

1

11

ψ ψ

ψφ

ln ln

L AL

z

Lz

L

mu

ht

=−

−

0

121

1

γ

γ

γ α ζ ζ γ α ζm m h h b bA A Ri Ri2 3 2 2 2 2 0− − + + =

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where

is the bulk Richardson number. It should be noted that noapproximating assumption has been made to obtain Eq. (19).However, in order to solve Eq. (19), the variables αh and αmmust still be parametrized.

Using Eqs (4) and (16) we obtain

(20)

which can be solved to obtain

(21)

Similarly using Eqs (4) and (17) it can be shown that

(22)

Using Eqs (12) and (21) we can obtain αh as a function of ζ.In order to simplify Eq. (19) we consider the asymptoticbehaviour of αh. It can be shown that the asymptotic behav-iour of αh as ζ → 0 is given by

(23)

where Similarly the asymptotic behaviour of αh for

small ζ is given by

(24)

where In order to approximate αh at large ζ (i.e.

ζ → – ∞), we first approximate L using Eq. (10) as

(25)

so that

(26)

where Similarly

(27)

where

The variables αh and αm are then chosen to approach theseasymptotic limits using the following expressions:

(28)

and

(29)

where s is a constant. While the approximation is not highlysensitive to the parameter s, s = 0.25 is chosen since this givesthe best approximations for α over a wide range of surfaceparameters.

Note that both the αh and αm parameters are analyticfunctions of z, zo, and zh. This is in contrast to the work ofDe Bruin et al. (2000) where the corrsponding α parametersare constants that are determined by matching the exact andthe approximate formulation at one selected value of L. Aspointed out in De Bruin et al. (2000), the assumption that theα parameters are constants is inconvenient and limits thegenerality and reduces the accuracy of the outlinedapproach.

By substituting Eqs (28) and (29) into Eq. (19) we obtainthe following bi-quadratic polynomial equation for ζ:

(30)

where

(31)

(32)

Since Eq. (30) is valid only for unstable conditions, negativereal roots are the only acceptable solutions. It can be shownusing the Intermediate Value Theorem and Descartes’ Rule ofSigns that, for a given set of surface parameters, there existsa unique negative real root of Eq. (30). This unique solutioncan be obtained using Cardan’s formulae and Ferrari’smethod (Uspensky, 1948) and is given by

(33)

252 / K. Abdella and Dawit Assefa

z

z

z

z

z

z

z

zRi

z

Lh

ht o

mu

b≤ = ≤ ≤ = ≤ =α α1 10

, , and

fz

L

z

L

z

Lh ht

hu

h h

=

= −

= −( )

− −φ γ γ α ζ1 1

12

12

αγ ζh

h hf= −

1

112 .

αγ ζm

m mf= −

1

114 .

αhh

h

b

b01= −

( )ln

bz

zhh= .

αmm

m

b

b01= −

( )ln

bz

zmo= .

L AL∞ = 0

αγh

h b h

A

Ri f∞∞

= −

1

12

f fRi

Ah hb

∞ =

.

αγm

m b m

A

Ri f∞∞

= −

1

14 .

f fRi

Am mb

∞ =

.

α α α ζζh

h hs

s= −

−∞0

α α α ζζm

m ms

s= −

−∞0

ζ ζ ζ ζ40

30

20 0 0+ + + + =a b c d

as

bRi sA

Am h

m h

h m b

m h0

00

2 2

21= − +

= −

−( )∞

∞

∞

γ αγ α

γ α

γ α,

cRi

A

sd

Ri

A

sb h m

m h

b

m h0

20

0

21=

+

= −

∞ ∞

γ αγ α γ α

.

ζ =

( ) −

−

− ( )

− −

sgn

sgn

b aa

ab a

qc

0

02

02

2

42

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where sgn represents the sign function and the other parame-ters are given by:

(34)

(35)

(36)

The fully detailed derivation as well as the proof of unique-ness of Eq. (33) may be found in Assafa (2004).

Equation (33) is the new approximation for ζ which can besubstituted into Eqs (5) and (6) to obtain the friction velocity u*and the temperature scale θ* respectively. The sensible heat andthe momentum fluxes are then computed using Eq. (2).

Note that in Eq. (33), ζ is a function of the roughness para-meters zo and zh which are related non-linearly to the turbu-lent fluxes of heat and momentum in the surface layer. Forexample, over the ocean surface zo is proportional to u*

2

according to the well-known Charnock’s formula (Charnock,1955). Due to the complex non-linearity, these relationshipscannot be solved analytically. Therefore, most surface fluxschemes, including the scheme presented in this paper,neglect these non-linear effects and assume the roughnessparameters to be known quantities in their derivation.

In the next section we will test the accuracy of this newformulation by comparing it with the fully iterated solution ofthe non-linear system.

4 Results and discussionsThe new non-iterative solution given by Eq. (33) is comparedwith the fully iterated solution of the coupled non-linear sys-tem of Eqs (1), (5), (6), (8) and (9). Although the new formu-lation is valid for any choice of k, γm, γh and prt, the valuessuggested by Dyer (1974) are used in this paper; k = 0.4, prt= 1, γm = 16 and γh = 16. Following Abdella and McFarlane(1996) and others, the comparison is carried out by comput-ing the normalized momentum and heat transfer coefficientsFm and Fh respectively which are given by

(37)

where Cmn and Chn are the neutral transfer coefficients formomentum and heat respectively.

A wide range of values is considered for both ∆θ and themean wind u(z) at the top of the surface layer. The effect ofthe roughness parameters bm, bh and their ratio

(38)

are also considered in order to test the performance of the newformulation under the full range of surface conditions. Overrough land surfaces r ≥ 1, and in some cases r >> 1, reflect-ing the more efficient transfer of momentum to the surfacecompared to the heat and mass transfer as reported byBeljaars and Holtslag (1991) for the Cabauw data and asreported by Beljaars (1995) for the Boundary LayerExperiment-1983 (BLX83) in Oklahoma. In contrast, over thesea surface and over smooth land surfaces r ≤ 1 (Brutsaert,1982; Guilloteau, 1998). Various typical surface conditionswere considered in order to illustrate the validity of the newnon-iterative formulation. Figure 1a and Fig. 1b respectivelydepict the momentum and heat transfer coefficients with r =1 over a smooth surface of roughness length zo = 10–5. This isa condition that is typical of a sea surface. As the figuresclearly show, the non-iterative solution is in excellent agree-ment with the fully iterated numerical solution. Figure 2 alsodepicts a smooth surface but with r < 1 representing a condi-tion in which the roughness length of heat exceeds the rough-ness length for momentum. Again the new formulation yieldsvery accurate results. For the sake of comparison with otherschemes, Fig. 3 also includes the solution of the Abdella andMcFarlane (1996) scheme as well as that of De Bruin et al.(2000) for the r < 1 case. Both the Abdella and McFarlane(1996) and De Bruin et al. (2000) schemes lead to an under-estimation of both the momentum and heat transfer coeffi-cients. The case of rough surfaces is depicted in Fig. 4 withzo = 0.1 and r = 1.0 where the non-iterative solution and thenumerical solution are in clear agreement. Similar accuracy isobtained for rough surfaces with r >> 1 as depicted in Fig. 5.However, as shown in Fig. 6 with r = 100, while the De Bruinet al. (2000) scheme slightly overestimates, the Abdella andMacFarlane (1996) scheme significantly overestimates boththe heat and the momentum transfer coefficients for theserough surface cases. Further investigation of the new formu-lation using a wide range of momentum roughness lengthsand the ratio r resulted in a similarly accurate approximationof the fluxes. Moreover, as the selected figures demonstrate,the new formulation works well for large and small Rib char-acterizing the success of the new formulation for a wide rangeof unstable surface conditions.

5 ConclusionThe surface flux equations for the unstable surface layer consistof several coupled non-linear equations obtained through theMonin-Obukhov similarity theory. Due to high computationalcosts associated with the fully iterated numerical solutions,

Non-iterative Surface Flux Parametrization for the Unstable Surface Layer / 253

aa

b q b q a c c q d= − + = − = −02

0 0 0 0 0 02

04

1

2

1

4, ,

qq q p

q q p b

0

2 313

2 313

0

2 4 27

2 4 27 3

= − + +

+ − + −

+ ,

p a c db

q b d a d c

ba c d b

= −( ) − = − −( )+ −( ) −

0 0 002

0 0 02

0 02

00 0 0 0

3

43

4

34

2

27

,

.

Fu

C u zF

pr w

C u zmmn

ht

hn=

( )= ′ ′

( )* ,2

2θ

θ∆

rb

b

z

zm

h

o

h= =

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254 / K. Abdella and Dawit Assefa

0

0.5

1

1.5

2

2.5

3

–10 –8 –6 –4 –2

Fm

Rib

(a)

Fh

Rib

0

0.5

1

1.5

2

2.5

3

–10 –8 –6 –4 –2

(b)

Fig. 1 a) Momentum and b) heat transfer coefficients versus Rib for zo = 10–5, and r = 1 computed using the fully iterated numerical solution (solid line) and the newnon-iterative formulation (dotted line).

Fm

Rib

0

0.5

1

1.5

2

2.5

3

–10 –8 –6 –4 –2

(a)

Fh

Rib

0

1

2

3

4

–10 –8 –6 –4 –2

(b)

Fig. 2 a) Momentum and b) heat transfer coefficients versus Rib for zo = 10–5, and r = 0.01 computed using the fully iterated numerical solution (solid line) and thenew non-iterative formulation (dotted line).

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Non-iterative Surface Flux Parametrization for the Unstable Surface Layer / 255

Fm

Rib

0

0.5

1

1.5

2

2.5

3

–10 –8 –6 –4 –2

(a)

Fh

Rib

0

1

2

3

4

5

6

–10 –8 –6 –4 –2

(b)

Fig. 3 a) Momentum and b) heat transfer coefficients versus Rib for zo = 10–5, and r = 0.005 computed using the fully iterated numerical solution (solid line), the newnon-iterative formulation (boxed symbol), the Abdella and McFarlane (1996) formulation (dashed line) and the de Bruin et al. (2000) formulation (dotted line).

Fm

Rib

0

1

2

3

4

5

–10 –8 –6 –4 –2

(a)

Fh

Rib

0

1

2

3

4

5

6

7

8

–10 –8 –6 –4 –2

(b)

Fig. 4 a) Momentum and b) heat transfer coefficients versus Rib for zo = 0.1, and r = 1 computed using the fully iterated numerical solution (solid line) andthe new non-iterative formulation (dotted line).

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256 / K. Abdella and Dawit Assefa

Fm

Rib

0

1

2

3

4

–10 –8 –6 –4 –2

(a)

Fh

Rib

0

1

2

3

4

–10 –8 –6 –4 –2

(b)

Fig. 5 a) Momentum and b) heat transfer coefficients versus Rib for zo = 0.1, and r = 1000 computed using the fully iterated numerical solution (solid line) and thenew non-iterative formulation (dotted line).

Fm

Rib

0

1

2

3

4

5

6

–10 –8 –6 –4 –2

(a)

Fh

Rib

0

1

2

3

4

5

6

–10 –8 –6 –4 –2

(b)

Fig. 6 a) Momentum and b) heat transfer coefficients versus Rib for zo = 0.1, and r = 100 computed using the fully iterated numerical solution (solid line), the newnon-iterative formulation (boxed symbol), the Abdella and McFarlane (1996) formulation (dashed line) and the De Bruin et al. (2000) formulation (dotted line).

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non-iterative solutions are highly desirable in large-scaleatmospheric models.

This paper proposes a new non-iterative formulation forapproximating the Obukhov stabilty parameter ζ needed forcomputing the surface fluxes of momentum and heat. In thisformulation, ζ is represented as a function of the bulkRichardson number Rib and makes no assumptions regardingthe ratio of the roughness length of momentum to that of heat.

The non-iterative solution is derived using the generalizedmean-value theorem which leads to a bi-quadratic equationfor ζ. The analytic solution of the bi-quadratic equation isobtained using Ferrari’s method and the existence anduniqueness of the unstable ζ in the new formulation is alsoestablished.

Investigations carried out to test the new formulationdemonstrate its validity. It is shown that the new formulationyields approximations that are in excellent agreement with the

fully iterated numerical solution for a wide range of surfacefluxes.

The possibility of implementing similar approaches for thestable surface layer will be investigated in the future. Furtherstudies will also involve the inclusion of free-convectioneffects for low wind surface conditions. For free-convectiveconditions, preliminary investigations demonstrate that theapproach used in this paper leads to equations that are toocomplex to simplify. Therefore, the present approach wouldneed to be modified in order to include free-convectioneffects.

AcknowledgementsThe general research in boundary layer modelling of one ofthe authors (KA) is supported by the Natural Sciences andEngineering Research Council of Canada (NSERC).

Non-iterative Surface Flux Parametrization for the Unstable Surface Layer / 257

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