THE PARAMETERIZATION OF STABLE BOUNDARY LAYERS

BASED ON CASES-99

THE PARAMETERIZATION OF STABLE BOUNDARY LAYERS

BASED ON CASES-99

Zbigniew SorbjanMarquette University, Milwaukee

Zbigniew SorbjanMarquette University, Milwaukee

i

Introduction:

Sw(z ) = w = w

Sh (z ) = w / (i) 1/2

S (z) = Sh i

z

Gradient-based scaling:

IL and SBL are both - turbulent- stably stratified - with negative heat fluxes- with wind shear- with waves

Hi

Ho Ho

Can the gradient-based scaling of the Interfacial Layer be applied in the Stable BL?

in the Stable BL:

Intermittent

Laminar

Turbulent

CASES-99

After Steeneveld et al

IntermittentTurbulent

Laminar

CASES-99

In the stable boundary layer, the Monin-Obukhov similarity predictions are:

dT/dz ~ T*/L* ~ Ho2/o

2

dq/dz ~ q*/L* ~ HoQo/o2

dU/dz ~ u*/L* ~ Ho/o

• Note 1: ”the z-less regime"

• Note 2: the fluxes H, Q, can be z-dependent (local scaling)

• Note 3: the gradient Richardson number is constant and sub-critical:

Ri = ( dT/dz) / (dU/dz) 2 ~ T* L* /u* 2 ~ 1.

Note 4: for Ho~ 0, o ~ 0 --> d/dz ~ Ho/o )2 ~ 0/0.

When the proximity to the surface is sufficiently small, turbulence can be controlled by radiative effects, and the long-wave flux cooling can exceed the sensible heat flux divergence. In this case: -the heat flux H should drop out from the list of the governing parameters (other fluxes are also small, and should not be taken into consideration as the governing parameters). -the radiative effects enter the list through the temperature gradient dT/dz.

Note 5:

Consequently, an alternate set of governing parameters can be considered:

• the scalar gradients: dT/dz, dq/dz, dU/dz, • the vertical velocity variance: w

2

• the buoyancy parameter: g

The 5 parameters above involve 4 independent units [m, s, K, kg]. Based on Buckingham's -theorem, the following 4 local scales can be derived:

un(z ) = w

Ln (z ) = w / (dT/dz) 1/2

Tn(z) = Ln dT/dz

qn (z) = Ln dq/dz

and one non-dimensional parameter: the gradient Richardson number

• Ri = (dT/dz)/(dU/dz)2

Dimensionless statistical moments are functions of the gradient Richardson number Ri:

€

XUnaTnbQncLnd=fx(Ri)

Buckingham's -theorem:

Gradient-based scaling:

un(z ) = w

Ln (z ) = w / (dT/dz) 1/2

Tn(z) = Ln dT/dz

qn (z) = Ln dq/dz

Flux-based scaling:

U*(z ) = w1/2

L* (z ) = w3/2/ [H(z)]

T*(z) = H(z)/U*(z)

q*(z) = Q(z)/U*(z)

€

X

UnaTn

bQncLn

d= fx(Ri)

€

X

U*aT*

bQ*cL*

d= f (z /L*) = const

in a “z-less” regime

A comparison:

Drawbacks of the M-O scaling in stable conditions:

• (1) The M-O similarity is invalid, when the Richardson number varies

• with height, and also when Ri is outside of the critical limit.

• (2) The M-O similarity fails in the intermittent case, when the fluxes are small.

• (3) When turbulence is weak, the fluxes are strongly contaminated by errors.

• (4) The M-O similarity introduces self-correlation errors,

i.e., the scaled variables and the stability parameter z/L* depend on surface fluxes.

• (5) Fluxes and variances can be influenced by non-turbulent motions,

which do not follow the Monin-Obukhov scaling laws.

Advantages of the gradient-based scaling:

(1) The velocity scale Un, defined by the vertical velocity variance, is less sensitive

to sampling problems, compared to the flux-based scale.

(2) The velocity scale Un is more robust, because the vertical velocity variance is

relatively less sensitive to the choice of an averaging time-scale, and its

probability distribution is nearly independent of Ri (e.g., Mahrt and Vickers, 2005).

(3) The length scale Ln does not inherit the difficulty of measuring fluxes.

(4) The measurements of temperature and humidity are more accurate

than the evaluation of their fluxes (even though an appropriate calculation

of gradients requires a sufficient vertical resolution of observations).

(5) Effects of non-stationarity and multiple layers within the SBL can be included and parametrically expressed in terms of the Richardson number Ri, which can vary with height, and can be larger than Ric.

(6) The evaluation of the SBL height h is irrelevant in this approach.

(7) The gradient-based scaling is valid in the stably-stratified interfacial layer above the CBL (Sorbjan, 2005 a, b, c).

EXPERIMENTAL DATA - CASES-99:

source: Mahrt, L. and D. Vickers, 2005: "Extremely weak mixing in stable conditions” (to appear in Bound.- Layer. Meteor.)

• Two composite cases:

W- very WEAK turbulence (strongly stable case) S- STRONG turbulence (weakly stable case)

_____________________________________

Compo- No. u* T* L site case records [m s-1] [K] [m] _____________________________________ W 22 0.04 0.07 0.51 _____________________________________ S 12 0.30 0.12 19.63 ______________________________________________

• 7x 2x 40x

• Profiles of: the potential temperature, in the composite case W (open circles) and S (filled circles). The potential temperature is the deviation from the surface value.

Weaker turbulence,stronger surface cooling

• Profiles of the wind velocity, in case W (open circles) and case S (filled circles).

Stronger turbulence,stronger wind

• Profiles of the temperature flux, in case W (open circles) and case S (filled circles).

10x

• Profiles of the Reynolds stress, in case W (open circles) and case S (filled circles).

100x

• Profiles of the temperature variance, in case W (open circles) and case S (filled circles).

• Profiles of the Richardson number Ri, in case W (open circles) and case S (filled circles).

Overcriticalregion

Gradient-based Scales:

• Profiles of length scales Ln, in case W (open circles) and case S (filled circles).

• Profiles of temperature scales Tn, in case W (open circles) and case S (filled circles).

• Profiles of velocity scales Un, in case W (open circles) and case S (filled circles).

Empirical similarity functions based on CASES-99

• The dependence of the Reynolds stress on the Richardson number Ri, in case W (open

circles) and case S (filled circles).

Ric

S

W

Neutral regime

The dependence of the temperature flux, scaled by the "gradient-based" local scales,

on the Richardson number Ri, in case W (open circles) and case S (filled circles).

Note: The dimensionless temperature flux in the SBL is bounded: H(z) ≥ - 0.3 UnTn = - 0.3 w

2 N/, for any z and any Ri (Derrbyshire, 1990: Ho ≥ -0.14 G2 f/ -- not valid in the neutral limit)

The dependence of the r, scaled by the "gradient-based" local scales, on the Richardson number Ri, in case W (open circles) and case S (filled circles).

Note: The dimensionless temperature variance in the SBL is

bounded :

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θ2(z) ≤ 1.5 Tn2, for any z and any Ri.

Conclusions:1) The "gradient-based" scaling produces consistent results in both, the weakly stable and very stable cases, with dimensionless parameters dependent on the Richardson number.

2) The "gradient-based" scaling is better suited for practical applications. It requires vertical profiles of wind and temperature, and also the vertical velocity variance. These characteristics of turbulence can be more accurately evaluated in the SBL, than the fluxes of heat, humidity, and momentum.

3) The "gradient-based" scaling seems to provide a useful framework for examining stably-stratified shear turbulence. Its general validity is being currently tested, based on a larger amount of empirical data.

• References:

• Sorbjan, Z., 2001: An evaluation of local similarity at the top of the mixed layer based on large-eddy simulations. Bound. Layer Meteor., 101, 183-207.

• Sorbjan, Z., 2004: Large-eddy simulations of the baroclinic mixed layer. Bound. Layer Meteor., 112, 57-80.

• Sorbjan, Z.:2005a, "Statistics of Scalar Fields in the Atmospheric Boundary Layer Based on Large-Eddy Simulations. Part I: Free convection". Bound.-Layer Meteorol. (in print)

• Sorbjan, Z.:2005b, "Statistics of Scalar Fields in the Atmospheric Boundary Layer Based on Large-Eddy Simulations. Part I: Forced convection". Bound.-Layer Meteorol. (in print)

• Sorbjan Z., 2005c: Similarity regimes in the stably-stratified surface layer. Submitted to Boundary-Layer Meteorology,

 • Sorbjan Z., 2005d: Comments of "Flux-gradient relationship, self-correlation and

intermittency in the stable boundary layer". Submitted to Quarterly Journal of the Royal Meteorological Society,

 • Sorbjan Z., 2005e: Local structure of turbulence in stably-stratified boundary layers.

Submitted to Journal of the Atmospheric Sciences.