A Simple Mixing Length Formulation for the Eddy-Diffusivity Parameterization of Dry Convection
ABSTRACT In this paper a simple mixing length formulation for the eddy-diffusivityparameterization of dry convection is suggested. The new formulation relates the mixinglength to the square root of the turbulent kinetic energy (e) and a time scale ( ):l = e. To close the parameterization the time scale is calculated as a functionof the boundary-layer height (h) and the convective velocity scale (w*), h/w*. Thesimpler approach of a constant time scale is also studied. The simulation of a case of dry atmosphericconvection with a one-dimensional boundary-layer model shows that the model with the new formulationreproduces quite well the main properties of the convective boundary layer. In particular,the entrainment is realistically represented by the new mixing length, which has the advantage of naturallydecreasing with the turbulent kinetic energy. Sensitivity studies to the surface flux and the lapserate, in the context of a simplified situation, show the robustness of the new formulation.
- Monthly Weather Review - MON WEATHER REV. 01/1997; 125(7).
- [show abstract] [hide abstract]
ABSTRACT: The present study introduces a parameterization for the mixing in the surface-generated free convection regime (including dry and moist convection cases). The parameterization explicitly represents the plumes that drive the convective turbulence. Ascending parcels that have similar properties in the surface layer are grouped into updrafts, with a determined fractional area. The updrafts ascend according to an entraining plume model, and may eventually overturn. The plume model can either use a time or a length scale for the lateral entrainment closure. The plumes' environment is assumed to be horizontally uniform.A one-dimensional framework is used to evaluate the parameterization in the case of a dry convective boundary layer (CBL). The predicted variances are underestimated (presumably due to the uniform environment assumption). The contrast between the strong upward motions and the compensating subsidence is simulated, and the skewness of w increases realistically with height. The plumes' dynamics agrees with experimental data throughout the entire CBL. The vertical entrainment is reproduced, and as expected, it is driven by the strongest motions. Sensitivity tests on atmospheric forcings and model parameters show the robustness of the results. The number of updrafts can be decreased to less than 10 without altering the heat transport prediction.Journal of The Atmospheric Sciences - J ATMOS SCI. 01/2003; 60(18):2313-2327.
- [show abstract] [hide abstract]
ABSTRACT: ABSTRACT A new parameterization for cumulus convection is formulated, that consists of an ensemble of small, rising parcels. Large eddy simulation (LES) results are used to parameterize the lateral mixing of such a parcel: for the mixing process a relaxation timescale is defined and its value is determined,by investigating individual LES clouds. The timescale is found to be nearly independent of cloud depth, which implies that the entrainment rate is inversely proportional to the vertical velocity. As a consequence, a dynamical feedback mechanism is estab- lished: the parcel dynamics influence the mixing rate, which, together with the environmental properties, feeds back on the parcel properties and therefore on the parcel dynamics. The multiparcel model is validated with LES fields. The characteristics of the buoyant part of the clouds are reproduced: the decreasing fractional cover and increasing liquid water content with height, the vertical dynamics and mass flux, and the conserved properties and the marginally buoyant state. The model also produces the variability typical for shallow cumulus.Journal of The Atmospheric Sciences - J ATMOS SCI. 01/2002; 59(10):1655-1668.
A SIMPLE MIXING LENGTH FORMULATION FOR THE
EDDY-DIFFUSIVITY PARAMETERIZATION OF DRY CONVECTION
JOÃO TEIXEIRA1,?and SYLVAIN CHEINET2
1UCAR/VSP, Naval Research Laboratory, 7 Grace Hopper Ave., Monterey, CA 93943, U.S.A.;
2Laboratoire de Météorologie Dynamique, Paris, France
(Received in final form 27 March 2003)
Abstract. In this paper a simple mixing length formulation for the eddy-diffusivity parameterization
of dry convection is suggested. The new formulation relates the mixing length to the square root of
the turbulent kinetic energy (e) and a time scale (τ): l = τ√e. To close the parameterization the
time scale is calculated as a function of the boundary-layer height (h) and the convective velocity
scale (w∗), τ ∝ h/w∗. The simpler approach of a constant time scale is also studied. The simulation
of a case of dry atmospheric convection with a one-dimensional boundary-layer model shows that
the model with the new formulation reproduces quite well the main properties of the convective
boundary layer. In particular, the entrainment is realistically represented by the new mixing length,
which has the advantage of naturally decreasing with the turbulent kinetic energy. Sensitivity studies
to the surface flux and the lapse rate, in the context of a simplified situation, show the robustness of
the new formulation.
Keywords: Convection, Eddy diffusivity, Mixing length.
The eddy-diffusivity (for heat) and eddy-viscosity (for momentum) approach to
representing turbulent motions can be traced back to Saint-Venant (1851) and
Boussinesq (1870) (see Frisch, 1995, for details). The mixing length concept was
apparently introduced by Taylor (1915) and Prandtl (1925) and has been used in
numerical modelling of the atmospheric boundary-layer turbulence and convection
for a long time (e.g., Priestley, 1959; Estoque, 1960).
In one way or the other, virtually all large-scale (climate and numerical weather
prediction – NWP) and mesoscale models use the eddy-diffusivity approach to
parameterize turbulent and convective motions in the atmospheric planetary bound-
ary layer (PBL). The physical parameterizations of global and mesoscale models
are essentially one-dimensional (1D) in the vertical and 1D modelling of the PBL
is an essential tool for the development and validation of turbulence parameteriz-
ations for a variety of PBL situations (e.g., Ayotte et al., 1996; Bretherton et al.,
1999) and for the improvement of weather prediction and climate models.
In order to parameterize the eddy-diffusivity (or eddy-viscosity) coefficients
(K), the most common approach in large-scale models is the one in which the
Boundary-Layer Meteorology 110: 435–453, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
JOÃO TEIXEIRA AND SYLVAIN CHEINET
coefficients are a function of the local Richardson number (e.g., Louis et al.,
1982). This type of parameterization is used in the European Centre for Medium-
range Weather Forecasts (ECMWF) model (ECMWF, 2000), the Navy Operational
Global Atmospheric Prediction System (NOGAPS) (Hogan and Rosmond, 1991)
and the Laboratoire de Météorologie Dynamique (LMD) climate model (LMD,
1997) among many others.
The major drawback of this parameterization is that it creates a direct depend-
ence between the diffusivity coefficients at a certain level and the local instability
(or stability) at that level, which is particularly unrealistic in PBL convective
situations where there is strong ‘non-local’ convective mixing. Also, due to this
dependence between the diffusivity coefficient and the local instability (or stabil-
ity), this scheme tends to be very sensitive to numerical stability problems, which
is another serious disadvantage (e.g., Girard and Delage, 1990). In Beljaars (1991)
and Teixeira (2000) more detailed discussions on the numerical stability problems
of this closure are presented.
Recently the K-profile closure (e.g., Troen and Mahrt, 1986) has been imple-
mented in some large-scale models (e.g., Beljaars and Betts, 1993) to parameterize
PBLdry convection. Inthis parameterization, a diffusivity coefficient profile, based
on mixed-layer scaling results, is imposed between the surface and the PBL top. A
major drawback of this type of scheme is that it relies heavily on the determination
of quantities (such as PBL height) that are often difficult to define precisely and
also hard to obtain accurately in a large-scale model. Different strategies for the
determination of the PBL height in large-scale models often lead to significantly
An important aspect in the physics of the convective PBL is the entrainment
that occurs at the PBL inversion. A major drawback of the Richardson number
closure is the fact that, due to its intrinsically local nature, it is not capable of
representing the mixing that occurs at the inversion, which is mainly due to eddies
that are not generated locally. In the K-profile closure used at ECMWF (Beljaars
and Betts, 1993) the entrainment flux is imposed, somewhat artificially into the
diffusion coefficient at the inversion height, as 20% of the surface heat flux. This is
done in order to have a realistic representation of entrainment and the PBL growth.
A third approach based on a predictive equation for the turbulent kinetic energy
(TKE) assumes that the diffusivity coefficient is a function of TKE (TKE closure).
This parameterization has been used in many mesoscale models (e.g., Golding,
1993; Hodur, 1997) but, with a few exceptions (e.g., Brinkop and Roeckner, 1995),
is not commonly used in large-scale models. TKE schemes have been tested in
NWP models, such as ECMWF, in the past (Manton, 1983; Dumenil, 1987), but
have never been implemented operationally. The major advantage of this closure
is that it takes into account all the sources and sinks of turbulence, and is able to
represent ‘non-local’ mixing.
A major obstacle facing the development of eddy-diffusivity closures based on
a balance equation for TKE is the specification of the mixing length. This fact has
A SIMPLE MIXING LENGTH FORMULATION
been recognized for a long time and has been prominently mentioned in Mellor
and Yamada (1982), for example.
In this paper we briefly discuss the mixing length problem and propose a simple
mixing length formulation for the convective PBL (Section 2). Preliminary results
with this new formulation have been reported in Teixeira and Cheinet (2001). In
Section 3, the 1D model that is used as the platform for testing this new formulation
is described. Results for a dry convective PBL test case are presented in Section 4,
followed by sensitivity studies (Section 5) and conclusions in Section 6.
2. The Mixing Length
2.1. DEFINITION AND FORMULATION
The eddy-diffusivity parameterization of the subgrid-scale turbulent fluxes in the
PBL relates the fluxes to the vertical gradient:
where ϕ is the mean value of a generic variable, the prime refers to the fluctuations
and w?ϕ?is the turbulent flux. The mixing length l relates the eddy-diffusivity
coefficient with a turbulent velocity wt
where Cϕis a constant.
Theoretically, the velocity scale can be determined based on a balance equa-
tion of the turbulent kinetic energy, as originally suggested independently by
Kolmogorov (1942) and Prandtl (1945). The predictive TKE equation can be
solved directly or it can be assumed that there is a local balance between the shear,
buoyancy and dissipation terms of the TKE equation (Richardson number closure).
In the surface layer, a linear relation between the mixing length and height has
been established for a long time (e.g., Von Karman, 1930; Prandtl, 1932). Black-
adar (1962) suggested the following formulation for the momentum mixing length
in a neutral atmosphere: 1/l = 1/kz + 1/λ, where k is the Von Karman constant
and λ is an asymptotic value often taken as constant. In large-scale models, Black-
adar’s (1962) mixing length formulation is often still used (e.g., ECMWF, 2000)
for wind, temperature and specific humidity. Recently more complex formulations
have been designed for boundary-layer and mesoscale models (e.g., Therry and
Lacarrère, 1983; Bougeault and Lacarrère, 1989), and although in some situations
these formulations lead to more realistic results, there is no such thing as a mixing
length formulation that is robust, flexible and simple enough to allow for a realistic
simulation of the variety of convective boundary layers that occur in the Earth’s
JOÃO TEIXEIRA AND SYLVAIN CHEINET
2.2. THE NEW FORMULATION
In this paper we propose an alternative formulation to diagnose the mixing length
for the convective boundary layer. We link directly the value of the mixing length
with the square root of the TKE (e) scaled by a variable τ with time dimension:
l = τ√e. The closure problem is now shifted from determining a length scale, to
the determination of a time scale τ. We suggest and test two different methods to
determine the time scale:
(1) τ = µh/w∗.
From mixed-layer analysis it can be argued that the time scale should be pro-
portional to h/w∗ , where h is the PBL height, w∗ is the convective velocity
scale (Deardorff, 1970) defined as w∗ =
kinematic buoyancy flux, w?θ?, g is the acceleration of gravity and θ0is a reference
temperature) and µ is a proportionality coefficient.
(where Q is the surface
(2) τ = const.
A simpler approach is to assume that the time scale can be considered constant
and it turns out that this simpler assumption produces in many situations results
that are comparable with the previous (more physically based) assumption.
Some compelling evidence that using a constant time scale may not result in a
substantial loss of accuracy is provided by the following analysis. Let us assume
that τ ∝ h/w∗is the correct solution. It can be seen that
where β = g/θ0.
Considering the simplified case of a boundary layer with a constant lapse rate
? down to the surface and a constant surface heat flux, mixed-layer scaling (e.g.,
Duan and Stevens, 2002) shows that
where t is the time, which leads to
This last equation shows that for this situation the time scale is independent of
the surface heat flux and has a rather weak dependence on time and on the lapse
rate. This suggests that using a constant time scale may not necessarily lead to
A SIMPLE MIXING LENGTH FORMULATION
much worse results than using τ = µh/w∗. A more detailed analysis of this issue
is presented in the following sections.
It should be noted that it has been suggested (e.g., Siebesma, 1996; Neggers
et al., 2002; Cheinet, 2003) to use a time scale instead of a length scale in the
context of the mass-flux closure. In the context of the stable PBL, there are some
similarities between this new formulation and an approach for the stable PBL (e.g.,
Deardorff, 1976), which uses the inverse of the Brunt–Väisälä frequency as the
time scale. However, we are not aware of the use of any similar approach for the
With the new formulation, the mixing length naturally decreases when the tur-
bulent kinetic energy decreases. In particular, at the PBL top this factor will lead,
when compared to more traditional formulations such as the one from Blackadar
(1962), to a more realistic representation of the PBL top entrainment, as will be
shown in the following sections.
3. 1D Model
The 1D boundary-layer model used in the present study has prognostic equations
for the mean potential temperature and the turbulent kinetic energy. Under hori-
zontally homogeneous conditions, assuming a zero mean vertical velocity and with
no diabatic terms, the energy conservation equation is
In the absence of wind and moisture, the prognostic equation for TKEis (e.g., Stull,
where ? represents the TKE dissipation.
The parameterization of the turbulent terms uses the eddy-diffusivity approach:
where Kθand Keare the eddy-diffusivity coefficients for potential temperature and