Analogies between the Mass-Flux and the Ensemble-Mean Second-Moment

Modelling Frameworks

Dmitrii V. Mironov

German Weather Service, Offenbach am Main, Germany

dmitrii.mironov@dwd.de

Workshop on Concepts for Convective Parameterisations in Large-Scale Models„Entrainment and Detrainment in Convective Plumes“

Czech Hydrometeorological Institute, Prague, Czech Republic, 25-27 March 2009

Outline

• Mass-flux convection schemes – a recollection • Analogies between the mass-flux and the ensemble-

mean modelling frameworks – analysis of the second-moment budgets

• Extended entrainment/detrainment formulation• Towards a unified description of turbulence and

shallow convection – possible alternatives • A three-equation model of dry convective PBL • Conclusions and outlook

Recall that …

Transport equation for a generic quantity f

...// ii xfudtfdSplitting the sub-grid scale flux divergence [artificial separations of processes and scales (Arakawa 2004)]

turbiiconviiii xfuxfudxfud )/()/(/

Convection (quasi-organised)

mass-flux closure

Turbulence (random)

ensemble-mean Reynolds-average closure

Environment black hole (dustbin)?!

Many closure assumptions that require careful reconsideration.

No time-rate-of-change terms!

Mass-Flux Convection Schemes (Basics)

Still no time dependency!(cf. Gerard and Geleyn 2005)

(Notice that the above equation is pertinent to the vertical velocity of a single buoyant plume rising in a resting environment, and LESS SO to the vertical velocity of a multitude of plumes in a grid-box of an atmospheric model.)

Equation for the Vertical Velocity in the Updraught

Second-Moment Budgets within the Mass-Flux and the Ensemble-Mean Modelling Frameworks

Aims of the analytical exercise

• to understand similarities (and distinctions) between two approaches and their potentials and limitations

• to help elucidate the physical meaning of “disposable” parameters of the mass-flux models, e.g. of E and D

• to (possibly) come up with improved formulations

Principally the same as the three-delta-functions framework, but the mathematics is easier (hereafter a=au, fd=fd+fe)

Two-Delta-Function Mass-Flux Framework

Two-Delta-Function Mass-Flux Framework (cont’d)

A top-hat representation of a fluctuating quantity

After M. Köhler (2005)

Updraught

Environment

Only coherent top-hat part of the

signal is accounted for

Compare with the definition of Mc used in MF convection schemes,

1-a1, i.e. a<<1 - too restrictive, pointed out already by Tiedtke (1988, page 21)

Zero in case of zero skewness, i.e. when a=1/2

Two-Delta-Function Mass-Flux Framework (cont’d)

Compare with conventional MF schemes! (However, see Gerard and Geleyn 2005)

No explicit pressure terms!

Two-Delta-Function Mass-Flux Framework (cont’d)

Useful result (helps to relate E and D to the dissipation length scale and the grid size), but …

Scalar Variance Budget (de Rode et al. 2000,

Lappen and Randall 2001)

Different from the scalar variance budget!

Vertical-Velocity Variance Budget

Different from the scalar-variance and velocity-variance budgets!

Scalar Flux Budget

What Do We Learn from the Analytical Exercise?

Limitations of the Mass-Flux Models • The term with E and D in the <X’2> budget describes the scalar

variance dissipation• Similar terms in the <w’2> and <w’X’> budgets describes the

combined effect of dissipation and pressure redistribution and the effect of pressure destruction, respectively

• It is not easy to describe all the above effects in terms of only two quantities, viz., the rates of lateral entrainment and detrainment

Useful Outcome • Extended formulations of E and D

Important in convective flows

Modelling Pressure- Scalar Covariance

Account for the buoyancy contribution to the pressure scrambling term

Extended Formulation of E and D

Bring SOC ideas into MFC

Extended formulation

Towards a Unified Description of Turbulence and Shallow Convection

Quoting Arakawa (2004, The Cumulus Parameterization Problem: Past, Present, and Future. J. Climate, 17, 2493-2525), where, among other things, “Major practical and conceptual problems in the conventional approach of cumulus parameterization, which include artificial separations of processes and scales, are discussed.”

“It is rather obvious that for future climate models the scope of the problem must be drastically expanded from “cumulus parameterization” to “unified cloud parameterization” or even to “unified model physics”. This is an extremely challenging task, both intellectually and computationally, and the use of multiple approaches is crucial even for a moderate success.”

The tasks of developing a “unified cloud parameterization” and eventually a “unified model physics” seem to be too ambitious, at least at the moment.

However, a unified description of boundary-layer turbulence and shallow convection seems to be feasible. There are several ways to do so, but it is not a priory clear which way should be preferred (see Mironov 2009, for a detailed discussion).

Towards a Unified Description of Turbulence and Shallow Convection – Possible Alternatives

• Extended mass-flux schemes built around the top-hat updraught-downdraught representation of fluctuating quantities. Missing components, namely, parameterisations of the sub-plume scale fluxes, of the pressure terms, and, to some extent, of the dissipation terms, are borrowed from the ensemble-mean second-order modelling framework. (ADHOC, Lappen and Randall 2001)

• Hybrid schemes where the mass-flux closure ideas and the ensemble-mean second-order closure ideas have roughly equal standing. (EDMF, Soares et al. 2004, Siebesma and Teixeira 2000)

• Non-local second-order closure schemes with skewness-dependent parameterisations of the third-order transport moments in the second-moment equations. Such parameterisations are simply the mass-flux formulations recast in terms of the ensemble-mean quantities!

A Three-Equation Model of Dry Convective PBL(Dmitrii Mironov and Ekaterina Machulskaya)

Salient Features

• “Convection” and “turbulence” are treated together (both are sub-grid scale motions!)

• Prognostic equations for <ui’2> (kinetic energy of SGS motions) and for <’2> (potential energy of SGS motions); convection = Potential Energy Kinetic Energy, no reason to prefer one form of energy over the other!

• Algebraic relations for <ui’uj’> and <ui’’> (computationally inexpensive)

• Skewness-dependent formulations for the third-order moments (accounts for non-local transport properties of convection – mass-flux ideas recast in terms of ensemble-mean quantities!)

Prognostic Equations

The TKE equation (both 1-Eq and 2-Eq models),

.21

21 2

2

w

zzw

t

,21

pwuuw

zwg

zvvw

zuuw

te

ii

The potential-temperature variance equation (2-Eq model only),

One-equation (1-Eq) closure model (transport equation for TKE only) is used as a reference. 1-Eq models are still draft horses of geophysical turbulence modelling.

Prognostic and Diagnostic Variables

wvwuwwwvvuu ,,,,,

2, e

Diagnostics variables (both models): components of the Reynolds stress and of the potential-temperature flux,

Prognostic variables: 1-Eq model 2-Eq model

iiuue 21

One-Equation Model vs. Two-Equation Model – Key Differences

Equation for <’2>,

22

21

21 w

zzw

tProduction = Dissipation (implicit in all models that carry the TKE equations only).

Equation for <w’’>,

No counter-gradient term (cf. turbulence models using “counter-gradient corrections” heuristically).

2

gCz

eCw bg

Third-Order Transport Terms

,135.0,21

dedeii CzeeCpwuuw

Transport of TKE:

.135.0,2

2

dd Cz

eCw

A higher value of Cd can also be tested, e.g. Cd=0.20.

Transport of temperature variance:

Accounts for non-local transport due to coherent structures (convective plumes or rolls) – mass-flux ideas!

2/32

32/12

22 ,

SuSx

Ku ii

i

Skewness-Dependent Parameterisation of Third-Order Transport

Down-gradient term (diffusion) Non-gradient term

(advection)

In order to determine skewness, we make use of the transport equation for the potential-temperature triple correlation

Using the mass-flux ideas, the fourth-order moment is closed through the temperature skewness (Gryanik and Hartmann 2002) – no need for equations of higher order!

ii uSu 223

3113

33223

31

31

ii

iii

ii

i ux

uxx

ux

ut

Skewness-Dependent Parameterisation of Third-Order Transport (cont’d)

Anisotropy in Stable Stratification

2

gCz

eCw bg

Cb is a constant only if the simplest linear parameterisation of the pressure gradient-temperature covariance is used. Actually, Cb is a (very complex) function of the governing parameters of the problem (departure-from-isotropy tensor, heat flux vector, etc.).

Cb = const is not acceptable in case of stable stratification.

It leads to a spurious generation of heat flux (Cb must be zero as w’0).

N largeat 0where,/ FNeFCC bb

Anisotropy in Stable Stratification (cont’d)

ikll

kiikikikb uu

uuaaC 322,

32

where aik is the departure from isotropy tensor.

Physically, TCL means that all terms in the <w’’> equation must go to zero as w’0, including the <’2> term!

A two-component limit (TCL) formulation (more physically justified):

Outline of Test Cases (CBL)Convective PBL • Shear-free (zero geostrophic wind)• Domain size: 4000 m, vertical grid size: dz_min = 10 m, time step: 70 s,

simulation length: 4 h • Lower b.c. for : constant surface temperature (heat) flux of 0.24 K·m/s • Upper b.c. for : constant temperature gradient of 3·10-3 K/m • Lower b.c. for U: no-slip, logarithmic resistance law to compute surface

friction velocity • Upper b.c. for U: wind velocity is equal to geostrophic velocity • Initial temperature profile: height-constant temperature within a 780 m

deep PBL, linear temperature profile aloft with the lapse rate of 3·10-3 K/m • Initial TKE profile: similarity relations in terms of z/h • Initial <’2> profile: zero throughout the domain • Turbulence moments are made dimensionless with the Deardorff (1970)

convective velocity scales h, w*=(g<w’’>sfc)1/3 and * =<w’’>sfc/ w*

Budget of TKE in Convective PBL One-Equation and Two-Equation Models vs. LES Data

Dotted curves – LES data, solid curves – model results. Left panel – one-equation model, right panel – two-equation model. Red – buoyancy production/destruction, green – third-order transport, blue – dissipation. The budget terms are made dimensionless with w*

3/h.

Budget of Potential-Temperature Variance in Convective PBL One-Equation and Two-Equation Models vs. LES Data

Dotted curves – LES data, solid curves – model results. Left panel – one-equation model, right panel – two-equation model. Red – mean-gradient production/destruction, green – third-order transport, blue – dissipation. The budget terms are made dimensionless with w**

2/h.

Counter-gradient heat flux

TKE and Potential-Temperature Variance in Convective PBL One-Equation and Two-Equation Models vs. LES Data

TKE (left panel) and <’2> (right panel) made dimensionless with w*2 and *

2, respectively Black dotted curves show LES data, red – one-equation model, blue – two-equation model.

Mean Temperature in Convective PBL

One-Equation and Two-

Equation Models

Red – one-equation model, blue – two-equation model. Black curve shows the initial temperature profile.

Mean Temperature in Convective PBL (cont’d)

One-Equation and Two-

Equation Models vs. LES Data

Potential temperature minus its minimum value within the PBL. Black dotted curve shows LES data (Mironov et al. 2000), red – one-equation model, blue – two-equation model.

Third-Order Transport Terms in Convective PBL One-Equation and Two-Equation Models vs. LES Data

Third-order transport terms 0.5<w’u’u’>+<w’p’> and <w’’2> made dimensionless with w*3 and w**

2, respectively. Left panel: black dotted curve shows LES data, red – one-equation model, blue – two-equation model. Right panel: black dotted curve shows LES data, green – two-equation model with down-gradient parameterisation of <w’’2>, blue – two-equation model with skewness-dependent parameterisation of <w’’2>.

• Wind forcing: 5 m/s geostrophic wind • Domain height: 2000 m, vertical grid size: dz_min =10 m, time step: 70 s,

simulation length: 12 h • Lower b.c. for <’2>: zero flux, <w’’2>sfc=0 K2·m/s• Lower b.c. for : radiation-turbulent heat transport equilibrium,

Tr4+Ts

4+<w’’>sfc=0, logarithmic heat transfer law to compute the surface heat flux as function of the temperature difference between the surface and the first model level above the surface

• Upper b.c. for : constant temperature gradient of 3·10-3 K/m • Lower b.c. for U: no-slip, logarithmic resistance law to compute surface friction

velocity • Upper b.c. for U: wind velocity is equal to geostrophic velocity• Initial temperature profile: log-linear with 15 K temperature difference across a

200 m deep PBL, linear temperature profile aloft with the lapse rate of 3·10-3 K/m

• Initial profiles of TKE and <’2>: similarity relations in terms of z/h

Outline of Test Cases (strongly stable PBL)

Stable PBL One-Equation vs. Two-Equation Models

TKE (left panel) and <’2> (right panel). Red shows one-equation model, green – two-equation model with down-gradient parameterisation of <w’’2>, blue – two-equation model with skewness-dependent parameterisation of <w’’2>.

Mean Temperature in (Strongly) Stable

PBL

One-Equation and Two-Equation Models

Potential temperature.

Red curve shows one-equation model, blue – two-equation model.

Conclusions• Analogies between the mass-flux and the ensemble-mean second-order models reveal

limitations of the mass-flux closures (at least as they are currently used in NWP and climate studies)

• Extended formulation for the rate of turbulent entrainment and detrainment is suggested

• A three-equation turbulence closure model (prognostic equations for the TKE, for the temperature variance and for the temperature skewness) is developed and favourably tested through single-column numerical experiments

Prospects • A unified description of turbulence and shallow convection (can be achieved in a

number of ways but understanding of the essential physics is crucial!)• In case different parameterisation schemes are kept (“separation of processes and

scales”), an improved understanding of physical processes at work is required to bring various schemes into harmony

Thanks for your attention!

Acknowledgements. Thanks are due to Peter Bechtold, Evgeni Fedorovich, Jean-François Geleyn, Vladimir Gryanik, Friedrich Kupka, Bodo Ritter, and Jun-Ichi Yano for discussions. The work is partially supported by the EU Commissions through the Project INTAS-05-1000007-431.

Stuff Unused

Mean Temperature in Convective PBL (cont’d)

One-Equation and Two-

Equation Models vs. LES Data

Potential temperature minus its minimum value within the PBL. Black dashed curve shows LES data (Mironov et al. 2000), red – one-equation model, blue – two-equation model with skewness-dependent parameterisation of <w’’2>, green – two-equation model with down-gradient parameterisation of <w’’2>.

TKE and Potential-Temperature Variance in Convective PBL One-Equation and Two-Equation Models vs. LES Data

TKE (left panel) and <’2> (right panel) made dimensionless with w*2 and *

2, respectively. Black dashed curves show LES data, red – one-equation model, green – two-equation model with down-gradient parameterisation of <w’’2>, blue – two-equation model with skewness-dependent parameterisation of <w’’2>.

Budget of Potential-Temperature Variance in Convective PBL Two-Equation and Three-Equation Models vs. LES Data

Dashed curves – LES data, solid curves – model results. Left panel – two-equation model with down-gradient parameterisation of <w’’2>, right panel – two-equation model with skewness-dependent parameterisation of <w’’2>.Red – mean-gradient production/destruction, green – third-order transport, blue – dissipation. The budget terms are made dimensionless with w**

2/h.