Moist aspects of TOUCANSJ.-F. Geleyn, I. Bašták-Ďurán, F. Váňa and P.

Marquet

A1WD, Ljubljana, Slovenia, 13-15/06/2012

Topics to be considered Recall about the time-step internal TOUCANS

organisation (‘dry’ or ‘moist’, indifferently) The respective roles of a ‘moist Ri’ and of a

‘moist Kh’ Consequences for the shallow-convection

parameterisation, especially at the level of the cloud-cover (SCC)

The specific moist entropy and the associated hope for parameterising moist turbulence

Moist TOMs terms and ql/i turbulent transport Plans for a ‘stand-alone’ specific SCC

computation

TOUCANS time-step organisation

The crucial choice was made at the time of the p-TKE design, for reasons having nothing to do with what is now (most crucially) at stake.

In ‘classical’ schemes, one uses e- (from the previous time-step, just advected) to compute vertical exchange coefficients Km/h, the latter are used to diffuse sl/i and qt, optionally the transport of ql/i is done, and finally e+ is obtained using the diffusive fluxes as input for the various source/sink terms.

In TOUCANS, ‘static’ estimates of Km/h are used to compute e+, the latter gives the final Km/h values for the full diffusion computations and e+ is updtated in case TOMs corrections were used.

TOUCANS time-step organisation

The key issues are then:– The inclusion of the stability dependency (via any

kind of Ri value) may simultaneously touch, the prognostic equation for e, the choice of the length scale L, the links of e+ with Km (as well as with Kh, may be independently) and the specification of the TOMs computations’ constants.

– Kh is in factor of all relevant terms for the sl/i and qt fluxes, be it without TOMs or with it!

– The belief that we might extend the dry case system via ‘moist Ri’ values for getting [e+ ,Km] and Kh !

TOUCANS time-step organisation

The latter ‘act of faith’ is prompted by the constatation that, even if ‘thermals’ need non-locality (e.g. TOMs’ terms inclusion) for their parameterisation, they are mostly associated with the clouds of the ‘shallow convection’ (and seldom with ‘dry’ PBL circulations). See the next slide as related parenthesis.

Pushing this ‘special logic’ to its ultimate consequences, one discovers that, if we may know a ‘shallow convection cloudiness’ (SCC) C on half levels even before computing the static Km/h coefficients, moist Ri values will follow, leading to consistent [e+ ,Km] and Kh derivations as well as to Kh compatible TOMs corrections, all at unchanged basic algorithms!

Towards a Unified Description of Turbulence and Shallow Convection (D. Mironov)

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

(D. Mironov) 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!

Respective roles of a ‘moist Ri’ and of a ‘moist Kh’

Even without knowing how to obtain them, we know the roles that these two quantities would play:– A ‘moist Ri’ would help obtaining the dynamical

aspects of the turbulence [e+,Km], via L, 3, 3, , etc. and hence also the momentum turbulence fluxes (by nature without significant TOMs contribution).

– A ‘moist Kh’ would allow to compute the true TKETPE ‘conversion term’ (different from the buoyancy flux of the e prognostic equation, as soon as we consider even only water vapour) and hence also the heat and moisture turbulent fluxes (via the ‘classical’ –untouched- use of Betts’ ‘moist conservative variables’ sl/i and qt).

About SCC From all the above it transpires that we need a

SCC value to infer the ‘moist values’ of Ri and Kh, and this prior for instance to the setting of L.

At that stage, contrary to what happens in the classical schemes derived from the ‘Sommeria-Deardorff’ proposal, we have very little idea about the characteristics of the ‘thermodynamical turbulence of the time-step’.

The Ri*/** methods of shallow-convection equation’s inversion go around this hurdle, but they are much heuristic and fibrillation-prone.

The use of entropic considerations is far more promising, but it will require an additional development, probably in the spirit of Tompkins (2002). For the time being we are using a proxy.

Recall: set of basic ‘dry’ equations

if

ifcifi

if

ifi

i

iiif

R

RRR

R

RRR

R

RRCR

1

/1)(

1

/1)(

)(

)(

3

3

3

33

C3 : inverse Prandtl number

at neutrality

R : parameter characterising the flow’s anisotropy

Rifc : critical flux-Richardson number (Rif at +)

Plus the ‘developed’ prognostic TKE equation (for ‘E’)

numberRichardsonfluxRparameterstuningCCscalelengthL

ELRRCCK

ERLCCKERLCK

L

EC

zK

g

z

v

z

uK

z

EK

zEA

t

E

ifK

iifKE

iKhiKm

hmEdv

&

)(/)1(

)()(

1)(

4/33

333

2/322

Moist entropic potential temperature

Having a ‘moist potential temperature’ both with good ‘Lagrangian’ and with good ‘intensive’ conservation properties has been the aim of many studies.

Recent new proposal, Marquet (2011):– Go to the most general moist entropy formulation in order to

implicitely define a s totally free from any dependency on a reference state;

– Make (at that stage only) a few approximations to get a relatively simple equivalent named (s )1 ;

– Find that the new quantity only combines the two famous ‘moist conservative variables’ of Betts (1973), l and qt :

).exp(.1 tls q (~5.87) is the novelty with

respect to past proposals. It is linked to the 2nd and 3rd laws of

thermodynamics.

Moist entropic potential temperature

verification on FIRE-I data

Cloud layer

Bett’s ‘moist conservative’ l

New proposal (s )1

More homogeneity between cloudy

and clear air parts in the new case

The ‘top of PBL discontinuity’ practically disappears when using the new quantity

The problem of the ‘ideal’ moist potential temperature

How to explain (if it is anything else than a coincidence) the observed link between moist entropy conservation and the known phenomenology of turbulent heat fluxes in the Sc case?

Buoyancy-oriented

versions of l

Entropy temperatures ‘without ’

Inversion ‘barrier’

Moist-static energy-linked E versions

‘Overshoot’

s

Well-mixed & continuous

Classical interpretations (dry ones)

Repetition of the prognostic TKE equation

L

EC

zK

g

z

v

z

uK

z

EK

zEA

t

EhmEdv

2/3221

)(

)1(

1)/(1

/1

2

2222

2222

ifm

im

hm

m

hm

m

hmhm

RSK

RK

KSKSN

K

KSK

Sz

g

K

KSK

zK

g

z

v

z

uK

Developement of terms of shear production and of production/destruction by buoyancy (‘conversion term’)

One establishes a direct link between the Richardson number, the Richardson-flux number, the conversion term

(<w’.’>) and the static stability (caracterised by N²). Should all this be conserved as such in the ‘moist’ case?

s within its related N² expansion (1/2)

For homogenous (non-saturated and fully-saturated) situations, one can compute the ‘squared’ BVF by using the idea that density is a function of moist entropy ‘s’, total water content ‘qt’ and pressure ‘p’ only.

Let us suppose that we know a ‘transition parameter’ (‘C’, which can be identified to SCC) and let us define:

F(C) ensures the transition between the non-saturated case (C=0) where moisture acts only through expansion (Rv/R) and the fully-saturated one (C=1) where it acts only through latent heat release (Lv(T)/(Cp.T)).

M(C) cares for the linked change of adiabatic gradient.

dT

Tde

Tep

TD

CFD

DCM

R

R

TC

TLCCF s

sC

C

C

vp

v )(

)()(1

1)(1

)(1)(

s within its related N² expansion (2/2)

Then, for any atmospheric condition, one gets:

Interpretation (following Pauluis and Held (2002)):

z

q

C

CCF

R

RrCM

z

q

z

s

CCMgCN t

p

pdvv

t

p

)()1()()1ln(1

)(/)(2

‘Classical’ TKE TPE conversion

Total water lifting effect (TKEPE)

-scaled differential expansion and latent heat effects (TKE ?)

A hint for a new way of looking at the d(TKE)/dt equation?One possible way: 2 Kh values [from N²(C) and from M(C).N²s], the 1st one for E+, the 2nd one for <w’.’> (in SOMs & TOMs)

…

Analytical equivalent (coding nightmare)

if

if

if

if

if

R

R

R

RR

R

1

/1)(

1

/1)(

)(

)(

3

3

3

3

'ifc*

i

*i

*i

*i*

i3

RR

R

R

RRC

C’3 : inverse Prandtl number at

neutrality times M(C)

R’ : parameter characterising the flow’s anisotropy modified in consistency with C3=>C’3

R’ifc : critical flux-Richardson number (Rif at +) modified in consistency with C3=>C’3

Plus the ‘developed’ prognostic TKE equation (for ‘E’)

z

q.K

M(C).T

TΛF(C)

R

RgW/SNR

ELRCCKERLCCKERLCK

WL

EC

z

θK

θ

g

z

v

z

uK

z

EK

zEA

t

E

th

v

vL

21

2s

*i

ifKE*iK3h

*iKm

L1s

h

1smEdv

)1()()(

1)(

33

2/322

Explanations Phenomenology: the obvious ‘target’ of (‘moist’ &

‘dry’) turbulence in the Sc case being to get a well-mixed s , one may say that it amounts to maximise the part due to qt transport in the production-destruction terms of E: Km S² (1-Rif (C));

Parameterisation: the proposed split of Rif (C) between (A) the s part interacting (alike in dry turbulence) with the shear-production term (under control of stability dependency considerations) and (B) the qt transport part (not contributing, in first approximation, to TPE) added, for conveniency, to the dissipation terms (in the equation only), tries to concretise the above maximisation principle;

Of course, all well-mixed underlying considerations should be considered as target-, non-imposed-states.

Digression: TOMs solver ! Like for classical mass-flux advection we need a

solver for the TOMs terms which is secured against linear instability (implicitness) and against non-linear instability (diagonal dominance).

We know that a certain class of ‘upstream differencing’ allows this. But for TOMs we do not know the direction (Upwards? Downwards?) of the ‘mass transfer’ …

The trick is to separate the matrix of the solver as the sum of two matrices, each with only one direction. The split of the diagonal requires a bit of attention (iterative trick), but things work out correctly (you have to believe me)!

Application to ql/i diffusion The turbulent diffusion of cloud condensate is

given by a linear combination of the fluxes of sl/i and qt (with or without TOMs effect, indifferently).

When used directly, there are many numerical problems appearing, which have made many teams (including for ALARO-0) get away from this parameterisation (going to no transport).

But the multi-upstream algorithm of the TOMs solver happens to be an excellent way to attack the problem (if complemented by a ‘non-negative end-state’ security).

The results are stunning (next slide).

Impact of the new algorithm for ql/i diffusion (qv

fibrillations)

No ql/i diffusion

New ql/i diffusion

Classical ql/i diffusion

SCC determination for entropic case

We are currently using the proxy of a ‘radiative type’ determination of C, as a temporary solution for testing.

We soon need something that:– Can be computed at the very beginning of the time

step, and on the model’s half-levels;– Concerns only the shallow convection

condensation/evaporation;– Allows replacing the spirit of the Sommeria-Deardorff

approach (link with the turbulent state of the atmosphere).

It appears that a targetted variant of the scheme of Tompkins (2002) might be answering these three challenges.

Proposal for a merge of Tompkins and ‘implicit Xu-

Randall’ (1/2) The variant of the Xu-Randall scheme currently

used in the ALARO thermodynamic adjustment possesses two degrees of freedom:– dc=1-HUcrit

But the first one is much alike the width of the

moisture statistical distribution of Tompkins and the second one can be put in correspondance with skewness (if we make it height-dependent).

Then we keep the spirit of the Tompkins scheme by simply replacing the static dc and values by prognostic ones, on the basis of the turbulent information at the end of the previous time-step.

Proposal for a merge of Tompkins and ‘implicit Xu-

Randall’ (2/2)

]).(,0[max~

)~

(1)(1)(

2

2222

z

qqwd

withddz

dK

zt

d

ttdc

ccc

Ec

withSt

)(1

0

/)1()()1()( t

vil qCF

R

RqqCMS

d (dimensionless) and (dimension of time) are the only two tuning coefficients of the proposed scheme. C will be recomputed at the next time-step on the basis of

advected dc2 and , as well as the local HU(qt).

Conclusion (for this part of TOUCANS only)

We solved the problem of ‘orthogonality’ between the moist part of turbulence and the ‘dry backbone’.

The phenomenology of the specific moist entropy gives us a track for parameterising the TKE exchange terms with ‘two Ri values’ depending on the target of such exchanges (TPE or not).

The quality of the results will however depend a lot on our capacity to handle a realistic SCC (for all its qualities, s is a bad tracer of cloudiness, to which it is mostly indifferent!).