Jean-François Geleyn, September 2001
Introduction
The surface and atmospheric turbulent exchange schemes are designed in continuity following the suggestion of Louis (1979). However the exact implementation of the basic algorithms is the one suggested in Louis et al. (1982) where a few additional "physical" constraints are put on the formulation and the interdependence of constants. Hence, apart from the rather arbitrary prescription of the asymptotic mixing lengths for momentum and temperature/moisture four "historical" constants always controlled the intensity of vertical exchanges and their functional dependency on stability:
· k : von Karman constant ;
· b : the slope of the Monin-Obukhov universal function for momentum at neutrality ;
· c : a free-convection limit adimensionalised constant ;
· d : a non-dimensional constant representing the divergence of the momentum and heat functions in the stable case, linked to the maximum flux Richardson number in the stable case.
The current names and tunings for these constants are :
k = VKARMN = 0.4 , b = EDB = 5. , c = EDC = 5. , d = EDD = 5.
The studies performed along the last two years resulted in a concerted reformulation of the vertical profiles of mixing lengths and the model representation of the effective Richardson number for stable cases, together with the correction of some inconsistencies in the management of momentum versus heat fluxes.
Description of mixing and roughness lengths
The prescription of the asymptotic mixing lengths for momentum (M) and heat (H) required several adjustments and it was found that a functional representation with 3 parameters for each of the two cases was necessary. The 3 parameters are the asymptotic mixing lengths respectively at top and bottom of the atmosphere (or rather the bottom one l and their ratio b) and a "transition height" H, the formula for the mixing lengths themselves finally being :
l increases from the ground (l » k z0 as z® 0 ) up to l » l /2.5 at z » H /2, then decreases towards l » b l as z® ¥ .
The number of independent parameters is in fact smaller, reduced to 3 through the relationships :
The current names and tunings for these independent parameters are :
lM =ALMAV = 300 m, bM = BEDIFV = 0.05 and HM = UHDIFV -1 = 1250 m .
The main changes, still in test in September 2001 (CYCORA-ter parallel suite), are :
· changes of the ratios lH /lM and bH /bM , now chosen according to the asymptotic behaviour as z® ¥ ;
· improved consistency in the definition of lH , now depending on z0,H instead of z0,M .
This consistency also modifies the exchange coefficients : KH = lM lH fH(...) and KM = lM 2 fM(...) . The figure hereafter shows the old (dashed lines) and new (full lines) profiles of lM (blue), lH (red), and the effective length now used in the exchange coefficient for heat : Ö(lMlH) (purple).
No change was introduced recently concerning roughness lengths. Values over land are constant, derived from orography, urbanisation and vegetation cover. The ratio z0H / z0M has been tuned to .10 here. Over sea we are however using the so-called Charnock formula with and additional "gustiness" stability dependent term. The latter has been introduced to ensure that, in low wind and unstable conditions the heat fluxes do not drop to zero, not for lack of a correct formula (the limit case is in principle taken into account by the Louis scheme), but simply for a vanishing roughness. Thus the formula now reads :
where :
· Chk is the Charnock constant (VCHRNK), currently set to 0.021 ;
· z0,cr is the critical "low wind" neutral roughness length (VZ0CM), set to 1.5 10-4 m ;
· u* is the friction velocity, computed from the wind at the lowest model level as : u*2 = uL2 ´CD ;
· CD and CDN are respectively the drag coefficient and its neutral counterpart .
We are currently investigating the possibility to replace this formula by another one in the case of moist convective activity, with a progressive transition between the two extremes of "dry" and "moist" gustiness.
Description of the Richardson number
The formulation of Ri*, the "effective" Richardson number used in computations for the stable case instead of the initial one Ri, is far more complicated now, with differences between the values used for heat and momentum fluxes.
The former parameterization of the vertical turbulent fluxes in the stable case (both at the surface and in the atmosphere) included a limitation of the Richardson number to a value Ricr which inverse was named USURIC in the code. It was introduced in May 1997 to avoid the negative feedback of the reinforcement of stability in the case of very cold surface, i.e. simulating the fact that less stable parts of an inhomogeneous flow always dominate the vertical flux computations owing to the strong nonlinearity of the fluxes with respect to the gradient of potential temperature. In practice the employed equation was trivially simple :
The new formulation reads :
A first new development consisted in replacing Ricr by Ricr /Ucl at the surface and designing a vertically varying profile between the top of the atmosphere (where Ricr keeps its original value) and the bottom part of the PBL (where it tends towards the new surface value). This moved to operations in October 1999, with the first CYCORA package. The chosen expression for this variation is controlled by an exponent Uce :
The former variable USURIC is now the inverse critical Richardson number at infinity, set to 1. , while Ucl =USURICL = 4. and Uce =USURICE = 0.5 .
Next, the tests carried out on the 1998 and 1999 storm cases, which have shown a very strong sensitivity to the values of the exchange coefficients in stable regime, led to enlarging the usage of the critical Richardson number through a scaling factor a, differing for heat and momentum. The new formulation is more general and has the same behaviour near the neutral state. But for a > 1, in the very stable cases, the exchange coefficients tend towards zero as it was the case in the operational situation before May 1997 (Ricr º 0). Theoretical considerations (and related computations) have shown that the values :
aM =1 (for momentum) and aH =3 (for temperature and humidity),
have the desired properties, i.e. an asymptotic value of the sensible heat flux and a convergent critical Richardson flux number, irrespective of the Ricr value, as Ri ® ¥ .
Due to the little influence of a near the neutrality it might have been sufficient to have a unique value for each of the two types of coefficients. Unfortunately no compromise is possible in this case and the positive effect of strong values of 1/Ricr for a realistic behaviour of the storms is systematically lost. To solve the problem a second transition was introduced for heat (temperature and humidity) :
, where Rid is a new tunable "parameter", while for momentum the "status quo" was kept :
In fact the above approach has only changed the state of the problem : a realistic simulation of the storm activity requires big values of Rid (~100 or even more) while to avoid the inversion erosion small values of Rid (around 10 or less) are necessary. The results of the experiments, showing values of Rid controlling the storm development at the top of PBL and inversion erosion near the surface, led to the introduction of a vertical variation for Rid too. Its vertical profile is determined by the asymptotic value in the upper atmosphere and an exponent characterizing the vertical variation :
Rid at the top of the atmosphere (USURID -1) is tuned to about 30 and Ude (USURIDE) to 1 . At the surface Rid is taken to 0.
The figure hereafter shows the vertical profiles of the intermediate Ricr, Rid and Ri* quantities, for a constant vertical profile of Ri (Ri º 10.).
Antifibrillation scheme
The scheme, described in details in Bénard et al. (2000), was updated consistently.
References
Bénard P., A. Marki, P.N. Neytchev and M.T. Prtenjak : Stabilisation of non-linear vertical diffusion schemes in the context of NWP models. Mon. Wea. Rev., vol. 128 n° 6, pp. 1937-1948.
Louis, J.F., 1979 : A parametric model of vertical eddy fluxes in the atmosphere. Bound.-Layer Meteor., 17, 197-202.
Louis, J.F., M. Tiedke and J.F. Geleyn, 1982 : A short history of the operational PBL parameterization at ECMWF. Workshop on planetary boundary layer parameterization, ECMWF, 25-27 nov 1981, pp 59-80.
and
Bellus, M., 2000 : Vertical turbulent transport parameterization (the vertical profile for USURID parameter)
Belo Pereira, M., 2000 : Diagnostic of the shape of the exchange coefficients used in vertical diffusion
Gérard, L., 2000. : Documentation of physical parameterizations - Part III : Turbulent fluxes and PBL processes. (to be updated)
Simon, A., 2001 : Horizontal and vertical diffusion parameters tests on ARPEGE and ALADIN on problematic situations.
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