A consistent vertical Bowen ratio profile in the planetary boundary layer

Transcription

1 Q. J. R. Meteorol. Soc. (2006), 132, pp doi: /qj A consistent vertical Bowen ratio profile in the planetary boundary layer By M. HANTEL and M. STEINHEIMER University of Vienna, Austria (Received 13 December 2005; revised 6 June 2006) SUMMARY It has recently been suggested that the integrand b = α ω of the subgrid-scale conversion rate between available and kinetic energy has a measurable impact upon the Lorenz energy cycle. Here we discuss a technique to estimate this quantity within the lower part of an atmospheric column by relating b to the subgrid-scale fluxes of sensible and latent heat in form of their sum (the total convective heat flux, c, to be diagnosed from the pertinent energy law) and their ratio (a generalized Bowen ratio, β, to be specified a priori). We focus on the frequently observed case that c vanishes at or above the top of the boundary layer, which implies that β must be minus unity at the same level (referred to as critical pressure ). β at the earth s surface is taken as measured. Observations suggest that the vertical curvature of the β profile is negative in the boundary layer. We specify an analytic vertical profile β(ζ) that interpolates these pieces of information; ζ is a non-dimensional vertical coordinate. The pertinent thermodynamic energy law from which the column profile c(ζ) is gained (referred to here as convection equation) is driven by the (observed) grid-scale budget; the solution c is over most of the boundary layerquiteinsensitivetoβ. Itisonlyintheimmediatevicinityofthecriticalpressure thatc(ζ) becomes sensitively dependent upon β(ζ); it actually turns infinite at this level (a pole of the convection equation). We remove the pole through adjusting the critical pressure by a uniquely determined (and actually quite small) amount. This makes the β profile consistent with the convection equation and with the other convective flux profiles, across the entire boundary layer. The remaining open parameter that cannot be fixed by our method is the curvature of the Bowen ratio profile. This exercise has implications for about a third of all atmospheric columns over the globe and thus may be relevant for the quantification of the global energy cycle. KEYWORDS: Buoyancy flux Convective heat flux Energy conversion Lorenz cycle 1. INTRODUCTION Convection in the atmosphere is a subgrid-scale phenomenon; it cannot be routinely observed and remains unresolved in grid-scale forecast equations. Yet the subgridscale is coupled to the grid-scale through the dynamic budgets; it deeply influences the resolved scales. The cut between grid-scale and subgrid-scale is not fixed by nature but is due to the limitations of data availability and model resolution. For this reason, the exchange rates between the global reservoirs of available potential and kinetic energy (Lorenz s quantities G, C) have traditionally been evaluated on the grid-scale (G grid, C grid +2.5 ± 0.4 Wm 2 ). Convective phenomena acting on the subgrid-scale (e.g. thunderstorms) have been treated as molecular (Lorenz 1955, 1967). With the availability of much improved datasets with respect to both completeness and accuracy (Kalnay et al. 1996; Uppala et al. 2005), it became possible to implement convection into the Lorenz energy cycle (Hantel and Haimberger 2000). In a recent study, the subgrid-scale conversion rate was estimated as C sub +2.2 ± 1.7Wm 2 corresponding to a net C = C grid + C sub +4.7 ± 2.0Wm 2 (Haimberger and Hantel 2000). This suggests that the intensity of the general circulation of the atmosphere, understood as the global mean conversion rate C (Lorenz 1960), may jump to about twice the traditional estimate if the subgrid-scale contribution is properly included. This has been the motivation for the present study. The global figures for the subgrid-scale conversion rate just cited have been based on indirect estimates of the local subgrid-scale conversion rate: b α ω, (1) Corresponding author: Department of Meteorology and Geophysics, University of Vienna, Althanstrasse 14, 1090 Vienna, Austria. michael.hantel@univie.ac.at c Royal Meteorological Society,

2 2460 M. HANTEL and M. STEINHEIMER where α is the specific volume, ω = dp/dt, the overbar represents a horizontal/time average on the pressure surface over a diagnostic grid cell (typically km and 1 6 hours), and b is the integrand of the global conversion rate C sub which is the subgridscale equivalent of the Lorenz conversion rate C. The textbook evaluations of C (Peixóto and Oort 1992) have traditionally been limited to the grid-scale component C grid. Estimating the global pattern of the local conversion rate is prerequisite for the calculation of C sub, and this requires an estimate of the vertical profile b(p) within an atmospheric column. The idea is to consider the thermodynamic grid-scale budget quantities as observed, notably local time tendencies of sensible and latent heat and their three-dimensional advection along with radiation forcing, and to use the energy conservation law plus the specified vertical Bowen ratio profile to solve for the vertical subgrid-scale flux of moist enthalpy, referred to in the following as total convective heat flux. This flux and the Bowen ratio together will eventually yield the profile of the local conversion rate. The key problem in this approach becomes the a priori specification of the Bowen ratio profile. In order to fix this profile, we restrict discussion to cases in which the total convective heat flux turns zero at or above the top of the boundary layer; this is the case in about a third of all atmospheric columns over the globe and implies that the Bowen ratio becomes minus unity at the zero-flux level. With the observed surface Bowen ratio we then have two measured values of the Bowen ratio profile; we interpolate between the two, which requires specification of the curvature of the Bowen ratio profile. It is the purpose of this paper to develop this method and to come up with an internally consistent Bowen ratio profile across the boundary layer; the only remaining free parameter will be the curvature of the Bowen ratio profile. 2. SUBGRID-SCALE VERTICAL HEAT FLUXES IN ATMOSPHERIC COLUMNS Haimberger and Hantel (2000) have related b to the subgrid-scale quantities sensible heat flux w g 1 c p T ω and latent heat flux f g 1 Lq ω expressed in pressure coordinates (T = temperature, q = specific humidity, g = acceleration due to gravity, c p = specific heat at constant pressure, L = specific condensation heat; the latter three are assumed constant). In following their approach, we shall use the vertical profiles w(p) and f(p)in the form of their sum, referred to as and in the form of their generalized total convective heat flux: c(p) w(p) + f(p), (2) Bowen ratio: β(p) w(p) f(p). (3) We shall consider the fluxes in Eq. (2) a function of the vertical pressure coordinate (to be transformed through Eq. (11) into a non-dimensional coordinate ζ ); implied is the sign convention that positive fluxes are directed downwards. The total convective heat flux is identical to the subgrid-scale vertical flux of moist enthalpy (for moist enthalpy, see e.g. Satoh 2004). Observing the gas law which replaces α by T (or virtual temperature T v ) in Eq. (1) and eliminating T ω with β brings the subgrid-scale conversion rate into the form b(p) = gκ p β(p) c(p), (4) 1 + β(p)

3 A BOWEN RATIO PROFILE IN THE BOUNDARY LAYER 2461 where κ = R/c p. Haimberger and Hantel (2000) have used Eq. (4) for estimating the local conversion rate. First, they specify the Bowen ratio profile β(p) across the entire atmosphere (Emeis and Hantel 1984). Second, they solve the pertinent energy equation (referred to in Eq. (9) as the convection equation)fortheprofilec(p). Their specification of β(p) was based on vertical profiles of the ratio between the variances of temperature and humidity (Emeis 1986 or Fig. 6.1 in Hantel et al. 1993). This recipe avoids negative values of β(p) and guarantees that the denominator of Eq. (4) cannot turn zero. Together with Eq. (4), the profiles β(p) and c(p) yield an estimate of the local conversion rate as a function of pressure. The limitation of this approach is that it causes a consistency problem in atmospheric columns in which c(p) vanishes somewhere in the vertical. Due to Eqs. (2) and (3), the statements c(p) = 0andβ(p) = 1 need to be equivalent; solutions for which one is true but the other is false are inconsistent (except for the unrealistic case, implicitly assumed by Haimberger and Hantel (2000), that both w(p) and f(p),and consequently c(p), vanish at exactly the same level). For example, at a level p with β(p) = 1butc(p) 0, Eq. (4) would yield an infinite b(p); this phenomenon will be called a pole of the solution, the corresponding level the critical pressure p crit.this inconsistency occurs when c(p) and β(p) are determined independently, as is the case for externally specified β(p). The case c(p) = 0 occurs typically at, or above, the top of the planetary boundary layer (PBL); this requires a β(p) which adopts the value 1 at the same level at which c vanishes. In order to identify and remove this inconsistency we shall take the following steps: introduce the convection equation and its driving quantity, the grid-scale budget; specify the generalized Bowen ratio profile β(p) across the PBL; solve the convection equation and identify the pole problem; remove the pole by fitting an appropriate height parameter; find consistent vertical profiles of b(p) across the PBL; discuss the implications of the method. This study will be based on the energy equation in p-coordinates, as is standard for the theory of the Lorenz cycle. However, we will introduce a non-dimensional coordinate ζ that will allow us to restrict the discussion to the vertical interval from ζ = 0(earth s surface) to ζ = 1 (critical pressure located somewhat above the top of the PBL). 3. ENERGY CONSERVATION: THE CONVECTION EQUATION The convection equation (CE) to be used in this study follows from the budget equation of moist enthalpy c p ϑ; the quantity ϑ = T + Lq/c p is the equivalent temperature. The moist enthalpy equation is a frequently used version of the energy conservation law (see textbooks of theoretical meteorology like Garratt 1992; Satoh 2004; Hantel and Haimberger 1998 ): c p dϑ dt + g R p αω = 0, (5) with R = vertical radiation flux component (positive downwards). Implicit in Eq. (5) is the assumption that phase changes of water are restricted to the gaseous and the Note that b = α ω is closely related to the buoyancy flux. In the exact definition of enthalpy for moist cloudy air, c p is the specific heat of the moist air and L is temperature dependent; also, the liquid water content has to be included (e.g. Satoh 2004). We shall consider these effects here as secondary. Most of them can be taken care of by replacing the actual through the virtual temperature (e.g. Stull 1988) while keeping c p valid for dry air and both c p and L constant.

4 2462 M. HANTEL and M. STEINHEIMER liquid phase; if freezing/melting and sublimation are involved, Eq. (5) needs some modification (Hamelbeck et al. 2001; Hantel et al. 2001). Expanding the total time derivative in Eq. (5) in hydrostatic pressure coordinates, averaging over a standard grid cell (overbar plus prime) and introducing the ( ϑ grid-scale budget: bud = c p + 2 ϑv 2 + ϑω ) + g R αω (6) t p p transforms the moist enthalpy equation into ϑ bud + c ω p p α ω = 0, (7) where V 2 is the horizontal wind, and 2 the horizontal nabla operator. The correlation in the second term of Eq. (7) may be expressed by the total convective heat flux c introduced in Eq. (2): c p ϑ ω = gc. (8) The last term in Eq. (7) is the subgrid-scale conversion rate b introduced in Eqs. (1) and (4). With these measures, the moist enthalpy equation becomes what we shall call the convection equation (CE): bud + g c p gκ β c = 0. (9) p 1 + β We shall refer to the classical Bowen ratio at the earth s surface (Bowen 1926) as β(p s ) = β s. The last term in Eq. (9), including the sign, is the local conversion term b. Throughout most of the atmosphere, it is quite small (e.g. Hantel and Haimberger 2000; see also Fig. 1). In fact, the governing balance in the CE is between the gridscale budget and the subgrid-scale total convective heat flux divergence. It is only in the immediate vicinity of a pressure level at which β(p) approaches 1 that the local conversion term b becomes critically large. The CE has been discussed by Hantel et al. (1993), Hantel and Hamelbeck (1998) and Steinheimer (2004), and applied to both the tropics and extratropics by Hantel and Haimberger (1998). We consider it a linear differential equation of first order in p; it yields, with the lower boundary condition c(p s ) = c s, the profile c(p) as solution. This interpretation implies that grid-scale budget bud(p) and Bowen ratio profile β(p) have been specified externally. Hantel et al. (1993), for finite bud(p) throughout the atmosphere, demonstrated that c(p) is forced by the CE to zero at the top of the atmosphere (TOA), no matter how β(p) is specified. Yet b(p), small throughout most of the atmosphere, becomes infinite at TOA; the reason is the pressure in the denominator of b according to Eq. (4). Likewise, the divergence of the total convective heat flux becomes infinite at TOA. They interpreted this double inconsistency as a mismatch between the specified budget profile and the boundary condition c s and removed it through modifying the grid-scale budget by a certain constant. Hantel et al. (1993) gained their results with β profiles that were nowhere negative in the atmosphere. They focused on cases with deep convection for which negative Bowen ratios do not seem to appear (see also Haiden 1997) and thus had no problem with any inconsistency of the CE away from TOA. However, at pressure levels with β = 1, the TOA inconsistency, now generated not by p but by 1 + β in the denominator of the expression for b, appears again; this phenomenon will be called the pole at the top of the PBL. Thus we shall now consider the state of affairs in the PBL; it is different from the free atmosphere. Particularly, the CE needs special treatment in the vicinity of the pole.

5 A BOWEN RATIO PROFILE IN THE BOUNDARY LAYER 2463 p [hpa] (a) bud g c/ p ' ' Australia, June E 29.5 S p [hpa] [ W kg 1 ] (b) Atlantic, August W 20.5 S [ W kg 1 ] Figure 1. Vertical profiles of the terms in the convection equation (9), for two locations. Observed grid-scale budgets of moist enthalpy (bud, solid curves) are evaluated according to Eq. (6) from ERA-40 data, and divergence of convective heat flux (g c/ p, dashed curves) and local conversion rate (b = α ω, dotted) are calculated as solutions of the CE; see also Eqs. (1) and (4). 4. GRID-SCALE BUDGET AND EASY SOLUTION Measurements and theoretical reasoning suggest that the convective flux should become small or zero near the top of the PBL in frequent and representative cases, (e.g. Lilly 1968; Kraus and Schaller 1978; Stull 1988; LeMone 2003; and many others). This is further supported by field experiments (e.g. Betts et al. 1990) as well as by large-eddy simulation experiments (Moeng and Sullivan 2004). We have reproduced this result with typical observed budget profiles gained from the 45-year re-analysis dataset ERA-40 (Kållberg 1997). The grid-scale budget as defined in Eq. (6) comprises quantities that are readily available from ERA-40. From the monthly averages of January 1993 we have, for each grid column, calculated the vertical budget profiles. The majority of these had predominantly positive bud throughout the PBL and an upward c = c s at the earth s surface as boundary condition with the result that about 36% of all grid columns over the globe exhibited c = 0 at a critical pressure p crit located somewhere around 900 to 600 hpa. Two typical examples for atmospheric columns with this property are reproduced in Fig. 1 (corresponding c flux not shown). This demonstrates that the vanishing of the total convective heat flux above the top of the PBL is a frequent phenomenon, both over land and ocean, not only for monthly averages as shown here, but also for daily budgets as found earlier (Hantel and Ács 1998). For this reason we shall run our experiments with an idealized top-hat profile with constant bud = Wkg 1 between 1000 and 800 hpa and zero above (see section 7). This will be sufficient for the arguments of this study. The profiles in Fig. 1 show that the conversion term is very small indeed. A first approximation of Eq. (9) without it yields the easy solution c e (p) of the CE: bud(p) + g ce (p) = 0. (10) p

6 2464 M. HANTEL and M. STEINHEIMER The level at which c e (p) vanishes gives a first estimate pcrit e of the critical pressure. Both approximations c e (p) and pcrit e should come quite close to the final result c(p) and p crit. It is for these reasons that the mean of the vertical bud profile and not its variance is central for the modifying theory to be developed next. Thus it will be sufficient to consider a top-hat profile for bud. The following considerations will only be relevant for atmospheric columns that have a pcrit e ; this will indicate that at this level there is a zero of the ce profile with the subsequent requirement for a consistency consideration concerning the β(p) profile. 5. INTERPOLATING THE β PROFILE IN THE PBL The observation that the total convective heat flux tends to vanish at the top of the PBL will now be used for analyzing the Bowen ratio profile in the PBL. c(p crit ) = 0 means that heat and moisture flux are equal with opposite sign which implies β(p crit ) = 1; we interpret this, in addition to the observed β(p s ) = β s at the earth s surface, as a second observed point of the β profile. In order to interpolate the two observed β values, we introduce the dimensionless vertical coordinate p s p ζ. (11) p s p crit Keeping the parameters p s, p crit fixed, ζ depends upon p. Specific values are: ζ(p s ) = 0; ζ(p crit ) = 1; ζ(0) ζ. (12) The parameter ζ represents the mass ratio of the entire atmosphere and the PBL between surface and critical pressure; it will be used as substitute for p crit. The observed values of the Bowen ratio can be written with the coordinate ζ : β(0) = β s ; β(1) = 1. (13) The simplest choice would be to connect them linearly. However, observations suggest that β(p) has negative curvature. A situation typical for fair-weather conditions (Fig. 2) is that both fluxes w(p) and f(p) start with negative (= upward) values at the earth s surface, implying β(p s ) = β s > 0. The latent flux remains negative throughout the PBL and into the free atmosphere, while the sensible flux changes sign before reaching the top of the PBL, at which it is positive (Lilly 1968). Guided by these observations we specify, with the positive parameter B, the analytic function [ ( ) { ( ) } ] 1 + βs 1 + βs β log (ζ ) = 1 + B log exp exp 1 ζ. (14) B B It has the following desired properties which represent the observable information for the otherwise unknown Bowen ratio profile: } β log (1) = 1; negative curvature controlled by the parameter B;and (15) β log (0) = β s, with β s observed at the earth s surface. The criteria (15) imply that, except for the critical pressure level pcrit e that is controlled by the vertical mean of the bud profile and forces the sheer existence of β(1) = 1, the further specification of β(ζ) is strictly independent of the bud profile. This justifies our focusing on the simplest implementation of bud, i.e. the top-hat profile.

7 A BOWEN RATIO PROFILE IN THE BOUNDARY LAYER (a) p [hpa] p [hpa] 900 f w [ W m 2 ] (b) Figure 2. (a) Measured vertical profiles of sensible (w) and latent (f ) heat fluxes, redrawn from Fig. 3.3 of Stull (1988), used here to define (b) a typical Bowen ratio profile β. For definitions of w, f see Eqs. (2) and (20) Figure STULL ( ) Bowen ratio profiles according to Eq. (14). The parameter of the curves β 2 β 6 is the curvature B (see Table 1). Profile β STULL refers to the interpolation curve shown in Fig. 2. The function (14) becomes equal to β s at ζ = 0and 1 atζ = 1. The parameter B defines the curvature of the function which for positive B is negative. The benefit of the non-dimensional coordinate ζ is that the specification (14) is independent of p crit ; fixing the critical pressure level has been delegated to the parameter ζ which will be recovered in the non-dimensional form of the CE. Figure 3 illustrates some of the logarithmic profiles defined in (14); they are in accord with the idealized PBL budget (see section 7). The corresponding values of p crit and ζ (gained with the modification procedure to be discussed next) are listed in Table 1. The indexing for the β profiles is identical to the corresponding curvature parameter B; for example, β 4 (ζ ) refers to the profile β log defined with B 4.

8 2466 M. HANTEL and M. STEINHEIMER TABLE 1. CURVATURE PARAMETERS B USED FOR DEFINING BOWEN RATIO PROFILES AND CORRE- SPONDING CRITICAL LEVELS B 1 B 2 B 3 B 4 B 5 B 6 B B (0.1) 2 (0.1) 1.5 (0.1) 1.0 (0.1) 0.5 (0.1) 0 (0.1) 0.5 = 1 = 3 = 0.10 = 0.33 = 1.00 = 3.33 p crit (hpa) ζ The parameter B has been changed in geometric progression. Very small B in the profiles β log corresponds to a step transition for β from β s to 1, while very large B approaches the limit of straight linear interpolation. β log profiles for extremes (i.e. B 1 and B ) are not drawn in Fig (a) ( ) 1.0 (b) ( ) Figure 4. (a) shows the linear interpolating Bowen ratio profile with parameter B in Table 1 (dash-dotted line) and an example with negative curvature (B 4, solid curve). (b) shows the corresponding convective height profiles π(ζ) according to Eq. (17). 6. IDENTIFYING AND REMOVING THE POLE AT THE TOP OF THE PBL Solving the complete CE (9), instead of its easy version (10), with the Bowen ratio profile (14) just specified yields the pole near the top of the PBL; this will be demonstrated as follows. We substitute the non-dimensional coordinate ζ for p in (9): p s bud(ζ ) c(ζ) gζ ζ κ ζ ζ and introduce the auxiliary vertical coordinate π(ζ) = κ ζ 0 β(ζ) c(ζ) = 0, (16) 1 + β(ζ) 1 β(ζ ) ζ ζ 1 + β(ζ ) dζ. (17) π can be interpreted as convective height, similar in some respects to a generalized optical path (Hantel et al. 1993). π approaches minus infinity when ζ approaches the limit 1, provided β(1) = 1, as is the case here; examples are drawn in Fig. 4. With the boundary condition c(0) = c s (i.e. the familiar sensible plus latent heat flux across the

9 A BOWEN RATIO PROFILE IN THE BOUNDARY LAYER 2467 (a) c( ) [ W m 2 ] c( ) ( ) (b) c( ) [ W m 2 ] = ( ) Figure 5. A close look at the pole problem; the pole is located at ζ = 1, and (b) is a magnification of (a). The figure shows selected profiles of the total convective heat flux (upper abscissae, solid curves) for a fixed Bowen ratio (lower abscissae, bold dashed curve). The flux becomes infinite (positive or negative) at the pole for arbitrary values of ζ which is the parameter of the c curves. There is just one choice of ζ for which c(ζ) stays finite at ζ = 1. This is the bold solid curve c(ζ ) in both plots, valued in this example as ζ = earth s surface), the solution of Eq. (16) is: { c(ζ) = e π(ζ) c s + p ζ } s bud(ζ ) e π(ζ ) dζ. (18) gζ 0 }{{} Z(B,ζ,ζ ) The validity of Eq. (18) can be verified by straight differentiation. The factor Z(B, ζ,ζ)is a functional depending on the entire profile β (specified through B)aswellas on the parameter ζ and the single level ζ. It is finite at all levels, including ζ =1. This has been shown rigorously by Steinheimer (2004) and is qualitatively plausible from noting that e π goes towards zero for ζ towards 1. However, the first factor of (18) becomes infinite for ζ towards 1. Thus both the total convective heat flux profile (18) and also its divergence c(ζ )/ ζ go generally towards infinity at ζ = 1. This is evidently not physical and represents the pole problem at the critical pressure. Figure 5 illustrates the pole problem. The preliminary parameter ζ e found from the easy solution, when inserted into formula (18), yields a Z(B, ζ e, 1) that is generally

10 2468 M. HANTEL and M. STEINHEIMER p e crit p crit [hpa] B Figure 6. Contours of Z(B, ζ, 1). For each B there is one corresponding ζ as a solution of the implicit equation Z(B, ζ, 1) = 0. This is the curve labelled 0. The horizontal dotted line denotes the easy estimate pcrit e for the top-hat budget profile in the present study. different from zero, equivalent to a flux profile that becomes infinite at the pole. By modifying ζ slightly, one approaches the consistent solution from the right (left) side if the first estimate of ζ was too small (big). Both the total convective heat flux and its divergence become finite at ζ = 1 for the only consistent value of ζ. Even then, the level ζ = 1 stays critical since the last term of the CE, no matter if in dimensional or non-dimensional form, still involves the ratio 0/0 so that the result cannot naively be found by determining the numerator and the denominator independently and then taking their ratio. The limit of the conversion term in the CE is finite but generally not zero. This justifies our pedantic treatment of this level when solving the CE. The condition for a finite c, both necessary and sufficient (Hantel et al. 1993; Steinheimer 2004), is to modify the parameter ζ such that Z(B, ζ, 1) = 0. (19) Figure 6 shows the function Z at ζ = 1inthe(B, ζ ) plane. The condition Z = 0, necessary for c(1) = 0 according to Eq. (18), is only observed on the curve labelled 0. Once B has been fixed, there is only one ζ that forces the model to obey (19). This unique ζ can be found in practice with a fast-converging iteration routine. For each budget profile there exists a plot of the type of Fig RESULTS Our method yields a series of consistent vertical profiles across the PBL plotted in Figs.7and8.Themainresultsaretheb profiles of Fig. 7; for example, consider the b profile that belongs to β 6, an almost linear β profile with little or no curvature (i.e. B towards ). b is here positive only close to the earth s surface but negative through most

11 A BOWEN RATIO PROFILE IN THE BOUNDARY LAYER Figure for B 2 for B 3 for B 4 for B 5 for B ' ' [ 10 2 W kg 1 ] Subgrid-scale local conversion rate b = α ω between eddy available and kinetic energy, estimated with Eq. (4) for various vertical Bowen ratio profiles. of the PBL; thus for large B the net contribution of the PBL to the column-integrated b will be negative. Conversely, for strong curvature of β (i.e. B towards zero) the b profile corresponding to β 2 is mostly positive and will yield a positive contribution to the conversion rate. Since we expect for the global atmosphere a positive conversion C sub from subgrid-scale available potential energy to subgrid-scale kinetic energy, the curvature parameters B 2, B 3, B 4 seem to be more realistic than B 5, B 6. This is supported by the observed profile β STULL of Fig. 3 which falls between our B 3 and B 4.Further, there are indications that a certain part of surface layer air rises almost undiluted through the depth of the convective boundary layer (as discussed by Haiden 1997), which means that β should decrease more slowly with height and would call for small values of our parameter B. Figure 8 shows all convective fluxes that follow from the solution of the CE for the profile β 4 (corresponding to the solid curves in Figs. 3 and 7). The combined knowledge of c and β has been used for calculating the individual convective heat fluxes w(ζ) β(ζ) 1 + β(ζ) c(ζ); f(ζ) 1 c(ζ). (20) 1 + β(ζ) Figure 8(a) shows the top-hat budget profile chosen for all cases of this study (compare with Fig. 1), Fig. 8(b) reproduces the Bowen ratio profile for B 4, and Fig. 8(c) shows the three convective heat fluxes w, f and w + f = c. Bothw and f are qualitatively similar to typical PBL profiles (Fig. 2); further, they are in accord with independent research (see several articles in Fedorovich et al. 2004, for example Moeng and Sullivan 2004). Figure 8(d) reproduces the profile of b. Note the smallness of b which is reflected in the minute difference between bud and g c/ p in Fig. 8(a). The profile of c is quite insensitive to the parameter B; in fact, c(p) in Fig. 8(c) is practically the same for all values of B in Table 1 and thus for all Bowen ratio profiles (not shown). Thus the present study reproduces the earlier result (Hantel et al. 2001) that the CE is a robust instrument for diagnosing the total convective heat flux. Concerning the idealized top-hat forcing chosen in this study, one might ask how different the solutions in Fig. 8 would be when using a non-constant bud profile. Suppose we repeat the experiment of Fig. 8 with the same vertical mean but a different variance of bud(ζ ) in the vertical. The profile β(ζ) would be the same in all cases. Clearly, c(ζ), and thus also w(ζ), f(ζ)and b(ζ), would be different.

12 2470 M. HANTEL and M. STEINHEIMER (a) 1.0 bud g c/ bud [ 10 2 W kg 1 ] (b) (c) 1 0 ( ) 1.0 c w 0.5 f c( ) [ W m 2 ] 1.0 (d) ' ' [ 10 3 W kg 1 ] Figure 8. (a) shows an idealized PBL budget (bud, solid), (b) the a priori chosen Bowen ratio profile (parameter B 4 ), (c) the final modified total convective heat flux (solid), with the subgrid-scale fluxes of sensible heat (dashed) and latent heat (dash-dotted), and (d) the local subgrid-scale conversion rate. (Note that the abscissa scale in (d) is ten times smaller than in (a)). Also in (a) is the convergence g c(p)/ p (dash-dotted) of the total convective flux that almost exactly balances bud. It may be important to note that the present theory is not a residual technique. A residual technique would try to determine the small b in the CE from the difference of the large budget and the large convective heat flux divergence which very closely balance each other; however, this would evidently be a questionable approach. In our approach, both the large flux divergence and the small conversion rate are simultaneously determined from the specified budget. This reflects the internal consistency of our theory. On the other hand, our results suggest that, for diagnosing profiles of the individual convective fluxes, the convection equation is not the proper instrument. All the CE can do is to make these fluxes consistent. This includes the main object of this study, the Bowen ratio. The CE cannot distinguish between different β profiles. While the CE can diagnose, with good accuracy, the c profile, it cannot at all diagnose the β profile. The parameter B cannot be found from the grid-scale budget but must be specified externally.

13 A BOWEN RATIO PROFILE IN THE BOUNDARY LAYER ERRORS OF THE METHOD Various assumptions and approximations have been used in the theory above which may spoil the quantitative results. Some of them are: The virtual temperature has been approximated by the actual temperature. This is of no principal concern and has been done here to simplify the argument. In the practical calculations of (e.g.) Haimberger and Hantel (2000), T has been replaced by the more exact T v. The condensation rate is not quantitatively removed by adding the dry enthalpy equation and the latent heat equation which has implicitly been done in deriving the CE (5). There is a small effect when processes with solid ice are included. Again, this effect is of no principal concern and has been quantitatively considered in the estimates of Haimberger and Hantel (2000). Dissipation has not been explicitly included in the CE. Since it is a small term it can be treated as being lumped into the diagnosed budget. The error of the grid-scale budget is unknown; this is no flaw of the present theory but a basic limitation of all subgrid-scale diagnostics. The principal results found here should not be touched by these limitations. Note that the ignorance in specifying the correct β profile, equivalent to the ignorance in fixing B, is not an error of our method. It is its principal limitation in that we are unable to diagnose b either directly or from Eq. (4). 9. CONCLUSIONS We have considered in this study the convective quantities sensible heat flux w g 1 c p T ω, latent heat flux f g 1 Lq ω, and local conversion rate b α ω between eddy available and kinetic energy. We have eliminated w and f in favour of their sum, the total convective heat flux c = w + f, and their ratio, the generalized Bowen ratio β w/f. We have considered the coupling of b, c and β through the moist enthalpy equation, referred to here as convection equation CE. The driving quantity of the CE is the grid-scale budget of moist enthalpy, the vertical profile of which is considered observed. Also observed are the values c s and β s at the earth s surface. This setting makes the CE a differential equation of first order in pressure (expressed in nondimensional form as ζ ) for the unknown profile c(ζ), driven by the grid-scale budget profile bud(ζ ); the Bowen ratio profile β(ζ) is specified externally. We have shown that the choice of this parameter function is however not critical for the solution c(ζ) since β influences the CE only through the conversion rate b which is a very small quantity; in fact, c(ζ) is practically independent of β(ζ) for most atmospheric levels. However, there is the not infrequent case in which β exerts a nontrivial (in fact a controlling) influence upon the solution of the CE; this is the case of vanishing c at the top of, or above, the PBL. For about 36% of all monthly atmospheric grid-scale columns over the globe in January 1993, the ERA-40 data have bud profiles that yield c = 0ata level around 800 hpa, a critical level p crit. We have focused in this study upon this case and have chosen for bud a top-hat profile with positive constant grid-scale budget in the PBL and zero above it. Since p crit, represented by ζ, is controlled by the vertical mean of the bud profile, this mean, as opposed to its variance, is the governing parameter of the present theory; thus the top-hat profile chosen has been sufficient for the main conclusions of this study. p crit is normalized in our theory as ζ = 1, the earth s surface as ζ = 0, the pressure level zero as ζ ;thevalueofp crit is expressed through ζ.

14 2472 M. HANTEL and M. STEINHEIMER The vanishing of c at ζ = 1 is advantageous since c(1) = 0 is equivalent to β(1) = 1. We have considered this a second observed value of the Bowen ratio profile, in addition to β s and have interpolated between the two values the logarithmic function (14) which is controlled by the curvature parameter B. We have shown that solving the CE in this way runs into a consistency problem in the immediate vicinity of ζ = 1 (a pole of the differential equation); we have removed the pole by adjusting the parameter ζ by a very small amount. The result of this exercise is a solution of the CE with the following characteristics: Theprofileofbud controls the profile of c; c vanishes at ζ = 1; consistency of the solution is enforced through fitting ζ properly; from c with β the conversion rate b follows; further, the convective fluxes w and f follow; all convective quantities b, w, f are very sensitive to B; c is quite insensitive to B. The simplification of this theory may be condensed into the following statements. Specify the vertical Bowen ratio profile through the curvature parameter B in the PBL and solve the CE, driven by the grid-scale moist enthalpy budget, in properly consistent form. This yields consistent profiles of all convective quantities total convective heat flux c, conversionrateb, sensible heat flux w, and latent heat flux f. The only profile insensitive to B is c; all others are very sensitive. The limitation of this theory is that B cannot be determined from the CE. It must come from an external specification. Thus our analysis has not yielded a better estimate of b in the PBL. It has only yielded a consistent relation between b, c, andβ. Thisis relevant for about a third of all atmospheric columns of the globe. There are other ways to more accurately specify the convective flux profiles in the PBL, different from the approach adopted here. However, any new approach would require additional data. Given the input data used in the present study, we feel that more information cannot be squeezed out. It appears that we have used the information contained in the vertical grid-scale budget profile to maximum benefit. ACKNOWLEDGEMENTS Leopold Haimberger (Vienna), Helmut Kraus (Bonn) and Stefan Emeis (Garmisch- Partenkirchen) read a preliminary version of the manuscript and gave valuable suggestions for improvement. The Austrian Academy of Sciences supported the work done within this research. Acknowledgment is also made for permission to use ECMWF s re-analysis data. Betts, A. K., Desjardins, R. L., MacPherson, J. I. and Kelly, R. D. REFERENCES 1990 Boundary-layer heat and moisture budgets from FIFE. Boundary- Layer Meteorol., 50, Bowen, I. S The ratio of heat losses by conduction and by evaporation from any water surface. Phys. Rev., 27, Emeis, S Subsynoptic vertical energy fluxes in midlatitude cyclones. Meteorol. Rundsch., 39, Emeis, S. and Hantel, M ALPEX-diagnostics: Subsynoptic heat fluxes. Beitr. Phys. Atmos., 57, Fedorovich, E., Rotunno, R. and Stevens, B. (Eds.) 2004 Atmospheric turbulence and mesoscale meteorology. Cambridge University Press

15 A BOWEN RATIO PROFILE IN THE BOUNDARY LAYER 2473 Garratt, J. R The atmospheric boundary layer. Cambridge University Press Haiden, T An analytical study of cumulus onset. Q. J. R. Meteorol. Soc., 123, Haimberger, L. and Hantel, M Implementing convection into Lorenz s global cycle. Part II: A new estimate of the conversion rate into kinetic energy. Tellus, 52A, Hamelbeck, F., Haimberger, L. and Hantel, M Convection in PIDCAP. Part I: Evaluating LAM convection. Meteorol. Atmos. Phys., 77, Hantel, M. andács, F Physical aspects of the weather generator. J. Hydrol., , Hantel, M. and Haimberger, L Diagnosing deep convection from global analyses. Meteorol. Atmos. Phys., 67, Implementing convection into Lorenz s global cycle. Part I: Gridscale averaging of the energy equations. Tellus, 52A, Hantel, M. and Hamelbeck, F Convective activity quantified by sub-gridscale fluxes. Phys. Chem. Earth, 23, Hantel, M., Ehrendorfer, M. and Haimberger, L A thermodynamic diagnostic model for the atmosphere. Part II: The general theory and its consequences. Meteorol. Z., N.F., 2, Hantel, M., Haimberger, L. and Hamelbeck, F Convection in PIDCAP. Part II: DIAMOD A standard for diagnosing convective quantities. Meteorol. Atmos. Phys., 77, Kållberg, P Aspects of the re-analysed climate. ERA-15 Project Report Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D., Gandin, L., Iredell, M., Saha, S., White, G., Woollen, J., Zhu, Y., Chelliah, M., Ebisuzaki, W., Higgins, W., Janowiak, J., Mo, K. C., Ropelewski, C., Wang, J., Leetmaa, A., Reynolds, R., Jenne, R. and Joseph, D. Series No. 2. ECMWF, Reading, UK 1996 The NCEP/NCAR 40-year reanalysis project. Bull. Amer. Meteorol. Soc., 77, Kraus, H. and Schaller, E Steady-state characteristics of inversions capping a well-mixed planetary boundary layer. Boundary-Layer Meteorol., 14, LeMone, M. A Convective boundary layer. Pp in Encyclopedia of Atmospheric Sciences Vol 1, Academic Press, Amsterdam, the Netherlands Lilly, D. K Models of cloud-topped mixed layers under a strong inversion. Q. J. R. Meteorol. Soc., 94, Lorenz, E. N Available potential energy and the maintenance of the general circulation. Tellus, 7, Energy and numerical weather prediction. Tellus, 12, The nature and theory of the general circulation of the atmosphere. WMO No.218. TP.115. World Meteorological Organization, Geneva, Switzerland Moeng, C.-H., Sullivan, P. P. and Stevens, B Large-eddy simulations of cloud-topped mixed layers. Chapter 5inAtmospheric turbulence and mesoscale meteorology. Cambridge University Press Peixoto, J. P. and Oort, A. H Physics of Climate. American Institute of Physics, New York, USA Satoh, M Atmospheric Circulation Dynamics and General Circulation Models. Springer-Verlag Steinheimer, M Grenzschichtkonvektion. Master s thesis, University of Vienna, Austria Stull, R. B An introduction to boundary-layer meteorology. Atmospheric Sciences Library, Kluwer Academic Publishers

16 2474 M. HANTEL and M. STEINHEIMER Uppala, S. M., Kållberg, P. W., Simmons, A. J., Andrae, U., da Costa Bechthold, V., Fiorino, M., Gibson, J. K., Haseler, J., Hernandez, A., Kelly, G. A., Li, X., Onogi, K., Saarinen, S., Sokka, N., Allan, R. P., Andersson, E., Arpe, K., Balmaseda, M. A., Beljaars, A. C. M., van den Berg, L., Bidlot, J., Bormann, N., Caires, S., Chevallier, F., Dethof, A., Dragosavac, M., Fisher, M., Fuentes, M., Hagemann, S., Hólm, E., Hoskins, B. J., Isaksen, L., Janssen, P. A. E. M., Jenne, R., McNally, A. P., Mahfouf, J.-F., Morcrette, J.-J., Rayner, N. A., Saunders, R. W., Simon, P., Sterl, A., Trenberth, K. E., Untch, A., Vasiljevic, D., Viterbo, P. and Woollen, J The ERA-40 re-analysis. Q. J. R. Meteorol. Soc., 131,