Coupling between CO 2, water vapor, temperature and radon and their fluxes in an idealized equilibrium boundary layer over land.

Transcription

1 Coupling between CO, water vapor, temperature and radon and their fluxes in an idealized equilibrium boundary layer over land. Alan K. Betts Atmospheric Research, Pittsford, VT 0 Brent Helliker and Joe Berry Dept. of Global Ecology Carnegie Institution of Washington Stanford, CA 0 December, 00 [JGR, submitted] Abstract We propose a simple equilibrium boundary layer model [an extension of Betts, J. Hydrometeorol., 000] to link the mixed layer equilibrium of potential temperature, water vapor, CO (and other tracers such as radon) with the surface energy and water budgets and net ecosystem exchange (and the surface radon flux). We show equilibrium solutions as a function of a set of external parameters: soil water content, which directly impacts respiration and photosynthesis, and hence transpiration; the surface net shortwave, directly linked to net radiation, which is also coupled to photosynthesis and transpiration; and the radiative cooling of the ML, which in the equilibrium model directly affects the surface sensible heat flux. We also show solutions where the net shortwave and radiative cooling are coupled to the cloud field. Our other variable model parameters are mid-tropospheric values of vapor mixing ratio, CO and radon, the properties of air entrained into the CBL, and the lapse-rate above cloud-base. We considered two idealized ecosystems, forest (based on observations in Wisconsin) and grassland (using parameter estimates from the literature) to show how the vegetation model affects the ML equilibrium. We show how the mass transport out of the sub-cloud layer and the mass exchange with the free troposphere couples the mixed layer equilibrium of water vapor, CO, and radon with the corresponding surface fluxes. We suggest that regional ML budgets may give useful constraints on regional carbon budgets.. Introduction There is a current initiative in the carbon cycle community to develop methods to resolve regional to continental scale net CO fluxes, largely in an effort to constrain larger scale estimates of the global carbon budget. Past efforts to obtain regional scale estimates of CO flux have concentrated on the atmospheric boundary layer (ABL) as the regional flux integrator (Raupach et al., ; Denmead et al., ; Levy et al., ; Kuck et al., 000; Lloyd et al., 00). The links between surface transpiration, ML depth and climate over the seasonal cycle are well established (Fitzjarrald et al., 00). Accurate ABL budgets are often constrained experimentally by accounting for flux divergence in the horizontal and through-cloud flux in the vertical (Fitzjarrald, 00). However, boundary layer budgets are difficult to close over the diurnal time scale, and hence do not represent practical means to obtain long-term regional

2 estimates of CO flux. Over the oceans, the diurnal cycle is small, and Betts and Ridgway () proposed an equilibrium approach to the ABL budgets of water and energy, in which the radiatively driven subsidence, radiative cooling and the surface fluxes are in balance. The tropical circulation has broad regions where large-scale subsidence caps a remarkably uniform convective boundary layer (CBL) of shallow non-precipitating cumulus clouds (the trade-winds ). Here the subsidence of dry air balances the large surface evaporation, while the radiative cooling of the moist CBL is balanced the subsidence of warm air though the inversion and a small surface sensible heat flux. This moist equilibrium CBL in turn feeds moisture to the precipitating convection in the ascending branches of the Hadley and Walker circulations, where the condensation heating lifts air in deep convective clouds to the upper troposphere. The circulation is closed in the subsiding trade-wind branch, where air sinks at about 0 hpa day -, as it cools radiatively. So for the tropics as a whole, there is a rough balance in energy terms between surface evaporation, condensation heating and net radiative cooling on time-scales of a week or so (Betts and Ridgway, ). Over the summer continents, away from the major monsoon circulations, there is also a rough balance between precipitation and evaporation. Consequently, Betts (000) suggested a similar approach to the surface-abl climate equilibrium problem over land by averaging over the diurnal cycle. He proposed an equilibrium model for the mixed sub-cloud layer (ML) over land, in which the diurnally averaged surface fluxes balance the fluxes through cloud-base, and the other diabatic terms in the ML, such as radiative cooling and the evaporation of falling precipitation. This is the simple assumption that, over integration periods longer than a day, the ABL approaches a steady-state or equilibrium between the surface and the overlying free troposphere. He showed the strong dependence of mean ML depth on vegetative resistance to evaporation, as well as the dependence on surface net radiative forcing. This idealized equilibrium boundary layer model (Betts 000), gave a reasonable representation of the observed trends with soil moisture of composite data from the First ISLSCP [International Land-surface Climatology Project] Field Experiment (FIFE), and the trends with soil moisture of daily mean surface data from the European Centre for Medium-Range Forecasts (ECMWF) reanalysis. Betts (000) simply specified vegetative resistance as a parameter. In this paper we extend this equilibrium model by adding a model for photosynthesis and respiration, largely as a function of soil moisture and incident radiation, and with an observational basis. This couples the CO and water vapor fluxes at the surface, and enables us to discuss the coupling of CO and mixing ratio, q, in the ML. In addition, we add a radon model with constant surface emission rate to show how this scalar is coupled to ML structure and fluxes of water and CO on time-scales longer than a day. Radon is a naturally produced radioactive gas emitted by soils, but not water bodies, and it has a short (. day) half-life in the atmosphere. This means there is a strong gradient of radon between the surface and free troposphere, which depends on the ML depth and the coupling of the ML to the layer above. The transports of radon, water vapor and CO out of the ML are coupled, but the surface fluxes are different, so we shall show that their different equilibria provide different insights into the ML budget. Betts (000) closed the ML solutions by specifying lapse rate and relative humidity just above the ML, and by requiring ML top to coincide with cloud-base. The model showed that in general the ML balance required a moisture flux into clouds, unless there was strong subsidence of rather dry air above the ML. Here we

3 make two further extensions. We will define an equilibrium shallow cloud layer, so that we can couple the entire convective boundary layer (CBL) to the entrainment or mass exchange at the CBL-top, where we specify free tropospheric values of mixing ratio, CO and radon. We can then determine the entrainment rate that will satisfy equilibrium of both the sub-cloud ML and the CBL as a whole. This corresponds quite closely to the radiative equilibrium subsidence rate, so the model is conceptually workable. The second extension will be to couple the surface net shortwave and the ML radiative cooling to the net cloud mass flux. The inspiration for the first extension to couple to the free troposphere came from observations. In a parallel paper, Helliker et al. (00) show that the monthly average vertical gradients of CO and mixing ratio between the ML and the free troposphere, coupled with a single vertical exchange velocity, were qualitatively representative of the corresponding surface fluxes: net ecosystem exchange (NEE) and evaporation, as measured by eddy covariance. In our equilibrium model, the CO and water vapor fluxes are coupled exactly, and the entrainment rate that we derive, has just the magnitude of the equivalent vertical exchange velocity of Helliker et al. (00). The coupling of the CO and water vapor structure is very tight in the ML (e.g. Betts, 00), and if this coupling can be extended to the entire CBL, then the vertical gradients, can be used to estimate the NEE from the surface evaporation. Precipitation of water from clouds complicates the use of the water budget, so we explore the information in the ML radon budget. The decay of radon in the ML and cloud layers introduces additional terms, but they can be estimated with some precision, so the radon budget may provide an additional independent estimate of the vertical mass exchange. Our equilibrium model provides a framework for interpreting data, and moves us closer to the goal of using the integrating effect of the CBL to improve regional-scale estimates net surface CO flux, and surface evaporation.. Idealized equilibrium model. Surface energy balance The model is an extension of Betts (000). The surface fluxes satisfy an energy budget. The daytime balance, suffix D, for time period t D, is t D *[SW netd +LW netd = R netd = H D + λε D + G D ] () where SW net, LW net, R net are the net short-wave, long-wave and net surface radiative fluxes, H is the sensible heat flux, E the evaporation [which we shall later assume is stomatally controlled], λ is the latent heat of vaporization, and G is the ground heat flux and heat storage in the canopy. We neglect the energy needed for photosynthesis. The night-time balance, suffix N, for time period t N, is t N *[LW netn = R netn = H N +λε N + G N ] () Add () and () to get -hour total, and neglect net storage and night evaporation by setting G D *t D +G N *t N + +λε N *t N =0 ()

4 to give R net *t = H *t + λε *t () Where we have defined -h mean fluxes as follows R net *t = R netd *t D + R netn *t N = SW netd *t D + LW netd *t D + LW netn *t N (a) H *t = H D *t D + H N *t N (b) λε *t = λε D *t D (c) We express everything in terms of -h fluxes, although we shall neglect evaporation at night by assuming E is photosynthetically controlled (see section.). We absorb the typically downward night-time sensible flux, which contributes to the stable BL at sunrise, into the -h mean. This introduces a small hidden approximation, since the model of Betts (000) couples H with an aerodynamic resistance to a skin-air temperature difference, so we are assuming a daytime, unstable transfer model for heat. One obvious extension would be to average over day and night separately. We have done this, but the solutions are not qualitatively different, so we do not present these solutions.. Radiative fluxes We shall use only a simplified representation of the -h short-wave and long-wave fluxes, which are needed to drive the energy balance (and in the next section, we couple photosynthesis to the shortwave flux). Figure shows the coupling in model data between total cloud cover (TCC) and SW net (upper panel) and LW net (lower panel) from the ECMWF 0-year reanalysis (ERA-0) averaged over three Mississippi sub-basins (see Betts et al., 00) for the months June, July and August, -. The SW relationship is independent of basin, and reasonably represented by the dashed line shown SW net = * TCC (a) For the long-wave flux there is some variation between the basins, related to temperature (the Mississippi is the coolest basin) and humidity (the Missouri is the driest basin), which we shall ignore, since we will not have a full model for the structure of the convective boundary layer (CBL). The dashed line fit shown is LW net = * TCC (b) Combining (a) and b) couples the LW to the SW flux with the simple relation LW net = 0. * (SW net - 0) (c) We shall show solutions in which we specify SW net as an external parameter. Our BL model does not give solutions for cloud cover, but it will give a net cloud mass flux, and this will be used to give solutions in which SW net is coupled to the cloud field (see.).

5 Photosynthetic uptake model We use a simplified canopy photosynthesis model, based initially on the work of Monteith (), Collatz et al. (), and then fitted to observed eddy-covariance data from temperate mixed forest in northern Wisconsin (Davis et al, 00), in which photosynthesis is a function of incoming absorbed photosynthetic photon flux density (PPFD), leaf area index (LAI), with stress factors for temperature and relative humidity at the leaf, and soil moisture. We use different light use efficiency parameters representing forests and grasslands to obtain the full range of vegetative resistance seen in grassland and forest systems (Kim and Verma 0, Betts et al. ). The stomatal resistance to the evaporation of water is then computed from the photosynthetic flux and CO gradient across the leaf. The equation set, which follows, were derived by fitting mean-daily average data from a forest site in Wisconsin (REF), so as to be consistent with the daily average equilibrium model. We obtain photosynthetic flux density (PPFD in µmol m - s - ) from SW net (in W m - ) PPFD = 0.*SW net /(0.*( -albedo)) () where the factor of 0. is the fraction of the SW spectrum in the photosynthetic band, and 0. J µmol - is the unit conversion. We specify a fixed albedo of 0., perhaps slightly high for forests, but other terms in the energy balance such as G have been neglected, so this partly compensates. The actual photosynthetic energy absorbed (APPFD) depends on LAI and was modeled as APPFD = PPFD * ( () with LAI= for grassland and for forest. We then compute the -h mean photosynthetic uptake in µmol m - s - from APPFD using PH_fd = - APPFD * epsilon * fstress * ftemp () where the light use efficiency coefficient (epsilon) incorporates the efficiency of conversion of photons captured to CO uptake, as well as the observation that light use efficiency increases as the ratio of direct to diffuse PPFD decreases. epsilon = E veg * PPFD -0.0 (0) with E veg = for forest (obtained by a fit to data) and 0 for grassland, to represent their inherently different light use efficiencies. A quadratic soil moisture stress term (fstress), which is zero at a permanent wilting point of 0. and unity at 0., is defined as a function of fractional soil water content (SWC) fstress = * SWC -. * SWC () A quadratic temperature stress term (ftemp), which is zero at 0 o C and has a maximum at o C, is defined as a function of surface temperature, T sf (units, o C)

6 ftemp = 0.0 * T sf * T sf (). Respiration The -h gross respiration is parameterised, using the same soil water stress function, as RESP_fd = fstress *[. * Q 0.*(Tsf - 0) 0 ] () Where. is the observed ecosystem respiration rate (µmol m - s - ) at 0 C and Q 0 is the coefficient by which ecosystem respiration increases with temperature. The Q 0 function (in brackets) is a basic kinetic function which accounts for the increase in molecular velocity (and hence respiration substrate velocity) as temperature increases by 0 C. A Q 0 of would represent a doubling of the respiration rate for a 0 C increase in temperature (Salisbury and Ross, ). We used Q 0 =. for forest and. for grassland. Table summarizes the three vegetation parameters used to represent the two idealized ecosystems.. Net ecosystem exchange Plant physiologists, and more recently some micrometeorologists, have adopted the use of flux density as opposed to velocity to describe trace gas fluxes. Flux densities are more desirable than velocities, because a calculated flux density is independent of pressure and changes considerably less with temperature as compared to a calculated velocity. Hence, flux densities are more immediately comparable across sites and experimental conditions. However, velocities are more easily translated into one dimensional calculations where all parameters (e.g. pressure) are prescribed and trace-gas flux can be expressed as a change in mixing ratio per unit time. We convert both photosynthesis and respiration to units of velocity (ppm CO m s - ) from flux density (µmol CO m - s - ) using the molar density conversion ρ mol = P sf /(.* (T sf +.) ) () where P sf is the surface pressure in Pa, and. is the gas constant in J mol - K - PH_vel = PH_fd /ρ mol (a) RESP_vel =RESP_fd /ρ mol (b) The net ecosystem exchange, which represents the diurnally averaged flux of CO at the surface in the ML CO budget is just the sum NEE = PH_vel + RESP_vel (). Stomatal resistance We compute leaf stomatal resistance from the photosynthetic flux and the substomatal-toambient CO gradient

7 R veg = (CO I - CO L )/ (. PH_vel ) () where the factor of. is the ratio of molecular diffusivities of H O to CO (water diffuses faster than CO, so the effective resistance is less), and CO I, CO L are the CO mixing ratios inside and outside the leaf. CO L will be computed from CO m in the mixed layer using an aerodynamic resistance at the surface, and CO I can be modeled as CO I = CO L * C RH () where C RH = * RH sf () is a function of the surface relative humidity. This relationship is based on the Ball-Berry model of stomatal resistance where CO I is linearly related to leaf surface relative humidity (assumed to be equal to surface relative humidity here; Ball ). In general, there is an interplay between the rate of CO diffusion into a leaf and the biochemical assimilation rate of CO. As relative humidity increases stomatal resistance decreases, and as a result CO I increases due to higher rates of diffusion into the leaf relative to the rate of biochemical uptake. The rate of biochemical uptake is assumed to always be greater than the rate of diffusion through wide-open stomata, such that at 00 % relative humidity CO I is still less than CO L. This gives R veg = CO L (- C RH )/ (. PH_vel ) (0) This photosynthetically determined stomatal resistance is then used to compute surface evaporation. We make the simplification that evaporation is entirely controlled by photosynthesis, which includes the approximation that there is no evaporation at night.. Method of solution at surface With R veg given by (0), the equilibrium model of Betts (000) is then solved by iteration. At the surface, we specify a range of SW net (or couple it to cloud mass flux, using (a): see., later) and find R net using (c) to give LW net. We also specify soil moisture over a range, 0. to 0.. This is the major factor controlling R veg (see later). The aerodynamic resistance and surface pressure are specified and the surface temperature (needed in (), () and ()) is found that satisfies the surface energy balance, (), as in Monteith (), using the following surface transfer equations. H = ρ sf C p g a (T sf T m ) (a) where ρ sf is the surface air density, C p is the specific heat, and g a is an aerodynamic conductance, which will be specified. T m is the temperature at the base of the ML, which we shall find from the mixed layer potential temperature, θ m, using the Poisson equation T m = θ m (P sf /00000) 0. The evaporation E = ρ sf {g a g v /(g a +g v )}(q s (T sf ) q m ) (b)

8 involves q s (T sf ), the saturation mixing ratio at T sf, as well the additional conductance across the leaf, which is the inverse of the resistance g v = / R veg For CO, we need to link the value outside the leaf to the mixed layer value NEE = g a (CO L CO m ) (c) For radon, we shall specify a constant surface flux. Mixed layer values, θ m, q m, CO m, as well as their values above the ML, are discussed in the next section, as they involve the cloud layer and its coupling to the free troposphere.. ML equilibium We consider a well-mixed layer with potential temperature, θ m, mixing ratio, q m, CO m, and radon mixing ratio, Rn m, with jumps at ML top to corresponding values just above in the shallow cloud layer θ b = θ cld - θ m (a) q b = q cld - q m (b) CO b = CO cld - CO m (c) Rn b = Rn cld - Rn m (d) Following Betts (000), ML-top, is specified as cloud-base, the lifting condensation level (LCL) of ML air with properties (θ m, q m ), so that ML-depth, h is given by h = h LCL (θ m, q m, p 0 ) () where p 0 is surface pressure. This subtle closure will be justified, when we solve the coupled system for the mass flux out of the ML (see section. and Figure b, later). At cloud-base (suffix b), we couple the fluxes with a mass flux exchange, ρ b W b (Betts, ), where we use the density of dry air at the base of the cloud layer. Note that the specification of ML-top as cloudbase fundamentally changes the equilibrium surface energy balance by introducing one component of the coupling to the cloud field. (We shall introduce a second radiative component later in section.). In contrast, traditional analyses of equilibrium evaporation (see reviews by Raupach, 000, 00) are essentially dry models, which do not include these direct impacts of the lifting condensation level. The mixed layer equilibrium for q and CO are just the balance of the surface and cloud-base fluxes. Ε = - ρ b W b q b = E b (a) ρ sf NEE = - ρ b W b CO b (b)

9 If the dry air densities are both converted to molar densities by multiplying by the constant factor (000/.), related to the conversion from Kg to g, and the molecular weight of dry air, we retrieve the CO flux density in µmol CO m - s -. The heat balance contains an additional term for the net radiative cooling of the ML, (dθ/dt) rad, which we specify as an external parameter (or couple to the cloud mass flux: see section. later). α sf H = - ρ b W b θ b ρ h LCL (dθ/dt) rad (c) where h LCL is the depth of the sub-cloud layer (with mean density ρ), and α sf =(θ/t) sf is approximately unity, unless the surface pressure is far from 000hPa. It is convenient to rewrite (c) as α sf H = - ρ b W b θ b + F rad () where the radiative flux divergence across the ML is defined positive as F rad = ρ h LCL (dθ/dt) rad = ( /g) (dθ/dt) rad () It is also convenient to define the pressure height of cloud-base = ρg h LCL (). Radon sub-model Radon is produced in the soil from decay of radium at a constant rate and the rate of emission from the soil is approximately constant. Radon decays with a half life of. days, so the ML equilibrium for radon contains an extra decay term. Equilibrium radon in the ML can be written as, F Rn = - ρ b W b Rn b + Rn m * ρw dm (d) where F Rn is the surface flux of radon, and W dm represents the radon decay velocity scale for the ML, given by ρw dm = ρh LCL * L d () where L d is the decay constant for radon,.0 * 0 - s -. Using (), replaces h LCL in () with the pressure thickness ρw dm = ( /g) * L d ( ) For our analysis we used a constant F Rn of 0.0 Bq m - s -. Typical concentrations of radon are about 0. Bq m at STP (standard temperature and pressure) in the free troposphere, and.. Bq m in the ML (Kritz et al. )..0 Upper boundary conditions for ML.

10 Closing the ML equations requires values for θ cld, q cld, CO cld and Rn cld just above cloud-base as a function of ML depth and the stratification above. As in Betts (000), we do not solve for the thermodynamic structure in the cloud layer, but make two simplifications. We define θ cld = + Γ ( - 0) () where Γ = -dθ/dp is a measure of the stratification in the cloud layer. We find q cld by constraining the sub-saturation or RH cld just above cloud-base. Betts (000) simply specified a range, but here, to reduce the number of free parameters, we shall couple subsaturation and RH in the cloud layer to ML depth using cld = ( - 0) (0) Figure shows that this constrains cloud layer RH cld to decrease from % to % as the depth of the sub-cloud layer increases from 00 to 00 hpa (roughly 000 to 000m). Appendix A summarizes the general relationship between RH and. Equations () and (0) are simplifications which replace the more difficult task of solving for the coupled thermodynamic structure in the cloud layer, which includes the solution for the radiation field in a partially cloudy CBL (recall that in (c) we specify the radiative term). CO and radon are transported out of the sub-cloud layer into clouds, and then mixed through the cloud layer, but they can be considered passive tracers during this process (allowing for the radon decay). We shall solve for the cloud base mass flux using (a), but we need a further closure to give values for CO cld and Rn cld in (c) and (d). For this we consider the equilibrium of the CBL as a whole.. CBL equilibrium The condensation, upward advection and evaporation of liquid water in shallow cumulus clouds is a destabilizing process (Betts,, ) and the CBL will deepen, entraining warm dry air from above, unless there is a compensating subsidence. In the subsiding branches of the atmospheric circulation, such as the trade-winds over the oceans, the radiatively driven subsidence, which is of order 0.00 m s - or 0.0 Pa s -, balance the CBL growth by entrainment (Betts and Ridgway,, ). Suppose such an equilibrium is established in the long-term over land with subsidence balancing an entrainment velocity, W E. If we suppose there is constant subsidence in the cloud layer (that is, all the mass divergence is in the ML), the equilibrium CO budget for the whole CBL is just ρ sf NEE = - ρ b W Eb (CO t - CO m ) () where CO t is the value in the free troposphere, which we shall consider fixed at ppm. We have again defined a mass flux at cloud-base corresponding to the subsidence, which being constant above cloud-base satisfies ρ b W Eb = ρ t W Et ()

11 Following conventional meteorological practice we shall solve the system in pressure coordinates to absorb the density variation with height. Eq. (0) gives a direct solution for CO m, as a balance between the surface NEE and the free tropospheric entrainment, given the vertical mass flux ρ b W Eb. The equivalent equilibrium equation for mixing ratio is E = - ρ b W Eb (q t - q m ) () where q t is a free tropospheric mixing ratio, above the CBL, typically in the range to g Kg -. This can then be inverted to give the mass exchange ρ b W Eb = E /(q m - q t ) ( ) Equations (a) and () can both be satisfied. We use (a) to determine the total cloud-base mass exchange as part of a fully coupled ML equilibrium, and then () to diagnose the smaller subsidence or mass exchange needed to maintain the CBL equilibrium. A little manipulation shows that the values above cloud-base satisfy q Cld = q t + E * ρ b W CLD /(ρ b W b * ρ b W Eb ) (a) CO Cld = CO t + ρ sf NEE * ρ b W CLD /(ρ b W b * ρ b W Eb ) (b) where we have defined a net cloud mass flux ρ b W CLD = ρ b W b - ρ b W Eb () as a measure of an additional cloud mass exchange needed to pump sufficient water vapor out of the ML to satisfy the ML component of the equilibrium model. Equations (a) and (b) show that q cld > q t and CO cld > CO t, provided W CLD > 0. The extra cloud flux modifies the layer above cloud-base, and balances the mean subsidence in this same layer. We shall find that if evaporation is very low, so that the ML becomes very deep, then W CLD goes to zero, and the conceptual cloud field disappears (see section.), and unmodified free tropospheric air is entrained into the ML. The radon balance is more complex as there are the additional decay terms. The equilibrium radon budget for the whole CBL is F Rn = - ρ b W Eb Rn b + Rn m * ρw dm + Rn CL * ρ CL W dcl () where the decay constant for the cloud layer (subscript CL) can be written ρ CL W dcl = (P CL /g) * L d () However, two additional assumptions are needed to compute the decay in the cloud layer. One is cloud layer depth, for which we simply assume the entire CBL is 0hPa deep, so that the cloud layer shrinks as the ML deepens, and P CL = 0 -. The second is that, unlike the ML, the cloud layer is generally not well mixed, so we assume a linear distribution between the value just

12 above cloud-base and the free troposphere, so that Rn CL = (Rn Cld + Rn t )/. Algebraic manipulation then gives these expressions for radon just above cloud-base and for the ML Rn Cld = [Rn t (+C)(-C) + F Rn ρ b W CLD /(ρ b W b * ρ b W Eb )]/[+C+C+CC] (a) Rn m = [Rn t (-C) + F Rn (ρ b W b +ρ b W Eb *(C+C+CC) /(ρ b W b * ρ b W Eb *(+C))] /[+C+C+CC] (b) where we have defined the ratios C= ρw dm /ρ b W b ; C= ρw dm /ρ b W Eb ; C = ρ CL W dcl /ρ b W Eb. For almost all ML solutions, C, C and C are <. Note that because of the additional decay in the cloud layer, (a) does not give Rnc ld = Rn t, when ρ b W CLD = 0. This would require a refinement to the model to ensure that P CL and therefore C also become zero. However the impact of our constant CBL depth assumption of Rn m is small.. Coupling of transports in CBL For an equilibrium CBL, () and () link the surface NEE and E to a common mass transport and a gradient between the ML and the free troposphere. Dividing () by () gives ρ sf NEE/E = (CO t - CO m )/ (q t - q m ) () In a companion paper, Helliker et al. (00) use () to estimate NEE over the annual cycle from measurements of surface evaporation and the vertical gradients of CO and q between the ML and mid-troposphere.. Closure and method of solution As in Betts (000) (based on the earlier papers of Betts () and Tennekes ()), we use a ML closure on the virtual potential heat flux H vb = - k H vsf with k = 0. (0) where the virtual heat potential flux can be approximated as H v = α H λe () The coefficient is 0.0 θ (C p /λ) = 0.0 at θ =0K, and it will be treated as a constant. Substituting (a) and () in (0) and rearranging gives H vb = -k F rad /(k+) () Since α b H b = H vb 0.0 λe b, we can, using (), (), (a) and (), solve for α b H b = - ρ b W b θ b = [-k F rad /(k+) - 0.0(R net F rad )]/(-0.0) () This form of the solution is useful because it does not depend on the surface energy flux partition. The similar solution for the surface heat flux is

13 α sf H = [F rad /(k+) - 0.0R net ]/(-0.0) () Since F rad is proportional to ML depth, h LCL, from (), equation () determines the surface sensible heat flux as a linear function of = ρ g H LCL, as long as the ML radiative cooling rate is a constant. The surface heat flux is primarily controlled by the depth of the ML and its radiative cooling rate (Betts, 000). In fact, all the fluxes at both top and bottom of the ML are then linear functions of, so we shall use this as a coordinate in many figures. Table shows the set of specified external parameters, their default values and the ranges used, where they were varied. We find equilibrium solutions iteratively. Starting with a guess of, we find θ cld and cld from () and (0), which also gives us q cld. A first guess of θ b gives θ m, and q m is found from RH m at the base of the ML, as it is a unique function of (see Appendix A). From θ m and q m, we can find a T sf that gives an H and λe that satisfy (), with R veg given by (0). An iteration is needed on R veg, since it is a weak function of T sf and RH sf. Then can be updated from () using (). θ b can be updated from () by finding ρ b W b from (a) and (b). Solutions for the CO structure follow from ( ), () and (b). CO m has a weak impact on R veg. Solutions for radon then follow from ().. Cloud coupled solutions The formation of clouds at the top of the ML introduces two feedbacks. ML-top is constrained to the LCL, as represented by (), and the cloud field has a radiative impact. For two variables, surface net short-wave and ML radiative cooling rate, which both depend on cloud-cover, in addition to using parameter ranges in Table, we also defined cloud coupled values, using these simple closures related to the net cloud mass flux, given by (). SW net = 0 00 ρ b W CLD /0.0 for ρ b W CLD > 0; and 0 for ρ b W CLD <0 (a) (dθ/dt) rad = - + ρ b W CLD /0.0 for ρ b W CLD > 0; and - for ρ b W CLD <0 (b) In the range of solutions, 0 <ρ b W CLD <0.0, (a) and (b) give the same ranges shown in Table, namely 0 > SW net >0 Wm - and - < (dθ/dt) rad < - K day -, that is the net shortwave and ML cooling rate decrease as the cloud mass flux increases. Over the same range, the surface LW net, which satisfies (c), increases from -0 to -0 Wm -.. Equilibrium solutions. We shall show the sensitivity of the equilibrium solutions to the parameter ranges shown in Table and in addition, some figures will also show the cloud-coupled solutions, using (). Since all fluxes are -h means, we will drop the suffix in the Figures.. Sensitivity to soil moisture and ML depth This sensitivity is the strongest control on vegetative (stomatal) resistance, so we first show the equilibrium solutions with varying SWC for three values of SW net (we could have shown R net instead, since the SW and LW fluxes are coupled through (c)), with the stratification, ML cooling rate and q t set at their default values, shown in Table.

14 Figure a shows that R veg increases sharply, as SWC drops towards the model permanent wilting point of 0., since R veg is affected by the stress function (fstress in ()). Surprisingly however, except at very low SWC, there is almost no dependence on SW net. This arises from a cancellation of effects in (0). Photosynthesis increases with SW net, but changes in CO and RH sf at the leaf largely compensate in (0). This simplification, which appears to be a characteristic of the Wisconsin forest, leads to a reduced variability in the CO budget in later figures. Figure b shows that the equilibrium ML depth increases sharply as SWC decreases (and stomatal resistance increases), as reduced evaporation leads to a warmer drier equilibrium ML; and also increases as the radiative driver, SW net, increases the surface energy fluxes. The left-hand-scale shows ML depth as a pressure thickness, which we shall use in subsequent figures; and the righthand-scale the corresponding depth to cloud-base in meters (with slight approximation). The heavy dashed line is the single cloud coupled solution, derived by doing a further iteration to satisfy as well (a, b). Figure shows the surface energy fluxes (upper panels (a) and (c)) as a function of SWC (left) and on the right. The lower panels, (b) and (d), show the corresponding daily mean evaporative fraction, defined as EF = λε /(H + λε) () We shall discuss the family of fixed SW net curves first. As expected, panel (a) shows that for fixed SW net evaporation increases and sensible heat flux decreases non-linearly as SWC increases, since R veg decreases (Figure a), while both fluxes increase as the surface radiative forcing SW net increases. Figure b shows that the increase of EF with SWC is almost independent of SW net (this follows from the very weak dependence of R veg on SW net in Figure a). Figure c shows that the surface fluxes are linear functions of ML pressure thickness as discussed in section.. As SW net increases (and R net with it using (c)), evaporation increases, but the surface sensible heat flux changes little, as () shows it depends primarily on F rad. Consequently, for a given ML depth, EF increases with surface radiative forcing (Figure d). Note that the left and right panels give quite a different perspective on the surface energy balance. As a function of soilwater, the fluxes are non-linear, but EF is only a weak function of SW net. As a function of ML depth, the fluxes become linear, because the energy balance terms are proportional to depth,. However, whereas at a given SWC, H increases with SW net, for a given ML depth, H decreases slightly with SW net, because of the large increase of equilibrium depth with solar forcing, shown in Figure. The upper panels show that our cloud coupled model (heavy dashes) has a significant impact on the fluxes. H becomes non-linear, primarily because the ML radiative cooling rate increases from wet to dry soils (the corresponding cloud mass flux will be shown later in Figure b); while the latent heat flux has little variation, as R net increases significantly from wet to dry soils from (a) and (c). However, the lower panels show that the corresponding impact on EF is rather small. Figure shows the corresponding set of plots as a function of SWC on the left and on the right for θ m and q m (upper panels) and CO m and radon (lower panels). The cloud coupled solutions, shown in panels (a), (b) and (d) intersect the family of fixed SW net solutions, because

15 of the variation of SW net given by (a). The cloud coupled solutions for panel (c) are indistinguishable from the plots for SW net =00 W m -, and are not shown. We see, as expected, that the ML gets warmer and drier as SWC decreases and the ML deepens. As a function of SWC (panel (a)), θ m and q m also get warmer and drier with SW net. However, when plotted as a function of (panel (c)), the ML gets slightly warmer and wetter as the surface SW net increases, as discussed in Betts (000). In terms of the model, equations () and (0) determine θ cld, q cld just above the ML from. The jump θ at cloud-base decreases slightly as the cloudbase mass flux increases with SW net (see section.), so that θ m increases slightly. In the moisture budget, the λe increases faster with SW net than the cloud-base mass flux, so that the jump q at cloud-base increases in magnitude, and so q m increases. Figures (b) and (d) shows the corresponding plots for ML CO and radon. For radon, where the surface flux is constant, the ML equilibrium depends strongly on the radiative forcing, since this affects the cloud-base flux (see next section), so Rn m decreases in both panels (b) and (d) as SW net increases. For CO m, the decrease with increasing SWC or decreasing ML depth at constant SW net follows from the increasing photosynthesis as fstress in () approaches unity. As a function of SWC, NEE depends rather weakly on SW net (see Figure a), so that CO m increases with SW net, as the cloud-base mass flux increases, transporting more CO out of the ML. However, as with q m, as a function of, we see the reverse; CO m decreases weakly with increasing surface solar forcing. Figure a shows the increase of both photosynthesis and respiration with increasing SWC and SW net. The difference between the two curves is the NEE, which increases over wetter soils, but has rather a weak dependence on SW net, except over dry soil, when NEE is reduced. This insensitivity of NEE to SW net is again related to the very weak dependence of R veg on SW net shown in Figure a. This in turn leads to the tight coupling between ML CO m and q m shown on the left-hand scale of Figure b, which is almost insensitive to net shortwave (as can be deduced from Figures (c) and (d)). However, the link between NEE and CO m, shown on the right-handscale is strongly dependent on SW net, because CO m is directly coupled to the vertical exchange, shown in the next section. The cloud coupled equilibrium solution is more non-linear.. Vertical mass fluxes We have determined two equilibrium mass fluxes: the mass exchange, ρ b W b, at the top of the mixed layer which balances the surface fluxes in (); and the mass exchange, ρ b W Eb, with the free troposphere above the CBL, which balances the surface fluxes in () and (). These are both shown in Figure (a) as a function of SW net, and for the cloud coupled solution. For the specified SW net family, the mass exchange, ρ b W Eb, with the free troposphere needed to give CBL equilibrium, increases weakly with ML depth, and sharply with SW net. The mass exchange, ρ b W b, which balances the ML budgets also increases with SW net, but falls steeply as the ML deepens, so that each pair crosses at a ML depth of hpa. The heavy dashed curves show the cloud coupled solutions, which have a similar intersection, but cut across the family of SW net curves. The difference is the additional cloud mass flux, ρ b W CLD, which we defined in (), and it is shown in the lower panel (b). This cloud mass flux does increases with SW net, but all the curves fall steeply as the ML deepens to cross zero at a depth of hpa. Above this ML depth, the total mass exchange at cloud-base is less than the equilibrium subsidence, and q CLd < q t, and

16 the model with a cloudy boundary layer becomes physically unrealistic. Below hpa, ρ b W CLD is a measure of the extra cloud mass exchange needed to pump sufficient water vapor out of the ML to satisfy the ML component of the equilibrium model. Above this ML depth threshold, we could consider that the solutions to the dry ML model would be appropriate, without the condition that the top of the ML also be cloud-base. The cloud coupled solution is almost linear with ML depth, so that (a, b) also give almost a linear variation of SW net and (dθ/dt) rad with ML depth. One general and important conclusion is that unless SWC becomes very low so that the ML becomes very deep, the assumptions of the model can be satisfied, with cloud-base at the top of the ML and a shallow cloud field above in balance with weak large-scale subsidence. For shallow MLs, much stronger subsidence (equal to ρ b W b in Figure (a)) is needed to suppress the cloud field, and give a BL without clouds. We see from Figure (d) that ML radon concentrations are rather sensitive to the cloud-base exchange, and therefore might be a useful way of estimating it, provided estimates of the surface radon flux are available.. Sensitivity to mixing ratio above the CBL. In the next three sections, we shall select one parameter range from Table, and keep the others at their defaults. For brevity we shall not show the cloud coupled solutions, because they give rather similar families of curves, which for the most part can be qualitatively inferred from the plots in the previous sections. The mixing ratio q t of the free troposphere above the CBL has no direct effect on the energy balance of the ML or on its thermodynamic solutions for θ m and q m, but it does impact the mass flux ρ b W Eb through (). Figure a shows that while the cloud-base exchange, ρ b W b, is constant (for default SW net =00 Wm - : it has a small variation for the cloud coupled solution, not shown), ρ b W Eb derived from () increases with q t, with the corresponding shift of the entire net cloud mass flux profile shown in the lower panel (b). Essentially the ML stays the same, but a larger cloud mass flux compensates for the drier air entrained at CBL top. This change in ρ b W Eb has a corresponding impact on the ML equilibrium for CO m and radon, shown in Figure. As passive scalars, both move closer to their values in the free troposphere as the ρ b W Eb increases. As shown in Figure 0, this affects the coupling between CO m and q m, which is not affected by q t ; and between CO m and NEE, since NEE is almost unaffected by these changes in CO m (not shown). We can conclude, not surprisingly, that if we are to relate the moisture structure and fluxes to the CO structure and NEE, we need to know both the mixing ratio and CO content of air entrained into the CBL. A typical radiative equilibrium subsidence rate above the CBL is 0.0 Pa s - (e.g. Betts and Ridgway, ), corresponding to a mass flux (dividing by. ms - ) of 0.00 kg m - s - in Figures (a) and (a), at the lower end of the range for ρ b W Eb, the mass exchange necessary to keep the CBL from moistening. In other words, a subsidence rate of ρ b W Eb is needed to keep the CBL from growing deeper, given that the air being entrained into the CBL has mixing ratio q t. Conversely, if the subsidence, whether radiatively or dynamically driven, is less then ρ b W Eb in Figure (a), then the CBL cannot be in equilibrium, and we could interpret the difference as a CBL growth rate. Such a growth rate would be small: for example, a growth of 0.0 Pa s- corresponds to only 0 hpa day -, so the ML equilibrium solutions could be quite well satisfied over timescales of several days. If the CBL does deepen beyond the middle troposphere, then the

17 growth of precipitation particles, and their evaporation as they fall, will introduce additional processes which we have not considered in our simple model.. Sensitivity to stability The stability parameter Γ controls potential temperature above cloud-base, θ cld, through () and also q cld, since (0) determines RH cld, as shown in Figure. Figure shows the impact on the equilibrium ML, keeping fixed SW net =00 Wm -, and q t = g kg -. Increasing stability (panel (a)) warms and moistens the ML significantly. The impact on CO m is relatively small, while ML radon increases (panel (b)). The response of the coupled system involves many factors. Figure shows the impact on the vertical mass exchanges. Increasing stability reduces both ρ b W b and ρ b W Eb, as a partial compensation for the increases in q m and q cld (and their difference at cloudbase, not shown), but when the ML is deep, W CLD increases. The increase in ML radon in Figure (b) is a direct consequence of the decrease of ρ b W b in Figure (a). In the CO budget, however, this drop in the mass exchange at the top of the ML is largely compensated by a fall in the NEE, primarily caused by warmer surface temperatures increasing respiration (see Figure b, later). As a function of ML depth, the surface H and λe are independent of Γ, but the change of θ m, and q m changes the surface energy budget and NEE in relation to the SWC, which could be considered an independent surface variable. Figure (a) shows on the left-hand-scale that, for the same SWC, gets shallower as Γ increases (giving a warmer and moister ML). The righthand-scale shows the corresponding increase of EF, which reflects the increase of λe and decrease of H with increasing stability. Figure (b) shows that the increase of respiration with increasing stability (and surface temperature) is larger than the very small increase in photosynthesis. NEE, the difference between the two sets of curves, therefore decreases. Figure summarizes the coupling between q m and CO m (left-hand-scale), and CO m and NEE (right-hand-scale) as Γ increases, for comparison with Figures and 0. The simple coupling between CO m and q m seen in Figure has gone, because q m is much more sensitive to Γ than is CO m. On the other hand the link between NEE and CO m is a little stronger than in Figure 0. One conclusion we can draw from the sensitivities shown in this section is that a better representation of the temperature structure of the CBL is needed, than our simple specification of a stability parameter. Essentially we have found that warmer temperatures in the cloud layer (which may be linked to warmer temperatures in the middle troposphere) warm and moisten the ML, lower cloud-base, increase evaporation and reduce the sensible heat flux and NEE. At the plant level, the increase in evaporation is related to a small drop in the stomatal resistance coming from an increase of C RH in (0), and an increase of ftemp in () over wetter soils.. Sensitivity to ML radiative cooling rate. The ML radiative cooling rate and the ML depth control the surface sensible heat flux through () and (). Figure shows the impact on the equilibrium ML as a function of SWC, keeping fixed the other parameters in Table (SW net =00 Wm -, q t = g kg -, Γ = 0.0 K hpa -). As the ML radiative cooling rate increases, the ML becomes cooler, H increases and λe decreases; and

18 both ML depth and EF decrease. Figure shows that the mass exchange terms, ρ b W b and ρ b W Eb, both decrease, because of the decrease of evaporation. As a function of ML depth, Figure (a) shows that the ML actually gets cooler and drier as ML radiative cooling increases, while both CO m and radon increase (Figure (b)). The fall of ML depth with increased cooling shown in Figure, means that, as in Figure, as a function of SWC, CO decreases, and q m increases with ML radiative cooling (not shown). Figure shows that while the coupling between CO m and NEE changes with ML radiative cooling (respiration decreases as the ML cools), that between CO m and q m is almost unaffected. Comparison of Figures to with Figures to, shows that the response of the ML equilibrium to increased surface radiative forcing from 00 to 0 Wm - is qualitatively similar to the response of reduced radiative cooling of the ML from - to - K day -, except for EF, where the impact of F rad on H in () is large.. Dependence on vegetation type We model two idealized ecosystems, forest and grassland, represented by the three different vegetation parameters given in Table. For grassland, both photosynthesis and respiration are greater for the same SWC, SW net and surface temperature. Figure shows that as a function of SWC (other parameters set at their defaults in Table ), EF increases for grassland and ML depth decreases, both a direct response to increased photosynthesis, and reduced R veg. Consequently as a function of SWC, for grassland, the ML is cooler, moister, and has lower CO m (not shown). However as a function of ML depth, the picture again changes and simplifies, as in Figure. The ML θ m, q m, radon, all the mass fluxes and energy fluxes are independent of ecosystem: the lower grassland R veg gives a lower equilibrium depth, but the thermodynamic structure and fluxes depend only on this depth. The cloud coupled solutions corresponding to Figure (not shown) are just slightly steeper (compare Figures (b) and (b)). Figure 0 shows that the coupling between CO and NEE does not depends on ecosystem, while the coupling between CO and q m has only a weak dependence. The larger NEE over grassland, and the shallower equilibrium depth, however, extends the curves to much lower equilibrium values of CO. This is clearly a useful result, since generally the landscape will have mixed ecosystems. In this final figure we show the cloud coupled solutions for CO and NEE. These extend to lower CO values, because of the lower SW net and vertical mass exchanges over moist soils, and are more curved, because of the internal variation of SW net (compare Figure ), but the dependence on ecosystem is very weak.. Discussion and Conclusions This simple equilibrium model gives insight into the coupling between the surface energy and water budgets and net ecosystem exchange, and the mixed layer equilibrium of potential temperature, water vapor, CO (and other tracers such as radon). Mixed layer parameters, such as θ m, q m, CO m and Rn m as well as ML depth (our in pressure coordinates) are relatively easy parameters to measure, so we expect that the solutions could be used to help constrain surface flux estimates averaged over the diurnal cycle. We have shown solutions as a function of a set of external parameters: soil water content, which directly impacts respiration and photosynthesis, and hence transpiration; the surface net shortwave, directly linked here to net radiation, which is also coupled to photosynthesis and transpiration; and the radiative cooling of the ML, which in