de Roo, F.

Mauder, M.

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Roo, F. de

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ATMOTheoretical considerations onthe energy balance closureFrederik De Roo and Matthias MauderKarlsruhe Institute of Technology, Campus AlpinAtmospheric Environmental Research (KIT/IMK-IFU)Garmisch-PartenkirchenOverviewThe energy balance closure problem• Heterogeneity at the landscape scale• Possible causes for the imbalance• Secondary circulationsHeat fluxes and averagesNear-surface energy budgetHeterogeneity affects boundary layer processesMesoscale circulations. . . what about smaller scales?Lake breeze, valley windOasis effect, urban heat islandLeading edge effectIn simulations, often idealized heterogeneities:Chessboards and zebra patternsHeterogeneity affects boundary layer processesMesoscale circulations. . . what about smaller scales?Lake breeze, valley windOasis effect, urban heat islandLeading edge effectWhy has a zebra stripes? (Ruxton, Mammal Rev 2002)Predator/parasite avoidance, social benefits, thermoregulation“rotary breezes could be created by differential heating”Heterogeneity affects boundary layer processesMesoscale circulations. . . what about smaller scales?Lake breeze, valley windOasis effect, urban heat islandLeading edge effectOf course, we aim for realistic simulations:How do these patched heterogeneous landscapesinfluence the surface energy budget?Heterogeneity affects boundary layer processesMesoscale circulations. . . what about smaller scales?Lake breeze, valley windOasis effect, urban heat islandLeading edge effectOf course, we aim for realistic simulations:How do these patched heterogeneous landscapesinfluence the surface energy budget?Turbulent fluxes of latent and sensible heat oftenadd up to only 70–90 % of the available energySurface energy budget: Rn − G = LE + HTurbulent fluxes measured by eddy covariance towers,scintillometry, aircraft dataClosure problem: eddy covarianceunderestimates turbulent fluxesLSM etc. use surface fluxesas boundary conditionsPartitioning of missing flux?(Stoy and Mauder 2011)Possible causes for the underestimation of turbulent fluxesStorage terms give phase lag (Leuning et al ’12)Small remainder of nonclosure possibly from flux-divergenceInstrumental errors (Frank et al ’12; Kochendorfer et al ’13)for non-orthogonal sonic anemometersAdvection effects (Foken 2008)quasi-stationary secondary circulationsin heterogeneous terrainEC towers cannot capture mean flowCorrelation between terrain characteristics and air circulationcreates systematic underestimation of the energy budgetSecondary circulations in heterogeneous terrainInagaki et al (2006)turbulent mesoscale circulationswhich carry part of the imbalanceStoy et al (2013)correlation between non-closureand terrain heterogeneityBaidya Roy et al (2002):Shift from turbulent to mean transport:Nonclosure issues:• At measurement heights?• QuantificationOverviewThe energy balance closure problemHeat fluxes and averages• Webb, Pearman and Leuning (1980)Near-surface energy budgetHeat fluxes as defined by Webb, Pearman & Leuning (1980)No net vertical transport of dry air: ρdw = 0Latent heat flux from mixing ratio: λE = λρw ′r ′H = cpd ρdw (T − Tb) + cpv ρvw (T − Tb) ≈ cp ρ̄w ′T ′“Here Tb, taken as constant at any given height, representsroughly an assumed initial “base” temperature from which eachelement of air is warmed (or cooled) during the verticaltransfer of heat supplied (or removed) at the underlyingsurface. Even though Tb is not amenable to precisespecification, it is included because the heat imparted to andcarried by each parcel of air is represented by the temperaturechange, T − Tb, not the temperature T itself.”Tb drops out in homogeneous terrain, what when heterogeneous?OverviewThe energy balance closure problemHeat fluxes and averagesNear-surface energy budget• Expression from first principle• Base temperature and base humidityEnergy balance closure in a formulaCommonly: Rn − G = LE + HYet experiments find Rn − G ≥ LE + HTrue expression above the surfaceRn − G = LE(t) + H(t) + ∆Additional advection and accumulation terms??6 6RnGLE H??6 6RnGLE H- . . .Control volume (no air parcels!)Air + canopyEnergy conservationBoussinesq approximationSurface layer (EC towers)An expression for the (time-averaged) imbalance ∆Hm + λEm mean upward fluxesH‖ + λE‖ laterally advected fluxes1δt[∫ρcvTdz]t+δtt thermal energy accumulationgzvEv potential energy accumulation1δt [Kv ]t+δtt kinetic energy accumulation−∫v · ∇p dz minor rest termLp(Fpg − Fp − Fp‖)CO2 flux due to photosynthesis1δt[∫ρ (λq − Lpqp) dz]t+δtt water and CO2 accumulationNecessary to estimate magnitude of these terms, with:w̄ ∼ 1 cm/s ; u ∼ 4 m/s ; ∇p ∼ 0.1 Pa/mzm ∼ 2m ; δT/δt ∼ 2 K/hr ; q̄ ∼ 3 g/m3Advection terms largely cancel each other outbut leave a remainder of the order of the imbalance1δt[∫ρcvTdz]t+δtt 1W/m2 in air∼ 50W/m2 (Leuning et al 2012)gzvEv 10−5 λE (cf Oncley et al 2007)1δt [Kv ]t+δtt same order as next term−∫v · ∇p dz 1W/m2Lp(Fpg − Fp − Fp‖ 10% (Rn − G ) under very productive− 1δt[∫ρqpdz]t+δtt)circumstances (Meyers & Hollinger 2004)1δt[∫ρλqdz]t+δtt 0.6W/m2Hm ∼ 4000W/m2 ; λEm ∼ 80W/m2 ; H‖ + λE‖Simulations: H· ∼ 104 W/m2 ; Hm + H‖ ∼ 10− 102 W/m2The base temperature incorporates advection effectsSum of sensible heatflux through upper boundary and advected fluxH =∫UρcpT (v · dA) +∫SρcpT (v · dA)Rewrite asH =∫Uρcp (T − Tb) (v · dA)with base temperature:Tb = −∫S ρcpT (v · dA)∫U ρcp (v · dA)From conservation of air and incompressibility: Tb ≈< T >SSimilar procedure for base humidity qbAdvection effects important when secondary air circulationdriven by local temperature differencesH = ρcpw (T − Tb) means correlation of• vertical wind• temperature difference between upward and advected airDifficult in practicecf. Mauder et al (’08, ’10)• spatiotemporal averageover nearby stations• better results for sensiblethan for latent heatFrom simulation: Var[qb(t)] stronger than Var[Tb(t)]Tb(t) different from constant Tb in WPL => w ′T ′bTheoretical considerations on the energy balance closureAdditional terms appearing in near-surface energy budgetImportance of advection and storageInterpretation to the base temperature of WPL (1980)Average temperature over lateral sides of control-volumeAllows – in principle – to account for advection effectsSecondary circulations in heterogeneous terrainas a cause for the non-closure of the energy balanceespecially when storage/NPP is lowOverviewThe energy balance closure problemHeat fluxes and averages• Webb, Pearman and Leuning (1980)• Kowalski (2012)Near-surface energy budgetAveraging procedures: a matter of taste?Kowalski (2012) seeks alternative “correct” averagesthat satisfy physical laws without corrections“Boundary layer meteorology [...] clearly suffers from a graveand persistent fault. The inability to close the surface energybudget [...] suggests possible errors in basic methodology,within which accurate averaging procedures are critical.”“For studies of eddy transport, and micrometeorology inparticular, [...] imprecisely determined averages of state andflow variables bias the perturbation variables over the entireaveraging domain and thereby skew estimates of mass, heat,and momentum exchange.”For example, ideal gas law (e.g. Stull ’88): p̄ = Rρ̄T̄ +R ρ′T ′Define T̃ = T̄ + ρ′T ′/ρ̄ such that p̄ = Rρ̄T̃Averaging procedures: a matter of taste?Kowalski raises valid remarks about sample & ensemble meansBUT(1) Impossible to satisfy multiple laws/definitions at onceT̃ = T̄ +ρ′T ′ρ̄6=T̄ + (ρcp)′T ′(ρcp)=> corrections always neededWhen corrections are taken into accounttraditional averages remain equally valid(2) Necessary to go beyond Boussinesq approximation?Otherwise only triple correlation H = cpρ′T ′w ′ + q-terms

Theoretical considerations on the energy balance closure

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Theoretical considerations on the energy balance closure

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