We perform analysis of direct numerical simulation (DNS) data of two flow cases: stratocumulus cloud-top (SCT) and convective boundary layer (CBL). We test different methods for turbulence kinetic energy dissipation rate (EDR) retrieval. Among others we investigate performance of a new, iterative method, proposed recently in Wacławczyk et al. (2017), where an analytical model for energy spectra in the dissipative range is needed. We argue, the new method has some advantages over the standard spectral retrieval techniques. To apply it, only the information on the signals’ cut-off wavelength is needed and it is not necessary to define the fitting range in the inertial part of the spectrum. With this, the new method could be a basis of a general algorithm for EDR retrieval, applicable to a wide range of different atmospheric data (e.g. from commercial aircrafts). Moreover, we investigate how the presence of anisotropy due to shear, buoyancy and external intermittency in the flow affects the EDR retrieval based on the classical K41 for isotropic turbulence (Kolmogorov, 1941). © 2019 International Symposium on Turbulence and Shear Flow Phenomena, TSFP. All rights reserved

Akinlabi, E.

Malinowski, S.

Mellado, J.

Wacławczyk, M.

MPG.PuRe

11th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11)Southampton, UK, July 30 to August 2, 2019ESTIMATING TURBULENCE KINETIC ENERGY DISSIPATIONRATES IN NUMERICALLY SIMULATED STRATOCUMULUSCLOUD-TOP AND CONVECTIVE BOUNDARY LAYER FLOW:EVALUATION OF DIFFERENT METHODS.Marta WacławczykFaculty of PhysicsUniversity of Warsaw, Warsaw, Polandmarta.waclawczyk@igf.fuw.edu.plEmmanuel O. AkinlabiFaculty of PhysicsUniversity of Warsaw, Warsaw, PolandEmmanuel.Akinlabi@fuw.edu.plJuan Pedro MelladoMax-Planck Institute for MeteorologyHamburg, Germanyjuan-pedro.mellado@mpimet.mpg.deSzymon P. MalinowskiFaculty of PhysicsUniversity of Warsaw, Warsaw, PolandSzymon.Malinowski@fuw.edu.plABSTRACTWe perform analysis of direct numerical simulation(DNS) data of two flow cases: stratocumulus cloud-top(SCT) and convective boundary layer (CBL). We test dif-ferent methods for turbulence kinetic energy dissipationrate (EDR) retrieval. Among others we investigate per-formance of a new, iterative method, proposed recently inWacławczyk et al. (2017), where an analytical model forenergy spectra in the dissipative range is needed. We ar-gue, the new method has some advantages over the standardspectral retrieval techniques. To apply it, only the informa-tion on the signals’ cut-off wavelength is needed and it isnot necessary to define the fitting range in the inertial partof the spectrum. With this, the new method could be a ba-sis of a general algorithm for EDR retrieval, applicable toa wide range of different atmospheric data (e.g. from com-mercial aircrafts).Moreover, we investigate how the presence ofanisotropy due to shear, buoyancy and external intermit-tency in the flow affects the EDR retrieval based on the clas-sical K41 for isotropic turbulence (Kolmogorov, 1941).TEST CASEAtmospheric turbulence is the key physical mechanismthat is behind the occurrence of many atmospheric phenom-ena. An important quantity which characterises smallestscales of such flows is the mean turbulence kinetic energydissipation rate ε . Still, the information on ε in atmosphericflows is scarce. The velocity time series obtained from air-borne experiments are characterized by the presence of ef-fective spectral cutoffs and are affected by measurement er-rors. Moreover, turbulence anisotropy, buoyancy and exter-nal intermittency affect the EDR estimates.The available DNS data allow to test performance ofdifferent methods for ε retrieval in atmospheric configura-tions. In this work we consider two systems. The first onemimics the stratocumulus cloud-top region. Longwave ra-diative cooling of the upper region of the cloud leads to con-vective instability, and this instability is a major source ofcloud turbulence. These radiative properties imply a ref-erence buoyancy flux B0 = 0.002 m2 s−3 and a referencevelocity scale U0 = (B0L0)1/3 = 0.3 m s−1, where L0 isthe radiative extinction length. The Reynolds number inthe simulation is U0L0/ν = 800, which is about 300 timessmaller than that in the atmosphere. Further details of thesettings and simulations can be found in Mellado (2017);Schulz & Mellado (2018).The second system a dry, shear-free convective bound-ary layer (CBL) that grows into a linearly stratified atmo-sphere. The flow is driven by a constant and homogeneoussurface buoyancy flux B0, and the buoyancy stratificationof the free atmosphere is N2, where N is the buoyancy fre-quency. This configuration is representative of midday at-mospheric conditions over land. Statistical properties areexpressed as a function of the buoyancy Reynolds num-ber Re0 = B0/(νN2), the normalized vertical distance tothe surface z/h, and the ratio h/L. Here, the variable h(t)provides a measure of the CBL depth and is defined ash' (2B0N−2t)1/2, the parameter L= (B0/N3)1/2 is the ref-erence Ozmidov scale. The ratio h/L increases as the CBLgrows into the linearly stratified atmosphere and attains aquasi steady regime beyond h/L ' 10− 15. We considerdata from a simulation with a buoyancy Reynolds numberRe0 = 117 and at a state of development h/L ' 21.5. Fur-ther details can be found in Mellado et al. (2016).DISSIPATION RATE RETRIEVALThe turbulence kinetic energy dissipation rate is de-fined as (Pope, 2000)ε = 2ν〈si jsi j〉, (1)111th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11)Southampton, UK, July 30 to August 2, 2019wheresi j =12(∂u′i∂x j+∂u′j∂xi),〈·〉 is the ensemble average and u′i = ui−〈ui〉 denotes thei-th component of fluctuating velocity. The two commonapproaches for EDR retrieval use the inertial-range scalingform of the power spectra and structure functions. In thehomogeneous and isotropic turbulence the following energyspectrum is assumed (Pope, 2000)E(k) =Cε2/3k−5/3 fL(kL) fη (kη) (2)where C ≈ 1.5 derived from experimental data, fL and fηare non-dimensional functions, which specify the shape ofenergy-spectrum in the energy-containing range and the dis-sipation range, respectively. L is the length scale of largeeddies andη =(ν3ε)1/4is the Kolmogorov length scale, which is connected with thedissipative scales. The function fη tends to unity for smallkη while fL tends to unity for large kL, such that in the iner-tial range, the formula E(k) =Cε2/3k−5/3 is recovered. Inthe homogeneous, isotropic turbulence the one-dimensionallongitudinal and transverse energy spectra E11 and E22, re-spectively, are related to the energy spectrum function E(k),such that in the inertial range they follow formulaeE11(k1) = αε2/3PS k−5/31 E22(k1) = α′ε2/3PS k−5/31 (3)where α ≈ 0.49 and α ′ ≈ 0.65 and εPS should approximateε . This allows to estimate the EDR from the 1D velocitysignal with spectral cut-off. Alternatively, the profiles ofthe second and third order longitudinal structure functionsin the inertial subrange can be used to calculate ε .The new, iterative method to estimate ε , proposed inWacławczyk et al. (2017) and Akinlabi et al. (2019) can beused for spectral cut-off’s placed in the inertial, as well asintermediate or dissipative range. In this method we makeuse of Eq. (2) and assume certain form of the function fη . Ifa velocity signal u′cut is measured up to a cut-off wavenum-ber kcut and we assume that the associated filter is rectangu-lar in the wavenumber space, the following relation between〈(∂u′cut/∂x)2〉 and 〈(∂u′/∂x)2〉 is obtained〈(∂u′∂x)2〉=〈(∂u′cut∂x)2〉 ∫ ∞0 k21E11dk1∫ kcut0 k21E11dk1︸ ︷︷ ︸CF, (4)where the fraction on the RHS is called the ”correcting fac-tor” CF . Using relations between E11(k1) and E(k) inisotropic turbulence (Pope, 2000)E11(k1) =∫ ∞k1E(k)k(1− k21k2)dk, (5)and using Eq. (2), the following formula for CF was de-rived in Wacławczyk et al. (2017)CF = 1+∫ ∞kcutβη ξ21∫ ∞k1 ξ−8/3 fη (ξ )(1− ξ 21ξ 2)dξdξ1∫ kcutβη0 ξ21∫ ∞ξ1 ξ−8/3 fη (ξ )(1− ξ 21ξ 2)dξdξ1. (6)If CF is known, the EDR can be estimated from the directrelationεNCR = 15ν〈(∂u′∂x)2〉= 15ν〈(∂u′cut∂x)2〉CF . (7)In order to calculate CF from Eq. (6) a value of η shouldfirst be specified, hence, an iterative procedure was pro-posed in Wacławczyk et al. (2017). It starts with an ini-tial guess of ε0. With this, the corresponding value of theKolmogorov length η0 = (ν3/ε0)1/4 is calculated and in-troduced into Eq. (6) for CF . The EDR after the first it-eration, ε1 is found from Eq. (7). The procedure can berepeated, i.e. the next approximation of η1 can be calcu-lated and substituted into Eq. (6). After several iterationsthe procedure converges to the final value of εNCR whichshould approximate ε with an error defined by a prescribedform ∆η = |ηn+1−ηn|.Moreover, in order to calculate CF from Eq. (6), cer-tain form of fη should be assumed. In this work we inves-tigated three different forms of fη (see Pope (2000)), theexponential spectrumfη = exp(−βkη), (8)where β = 2.1, the Pao spectrumfη = exp(−β (kη)4/3), (9)with β = 2.25 and the Pope modelfη = exp(−β{[(kη)4 + c4η ]1/4− cη}), (10)where β = 5.2, cη = 0.4.RESULTSFor the startocumulus-top simulations we investigated1D spectra of three velocity components at four horizontalplanesz=−5.2L0,−3.5L0,−1.7L0,z= 0.1L0.The plane z = −3.5L0 indicates the height of maximumbuoyancy flux and the plane z= 0.1L0 is placed in the upperpart of stratocumulus cloud and indicates the position of theminimum buoyancy flux. This region of the flow is affectedby the presence of shear, stable stratification as well as ex-ternal intermittency and the spectra calculated at this planedeviate from K41 theory.211th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11)Southampton, UK, July 30 to August 2, 2019For the CBL case we perform analogous analysis forfive horizontal planes, namely,z= 0.25h, 0.5h, 0.75h, 1.0h, 1.14h.Apart from the layer z = 0.1L0 in the SCT we found closesimilarities between spectra in both flow cases. The inertialrange scaling ∼ k−5/31 can be best recognised for the lon-gitudinal spectra of horizontal velocity components u (seeFig. 1) and v and for the transverse spectra of the verti-cal velocity component w (see Fig. 2). In this latter case,however, the proportionality constant exceeds the isotropicvalue α ′ = 0.65. We can conclude that anisotropy due tobuoyancy will affect EDR estimates based on the verticalvelocity component. They will be overpredicted in compar-ison to the true εDNS values inside the cloud and in the CBL,where buoyancy is a source of turbulence generation andundepredicted in the STC case at plane z= 0.1L0 where thenegative buoyancy damps the vertical velocity fluctuations.Hence, in the further part of the work the EDR estimateswere based on the horizontal components of velocity u andv.Deviations from the K41 scaling were observed for thetransverse spectra of u and v, where∼ k−a1 or∼ k−b1 scaling,with a and b smaller than 5/3, was found, see Figs 3 and4. As we expect, the EDR estimates based on atmosphericmeasurements may also be biased due to these effects.To test the performance of the new, iterative methodfor ε retrieval given by Eqs. (6) and (7) we compared dif-ferent analytical forms of fη , given in Eqs. (8–10) for theSCT case in Fig. (5) with kcut = 5.3m−1 which is withinthe dissipative range. In this plot, εDNS is the exact EDRvalue calculated from the formula (1) from the DNS data.As it is seen, the Pope formulation provides better fit withDNS than the Pao or exponential models. All results areunderpredicted in comparison to εDNS at plane z= 0.1L0.Next, we compared results with standard, power spec-trum EDR retrieval method, as given in Eq. (3). The valueof εNCR from Eq. (7) was calculated with the Pope’s model(10) with kcut = 5.3m−1 in the STC case and kcut = 614m−1in the CBL case. As it is seen in Fig. 6, predictions withthe new, iterative model are comparable with εPS calculatedfrom the power spectrum. 10 iterations were sufficient tocalculate εNCR with the error ∆η = 10−6.We argue, the iterative method have certain advantagesover the standard spectral retrieval estimates. In the lattercase, it is necessary identify the inertial-range part of thespectrum and choose the optimal fitting range to find εPSfrom Eq. (3). This is not easy when large sets of atmo-spheric data (e.g. from commercial aircrafts) are to be anal-ysed. To apply the iterative method only the information onthe cut-off wavelength are required. With this, writing analgorithm for EDR retrieval applicable for a wide range ofdata is easier.Some regions of the considered flows are affected bythe presence of external intermittency, i.e. spots of non-turbulent fluid within cloud or boundary layer. The inter-mittency ratio γ is the volume fraction of turbulent fluid,that is, γ = 0 in purely laminar and γ = 1 in purely turbu-lent flow. The presence of external intermittency may af-fect EDR retrieval, especially in the top region of the cloud(at z = 0.1L0) and the top of the CBL. As it was argued inAkinlabi et al. (2019), the external intermittency can quali-tatively be estimated by the ratio of the Liepmann and Tay-10 -1 10 0 10 1k1  [m-1 ]10 -210 -110 0-2/3k 15/3 E 11(k1)z = -5.2L 0z = -3.5L 0z = -1.7L 0z = 0.1L 0Figure 1. Compensated longitudinal spectra E11(k1) forSCT (top figure), CBL (bottom figure)lor microscales. The Liepmann scale is defined asΛ=1piNL,where NL is the number of times per unit length the fluctu-ating signal crosses the threshold 0. The longitudinal Taylormicroscale equals (Pope, 2000)λl =[2〈u′2l 〉〈(∂u′l/∂x)2〉]1/2(11)and the transverse microscale is λn = λl/√2. It was shownby Sreenivasan et al. (1983) that in fully turbulent signalsλn/Λ= 1.The case of externally intermittent flow was consideredby Akinlabi et al. (2019). Therein, we assumed as a firstapproximation, that in the externally intermittent flow, the311th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11)Southampton, UK, July 30 to August 2, 201910 -1 10 0 10 1k1  [m-1 ]10 -110 0-2/3k 15/3 E 33(k1)z = -5.2L 0z = -3.5L 0z = -1.7L 0z = 0.1L 0Figure 2. Compensated transverse spectra E33(k1) forSCT (top figure), CBL (bottom figure)statistics will change to γ〈u′2〉 and γ〈(∂u′/∂x)2〉. More-over, the laminar part of the signal does not significantlycontribute to the number of crossings, hence, we will de-tect γNL crossings per unit length. With this, the Taylormicroscale, the Liepmann scale and their ratio will changetoλnI =[γ〈u′2〉γ〈(∂u′/∂x)2〉]1/2= λn,andΛI =1γpiNL=1γΛ, henceλnIΛI= γλnΛ,where the subscripts I are related to the statistics in the in-termittent flow. If λn/Λ ≈ 1, then in the intermittent flow,λnI/ΛI ≈ γ . We note, however, that both in the STC andCBL case λn/Λ was around 0.8 even in the core region10 -1 10 0 10 1k2  [m-1 ]10 -210 -110 0-2/3k 25/3 E 11(k2)z = -5.2L 0z = -3.5L 0z = -1.7L 0z = 0.1L 0Figure 3. Compensated transverse spectra E11(k2) forSCT (top figure), CBL (bottom figure)of the flow. The reason for this may be the strong non-Gaussianity of the velocity derivatives. Hence, in Akinlabiet al. (2019) the ratioλnI/ΛI(λn/Λ)T(12)(the subscripts T is used to denote mean value in the turbu-lent, core region of the flow) was compared with γ estimatedfrom DNS based on the enstrophy values. In this work weshow analogous results for the CBL case in Fig. 7. As it isobserved, λnI/ΛI decreases in the upper part of the CBL,similarly as γ .CONCLUSIONSWith this study, we investigated how the anisotropy ofturbulence in startocumulus clouds and convective bound-ary layer affects estimates of ε based on 1D velocity timeseries. Moreover, we tested the new, iterative method for411th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11)Southampton, UK, July 30 to August 2, 201910 -1 10 0 10 1k1 [m-1]10 -210 0-2/3k 15/3 E 22(k1)z = -5.2L 0z = -3.5L 0z = -1.7L 0z = 0.1L 0Figure 4. Compensated transverse spectra E22(k1) forSCT (top figure), CBL (bottom figure)the EDR retrieval. In the new method a universal formulafor the dissipative part of the spectrum is employed. Weshow that this assumption works well, at least in the coreregion of the considered flow cases, not affected by the ex-ternal intermittency. Hence, the new method can provide aninteresting alternative to the standard approaches.ACKNOWLEDGEMENTSMW acknowledges the financial support ofthe National Science Centre, Poland (project No.2014/15/B/ST8/00180). EA received funding from theEuropean Union Horizon 2020 Research and InnovationProgramme under the Marie Sklodowska-Curie Actions,Grant Agreement No. 675675. SPM acknowledge match-ing fund from the Polish Ministry of Science and HigherEducation No. 341832/PnH/2016.-6 -5 -4 -3 -2 -1 0z/L00.10.20.30.40.50.60.70.8/B0 Pope /B0Exp /B0Pao/B0PS/B0DNS/B0Figure 5. EDR estimates from Eq. (7) for different formsof fη in Eq. (6) compared with εDNS-6 -5 -4 -3 -2 -1 0z/L00.10.20.30.40.50.60.70.8/B0PS/B 0NCR/B 0DNS/B 00.2 0.4 0.6 0.8 1 1.2z/h0.10.150.20.250.30.350.40.45/B0PS/B0NCR/B0DNS/B0Figure 6. EDR estimates with the power spectrum and theiterative method, STC (top figure), CBL (bottom figure).REFERENCESAkinlabi, E. O., Wacawczyk, M., Mellado, J. P. & Mali-nowski, S. P. 2019 Estimating turbulence kinetic energydissipation rates in the numerically simulated stratocu-mulus cloud-top mixing layer: Evaluation of differentmethods. Journal of the Atmospheric Sciences 76 (5),1471–1488.Kolmogorov, A. N. 1941 Dissipation of energy in locallyisotropic turbulence. Dokl. Akad. Nauk SSSR 434 (1890),15–17.511th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11)Southampton, UK, July 30 to August 2, 20190 0.2 0.4 0.6 0.8 1 1.2z/h0.20.40.60.811.2GammanI / I(nI / I)/(( n/ )T)Figure 7. Values of the intermittency factor calculatedfrom the enstrophy and λn/Λ as a function of z/h for theCBL case.Mellado, J. P. 2017 Cloud-top entrainment in stratocumulusclouds. Annu. Rev. Fluid Mech. 49, 145–169.Mellado, J. P., van Heerwaarden, C. C. & Garcia, J. R. 2016Near-surface effects of free atmosphere stratification infree convection. Bound-layer Meter 159, 69–95.Pope, S. B. 2000 Turbulent flows. Cambridge .Schulz, Bernhard & Mellado, Juan Pedro 2018 Wind sheareffects on radiatively and evaporatively driven stratocu-mulus tops. Journal of the Atmospheric Sciences 75 (9),3245–3263.Sreenivasan, K., Prabhu, A. & Narasimha, R. 1983 Zero-crossings in turbulent signals. J. Fluid Mech. 137, 251–272.Wacławczyk, M., Ma, Y-F, Kopec´, J. M. & Malinowski,S. P. 2017 Novel approaches to estimating turbulent ki-netic energy dissipation rate from low and moderate res-olution velocity fluctuation time series. Atmos. Meas.Tech. 10, 45734585.6

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Estimating turbulence kinetic energy dissipation rates in numerically simulated stratocumulus cloud-top and convective boundary layer flow: Evaluation of different methods.

Estimating turbulence kinetic energy dissipation rates in numerically simulated stratocumulus cloud-top and convective boundary layer flow: Evaluation of different methods.

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