Theories of Mixing in Cumulus Convection A. Pier Siebesma Royal Netherlands Meteorological Institute (KNMI) De Bilt The Netherlands 1. Motivation 2. Essential Thermodynamics 3. Phenomenology and Observations 4. Cloud Mixing Models 5. Parameterizations 6. Remaining Problems. Many thanks to: Roel Neggers (KNMI) Harm Jonker, Stephaan Rodts (TU Delft)

See also: http://www.knmi.nl/~siebesma References A.P. Siebesma and J.W.M. Cuijpers, Evaluation of parametric assumptions for shallow cumulus convection, J. Atmos. Sci., 52, 650-666, 1995 A.P. Siebesma and A.A.M. Holtslag, Model impacts of entrainment and detrainment rates in shallow cumulus convection, J. Atmos. Sci., 53, 2354-2364, 1996 A.P. Siebesma , ‘’Shallow Cumulus Convection” published in: Buoyant Convection in Geophysical Flows, p441-486. Edited by: E.J. Plate and E.E. Fedorovich and X.V Viegas and J.C. Wyngaard. Kluwer Academic Publishers. R.A.J. Neggers,A.P. Siebesma and H.J.J. Jonker. A multiparcel method for shallow cumulus convection. Accepted for J. of Atm Sci. 2002. R.A.J. Neggers,A.P. Siebesma and H.J.J. Jonker. Size statistics of cumulus cloud populations in large-eddy simulations. Submitted to J. of Atm Sci. 2002. See also: http://www.knmi.nl/~siebesma 21/09/2018 theories of cloud mixing

Motivation and Objectives 21/09/2018 theories of cloud mixing

Cartoon of Hadley Circulation Subsidence ~0.5 cm/s 10 m/s inversion Cloud base ~500m Tropopause 10km Deep Convective Clouds Precipitation Vertical turbulent transport Net latent heat production Engine Hadley Circulation Shallow Convective Clouds No precipitation Vertical turbulent transport No net latent heat production Fuel Supply Hadley Circulation Stratocumulus Interaction with radiation 21/09/2018 theories of cloud mixing

theories of cloud mixing 50 km Shallow cumulus not resolved by “state of the art” global atmospheric models. 21/09/2018 theories of cloud mixing

Grid Averaged Budget Equations Large scale advection Large scale subsidence Vertical turbulent transport Net Condensation Rate 21/09/2018 theories of cloud mixing

theories of cloud mixing Schematically: Objectives Understand Cumulus Convection…. Design Models….. But ultimately design parameterizations of: 21/09/2018 theories of cloud mixing

theories of cloud mixing 2. Thermodynamics 21/09/2018 theories of cloud mixing

theories of cloud mixing 2.1 Moisture Variables qv :Specific Humidity ql :Liquid Water qt = qv + ql :Total water specific humidity (Conserved for phase changes 21/09/2018 theories of cloud mixing

theories of cloud mixing Remark 1: In thermodynamic equilibrium: qt = qv if qv < qsat: undersaturation qt = qsat + ql if qv > qsat: oversaturation Remark 2: qsat = qsat (p,T) is a state function (Clausius-Clayperon) T qt qs(p,T) ql qv 21/09/2018 theories of cloud mixing

2.2 Used Temperature Variables Potential Temperature Conserved for dry adiabatic changes Liquid Water Potential Temperature Conserved for moist adiabatic changes Virtual Potential Temperature Directly proportional to the density Measure for buoyancy 21/09/2018 theories of cloud mixing

Grid averaged equations for conserved variables: Parameterization issue reduced to a turbulent mixing problem! 21/09/2018 theories of cloud mixing

theories of cloud mixing 3. Phenomenology and Observations 21/09/2018 theories of cloud mixing

Typical Tradewind Cumulus Strong horizontal variability ! 21/09/2018 theories of cloud mixing

Horizontal Variability and Correlation Mean profile height Horizontal Variability and Correlation 21/09/2018 theories of cloud mixing

Poor man’s cloud model: adiabatic ascent Mean profile height “Level of zero kinetic energy” Level of neutral buoyancy (LNB) Level of free convection (LFC) well mixed layer Inversion non-well mixed layer Lifting condensation level (LCL) 21/09/2018 theories of cloud mixing

Schematic picture of cumulus convection: more intermittant more organized than Dry Convection. 21/09/2018 theories of cloud mixing

Mixing between Clouds and Environment (SCMS Florida 1995) adiabat Due to entraiment! Data provided by: S. Rodts, Delft University, thesis available from:http://www.phys.uu.nl/~www.imau/ShalCumDyn/Rodts.html 21/09/2018 theories of cloud mixing

Virtual potential temperature Entrainment Influences: Liquid water potential temperature Virtual potential temperature Entrainment Influences: Vertical transport Cloud top height 21/09/2018 theories of cloud mixing

theories of cloud mixing 4. Cloud Mixing Models 21/09/2018 theories of cloud mixing

4.1 lateral mixing bulk model Fractional entrainment rate hc Fractional entrainment rate 21/09/2018 theories of cloud mixing

Diagnose through conditional sampling: Typical Tradewind Cumulus Case (BOMEX) Data from LES: Pseudo Observations 21/09/2018 theories of cloud mixing

Trade wind cumulus: BOMEX Cumulus over Florida: SCMS LES Observations Cumulus over Florida: SCMS 21/09/2018 theories of cloud mixing

Implementation simple bulk model: Updraft Calculation in conserved variables: continue Stop (= cloud top height) B>0 3. Check on Buoyancy: 2. Reconstruct non-conserved variables: 21/09/2018 theories of cloud mixing

theories of cloud mixing Criticism: No correct simultaneous prediction of cloud top height (=zero buoyancy level) and cloud fields (Warner paradox) Due to: Bulk model 21/09/2018 theories of cloud mixing

4.2 Multiparcel Mixing Models Ensemble of parcels (cloud elements) Each parcel has a different mixing fraction with environment Are send to their zero buoyancy level Spectral mass flux models: (Arakawa Schubert 1974) Stochastic versions: Raymond, Blyth, Emanuel) 21/09/2018 theories of cloud mixing

4.3 Example: Lateral mixing multiparcel model Ensemble of parcels (cloud elements) Parcels are send to their zero vertical velocity level. All parcels obey the same dynamical equations. All parcels only interact with a background (mean) field. 21/09/2018 theories of cloud mixing

2. Fractional Entrainment rate e 1. Parcel equations: 2. Fractional Entrainment rate e 21/09/2018 theories of cloud mixing

Test: Test their properties in the cloud layer BOMEX, LES data Initialise core parcels at cloud base 21/09/2018 theories of cloud mixing

Other results: ql, qt, and ql in the cloud core: 21/09/2018 theories of cloud mixing

Other results: vertical velocity cloud core cover entrainment 21/09/2018 theories of cloud mixing

Other results: variance of qt, ql 21/09/2018 theories of cloud mixing

5. Turbulent Flux Parameterizations 21/09/2018 theories of cloud mixing

5.1 Mass Flux Approximation wc 21/09/2018 theories of cloud mixing

theories of cloud mixing No observations of turbulent fluxes and mass flux Use Large Eddy Simulation (LES) based on observations BOMEX ship array observed observed To be modeled by LES 21/09/2018 theories of cloud mixing

theories of cloud mixing 10 different LES models Initial profiles Large scale forcings prescribed 6 hours of simulation Is LES capable of reproducing the steady state? 21/09/2018 theories of cloud mixing

Mean profiles after 6 hours Use the last 4 simulation hours for analysis of ……. 21/09/2018 theories of cloud mixing

theories of cloud mixing Cloud cover 21/09/2018 theories of cloud mixing

theories of cloud mixing Turbulent Fluxes 21/09/2018 theories of cloud mixing

theories of cloud mixing Mass Flux Decreasing with height Also observed for other cases Obvious reason……….. 21/09/2018 theories of cloud mixing

theories of cloud mixing Conditional Sampling of: Total water qt Liquid water potential temperature ql liquid water virtual pot. temp. 21/09/2018 theories of cloud mixing

Test of Mass flux approximation 21/09/2018 theories of cloud mixing

Simple Bulk Mass flux parameterization Where: “Empty” equation detrainment 21/09/2018 theories of cloud mixing

theories of cloud mixing 6. Open Problems 21/09/2018 theories of cloud mixing

6.1 Issues within the mass flux parameterization Entrainment Formulation (relatively easy) Mass Flux Formulation (hard) Closure Problem, i.e. boundary values at cloud base 21/09/2018 theories of cloud mixing

theories of cloud mixing Boundary layer equilibrium subcloud velocity closure CAPE closure: based on 21/09/2018 theories of cloud mixing

6.2 Issues beyond the mass flux parameterization 21/09/2018 theories of cloud mixing

K-diffusion versus Mass flux 21/09/2018 theories of cloud mixing

theories of cloud mixing K-diffusion 21/09/2018 theories of cloud mixing

theories of cloud mixing OPTIONS Do all mixing processes with K-diffusion Do all mixing with mass flux (Randall and coworkers) Design a blend between mass flux and K-diffusion (two-scale approach) 21/09/2018 theories of cloud mixing

theories of cloud mixing 21/09/2018 theories of cloud mixing

Find equilibrium solutions for f={ql,qt} To be done: Find equilibrium solutions for f={ql,qt} 21/09/2018 theories of cloud mixing

theories of cloud mixing 21/09/2018 theories of cloud mixing

theories of cloud mixing Cloud Boundaries Cloud size distributions 21/09/2018 theories of cloud mixing

theories of cloud mixing Bulk means: Cloud ensemble: approximated by 1 effective cloud: 21/09/2018 theories of cloud mixing

Determination of the relaxation time: w Use LES Determine for each cloud: cloud height h and vert. vel. w Estimate t by t=1/wh h(m) t(sec) Conclusion: Relaxation time ~ constant 21/09/2018 theories of cloud mixing

Due to decreasing cloud (core) cover 21/09/2018 theories of cloud mixing

Virtual potential temperature: qv 21/09/2018 theories of cloud mixing

theories of cloud mixing Cloud Liquid water 21/09/2018 theories of cloud mixing

Prescribe non-dimensionalised mass flux profile 21/09/2018 theories of cloud mixing

Case studies - The GEWEX Cloud System Study CRMs GCSS WG4 TOGA/COARE 6-day average cloud cover SCMs 21/09/2018 theories of cloud mixing

theories of cloud mixing 21/09/2018 theories of cloud mixing

theories of cloud mixing 21/09/2018 theories of cloud mixing

theories of cloud mixing 21/09/2018 theories of cloud mixing

theories of cloud mixing 21/09/2018 theories of cloud mixing