Atmospheric dry and shallow moist convection Pier Siebesma siebesma@knmi.nl KNMI, The Netherlands Het komende uur zal ik proberen antwoord te geven op een 4-tal vragen: Waarom zijn wolken belangrijk om mee te nemen in het klimaatsysteem Wat zijn de problemen om wolken goed te representeren in klimaatmodellen Hoe goed zijn klimaatmodellen momenteel in staat om wolken correct te representeren. Wat zijn de toekomstige ontwikkelingen. Motivation Fundamentals, Models, Equations Dry Convective Boundary layer Shallow Moist Convection (Friday) Parameterizations of moist and dry convection (Saturday)

1. Motivation

Deep Convective Clouds Precipitation Vertical turbulent transport Tropopause 10km Subsidence ~0.5 cm/s inversion 10 m/s Cloud base ~500m 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

ECMWF IFS overestimates Tradewind cumulus cloudiness: The GCSS intercomparison project on cloud representation in GCM’s in the Eastern Pacific ECMWF IFS overestimates Tradewind cumulus cloudiness: Deep cu scu Shallow cu Siebesma et al. (2005, QJRMS)

building material for clouds Water vapour is the building material for clouds name Symbol Units Definition Near surface values Atmospheric column spec. humidity qv [g/kg] amount of water vapour in 1kg dry air 10 g/kg 20 kg/m2 Saturation spec. hum. qs [g/kg] Max. amount of water vapour in 1kg dry air 15 g/kg Liquid water ql [g/kg] amount of liquid water in 1kg dry air 1 g/kg 200 g/m2 Tip of the iceberg qsat qt

water vapor clouds albedo lapse rate total

Sensitivity of the Tropical Cloud Radiative Forcing to Global Warming in 15 AR4 OAGCMs High-sensitivity GCMs Low-sensitivity GCMs (Bony and Dufresne, GRL, 2005)

Fundamentals , Models and Equations

Some fundamental notions on Turbulence (1) Conservation of momentum: Navier Stokes equations: storage term advection term gravity term pressure gradient term viscosity term In order to discuss the non-linearity consider a simpler 1d version: The Burgers Equation: And treat both physical processes seperately:

Some fundamental notions on Turbulence (2) 1) The diffusion equation: dissipation  gradient weakening  stabilizing 2) The advection equation: General solution: Advection term  gradient sharpening  instability

Some fundamental notions on Turbulence (3) Competition between both processes determines the solution. Compare both terms by making the equation dimensionless: Reynolds Number measures the ratio between the 2 terms. dissipation dominates flow is stable  laminar Non-linear advection term dominates  flow is unstable turbulent

Turbulence made by convection in the atmospheric boundary layer Some fundamental notions on Turbulence (3): Turbulence made by convection in the atmospheric boundary layer Large eddy simulation of the convective boundary layer u = 10-5m2s-1 U=10m/s L=1000m Macrostructure dominated by non-linear advection!! Poor man’s artist impression of the convective boundary layer Potential temperature profile Heat flux

Energy Cascade Some fundamental notions on Turbulence (4): Energy injection through buoyancy at the macroscale dominated by non-linear processes Hence, Large eddies break up in smaller eddies that have less kinetic energy: and a lower “local” Reynold number until they are so small that : and viscosity takes over and the eddies dissipate.

J.L Richardson (1881-1953) Some fundamental notions on Turbulence (4): free after Jonathan Swift (1733):

(Sloppy) Kolmogorov (1941) Some fundamental notions on Turbulence (4): (Sloppy) Kolmogorov (1941) Kinetic Energy (per unit mass) : E Dissipation rate : e eddy size: eddy velocity: eddy turnover time: Kolmogorov Assumption: Kinetic Energy transfer is constant and equal to the dissipation rate 

Consequences of Kolmogorov (1941) Some fundamental notions on Turbulence (5): Consequences of Kolmogorov (1941) Structure functions: Fourier transform of kinetic energy Famous 5/3-law!! Largest eddies are the most energetic Kolmogorov scale : the scale at which dissipation begins to dominate:

Energy Spectra in the atmosphere data: van Hove 1957 cyclones microscale turbulence diurnal cycle spectral gap 100 hours 1 hour 0.01 hour

energy spectra at z=150m below stratocumulus U Spectrum V Spectrum W Spectrum 500m Duynkerke 1998

Governing Equations for incompressible flows in the atmosphere Continuity Equation (incompressible) with NS Equations gravity term coriolis term Heat equation Moisture equation Condensed water eq. Gas law

The Zoo of Atmospheric Models 1 km 10 km 100 km 1000 km 10000 km  turbulence Cumulus clouds Cumulonimbus clouds Mesoscale Convective systems Extratropical Cyclones Planetary waves Subgrid Large Eddy Simulation (LES) Model Cloud System Resolving Model (CSRM) Numerical Weather Prediction (NWP) Model Global Climate Model

Global Climate and NWP models (Dx>10km) Subgrid To be parameterized

Large Eddy Simulation (LES) Model (Dx<100m) High Resolution non-hydrostatic Model: 10~50m Large eddies explicitly resolved by NS-equations inertial range partially resolved Therefore: subgrid eddies can be realistically parametrised by using Kolmogorov theory Used for parameterization development of turbulence, convection, clouds Inertial Range Resolution LES 5 3 ln(Energy) DissipationRange ln(wave number)

Dynamics of thermodynamical variables in LES Resolved turbulence subgrid turbulence Subgrid turbulence: Remark: Richardson law!!

Cloud Scheme in LES Simple: All or Nothing: { {

Turbulent Kinetic Energy (TKE) Equation Definition: Assume: No mean wind No horizontal flux terms Reynolds Averaged budget TKE-equation: Shear production Buoyancy production Transport Dissipation S B T D Laminar flow Richardson Number: Shear driven turbulence Buoyancy driven turbulence

Mixed layer turbulent kinetic energy budget dry PBL Stull 1988 normalized

Conditions for Atmospheric Convection Reynolds Number Condition for fully developed turbulence Richardson Number Condition for buoyancy drive turbulence Atmospheric Convection = Turbulence driven by Buoyancy

Objectives Tools Methods Application To “understand” the various aspects of atmospheric convection To find closures (for the turbulent fluxes and variances) Tools Observations, Large Eddy Simulation (LES) models Methods Dimension analysis, Similarity theory, common sense Application Climate and Numerical Weather Prediction (NWP) Models

(Simplified) Working Strategy Versions of Climate Models See http://www.gewex.org/gcss.html Large Eddy Simulation (LES) Models Cloud Resolving Models (CRM) Single Column Model Versions of Climate Models 3d-Climate Models NWP’s Global observational Data sets Observations from Field Campaigns Development Testing Evaluation

Dry Convective Boundary Layer Phenomenology Properties Models and Parameterization for Convective Transport

The Place of the Atmospheric Turbulent Boundary Layer Depth of a well mixed layer: 0~5km Determined by: Turbulent mixing in the BL Large Scale Flow (convergence, divergence) tropopause we z Q0 q

Can we see the convective PBL? July 2001 Downtown LA PBL top 10km (Courtesy Martin Kohler)

Typical Profiles of the convective BL Entrainment Zone Mixed layer Surface layer Stull 1988

LES View of the Dynamics: potential temperature Courtesy: Chiel van Heerwaarden, Wageningen University, Netherlands

LES View of the Dynamics: vertical velocity Courtesy: Chiel van Heerwaarden, Wageningen University, Netherlands

Horizontal Crosscut Inversion layer Mixed Layer Surface layer Irregular polygonal structures! Moeng 1998

Dry Convective Boundary Layer 2. Properties Surface Layer Mixed Layer Inversion Layer

Monin-Obukhov Similarity Construct dimensionless gradient terms: and evaluate this as a function of the stability parameter unstable stable Fleagle and Businger 1980

MO theory allows to formulate the turbulent fluxes in a diffusivity form: Diffusion Eq.

Dry Convective Boundary Layer 2. Properties Surface Layer Mixed Layer Inversion Layer

Scaling Parameters for the convective mixed layer Relevant parameters: form units 1) TKE production through buoyancy: 2) Depth of the boundary layer: Construct a convective velocity scale: Interpretation: velocity that results if all potential energy is converted into kinetic energy in an eddy of size z*

Typical Numbers

Dimensionless vertical velocity variance (in the free convective limit) Garrat 1992)

Mixed layer turbulent kinetic energy budget (LES) Shear production Buoyancy production Dissipation Transport D B T S Pino 2006

Dry Convective Boundary Layer 2. Properties Surface Layer Mixed Layer Inversion Layer

Turbulent Entrainment quiet non-turbulent air turbulent air One-way entrainment: less turbulent air is entrained into more turbulent air Mixed layer erodes into the Free atmosphere and is growing as a result of the entrainment proces

Entrainment Flux Free Convection: Entrainment flux directly related to surface buoyancy flux Observations suggest (Tennekes 1972)

Dry Convective Boundary Layer Phenomenology Properties Models and Parameterization for Turbulent Transport

Prototype: Dry Convection PBL Case Initial Stable Temperature profile: qs=297 K ; g = 2 10-3 K m-1 No Moisture ; No Mean wind. Prescribed Surface Heat Flux : Qs = 6 10-2 K ms-1 h (km) 1 5 x(km)

Mean Characteristics of LES (virtual truth) ^ Non-dimensionalise : z  z/z* w  w/w* t  t/t* Q  Q/Qs

Growth of the PBL PBL height : Height where potential temperature has the largest gradient

Mixed Layer Model of PBL growth Assume well-mixed profiles of Q. Use simple top-entrainment assumption. q Boundary layer height grows as:

Simplest Model of PBL growth: Encroachment Assume well-mixed profiles of Q. No top-entrainment assumed. Boundary layer height grows as: time

Internal Structure of PBL Rescale profiles

Classic Parameterization of Turbulent Transport in de CBL Eddy-diffusivity models, i.e. Natural Extension of MO-theory Diffusion tends to make profiles well mixed Extension of mixing-length theory for shear-driven turbulence (Prandtl 1932)

K-profile: The simplest Practical Eddy Diffusivity Approach (1) The eddy diffusivity K should forfill three constraints: K-profile should match surface layer similarity near zero K-profile should go to zero near the inversion Maximum value of K should be around: z/zinv 1 0.1 K w* /zinv (Operational in ECMWF model)

“flux against the gradient” A critique on the K-profile method (or an any eddy diffusivity method) (1) Diagnose the K that we would need from LES: K>0 Forbidden area “flux against the gradient” K<0 K>0 Down-gradient diffusion cannot account for upward transport in the upper part of the PBL

Physical Reason! In the convective BL undiluted parcels can rise from the surface layer all the way to the inversion. Convection is an inherent non-local process. The local gradientof the profile in the upper half of the convective BL is irrelevant to this process. Theories based on the local gradient (K-diffusion) fail for the Convective BL.

“Standard “ remedy Add the socalled countergradient term: Long History: Ertel 1942 Priestley 1959 Deardorff 1966,1972 Holtslag and Moeng 1991 Holtslag and Boville 1993 B. Stevens 2003 And many more……………. zinv

Can we understand the characteristics of this system? B. Stevens Monthly Weather Review (2000) Non-dimensionalise: (leave the ^ out of the notation from now on) And let’s find quasi-steady state solutions for

Quasi-Steady Solutions (1) That is to say to find steady state solutions of: Which is to say solutions for which the shape of is not changing with time. This implies a linear flux! Use as boundary conditions: (Remember we work in non-dimensionalised variables)

Quasi-Steady Solutions (2) Solution for the gradient Where K-profile is given by: and g is constant Solution of Q: Top-entrainment Surface fluxes Non-local processes Surface fluxes Top-entrainment B A

Quasi-Steady Solutions without countergradient (1) No countergradient: A=0 no top-entrainment A=-0.2 typical top entrainment value K-profile without countergradient LES

Quasi-Steady Solutions without countergradient (2) The system tends to make quasi-steady solutions (in the absence of large scale forcings) So it allways produces linear fluxes It will find a quasi-steady profile that along with the K-profile provides such a linear flux So it is the dynamics that determines the profile (not the other way around!!!) K-profile without countergradient LES

Quasi-Steady Solutions with countergradient term Height where A = -0.2, k = 0.675, gk =1.6, 3.2, 4.8

Countergradient: Conclusions Addition of a countergradient gives an improved shape of the internal structure But… How does it affect the interaction with the free atmosphere, i.e. what happens if we do not prescribe the top-entrainment anymore.? Can it be used in the presence of a cloud-topped boundary layer? Are there other ways of parameterizing the non-local flux?