PBL parameterizaon review

Haroldo Fraga de Campos Velho E-mail: [email protected] Web-page: www.lac.inpe.br/~haroldo

Research interest

Haroldo F. de Campos Velho (LAC-INPE) Senior Researcher / Scientific computing

RESEARCH FIELDS: • Scientific computing and Inverse problems • Data science and data assimilation: New method: based on neural network • Atmospheric turbulence parameterization: Some results: Taylor’s approach for turbulence in clouds New model for convective boundary layer growth Cosmological evolution as turbulent-like dynamics Scienfic compung

Solving (integral and/or) partial differential equations by numerical methods COMPUTATIONAL FLUID DYNAMICS: • Cavity Flow

Scienfic compung

Cavity flow under fractal domain

GENERATING PRE-FRACTAL WITH L-SYSTEM: 1. Start with a inital configuration 2. A rule to change de configuration 3. Iteration for Step 2.

Koch square curve Postulate: F rules: F ==> F+F-F-FF+F+F-F θ = 90o Scienfic compung

Cavity flow under fractal domain

Koch square curve Postulate: F rules: F ==> F+F-F-FF+F+F-F θ = 90o Scienfic compung

Cavity flow under fractal domain

Scienfic compung

Cavity flow under fractal domain

Scienfic compung

Cavity flow under fractal domain

Scienfic compung

Cavity flow under fractal domain (phase diagrams)

Fractal: Level-zero Fractal: Level-two Inverse problems

Damage in structural systems ISS: International Space Station

Inverse problems

Damage in structural systems ISS: International Space Station

Inverse problems

A.P. Calderon - Seminar on Inverse Problems and Applications LNCC, Rio de Janeiro (RJ), March-2006 Forecasts Scores ECMWF Inverse problems

Atmospheric profiles from satellite data

Inverse problems

Atmospheric profiles from satellite data

Inverse problems

Atmospheric profiles from satellite data

Causes Forward problem Effects

Temperature Profile Radiance

Inverse problem More important for Inverse problems numerical weather prediction Atmospheric profiles from satellite data

1 10 1 1 10 10 Radiossonde Radiossonde Radiossonde Neural Network Neural Network Neural Network Tikhonov-1 Tikhonov-1 Tikhonov-1 MaxEnt-2 MaxEnt-2 MaxEnt-2

2 2 2 10 10 10 Pressure Pressure (hPa) Pressure Pressure (hPa) Pressure Pressure (hPa)

3 3 3 10 10 10 180 200 220 240 260 280 300 320 180 200 220 240 260 280 300 320 180 200 220 240 260 280 300 320 Temperat ure (K) Temperat ure (K) Temperat ure (K) Retrievals using SDB1 Retrievals using TIGR Retrievals combining for training phase. for training phase. SDB1+TIGR. Inverse problems Inverse problems

Calibration of models (parameter identification)

Inverse problems

Preparing the models: calibration (IBIS model)

Inverse problems

Preparing the models: calibration (IBIS model)

Inverse problems

Preparing the models: calibration (IBIS model)

Calibraon: sensivy analysis (2) n Morris’ method ¨ Sensivity analysis

¨ Trajectories in the search space (2D (a) / and 3D (b)) Numerical results (1) n Idenfying faster to slower processes (Figure 1) Inverse problems

Preparing the models: calibration (IBIS model)

Data assimilaon

Very important inverse problem Determining the initial condition Forecasts Scores ECMWF Data assimilaon

Very important inverse problem Determining the initial condition

29 Data assimilaon

Very important inverse problem Determining the initial condition

30 Data assimilaon

§ Data assimilation: data fusion

31 Impact with the exponential growth for the available data

Numerical models with very high resolution

Number of observation are increasing: different satellites with thousands of bands, sensor cost decreasing. Data assimilation: an essential issue § Temperature: ANN assimilation experiment

LETKF neural network

True

Results from Rosangela Cintra PhD thesis (2011) Data assimilation: an essential issue § Moisture: 1 month assimilation experiment

LETKF neural network

True

Results from Rosangela Cintra PhD thesis (2011) Numerical experiment: LEKF and ANN

Execuon me LETKF method ANN method 04:20:39 00:02:53 hours : minutes : seconds

Atmospheric general model circulaon (spectral model): 3D SPEEDY (Simplified Parameterizaons primivE Equaon DYnamics)

Gaussian grid: 96 x 48 (horizontal) x 7 levels (vercal) = T30L7 Total grid points: 32,256 Total variables in the model: 133,632 Observaons: (00, 06, 12, 18 UTC) – radiosonders “OMM staons” Observaons: 12035 (00 and 12 UTC) = 415 x 4 x 7 + 415 Observaons: 2075 (06 and 18 UTC) = 415 x 5 (only surface) Errors: (Kalman, Particles, Variational) x Neural Networks Atmospheric turbulence modeling

Boundary Layer theory Why “boundary layer theory”?

1. The “happy world”: ideal fluid dynamics 2. The end of the “happy world”: the D’Alembert’s paradox 3. Solving the D’Alembert’s paradox: boundary layer 4. Boundary layer theory: a place to find turbulence

Atmospheric turbulence modeling

The “happy world”: ideal fluid dynamics Why “boundary layer theory”?

1. Ideal fluid (mechanical): incompressible 2. Ideal fluid (mechanical): constant density 3. Ideal fluid (mechanical): invisity

Atmospheric turbulence modeling

Ideal fluid dynamics: incompressible fluid Why “boundary layer theory”?

Atmospheric turbulence modeling

Ideal fluid dynamics: incompressible fluid Why “boundary layer theory”?

Atmospheric turbulence modeling

Ideal fluid dynamics: Rankine oval Why “boundary layer theory”?

Atmospheric turbulence modeling

Ideal fluid dynamics: Rankine oval Why “boundary layer theory”?

Atmospheric turbulence modeling

Ideal fluid dynamics: Rankine oval Why “boundary layer theory”?

Atmospheric turbulence modeling

Ideal fluid dynamics: Rankine oval Why “boundary layer theory”?

Atmospheric turbulence modeling

Ideal fluid dynamics: Rankine oval Why “boundary layer theory”?

Atmospheric turbulence modeling

Ideal fluid dynamics: Rankine oval Why “boundary layer theory”?

Atmospheric turbulence modeling

Ideal fluid dynamics: Rankine oval Why “boundary layer theory”?

Atmospheric turbulence modeling

The end of “happy world”: D’Alembert paradox Why “boundary layer theory”?

“The drag force on a submerged body is zero!” Atmospheric turbulence modeling

Solution for the paradox: Boundary layer theory

Ludwig Prandtl (German physicist) suggested in 1904 that the effects of a thin viscous boundary layer could be the reason for the drag. Atmospheric turbulence modeling

Boundary layer theory

Theodore von Kármán (Hungarian): von Kármán vortex street Atmospheric turbulence modeling

Boundary layer theory

Theodore von Kármán (Hungarian): von Kármán vortex street

Ocean: coast of Chile (near the islands Juan Fernandez) Atmospheric turbulence modeling

• Leonardo da Vinci (~ 1510): ‘‘... thus the water has eddying motions, one part of which is due the principal current, the other to the random and reverse motion.’’

• Osborne Reynolds (1883): flow decomposition ! ! ! ⎧ v ( r , t ) = v(r,t) + v'(r,t) ⎨ ! ⎩v(r,t) : average flow (similar to laminar flow)

• Lewis Fry Richardson (1922): - Cascade: ‘‘Big whirls have little whirls ...’’ - Universality: statistical behaviour is independent of external geometry, fluid nature, mecanism of injection of energy, ...

• G.I. Taylor (1921): statistical theory Atmospheric turbulence modeling Old ideas on turbulence

• Leonardo da Vinci (~ 1510):

‘’... thus the water has eddying motions, one part of which is due the principal current, the other to the random and reverse motion..” Atmospheric turbulence modeling Old ideas on turbulence

• Osborne Reynolds (1883): flow decomposition

Atmospheric turbulence modeling Old ideas on turbulence

• Osborne Reynolds (1883): flow decomposition

Atmospheric turbulence modeling Atmospheric turbulence modeling Old ideas on turbulence • Lewis Fry Richardson (1922):

o Energy cascade: “Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity.

Turbulence Universality: o - statístical properties do not depend on geometry - turbulence does not depend on fluid nature - mechanism of energy injection ...

L. F. Richardson: English scientist, matematician, physisit, meteorologist, psicologist, and paece activist. Proponent of the numerical weather prediction and suggested similar approach for studying war causes for avoiting the war. Pionering in the fractal technics, linear equation solution, extrapolation method.

Atmospheric turbulence modeling Old ideas on turbulence • Lewis Fry Richardson (1922): o Modern numerical weather prediction follows the Richardon’s scheme

o Computers: Human beings to do the computation Atmospheric turbulence modeling Old ideas on turbulence

• Lewis Fry Richardson: o Richardson’s Number (ratio between potential and kinetic energies): gh Ri = 2 u

o Richardson’s extrapolation: - Numerical analysis: a method for sequential speed-up - Aplications: (i) Romberg’s integration method (ii) Bulirsch–Stoer’s algorithm for EDO solution

From the wikipedia Atmospheric turbulence modeling The (Geoffrey Ingram) Taylor’s theory • Associated phenomena with G. I. Taylor name

* Taylor cone * Taylor dispersion * Taylor number * Taylor vortex * Taylor–Couette flow * Rayleigh–Taylor instability * Taylor-Proudman theorem * Taylor-Green vortex * Taylor microscale

Atmospheric turbulence modeling Our challenge: representing Planetry Boundary Layer Introduction to the Taylor’s theory The (Geoffrey Ingram) Taylor’s theory Turbulence and Reynolds Hypothesis:

The dynamical (turbulent) variable can be understood a sum of a mean stream and a deviation: ϕ = ϕ + ϕ ' ; being ϕ a mean stream (similar to laminar flow), and ϕ' a fluctuation (deviation) with zero mean: ϕ ' = 0 .

Dealing with average process the Navie-Stokes and the diffusion equations become

dvi ∂p ∂ ρ = − − 2εikjΩk v j + ρgδ 3i + ∑ [σ ij − ρ vi 'v j '] dt ∂xi j ∂x j Reynolds’ fluxes

dα ∂α ∂(v jα ) ∂ ⎡ ∂α ⎤ = + ∑ = ∑ ⎢K − v 'α'⎥ dt t x x m x j ∂ j ∂ j j ∂ j ⎣⎢ ∂ j ⎦⎥ eddy diffusivity

⎡ ∂α ⎤ ∂α First order closure: ⎢K − v 'α'⎥ ≈ − v 'α' = K m x j j ij x ⎣⎢ ∂ j ⎦⎥ ∂ j Introduction to the Taylor’s theory

Fick’s Law: From a mass differential balance: ∂c ε c(x + ε) − c(x) ≈ ε ∂x considering ε as a small length. Assuming ΔA ε ≈ lm, where lm is the characterisc length, the mean free path in Brownian movement u and gas theory. For net flux F: 1 u2 F = F2 − F1 = −um [c(x + lm ) − c(x)] ∂c = −(umlm ) ∂x mass diffusivity concentration : c ≡ g cm-3 volume [ ] mass time m Flux : F = = = u.c g s-1 cm-2 Generalizing: area At [ ] net flux : F = F1 − F2 = [u1c1 − u2c2 ]≈ um [c1 − c2 ] ! ! F = −D ∇c (mass or particles) above it was assumed : u1 ≈ u2 Taylor’s equation

dx(t) t Particle moviments: v(t) = ⇒ x(t) = ∫ v(t') dt' dt 0

Remembering: diffusivit ty = v ( t ). x ( t ) - then: dx(t) d 1 v(t)x(t) x(t) ⎡ x2 (t)⎤ t v(t)v(t') dt' = = ⎢ ⎥ = ∫0 dt dt ⎣2 ⎦ (self-)correlation function d 1 taking the average: ⎡ x2 (t)⎤ t v(t)v(t') dt' ⎢ ⎥ = ∫0 dt ⎣2 ⎦

The correlation function R(τ) in a homogeneous statistical process is independent of the parameter ‘‘t’’: t = t’ + τ. In this case (homogeneous) R(τ) is given by R(τ) = v(t')v(t'+τ) = v2 ρ(τ) (Obs.: R(0) = v2 ⇒ ρ(0) =1)

Substituting and integrating by parts: 2 2 t x (t) = 2v ∫0 (t −τ )ρ(τ ) dτ Deriving the Taylor’s equation

From the average displacement:

t τ t τ x2(t) = 2 v(t)v(t') dt'dτ = 2v2 ρ(t')dt'dτ ∫0 ∫0 ∫0 ∫0

τ Integrating by parts (being: dw = d τ and u ( τ ) = ρ ( t ' ) dt ' ⇒ du d τ = ρ ( τ )): ∫0 t 2 2 t τ 2 ⎪⎧ τ t ⎪⎫ x (t) = 2v ρ(t')dt'dτ = v ⎡τ ρ(t')dt'⎤ − τρ(τ )dτ ∫0 ∫0 ⎨⎢ ∫0 ⎥ ∫0 ⎬ ⎩⎪⎣ ⎦0 ⎭⎪ 2 ⎧⎡ t ⎤ t ⎫ = 2v ⎨ t ρ(τ )dτ − 0 − τρ(τ )dτ ⎬ ⎩⎣⎢ ∫0 ⎦⎥ ∫0 ⎭ t = 2v2 (t −τ )ρ(τ )dτ ∫0 Taylor’s equation (spectral version)

1 +∞ Fourier transform: Φ(ω) = R (τ ) e iωτ d τ π ∫ −∞ 1 +∞ 1 +∞ R (τ ) = Φ(ω) e −iωτ d ω ⇒ R (0) = Φ(ω) d ω 2 ∫ −∞ 2 ∫ −∞

If the correlation is independent of parameter τ (homogeneous turbulence):

v(t)v(t +τ) = v(0)v(τ) = v(−τ)v(0) ⇒ R(τ) = R(−τ) (even function) odd function the spectral equation becomes:

1 +∞ 2 ∞ Φ(ω) = ∫ R(τ )[cos ωτ + sin ωτ]dτ = ∫ R(τ ) cos ωτ dτ π −∞ π 0

In meteorology, the frequency is given in Hz (s-1) instead rad. s-1: n = ω/2π. Hence, the spectra will be: S(n) = 2π Φ(2πn). Then, the velocities variances are:

v2 2 R(0) ∞ S (n) dn ∞ nS (n) d(ln n) i = σ i = = ∫0 Li = ∫0 Li Considering the 1st form of Taylor’s equation

t ⎡∞ S (n) ⎤ x2 2 2 t Li cos 2 n dn d i = σ i ∫ ( −τ )⎢∫ 2 ( π τ ) ⎥ τ 0 ⎣⎢0 σ i ⎦⎥ 2 2 ∞ sin (nπ t) = σ i ∫0 FL (n) dn t i (nπ )2 x 2 = 2v 2 (t −τ )ρ(τ )d τ i ∫ 0 From the Batchelor’s relation:

d 1 ⎛α = x, y, z⎞ K ⎡ x2 ⎤ αα = ⎢ i ⎥ ⎜ ⎟ dt ⎣2 ⎦ ⎝ i = u,v, w ⎠ the fundamental equation is obtained:

2 σ i ∞ sin(2π nt) K = ∫ FL (n) dn αα 2π 0 i n2

The Lagrangian form for the eddy diffusivity. ’ ( ) ( ) ; T T Using the Gifford-Hay & Pasquill s assumption: ρLi βiτ = ρi τ βi = Li i

a relation can be obtained for the spectra: nF (n) nF ( n) Li = βi i βi and a Eulerian form for eddy diffusivity follows:

2 2 σ i βi ∞ sin(2π nt βi ) K = ∫ Fi (n) dn αα 2π 0 n2

An asymptotic form can also be derived (long travel times, t → ∞):

σ 2 β F (0) K = i i i αα 4 Deriving the Taylor’s equation

2 2 2 ∞ sin (nπ t) xi = σ i FL (n) dn ∫0 i (nπ )2

Análise assintótica: t → 0 t → ∞

Brownian motion: Einstein’s relation distance between two particles ~ t Turbulence parameters:

x2 = v2 t 2 for t → 0 From the Taylor’s theory is easy to show that: i i x2 v2 T t for t i = i Li → ∞

d ⎡1 2 ⎤ 2 Now, using well establishied expressions: K = x (t) = σ T = σ l αα ⎢ i ⎥ i Li i i dt ⎣2 ⎦ ∞ 2 σ i = ∫ Si (n) dn 0

⎡σ i βi Fi (0)⎤ Eddy diffusivity : K zz = σ i ⎢ ⎥ Degrazia et al. (1998): ⎣ 4 ⎦ BLM, 86, 525-534 β F (0) Lagrangian time -scale : T = i i Li 4 σ β F (0) Mixing length : l = i i i i 4 Kolmogorov’s hypothesis (K-41):

• Hypothesis 1: For high Re number, the small scale movements of the turbulence are statistilly isotropics.

• Hypothesis 2: For high Re number, the small scale statistics are determined only by viscosity (ν) and by dissipation rate of energy (ε). From theses parameters a length scale is definided: 1 4 η = (ν 3 ε ) the Kolmogorov’s length scale (η).

• Hypothesis 3: For high Re number, statistics for the scales in the interval η << r << L are determined only by r e ε.

Kolmogorov’s hypothesis (K-41):

• From Hipothesis 3: The space scale r (or k, for the Fourier transform) and the dissipation function are the only scales to be considered for describing turbulence process in the interval: η << r << L.

Below the Kolmogorov’s scale η the viscosity is relevant.

From the dimensional analysis, the espectra for turbulent kinetic energy is defined by:

e1 e2 E(k) ~ ε k (dimensional analysis) ⇓ E(k) ~ ε 2/3k −5/3 Similarity theory

Goal: describing turbulence with few parameters

Parameters: u * (friction velocity) L (Obukhov’s length)

q * (heat flux)

u* = τ s ρ (τ s : tensão de cizalhamento (shear))

3 R c p u* ⎛ p0 ⎞ L = − , q* = w'θ' , θ = T⎜ ⎟ , z=0 ⎜ ⎟ κβcq* ⎝ p ⎠

q* θ* = − . (θ : temperatura potencial) κu* Similarity theory

• Computing the gradient by universal functions

Here “universal” means the property is the same for all fluxes and the universal function ξ ( z / L ) is empirically determined.

'κ z dU ! z $* ' dU * ) = ξ # &, mas: )−u 'w ' = K , ( u* dz " L %+ ( dz + 

⇓ κu z K = * ξ(z L) Local similarity theory (SBL)

• Convective boundary layer (CBL): parameters are computed for entire PBL.

• Stable boundary layer (SBL): local similarity

2 τ (z) U* (z) α1 = 2 = (1− z h) τ 0 u*

w'θ '(z) α = (1− z h) 2 (w'θ ')0

Λ(z) (3α 2)−α = (1− z h) 1 2 L

From: F.T.M. Nieuwstadt, J. Atm. Sci., 41, pp. 2202-2216 (1986). Cabauw, Netherlands • central NL 45km east of N.Sea • short grass site, clay soils • 213m tower multiple levels of wind, temperature, moisture • micromet site surface fluxes, soil moisture & temp, radiation • radiosondes → Cabauw & DeBilt • 31 May 1978 synoptically ‘quiet’ Sketch of physical processes on the atmospheric boundary layer (cartoon extract from Stull’s book) Degrazia et al. (2003): BLM Degrazia et al. (2002): N. Cimento Almeida et al (2006): Atmospheric Research

Nunes et al (2010) Degrazia, Campos Velho, Carvalho (1997), BzPA

???? Degrazia-Moraes (1992), BLM Applicaon of G. I. Taylor theory on PBL The key issue in the Taylor’s approach is the spectra analytical model

The model for the nondimensional turbulent velocity spectra is:

m3 nSi (n) Af

2 = m m1 2 u* (1+ Bf )

2 where: u* friction velocity in the surface layer f nz /U nondimensional frequency =

For deriving spectral models, some considerations should be assumed:

1. S(k) ≈ AB −m1m2 k m3 −m1m2 −1 for k → ∞

2. Emodel (k) → EKolmogorov for k ∈ inertial subrange

The spectral model should agree with the spectra in the inertial sub-range:

E(k) = d ε 2 3k −5 3 2

The Kolmogorov’s equation for the inertial subrange.

STABLE BOUNDARY LAYER (SBL)

Changing variables for the Kolmogorov’s law in the inertial subrange:

2 3 5 3 E(k) = d ε k − ⎧k = (2πn U ) (hipótese de Taylor da ⎫ i ⎨ ⎬ ⎩ turbulência congelada)⎭ ⇓ 2 3 2 3 − 2πn 1 ⎛ 2πn ⎞ ⎛ εκz ⎞ ⎛ 2πnz ⎞ −2 3 E d ⎜ ⎟ 2 ⎜ ⎟ = i ⎜ 3 ⎟ ⎜ ⎟ κ U u* ⎝ U ⎠ ⎝ u* ⎠ ⎝ U ⎠ ⇓ nS (n) 2 3 −2 3 2 = diΦε f ( f = nz /U) U*

⎛ z ⎞ Φε = cε ⎜1+ 3.7 ⎟ : função de dissipação (CLE) ⎝ Λ ⎠ (Z. Sorbjan, BLM, 34, p. 377 -397, 1986) STABLE BOUNDARY LAYER (SBL)

For deriving a spectral model, some considerations should be assumed: 2 1. for k → ∞ : m1 − m2m3 = − 3 A −2 3 2. di = m Φε B 3

d −m3 1.5m 3. m1 m2 1 Af 1+ Bf f ( f ) = 0 ⇒ B = [ ( ) ] = m i m2 df ( fm )i

4. De experimentos CLE : m2 = 5/ 3

m1 = m3 =1

STABLE BOUNDARY LAYER (SBL)

For deriving a spectral model, some considerations should be assumed:

1. B 1.5( f )−5 3 A 1.5d ( f )−5 3 2 3 = m i = i m i Φε

2 3 nS (n) 1.5di fΦε 2. = U 2 5 3 5 3 * ( fm )i [1+1.5( f ( fm )i ) ]

2 3 2 ∞ 2 2.32diΦε U* 3. σ i = Si (n) = ∫0 2 3 ( fm )i

⎧0.058 i = u

⎡ ⎛ z ⎞⎛ h ⎞⎤ ( f ) ⎪ 0.22 i v 4. ( f ) ( f ) 1 3.7 m n,i = ⎨ = m i = m n,i ⎢ + ⎜ ⎟⎜ ⎟⎥ ⎪ ⎣ ⎝ h ⎠⎝ Λ ⎠⎦ ⎩ 0.33 i = w

Z. Sorbjan: Structure of the Amospheric Boundary Layer, Prentice Hall (1989). STABLE BOUNDARY LAYER (SBL)

For deriving a spectral model, some considerations should be assumed: 0.64z 1. S (0) = i U ( fm )i

S (n) 1.5 f ( f ) 2. i [ m i ] Fi (n) ≡ 2 = 5 3 σ* 1+1.5[f ( fm )i ]

α1 2 K zz 0.32(1− z h) (z h) 3α1 2+α2 3. = Λ ⎛ z ⎞ = ⎜1− ⎟ . u*h 1+ 3.7(z Λ) LMO ⎝ h ⎠

From: G.A. Degrazia, O.L.L. Moraes, BLM, 58, pp. 205-215 (1992). CONVECTIVE BOUNDARY LAYER (CBL)

Changing the velocity scaling:

1 3 −1 3 w ⎛ −κ ⎞ L = * ⎜ ⎟

u* ⎝ h ⎠

2 3 m1 nSi (n) Af ⎛ −κL ⎞ 2 = m ⎜ ⎟ m2 3 w* 1+ Bf ⎝ h ⎠ ( ) nS (n) z 2 3 i 2 3 −2 3 2 3⎛ ⎞ ( h / w3 ) 2 = diκ f Ψ ⎜ ⎟ Ψ = ε * w* ⎝ h ⎠ CONVECTIVE BOUNDARY LAYER (CBL)

Changing the velocity scaling:

2 3 2 3 nSi (n) 0.98( f q)(Ψ q) (h) = 5 3 w2 5 3 * ( fm )i [1+ (1.5 ( fm )i )( f q)]

CONVECTIVE BOUNDARY LAYER (CBL)

Changing the velocity scaling:

Degrazia, Campos Velho, Carvalho, BzPA, 70, pp. 57-64 (1997). SUMMARIZING: similarity + statistical (Taylor) theories Deriving the spectra: (a) maximum spectral should be the same (experimental and analytical spectra) (b) fitting the model with generalized Kolmogorov's energy spectra.

Exponents CBL: m1 = m3 = 1 and m2 = 1 + 2/3 SBL: m1 = 1 + 2/3 and m2 = m3 = 1

Therefore expressions for turbulent spectra are -5 3 nS (n) 1 ⎛ z ⎞⎡ f ⎤ i = ⎜ ⎟ 1+ for SBL 2 ⎢ ⎥ u* ( fm )i ⎝U ⎠⎣ (5 3)( fm )i ⎦ -1 ⎡ 5 3 ⎤ nSi (n) −3 5 ⎛ z ⎞ −1 1 ⎛ f ⎞ = c 2 3 ⎜ ⎟ f ⎢1+ ⎜ ⎟ ⎥ for CBL 2 ζ ( ) ⎜ ⎟( m )n,i 5 3 ⎜ ⎟ w* ⎝ ρiU ⎠ ⎢ (2 3)( fm )n,i ⎝ ρi ⎠ ⎥ ⎣ ⎦ where: cζ = [sin(π (5 3))] [π (5 3)] ; n = kU 2π −1 −1 ρi = ( fm )i ( fm )n,i = [z (λm )i ]( fm )n,i

(λm )i wavelength value at spectral peak An analytical integration of the spectra for whole freqency domain and the Taylor’s theory produce expressions for eddy diffusivities:

π σ z π K = i = σ λ for CBL αα i i 16 ( fm )i 16

σ z π i π Kαα = = σ iλi for SBL 25 ( fm )i 25

For example: vertical eddy diffusivities for CBL

K zz * 1 3 −4(z h) −4 8(z h) 5 3 = β Ψ [1− e − 3×10 e ] w*h

* * 4 3 Ψ z w3 where: β = πc 16 (1.8 ) ; = ε i * vertical eddy diffusivities for SBL

3α1 2+α2 α1 2 Λ z K zz 0.32(1− z h) (z h) ⎛ ⎞ = ; = ⎜1− ⎟ . u*h 1+ 3.7(z Λ) LMO ⎝ h ⎠ Planetary boundary layer modeling n Similarity and Taylor’s theories ¨ Important parameters from similarity theory ¨ Parameterizaon from Taylor’s theory for the PBL ¨ Taylor’s theory: applicaon for all stability condions n Next presentaon ¨ Applicaon to the atmospheric numerical models ¨ Applicaon to the turbulence inside the clouds ¨ Transion boundary layers parameterizaon ¨ Intermiency parameterizaon ¨ Cosmological evoluon: turbulent dynamics? Planetary boundary layer modeling

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