Spring Semester 2011 March 1, 2011

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1 METR 130: Lecture 3 - Atmospheric Surface Layer (SL - Neutral Stratification (Log-law wind profile - Stable/Unstable Stratification (Monin-Obukhov Similarity Theory Spring Semester 011 March 1, 011

2 Reading Arya, Chapter 10 ( log-law, neutral SL Arya, Chapter 11 (MO theory, stable/unstable SL Review Arya, Chapters 5 through 7 (review material covered so far Arya, Chapter 8.1 (review turbulence & Reynolds number

3 Before we begin (Diagnosing surface fluxes from routine observations Example: Observational platform for California DWR CIMIS Network 1. Total solar radiation (pyranometer. Soil temperature (thermistor 3. Air temperature/relative humidity (HMP35 4. Wind direction (wind vane 5. Wind speed (anemometer 6. Precipitation (tipping-bucket rain gauge Can compute ( diagnose surface fluxes from these routine near-surface meteorological observations coupled with certain assumptions (e.g. on C H, R L. See Lecture and Exam #1 HW calculation for details on this procedure.

4 Why study surface layer theory? 1. To understand often used expressions used for turbulent exchange coefficients (C M, C H, C Q, etc when diagnosing fluxes, boundary layer heights, and other things from measured data.. When predictions of near-surface met variables are needed (e.g. in models. Expressions resulting from surface layer theory are used in models to make predictions of near-surface mean profiles as well as surface fluxes.

5 Surface Layer: Definitions Let H = surface layer depth, h = boundary layer depth H lowest 10% of boundary layer H 0.1h H times roughness length (z 0 10z 0 < H < 100z 0 z < 10z 0 is roughness sublayer z > 100z 0 is outer layer In models, surface layer is effectively the lowest model grid layer adjacent to surface (typically lowest grid layer spacing is around 10 meters.

6 Surface Layer

7 Surface Layer is an inherent part of a turbulent boundary layer boundary layer height (h Surface Layer ( 0.1h ~ 100 z 0 ~ 10 z 0 laminar BL over rough surface Transition to turbulence turbulent BL over rough surface

9 Neutral SL Theory (Log-Law Wind Speed Profile

10 Derivation: Log-Law Wind Speed Profile (Neutral SL Dimensional analysis Let friction velocity be velocity scale, i.e. u = (τ 0 /ρ a 1/ Let height above surface (z be length scale Therefore du dz = u kz where k = 0.4 is the von karman constant Integrate with respect to z U = ( u* / k ln ( z + constant Define constant of integration in terms of height where extrapolated wind profile equals zero. Define this height as the momentum roughness length, z 0,m This leads to constant = - (u * /kln(z 0,m, and therefore to the log-law wind speed profile U = ( u / k ln ( z / z0, * m

11 What about wind direction? Near the surface p p + p Friction PGF Co V < G α 0 Wind at angle α 0 to isobars ( cross isobaric flow angle. Can be shown that α 0 does not change with height in surface layer Value of α 0 for neutral conditions a function of surface Rossby Number, Ro = G/fz 0 See handout for Assignment 1 for reminder α 0 also depends on stability, will be covered later

12 U(z Modified Form for Flow over Forests and Buildings u ln z - D = k z 0, m D defined as zero-plane displacement height. Some rule-of-thumb relationships - z 0 = 0.1 mean canopy (or building height ( mch or mbh - D = 0.7 mean canopy (or building height Surface Layer (SL log law profile z Roughness element Roughness sub-layer Canopy Layer z = D 0.7 mbh figure not drawn to scale

13 Calculating wind speed at one height given wind speed at another (applying log-law U(z U(z 1 = ln(z ln(z 1 /z /z 0,m 0,m U ( z = ( u* / k ln ( z1 / z0, 1 m U ( z = ( u* / k ln ( z / z0, m or with zero-plane displacement U(z U(z 1 = ln[(z ln[(z 1 D/z D/z 0,m 0,m ] ]

14 Momentum Flux = τ 0 = -ρ a u * Surface Drag Coefficient (Momentum From log-law and given mean wind speed U a at height z a u = * ln ku a ( / z z a 0, m Substituting for u * k τ 0 = ρau* = ρa ln ( z / z where a 0, m U a = ρ C a m U a C M = ln k z ( / z a 0, m

15 A passive scalar is a scalar variable that is purely transported around with the wind, and does not affect the dynamics of the wind field or the thermodynamics of the temperature field. Examples include: concentration of an air pollution species in many cases, humidity (provided that air is not saturated and that virtual temperature effects are small, temperature (provided very close to neutral stability and radiation effects are neglected. Dimensional analysis Profiles for Passive Scalars (Neutral SL Let passive scalar = A, and F A,0 be the surface flux of A Let surface layer scale for this quantity then equal A * = -F A,0 /(ρ a u * Let height above surface, z, be length scale Therefore da dz = A kz where k = 0.4 can be assumed (based on most recent data Will finish derivations for profile and exchange coefficients (e.g. C Q in Assignment #

16 Passive Scalar Profiles in Neutral Surface Layer ( From Assignment (Problem 1, can be shown A ( z = A + ( A* / k ln ( z / z0, 0 A where z 0,A is the roughness length for A, determined by extrapolating the log profile to the point where A = A 0, the surface value of A. In general, z 0A z 0m. A common relationship is z 0a = 0.1z 0m.

17 Surface Drag Coefficient (Passive Scalar Scalar Flux = F A = -ρ a u * A * Substituting into the above u * and A * from log-law relationships mean wind speed (U r and scalar (A r at reference height z r within SL Surface value of scalar A 0 leads to F A k k = ρau* A* = ρa U r ( Ar A0 = ρacau r ( Ar A ln( zr / z0, m ln( zr / z0, A 0 where C = A ( z / z ln( z / z ln 0, 0, r m k r A

18 Data confirming log-law (Figures 10.4 and 10.9; Arya Tables & Figures for Roughness Length & Displacement Height values (Figures 10.5, 10.6, and 10.8, Arya

19 Stable/Unstable Surface Layer (Monin-Obukhov Similarity Theory

20 Some Background on Turbulence Mechanical Turbulence Caused by shear instability (i.e. instabilities in wind shear Friction velocity, u *, appropriate SL scale to characterize this. Possible in all static stabilities (neutral, stable or unstable. Buoyant Turbulence Caused by positive buoyancy (buoyant instability Associated with unstable air ( θ/ z < 0 More generally, when turbulent heat flux > 0 Buoyant Suppression of Turbulence Caused by negative buoyancy Associated with stable air ( θ/ z > 0 More generally, when turbulent heat flux < 0

21 Derivation (Buoyant Turbulence and Heat Flux Summary of derivation done on white-board dw dt g = ( θ θ p θ dw dt g = l θ dθ dz d( w / dt g = wl θ dθ dz parcel theory letting θ p = θ l dθ dz dθ θ = θ p θ = l dz d( w / dt g = K θ H dθ dz d( w / dt g H = θ ρac S = p B θ mean and parcel potential temperature at some height θ p l dθ/dz = constant

22 Key Points Heat flux affects vertical turbulence (dynamically active. Positive Heat Flux (H S > 0 Vertical turbulent energy increases Associated with positive buoyancy; unstable air Negative Heat Flux (H S < 0 Vertical turbulent energy decreases Associated with negative buoyancy; stable air Quantity B (g/θ(h s /ρ a c p called buoyancy flux Let H S,0 be the surface heat flux (H S in Lecture. Let B 0 = (g/θ(h s,0 /ρ a c p therefore be the surface buoyancy flux. B 0 is new scaling variable for stratified conditions. Used to extend neutral SL layer theory to stratified conditions. Modified theory called Monin-Obukhov (MO theory

23 Monin-Obukhov Length, L L u kb 3 * 0 = 3 u* θ kg( H s / ρ c a p Use to define a non-dimensional stability parameter ζ z/l ζ > 0 (stable ζ < 0 (unstable ζ = 0 (neutral / ρ c kzg( H s, 0 a p ζ = z / L = 3 u* θ

24 Physical Meaning of z/l z/ L large (> 1 Buoyancy effects start to affect turbulence and profiles (or even dominate L > 0 (stable, buoyant suppression of turbulence and mixing L < 0 (unstable, buoyant enhancement of turbulence and mixing z z = L i.e. z/l = 1 z/ L small (<< 1 Buoyancy effects relatively small Mechanical turbulence dominates (i.e. due to wind shear surface (z = 0

25 Wind Speed Profile in Stratified Surface Layer (1 Applying stability parameter ζ = z/l du dz = u φ m kz ( ζ where φ m is a stability function of z/l. Following standard equations determined from theory and field measurements ( ζ = 1+ β ζ φm 1 ζ > 0 (stable (typical value β 1 = 4.7 φ m ( ζ = (1 γ ζ 1/ 4 1 ζ < 0 (unstable (typical value γ 1 = 15 See also Arya equations 11.6 and 11.7

26 Wind Speed Profile in Stratified Surface Layer ( Upon vertically integrating du/dz from previous slide, mean wind speed profile is a modified version of log-law as follows U( z = ( u* / k[ln ( z / z0, m -ψ m( z / L] where ψ m is a function of z/l determined from vertically integrating the ϕ m (z/l functions on the previous slide. Exact form of ψ m is different in stable (z/l > 0 and unstable (z/l < 0 conditions, since ϕ m functions are different in stable and unstable conditions. See Arya Section 11.3 (equations for further details, and for exact form of ψ m functions.

27 Applying stability parameter ζ = z/l dθ = dz θ φh( ζ kz Scalar Profiles in Stratified Surface Layer (1 For potential temperature, for example where φ h is a stability function of z/l. Following standard equations determined from theory and field measurements φh ( ζ = 1+ β ζ ζ > 0 (stable (typical value β = β 1 = 4.7 φ h ( ζ = (1 γ ζ 1/ ζ < 0 (unstable (typical value γ = 9 See Arya equations 11.6 and 11.7

28 Scalar Profiles in Stratified Surface Layer ( For potential temperature, for example Upon vertically integrating dθ/dz from previous slide, mean potential temperature profile is a modified version of log-law as follows θ ( z = θ + ( θ* / k[ln ( z / z0, h -ψ h ( z 0 L / ] where ψ h is a function of z/l determined from vertically integrating the ϕ h (z/l functions on the previous slide. Exact form of ψ h is different in stable (z/l > 0 and unstable (z/l < 0 conditions, since ϕ h functions are different in stable and unstable conditions. See Arya Section 11.3 (equations for further details, and for exact form of ψ h functions.

29 Drag Coefficients (Modified to Account for Stability via MO Theory Redoing derivation for neutral expressions (previous slides, except this time accounting for ψ functions C M = [ln ( z / z ψ ( z / L] r 0, m k m r C H = k [ln( z / z0, ψ ( z / L][ln( z / z0, ψ ( z r m m r r h h r / L] where z r is the reference (measurement height C H = C Q = C A see Arya Equations 11.17

30 Calculating wind speed at one height given wind speed at another (extended for MO theory U(z U(z 1 = ln(z ln(z 1 /z /z 0,m 0,m ψ ψ m m ( z ( z 1 / L / L ( z1 = ( u* / k[ln ( z1 / z0, m -ψ m( z / L] U 1 U ( z = ( u* / k[ln ( z / z0, m -ψ m( z / L] or with zero-plane displacement U(z U(z 1 = ln[(z ln[(z 1 - D/z - D/z 0,m 0,m ] ψ ] ψ m m [( z [( z 1 D / L] D / L]

31 Wind Speed Profiles Computed for Neutral, Stable and Unstable Stratification using log-law (neutral and MO Theory (stable and unstable (z 0m = 0.05, 10-meter wind speed = 4.5 m/s = 10 mph, specified values for L 10 Neutral 8 Unstable (L = -4 meters z (m 6 Stable (L = 40 meters stable: less mixing more shear 4 unstable: more mixing, less shear (a fuller profile Wind speed (m/s

32 Wind Speed Profiles Computed for Neutral, Stable and Unstable Stratification using log-law (neutral and MO Theory (stable and unstable (z 0m = 0.05, 10-meter wind speed = 4.5 m/s = 10 mph, specified values for L SAME PROFILE AS PREVIOUS SLIDE, BUT NOW PLOTTED UP TO Z = 50 METERS 50 Neutral 40 Unstable (L = -4 meters z (m Stable (L = 40 meters unstable: more mixing, less shear. Does not require as strong a wind aloft to produce a 10-meter wind speed = 10 mph since there is enhanced vertical mixing due to buoyant instability (z/l < 0 stable: less mixing, more shear. Requires a much stronger wind aloft to produce a 10-meter wind speed = 10 mph since there is suppressed vertical mixing due to buoyant stability (z/l > Wind speed (m/s

1 The Richardson Number 1 1a Flux Richardson Number b Gradient Richardson Number c Bulk Richardson Number The Obukhov Length 3

Contents 1 The Richardson Number 1 1a Flux Richardson Number...................... 1 1b Gradient Richardson Number.................... 2 1c Bulk Richardson Number...................... 3 2 The Obukhov