Spring Semester 2011 April 5, 2011

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Domenic Stafford
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1 METR 130: Lectre 4 - Reynolds Averaged Conservation Eqations - Trblent Flxes (Definition and typical ABL profiles, CBL and SBL) - Trblence Closre Problem & Parameterization Spring Semester 011 April 5, 011

2 Reading from Arya Chapters 5.4 & 5.5 Chapter throgh 6.3 (Review) 6.4 & 6.5 (note Richardson nmber vs. height in stable BL) Chapter 8.1 throgh 8.5 Chapter 9.1 & 9. Chapter 13 Not reqired, bt maybe helpfl. Some advanced topics related to parameterization. Page 87 ( integral models ) has some material relevant to Assignment #3 Problem 3.

3 Trblence Decomposition of Velocity (See also 8.4 of Arya) U i (t) i (t) trblent flctation U i mean velocity Similar decomposition for other variables 1) Potential Temperatre ) Specific Hmidity 3) Species Concentration 4) Pressre 5) Density (althogh, can relate to P & T throgh IGL)

4 Reynolds Averaging Postlates (or reslts based on these ) Let A and B be variables, and c be a constant A A = = A 0 ca = ca A + B = A + B AB AB A x = = AB AB A x Space for any derivations, math to show that these are tre

5 Starting Point (-momentm eqation) Combine U momentm eqation and incompressible form of continity eqations t = ( ) x ( v) y ( w) z 1 ρ 0 p x + fv + x ν x + y ν y + z ν z Advection (written in flx form) Pressre Gradient Force coriolis force ρ ρ 0 = constant via Bossenesq assmption Eqation above is for instantaneos flow. Viscosity

6 Ending Point (Reynolds Averaged -momentm eqation) Decompose variables as ( ) + ( ) Reynolds Average both sides of eqation t ( ) ( v) ( w) 1 = x y z ρ Mean Advection (written in flx form) 0 p x Pressre Gradient Force (Mean) + f v + ν x + y ( ) ( v) ( w) + z x y z Coriolis Force (mean) Mean Viscosity Divergence of trblent momentm flx. (NEW TERMS) Above eqation is for the Reynolds-averaged (or mean) velocity. ρ ρ 0 = constant via Bossenesq assmption. Viscosity term can be shown to be small in most flows of geophysical interest (meteorological, oceanographic) Above eqation w/ot viscosity term is essentially the form of the -momentm eqation sed in 3-D weather & climate models

7 Bondary Layer Form of Eqation (i.e. after making bondary-layer assmption) ( ) ( w = f v vg ) t z z >> x and y Pressre Gradient & Coriolis Forces Divergence of vertical trblent momentm flx. Wrote PGF in above eqation in terms of geostrophic wind BL Assmption alternatively can be viewed as an assmption of horizontal homogenieity Horizontal Homogeneity statistics of variables do not vary horizontally. Horizontal homogeneity implies throgh incompressible continity eqation that w = 0. Above eqation is the form of the -momentm eqation sed for the basic bondary layer research and testing of parameterizations.

8 Closre Problem z w y v x z y x v f x p z w y v x t = ) ( ) ( ) ( 1 ) ( ) ( ) ( 0 ν ρ Divergence of trblent momentm flx. (NEW TERMS!) New terms involving trblence flxes introdce additional nknowns Similar terms get introdced when going throgh the procedre for other eqations (e.g. v, θ, q) However since no new eqations have been introdced into the system system is nclosed An nclosed system of eqations cannot be solved Need to represent nclosed terms in terms of known variables (i.e. those that we have eqations for) in order to solve system i.e. we reqire trblence parameterizations for the trblence flxes. Will be seen how to do this later on

9 Also Remember from Lectre 1 (Also see Arya, Chapter 6) Above the bondary layer (two main forces: PGF and CO) Near the srface (three main forces: PGF, CO & Friction) p p + p PGF Co V = V G p p + p Friction PGF Co V < V G Wind is geogrophic (or perhaps gradient flow or something in between); main point: no friction, wind parallel to isobars Wind slowed de to friction. Wind flow at angle α 0 to isobars ( cross isobaric flow angle )

10 Momentm Eqations: ABL Eqations for mean velocity (Note three forces, which is friction?) Divergence of vertical trblent shear stress per nit mass, where τ x = x-component of vertical trblent shear stress and τ y = y-component of vertical trblent shear stress. These are the F terms sed in MET11 for the friction force. Magnitde of shear stress = (τ x + τ y ) 1/ τ Srface vale τ(z=0)/ρ = τ 0 /ρ = * The implied key velocity scale * is called the friction velocity NEW UNDERSTANDING: τ x /ρ = - w and τ y /ρ = -v w (i.e. stress = flx)

11 Fll Bondary Layer Eqations t v t q t = f = ( v v ) f ( ) θ ( w θ ) = + S t z ( w q) = z χ ( w χ) = t z g ( w) z g ( v w) z + S + θ + q + + S + χ S S θ q Divergence of vertical trblent flxes of heat (θ), moistre (q) and a polltant species (χ). S χ Divergence of vertical trblent flxes of and v velocity. HOMEWORK: Derive one of these three eqations. In above eqations g and v g are geostrophic wind speed components S + and S - are sorce and sink terms, respectively

12 Reynolds Stress Tensor w w v w w v v v w v j i = = i = 1, and 3 & j = 1, and 3 are components of flctating velocity vector i and j Far RHS: set 1 =, = v and 3 = w (typical meteorological coordinates) Sm of diagonal components = = trblent velocity variance = trblent kinetic energy σ = + + w v ( ) / / = σ + + w v

13 Trblence Flxes (Random Example) Again, se w as an example In this example, I have drawn the and w traces completely Randomly and completely Independent of each In this case, we say that and w are ncorrelated And w = 0. t w R w w = σ σ w = 0 t correlation coefficient for w. In this case, R w = 0.

14 Flxes tend to be correlated in ABL de to non-niform mean profiles (e.g. mean wind shear) (z) w ( ) > z f 0 final height of parcel (z f ) initial height of parcel (z i ) ( z ) ( z ) ( z ) = ( z ) ( z ) < f = f f i f 0 therefore w > 0 associated with < 0 (negatively correlated). Can be shown (diagram for yorself) that, likewise, w < 0 associated with > 0.

15 Trblence Flxes (Negatively Correlated, Typical of ABL) Again, se w as an example In this example, I have drawn the and w traces to reflect that most of time and w are negatively correlated (i.e. > 0 with w < 0, and < 0 with w > 0). t w R w w = σ σ w < 0 > 0 with w < 0 (and vice-versa) t correlation coefficient For w. In this case < 0.

16 Compare previos slides (Random example vs. negatively correlated example) Negatively correlated both appear random (bt they aren t ) t w t

17 Compare previos slides (Random example vs. negatively correlated example) Random example both appear random (and they are ) t w t

18 Daytime: Convective Bondary Layer (CBL)

19 Fair-weather cmls (Cmlis Hmlis) Clody regions indicate Regions of pdrafts in CBL moving moistre pwards with evental condensation and clod formation.

20 free troposphere mixed layer srface layer

21 Convective Updrafts & Downdrafts (Convective Bondary Layer generated from LES compter simlation) vertical cross section Horizontal cross section White: Updrafts Grey and darker: Downdrafts

22 Typical flx profiles in the daytime ABL (Stll Figre.15, two lines are two typical cases) NOTE THREE POINTS ALONG FLUX PROFILES (see labeling above on far left) 1. Srface flx (vales above at z = 0). Entrainment flx (vale in middle of entrainment zone, point where profile breaks from linear) 3. Point where flx eqals zero atop ABL *** Profiles tend to be linear between points 1 and ***

23 Dirnal Potential Temperatre on Wangara Day 33 (classic ABL field experiment, Astralia) Daytime ABL convective Heating. Height (m) Note more or less niform heating rate with height. midnight (day 33) Time (hr) midnight (day 34)

24 Mean Potential Temperatre Profiles vs. Time (Daytime ABL heating; Wangara Day 33) model observed Uniform warming with height Lines indicate hor of day

25 Explanation θ ( w θ = ) t z constant ( since flx varies ~ linearly with height within ABL) θ = t [( w θ ) ( ) ] e w θ [( < 0)] = = ( 0) ( w θ ) 0 > z h h ( since flx decreases linearly with height, Therefore flx-divergence is greater than zero.) Reslt warming rate within daytime ABL tends to be niform with height.

26 Entrainment entrainment zone Note: Point is point where trblent flx breaks from linear (inflection point in flx profile). ( w θ ) < 0 e Downward directed entrainment flx ABL warming from below and (more weakly) from above, the latter de to entrainment. h ( w θ ) > 0 0 Upward directed Srface heat flx

27 Bt what abot at top of entrainment zone? entrainment zone, ( h) e Note: Point is point where trblent flx breaks from linear (inflection point in flx profile). ( w θ ) = 0 Zero top of ABL Cooling as heat from entrainment zone being flxed in ABL, and relatively cool air from ABL being flxed into EZ. i.e. heat exchange & mixing between ABL and EZ. ( h) e ( w θ ) < 0 e Downward entrainment flx

28 Mean Potential Temperatre Profiles vs. Time (Daytime ABL heating; Wangara Day 33) model observed Cooling de to entrainment Also note growth of ABL with time. This is another conseqence of entrainment i.e. mixing between ABL/EZ brings θ profile in this region qasi-netral (constant θ with z) Lines for different hor of day

29 Corresponding Heat Flx (w θ ) Profiles vs. Time (Note ABL growth in time bt flx profile still has same basic shape)

30 Bt what abot observations? Right side of plot model Cooling & ABL growth de to entrainment observed Cooling & ABL growth not as evident in observations, why? Lines for different hor of day

31 ABL Growth Rate (1) (daytime ABL) Can be shown, assming linear flx profiles in ABL, that h = w e + w sb t ( w θ ) where we ( θ ) e e w e is termed the entrainment velocity, with ( θ) e = θ h - θ abl the mean potential temperatre jmp from bottom to top of entrainment zone. and w sb is the large-scale (synoptic, general circlation) mean vertical velocity (called w sb becase often < 0 de to large-scale sbsidence)

32 ABL Growth Rate () (daytime ABL) h t = w e + w sb w e ( w θ ) ( θ ) e e = [ < 0] [ > 0] = [ > 0] Entrainment velocity > 0. Leads to ABL growth, as expected. w sb on the other hand is negative dring large-scale sbsidence. (Fair-weather, synoptic scale high pressre sitation). Therefore we and wsb often conter each other. Daytime ABL growth therefore often capped as a reslt of sbsidence.

33 Synoptic Scale Vertical Velocity (Stll Figre 1.6)

34 Mean Specific Hmidity Profiles vs. Time (Daytime ABL; Wangara Day 33) model observations Lines indicate hor of day

35 Corresponding Moistre Flx (w q ) Profiles vs. Time (Note ABL growth in time bt flx profile still has same basic shape)

36 Nighttime: Stable Bondary Layer (SBL)

37 Before we start (rate eqation for Trblent Kinetic Energy, TKE) Let E = trblent kinetic energy = ( ) + v + w / Then, a rate eqation for TKE can be derived E t = w z v w v z + g θ a w θ ε + z K m E z Shear prodction boyancy prodction (or destrction) moleclar dissipation Vertical trblent diffsion ( transport )

38 Rewritten sing K-theory E t = K m z v z g θ a K h θ ε + z z K m E z Shear prodction boyancy prodction (or destrction) moleclar dissipation Shear prodction Positive Generates trblence along direction of mean wind (i.e. and v, not w ) mechanically driven trblence Boyancy prodction (or destrction) Positive or negative (depending on stability) Generates (or destroys) trblence along vertical component (w ) boyantly driven trblence (or sppressed) Vertical trblent diffsion ( transport )

39 Stable Bondary Layer Schematic notice trblent eddies are more horizontally oriented than vertical. A conseqence of stable stratification (boyant destrction of TKE) inhibiting vertical trblent kinetic energy, and therefore vertical length of eddies. Compare with corresponding pictre for daytime bondary layer in daytime BL vertical trblence is enhanced, therefore eddies are large and vertically encompass entre bondary layer.

40 Richardson Nmber (Ri) Flx Richardson Nmber (Ri f ) Ri f = Boyancy Destrction of TKE Shear Prodction of TKE = K K H M g θ ( ) a θ / z ( / z) / Gradient Richardson Nmber (Ri g ) Ri g = g θ ( ) a θ / z ( / z) / K M = Ri f = K H Pr t Ri f

41 Critical Richardson Nmber (Ri c )... A critical Richardson nmber exists in which trblence generation cannot be sstained. That is boyant sppression of TKE is sfficiently strong to offset shear prodction Has been shown theoretically and experimentally Ri c 0.5 (=1/4). Ri < or > Ri c in stable bondary layer is a likely divider between continosly trblent ( trblent, Ri < Ri c ) and intermittently trblent or non-trblent ( intermittent or laminar, Ri > Ri c ) stable bondary layers observed in natre.

42 After Steeneveld et al Srface sensible Heat flx CASES-99 Laminar Intermittent Trblent Three different days dring CASES-99 experiment (Kansas-Oklahoma)

43 Srface friction velocity (-star) CASES-99 Intermittent Trblent Laminar

44 Stronger trblence, stronger wind Profiles of the wind velocity, in case W (open circles) and case S (filled circles).

45 Weaker trblence, stronger srface cooling Profiles of: the potential temperatre, in the composite case W (open circles) and S (filled circles). The potential temperatre is the deviation from the srface vale.

46 Profiles of the temperatre flx, in case W (open circles) and case S (filled circles).

47 Profiles of the Reynolds stress, in case W (open circles) and case S (filled circles). 100x

48 Overcritical region Profiles of the Richardson nmber Ri, in case W (open circles) and case S (filled circles).

49 Focs on trblent stable bondary layer (Ri < Ri c throghot most stable bondary layer) Two isses will be investigated - Trblence vs. Radiation in potential temperatre profile - Noctrnal ( low-level ) jet development in wind speed profile

50 Mean Potential Temperatre Evoltion (Wangara Day 33 simlation) Cooling with time. Different lines are different hors of night, starting arond snset (far right line) ending arond snset (far left line)

51 Potential Temperatre (Stable Bondary Layer) θ ( w θ = ) t z - F z RAD Vertical divergence of net IR radiative flx, F RAD F RAD = F IR - F IR (Upward mins downward IR flx) k+1 k F IR,k+1 F IR,k+1 Upward and downward ir radiative flx across two vertical levels k & k+1. Divergence (convergence) of these flxes leads to radiative cooling (warming) of this layer. This is an important process in the nderstanding cooling profiles in the nighttime, stable bondary layer over land. F IR,k F IR,k

52 Trblent (solid) vs. Radiative (dashed) cooling Upper part of SBL (Raditive cooling dominant) Lower part of SBL (Trblent Cooling dominant)

53 Stable Bondary Layer Depth (Wangara Day 33 simlation) Top of noctrnal temperatre inversion layer Depth of trblence within Bondary layer (i.e. top of trblent layer) snset Arond snrise

54 Noctrnal Jet (From Garratt Chapter 6..7) Basic Explanation Abrpt decrease in trblence in ABL dring transition from daytime to nighttime (de to switch from nstable to stable conditions). Trblent flx divergence in pper SBL (and RL) becomes practically zero. Wind accelerates (and rotates) towards geostrophic in pper SBL and RL. Overshoots geostrophic slightly, leading to sper-geostrophic wind in SBL and RL. Noctrnal wind maximm reslts in pper SBL and RL ( low-level jet, LLJ). Time period over which this occrs arond 9 hors (althogh depends on latitde) Wind max occrs late night/early morning hors (3 to 6am-ish). Mathematical illstration of above process shown in white-board notes See also handots in class for typical wind speed profiles showing LLJ

55 Some amplifying effects in Sothern U.S. Great Plains (Kansas, Oklahoma) Glf High Pressre forces sotherly flow in area (sotherly geostrophic wind) Sloping terrain pwards towards Rockies provides amplification of sotherly geostrophic wind (as slope cooling occrs at night). Lee-side low development at times east of Rockies also amplifies sotherly geostrophic wind. Reslt: Sothern U.S. Great Plains very (!) condcive to LLJ development.

Turbulence and boundary layers

Turbulence and boundary layers Trblence and bondary layers Weather and trblence Big whorls hae little whorls which feed on the elocity; and little whorls hae lesser whorls and so on to iscosity Lewis Fry Richardson Momentm eqations