A new source term in the parameterized TKE equation being of relevance for the stable boundary layer - The circulation term

Size: px

Start display at page:

Download "A new source term in the parameterized TKE equation being of relevance for the stable boundary layer - The circulation term"

Peregrine McBride
5 years ago
Views:

A new source term in the parameteried TKE equation being of releance for the stable boundary layer - The circulation term Matthias Raschendorfer DWD Ref.

1 A new source term in the parameteried TKE equation being of releance for the stable boundary layer - The circulation term Matthias Raschendorfer DWD Ref.: Principals of a moist non local roughness layer extension of atmospheric turbulence closure M. Raschendorfer, 006, DWD internal document, going to be published NetFam Matthias Raschendorfer Toulouse 007

2 Single column solution for turbulent flux densities:. Using closure assumptions: -nd order budgets reduce to a 5X5 linear system of equations built of all second order moments of the ariable set {, q, u,, w }. Using horiontal boundary layer approximation: Flux gradient representation of the only releant ertical flux densities: turbulent master length scale ρφ w ρφ w ρk φ φˆ K φ : = ql S φ turbulent diffusion coefficient q TKE stability function NetFam Matthias Raschendorfer Toulouse 007

3 The limit of the pure turbulent BL approximation (here of the ordinary TKE-equation) for stabile stratification: D t ρq anishing mean wind stable thermal stratification ρu w ρk M u + g g H ( u) ρk ε < 0 0 ρ w ε < 0 free atmospheric turbulence closure NetFam Matthias Raschendorfer Toulouse 007

4 What processes can preent TKE form decreasing to ero (what is missing)? 0. Prognostic extension Sub grid scale transport. Moist extension Sub grid scale condensation. Roughness layer extension Canopy terms. Non turbulence scale (non local, multi scale) extension Circulation terms NetFam Matthias Raschendorfer Toulouse 007

5 The moist extension - sub grid cloud processes: Inclusion of sub grid scale condensation: -Using conseratie ariables with respect to condensation: c w = ql q l l c p q + = Correlations with condensation source terms are considered implicitly for non precipitating and not icing clouds. Soling for cloud water and cloud fraction by using the statistical condensation scheme (according to SD): L d q -Normal distribution of saturation deficiency -Expressing ariance of by ariance of and Δq sat Δq sat l qw, both generated from the turbulence scheme Δq sat 0 x NetFam Matthias Raschendorfer Toulouse 007

6 Mathematical reason for the canopy terms: Sub grid scale slope of the model layers or Intersections of roughness elements Aeraging and differentiation in space (flux diergence) can no longer be commuted ( ) x σ Canopy terms in the aeraged model equations 0 x σ D g x Wake turbulence production for TKE NetFam Matthias Raschendorfer Toulouse 007

7 The general budget of sub grid scale kinetic energy SKE: grid olume reduction effect by intersections molecular friction Dt ρq : = t ρq + E + E total flux conergence lnr a = Q bod molecular strain on intersected bodies 0 i=, ( ) ρ ~ + e~ ˆ μ Q i i i i + total shear stress including effect of sub grid scale surface slopes ( ~ ) 0 i=, molecular dissipation < 0 prs pressure source NetFam Matthias Raschendorfer Toulouse 007

8 ρk g ˆ Q prs ρ w = pure atmospheric terms 0 gρw for stable stratification only the turbulent part ˆ is necessarily negatie ρ w + ρ w turb circ p = p ~ total pressure transport + p expansion work p buoyant production p ~ lnr roughness layer terms ( ) ( ) p NetFam Matthias Raschendorfer Toulouse 007 a + + p G h ˆ h + s B p n d form drag wake and buoyant production due to intersections σ h σ ˆ form drag wake production due to sub grid scale slopes of the grid box in case of down hill density flows 0 buoyant production due to sub grid scale slopes of the grid box h s 0 0

9 w turb = K w circ x 0 Δx Een for anishing mean wind and negatie turbulent buoyancy there remains a positie definite source term SKE will not anish if there are thermal surface patterns 0 w NetFam Matthias Raschendorfer Toulouse 007

10 How to deal with the multi-scale problem? In general there are non turbulent, arranged circulation structures present as well: -different length scales of sub grid structures -turbulent scales -circulation scales (conection, down hill density flows, body wakes) More general closure assumptions alid for both, turbulence and circulations Impossible without knowing moments of as much classes as we hae length scales for Separation of turbulence and circulations with adopted closure assumptions for each class and related quite easy particular solutions NetFam Matthias Raschendorfer Toulouse 007

11 What are the postulates of turbulence: - Equilibrium of the source terms in all second order budgets (except that for TKE): - neglect of grid scale and sub grid scale transport - spectral density of contributing modes follows a power law in terms of wae length in each direction: inertial sub range spectrum - whole spectrum in a gien direction is determined by a single peak wae length - the peak wae length is the same for samples in all directions: isotropic length scale - pressure correlation and dissipation can be closed using a single turbulent master length scale for each location Turbulence is that class of sub grid scale structures being in agreement with turbulence closure assumptions! NetFam Matthias Raschendorfer Toulouse 007

12 What is the remaining difficulty with circulations (beside sub grid icing and precipitation)? - they are related with at least one additional spectral peak - or they cause different peak waelengths in ertical direction compared to the horiontal directions: ρφ ψ ln k dk anisotropic peak wae length - larger peak wae length in ertical direction in case of labile stratification ( dk) conectie peak in ertical spectrum at least a two scale problem Resoled Structures Circulations - slope in case of TKE labile neutral stabile Turbulence catabatic peak Microphysics ln D g ln L p L p : largest turbulent wae length lnk NetFam Matthias Raschendorfer Toulouse 007

13 How to find a particular solution for turbulence? turbulence closure is only alid for scales not larger than -the smallest peak wae length L p, in any direction -and the largest dimension D g of the control olume Spectral separation by considering budgets with respect to the separation scale L = min{, } and aeraging these budgets along the whole control olume -Additional scale interaction term in TKE equation due to non linear shear term L p D g D t ρ q L = ρk M ( uˆ ) + ˆ i L i=, shear turbulent buoyancy g + ρ w ˆ shear by circulation motions S C L dissipation q ρ α L MM l NetFam Matthias Raschendorfer Toulouse 007

14 Budgets for the circulation structures: Physical meaning of the circulation term: Dt ρ φ ψ ρφ ψ L = ( ˆ ) ( ˆ ψ +ρψ φ) ρφ L L L L + φψ S C Circulation term is the scale interaction term shifting SKE or any other ariance form the circulation part of the spectrum (CKE) to the turbulent part (TKE) by irtue of shear generated by the circulation flow patterns. CKE S C TKE NetFam Matthias Raschendorfer Toulouse 007

15 Parameteriation of the buoyant circulation term for thermal drien (direct) circulations: - a first attempt -. In all budgets: circulation structures are negligible during neutral stratification Since local shear production of circulation motions acts also for neutral stratification, this should be negligible: a ertical constant circulation time scale for expressing scale interaction loss and pressure destruction a mass flux approach for circulation patterns. In the CKE budget: scale interaction loss = buoyant production. In the budget for circulation scale heat and moisture flux : scale interaction loss + pressure destruction = buoyant production 4. In the budget for circulation scale temperature ariance : scale interaction loss = ertical flux diergence from the surface flux gradient form of temperature ariance flux with a ertical constant circulation scale diffusion coefficient NetFam Matthias Raschendorfer Toulouse 007

16 () ρ ρ ρ ρ pat C C V CB V g L 0 H w g w g S L L L L L ˆ ˆ exp ˆ ˆ ˆ ˆ ˆ ˆ Δ ( ) ( ) C C C H g a a a w = γ ˆ Δˆ circulation boundary layer height horiontal updraft fraction difference between updraft and downdraft other circulation scale -nd order moments hae a similar representation turbulent and circulation scale flux densities can be added: - in the first order budgets - as input for the statistical condensation scheme Circulation term ~ circulation scale temperature ariance ~ circulation scale buoyant heat flux pattern length NetFam Matthias Raschendorfer Toulouse 007 in general a form drag circulation term has to be considered as well: S CD CD CB C S S S + =

17 Simulated midnight profile of potential temperature NetFam Matthias Raschendorfer Toulouse 007

18 Conclusions:. Een for anishing mean wind shear and stable thermal stratification there are source terms for TKE in general: transport from other locations (by the mean wind, by turbulent transport e.g. from the surface layer) sub grid scale condensation wake turbulence production shear production from circulation scale sub grid motions (circulation term) - here: direct thermal circulations, drien by differential heating or cooling. Volume aeraging along a grid box with solid intersections or sub grid scale slopes along the earth surface leads to the wake turbulence terms: non commutability of differentiation in space and aeraging. Scale separation approach leads to a parameteriation of the circulation term: transport of circulation scale temperature ariance from the surface ~ circulation scale upward heat flux ~ thermal production of CKE ~ shear production of TKE by circulation motions NetFam Matthias Raschendorfer Toulouse 007

19 Some remaining problems:. Inclusion of deep conection seems to be impossible sub grid scale precipitation, - radiation, further different length scales We do it with a separate deep conection scheme, unless this process is already grid scale. Closure of circulation scale budgets is not yet ery sophisticated Introduction of at least an equation for the skewness in order to sole for the horiontal updraft fraction or without an additional mass flux approach and: A full set of -nd order equations Soling of at least one prognostic equation (circulation scale temperature ariance) Proper parameteriation of third order moments. Adaptation of the statistical condensation scheme to circulations Other distribution function of saturation deficiency for circulation class 4. Adaptation of deep conection and microphysical processes Separation of turbulent and circulation condensation from -Condensation due to deep conection: -Microphysical processes, like cloud ice formation and precipitation: NetFam Matthias Raschendorfer Toulouse 007

20 Thanks for your attention! NetFam Matthias Raschendorfer Toulouse 007

21 NetFam Matthias Raschendorfer Toulouse 007

22 ρζ ζ ˆ := ζ = 0 ρ ~ r h h σ ρ V : = V V ( ) e~ φ ( ) φ = a φ φ h h σ e kin = ρ = ρ q e kin ii : = = ρ E = ρ ρ E = ρq ˆ~ + ρ ~ + i=, i=, q : ~ e i i Q prs : = Q i=, ii prs = p Q bod : = i=, Q ii bod = G s B i=, e i i n d s = G ˆ i e i=, s B i n d s Dt ρq : = t ρq + E + E lnr a Q bod = + i=, i=, i ( ρ ~ i + e~ ) Q i i ˆ i μ i=, i + Q prs NetFam Matthias Raschendorfer Toulouse 007

ESCI 485 Air/sea Interaction Lesson 3 The Surface Layer

ESCI 485 Air/sea Interaction Lesson 3 he Surface Layer References: Air-sea Interaction: Laws and Mechanisms, Csanady Structure of the Atmospheric Boundary Layer, Sorbjan HE PLANEARY BOUNDARY LAYER he atmospheric