ESS Turbulence and Diffusion in the Atmospheric Boundary-Layer : Winter 2017: Notes 1

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1 ESS Turbulence and Diffusion in the Atmospheric Boundary-Layer : Winter 2017: Notes 1 Text: J.R.Garratt, The Atmospheric Boundary Layer, Cambridge Also some material from J.C. Kaimal and J.J. Finnigan, 1994, Atmospheric Boundary-Layer Flows - Oxford. 1a) General Introduction. Laminar and Turbulent flow, averaging, the atmospheric boundary-layer - diurnal cycle, role of density stratification. G1 1b) Review of governing equations for incompressible flow, continuity, Navier Stoes, equation of state, thermodynamic equation. Vorticity (G2) Boundary-Layer definitions. Aerodynamics - laminar flow, boundary-layer approximations to N-S equns. In the absence of any body forces, the Navier - Stoes equations for velocity components, u i in a viscous fluid can be written, inertial frame of reference. Cartesian tensor notation. ( u t i + u u i p )= - i + 2 ui while the continuity equation for an incompressible fluid is, u j j = 0 For air μ 1.8 x10-5 Nm -2 s, ρ 1.2 gm -3 then ν = μ/ρ = 1.5 x 10-5 m 2 s -1. (vary with T and p) ABL or PBL, Surface layer. Roughness sublayer. Flow above a horizontally homogeneous, infinite, flat, plane. Surface roughness z 0. Geostrophic and gradient wind level. Coriolis force, stratification. Constant Flux layer. Hodograph and profiles. Eman spiral (constant eddy viscosity ~ 10m 2 s -1 ) - see Garratt p43. more realistic see p46. and profiles below (Weng and Taylor, 2003) 1

2 Navier-Stoes and continuity equations. Role of molecular viscosity. Laminar and turbulent flows. Range of scales mm to m. Thermodynamics, potential temperature. Turbulent fluxes, momentum, heat, water vapour, etc. Surface energy budget. U = U + u'. Averages! Reynolds rules. u * 2 = -u'w' (with x-axis aligned with surface wind). The friction velocity. PBL depth, of order 1 m. Stresses, fluxes drop by about 10% over lowest 100m, consider them approximately constant up to? 50m, 100m... Bucingham's pi theorem, see 2

3 Used in several places in boundary-layer turbulence. A ey application is: Constant stress layer, unidirectional flow, distance from flat, uniform, rough surface, z, Neutral stratification. Considser du/dz = f(u *, z,???), Assume far enough from surface that details of the roughness elements are not important. Dimensional considerations, [du/dz] = T -1, [u * ] = LT -1, [z] = L. So 3 variables, 2 dimensions, one π 1 = zdu/dz/u *. so must have π 1 = constant, (1/) - by convention. du/dz = u * /z. Has a problem at z = 0 stay away from roughness elements. Suppose we integrate the equation? U = (u * /) ln z + Constant Suppose U = 0 at z = z 0.- the roughness length - U = (u * /) ln (z/z 0 ).? Observations. See also Modified form for convenient model application, U = (u * /) ln (1+z/z 0 ). Then U = 0 at z = 0. 3

4 Typical z 0 values, Terrain description (m) Open sea, Fetch at least 5 m Mud flats, snow; no vegetation, no obstacles Open flat terrain; grass, few isolated obstacles 0.03 Low crops; occasional large obstacles, x/h > High crops; scattered obstacles, 15 < x/h < parland, bushes; numerous obstacles, x/h Regular large obstacle coverage (suburb, forest) 1.0 City centre with high- and low-rise buildings 2 x is fetch over surface. Footprint issues. Note that ln z is the relevant quantity. Small variations in z0 not critical, variations by factor 10 or more affect flow, e.g water - land. Rough guide, approximately 1/30-1/10 of size of roughness elements. Sand grain roughness, d/30. Navier-Stoes Equations. Cartesian co-ordinates on an f-plane, Boussinesq Approximation Du/Dt = -(1/ρ) p/ x + fv + υ( 2 u/ x u/ y u/ z 2 ) Dv/Dt = -(1/ρ) p/ y - fu + υ( 2 v/ x v/ y v/ z 2 ) Dw/Dt = -(1/ρ 0 ) p p / z -(ρ p /ρ 0 ) g + υ( 2 w/ x w/ y w/ z 2 ) p p and ρ p pressures and densities as perturbations from a bacground hydrostatic state. Noting that, Du/Dt = u/ t + u u/ x + v u/ y + w u/ z etc. Continuity ρ/ t + (ρu) = 0 or, equivalently, Dρ/Dt + ρ (U) = 0 Incompressible fluid, Dρ/Dt = 0 so (U) = 0. Mean + turbulence perturbation u = U + u' etc, p = p b + p p + p' Means can be 1-D, 2-D, 3-D, steady state or time dependent. Perturbations are always 3D and time varying. Reynolds number R = UL/υ. High R means molecular viscous termd neglected. Garratt, p20 re Boussinesq approximatioms. Note p'/p 0 << T'/T 0, ρ'/ρ 0. Why? 4

5 Reynolds Averaged Navier Stoes, RANS equations, b-layer approx. U/ t + U U/ x + V U/ y + W U/ z = -(1/ρ) p b / x + fv - <u'w'>/ z V/ t + U V/ x + V V/ y + W V/ z = -(1/ρ) p b / y - fu - <v'w'>/ z U/ x + V/ y + W/ z = 0 : Note u'/ x + v'/ y + w'/ z = 0 as well Assume p b (x,y) specified and replaced by Geostrophic wind term. -(1/ρ) p/ y - fu g = 0, -(1/ρ) p/ x + fv g = 0 1-D case: U/ t = + f(v-v g ) - <u'w'>/ z ; V/ t = -f(u - U g ) - <v'w'>/ z Since U/ x + V/ y = 0, W = const = 0 (flat ground). Two equations 4 unnowns Eddy Viscosity τ x /ρ = <-u'w'> = K U/ z : τ y /ρ = <-v'w'> = K V/ z A LOCAL gradient hypothesis - non-local closure schemes? K = constant? Mixing length K = u * l(z) or u * z (from surface layer, log profile) Eman layer solution, constant K, steady state. Let W = U + iv, V g = 0 leads to U = U g [1-e -αz cos(αz)] ; V = (f/ f )U g e -αz sin(αz) where α 2 = f /(2K). Other closures. K = velocity scale x length scale. May depend on stability (dθ 0 /dz ). Velocity scales, u * or E = ((u' 2 + v' 2 + w' 2 )/2) 1/2, TKE equation needed. (E 2 = TKE per unit mass) TKE equation includes ε - dissipation rate. Usually αe, Length scales? z, (z+z 0 ), 1/l = 1/(z+z 0 ) + 1/λ. Blacadar, λ = U g /f cf. Taylor (1969). 5

6 With stratification can use (Delage, 1974) TKE equation derivation. Full equn for u i / t, subtract U i / t equation and get equation for u i '/ t, Multiply by u j ', add u i ' u j '/ t, average to get equn for <u j 'u i '>/ t, let i =j and apply summation. Include viscous terms which lead to dissipation. For stratification also need thermodynamic equation, for potential temperature, θ θ/ t + u θ/ x + v θ/ y + w θ/ z 0 but there may be heat sources. Mean + perturbation and average, and the assumption that <w'θ'> = -K h θ/ z. 6

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