Time and length scales based on the Brunt Vasala frequency N BV. (buoyancy) length scale l B = σ w / N BV

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1 Time and length scales based on the Brunt Vasala frequency N BV time scale: t BV = 1/N BV (buoyancy) length scale l B = σ w / N BV period of oscillation of a parcel in a statistically stable environment: stratification induce vertical oscillations that prevent the air parcel from moving freely in the SBL buoyancy waves with scale l B ~O(100m) in weak stratification l B ~O(1m) in strong stratification (and weak ): more attenuation! N BV 2 = g θ v θ v z Brunt Vasala frequency N BV for a fluid parcel in the presence of a stable stratification (in which the density decreases with height). The parcel, perturbed by vertical turbulent fluctuations, from its starting position, experiences a vertical acceleration. If the acceleration is back towards the initial position, the stratification is said to be stable and the parcel oscillates vertically with the frequency of oscillation is given by N BV

2 Buoyancy waves N BV 2 = g θ v θ v z the frequency of buoyancy waves must be smaller than N BV why? the vertical velocity fluctuation acts as a square wave perturbation (remember hotwire square wave test) the response time of an air parcel is defined by 1/N BV any faster oscillation would not be affected by buoyancy, it would be purely Slower oscillation are instead amplified (the stable BL act as a low pass filter for the generation of buoyancy/gravity waves) spring analogy typically wave periods ranges 1-40minues wave displacement m, λ > h

3 Monin Obhukov length scale u 3 L = k g w θ θ s Another scale of interest : Ozmidov scale ε l O = N BV 3 Buoyancy wavenumber K B =1/l O 1/2 the vertical length scale at which the buoyancy force is of the same order of magnitude as the inertial forces. In stably stratified ~10-50m The Ozmidov scale is the largest scale of that can overturn (e.g. contributing to energy cascade) in a stratified flow. above l O overturning is inhibited by stratification i.e. buoyancy has only a minor effect at smaller scales but dominates at larger ones Taylor hyp. Note : Corrsin scale l c =(ε/s 3 ) 1/2 : above which eddies are deformed by shear (S) Ozmidov scale l c =(ε/n BV3 ) 1/2, above which eddies are deformed by stratification not affected by buoyancy

4 In the buoyancy subrange the relevant scales for the power spectrum S(k) [m 3 s -2 ] (integrated in the wave number domain) are: N BV [s -1 ] and k [m -1 ] f S(f) ~ f -2/3 S(f) ~ f -5/3 ok Dimensional analysis S(k) ~ k -3 N BV 2 k S(k) ~ k -2 N BV 2

5 weak stratification strong stratification strong Quadrant II Quadrant I Quadrant I,II most of geophys. flows strong weak Quadrant III Quadrant IV weak suppresed isotropic Quadrant III: low Re l (~laminar ) transport of momentum and scalar(temperature) are by molecular diffusione Quadrant IV: suppression of by strong stratification

6 Note: profiles are obtained to estimate the Thorpe displacements to linear stratification at low Ri 0 is strong and mixing is evident by large density fluctuations, at large Ri 0, fluctuations in density are minimal due to buoyancy stabilizing effects

7 3a) Limited effects from non overturning motions L T3D ~ L T 3b) For constant density gradients L T = L E (Ellison length scale)

8 strong L O = 0.8 L T weak suppressed 4a) Non linear dependency with L O varying significantly at very low Ri o (strong turb Q1) 4b) Power law dependency with L T / L O depending on (NT L ) exponent

9 strong weak suppressed L kn works for strong stratification, suppressed (NT L ) >1, Q4

10 strong weak suppressed L ke works for weak stratification (NT L ) < 1, Q1-2

11 strong slope 1 slope 2 weak suppressed L kn works for strong stratification, suppressed (NT L ) >1, Q4 L kn / L T constant implies E p / k = constant MG however the slope -1 implies that L T /L ke still keeps a weak dependency on (NT L ) -1/2

12 EXP-DNS agreement Remember L E =L T under constant density gradient