The Stable Boundary layer the statistically stable or stratified regime occurs when surface is cooler than the air The stable BL forms at night over land (Nocturnal Boundary Layer) or when warm air travels over a cold surface (e.g. when south wind in spring blows over an icy lake) z L = (g/θ )w θ s u 3 /kz note that z/l is governed by two opposing contributions: high shear (low z close to the ground) but low u * (attenuated mechanical production) and strong negative heat flux. As compared to the flux Richardson number, in z/l the stratification intensity is governed by surface processes. z is only amplifying those processes high z, thermal effects dominates Based on the actual temperature the stable BL is also called a nocturnal inversion (dt/dz changes sign) nocturnal jet or supergeostrophic wind
Definition of the stability functions
decaying turbulence from the ML vertical turbulent motion are suppressed by buoyancy, however buoyant oscillation may occur as gravity waves. Breaking of gravity waves may generate patches of turbulence at the interface between the SBL and the Residual layer RL
400 Example of fanning : horizontal spread of aerosols which accumulates in thin layers with different concentration : layer cake pattern z [m] 0 z/h z/h typical scaling for the stable BL. Note that for strong stratification local shear + gravity waves becomes the major source of turbulence (not this case on the left). If turbulent structures form and develop at some height as a result of breaking gravity waves, it means that the shear stress at the ground (and thus u*) and the elevation z are not any more key scaling parameters. z-less stratification decouple ground turbulence by gravity wave turbulence turbulent stats do not depend on z/l anymore
SBL strength SBL strength is a bulk measure of the buoyancy effect: θ s = θ 0 θ s virtual potential temperature of the residual layer air ~ temperature of the surface at virtual potential temperature of the surface air the formation of the SBL (neutral regime) Note that the SBL strength is not a measure of stability (no velocity or mechanical production of turbulence) θ s is a measure of the cumulative cooling of the surface since sunset
T θ z =0 =0 z TKE=0 TKE=5%TKE s u w =0 u w =5% u w s M is maximum M= G bottom of free atmosphere: geostrophic wind SBL depth h (possible choices) Definition of an integral depth scale based on θ (z) = θ 0 θ(z) H θ = h z θ s θdz The ratio B = H θ (ranging 3 15 [m/k]) is a measure of how diluted θ s or compact is the strong gradient region of the SBL
B = H θ is a measure of how diluted or compact is the θ s strong gradient region of the SBL. Large B deep boundary layer over a small temperature difference ( neutral regime) Small B shallow boundary layer over a large temperature difference ( strong stable stratification) Different models for the temperature profiles are available for each each model the integral vertical scale has a different expression for a given θ s
EVOLUTION OF THE STABLE BOUNDARY LAYER 1)RADIATION Stability is maintained by surface cooling at night which is governed by net longwave radiation flux (positive upward so removing heat from the ground and near wall layers ~ 100W/m 2 ) e.g. cooling rate is: 0.1K/h (hour) for z > 500m 0.2K/h for z ~ 50m 1-3K/h for z ~ 1m the radiative flux is not constant but rather increases with z it means that the flux is divergent and that different layers are cooled at a different rate. θ t radiative= - I z radiative cooling flux divergence I * increases with z, implies θ t <0 temperature decrease in time, changing the BC at the surface and increasing θ s on a clear sky night colder air near the ground radiates less than warmer air in the upper layers : the difference represent the bulk (-) heat flux
SUBSIDENCE clear sky nights are associated to high pressure at a synoptic scale and thus (divergence and subsidence) subsidence is bringing warm air from the residual layer or FA into the stable boundary layer warming it θ t subsidence= - w θ z w is negative (downward), θ z > 0 as the mean temperature increases with z, thus θ increases with time ( ~ mitigate the radiation effect) ADVECTION θ θ t advection= - U j x j related to spatial heterogeneity?><0 From each of these contributions, including radiation, we can define a heat flux that will be useful for the evolution equation : ( θ s H θ ) t H θ = h z θ s θdz = Q T bulk cooling including the effect of advection, subsidence, radiation and sensible heat flux
sensible heat flux Evolution: from sunset on 18pm, cooling (by radiation, subs, adv. etc) induce a weak shallow stable BL time Temperature profile becomes deeper as cooling affects the lower region of the residual layer. cooling contribution shift the surface temperature to low values, while the whole profile adjust itself eroding the Residual layer above H SBL increases (mostly due to mixing)
LOW LEVEL JETS (Nocturnal jets or katabatic winds) stream/sheets of fast moving air with maximum speed of 10-20m/s at around z=100-300m very wide (y) and long (x) O(100Km) (detectable with weak synoptic wind) Origins: 1 )synoptic scale pressure grad. 2) sloping terrain/orographic winds 3) valley winds 4) cold fronts TERRAIN SLOPE and DRAINAGE WIND (gravity flows or katabatic winds) very sensitive to slope even down to 0.1% but weak winds 1-2m/s 5) land and sea breeze (thermal response heterogeneity)
Turbulence in the SBL Different mechanisms of generation : gravity waves, low level jet, standard mechanical production at the wall Time scales: τ r = h 0.01u proposed to quantify the time for a fluid parcel to reach the top of the SBL and come back (integral time for the SBL) τ r = h ~ =O(10hours) >> t 0.01u ConvBL ~ 15 minutes what does it mean? The stable boundary layer is rarely in equilibrium. The nocturnal diurnal cycle is such that thermal forcing varies on the same scale of the response time of the stable boundary layer. In the convective BL the response time was much shorter, implying that any change in the boundary conditions (say variation of surface temperature) is almost instantaneously felt by the flow. Equilibrium in the stable BL impacts the definition of neutral condition in the night day transition. Stronger winds helps.
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 turbulence): 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
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 turbulence 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 10-100m, λ > h
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 turbulence 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
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
structural buoyancy length scale D ww (r) l SB (r) = g/θ v D θθ r 1/ 2 l B ~O(1m) in weak stratification l B ~O(50-100m) in strong stratification be careful that l SB is a scale dependent (r) length scale dissipation length scale l ε = 0.4 σ w 3 / ε l ε ~O(10-100m), comparable with an integral length scale (remember that ε ~ u 3 /l ) It is important to replace u with w, as vertical velocity component is more affected by stratification anisotropy