Physical Basis of Boundary Layer

Pabitra Gurung
CHAPTER 2
PHYSICAL BASIS OF
BOUNDARY LAYER

CONTENTS
1. Exchanges and climatic response near surfaces
(a) The ‘active’ surface
(b) Exchange in a volume
(c) Exchange in boundary layers
2. Sub-surface climates
(a) Soil heat flux and soil temperature
(b) Soil water flow and soil moisture
3. Surface layer climates
(a) Lapse rates and stability
(b) Momentum flux and air temperature
(c) Sensible heat flux and air temperature
(d) Water vapour and latent heat fluxes and atmospheric humidity
(e) Further remarks on convective exchange
4. Outer layer climates

1. EXCHANGES AND CLIMATIC RESPONSE NEAR SURFACES
 A plane that separate two different media; it contains no energy &
mass; important for energy & mass exchange and conversion
 The surface may be in motion, semi-transparent, undulation
 Principal plane of climatic activities in a system, where
omajority of the radiant energy is absorbed, reflected, and emitted;
omain transformation of energy and mass occur;
oprecipitation is intercepted;
othe major portion of drag on airflow is exerted
 Energy absorption by day and depletion by night
(a) The ‘active’ surface

(b) Exchange in a volume
Convergence Divergence
Advection: Convergence and divergence of
horizontal fluxes (warming and cooling)
 Radiation budget and surface energy balance, 𝑄∗ = 𝑄 𝐻 + 𝑄 𝐸 + 𝑄 𝐺
 Energy balance in a volume or layer,
𝑄∗ = 𝑄 𝐻 + 𝑄 𝐸 + 𝑄 𝐺 + ∆𝑸 𝑺, Where, ∆𝑄 𝑆 is net energy storage
 In a soil layer (∆𝑧):
∆𝑄 𝑠
∆𝑧
= 𝐶𝑠
∆ 𝑇𝑠
∆𝑡
 𝑄𝑖𝑛 = 𝑄 𝑜𝑢𝑡 ± ∆𝑄 𝑆
 Flux convergence (warming) and Flux
divergence (cooling); and advection
 In a time period ∆𝑡,
Temperature change (∆ 𝑇) ∝ Change of
heat flux (∆𝑄 𝑆)
 Specific heat (c) (J kg-1 K-1) and heat
capacity (C) (J m-3 K-1)

 Division of the soil-atmosphere system (parallel to the active
surface)
o Sub-surface layer
o Laminar boundary layer
o Roughness layer
o Turbulent surface layer
o Outer layer

 Sub-surface layer
o Soil heat fluxes, 𝑄 𝐺 = −𝑘 𝐻 𝑠
𝐶𝑠
𝜕 𝑇
𝜕𝑧
Where,
𝑘 𝐻 𝑠
= Soil thermal diffusivity (m2 s-1)
[It measures the ability of a material to conduct
thermal energy relative to its ability to store
thermal energy]
o The energy transfer occurs due to
molecular collisions transferring kinetic
energy (molecular exchange)
Exchange in boundary layers
𝐹𝑙𝑢𝑥 𝑜𝑓 𝑎𝑛 𝐸𝑛𝑡𝑖𝑡𝑦
= 𝐴𝑏𝑖𝑙𝑖𝑡𝑦 𝑡𝑜 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 × 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝑎 𝑅𝑒𝑙𝑒𝑣𝑒𝑛𝑡 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦
Idealized mean profiles of air and soil temperature
near the soil-atmosphere interface in fine weather
1. SUB-SURFACE CLIMATES & EXCHANGES

Soil heat flux (𝑸 𝑮) and soil temperature (𝑻 𝑺)
HEAT
HEAT
Idealized mean profiles of air and soil temperature near
the soil-atmosphere interface in fine weather
 Soil Thermal Properties:
o Thermal conductivity ( 𝓀 𝑠), [W m-1 K-1]
o Heat capacity (𝐶𝑠), [J m-3 K-1]
o Thermal diffusivity (𝑘 𝐻 𝑠
), [m2 s-1]
o Thermal admittance (𝜇 𝑠), [J m-2 s-1/2 K-1]
(a measure of the ability of a surface to
accept or release heat)
 𝑄 𝐺 = −𝓀 𝑠
𝜕 𝑇𝑠
𝜕𝑧
≈ −𝓀 𝑠
( 𝑇2− 𝑇1)
(𝑧2−𝑧1)
 𝑘 𝐻 𝑠
=
𝓀 𝑠
𝐶 𝑠
 𝜇 𝑠 = 𝐶𝑠 𝑘 𝐻 𝑠
= 𝓀 𝑠 𝐶𝑠

𝜇 𝑠
𝜇 𝑎
=
𝑄 𝐺
𝑄 𝑎

), [m2 s-1]
 𝑄 𝐺 = −𝓀 𝑠
𝜕 𝑇𝑠
𝜕𝑧
≈ −𝓀 𝑠
( 𝑇2− 𝑇1)
(𝑧2−𝑧1)
 𝑘 𝐻 𝑠
=
𝓀 𝑠
𝐶 𝑠

𝜇 𝑠
𝜇 𝑎
=
𝑄 𝐺
𝑄 𝑎
Relationship between soil moisture content: (a) thermal conductivity,
(b) heat capacity, (c) thermal diffusivity and (d) thermal admittance

1. SUB-SURFACE CLIMATES
), [m2 s-1]
 𝑄 𝐺 = −𝓀 𝑠
𝜕 𝑇𝑠
𝜕𝑧
≈ −𝓀 𝑠
( 𝑇2− 𝑇1)
(𝑧2−𝑧1)
 𝑘 𝐻 𝑠
=
𝓀 𝑠
𝐶 𝑠

𝜇 𝑠
𝜇 𝑎
=
𝑄 𝐺
𝑄 𝐻
Generalized cycles of soil temperature at different
depths for (a) daily and (b) annual periods

 Soil moisture potential (Ψ): Energy required to
extract water from the soil matrix (displaced water
head).
 Ψ ∝
1
S
, and Ψ ∝ 𝐸𝑣𝑎𝑝𝑜𝑡𝑟𝑎𝑛𝑠𝑝𝑖𝑟𝑎𝑡𝑖𝑜𝑛
 Darcy’s Law for the vertical flux of liquid in the
soil matrix: 𝐽𝑙 = −𝐾𝑓
𝜕Ψ
𝜕𝑧
, where 𝐾𝑓 is hydraulic
conductivity
 The flux-gradient relationship for the vertical flux
of vapour in the soil matrix: 𝐽 𝑉 = −𝑘 𝑉𝑎
𝜕 𝜌 𝑉
𝜕𝑧
 The saturation vapour concentration is directly
related to temperature
 Vapour flow: Day (↓) and Night (↑)
Soil water flow (𝐉) and Soil moisture (𝐒)
Particularly in this site, soil-air exchange is occurred to a depth of
nearly 0.8 m (in a sandy loam during a drying phase)

 Laminar boundary layer
o Speed of flow, distance and viscosity
creates turbulence flow
o Thickness of Laminar sub-layer
depend on the surface roughness and
the external wind speed
o No convection: All non-radiative
transfer is by molecular diffusion
o Sensible heat flux: 𝑄 𝐻 =
− 𝜌𝑐 𝑝 𝑘 𝐻 𝑎
𝜕 𝑇
𝜕𝑧
= −𝐶 𝑎 𝑘 𝐻 𝑎
𝜕 𝑇
𝜕𝑧
o For water vapour: 𝐸 = −𝑘 𝑉𝑎
𝜕 𝜌 𝑉
𝜕𝑧
o For momentum: 𝜏 = 𝜌𝑘 𝑀 𝑎
𝜕 𝑢
𝜕𝑧
(a) Development of a laminar boundary layer over a flat plate and its transition to
turbulent flow, (b) The vertical variation of the flux of any entity, the associated
diffusion coefficients and the concentration of its property.
2. SURFACE LAYER CLIMATES & EXCHANGES

 Roughness layer
o Complex flows; and formed eddies and vertices (Due to surface roughness elements)
o 3-D effects: Depends on characteristics of elements; and their shape, plan density, flexibility
o Difficult to express: Exchange of heat, mass and momentum
 Turbulent surface layer
o Small-scale turbulence above the
surface
o Constant flux layer
o Depth about 10% of planetary boundary
layer
o Day: Lapse profile & -ve gradient
o Night: Inversion profile & +ve gradient

 Dry adiabatic lapse rate (Γ): Constant (9.8 ℃ 𝑘𝑚−1) for dry/unsaturated air
 Environmental lapse rate (ELR): Based on actual observed temperature structure
above a given location
 (a) Unstable (ELR > Γ), (b) Stable (ELR < Γ), & (c) Neutral (ELR = Γ)
Lapse rates and stability
Warmer
Colder
Warmer
Colder
 With fine weather: Unstable by day
and Stable by night
 Over high latitude snow surfaces in
winter: Stable boundary layer for
longer period
 Over tropical ocean surfaces:
Unstable boundary layer for longer
period
Height vs temperature (a) unstable atmosphere on
sunny days and (b) stable atmosphere at night

 Wind field is largely controlled by the frictional drag imposed on
the flow by the underlying rigid surface
 Above Zg, wind speed is approximately constant due to negligible
frictional drag
 Depth of Zg will increase with strong surface heating and will
decrease with cooling
 Surface shearing stress (τ): The force exerted on the surface by
the air
 𝜏 = 𝜌𝐾 𝑀
𝜕 𝑢
𝜕𝑧
, where 𝐾 𝑀 is eddy viscosity (m2 s-1)
 𝑢 𝑧 =
𝑢∗
𝓀
𝑙𝑛
𝑧
𝑧0
, where 𝑢 𝑧 is mean wind speed (m s-1) at the height
𝑧, 𝑢∗ is friction velocity (m s-1),𝓀 is von Karman’s constant (≈
0.40), 𝑧0 is roughness length (m)
 Surface sharing stress, 𝜏 = 𝜌 𝑢∗
2
(the shearing stress is
proportional to the square of the wind velocity at some arbitrary
reference height)
Momentum flux (𝛕) and wind speed (𝐮)
The wind speed profile near the ground. (a) the effect of terrain
roughness; and (b, c, d, e) the effect of stability on the profile
shape and eddy structure

 In the turbulent surface layer, 𝑄 𝐻 = −𝐶 𝑎 𝐾 𝐻
𝜕 𝑇𝑠
𝜕𝑧
+ Γ = −𝐶 𝑎 𝐾 𝐻
𝜕 𝜃
𝜕𝑧
Where, 𝐾 𝐻 is eddy conductivity (m2 s-1)
 Vertical turbulent transfer
Sensible heat flux (𝑸 𝑯) and air temperature (𝑻 𝒂)
Results from fast response instruments at a height of 23 m over grass in
day time unstable conditions (at Edithvale, Australia)
updraft
downdraft
Generalized daily cycle of air temperature in the
atmosphere on a cloudless day
and 𝜃 is a potential temperature (temperature of an
air parcel at the arbitrary pressure value of 100 kPa)

 Idealized weather conditions
o Profile 1: Before sunrise
o Profile 2: Soon after sunrise
o Profile 3: Mid-day
o Profile 4: Near sunset
Sensible heat flux (𝑸 𝑯) and air temperature (𝑻 𝒂)
Generalized form of the air temperature profile in the lowest 150 m
of the atmosphere at different times on a day with fine weather.
[Where, h* is the depth of the mixed layer]
Potential
Temperature
Environmental
Temperature
Profile SRB ST 𝑸 𝑯 h*
1 − ↓ ↓ ~ 0
2 + ↑ ↑ shallow
3 + ↑ ↑ deep
4 − ↓ ↓ ~ 0
SRB = Surface radiation budget
ST = Surface temperature
 This idealized conditions will effect by cloud cover
and wind speed due to impact on radiation and
turbulence respectively.
 Due to cloud and wind, the daily range of
temperature (lower maxima and higher minima) and
extremes of stability (more neutral) will reduce.

 Vapour content (humidity)
 Vapour density (𝜌 𝑉) [kg m-3], and Vapour
pressure (𝑒) [Pa]
 The Ideal Gas Law: 𝑒 = 𝜌 𝑉 𝑅 𝑉 𝑇
 Saturation vapour density (𝜌 𝑉
∗
) and Saturation
vapour pressure (𝑒∗
)
[Water molecules escaping to the air = Water
molecules returning to the liquid]
 Vapour pressure (or density) deficit: 𝑣𝑝𝑑 =
(𝑒∗
− 𝑒), 𝑎𝑛𝑑 𝑣𝑑𝑑 = (𝜌 𝑉
∗
− 𝜌 𝑉)
 Greater the vpd/vdd → Greater the evaporation
at the surface (at same temperature)
 Dew-point (or frost-point)
[Useful to consider condensation of fog or dewfall
due to cooling]
Water vapour and latent heat fluxes (𝑬, 𝑸 𝑬) and atmospheric humidity (𝝆 𝑽 𝒐𝒓 𝒆)
17
157
Relationship between saturation vapour pressure
and temperature over a plane surface of pure water

 In the turbulent surface layer, Evaporation mass
flux, 𝐸 = −𝐾 𝑉
𝜕 𝜌 𝑉
𝜕𝑧
and the flux of latent heat,
𝑄 𝐸 = 𝐿 𝑉 𝐸 = −𝐿 𝑉 𝐾 𝑉
𝜕 𝜌 𝑉
𝜕𝑧
 Evaporation process depends on;
o The availability of water and energy
o The existence of a vapour concentration gradient
o A turbulent atmosphere to carry the vapour away
Water vapour and latent heat fluxes (𝑬, 𝑸 𝑬) and atmospheric humidity (𝝆 𝑽 𝒐𝒓 𝒆)
(a) Idealized mean profiles of water vapour concentration near the ground’s surface, and (b) the
diurnal variation of vapour pressure on cloudless days in May (Quickborn, Germany)

 Convection is the principal means of transporting the daytime energy surplus of
the surface away from the interface
 The relative importance of sensible versus latent heat is mainly governed by the
availability of water for evaporation
 Bowen’s ratio, 𝛽 =
𝑄 𝐻
𝑄 𝐸
;
If 𝛽 > 1: relatively warming of the lower atmosphere, and if 𝛽 < 1: may increase the
humidity of the lower atmosphere, and negative 𝛽 indicates that the two fluxes (𝑄 𝐻 & 𝑄 𝐸)
have different signs (especially at night due to evaporation)
Further remarks on convective exchange

o Mixed layer: Day time convective layer
(mixing of airborne materials: dust,
pollutants, spores, etc.)
o Heat flux ≈ 0 (near the inversion base)
o Convective entrainment: Heat
transport downwards
o Potential temperature,
θ = T
P0
P
R
cp
[Temperature of an air parcel at the
arbitrary pressure value of 100 kPa
(P0). Where, R is gas constant]
 Outer layer
o Turbulent layer to the top of the planetary boundary layer (≈ 90% of its depth)
Schematic representation of airflows in the outer
layer
3. OUTER LAYER CLIMATES AND EXCHANGES

(a) Daily variation of the boundary layer on an ‘ideal’ day. (b) Idealized mean profiles of potential temperature ( 𝜃), wind
speed ( 𝑢) and vapour density ( 𝜌 𝑉) for the daytime convective boundary layer (c) same as (b) for nocturnal stable layer
 The depth of mixed layer (ℎ∗
) starts to rise when the surface sensible heat flux density becomes positive
 When sunrise, ℎ∗
rapidly increases by eliminating the previous night’s inversion and reaches maximum in
mid-afternoon and the complete layer is convectively unstable
The temporal dynamics of the boundary layer under ‘ideal’ weather conditions
 The mixing equalizes temperature, wind speed,
humidity and other properties throughout the layer
Day
Night

(a) Stages in the temporal development of a
thermal
(b) Initiation of a thermal by a hill and
cumulus clouds
(c) Formation of cloud streets
Convective structures associated with instability
Little or no horizontal wind
Surface wind and hills/islands
Less convection and
high surface wind speed

Physical Basis of Boundary Layer

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Physical Basis of Boundary Layer