It turns out that this new term: − ′ρρg
has major significance for urban microclimates, and many environmental flows. Recall, small deviations ′ρ from ρ,are often due to heating/cooling of air, mainly at the surface.Let's now study the heat conservation equation to understand this.
How do the heat and momentum equation couple?How and where are the heating/cooling bouyant forces generated?What do they imply for the urban environment?... in the next few lectures
Buoyancy effect on atmospheric flow
Buoyancy effect on atmospheric flow
Rn
G
H LE
Surface (of earth, city water, ...)
n
n down down up up
R H LE GR S L S L
= + += + − −
Fourier's law (as with diffusion)
in 1D: q = −k dTdx
general form: q = −k∇Twhere k is the thermal conductivity (Wm−1K−1)
Again, as with diffusion, heat_in − heat_out:
∂T∂t
= kρc
∂2T∂x2
+ ∂2T∂y2
+ ∂2T∂z2
⎛⎝⎜
⎞⎠⎟
2 4
8 2 4
radiation emitted from a black body (Wm )Boltzman Constant = 5.67 10 (Wm ) Body temperature (K)
F TK
T
σσ
−
− − −
= == ×=
Fgrey = εσT 4
ε = emissivity of the grey body (units?)ε = absosptivity at thermal equilibrium and more
ε =1−α
Global average of the SEB
What is G on average?
(32 23) 96 4 117 30
n
n down down up up
R H LE GR S L S L
= + += + − −
= + + − − =
32
Spatial Variation of Rn
Rn = H + LE +G < 0 Rn = H + LE +G < 0Rn = H + LE +G > 0
Diurnal Variation of Rn
High Rn: clear summer days
Low Rn: cloudy fall days
4-component radiometers
Radiation on Princeton roof
Which is lighter?
Moisture and stability
Which is lighter?
Moisture and stability
Virtual temperature Tv
Virtual temperature Tv TV = (1+0.61q) T Proof as exercise
q = ρWV /ρ is the specific humidity (kgWV/kgMA)
ρWV = density of water vapor (WV) alone (kgWV/m3)
ρDA = density of dry air alone (DA) (kgDA/m3)
ρ = ρWV+ ρDA density of mixture (kgDA/m3)
m = mixing ratio = ρWV/ρDA
RH = m/m* ≅ e/e*
where e is the vapor pressure, and * denotes the value at saturation
ρ = pRT
= pRdTv
R = gas constant for moist air , Rd = gas constant for dry air
Height and stability: Potential Temperature
Potential Temperature
dP gdz
ρ= −
dpdz
= −ρg
p − p0 = −ρg(z − z0 )
with ideal gas law p=ρRdT
⇒ pp0
= TT0
⎛
⎝⎜⎞
⎠⎟
gRdΓ
(Poisson Equation)
Γ = −dT / dz is the environmental lapse rateGiven the difference of p (T) between 2 levels, one can directly get the difference in T (p)
Potential Temperature The poisson equation applies to an ideal gas in a static atmosphere,
pp0
= TT0
⎛
⎝⎜⎞
⎠⎟
gRdΓ
relates temperature and pressures at 2 elevations
Now we can also apply it to a single parcel of air rising and falling to see howT and P change with height, but in this case we need to use the adiabatic lapse rateΓd = g / cp,d (from the first law of thermodynanics, exercise) Why?
Hence if a parcel of air moves from a level (P,T) to a level (P0,T0 =θ )
pp0
= Tθ
⎛⎝⎜
⎞⎠⎟
gΓdRd →θ = T
p0
p⎛⎝⎜
⎞⎠⎟
ΓdRdg
→θ = Tp0
p⎛⎝⎜
⎞⎠⎟
Rdcp ,d
AND θv ≈ Tvp0
p⎛⎝⎜
⎞⎠⎟
Rdcp ,d
ANDNN θv
Lapse rates in the atmosphere
Lapse rates in the atmosphere Γ =
Γ =
Vertical Stability of the atmosphere
Γ > Γd Γ < Γd
Neutral Γ = Γd
Vertical Stability of the atmosphere
Vertical Stability of the atmosphere: why θ
z
θv
Absolute Stability
Absolute Instability
Conditional Stability
Dry adiabatic lapse rate of θv = 0 Moist adiabatic lapse
rate of θv
why θ
θ θ
θ, which only changes due to heat exchanges rather than work
ρ = pRT
= pRdTv
=p
1−Rd cp ,d( )p0Rd cp ,d
Rd θv= cstθv
⇒′θv
θv≈ − ′ρ
ρ(for a given parcel at an initially defined p)
∂θ∂t
+ u.∇θ =α∇2θ − 1ρcp
∇.R( )− Leρcp
E
Static Stability of the Troposphere
41
On average, the troposphere is statically stable with a potential temperature gradient of about 3.3 K/km Surface mixing inversion
θ
Troposphere is statically stable, but daytime surface heating creates an unstable lower layer: the atmospheric boundary layer
Actual height can range from: 100 m (or even less) up to 4 km
Inertial ForceCoriolis Force
= u.∇u2Ω × u
~ UU / LΩU
= UΩL
= Rossby Number = Ro >> in ABL
U : characteristic velocity of the flow ~ 10m/s in the ABLL: characteristic length of the flow ~ 1000m in the ABL