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