Boundary condition and surface forcing

the first few mm of the ABL obey to viscous mechanisms as opposed to pure turbulent ones. To avoid the parameterization of the viscous/roughness layer an effective turbulent flux is proposed

molecular thermal diffusivity remember Pr=0.7 > and ν = 1.5 10-5 m2/s we have a a diffusivity = 1.9 ~ 2.0 10-5 m2/s

assuming QH=0.2 Kms-1

if dz ~ 1mm, then dT ~ 100 =10K (very strong gradient in the microlayer)

The effective turbulent flux=molecular flux in the first few mm

net upward radiation at the surface (forcing term) the – sign indicates downward flux upward sensible

heat flux out of the top of the control volume QH=<w’θ’>surf

upward latent heat flux (evaporation of water from surface) QE=Lv /Cp <w’q’>surf

upward molecular heat flux into the bottom of the CV (to the ground)

storage of internal energy

note that for the infinitesimal layer (b) there is no storage: there is no mass to accumulate energy in. This option works for very flat terrains and the equation is a surface heat budget (Tsurf = Tair)

response terms

Energy balance can be applied on a finite (a) or infinite (b) layer, in both cases however we consider effective turbulent fluxes 1) Radiation

positive direction

Note that these are energy fluxes, i.e. amount of heat per unit square area per second [ J/m2s1 ] = [W/m2] or kinematic fluxes [K m/s]

IN OUT

a) day time over land: sun is heating the surface

–Q*s > 0 , QH,E>0 (heat and moisture are transported away /lifted up by convective motions -QG > 0 (heating the ground)

b) night time over land: long wave radiative cooling –Q*s < 0 , QH< 0 (downward heat and moisture flux > turb is warming the ground) QE< 0 frost/dew formation is absorbing heat from the air -QG < 0 (air is warming the ground: heat A>G)

c) oasis effect during day: sun is heating the surface –Q*s > 0 , QE> 0 heat is extracted from warm air due to moisture evaporation (dominant latent heat flux). This keeps the oasis air cool; as a consequence, we are in a stable BL and turbulence favor heat exchange mixing the cool air with the warm air above QH> 0 <w’t’> negative Note that, with too much humidity (close to saturation) we have to include heat storage in a “finite thickness” layer

QE: water evaporates from the ground, takes heat from the surface air layer. (dominant contribution, evapotranspiration QE is used to water liquid>gas state) QH turbulence mix the warm surface layer taking out heat towards cooler air above (convective motions) QG : surface air is warming the ground (downward heat flux)

QH QE

-QG

-QS*

daytime conditions in summer

reflected shortwave (sun)

shortwave (sun) transmitted through air

longwave (ground) emitted

longwave diffusive (air)

the net radiation is positive from the ground during night and negative from the sun mostly during the day

d

d

d

d

Remember the smaller the temperature of the emitting object, the longer is the wavelength (and the lower the energy)black body radiation

solar irradiance at the top of the atmosphere

with no clouds and sun at the zenit TK=0.8

with clouds everywhere (σ=1) and sun on the zenit TK=0.086 (0.8*0.6*0.3*0.6)

Ks=S TK sin(Ψ) with transmissivity TK , S solar irradiance, Ψ solar elevation angle

(downward shortwave radiation of the previous slide)

d

d

tropic of cancer

equator

tropic of capricorn

opposite (in sign) of the incoming s.w. downwelling radiation

a=0.95 fresh snow a=0.2 grass a=0.1 coniferous forest a=0.05 over dark wet soil

albedo over water depends on the sun inclination and other factor: note that the color of sea and lake water changes a lot.

note that the radiation depends on the temperature of the emitting object high T (sun)>shortwave low T (air/clouds/ground) >longwave

fractional area cloud cover

A black body in thermal equilibrium (that is, at a constant temperature) emits electromagnetic radiation called black-body radiation. The radiation is emitted according to Planck's law, meaning that it has a spectrum that is determined by the temperature alone (see figure at right), not by the body's shape or composition.

“gray body” Realistic emission

d

includes air and ground (opposite) contributions

April 2001 16 days averaged albedo NASA http://earthobservatory.nasa.gov/IOTD/view.php?id=2599

Downward longwave radiation

reflected shortwave (sun)

shortwave (sun) transmitted through air

longwave (ground) emitted

longwave diffusive (air)

Each air layer gives a contribution that depends on its temperature & emissivity, and on the transimissivity of the various other layers. The full contribution is the integral of the radiative flux equation

Q*=(1-a) S TK sin(Ψ)+I* (daytime)

Net radiation

albedo

irradiance

transmissivity

Net longwave radiation (to the ground & from the air)

Q* = I* (nightime)

Ψ solar elevation angle

2) Modeling of fluxes across an interface (e.g. for downward long wave radiation models)

Flux Fϕ= - Ut(ϕtop - ϕbottom)

top (just above the interface)

bottom (just below the interface)

ϕ (x,y,z)

Where Ut transport velocity across the interface

interface

Note that: Ut must be parameterized : 1) Ut = CD M with M : mean horizontal wind magnitude above the surface (solid-air interface)

CD : bulk transfer coefficient (takes into account the shear at the surface and the production of turbulence

2) Ut = we entrainment velocity (difficult to scale/model) e.g. top of the mixed layer or between turb and lam regions in a jet)

M

we

Example : let us consider the mixing ratio r= mvap / mdry 1 )entrainment of dry air in the Mixed layer 2) diffusion of humid air into the Free atmosphere

We

Free atmosphere

Mixed layer

interface

−𝑊𝑒 ∆𝑟 = 𝑤′𝑟′|interface

In equilibrium conditions, the diffusivity of any quantity due to turbulence can be represented as a mean flux using an entrainment velocity We

Entrainment mechanisms is very important for jet and wake evolution turbidity and density current, katabatic winds, suspensions...

∆𝑟 = rtop – rMixed layer ~ 0 - rMixed layer < 0

Ut = CD M with M : mean horizontal wind magnitude above the surface (solid-air interface) CD : drag coefficient

M Momentum flux Boundary conditions: <u’w’>surface ?

Fmom= - Ut(Utop - Ubottom ) = - CD M2 = u*2

~ 0 M

<u’w’>surface = -CD M U (@z= 5-10m surface reference height or measurement height)

<w’ θ’>surface = CH M (θ - θG) (with θ (z= 5-10m) and θG (z= 1mm real ground level) <w’ q’>surface = CE M (q - qG) (with qG often saturated over sea)

Remember the definition of effective fluxes (lam+turb) at the first grid point , not at ground; for practical reason we measure or simulate U at some finite elevation (where x-homog is ok)

either needs to be measured or requires a viscous model for the near wall layer

qG over land

Modeling CD (as well as CM CH )

CD depends on surface roughness, accounts for skin drag, form drag and wave drag

pressure gradient effects related to acceleration, deceleration and wakes around buildings and mountains

transport of momentum by buoyancy/gravity waves in the stable BL (~ very permeable complex topography / hills)

mean continental (!!!) values other sources of drag: canopies (forests, grass, corals, turbines, buildings, etc.) drag is created by the wakes of the roughness elements: CD accounts for the cumulative effects of all wakes (not discernible) canopy similarity law: Raupach, Finnigan 80s-90s

alternative formulation: fitting (if possible) the fully rough log law and estimate the aerodynamic roughness length z0

vegetation+ mountains

Effects on Drag : stability, reference (surface) height , sea waves

10m wind, means wind speed measured at z=10m

z0=0.015 u*2/g

(never a fully rough regime)

the bulk transfer , or drag, coeff. CD depends on the height of the reference surface. If zsurf then CD indeed as U(z=10m)>>U(z=1m) we must compensate with CD (10m)<< CD (1m) to get the same correct surface flux (e.g. u*)

𝐶𝐷𝑁 = 𝑘2[ lnz

𝑧0 ]−2 for neutral stability

𝐶𝐷 = 𝑘2[ lnz

𝑧0 − Ψ𝑀(

𝑧

𝐿) ]−2

for un/stable conditions

Remember the stability correction functions for z/L > 0 (stable) u* is attenuated and so the momentum transfer (except for gravity waves and KH instabilities>wave drag) for z/L < 0 (unstable) u* is enhanced and so the momentum transfer

z/L

stop

B.C. are defined by the surface fluxes. To estimate such fluxes we need : 1) T,q at the surface 2) Bulk transfer coeff. CD , CH CE

Or a model to derive the fluxes at equilibrium using the energy equation:

Partitioning of fluxes into sensible and latent portions

no sun T air ~ Tground

sun Tground increases

Q* incoming radiation: external forcing

heat flux w’t’ vapor flux w’q’ (evaporation) remove heat from the surface

Main point: if the fluxes cannot balance the net incoming radiation then the surface temperature and humidity will increase, which implies that the fluxes will increase too. Thus, surface temperature is not critical to estimate the fluxes as they result from the surface energy balance (at whatever Tsurf) Question: is it possible to obtain QH and QE from a radiation balance?

−QS + QG = QH +QE + (ΔQstorage)

forcing terms

used to define the contribution of sensible and latent heat flux. β depends on the vegetation and micrometeorological conditions: over moist surfaces, evaporation ^ latent heat ^ , β decreases down to 0.5 Over desert (almost no humidity) β up to ~5

if the surface characteristics are known, then from the balance eq. + definition...

however, history effects determine the water content of a surface, so it is complicate...

Remember : the goal is to estimate the surface fluxes, as a boundary conditions for your simulations, knowing forcing and avoiding complex near wall models

(sun-ground)=

−Q*S + QG = QH +QE + (ΔQstorage)

forcing terms

grass field

urban canyon system

lawn Vancouver

suburban area Vancouver

Measure of energy balance under different conditions

with

note that slope of the clausius clapeyron equation defining the vapour pressure for varying T

simplifications to explicitly derive dq/dT qsat specific humidity at saturation

esat vapur pressure at saturation

given: the curve Scc(T) the temperature gradient dT/dz

Again, we estimate fluxes given forcing terms −Q*S + QG

eventually estimate

or directly the two heat contributions (latent +sensible)

Flux to the ground

1) simple model: QG = X QS* with X = 0.1(day) 0.5(night)

2) multilayer soil model, governed by molecular conduction:

𝜕𝑇

𝜕𝑡= −

1

𝐶𝑔

𝜕𝑄𝐺

𝜕𝑡 with Cg soil heat capacity, kg thermal conductivity

𝑄𝐺 = −𝑘𝑔

𝜕𝑇

𝜕𝑧

𝜕𝑇

𝜕𝑡= −𝜈𝑔

𝜕2𝑇

𝜕𝑧2 substituting: heat conduction eq. > with 𝜈𝑔 soil thermal diffusivity

𝜈𝑔= 2.7 10-7 m2/s (snow)

𝜈𝑔= 1.0 10-6 m2/s (farm soil)

𝜈𝑔= 1.5 10-7 m2/s (water)

Higher values of 𝜈𝑔 imply

smaller response of the ground to thermal fluctuations in the air ref:

https://archive.org/stream/climatenearthegr032657mbp#page/n53/mode/2up

∆𝑇 𝑧 = ∆𝑇𝑠𝑢𝑟𝑓 exp −𝑧𝜋

𝜈𝑔𝑃

1/2

it is possible to define a layer where diurnal variation are confined d=[𝜈𝑔P/(4 𝜋)]1/2

assuming the lag is Δt = P, (?)

?

An interesting solution is showing the range of temperature fluctuations ΔT(z) within a period P, and the time lag Δt between air temperature and progressively deep soil temperature

phase shift

P=86400 for daily cycle

d : upper slab with T variation in time that feels the diurnal cycle

sun TG TM (fixed)

QS*

-QG

constant temperature layer

𝐶𝐺𝐴𝜕𝑇𝐺

𝜕𝑡= −𝑄 ∗ 𝑆 − QH - QE + QG

2 layers model: force restore model energy balance rewritten as a prognostic equation

Flux

𝜕𝑇𝐺

𝜕𝑡= −

𝑄 ∗ 𝑆

𝐶𝐺𝐴+

2𝜋

𝑃𝑇𝑀 − 𝑇𝐺 − 𝑎𝐹𝑅 𝑇𝐺 − 𝑇𝐴𝑖𝑟

forcing sensible heat flux

heat flux to the ground

−𝑄𝐺= −𝜕𝑇𝐺

𝜕𝑡+

2𝜋𝐶𝐺𝐴

𝑃𝑇𝑀 − 𝑇𝐺

sun TG TM (fixed)

QS*

-QG

QH

QH

These equations can be solved integrating forward with proper B.C. unknown Tg (t) Tair (t)

TG(t=0), QS* (coordinates, albedo, cloud cover), TM , TAir(t=0) , and given soil characteristics Cg

“conductivity” 3 10-4 day 1 10-4 night

the flux from the deep soil tends to restore the upper slab temperature (response) under radiative forcing (force)

CGA=CGd

Global energy balance: Earth’s Annual Global Mean Energy Budget J. T. Kiehl and Kevin E. Trenberth National Center for Atmospheric Research, Boulder, Colorado

Remote sensing, satellite data

Surface (blue) and top-of-atmosphere (red) upward longwave flux (W m-2 mm-1) for global cloudy conditions. Various gases contributing to the absorption and emission of longwave radiation are denoted