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vacuum. h. A. B. CHAPTER 2. VERTICAL STRUCTURE OF THE ATMOSPHERE. Measurement of atmospheric pressure with the mercury barometer:. Atmospheric pressure P = P A = P B = r Hg gh. Mean sea-level pressure: P = 1.013x10 5 Pa = 1013 hPa

CHAPTER 2. VERTICAL STRUCTURE OF THE ATMOSPHERE

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vacuum

h

A

B

Measurement of atmospheric pressure with the mercury barometer:

Atmospheric pressure P = PA = PB = rHggh

Mean sea-level pressure:

P = 1.013x105 Pa = 1013 hPa

= 1013 mb

= 1 atm

= 760 mm Hg (torr)

“SEA LEVEL” PRESSURE MAP (2/5/13, 13Z)

weather.unisys.com

Consider a pressure gradient at sea level operating on an elementary air parcel dxdydz:

P(x)

P(x+dx)

Pressure-gradient force

Vertical area

dydz

Acceleration

For DP = 10 hPa over Dx = 100 km, g ~ 10-2 m s-2 a 100 km/h wind in 3 h!

Effect of wind is to transport air to area of lower pressure a dampen DP

On mountains, however, the surface pressure is lower, and the pressure-gradient force along the Earth surface is balanced by gravity:

P(z+Dz)

P-gradient

- This is why weather maps show “sea level” isobars even over land; the fictitious “sea-level” pressure assumes an air column to be present between the surface and sea level

gravity

P(z)

Mean pressure at Earth's surface:

984 hPa

Radius of Earth:

6380 km

Total number of moles of air in atmosphere:

Mol. wt. of air: 29 g mole-1 = 0.029 kg mole-1

CLAUSIUS-CLAPEYRON EQUATION: PH2O, SAT = f(T)

A = 6.11 hPa

B = - 5310 K

To = 273 K

PH2O,SAT (hPa)

T (K)

Stratopause

Tropopause

- Consider elementary slab of atmosphere:

P(z+dz)

P(z)

hydrostatic

equation

unit area

Ideal gas law:

Assume T = constant, integrate:

Barometric law

The atmospheric evolution of a species X is given by the continuity equation

deposition

emission

transport

(flux divergence;

U is wind vector)

local change in concentration

with time

chemical production and loss

(depends on concentrations

of other species)

This equation cannot be solved exactly e need to construct model (simplified representation of complex system)

Improve model, characterize its error

Design observational system to test model

Design model; make assumptions needed

to simplify equations and make them solvable

Evaluate model with observations

Define problem of interest

Apply model:

make hypotheses, predictions

Atmospheric “box”;

spatial distribution of X within box is not resolved

Chemical

production

Chemical

loss

Inflow Fin

Outflow Fout

X

L

P

D

E

Deposition

Emission

Lifetimes add in parallel:

Loss rate constants add in series:

Satellite observations of NO2 columns

Satellite observations

Assimilated observations

IPCC [2001]

IPCC [2001]

Steady state solution (dm/dt = 0)

Initial condition m(0)

Characteristic time t = 1/k for

- reaching steady state
- decay of initial condition

If S, k are constant over t >> t, then dm/dt g0 and mg S/k: quasi steady state

F12

m2

m1

F21

Mass balance equations:

(similar equation for dm2/dt)

If mass exchange between boxes is first-order:

e system of two coupled ODEs (or algebraic equations if system is assumed to be at steady state)

LATITUDINAL GRADIENT OF CO2 , 2000-2012

Illustrates long time scale for interhemispheric exchange;

use 2-box model to constrain CO2 sources/sinks in each hemisphere

http://www.esrl.noaa.gov/gmd/ccgg/globalview/

The mass balance equation is then the finite-difference approximation of the continuity equation.

Solve continuity equation for individual gridboxes

- Models can presently afford
~ 106 gridboxes

- In global models, this implies a horizontal resolution of 100-500 km in horizontal and ~ 1 km in vertical
- Drawbacks: “numerical diffusion”, computational expense

Fire plumes over

southern California,

25 Oct. 2003

A Lagrangian “puff” model offers a much simpler alternative

CX(x, t)

In the moving puff,

wind

CX(xo, to)

…no transport terms! (they’re implicit in the trajectory)

Application to the chemical evolution of an isolated pollution plume:

CX,b

CX

In pollution plume,

Temperature inversion

(defines “mixing depth”)

Emission E

In column moving across city,

CX

x

0

L

C(x, to+Dt)

Individual puff trajectories

over time Dt

ADVANTAGES OVER EULERIAN MODELS:

- Computational performance (focus puffs on region of interest)
- No numerical diffusion
DISADVANTAGES:

- Can’t handle mixing between puffs a can’t handle nonlinear processes
- Spatial coverage by puffs may be inadequate

C(x, to)

Concentration field at time t defined by n puffs

Run Lagrangian model backward from receptor location,

with points released at receptor location only

backward in time

- Efficient cost-effective quantification of source influence distribution on receptor (“footprint”)