Download Chapter 1. Introduction

Chapter 1. Introduction
• 1.1 Atmospheric Continuum
– Continuous fluid medium
– “mass point” = air parcel (or air particle)
– Field variables: p, r, T and their derivatives (all
continuous functions)
• 1.2 SI Units
–
–
–
–
SI base units: m, kg, s, K
SI derived units: Hz, N, Pa, J, W
Exception: mb=10-3b  1mb = 100 Pa = 1hPa
pSL = 101,325 Pa = 1013.25 mb
1.3 Scale Analysis
• Estimate the significance of various terms in the
governing equations
– Magnitude of field variables
– Amplitude of their fluctuations
– Characteristic length and time duration of
fluctuations
• Examples of horizontal scale sizes: Table 1.4
1.4 Fundamental Forces
• Laws of conservation of mass, energy, and
momentum govern the motions of the
atmosphere
• Fundamental forces
– Pressure gradient force
– Gravitational force
– Viscous force (important only in boundary layer)
• Apparent forces (due to Earth rotation)
– Centrifugal force
– Coriolis force
Fundamental Forces
Pressure gradient force:
FPG
1
  p
m
r
Gravitational force:
Fg
(1.1)
GM r
g  2
(1.3)
m
r r
*
g
GM
r
1
GM
r
0
g*  



2
2
2
2
r
a
r
a  z
1  z a 
1  z a 
*
g*0  gravitational force at sea level
Fundamental Forces cont’d
Shearing stress  xi x j = viscous force per unit area
Cartesian coordinates:
x, y, z  x1 , x2 , x3
x x  
i j
U j
xi
u , v, w  U 1 , U 2 , U 3
 = dynamic viscosity
Fundamental Forces cont’d
Viscous force per unit mass
x component:
x-stress ~ z-gradient of u
Frx 1   xx  yx  zx
 


m
r  x
y
z
  2u  2u  2u 
Frx
  2  2  2 
m
y
z 
 x
or generally:
3
Frxi
 2U i
  2
m
j 1 x j
    2u  2u  2u 
  2  2  2 
y
z 
 r  x
1.5
  kinetic viscosity level = 1.46 10-5 m 2 s 1 at SL
 2ui
is negligibly small, except on Earth surface
2
x j
1.5 Noninertial Reference Frames
Angular velocity of Earth's rotation
d
2
2



 7.29  105 s 1
dt day 86,164 s
( =longitude)
A mass point fixed at the surface is rotating with the
velocity V = R when observed from an inertial frame.
R is the distance from the axis of rotation, R = a cos
( = latitude).
The magnitude of this velocity is constant, but its
direction in the inertial frame changes (Fig. 1.5):
d
V   2 R
dt
1.6 
1.5.1 Centrifugal Force
d
V   2 R is the centripetal force per unit mass,
dt
a component of the gravitational force, that acts towards the
axis of rotation (Fig. 1.5). Consider a mass point fixed at the surface.
In a coordinate system rotating with the Earth, the mass
point is resting. We introduce an apparent force +2 R, the
centrifugal force, that cancels the centripetal force,
i.e. Ftotal = 0 in the rotating coordinate system.
1.5.2 Gravity Force
Gravitational force per unit mass, g * (directed toward center of mass)
Gravity force per unit mass, g :
g  g *  2 R
R  a cos 
g 0  g*0   2 R 0 (Fig. 1.6)
g 0*  9.8 ms 2 , 2 R0  7.292 1010  6.375 106 cos 0  0.34ms 2
g 0*
By using g 0  g 0 k  9.81k ms 2 , we can forget the centrifugal force!
g is a conservative force, i.e., it can be expressed as the gradient
of a potential function :
z
g        g  dr   gdz
0
(1.8)
1.5.3 Coriolis Force
When the mass point moves with constant velocity U
relative to the rotating coordinate system, this constitutes
an accelerated motion in the inertial system. We introduce
an apparent force, the Coriolis force, in the opposite
direction to null the force in the rotating sytem. Chapter 2
derives the Corioilis force per unit mass:
FCo
 2Ω  U
m
Cartesian Components of FCo
FCo
 2Ω  U; Ω    j cos   k sin   , U  ui  vj  wk
m
i
j
k
2Ω  U  2 0 cos  sin 
u v
w
Cartesian components:
 2Ω  U  x  2  w cos   v sin  
 2Ω  U  y  2  u sin  
 2Ω  U  z  2  u cos  
Horizontal Component of FCo
For w
FCo ,hor
u, v, the horizontal Coriolis force is
  2Ω  U  x   2Ω  U  y
m
 2  iv sin   ju sin  
 2  iv  ju  sin   2 sin   k  U 
FCo ,hor
m
 2 sin   k  U 
Toward the right of U
in northern hemisphere
k
FCo
U
1.6 Structure of the Static Atmosphere
Equation of state of a (ideal) gas:
p  r RT R  287 J / kg  K for dry air
(1.14)
Hydrostatic equation:
dp   r gdz, or
(1.15)

p( z )   r gdz
z
Since gdz  d , we can write
1
RT
d   - dp  
dp   RTd (ln p)
r
p
(1.16)
Hypsometric Equation
Integrate from altitude z1 to z 2 :
1
RT
d   - dp  
dp   RTd (ln p)
r
p
p1
( z2 )  ( z1 )  R  Td (ln p)  R T
p2
p1
 d (ln p)
p2
where the layer mean temperature T is:
p1
T   Td (ln p)
p2
p1
 d (ln p)
p2
Geopotential Height Z
Definition
1
1
1
1
dZ  d  
gdz , or Z   
gz.
g0
g0
g0
g0
Layer thickness ZT :
p1
p1
R
T
R
ZT  Z 2  Z1 
Td (ln p) 
d (ln p)


g 0 p2
g 0 p2
R T
Layer mean scale height H 
. Then
g0
p1
p1
ZT   d (ln p)  H ln
p2
p2
p  p0e Z H
(1.19)
Generalized Vertical Coordinate
From Fig. 1.12, on the surface s=const:
p C  p A p C  pB p B  p A p C  pB  z p B  p A




x
x
x
z x
x
For   :
 p   p  z   p 
        
 x  s  z  x  s  x  z
1.22 
Pressure as a Vertical Coordinate
For the surface s = p:
 p   z   p 
0         or
 z   x  p  x  z
 p   p   z 
     
 x  z  z   x  p
 z 
  
 r g    r 


x

x
 p

p
Horizontal pressure gradient force:
1  p    
   

r  x  z  x  p
1.20 