Download Pressure Gradient Force

Pressure Gradient Force
δV = δxδyδz
B
δy
FBx
δz
·C
x0,y0,z0
δx
A
Element of air at C.
FAx
Due to random molecular motions, momentum is continually
imparted to the walls of the volume element by surrounding air.
Pressure exerted on the walls of the volume element =
momentum transfer per unit time per unit area
Pressure at C is p0
By Taylor series pressure on the right wall A
2
∂p δx 1 ∂ 2p ⎛ δx ⎞
= p0 +
+
⎟ +−−−−−−
2 ⎜
∂x 2 2 ∂x ⎝ 2 ⎠
Keeping upto second term, the force on wall A along x-axis
∂p ∂x ⎞
⎛
FAx ≈ −⎜ p 0 +
⎟δyδz
∂x 2 ⎠
⎝
∂p ∂x ⎞
⎛
FBx ≈ ⎜ p 0 −
⎟δyδz
∂x 2 ⎠
⎝
Net x-component force on the volume element
Fx = FAx + FBx
Fx
Similarly
Fy
FZ
⎡ ⎛
∂p ∂ x⎤
∂p ∂x⎞
⎟⎟ + p 0 −
= ⎢− ⎜⎜ p 0 +
⎥δy δz
x
2
x
2
∂
∂
⎠
⎣ ⎝
⎦
∂p
= - δx δy δz
∂x
∂p
= - δx δy δz
∂y
∂p
= - δx δy δz
∂z
Total pressure gradient force per unit mass
F
m
=
(i F
x
+ j Fy + k Fz )
ρ δx δy δz
=
⎛ ∂p
1
∂p
∂p ⎞ 1
= − ∇p
− ⎜⎜ i
+ j + k ⎟⎟
ρ
∂y
∂z ⎠ ρ
⎝ ∂x
Hydrostatic equilibrium
Let us consider atmospheric/oceanic pressure along with gravity.
The decrease in pressure with height in the atmosphere and the
increase with depth in the ocean give rise to a vertical pressure
gradient force −
1 ∂p
where z is the height or depth from the surface
ρ ∂z
and ρ is the density.
The vertical pressure gradient force results in a vertical acceleration
in the direction of decreasing pressure i.e. upwards.
The vertical pressure gradient force is closely in balance with the
downward gravity.
1 ∂p
g =−
ρ ∂z
i.e.
This is called hydrostatic
(1)
balance.
This is applicable to most situations in the atmosphere and ocean,
except in the case of large vertical accelerations as in thunderstroms.
Equation (1) can be integrated and written as
− ∫ dp = ∫ ρgdz
(2)
In the above integration g is usually assumed to be constant. However,
g varies with height and latitude due to nonsphericity of earth.
Gravitational Force:
r
Fg
r
r*
GM r
≡ g =− 2 ,
m
r r
G = 6.673 ×10 −11 Nm 2 kg − 2
r
r*
GM r
g0 = − 2
a r
r*
a2
*
g = g0
(a + z )2
a = 6.37 ×106 m
a
Effective Gravity:
r r r r
g ≡ g +Ω R
*
2
r=a+z
z
Fundamental forces in the atmosphere
• Real Forces:
Pressure gradient force
Gravitational force
Frictional force
• Apparent Forces:
Centrifugal force
Coriolis force
[An Introduction to Dynamic Met,
J.R. Holton
Inertial or Newtonian motion
Newton’s first law of motion
A mass in uniform motion relative to a coordinate system
fixed in space will remain in uniform motion in the absence of
any forces.
Non-Newtonian Motion
An object at rest w. r. t. the rotating earth is not at rest or in
uniform motion relative to the coordinate system fixed in
space.
Apparent forces are the inertial reaction terms which arise
because of the coordinate acceleration.
Centrifugal force
For an observer in fixed space, earth rotates with constant speed
but there is change in direction and hence velocity is not constant.
Ω → constant angular velocity
δ v = v δθ
δv
is directed towards the axis of rotation.
dv
dθ ⎛⎜ R ⎞⎟
=
= v
−
dt
dt ⎜⎝ R ⎟⎠
dθ
v = Ω R and
=Ω
dt
δv
Lim
δ t →0 δt
∴
dv
= − Ω 2R
dt
Ω
R
Centripetal acceleration
Viewed from fixed coordinates, the motion is one of uniform acceleration
directed towards the axis of rotation.
Coriolis Force:
Consider particle moving in the eastward direction. The particle rotates faster
than earth. Let u be eastward velocity relative to earth.
2
→
Centrifugal force = ⎛⎜ Ω + u ⎞⎟ R
R⎠
⎝
→
Ωu → u 2 →
2
R+ 2R
=Ω R+2
R
R
For synoptic motions (horizontal scale of 1000km or so):
u p p ΩR
Ω = 7.292 x 10 −5 sec −1 , R ≈ a = 6.37 x 10 6 m, u = 10 m / sec
3rd term u 2 →
R
u
R
pp 1
=
×
=
→
2 nd term R 2
2
R
Ω
2Ωu R
Hence 3rd term can be neglected compared to the second.
Centrifugal force of a parcel rotating in the eastward direction with relative
velocity u
→
→
Ωu
= Ω2 R + 2
R
R
= Centrifugal force of stationary parcel due to rotating earth + Coriolis force.