Hydrostatics

1
Hydrostatics
g
2
Reading

• Hess
• Chapter 6
• pp 75 80
• Wallace Hobbs
• pp 67 72
• Bohren Albrecht
• pp 54 59

3
Objectives

• Be able to write the vertical equation of motion
• Be able to state the assumptions made for
  hydrostatic balance
• Be able to describe hydrostatic balance

4
Objectives

• Be able to derive the hydrostatic equation from
  the vertical equation of motion
• Be able to provide the definition of geopotential
• Be able to calculate geopotential height given
  geopotential

5
Objectives

• Be able to perform calculations using the
  hypsometric equation
• Be able to describe the relationship between
  average temperature in a layer and geopotential
  height
• Be able to describe the analysis performed on
  constant pressure charts

6
Objectives

• Be able to describe the relationship between
  pressure and height on constant pressure charts
• Be able to explain the reason for the slope of
  pressure surfaces from the equator to the poles

7
Objectives

• Be able to provide the definition of thickness
• Be able to describe the relationship between
  average temperature in a layer and thickness

8
Objectives

• Be able to calculate thickness given the average
  temperature of a layer
• Be able to perform calculate the average
  temperature of a layer given the thickness

9
Vertical Equation of Motion

• Which forces are most important in the vertical?
• Coriolis
• Friction
• Pressure Gradient
• Gravity

10
Coriolis Force

• Mostly Horizontal

PGF
NP
Co
11
Frictional Forces

• Friction
• Mostly Horizontal

Wind
Friction
12
Pressure Gradient

• Change in pressure over a given distance

13
Pressure Gradient

• Three Dimensional Pressure Gradient

14
Pressure Gradient

• Lets evaluate the vertical horizontal pressure
  gradient

8 mb
150 mi
500 mb
3 mi
15
Pressure Gradient Force

• Vertical Pressure
• Cartesian (z)

z
16
Gravity

• Gravity
• Ability of objects to attract each other

Gravity
17
Gravity
m

• Gravitational Force
• Function of mass of each object
• Inversely proportional to distance

r
ME
18
Gravity
G 6.67 x 10-11 Nm2kg-2

• Gravitational Acceleration (g)
• Varies with
• Mass
• Radius
• Earths
• Height Above Ground

19
Gravity

• Gravitational Acceleration
• Assumed Constant

g 9.8 meters/sec2 32 ft/sec2

• Variation must be accounted

20
Equation of Motion

• Vertical Equation of Motion

21
Equation of Motion

• Vertical Acceleration
• Important Consideration in Thunderstorms

22
Equation of Motion

• Vertical Acceleration
• Not So Important in Synoptic Meteorology

23
Vertical Equation of Motion
Vertical Pressure Gradient
Gravity

• Only Two Forces
• Vertical Pressure Gradient
• Gravity

24
Vertical Equation of Motion
Vertical Pressure Gradient
Gravity

• Vertical Pressure Gradient is equal to Gravity!

25
Vertical Equation of Motion

• The Vertical Pressure Gradient is Balanced by
  Gravity!

z
g
26
Hydrostatic Equation

• This relationship is known as the Hydrostatic
  Equation

z
g
27
Hydrostatic Equation

• Hydro - fluid
• Static - not moving
• Balance!

z
g
28
Hydrostatic Equation

• Rearrange a few terms

dp change in pressure dz change in height
r density g gravity
29
Geopotential (F)

• The potential energy of a unit mass relative to
  sea level
• Numerically equal to the work that would be done
  in lifting the unit mass from sea level to the
  height at which the mass is located

30
Geopotential (F)

• The work that must be done against the Earths
  gravitational field in order to raise a mass of 1
  kg from sea level to that point

Glossary of Meteorology
31
Geopotential (F)
g
32
Geopotential (F)
g f(z)
g
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Geopotential Height (Z)
g f(z) go 9.8ms-2

• The height of a given point in the atmosphere in
  units proportional to the potential energy of
  unit mass (geopotential) at this height relative
  to sea level

Glossary of Meteorology
34
Geopotential Height (Z)
g f(z) go 9.8ms-2

• Used in upper air calculations
• Small difference between height (z) and
  geopotential height (Z) in lower atmosphere

35
Hydrostatic Equation

• We dont normally measure density.

Im too young to die!

• Eliminate density.

r
36
Hydrostatic Equation

• Using the Ideal Gas Law

• Substitute into Hydrostatic Equation

37
Hydrostatic Equation

• Rearrange terms

38
Hydrostatic Equation

• Remember ...

• Substitute

39
Hydrostatic Equation

• Integrate between two pressure levels

40
Hydrostatic Equation

• Divide both sides by go and reverse the limits

41
Hydrostatic Equation

• Remember Substitute!

42
Hypsometric Equation

• The height difference between two pressure
  surfaces depends on
• Virtual Temperature
• Pressure

43
Hypsometric Equation
p1 sea level pressure Z1 0 m

• At Sea Level

44
Hypsometric Equation
Z2
p2 700 mb

• Height of a pressure surface

p1 SLP
45
Constant Pressure Chart

• Height of a pressure surface

46
Constant Pressure Chart

• Contours - lines of constant height

47
Constant Pressure Chart

• Height of a pressure surface
• Function of Temperature

700 mb
H
L
z 3120 m
z 2850 m
SL
48
Constant Pressure Chart

• How does height compare to pressure?

H
L
690 mb
700 mb
p
z
710 mb
720 mb
10,000 ft
SL
49
Constant Pressure Chart
690 mb
700 mb
L
H
p
710 mb
10,000 ft
720 mb
SL
50
Constant Pressure Chart

• High Height
• High Pressure
• Low Height
• Low Pressure

700 mb
H
L
z 3120 m
z 2850 m
SL
51
Hypsometric Equation

• Hypsometric Equation
• Relates the distance between pressure surfaces

z2
p2 500 mb
5480 m
p1 1000 mb
z1
60 m
52
Thickness

• Thickness (DZ)
• Distance between pressure surfaces

z2
p2 500 mb
dp p2 - p1
DZ Z2 - Z1
5480 m
p1 1000 mb
z1
60 m
53
Thickness

• Calculate Thickness

Integrate!?

• Problem
• Temperature varies with height

54
Thickness

• What are we going to do?

55
Thickness

• Two methods
• 1.) New-Miracle Analysis
• 2.) Fudge

56
Thickness

• Take the average temperature of the layer

57
Thickness

• Substitute average temperature
• Mathematically ....

where
weighted average
58
Thickness

• Simplify

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Hypsometric Equation
DZ height between pressure surfaces p1 lower
pressure surface p2 upper pressure surface Tv
average temperature in layer Rd Gas
Constant go gravity
60
Hypsometric Equation

• Pressure decreases logarithmically

61
Hypsometric Equation

• Pressure decreases logarithmically

62
Hypsometric Equation

• Thickness of a layer depends on temperature

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Hypsometric Equation

• Thickness depends on temperature

500 mb
DZ
Warm
DZ
Cold
1000 mb
64
Hypsometric Equation

• Thickness depends on temperature

65
Hypsometric Equation
66
Thickness

• 1000-500 mb common

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Thickness
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