1. The simplifed momentum equations
Height coordinates
Pressure coordinates

2. In this section we will derive the four most important relationships describing atmospheric structure
Hydrostatic Balance: the vertical force balance in the atmosphere
The Hypsometric Equation: the relationship between the virtual temperature of a layer and the layer’s thickness
Geostrophic Balance: the most fundamental horizontal force balance in the atmosphere
4. Thermal wind Balance: a relationship between the wind at an upper level of the atmosphere and the temperature gradient below that level

3. We will then expand our understanding of balanced flow to include more complicated force balances in horizontal flows
Geostrophic Flow: the flow resulting from a balance between the PGF and Coriolis force in straight flow
Inertial Flow: curved flow resulting when the Coriolis force balances the “Centrifugal” force
Cyclostrophic Flow: Flow that results when the pressure gradient force balances the Centrifugal force
4. Gradient flow: Flow that results when the pressure gradient force and Coriolis force balance the Centrifugal force

4. Virtual Temperature: The temperature that a parcel of dry air
would have if it were at the same pressure and had the same density
as moist air.
P = pressure
rd = dry air density
rv = vapor density
= air density
R= gas constant
Rv = vapor gas constant
Rd = dry air gas constant
T = Temperature
Derivation:
Start with ideal gas law for moist air:
Now treat moist air as if it were dry by introducing the virtual temperature Tv
What is the relationship between the temperature, T
And the virtual temperature Tv?

5. Multiply and divide second term by Rd
M = mass of air
Md = mass of dry air
Mv = mass of vapor
V = volume
Write:
Multiply and divide second term by Rd
Cancel Rd and rearrange

6. From last page:
Cancel V, and divide top and bottom terms by Md:
Introduce mixing ratio: rv = Mv/Md and let e = Rd/Rv

7. From last page:
Approximate (1+rv)-1 = 1- rv and use 1/e = 1.61
Neglect term

8. HYDROSTATIC BALANCE
The atmosphere is in hydrostatic balance essentially everywhere except in core regions of significant storms such as hurricanes and thunderstorms

9. The Hypsometric Equation
Consider a column of atmosphere that is 1 m by 1 m in area and extends from sea level to space
Let’s isolate the part of this column that extends between the 1000 hPa surface and the 500 hPa surface
How much mass is in the column?
How thick is the column? That depends…………

10. Hydrostatic equation Ideal Gas Law
Substitute ideal gas law into hydrostatic equation
Integrate this equation between two levels (p2, z2) and (p1, z1)

11. This equation is called the Hypsometric Equation
From previous page
Problem: Tv varies with altitude. To perform the integral on the right we have to consider the pressure weighted column average virtual temperature given by:
We can then integrate to give
This equation is called the Hypsometric Equation
The equation relates the thickness of a layer of air between two pressure levels to the average virtual temperature of the layer

12. Geopotential Height
We can express the hypsometric (and hydrostatic) equation in terms of a quantity called the geopotential height
Geopotential (): Work (energy) required to raise a unit mass a
distance dz above sea level
Meteorologists often refer to “geopotential height” because this quantity is directly associated with energy to vertically displace air
Geopotential Height (Z):
g0 is the globally averaged
Value of gravity at sea level
For practical purposes, Z and z are about the same in the troposphere

13. Atmospheric pressure varies exponentially with altitude, but very slowly on
a horizontal plane – as a result, a map of surface (station) pressure looks like
a map of altitude above sea level.
observed surface pressure
over central North America
variation of pressure
with altitude

14. Station pressures are measured at locations worldwide
Analysis of horizontal pressure fields, which are responsible for
the earth’s winds and are critical to analysis of weather systems,
requires that station pressures be converted to a common level,
which, by convention, is mean sea level.

15. Reduction of station pressure to sea level pressure:
Integrate hypsometric equation
sea level pressure (PSL)
to station pressure (PSTN)
and right side from z = 0
to station altitude (ZSTN).
(4)
(5)
BUT WHAT IS Tv? WE HAVE ASSUMED A FICTICIOUS
ATMOSPHERE THAT IS BELOW GROUND!!!

16. National Weather Service Procedure to estimate Tv
Assume Tv = T
2. Assume a mean surface temperature = average of
current temperature and temperature 12 hours earlier
to eliminate diurnal effects.
3. Assume temperature increases between the station and sea level of
6.5oC/km to determine TSL.
4. Determine average T and then PSL.
In practice, PSL is determined using a table of “R” values, where
R is the ratio of station pressure to sea level pressure, and the table
contains station pressures and average temperatures.
Table contains a “plateau correction” to try to compensate
for variations in annual mean sea level pressures calculated for
nearby stations.

17. Consider the 1000—500 hPa thickness field. Using
Implications of hypsometric equation
Consider the 1000—500 hPa thickness field. Using
Ballpark number: A decrease in average hPa layer temperature of 1K leads to a reduction in thickness of the layer of 60 hPa

18. Implications of Hypsometric Equation
A cold core weather system (one which has lower temperature at its center) will winds that increase with altitude

19. How steep is the slope of
The 850 mb pressure surface
in the middle latitudes? oN oN
40o  60 nautical miles/deg  1.85 km/nautical mile = 4452 km
Slope = 430 m/ m ~ 1/10,000 ~ 1 meter/10 km

20. How steep is the slope of
The 250 mb pressure surface
in the middle latitudes?
9,720 80oN
10,720 40oN
40o  60 nautical miles/deg  1.85 km/nautical mile = 4452 km
Slope = 1000 m/ m ~ 1/4,000 ~ 1 meter/4 km

21. GEOSTROPHIC BALANCE A state of balance between
the pressure gradient force and the Coriolis force
Geostrophic balance
Air is in geostrophic balance if and only if air is not accelerating (speeding
up, slowing down, or changing direction).
For geostrophic balance to exist, isobars (or height lines on a constant pressure
chart) must be straight, and their spacing cannot vary.

22. Geostrophic wind
The wind that would exist if air was in geostrophic balance
The Geostrophic wind is a function of the pressure gradient and latitude
In pressure coordinates, the geostrophic relationships are given by

23. Where on this map of the 300 mb surface is the air in geostrophic balance?

24. Geostrophic Balance and the Jetstream
Take p derivative:
Substitute hydrostatic eqn
THE RATE OF CHANGE OF THE GEOSTROPHIC WIND WITH
HEIGHT (PRESSURE) WITHIN A LAYER IS PROPORTIONAL TO
THE HORIZONTAL TEMPERATURE GRADIENT WITHIN THE LAYER

25. We can write these two equations in vector form as
or alternatively

26. The “Thermal Wind” vector
The “Thermal Wind” is not a wind! It is a vector that is parallel to the mean isotherms in a layer between two pressure surfaces and its magnitude is proportional to the thermal gradient within the layer.
When the thermal wind vector is added to the geostrophic wind vector at a lower pressure level, the result is the geostrophic wind at the higher pressure level

27. A horizontal temperature
gradient leads to a greater
slope of the pressure
surfaces above the
temperature gradient.
More steeply sloped pressure
surfaces imply that a stronger
pressure gradient will exist
aloft, and therefore a stronger
geostrophic wind.

28. Note the position of the front at 850 mb……

29. ….and the jetstream at 300 mb.

30. Winds veering with height – warm advection
Implications of geostrophic wind veering (turning clockwise) and backing (turning counterclockwise) with height
Warm air
Cold air
Winds veering with height – warm advection

31. Implications of thermal wind relationship for the jetstream
Note the position of the jet over the strongest gradient in potential temperature (dashed lines) (which corresponds to the strongest temperature gradient)

32. Implication of the thermal wind equation for the general circulation
Upper level winds must be generally westerly in both hemispheres since it is coldest at the poles and warmest at the equator

33. Natural Coordinates and Balanced Flows
To understand some simple properties of flows, let’s consider atmospheric flow that is frictionless and horizontal
We will examine this flow in a new coordinate system called “Natural Coordinates”
In this coordinate system:
the unit vector is everywhere parallel to the flow and positive along the flow
the unit vector is everywhere normal to the flow and positive to the left of the flow

34. The velocity vector is given by:
The magnitude of the velocity vector is given by: where s is the measure of distance in the direction.
The acceleration vector is given by:
where the change of with time is related to the flow curvature.

35. We need to determine
The angle is given by
Where R is the radius of curvature following parcel motion
directed toward center of curve (counterclockwise flow)
directed toward outside of curve (clockwise flow)

36. Note here that points in the positive
direction in the limit that approaches 0.

37. Since the Coriolis force is always directed to the right of the motion
Let’s now consider the other components of the momentum equation, the pressure gradient force and the Coriolis Force
Coriolis Force:
Since the Coriolis force is always
directed to the right of the motion
Since the pressure gradient force
has components in both directions
PGF:
The equation of motion can therefore be written as:

38. Let’s break this up into component equations:
To understand the nature of basic flows in the atmosphere we will assume that the speed of the flow is constant and parallel to the height contours so that
Under these conditions, the flow is uniquely described by the equation in the yellow box

39. Geostrophic Balance: PGF = COR Inertial Balance: CEN = COR
Centrifugal Force
PGF
Coriolis Force
Geostrophic Balance: PGF = COR
Inertial Balance: CEN = COR
Cyclostrophic Balance CEN = PGF
Gradient Balance CEN = PGF + COR

40. Geostrophic flow
Geostrophic flow occurs when the PGF = COR, implying that R
For geostrophic flow to occur the flow must be straight and parallel to the isobars
Pure geostrophic flow is uncommon in the atmosphere

41. Inertial flow
Inertial flow occurs in the absence of a PGF, rare in the atmosphere but common in the oceans where wind stress drives currents
This type of flow follows circular, anticyclonic paths since R is negative
Time to complete a circle:
is one half rotation
 is one full rotation/day
called a half-pendulum day

42. Power Spectrum of kinetic energy at 30 m in the ocean near Barbados (13N)

43. Cyclostrophic flow
Flows where Coriolis force exhibits little influence on motions (e.g. Tornado)

44. Venus (rotates every 243 days):
Cyclostrophic flows can occur when the Centrifugal force far exceeds the Coriolis force
, a number called the Rossby number
A synoptic scale wave: NO
A tornado:
YES
Venus (rotates every 243 days):
YES

45. In cyclostrophic flow around a low, circulation can rotate clockwise or counterclockwise (cyclonic and anticyclonic tornadoes and smaller vortices are observed)
WHAT ABOUT A HIGH???

46. Venus has a form of cyclostrophic flow, but more is going on that we don’t understand

47. Gradient flow
This expression has a number of mathematically possible solutions, not all of which conform to reality

48. the unit vector is everywhere normal to the flow and positive to the left of the flow
 is the geopotential height
is the height gradient in the direction of
R is the radius of curvature following parcel motion
directed toward center of curve (counterclockwise flow)
directed toward outside of curve (clockwise flow)

49. V is always positive in the natural coordinate system
Solutions for
For radical to be positive
Therefore: must be negative.
V = negative = UNPHYSICAL

50. Solutions for
Radical >
Positive root physical
Negative root unphysical
Anticyclonic
Called an “anomalous low” it is rarely
observed (technically since f is never 0 in
mid-latitudes, anticyclonic tornadoes are
actually anomalous lows
outward
Increasing in direction (low)

51. Solutions for
Radical >
Positive root physical
Negative root unphysical
Cyclonic
inward
Called an “regular low” it is commonly
observed (synoptic scale lows to cyclonically
rotating dust devils all fit this category
decreasing in direction (low)

52. Solutions for
Radical >
Positive root physical
Negative root unphysical
Cyclonic
inward
Called an “regular low” it is commonly
observed (synoptic scale lows to cyclonically
rotating dust devils all fit this category
decreasing in direction (low)

53. Positive Root
Solutions for
or radical is imaginary
Antiyclonic
therefore or
outward
decreasing in direction (high)
Called an “anomalous high” : Coriolis force never
observed to be < twice the centrifugal force

54. Negative Root
Solutions for
or radical is imaginary
Antiyclonic
outward
decreasing in direction (high)
Called a “regular high” : Coriolis force exceeds
the centrifugal force

55. Condition for both regular and anomalous highs
This is a strong constraint on the magnitude of the pressure gradient force in the vicinity of high pressure systems
Close to the high, the pressure gradient must be weak, and must disappear at the high center

56. Note pressure gradients in vicinity of highs and lows

57. The ageostrophic wind in natural coordinates
For cyclonic flow (R > 0) gradient wind is less than geostrophic wind
For anticyclonic flow (R < 0) gradient wind is greater than geostrophic wind

58. Convergence occurs downstream of ridges
and divergence downstream of troughs