1. Leila M. V. Carvalho Dept. Geography, UCSB
Atmospheric Dynamics
Leila M. V. Carvalho
Dept. Geography, UCSB

2. Review: Kinematic of the horizontal flow
Streamlines: lines parallel to the horizontal velocity V at a particular level and at a particular instant in time

3. Natural Coordinates:
n and s are natural coordinates (perpendicular and parallel to the flow Y n n s s X

4. Definitions

5. Rotation with cyclonic curvature (NH) and cyclonic shear, no diffluence or stretching (and divergence
Sheared with no curvature, no diffluence, stretching or divergence
Radial flow with velocity directly proportional to radius. Diffluence, stretching, divergence and NO CURVATURE (or vorticity)
Hyperbolic flow: difluence and straching, no divergence (terms cancel). Shear and curvature cancel (vorticity free)

6. What is going on here? Y n n s s X

7. Forces in the Atmosphere
Equation of motion: (First and second Laws of Newton)
Real forces (independent on the rotating system): gravity, pressure gradient force and frictional force
Apparent forces due to rotation: apparent centrifugal force (affects gravity) and Coriolis (correction for horizontal movements).

8. Apparent forces: Centrifugal force:
Where RA is vector perpendicular to axis of rotation and  is angular velocity of earth
Combine with gravity to define "effective" gravity

9. Coriolis force:
Coriolis force takes care of rotational effects caused by motion relative to surface Ω
At rest over the Earth surface will have cetrifugal acceleration= Ω2R.
Suppose it moves eastward with speed u: the centrifugal force would increase to:
Centrifugal force= R

10. Coriolis force: Expanding the equation we have now: Coriolis Force
Synoptic scale motions u<< ΩR:
Last term can be neglected in a first approximation
Centrifugal force due to rotation of the Earth (independent of the relative velocity
Deflecting forces that act outward along the vector R

11. Coriolis force can be divided into vertical and meridional components :φ
To the right of the movement in the NH φ
A relative motion along the east-west coordinate will produce an acceleration in the north-south direction given by:
And vertical acceleration given by:

12. Suppose now that a particle initially at rest on the Earth is set in motion equatoward by impulsive forces
As it moves equatorward it will conserve its angular momentum in the absence of torques: a relative westward velocity must develop Ω R a
R + δR
If we expand the right hand side and neglect second order differentials (and assume that δR<<R and solve for δu, we get:
a= Earth’s Radius
Northward velocity component

13. Real forces in the Atmosphere

14. Pressure gradient Force

15. z y x High Pressure p1 Low Pressure p2 >0 for sure wind direction
REMEMBER THAT A “GRADIENT” ALWAYS POINT TOWARD THE HIGHEST MAGNITUDES OF THE SCALAR.
wind direction
Pressure Gradient Force z
Pressure Gradient y
High Pressure p1
Low Pressure p2
>0 for sure x
Hydrostatic Equation:
Definition of Geopotential
Geopotential Height
See Holton, 1979, second Ed. Chap1, pg. 21

16. Surface of constant Pressure
Changes in geopotential height
Changes in geopotential height imply in the existence of pressure gradient forces

17. Winds and geopotential height: example: sea breeze
High Pressure
LAND
OCEAN W E

18. Friction or Viscosity Force
τ is the shear stress and is the rate of vertical exchange of horizontal momentum N/m2
τs at the surface

19. Friction or Viscosity Force
τzx is the shear stress in the horizontal direction x due to the stress acting vertically
subscripts indicate that τzx is the shear stress in x direction due to vertical shear and μ is the dynamic viscosity coefficient

20. In summary Friction Very small outside boundary layer
Depends on vertical gradient in vertical component of shear stress
ν = viscosity coefficient = μ/ρ
~10-5m2s-1
Actual processes very complex, with turbulence playing key role
Approximate shear stress in surface boundary layer:
Shear stress depends on strength of vertical shear in horizontal wind. Empirically:
Drag coefficient, CD, depends on surface roughness and static stability

21. Horizontal Equation of Motion Newton’s Law in vectorial form per unity of mass:

22. In a tangent plan we have (this is important to remember):
Remember that Friction is defined as a negative component that is supposed to decrease (decelerate) the speed
We can eliminate density by using the relationship between pressure gradient and geopotential

23. Test your understanding:
Estimate pressure gradient and, coriolis parameter in Kansas ~38oN
Represent winds around the Low and High pressure systems

24. Geostrophic wind
By using scale analysis of horizontal equations it can be shown that:
Horizontal velocity scale:
Length scale:
Depth scale:
Horizontal pressure fluctuation scale:
Time scale (advective): 
Coriolis Scale: 

25. In the free atmosphere Coriolis balances with Gradient Force

26. Friction effect:

27. Geostrophic winds
5460m
5560m
5640m L H
FGP
FGP
FGP
wind
FGP
FGP CF CF CF CF
Estimate the geostrophic winds given this distribution of geopotential height, assuming that the spatial interval between the two lines is equal 100km. Assume this region is in midlatitudes of the NH

28. Tridimensional view
Northern Hemisphere

29. Gradient Wind The signs of these terms depend on the curvature
Curved trajectories when the wind direction is changed: the centripetal (or centrifugal) acceleration needs to be considered in the balance of forces
Centripetal acceleration is given by: V2/RT, where RT is the local radius of curvature of the air trajectories.
The signs of these terms depend on the curvature

30. Since Co is dependent on the wind speed, and since the centrifugal force is in the same direction as Co, the balance of forces can be achieved at slower speeds compared with a geostrophic one : SUBGEOSTROPHIC
The centrifugal force is opposite to Co: Balance is achieved at higher speeds compared with the geostrophic balance:
SUPERGEOSTROPHIC WINDS
The centrifugal force Acts in the same direction as Coriolis

31. Where would you expect to observe geostrophic balance, gradient balance, supergeostropic and subgeostrophic winds? Show the balance of forces in these regions

32. Thermal Wind
Is not an actual wind, as it does not blow the dust from the ground or rocks the leaves in the trees.
The purpose of thermal winds is to indicate a relationship between vertical shear in the geostrophic wind and temperature gradient that will help in weather forecast.

33. It can be obtained by writing the geostrophic equation for two different pressure surfaces and subtracting them (to calculate the shear in the intervening layer) z 2
In terms of geopotential height 1 x

34. In component form: z 2 1 x
In other words, the thermal wind equation states that the vertical shear of the geostrophic wind within the layer between any two pressure surfaces is related to the horizontal gradient of thickness

35. Interpretation z x x The wind shear (red arrow)
In this example in the NH, f >0 the atmospheric thickness is decreasing or increasing as we move north? Is it increasing or decreasing as we move westward?

36. In component form: Answer: thickness decreases northward and eastward
The thermal wind is parallel to thickness contours
The wind shear (red arrow)

37. Relationships with horizontal temperature gradient
Barotropic atmosphere: density depends only on the pressure (isobaric surfaces are also surfaces of constant density). Isobaric surfaces will be also isothermals (law of gases ):
Geostrophic winds is independent of height in a barotropic atmosphere (geopotential heights are stacked on the top of one another like dishes)

38. Baroclinic atmosphere:
Baroclinic atmosphere: Density depends on both the temperature and pressure. In a baroclinic atmosphere the geostrophic wind generally has vertical shear related to the horizontal temperature gradient by the thermal wind equation.

39. Equivalente Barotropic:
Horizontal temperature gradients are such that the thickness contour are parallel to the geopotential height contours. In this case, the thermal wind equation states that the wind shear should be parallel to the wind itself: there is no change in direction of the wind.

40. Exercise 7.3
During the winter in the troposphere ~ 30oN, the zonally averaged temperature gradient is ~0.75K per degree of latitude and the zonally averaged component of the geostrophic wind at the Earth’s surface is close to zero. Estimate the mean zonal wind at the jet stream level ~ 250hPa
Solution: take the zonal component and average:

41. Cold and warm temperature advection
Cold advection: flow across the isotherms from a colder to a warmer regions
Warm advection (opposite)
The thermal wind theory tells us that VT ‘blows’ parallel to the thickness with the colder (warmer) air to the left of the wind in the NH (SH)
If you know the geostrophic wind (the one that can blow your hair) between two levels you can estimate the mean wind direction in that layer.
Joining both info will tell you if the present wind configuration will advect cold or warm air, and therefore, you can use that to forecast the weather!

42. Discussion: are the regions marked in the map cooling or warming and why?1 5 2 3

45. 1 5 2 3

47. Discuss the advection of temperature in the regions marked with a star
Discuss the advection of temperature in the regions marked with a star . Plot the thermal wind, the temperature gradient (vectors). Assume that the 700hPa winds represent the mean wind between 1000 and 500 hPa.
Thickness ( )
700hPa height/temperature/winds) 4 3 2 1

48. 850mb
925 mb
500mb
700 mb