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Atmospheric Dynamics. Leila M. V. Carvalho Dept. Geography, UCSB. 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. http://weather.unisys.com/surface/sfc_con_stream.html.

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atmospheric dynamics

Atmospheric Dynamics

Leila M. V. Carvalho

Dept. Geography, UCSB

review kinematic of the horizontal flow
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

http://weather.unisys.com/surface/sfc_con_stream.html

natural coordinates
Natural Coordinates:

n and s are natural coordinates (perpendicular and parallel to the flow

Y

n

n

s

s

X

slide5

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)

forces in the atmosphere
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).
apparent forces
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
coriolis force
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

coriolis force10
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

coriolis force can be divided into vertical and meridional components
Coriolis force can be divided into vertical and meridional components :

R

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:

slide12
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

slide15

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

slide16

Surface of constant Pressure

Changes in geopotential height

Changes in geopotential height imply in the existence of pressure gradient forces

friction or viscosity force
Friction or Viscosity Force

τis the shear stress and is the rate of vertical exchange of horizontal momentum N/m2

τs at the surface

friction or viscosity force19
Friction or Viscosity Force

τzxis 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

in summary
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

in a tangent plan we have this is important to remember
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

test your understanding
Test your understanding:

Estimate pressure gradient and, coriolis parameter in Kansas ~38oN

Represent winds around the Low and High pressure systems

geostrophic wind
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: 

geostrophic winds
Geostrophic winds

L

5460m

FGP

FGP

FGP

wind

FGP

FGP

5560m

CF

CF

CF

5640m

CF

H

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

tridimensional view
Tridimensional view

Northern Hemisphere

gradient wind
Gradient Wind
  • 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

slide30

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

slide31

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

http://www.opc.ncep.noaa.gov/Loops/pac500satf00/Pacific_500mb_Analysis_07_Day.shtml

thermal wind
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.
slide33

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

slide34

z

  • In component form:

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
interpretation
Interpretation

z

The wind shear (red arrow)

x

x

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?

slide36

Answer: thickness decreases northward and eastward

  • In component form:

The thermal wind is parallel to thickness contours

The wind shear (red arrow)

relationships with horizontal temperature gradient
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)
baroclinic atmosphere
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.
equivalente barotropic
EquivalenteBarotropic:
  • 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.
exercise 7 3
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:

cold and warm temperature advection
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!
slide45

1

5

2

3

slide47

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 (1000-500)

700hPa height/temperature/winds)

4

3

2

1

slide48

850mb

925 mb

500mb

700 mb