Atmospheric Dynamics Leila M. V. Carvalho Dept. Geography, UCSB.
-
date post
19-Dec-2015 -
Category
Documents
-
view
243 -
download
4
Transcript of Atmospheric Dynamics Leila M. V. Carvalho Dept. Geography, UCSB.
Atmospheric Dynamics
Leila M. V. CarvalhoDept. Geography, UCSB
Review: Kinematic of the horizontal flowStreamlines: lines parallel to the horizontal velocity V at a particular level and at a
particular instant in timehttp://weather.unisys.com/surface/sfc_con_stream.html
Natural Coordinates:
Y
X
n n
ss
n and s are natural coordinates (perpendicular and parallel to the flow
Definitions
Sheared with no curvature, no diffluence, stretching or divergence
Rotation with cyclonic curvature (NH) and cyclonic shear, no diffluence or stretching (and 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)
What is going on here?
Y
X
n n
ss
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:
• 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 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=
RR
u2
R
Coriolis force:
2
22
22
R
Ru
R
uRRR
R
u
Expanding the equation we have now:
Centrifugal force due to rotation of the Earth (independent of the relative velocity
Deflecting forces that act outward along the vector R
Synoptic scale motions u<< ΩR:Last term can be neglected in a first approximation
Coriolis Force
Coriolis force can be divided into vertical and meridional components :
R
φR
uR2φ
cos2 u
sin2 uA relative motion along the east-west coordinate will produce an acceleration in the north-south direction given by:
sin2 udt
dvCo
And vertical acceleration given by: cos2 udt
dwCo
To the right of the movement in the NH
Suppose now that a particle initially at rest on the Earth is set in motion equatoward by
impulsive forcesΩ
RAs it moves equatorward it will conserve its angular momentum in the absence of torques: a relative westward velocity must develop
R + δRIf we expand the right hand side and neglect second order differentials (and assume that δR<<R and solve for δu, we get:
oaRu sin22
a
a= Earth’s Radius
oo vdt
da
dt
du sin2sin2
dt
dav
Northward velocity component
Real forces in the Atmosphere
Pressure gradient Force
Low Pressurep2
High Pressurep1
wind direction
Pressure Gradient
REMEMBER THAT A “GRADIENT” ALWAYS POINT TOWARD THE HIGHEST MAGNITUDES OF THE SCALAR.
Pressure Gradient Force
x
pP 1
)8.7(1
;1
y
pP
x
pP yx
Hydrostatic Equation:
gz
p
gdzd
Definition of Geopotential Geopotential Height
z
oo
gdzgg
zZ
0
1)(
ZgzgpP o1
z
y
>0 for sure
See Holton, 1979, second Ed. Chap1, pg. 21
ZgzgpP o1
Surface of constant Pressure
Changes in geopotential height
1
2
12
p
p
vo
d
p
dpT
g
RZZ
Changes in geopotential height imply in the existence of pressure gradient forces
Zgo
ppzppz yy
zg
y
p
xx
zg
x
p
1
;1
Winds and geopotential height: example: sea breeze
ZgPForceGradient o:
Z
LAND OCEAN
High Pressure
W E
ppz xx
zg
x
p
1
Friction or Viscosity Force
zF
1 τ is the shear stress and is the
rate of vertical exchange of horizontal momentum N/m2
τs at the surface
Friction or Viscosity Force
zF
1 τzx is the shear stress in the
horizontal direction x due to the stress acting vertically
z
uzx
subscripts indicate that τzx is the shear stress in x direction due to vertical shear and μ is the dynamic viscosity coefficient
In summaryFriction
Very small outside boundary layerDepends on vertical gradient in vertical component of shear stress
•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:
ν = viscosity coefficient = μ/ρ ~10-5m2s-1
Drag coefficient, CD, depends on surface roughness and static stability
Horizontal Equation of MotionNewton’s Law in vectorial form per unity of mass:
FVkV
fpFCPdt
d
1
In a tangent plan we have (this is important to remember):
xFfvx
p
dt
du
1
yFfuy
p
dt
dv
1
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
FVkV
fdt
d
Test your understanding:
Estimate pressure gradient and, coriolis parameter in Kansas ~38oNRepresent winds around the Low and High pressure systems
Geostrophic windBy 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:
In the free atmosphere Coriolis balances with Gradient Force
FVkV
fdt
d
Friction effect:
Geostrophic winds
5460m
5560m
5640m
L
H
FGP
CF
CF CF
FGP
CF
wind
FGP
FGPFGP
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
kVfg
1
yfug
1
xfug
1
Tridimensional view
Northern Hemisphere
Gradient WindCurved trajectories when the wind direction is changed: the centripetal (or centrifugal) acceleration needs to be considered in the balance of forcesCentripetal acceleration is given by: V2/RT, where RT is the local radius of curvature of the air trajectories.
FVkV
fdt
dVkn f
R
V
T
2
The signs of these terms depend on the curvature
The centrifugal force Acts in the same direction as Coriolis
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
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
• 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.
• 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)
1212
1 kVV
fgg
x
z2
1 1212ZZ
f
gogg kVV
In terms of geopotential height
• In component form:
y
ZZ
f
guu o
gg
1212
x
z2
1 x
ZZ
f
gvv o
gg
1212
• 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
xx
z
1212 gggg VVVV
12 gg VV
The wind shear (red arrow)
012 gg uu
012 gg vv
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?
• In component form:
y
ZZ
f
guu o
gg
1212
x
ZZ
f
gvv o
gg
1212
The wind shear (red arrow)
012 gg uu
012 gg vv
Answer: thickness decreases northward and eastward
The thermal wind is parallel to thickness contours
Relationships with horizontal temperature gradient
Tp
p
f
Rgg
kVV
2
112
ln
• 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 ):
0ln
0
p
VT g
p
• 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: 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:
• 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• 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:
y
TRuu gg
250
1000ln
sin21000250 T
p
p
f
Rgg
kVV
2
112
ln
1515
11
2508.36
1011.1
75.04ln
30sin1029.72
deg287
ms
m
Ku
og s
kgJ
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!
Discussion: are the regions marked in the map cooling or warming and why?
2 3
1 5
2 31 5
Thickness (1000-500) 700hPa height/temperature/winds)
2 31
4
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.
925 mb
700 mb
850mb
500mb