aoss 321, winter 2009 earth system dynamics lecture 4 1/20/2009
DESCRIPTION
AOSS 321, Winter 2009 Earth System Dynamics Lecture 4 1/20/2009. Christiane Jablonowski Eric Hetland [email protected] [email protected] 734-763-6238 734-615-3177. Today’s lecture. We will discuss Divergence / Curl / Laplace operator Spherical coordinates - PowerPoint PPT PresentationTRANSCRIPT
AOSS 321, Winter 2009 Earth System Dynamics
Lecture 41/20/2009
Christiane Jablonowski Eric Hetland
[email protected] [email protected]
734-763-6238 734-615-3177
Today’s lecture
We will discuss
• Divergence / Curl / Laplace operator
• Spherical coordinates
• Vorticity / relative vorticity
• Divergence of the wind field
• Taylor expansion
Divergence(Cartesian coordinates)
• Divergence of a vector A:
• Resulting scalar quantity, use of partial derivatives
€
∇• r
A =
∂
∂x∂
∂y∂
∂z
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
•
Ax
Ay
Az
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟=
∂Ax
∂x+
∂Ay
∂y+
∂Az
∂z
Curl(Cartesian coordinates)
• Curl of a vector A:
• Resulting vector quantity, use of partial derivatives
€
∇× r
A =
∂
∂x∂
∂y∂
∂z
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
×
Ax
Ay
Az
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟=
∂Az
∂y−
∂Ay
∂z∂Ax
∂z−
∂Az
∂x∂Ay
∂x−
∂Ax
∂y
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
Laplace operator(Cartesian coordinates)
• The Laplacian is the divergence of a gradient
• Resulting scalar quantity, use of second-order partial derivatives€
∇• ∇T( ) =∇ 2T =
∂
∂x∂
∂y∂
∂z
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
•
∂T
∂x∂T
∂y∂T
∂z
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
=∂ 2T
∂x 2+
∂ 2T
∂y 2+
∂ 2T
∂z2
Spherical coordinates• Longitude (E-W) defined from [0, 2]: • Latitude (N-S), defined from [-/2, /2]: • Height (vertical, radial outward): r
vr
Spherical velocity components• Spherical velocity components:
• r is the distance to the center of the earth, which is related to z by r = a + z
• a is the radius of the earth (a=6371 km), z is the distance from the Earth’s surface
€
u = rcosφDλ
Dt
v = rDφ
Dt
w =Dr
Dt=
D(a + z)
Dt≈
Dz
Dt
Unit vectors in spherical coordinates
• Unit vectors in longitudinal, latitudinal and vertical direction:
• Unit vectors in the spherical coordinate system depend on the location (, )
• As in the Cartesian system: They are orthogonal and normalized
€
ri =
−sinλ
cosλ
0
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
r j =
cosλ sinφ
sinλ sinφ
−cosφ
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
r k =
cosλ cosφ
sinλ cosφ
sinφ
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
Velocity vector in spherical coordinates• Use the unit vectors to write down a vector in
spherical coordinates, e.g. the velocity vector
• In a rotating system (Earth) the position of a point (and therefore the unit vectors) depend on time
• The Earth rotates as a solid body with the Earth’s angular speed of rotation = 7.292 10-5 s-1
• Leads to
€
rv = u
r i + v
r j + w
r k
€
⇒ δλ = Ωδt, δφ = 0, δz = 0
€
δ r
i
δt=
δr i
δλ
⎛
⎝ ⎜
⎞
⎠ ⎟δλ
δt
⎛ ⎝ ⎜
⎞ ⎠ ⎟ →
Dr i a
Dt=
Dr i
DλΩ
Time derivative of the spherical velocity vector
• Spherical unit vectors depend on time• This means that a time derivative of the velocity
vector in spherical coordinates is more complicated in comparison to the Cartesian system:
€
Dr v
Dt=
D
Dtu
r i + v
r j + w
r k ( )
=Du
Dt
r i +
Dv
Dt
r j +
Dw
Dt
r k + u
Dr i
Dt+ v
Dr j
Dt+ w
Dr k
Dt
€
ri ,
r j ,
r k
Spherical coordinates , , r
• Transformations from spherical coordinates to Cartesian coordinates:
€
x = rcosλ cosφ
y = rsinλ cosφ
z = rsinφ
Divergence of the wind field• Consider the divergence of the wind vector
• The divergence is a scalar quantity• If (positive), we call it divergence • If (negative), we call it convergence• If the flow is nondivergent • Very important in atmospheric dynamics
€
∇• r
v =
∂
∂x∂
∂y∂
∂z
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
•
u
v
w
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟=
∂u
∂x+
∂v
∂y+
∂w
∂z
€
rv =
u
v
w
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
€
∇• r
v > 0
€
∇• r
v < 0
€
∇• r
v = 0
Divergence of the wind field (2D)• Consider the divergence of the horizontal wind
vector
• The divergence is a scalar quantity
• If (positive), we call it divergence
• If (negative), we call it convergence
• If the flow is nondivergent
€
∇• r
v h =
∂
∂x∂
∂y
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟•
u
v
⎛
⎝ ⎜
⎞
⎠ ⎟=
∂u
∂x+
∂v
∂y
€
rv h =
u
v
⎛
⎝ ⎜
⎞
⎠ ⎟
€
∇• r
v > 0
€
∇• r
v < 0
€
∇• r
v = 0
http://www.atmos.washington.edu/2005Q1/101/CD/MAIN3.swfUnit 3, frame 17:
Vorticity, relative vorticity
• Vorticity of geophysical flows: curl of
(wind vector)
• relative vorticity: vertical component of
defined as
• Very important in atmospheric dynamics
€
rζ =∇×
rv =
∂
∂x∂
∂y∂
∂z
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
×
u
v
w
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟=
∂w
∂y−
∂v
∂z∂u
∂z−
∂w
∂x∂v
∂x−
∂u
∂y
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
€
ζ rel =r k •
r ζ =
∂v
∂x−
∂u
∂y
€
rζ
€
rv =
u
v
w
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
Relative vorticity
• Vertical component of
• Relative vorticity is
– (positive) for counterclockwise
rotation
– (negative) for clockwise rotation
– for irrotational flows
• Great web page: http://my.Meteoblue.Com/my/
€
ζ rel =r k •
r ζ =
∂v
∂x−
∂u
∂y
€
rζ
€
ζ rel > 0
€
ζ rel < 0
€
ζ rel = 0
Real weather situations500 hPa rel. vorticity and mean SLP
sea level pressure
Positive rel.vorticity, counterclock-wise rotation,in NH: lowpressure system
Divergent / convergent flow fields
• Compute the divergence of the wind vector
• Compute the relative vorticity
• Draw the flow field (wind vectors)
€
rv h =
1
2x
1
2y
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
x
y
Rotational flow fields
• Compute the rel. vorticity of the wind vector
• Compute the divergence
• Draw the flow field (wind vectors)
x
y
€
rv h =
−1
2y
1
2x
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
Taylor series expansion• It is sometimes convenient to estimate the value of a
continuous function f(x) about a point x = x0 with a power series of the form:
• In the last approximation, we neglected the higher order terms
€
f (x) = f (x0) +df (x0)
dx(x − x0) +
1
2!
d2 f (x0)
dx 2(x − x0)2
+ ...+1
n!
dn f (x0)
dx n(x − x0)n
≈ f (x0) +df (x0)
dx(x − x0)