AOSS 321, Winter 2009 Earth System Dynamics

Lecture 41/20/2009

Christiane Jablonowski Eric Hetland

cjablono@umich.edu ehetland@umich.edu

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)