03 kinematics of horizontal wind fieldsnesbitt/atms505/stuff... · kinematics of the wind field in...

35
Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an intuitive understanding of the kinematic properties of the wind field and 2) learn how these kinematic properties can be used to understand the evolution of weather systems See Bluestein Vol. I, Sections 3, 5.5 for more information (Kinematics: from the Greek word for ‘motion’, a description of the motion of a particular field without regard to how it came about or how it will evolve)

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Page 1: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Kinematics of the wind field

In this section of the course, our goal will be to

1)  develop a mathematical and an intuitive understanding of the kinematic properties of the wind field

and

2) learn how these kinematic properties can be used to understand the evolution of weather systems

See Bluestein Vol. I, Sections 3, 5.5 for more information

(Kinematics: from the Greek word for ‘motion’, a description of the motion of a particular field without regard to how it came about or how it will evolve)

Page 2: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

y

N

W

S

E

x

V vj

ui

To derive a mathematical expression for the key kinematic properties of the wind field we will use the coordinate system on the right.

y

x0, y0

x, y

We will use a technique called Taylor Expansion to estimate the wind field at an arbitrary point x,y from the wind at a nearby point x0, y0

Page 3: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

yyu

xvx

yv

xuy

yu

xvx

yv

xuuu ⎟⎟

⎞⎜⎜⎝

∂+

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂−⎟⎟

⎞⎜⎜⎝

∂+

∂+=

21

21

21

21

0

xyu

xvy

yv

xux

yu

xvy

yv

xuvv ⎟⎟

⎞⎜⎜⎝

∂+

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂−⎟⎟

⎞⎜⎜⎝

∂−

∂+⎟⎟

⎞⎜⎜⎝

∂+

∂+=

21

21

21

21

0

Divergence Relative Vorticity

Stretching Deformation

Translation Shearing Deformation

Any wind field that varies linearly can be characterized by these five distinct properties. Non-linear wind fields can be closely characterized by these properties.

Page 4: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

x

y

yyu

xvx

yv

xuy

yu

xvx

yv

xuuu ⎟⎟

⎞⎜⎜⎝

∂+

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂−⎟⎟

⎞⎜⎜⎝

∂+

∂+=

21

21

21

21

0

xyu

xvy

yv

xux

yu

xvy

yv

xuvv ⎟⎟

⎞⎜⎜⎝

∂+

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂−⎟⎟

⎞⎜⎜⎝

∂−

∂+⎟⎟

⎞⎜⎜⎝

∂+

∂+=

21

21

21

21

0

Translation

The effect of translation on a fluid element:

Change in location, no change in area, orientation, shape

Page 5: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

x

y

u = u

0+

12∂u∂x

+∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟x −

12∂v∂x−∂u∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟y +

12∂u∂x−∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟x +

12∂v∂x

+∂u∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟y

v = v

0+

12∂u∂x

+∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟y +

12∂v∂x−∂u∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟x −

12∂u∂x−∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟y +

12∂v∂x

+∂u∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟x

Divergence (δ > 0) Convergence (δ < 0)

The effect of convergence on a fluid element:

Change in area, no change in orientation, shape, location

convergence

Page 6: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

x

y

yyu

xvx

yv

xuy

yu

xvx

yv

xuuu ⎟⎟

⎞⎜⎜⎝

∂+

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂−⎟⎟

⎞⎜⎜⎝

∂+

∂+=

21

21

21

21

0

xyu

xvy

yv

xux

yu

xvy

yv

xuvv ⎟⎟

⎞⎜⎜⎝

∂+

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂−⎟⎟

⎞⎜⎜⎝

∂−

∂+⎟⎟

⎞⎜⎜⎝

∂+

∂+=

21

21

21

21

0

Positive (cyclonic) vorticity (ζ > 0). Negative (anticyclonic) vorticity (ζ < 0)

The effect of negative vorticity on a fluid element:

Change in orientation, no change in area, shape, location

anticyclonic vorticity

Page 7: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

x

y

yyu

xvx

yv

xuy

yu

xvx

yv

xuuu ⎟⎟

⎞⎜⎜⎝

∂+

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂−⎟⎟

⎞⎜⎜⎝

∂+

∂+=

21

21

21

21

0

xyu

xvy

yv

xux

yu

xvy

yv

xuvv ⎟⎟

⎞⎜⎜⎝

∂+

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂−⎟⎟

⎞⎜⎜⎝

∂−

∂+⎟⎟

⎞⎜⎜⎝

∂+

∂+=

21

21

21

21

0

E-W Stretching Deformation (D1 > 0). N-S Stretching Deformation (D1 < 0).

The effect of stretching deformation on a fluid element:

Change in shape, no change in area, orientation, location

Axis of

dilatation

Axi

s of

cont

ract

ion

E-W stretching deformation

Page 8: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

x

y

yyu

xvx

yv

xuy

yu

xvx

yv

xuuu ⎟⎟

⎞⎜⎜⎝

∂+

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂−⎟⎟

⎞⎜⎜⎝

∂+

∂+=

21

21

21

21

0

xyu

xvy

yv

xux

yu

xvy

yv

xuvv ⎟⎟

⎞⎜⎜⎝

∂+

∂+⎟⎟

⎞⎜⎜⎝

∂−

∂−⎟⎟

⎞⎜⎜⎝

∂−

∂+⎟⎟

⎞⎜⎜⎝

∂+

∂+=

21

21

21

21

0

SW-NE Shearing Deformation (D2 > 0). NW-SE Shearing Deformation (D2 < 0).

The effect of shearing deformation on a fluid element:

Change in shape, no change in area, orientation, location

NW-SE shearing deformation

Page 9: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Why are we interested in these properties?

We can use and divergence/convergence and the continuity equation to diagnose synoptic scale vertical motion within a column.

Vertical motion is what we are interested in in weather forecasting!

L H

Page 10: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

The continuity equation and vertical motion

Since synoptic scale vertical motions are difficult to observe, a conceptual model of vertical motions and circulations is needed to properly diagnose synoptic scale vertical motion.

The continuity equation states that

where &

We can also write

where &

∇ • V = 0 ∇ =

∂∂x

i +∂∂y

j +∂∂p

k

H• V

H+∂ω∂p

= 0 ∇

H=∂∂x

i +∂∂y

j

V = ui + vj + ωk

VH= ui + vj

Page 11: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Note &

From the previous page

Let’s diagnose the vertical motion (ω) by examining the continuity equation. Integrating the above downward from the top of the atmosphere (p0) to some pressure level p yields

At p0 (top of atmosphere), p=0 and ω=Dp/Dt=0 (no vertical motion), so

∇ • V =

∂u∂x

+∂v∂y

+∂ω∂p

∇H

• VH

=∂u∂x

+∂v∂y

3-D divergence Horizontal divergence

∂u∂x

+∂v∂y

+∂ω∂p

= 0

ω(p)−ω(p0) = −

∂u∂x

+∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟

p0

p

∫ ∂p

ω(p) = −∂u∂x

+∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟

p0

p

∫ ∂p

Page 12: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

ω(p) = −∂u∂x

+∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟

p0

p

∫ ∂p

What does this equation tell us?

It says that omega at a given pressure level is proportional to the integral of divergence or convergence above a given pressure level.

0

p

pressure

δ + –

0

p

pressure

ω + –

Page 13: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Assuming a mean divergence, we may solve the integral and write

Since p0=0, we obtain

Interpreting this result: 1) If we have mean divergence aloft,

>0, since p>0, then ω(p)<0, thus rising motion.

2) If we have mean convergence aloft,

<0, since p>0, then ω(p)>0, thus subsidence.

ω(p) = −

∂u∂x

+∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟

p− p0( )

ω(p) = −

∂u∂x

+∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟p

∂u∂x

+∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟

∂u∂x

+∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟

Page 14: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Now, considering that ω=0 at the surface and at the tropopause (a good approximation at synoptic scales unless there is a strongly sloped surface), then the continuity equation tells us that the sign of divergence must change at least once in the column given these boundary conditions. This can be shown by rearranging the continuity equation:

Since w must return to 0 at the boundaries, then the sign of the left side of the equation must change at least once in the equation, meaning that divergence must be 0 at some point in the profile. This level is known as a level of non-divergence (LND) since there is no divergence or convergence. At LND, there is a local maximum or minimum of ω ( ).

From synoptic experience, the LND exists in the mid-troposphere, around 500-600 hPa.

∂u∂x

+∂v∂y

⎝⎜⎜⎜⎜

⎠⎟⎟⎟⎟⎟

= −∂ω∂p

∂ω∂p

= 0

Page 15: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

This has several implications:

• Except over sloping terrain, the divergence must change signs at least once in a column (even over sloping terrain this is common). • There is a maximum or minimum in ω at the LND. • Rising motion is almost always accompanied by divergence aloft and convergence below • Subsidence is almost always accompanied by convergence aloft and divergence below.

Page 16: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

VORTICITY

Vertical vorticity (spin about a vertical axis) arises from three sources: Horizontally sheared flow, flow curvature, and the rotation of the earth.

Relative vorticity: shear and curvature.

Absolute vorticity: shear, curvature and earth rotation. fyu

xv

+∂

∂−

yu

xv

∂−

ζ > 0

ζ < 0 ζ > 0

ζ < 0

Page 17: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Absolute vorticity allows us to identify short waves and shear zones within the jetstream. Short waves trigger cyclogenesis and, in the warm season, can help trigger deep convection.

Page 18: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Positive Vorticity Advection on a 500 mb map can be used as a proxy for the development of low surface pressure and upward air motion.

Page 19: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

3D relative vorticity vector

Cartesian expression for vorticity

Vertical component of vorticity vector (rotation in a horizontal plane

Absolute vorticity (flow + earth’s vorticity)

Absolute vorticity

Page 20: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

The vorticity equation in height coordinates

(1) (2)

Expand total derivative

Page 21: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Rate of change of relative vorticity Following parcel

Divergence acting on Absolute vorticity (twirling skater effect)

Tilting of vertically sheared flow

Gradients in force Of friction

Pressure/density solenoids

Page 22: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Rate of change of relative vorticity Following parcel

Divergence acting on Absolute vorticity (twirling skater effect)

Tilting of vertically sheared flow

Gradients in force Of friction

Pressure/density solenoids

Page 23: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Rate of change of relative vorticity Following parcel

Divergence acting on Absolute vorticity (twirling skater effect)

Tilting of vertically sheared flow

Gradients in force Of friction

Pressure/density solenoids

geostrophic wind

Cold advection pattern

m (or ρ) large acceleration small

m (or ρ) small acceleration large

Solenoid: field loop that converts potential energy to kinetic energy

Page 24: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Rate of change of relative vorticity Following parcel

Divergence acting on Absolute vorticity (twirling skater effect)

Tilting of vertically sheared flow

Gradients in force Of friction

Pressure/density solenoids

Geostrophic wind = constant

N-S wind component due to friction

Page 25: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

The vorticity equation in pressure coordinates

(1) (2)

Expand total derivative

Page 26: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Local rate of change of relative vorticity

Horizontal advection of absolute vorticity on a pressure surface

Vertical advection of relative vorticity

Divergence acting on Absolute vorticity (twirling skater effect)

Tilting of vertically sheared flow

Gradients in force Of friction

The vorticity equation

In English: Horizontal relative vorticity is increased at a point if 1) positive vorticity is advected to the point along the pressure surface, 2) or advected vertically to the point, 3) if air rotating about the point undergoes convergence (like a skater twirling up),

4) if vertically sheared wind is tilted into the horizontal due a gradient in vertical motion 5) if the force of friction varies in the horizontal.

Solenoid terms disappear in pressure coordinates: we will work in P coordinate from now on

Page 27: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Deformation flow is fundamental to the development of fronts

x

y

x

y Time = t Time = t + Δt

T

T- ΔΤ

T- 2ΔT

T- 3ΔT

T- 4ΔT

T- 5ΔT

T- 6ΔT

T- 7ΔT

T- 8ΔT

T T- ΔΤ T- 2ΔT T- 3ΔT

T- 4ΔT T- 5ΔT

T- 6ΔT T- 7ΔT T- 8ΔT

Axis of dilatation

Page 28: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

EXAMPLES OF DEFORMATION

Axis of Dilatation

Page 29: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Axis of Dilitation

EXAMPLES OF DEFORMATION

Page 30: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

CONFLUENT and DIFLUENT FLOW

Is this flow convergent?

NO: The areas of the two boxes are identical. The flow is a combination of translation and deformation.

Is this flow divergent?

Difluent flow Confluent flow

Page 31: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

The terms for divergence, relative vorticity, and deformation strictly apply on a plane tangent to the earth’s surface. If we take earth’s curvature into account, we have to add an additional term.

Page 32: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an
Page 33: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Red = wind Blue = wind component

y y y

x x x

Suppose the wind is southerly and uniform. Is the wind convergent?

Yes!

φδ tanav

yv

xu

+∂

∂+

∂=

Convergence of meridians toward north leads to convergence. This is the earth curvature term (the last term) in the expression for convergence (δ).

Page 34: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

Suppose the wind is westerly and uniform. Does vorticity exist?

Yes!

Convergence of meridians toward north creates vorticity. This is the earth curvature term (the last term) in the expression for vorticity (ζ).

φζ tanau

yu

xv

+∂

∂−

∂=

Page 35: 03 Kinematics of horizontal wind fieldsnesbitt/ATMS505/stuff... · Kinematics of the wind field In this section of the course, our goal will be to 1) develop a mathematical and an

In a similar way, convergence of the earth’s meridians toward the north leads to deformation in otherwise uniform flow

φtan1 av

yv

xuD −

∂−

∂=

φtan2 au

yu

xvD +

∂+

∂=

Earth’s curvature terms are an order of magnitude smaller than other terms, but cannot be ignored in models, at least in the middle and high latitudes.