Kinematics of the horizontal wind field
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Kinematics of the horizontal wind field. (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). y. N. V. v. W. E. u. x. S. To derive a mathematical expression for the

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Kinematics of the horizontal wind field

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Kinematics of the horizontal wind field

Kinematics of the horizontal wind field

(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)


Kinematics of the horizontal wind field

y

N

V

v

W

E

u

x

S

To derive a mathematical expression for the

key kinematic properties of the wind field

we will use the coordinate system on the right.

y

We will use Taylor Expansion

to estimate the wind field at an arbitrary point

x,y from the wind at a nearby point x0, y0

x, y

x0, y0


Kinematics of the horizontal wind field

Peform a 2D Taylor expansion:

For simplicity, lets assume that x0, y0 is the origin 0,0

And that we can obtain an adequate estimate of u,v by retaining

only the first derivatives. We are assuming that over the small distance

the u and v field vary linearly. Then…


Kinematics of the horizontal wind field

Let’s take a simple step and write each derivative term as (for example) :


Kinematics of the horizontal wind field

From before:

(1)

(2)

Now we will write two nonsense equations

(3)

(4)

Now we add (1) and (3). We also separately add (2) and (4). Then we

rearrange the terms and get…………


Kinematics of the horizontal wind field

Translation

Divergence

Shearing

Deformation

Relative

Vorticity

Stretching

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.


Kinematics of the horizontal wind field

y

x

Translation

The effect of translation on a fluid element:

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


Kinematics of the horizontal wind field

y

Divergence (d > 0)

Convergence (d < 0)

The effect of convergence on a fluid element:

x

Change in area, no change in

orientation, shape, location


Kinematics of the horizontal wind field

y

Positive (cyclonic) vorticity ( > 0).

Negative (anticyclonic) vorticity ( < 0)

The effect of negative vorticity on a

fluid element:

x

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


Kinematics of the horizontal wind field

y

E-W Stretching Deformation (D1 > 0).

N-S Stretching Deformation (D1 < 0).

The effect of stretching deformation on a

fluid element:

x

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


Kinematics of the horizontal wind field

y

SW-NE Shearing Deformation (D1 > 0).

NW-SE Shearing Deformation (D1 < 0).

The effect of shearing deformation on a

fluid element:

x

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


Kinematics of the horizontal wind field

Why are we interested in these properties?

Net Divergence in an air column leads to the development of low surface pressure

Net Convergence in an air column leads to the development of high surface pressure

L

H


Kinematics of the horizontal wind field

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.

z

< 0

z

> 0

z

< 0

z

> 0


Kinematics of the horizontal wind field

Absolute vorticity allows us to identify short waves and shear zones within the

jetstream. Short waves trigger cyclogenesis and can help trigger deep

convection in the warm season.


Kinematics of the horizontal wind field

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


Kinematics of the horizontal wind field

T- 8DT

T- 8DT

T- 7DT

T- 7DT

T- 6DT

T- 6DT

T- 5DT

T- 5DT

T- 4DT

T- 4DT

T- 3DT

T- 3DT

T- 2DT

T- 2DT

T- DT

T- DT

T

T

Deformation flow is fundamental to the development of fronts

Time = t + Dt

Time = t

y

y

x

x


Kinematics of the horizontal wind field

EXAMPLES OF DEFORMATION

Axis of Dilitation


Kinematics of the horizontal wind field

EXAMPLES OF DEFORMATION

Axis of Dilitation


Kinematics of the horizontal wind field

CONFLUENT and DIFLUENT FLOW

Is this flow convergent?

Is this flow divergent?

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


Kinematics of the horizontal wind field

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.


Kinematics of the horizontal wind field

Suppose the wind is southerly and

uniform. Is the wind convergent?

Red = wind

Blue = wind component

y

Yes!

y

y

x

x

x

Convergence of meridians toward

north leads to convergence. This

is the earth curvature term (the last term)

in the expression for convergence (d).


Kinematics of the horizontal wind field

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 ().


Kinematics of the horizontal wind field

In a similar way, convergence of the earth’s meridians toward

the north leads to deformation in otherwise uniform flow

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.


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