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

(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

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


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…



From before: (for example) :

(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…………


Translation (for example) :

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.


y (for example) :

x

Translation

The effect of translation on a fluid element:

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


y (for example) :

Divergence (d > 0)

Convergence (d < 0)

The effect of convergence on a fluid element:

x

Change in area, no change in

orientation, shape, location


y (for example) :

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


y (for example) :

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


y (for example) :

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


Why are we interested in these properties? (for example) :

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


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


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.


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.


T- 8 a proxy for divergence aloft, and is related to the development of low surface pressure and upward air motion.DT

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


EXAMPLES OF DEFORMATION a proxy for divergence aloft, and is related to the development of low surface pressure and upward air motion.

Axis of Dilitation


EXAMPLES OF DEFORMATION a proxy for divergence aloft, and is related to the development of low surface pressure and upward air motion.

Axis of Dilitation


CONFLUENT and DIFLUENT FLOW a proxy for divergence aloft, and is related to the development of low surface pressure and upward air motion.

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.


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.


Suppose the wind is southerly 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.

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).


Suppose the wind is westerly 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.

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 ().


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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