VORTICITY AND CIRCULATION

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AMERICAN RESEARCH THOUGHTS ISSN: 2392 – 876X Available online at: www.researchthoughts.us http://dx.doi.org/10.6084/m9.figshare.1424786

Volume 1 │ Issue 7 │ May 2015

Impact Factor: 2.0178 (UIF)

VORTICITY AND CIRCULATION

Mohamed H. A. Elkanzi and Samah O. M. Elmaki

Department of Astronomy and Meteorology, Faculty of Science and Technology,

Omdurman Islamic University, Sudan

Abstract: Vorticity is a curl of fluid velocity and the Absolute ξ_a = ξ +f where ξ is the relative

vorticity while f is the coriolis parameter. North word moving air acquires and increasing value for f

and hence decreasing value for ξ. The concept of circulation is related that of vorticity, and it has a

number of important applications horizontal closed curves in either hemisphere will have positive

circulation vorticity is negative while the curvature is negative the vorticity is positive.

Key Words: vorticity, corioles parameter, cyclone, anticyclone

Vorticity defined as the curl of the fluid veloticity, x V; however, it remains to

determine the physical significance of this fluid property. Velocity may be expressed in

component form as:

x V = i + j + k

Where V = ui + vj + wk. Fundamentally, vorticity is an extension of the concept of

angular velocity of solid rotation and is essentially the angular velocity of a fluid

particle about a local axis through the particle.

For horizontal flow relative to the spherical earth, it is convenient to use the

expression for vorticity in spherical curvilinear coordinates.

= - +

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Mohamed H. A. Elkanzi and Samah O. M. Elmaki- VORTICITY AND CIRCULATION

1529 AMERICAN RESEARCH THOUGHTS- Volume 1 │ Issue 7 │2015

u eastwords component of the wind velocity over the spherical earth.

v north words component of the wind velocity over the spherical earth.

( y and x ) the corresponding displacement is the horizontal curvature of the

latitude circle and is the vorticity of uniform zonal flow.

However, a normally has the order of magnitude (10-5) per second, while =

- has the order (10-4) per second in regions of maximum vorticity. Thus, the term

usually may be omitted. The descriptive term cyclonic vorticity implies that the

resultant rotation in similar to that about a cyclone.

RELATIVE AND ABSOLUTE VORTICITY

Relative vorticity and absolute vorticity are obtained by using relative and absolute

velocity, respectively.

Va = V + x r

Va absolute velocity , V relative Velocity.

angular velocity of the earth

r position vector .

hence x Va = x V+ x ( x r).

Expanding the term gives

x Va = x V+ 2

Thus, the vector absolute vorticity is equal to the vector relative vorticity plus the

vector vorticity of the earth (twice the angular velocity) taking the dot product of the

both sides of the equation with K leads to:

= +f

K unit vector perpendicular to the plane.



Note that from this equation the absolute vorticity at any of the earth is equal to

the sum of relative vorticity and coriolis parameter. So absolute vorticity is conserved

following the flow. North word moving air acquires an increasing value for f and hence

deceasing value for , the air must curve ant cyclonically following a constant absolute

vorticity trajectory. Similarly, south word moving air must curve cyclonically.

Vorticity equation, is simplified form, and related to an isobaric surface, relates

the change of vorticity with time to divergence.

= -( +f) divp V

CIRCULATION AND ITS RELATIONSHIP TO VORTICITY

The concept of circulation is related to that of vorticity, and it has a number of

important applications. The circulation of the velocity vector "V" about a closed curves

Q is defined as:

C = Vs ds

Here ds are an element of arc length about the closed curve Q, and Vs is the tangential

component of the velocity vector. If Vs refers to the relative wind, C is called relative

circulation.

It is clear from C = Vs ds , that in either hemisphere:

1. The relative Circulation C is positive for cyclones

2. The relative Circulation C is negative for anticyclones

In the case of Horizontal Curves

= K- x Va, K - x V

K unit vector perpendicular to the plane.,

C = , and Ca = = CfA

Where f = 2 sin ; da is an element of area within the CurveQ.



Note that while C is Positive for both Northern and Southern Hemisphere, vorticity ( )

is negative, this follows since the area A is, by convention, negative when swept out

clockwise.

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VORTICITY AND CIRCULATION