Vorticity and Circulation Questions and Answers

Q: Which conclusion follows when the vorticity maximum becomes vertically stacked with the surface low?

Vertical alignment removes the westward tilt required for the conversion of available potential energy, so the system stops deepening and begins to decay under friction. This is the classical occlusion signature. Reinforcement requires the tilt rather than the alignment, and the movement of a system is governed by the steering flow, which does not reverse simply because the structure has become upright.

Q: Which term describes the wrapping of the vorticity maximum around the low centre?

Occlusion is the stage at which the cold front overtakes the warm front and the vorticity maximum becomes vertically aligned with the surface low. Cyclolysis is the subsequent decay, frontogenesis is the sharpening of a temperature gradient, and warm seclusion is a particular structure occurring in some very intense maritime cyclones. The terms describe successive or distinct stages and are worth keeping in order.

Q: Which relationship exists between vorticity and the curvature of a trajectory?

In natural coordinates the curvature contribution to vorticity is \( V/R \), where the square of the speed over the radius belongs to the centrifugal acceleration rather than to the vorticity. Confusing the two is a common error, since both involve speed and radius. The shear contribution completes the decomposition, and the Coriolis parameter enters absolute vorticity separately rather than through the curvature.

Q: Which estimate of vorticity does a strongly curved trough with 20 metres per second flow give?

Dividing 20 metres per second by a radius of \( 5 \times 10^{5} \) metres gives \( 4 \times 10^{-5} \) per second from curvature alone, which is a substantial fraction of the Coriolis parameter and a clearly significant vorticity maximum. Shear would add to this but the curvature term alone gives a useful estimate, which is exactly why the natural coordinate decomposition is so handy for reading a chart by eye.

Q: For an inviscid and barotropic ocean of constant depth (D), a water parcel with initial vorticity 2Ω is displaced from the equator to the north pole. Latitudinal variation of the parcel vorticity (ζ) is well represented by the curve

The curve is S.For an inviscid barotropic ocean of constant depth, potential vorticity conservation reduces to conservation of absolute vorticity, \(\zeta + f\), where \(f = 2\Omega\sin\phi\). At the equator f = 0, so \(\zeta + f = 2\Omega\) throughout, and\[ \zeta = 2\Omega(1 - \sin\phi) \]This falls from \(2\Omega\) at the equator to 0 at the pole. It falls fastest near the equator, where \(\sin\phi\) grows quickly, and flattens near the pole, so it bends below the straight line: curve S.R falls linearly with latitude, which would need \(\sin\phi\) to grow uniformly. Q keeps the relative vorticity constant, ignoring the change in f, and P adds f instead of subtracting it.

Q: A fluid is in solid body rotation in a cylindrical container of radius \(R\) rotating with an angular velocity \(\boldsymbol{\Omega} = (0, 0, \Omega)\). The circulation per unit area around a circular loop in the horizontal plane of radius \(r\) (\(r \lt R\)), whose center coincides with the axis of rotation is

The circulation per unit area is \(2\Omega\).In solid body rotation the speed at radius r is \(v = \Omega r\). Around the circle of radius r:\[ \begin{aligned} \Gamma &= \oint \mathbf{v}\cdot d\mathbf{l} = \Omega r \cdot 2\pi r = 2\pi\Omega r^{2} \\ \frac{\Gamma}{\pi r^{2}} &= 2\Omega \end{aligned} \]This is the vorticity of solid body rotation, twice the angular velocity (Stokes' theorem). \(\Omega/2\) and \(\Omega/4\) divide instead of multiply, and \(\Omega^{2}\) has the wrong dimensions for a vorticity.

Q: The vertical component of relative vorticity is given by :

The vertical component of relative vorticity is \(Y = \frac{\delta v}{\delta x} - \frac{\delta u}{\delta y}\) (the paper writes the vorticity as Y).Relative vorticity is the curl of the wind, and its vertical component, usually written \(\zeta\), is\[ \zeta = \partial v/\partial x - \partial u/\partial y \]where \(u\) and \(v\) are the eastward and northward wind components. It is positive for anticlockwise rotation in the northern hemisphere.The sum of \(\partial u/\partial x\) and \(\partial v/\partial y\) is the horizontal divergence, not vorticity. The difference \(\partial u/\partial x - \partial v/\partial y\) is a stretching deformation, and the sum of second derivatives is not a vorticity expression.

Q: The angular velocity of the Earth is \(\Omega\), then the vorticity of the earth is :

The vorticity of the Earth is 2 \(\Omega\).A body rotating as a solid with angular velocity \(\Omega\) has velocity \(\mathbf{v} = \boldsymbol{\Omega} \times \mathbf{r}\), and its vorticity is\[ \begin{aligned} \zeta &= \nabla \times (\Omega \times r) \\ &= 2\Omega \end{aligned} \]So the planetary vorticity is twice the angular velocity, and its local vertical component is \(2\Omega\sin\phi\), the Coriolis parameter f.Taking the vorticity equal to \(\Omega\) confuses vorticity with angular velocity, and the factors 3 and 1/2 have no basis in solid-body rotation.

101.

Which conclusion follows when the vorticity maximum becomes vertically stacked with the surface low?

A.Deepening has ceased
B.Deepening is about to accelerate because the two circulations are now reinforcing one another directly
C.The system is about to move rapidly
D.The system will begin to travel westward against the mean flow in which it has been embedded so far

Answer: A

Vertical alignment removes the westward tilt required for the conversion of available potential energy, so the system stops deepening and begins to decay under friction. This is the classical occlusion signature. Reinforcement requires the tilt rather than the alignment, and the movement of a system is governed by the steering flow, which does not reverse simply because the structure has become upright.

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102.

Which term describes the wrapping of the vorticity maximum around the low centre?

A.Seclusion, in which a pocket of warm air becomes cut off at the centre of a rapidly deepening storm
B.Occlusion
C.Frontogenesis, in which a deformation field concentrates the isotherms into a narrow sloping zone
D.Cyclolysis

Answer: B

Occlusion is the stage at which the cold front overtakes the warm front and the vorticity maximum becomes vertically aligned with the surface low. Cyclolysis is the subsequent decay, frontogenesis is the sharpening of a temperature gradient, and warm seclusion is a particular structure occurring in some very intense maritime cyclones. The terms describe successive or distinct stages and are worth keeping in order.

Discuss in Forum

103.

Which relationship exists between vorticity and the curvature of a trajectory?

A.Curvature makes no contribution
B.Curvature contributes an amount that depends on the latitude through the Coriolis parameter as well
C.Curvature contributes the speed divided by the radius of curvature
D.Curvature contributes the square of the speed divided by the radius of curvature of the path followed

Answer: C

In natural coordinates the curvature contribution to vorticity is \( V/R \), where the square of the speed over the radius belongs to the centrifugal acceleration rather than to the vorticity. Confusing the two is a common error, since both involve speed and radius. The shear contribution completes the decomposition, and the Coriolis parameter enters absolute vorticity separately rather than through the curvature.

Discuss in Forum

104.

Which estimate of vorticity does a strongly curved trough with 20 metres per second flow give?

A.About 4 times 10 to the power minus 3 per second, which would be comparable with a hurricane core
B.About 4 times 10 to the power minus 7 per second
C.A value that cannot be estimated without knowing the cross-stream shear of the wind at that location
D.About 4 times 10 to the power minus 5 per second for a radius of 500 kilometres

Answer: D

Dividing 20 metres per second by a radius of \( 5 \times 10^{5} \) metres gives \( 4 \times 10^{-5} \) per second from curvature alone, which is a substantial fraction of the Coriolis parameter and a clearly significant vorticity maximum. Shear would add to this but the curvature term alone gives a useful estimate, which is exactly why the natural coordinate decomposition is so handy for reading a chart by eye.

Discuss in Forum

105.

For an inviscid and barotropic ocean of constant depth (D), a water parcel with initial vorticity 2Ω is displaced from the equator to the north pole. Latitudinal variation of the parcel vorticity (ζ) is well represented by the curve

Vorticity zeta against latitude from 0 to 90 N, all curves starting at 2 Omega: P rises to 4 Omega, Q stays at 2 Omega, R falls linearly to 0 and S falls to 0 along a curve that bends below R and flattens near the pole

Exam Year(s)

GATE XE - 2018,

Answer: A

The curve is S.

For an inviscid barotropic ocean of constant depth, potential vorticity conservation reduces to conservation of absolute vorticity, \(\zeta + f\), where \(f = 2\Omega\sin\phi\). At the equator f = 0, so \(\zeta + f = 2\Omega\) throughout, and

\[ \zeta = 2\Omega(1 - \sin\phi) \]

This falls from \(2\Omega\) at the equator to 0 at the pole. It falls fastest near the equator, where \(\sin\phi\) grows quickly, and flattens near the pole, so it bends below the straight line: curve S.

R falls linearly with latitude, which would need \(\sin\phi\) to grow uniformly. Q keeps the relative vorticity constant, ignoring the change in f, and P adds f instead of subtracting it.

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106.

A fluid is in solid body rotation in a cylindrical container of radius \(R\) rotating with an angular velocity \(\boldsymbol{\Omega} = (0, 0, \Omega)\). The circulation per unit area around a circular loop in the horizontal plane of radius \(r\) (\(r \lt R\)), whose center coincides with the axis of rotation is

A.\(2\Omega\)
B.\(\Omega^{2}\)
C.\(\Omega/2\)
D.\(\Omega/4\)

Exam Year(s)

GATE XE - 2022,

Answer: A

The circulation per unit area is \(2\Omega\).

In solid body rotation the speed at radius r is \(v = \Omega r\). Around the circle of radius r:

\[ \begin{aligned} \Gamma &= \oint \mathbf{v}\cdot d\mathbf{l} = \Omega r \cdot 2\pi r = 2\pi\Omega r^{2} \\ \frac{\Gamma}{\pi r^{2}} &= 2\Omega \end{aligned} \]

This is the vorticity of solid body rotation, twice the angular velocity (Stokes' theorem). \(\Omega/2\) and \(\Omega/4\) divide instead of multiply, and \(\Omega^{2}\) has the wrong dimensions for a vorticity.

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107.

The vertical component of relative vorticity is given by :

A.\(Y = \frac{\delta u}{\delta x} + \frac{\delta v}{\delta y}\)
B.\(Y = \frac{\delta v}{\delta x} - \frac{\delta u}{\delta y}\)
C.\(Y = \frac{\delta u}{\delta x} - \frac{\delta v}{\delta y}\)
D.\(Y = \frac{\delta^{2} u}{\delta y^{2}} + \frac{\delta^{2} v}{\delta x^{2}}\)

Exam Year(s)

MH SET - 2018, SET Exam - 2018,

Answer: B

The vertical component of relative vorticity is \(Y = \frac{\delta v}{\delta x} - \frac{\delta u}{\delta y}\) (the paper writes the vorticity as Y).

Relative vorticity is the curl of the wind, and its vertical component, usually written \(\zeta\), is

\[ \zeta = \partial v/\partial x - \partial u/\partial y \]

where \(u\) and \(v\) are the eastward and northward wind components. It is positive for anticlockwise rotation in the northern hemisphere.

The sum of \(\partial u/\partial x\) and \(\partial v/\partial y\) is the horizontal divergence, not vorticity. The difference \(\partial u/\partial x - \partial v/\partial y\) is a stretching deformation, and the sum of second derivatives is not a vorticity expression.

Discuss in Forum

108.

The angular velocity of the Earth is \(\Omega\), then the vorticity of the earth is :

A.\(\Omega\)
B.2 \(\Omega\)
C.3 \(\Omega\)
D.1/2 \(\Omega\)

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: B

The vorticity of the Earth is 2 \(\Omega\).

A body rotating as a solid with angular velocity \(\Omega\) has velocity \(\mathbf{v} = \boldsymbol{\Omega} \times \mathbf{r}\), and its vorticity is

\[ \begin{aligned} \zeta &= \nabla \times (\Omega \times r) \\ &= 2\Omega \end{aligned} \]

So the planetary vorticity is twice the angular velocity, and its local vertical component is \(2\Omega\sin\phi\), the Coriolis parameter f.

Taking the vorticity equal to \(\Omega\) confuses vorticity with angular velocity, and the factors 3 and 1/2 have no basis in solid-body rotation.

Discuss in Forum

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