Vorticity and Circulation Questions and Answers

Answer: A

Vorticity is the curl of the velocity, a vector whose vertical component is what matters for large-scale atmospheric flow. It measures the local rotation of a fluid element about its own axis, and it is a point quantity rather than an integral one. Divergence is the dot product of the gradient operator with the velocity, circulation is the line integral of velocity around a closed loop, and deformation describes the straining of an element without rotation. The four together provide a complete local description of how a fluid element is being distorted by the flow around it.

Answer: B

Because large-scale motion is nearly horizontal, the rotation that matters is about a vertical axis, so only the vertical component of the curl appears in the synoptic vorticity equation. The horizontal components describe rotation about horizontal axes, which is important in convective storms and in the tilting term but not in the leading order balance. Streamwise vorticity, the projection along the flow, is central to supercell dynamics and is a different combination again.

Answer: C

Relative vorticity is the curl of the wind as measured in the rotating frame, so it describes the rotation of the air relative to the ground beneath. Adding the planetary vorticity, which is the Coriolis parameter, gives the absolute vorticity, and dividing that by a measure of layer depth or stratification gives potential vorticity. Keeping the four straight is essential because each is conserved under different conditions and confusing them leads to wrong conclusions.

Answer: D

The planetary vorticity is the vertical component of twice the Earth's rotation vector, which is exactly \( f = 2\Omega\sin\phi \). It is therefore zero at the equator and about \( 1.45 \times 10^{-4} \) per second at the pole. Since typical relative vorticity in a synoptic system is around \( 10^{-5} \) per second, planetary vorticity usually dominates in mid-latitudes, which has important consequences for the conservation arguments that follow. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Answer: A

Absolute vorticity is the vorticity seen from an inertial frame, which is the relative vorticity plus the planetary vorticity. It is the quantity conserved for frictionless barotropic flow without divergence, and it is what makes air moving poleward turn anticyclonically. The final option is a correct statement of the relationship between circulation and mean vorticity, but it defines mean vorticity rather than absolute vorticity specifically. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.

Answer: B

Anticlockwise rotation gives positive vorticity in the standard convention, and cyclonic means anticlockwise in the northern hemisphere, so cyclonic vorticity is positive there. In the southern hemisphere cyclonic means clockwise and therefore negative vorticity, which is why the word cyclonic is safer than any statement about sign. A circular flow certainly has vorticity, since the fluid elements themselves are rotating as they go round. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Answer: C

A wind of 10 metres per second varying over a thousand kilometres gives a shear of \( 10^{-5} \) per second, which is the characteristic relative vorticity of a synoptic system. That is about a tenth of the Coriolis parameter, which is another way of stating that the Rossby number is small. Tornadoes reach \( 10^{-1} \) per second and dust devils more, so the range across atmospheric phenomena spans several orders of magnitude. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.

Answer: D

Written in natural coordinates, vorticity is the sum of a curvature term, the speed divided by the radius of curvature, and a shear term, minus the cross-stream derivative of the speed. This decomposition is extremely useful in synoptic analysis because it lets vorticity be estimated by eye from the shape and spacing of the streamlines. Divergence and deformation are the other two components of the velocity gradient tensor. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

Answer: A

A perfectly straight jet has cyclonic shear vorticity on one flank and anticyclonic on the other, even though no streamline is curved. This is why the northern flank of a westerly jet in the northern hemisphere is a region of positive vorticity. Uniform flow has neither shear nor curvature, and along-flow acceleration produces divergence rather than vorticity, which is a different component of the velocity gradient. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Answer: B

On the poleward side of a westerly jet the wind increases towards the core as one moves south, and the resulting shear gives cyclonic vorticity there. The equatorward flank has the opposite sign. This is why cyclogenesis is favoured on the poleward side of a jet and why the vorticity maximum on a chart so often lies along that flank rather than at the wind maximum itself.