Vorticity and Circulation Questions and Answers

Answer: A

Frontogenesis is greatest when the dilatation axis is parallel to the isotherms, since the flow then converges most directly across the temperature gradient, and it changes sign beyond forty five degrees. Beyond that angle deformation acts to spread the isotherms apart instead. Knowing the critical angle allows a forecaster to judge from the flow pattern alone whether a temperature gradient is being sharpened or weakened. The quantity that varies and the quantity that does not are worth naming before any substitution is attempted.

Answer: B

The two by two horizontal velocity gradient tensor has exactly four independent components, conventionally taken as divergence, vertical vorticity and the two deformations. Every local distortion of a fluid element is a combination of these four. The vorticity equation terms describe how vorticity changes rather than how the flow is decomposed, and the Helmholtz split is a different and coarser division into two parts. Checking the units of each quantity catches a wrong formula before the arithmetic begins.

Answer: C

The decomposition describes the gradient of the velocity, so a uniform translation, which has zero gradient, contributes nothing to any of the four components. That is why the quantities are all frame-independent under a change of the moving observer. Rotation, area and shape are each altered by particular components, which is exactly what the decomposition sets out to quantify.

Answer: D

For solid body rotation the vorticity is uniform and equal to twice the angular velocity, which is why the planetary vorticity is \( 2\Omega\sin\phi \) rather than \( \Omega\sin\phi \). The factor of two catches many people out. Vorticity is uniform throughout a solid body rotation, so it certainly does not depend on radius, and it is emphatically not zero.

Answer: A

In a potential or irrotational vortex the tangential speed falls as one over the radius, and the curvature and shear contributions to vorticity cancel exactly everywhere except at the singularity. This is the outer flow of a bathtub vortex or a tornado. Solid body rotation has uniform non-zero vorticity, uniform flow has none anywhere including any centre, and a jet has shear vorticity on both flanks. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.

Answer: B

The Rankine combined vortex, with solid body rotation inside the radius of maximum wind and an irrotational profile outside it, is the standard idealisation and matches observation reasonably well. The vorticity is therefore concentrated in the core. An irrotational core would be singular at the axis, uniform vorticity would give solid body rotation at all radii, and annular structures occur transiently in some intense vortices but are not the basic model.

Answer: C

An inverse radius profile is the signature of irrotational flow, and it is observed in the outer parts of tornadoes and hurricanes. The final option is in fact an equivalent statement, since constant circulation with radius is exactly what zero vorticity outside the core implies, so it is a consequence rather than independent evidence. The pressure deficit follows from the balance and says nothing about the radial profile. Naming the cost of a choice alongside its benefit is what turns a fact into a design decision.

Answer: D

Kelvin's theorem gives conservation of circulation for the material loop threading the ring, which is why smoke rings persist so remarkably far. The speed decreases slowly as viscosity diffuses the core, the diameter grows as it does so, and the peak vorticity falls correspondingly. Circulation is the robust invariant, and its conservation is what holds the structure together for so long.

Answer: A

The descending air of a microburst spreads out at the surface and rolls up into a ring vortex, whose leading edge is the damaging gust front. The same structure appears in thunderstorm outflows generally. Fronts, jet streaks and Rossby waves are all essentially two-dimensional large-scale features with no ring structure, so none of them offers a comparable analogy.

Answer: B

An aircraft entering the outflow meets an increasing headwind, then a downdraught, then a tailwind, and the resulting loss of airspeed can exceed the aircraft's ability to recover close to the ground. Several major accidents established this. Icing and density changes are unrelated hazards, and radio contact is unaffected, so the wind shear is the whole of the danger.