Vorticity and Circulation Questions and Answers

Answer: C

Divergence measures the fractional rate of increase of the area of a horizontal fluid element, so positive divergence means the element is spreading out. Rotation is vorticity, circulation is the loop integral of velocity, and shape change at constant area is deformation. Together these four describe every way a fluid element can be affected by the local velocity gradient.

Answer: D

Surface pressure measures the weight of the overlying column, so if more mass leaves the top than enters at the bottom the column lightens and the pressure falls. This is the mass budget argument behind the deepening of surface lows beneath upper divergence. Static stability affects the vertical motion profile but not the sign of this relationship.

Answer: A

In a typical developing system convergence occupies the lower troposphere and divergence the upper, with the change of sign at a level of non-divergence around 500 to 600 hectopascals. That level is where the barotropic vorticity equation is most nearly valid, which is why 500 hectopascals was chosen for the first numerical forecasts. The tropopause is well above the divergence maximum rather than at the transition. A single measurement chosen to respond to the one asymmetry beats several that respond to shared behaviour.

Answer: B

The barotropic model assumes non-divergent flow, so it is most nearly valid at the level where the divergence actually vanishes, and 500 hectopascals is close to that level. It is also the level with half the atmospheric mass beneath it, which reinforces the choice. The strongest winds are near 250 hectopascals, and observational convenience did not drive the decision.

Answer: C

Convergence in the frictional boundary layer forces air upward out of it, and the associated divergence within the layer reduces the vorticity, spinning the vortex down over about four days. The inward turning of the wind is the mechanism by which this happens rather than a source of vorticity. Planetary and relative vorticity are exchanged by the beta effect rather than by friction.

Answer: D

The spin-down time is the depth of the troposphere divided by the Ekman pumping velocity, which works out at roughly 4 days for typical mid-latitude values. That is far shorter than molecular diffusion would give and matches the observed lifetime of decaying systems, which is the empirical check that makes the theory convincing. Boundary layer adjustment and turbulent mixing operate on much shorter timescales. The shared properties explain why these belong to one family, and the differences explain why more than one exists.

Answer: A

The vertical velocity at the top of the Ekman layer is proportional to the geostrophic relative vorticity multiplied by the square root of the eddy viscosity divided by twice the Coriolis parameter. So a cyclonic vortex pumps upward and an anticyclonic one pumps downward. Temperature does not enter, and the dependence is on the curl of the wind rather than on its speed or on the gradient directly. Most errors here come from carrying a result across from a neighbouring case where it does not hold.

Answer: B

Frictional turning makes the flow diverge outward from an anticyclone at the surface, so mass must descend into the boundary layer to replace it, giving subsidence above. That subsidence warms adiabatically and suppresses cloud, which is why anticyclones bring settled weather. The vertical velocity is proportional to the relative vorticity, so it is uniform in sign around a circular vortex rather than alternating, and its magnitude is largest where the vorticity is greatest.

Answer: C

Deformation has two components, stretching and shearing, and together they describe how a fluid element is distorted in shape without any change of area or any net rotation. It is the component of the velocity gradient tensor responsible for frontogenesis, since it can concentrate isotherms into a narrow zone. Divergence changes area, vorticity produces rotation, and circulation is an integral measure rather than a local one. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Answer: D

A deformation field whose axis of dilatation lies at less than forty five degrees to the isotherms concentrates them, sharpening the temperature gradient into a front. This is the classical frontogenesis mechanism and it is why fronts form preferentially in confluent flow regions such as cols. Vertical mixing and radiation are thermodynamic processes, and the energy conversion is the consequence of the ageostrophic circulation that frontogenesis forces.