Concept

Baroclinic — where it appears

Describing a fluid whose surfaces of constant pressure and constant density cross, so that pressure and gravity together exert a torque that creates vorticity. A baroclinic flow cannot keep its circulation constant, and the torque is what drives sea breezes, fronts and the vortex rings that form behind shocks.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Wound up, and worth exactly what it started with. A material loop in a steady cellular flow — an exact solution of Euler's equations — drawn at four times. Each streamline in the cell has its own period, so the loop is stretched steadily into a spiral: by the last frame its perimeter is 5.7 times what it started as. The circulation round it is 0.903741 at the start and 0.903666 at the end. Nothing about the curve survives except the number.

What survives being wound up

Draw a loop of marked fluid particles and let the flow carry it. It will be stretched, folded and wound into a spiral until nothing about its shape is recognisable, and the circulation round it will not have moved at all — provided three conditions hold, each of which can be broken on purpose.

inviscid · Kelvin

Over a day the breeze turns right round. The tip of the surface wind vector over one day, onshore to the right and along the coast upwards, at 15°, 30° and 45° north, with a spin-down time of twelve hours, in units of the push over the daily frequency. Without rotation the wind would swing on and off shore along one line. The Coriolis force turns it clockwise through the day into an ellipse; at 30°, where the Earth's inertial period is exactly a day, the turning keeps pace with the push and the ellipse is nearly a circle, of radius 1.8 against the ellipses' 1.2 at 15° and 1.2 at 45°.

At thirty degrees the Earth keeps time with the sea breeze

A sea breeze is Kelvin's circulation theorem failing: where warm air over land meets cool air over the sea, pressure and density surfaces cross and a circulation grows out of still air. Bjerknes' theorem prices the push, and the textbook's numbers give twenty-six metres a second in an hour — far more than any sea breeze blows. What stops it first is not friction but the evening: the push reverses before the wind can grow, and the day, not the drag, sets the breeze's size. Friction sets its hour, the Earth's rotation turns it through the day, and at thirty degrees, where the inertial period is a day, the turning keeps time with the push.

inviscid · Kelvin

Named alongside it

The objects these essays reach for when they reach for this one.

Circulation Model limit Rotating frame Barotropic Conservation Coriolis Eulerian and Lagrangian Inertial oscillation Inviscid Kelvin circulation theorem Kelvin's circulation theorem Material derivative

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