Boxology: the isobar–thickness box count as an exact advection integral, and its rendering

Evan James Lane 

"Boxology" is the hand-analysis practice of reading thermal advection from the angle at which mean-sea-level-pressure isobars cross 1000-500 hPa thickness contours: steep crossings cut the chart into small quadrilaterals and imply strong advection, parallel families imply none. This note shows the practice is an identity within geostrophic balance rather than a rule of thumb, and implements it as a rendered chart product. Splitting the layer-mean geostrophic wind into its bottom-surface and thermal parts removes the thermal wind exactly, since it blows along the thickness contours, so layer-mean thickness advection reduces to a Jacobian of the two plotted fields. Treating those fields as coordinates makes that Jacobian a change of variables, so the advection integrated over any quadrilateral bounded by consecutive isobars and consecutive thickness lines is a fixed quantum, independent of the box's shape, a specialization of the solenoid count in Bjerknes' circulation theorem. Box area and advection intensity are therefore reciprocal, which is what the eye is measuring. A renderer is described that shades the actual quadrilaterals at their own quantum-over-area rate, and the limits of the geostrophic premise are documented. In 48 GFS runs spanning a year, the box field correlates at 0.95 with the model's own full-wind thickness advection; about half of that advection, two thirds over the United States, is cancelled adiabatically, so the product is a diagnostic and not a forecast.