  When studying tropical cyclones using the $f$-plane, axisymmetric, gradient
balanced model, there arises a second-order elliptic equation for the
transverse circulation. Similarly, when studying zonally symmetric meridional
circulations near the equator (the tropical Hadley cells) or the katabatically
forced meridional circulation over Antarctica, there also arises a second order
elliptic equation. These elliptic equations are usually derived in the pressure
coordinate or the potential temperature coordinate, since the thermal wind
equation has simple non-Jacobian forms in these two vertical coordinates.
Because of the large variations in surface pressure that can occur in tropical
cyclones and over the Antarctic ice sheet, there is interest in using other
vertical coordinates, e.g., the height coordinate, the classical
$\sigma$-coordinate, or some type of hybrid coordinate typically used in global
numerical weather prediction or climate models. Because the thermal wind
equation in these coordinates takes a Jacobian form, the derivation of the
elliptic transverse circulation equation is not as simple. Here we present a
method for deriving the elliptic transverse circulation equation in a
generalized vertical coordinate, which allows for many particular vertical
coordinates, such as height, pressure, log-pressure, potential temperature,
classical $\sigma$, and most hybrid cases. Advantages and disadvantages of the
various coordinates are discussed.
