The Energy Balance, Available Potential Energy, and Rossby Waves in a Baroclinic Atmosphere

Abstract

We note one more important conclusion which follows from comparison of the energy relations. The ratio of the available potential energy to the kinetic one has the order of magnitude $L^{2}/L_{0}^{2}$ for barotropic flows and $L^{2}/L_{R}^{2}$ for baroclinic ones, where the quantity

$$ L_{R} \doteq \frac{NH_{0}}{f}=\frac{\sqrt{g^{\prime}H_{0}}}{f}\quad \bigl(g^{\prime}=N^{2}H_{0} \bigr) $$

of the length dimension is called the internal Rossby deformation radius. In the stability theory of baroclinic flows this fundamental parameter plays the role which is as important as that of $L_{0}=\sqrt{gH_{0}}/f$ in the stability theory of barotropic motions. In particular, L _R is the typical size of cyclones and anticyclones generated by instability of the vertical shear of the wind, i.e., by the instability of an inhomogeneous horizontal distribution of buoyancy forces, according to the thermal wind relation (9.37). Note that for L=L _R , the kinetic and potential energies of the vortex have the same order of magnitude. For the Earth’s atmosphere (or ocean) L _R and L ₀ are quantities of the same order of magnitude, equal to 1000 km (or 100 km respectively), although the square of their ratio (L _R /L ₀)² is usually assumed to be equal to 0.1 with some reservations. This circumstance is related to one of the main difficulties in constructing an analytical theory of the general atmosphere and ocean circulation.