An Analytical Solution of the Effective-Buoyancy Equation

Abstract

Abstract The effective buoyancy per unit volume is the statically forced part of the local nonhydrostatic upward pressure-gradient force. It is important because it does not depend on the basic-state density defined with the anelastic approximation. Herein, an analytical solution is obtained for the effective buoyancy associated with an axisymmetric column of less dense air. In special cases where the radial profiles of density are step functions, the analytical solutions replicate qualitatively several features in a recently published numerical solution as follows. The effective buoyancy is positive within the column of lighter air and negative outside. It increases from the axis to the inner edge of the column, then jumps discontinuously to a negative value and thereafter increases until it reaches zero at radial infinity. As the column radius increases, the effective buoyancy on the axis decreases and the change in effective buoyancy between the axis and the inner edge increases, but the jump magnitude is unaltered. For continuous radial density distributions that resemble step functions, the solutions are similar except the cusps are rounded off and the jumps become smooth transition zones. Significance Statement In atmospheric convection, vertical accelerations are due to buoyancy forces and vertical perturbation pressure-gradient forces. Separately, these forces depend on the choice of a basic state. To avoid the ambiguity of an arbitrary reference atmosphere defining which parcels are buoyant, we define an effective-buoyancy force per unit volume that is independent of any basic state. It is the part of the vertical nonhydrostatic pressure-gradient force that depends solely on horizontal density variations. The remaining part of the vertical force is dynamical in origin; it depends only on inertial forces. An analytical solution demonstrates that, for an axisymmetric column of lighter air, effective buoyancy is greatest just inside the column edge and is most negative just outside the edge.