### Inertial Oscillations and Phugoid Flight

The English aviation pioneer Frederick Lanchester (1868–1946) introduced many important contributions to aerodynamics. He analysed the motion of an aircraft under various consitions of lift and drag. He introduced the term “phugoid” to describe aircraft motion in which the aircraft alternately climbs and descends, varying about straight and level flight. This is one of the basic modes of aircraft dynamics, and is clearly illustrated by the flight of gliders.

Glider in phugoid loop [photograph by Dave Jones on website of Dave Harrison]

Phugoid Motion and Inertial Oscillations

A mathematical analysis shows that there is a close similarity between the dynamics of phugoid flight and the behaviour of atmospheric and oceanic motion known as inertial oscillations. In both cases, there is a steady state motion about which the motion varies periodically. In the simplest cases, the trajectories are trochoids (cycloids or curtate and prolate cycloids).

Geostrophic balance (left) and straight-and-level flight (right).

We illustrate the steady states in the figure here. The Coriolis force in the atmosphere and ocean acts to the right of the motion (in the Northern hemsiphere) and is proportional to the speed. If the speed is such that the Coriolis force is equal in magnitude to the pressure gradient force, there is no nett acceleration. The motion achieved in this case is known as geostrophic flow (turning Earth flow). The large-scale flow is normally close to geostrophic balance. However, small variations about it are commonly observed.

The equations governing phugoid motion in the case where the lifting force is proportional to the speed are mathematically equivalent to the equations for inertial oscillations in the atmosphere and ocean. Therefore, the same trochoidal trajectories are found.

Inertial Motion in the Atmosphere and Ocean

For horizontal motion of limited latitudinal extent, the Coriolis parameter ${f = 2\Omega\sin\varphi}$ at latitude ${\varphi}$ may be considered to be a constant, ${f_0}$. We assume that the background pressure field ${P}$ is constant in time, and is coupled with a constant zonal geostrophic flow

$\displaystyle \mathbf{V}_{\rm G} = \frac{1}{f_0}\mathbf{k\times \nabla}P$

The velocity can be separated into two components

$\displaystyle \mathbf{V} = \mathbf{V}_{\rm G} + \mathbf{V^\prime}$

where ${\mathbf{V^\prime}}$ is the varying ageostrophic component of the flow. The equations for the components of ${\mathbf{V^\prime}=(u^\prime,v^\prime)}$ are

$\displaystyle \frac{d u^\prime}{d t} - f_0 v^\prime = 0 \qquad\qquad \frac{d v^\prime}{d t} + f_0 u^\prime = 0$

The full solution is a sum of the geostrophic flow ${\mathbf{V}_{\rm G}=(u_{\rm G},v_{\rm G})}$ and the ageostrophic flow ${\mathbf{V^\prime}=(u^\prime,v^\prime)}$ which oscillates with frequency ${f_0}$:

$\displaystyle \begin{array}{rcl} u &=& u_{\rm G} + V_0 \sin (f_0 t - \alpha) \\ v &=& v_{\rm G} + V_0 \cos (f_0 t -\alpha) \end{array}$

The vector ${\mathbf{V}}$ rotates clockwise (Northern hemisphere) in the velocity plane, in a circle of radius ${V_0}$ about the point ${\mathbf{V}_{\rm G}}$, with period ${2\pi/f_0}$. The trajectory of an air parcel is a trochoid, a combination of translation due to the geostrophic flow and circular motion due to the ageostrophic flow.

Trochoidal trajectories for inertial flow with (top) V0 < ug, (middle) V0 = ug and (bottom) V0 > ug.

Phugoid Motion

The lift is given as a function of the speed, ${f(V)}$, and is assumed perpendicular to the velocity. The phugoid equations in the case of linear lift ${f(V)=\lambda_0 V}$ can be written

$\displaystyle \frac{d\mathbf{V}}{d t} + \lambda_0 \mathbf{j\times V} + g\mathbf{k} = \mathbf{0}$

or, in component form,

$\displaystyle \frac{d u}{d t} = - \lambda_0 w \qquad\qquad \frac{d w}{d t} = \lambda_0 u - g$

The steady-state solution for straight-and-level flight is ${\mathbf{V}_{\rm S} = u_{\rm S}\mathbf{i}+w_{\rm S}\mathbf{k}}$ with

$\displaystyle u_{\rm S} = g / \lambda_0 \qquad\qquad w_{\rm S} = 0$

The velocity can be separated into two components $\displaystyle \mathbf{V} = \mathbf{V}_{\rm S} + \mathbf{V^\prime}$

The pure phugoid component ${\mathbf{V^\prime}}$ satisfies equations isomorphic to the equations for ageostrophic flow. The solutions are circular trajectories in an anticlockwise sense. The period of the phugoid motion is ${2 \pi/\lambda_0}$.

Trochoidal trajectories for phugoid motion.

Summary

Since the aerodynamic lift is analogous to the Coriolis force, with both acting orthogonally to the motion, there is a close correspondence between phugoid flight and inertial oscillations. The oscillating and looping motions that have been observed in the atmosphere and oceans closely resemble the patterns of a phugoid.

Sources

${\bullet}$ Doering, Charles R., 2017: Hamiltonian equations of motion and the `extra’ conserved quantity for the Lanchester-Joukowski glider. Conference presentation: New Trends in Applied Geometric Mechanics. ICMAT (Madrid, Spain), July, 3–7 2017.