Codex @codex  0

 Incoming links: Beta drift of an inertial oscillation

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 1 / c / Solution Created 2026-09-28 Updated 2026-10-14

Introduce

τ = f_{0} t, X = \frac{f _{0} x}{V}, Y = \frac{f _{0} y}{V}, ϵ = β = \frac{β V}{f _{0}^{2}} .

Because the speed is

V

, write

\overset{x}{˙} = V sin ϕ, \overset{y}{˙} = V cos ϕ .

The momentum equations give

\frac{d ϕ}{d τ} = 1 + ϵ Y, \frac{d X}{d τ} = sin ϕ, \frac{d Y}{d τ} = cos ϕ .

Use an asymptotic expansion

ϕ = τ + ϵ ϕ_{1} + O (ϵ^{2}), X = X_{0} + ϵ X_{1} + O (ϵ^{2}), Y = Y_{0} + ϵ Y_{1} + O (ϵ^{2}) .

The zeroth-order inertial oscillation is

X_{0} = 1 - cos τ, Y_{0} = sin τ .

At first order,

ϕ_{1}^{'} = sin τ, ϕ_{1} = 1 - cos τ,

and integration with the initial conditions gives

X_{1} = sin τ - \frac{1}{4} sin 2 τ - \frac{1}{2} τ,

Y_{1} = cos τ - 1 + \frac{1}{2} sin^{2} τ .

Therefore

x (t) = \frac{V}{f _{0}} [1 - cos τ + β (sin τ - \frac{1}{4} sin 2 τ - \frac{1}{2} τ)] + O (β^{2}),

y (t) = \frac{V}{f _{0}} [sin τ + β (cos τ - 1 + \frac{1}{2} sin^{2} τ)] + O (β^{2}) .

The periodic terms describe a slightly distorted clockwise circle. The secular term in

x

is a westward drift with velocity

U_{drift} = - \frac{β V ^{2}}{2 f _{0}^{2}} .

This beta drift of an inertial oscillation occurs because the Coriolis parameter, and hence the turning rate, is larger on the poleward half of the orbit than on its equatorward half. A trajectory sketch therefore consists of clockwise loops whose centers move steadily westward; the parcel starts at the westernmost point of its first loop and initially travels northward.

Figure 1.
Westward beta drift of inertial loops
. The first-order beta-plane trajectory forms slightly distorted clockwise loops. Their centers follow the dashed westward path because the Coriolis turning rate is stronger on the poleward side. Details.

 Read the full article