Ch 3. Effects of rotation

Overview

Two questions about what rotation alone does to a fluid. Give a parcel a push and remove every other force — what does it do? Not travel in a straight line: it coasts around a circle at the inertial period $T=2\pi/f$, the signature seen in nearly every filtered current-meter record from the upper ocean. Force a rapidly-rotating, homogeneous fluid at one level only — does the rest of the column notice? It does, rigidly: the Taylor-Proudman theorem says a steady, balanced, homogeneous rotating flow cannot vary along the rotation axis. This chapter derives both exactly, and shows the second is not a separate calculation from Ch. 5’s thermal wind — it is that relation’s zero-buoyancy limit.

The model

Notebook: notebooks/ch03_rotation.py → exported to /nb/ch03_rotation.html. Governing equations use the symbols in notation.

gfdlib.rotation.inertial_trajectory is the closed-form circular solution of the first equation, cross-checked against an independent gfdlib.timestep.rk4 integration; Part B reuses gfdlib.balance’s exact thermal-wind machinery from Ch. 5 unchanged, sweeping the buoyancy contrast toward zero.

Both fluids

• Ocean: inertial oscillations are the dominant signal in storm-generated near-surface currents once tides and the mean flow are filtered out; Taylor columns are directly observed steering deep, weakly-stratified abyssal currents around seamounts and ridges far more than the overlying wind or buoyancy forcing would predict.
• Atmosphere: inertial oscillations appear as the low-level jet’s nocturnal acceleration once daytime turbulent friction relaxes at sunset — a textbook case of a “sudden push” (removal of drag) exciting the same circular response derived here; strongly rotating, weakly stratified planetary atmospheres (e.g. the gas giants’ deep interiors) show Taylor-column-organized convection.

Exercises

1. (analytic) Verify by direct substitution that $w(t)=w_0e^{-ift}$ solves $dw/dt=-ifw$ with $w=u+iv$, and that integrating once more gives a circle of radius $|w_0|/f$ centered at $z_0+w_0/(if)$ (matching gfdlib.rotation.inertial_trajectory and its test in tests/test_gfdlib.py).
2. (computational) Modify ch03_rotation.py’s Part B to plot the shear range as a function of $L_y$ instead of $\Delta b$ (holding $\Delta b$ fixed) — does narrowing the front also drive the flow toward Taylor-Proudman columnarity, or only $\Delta b\to0$ does?
3. (exploratory) Using Part A’s sliders, estimate how many inertial periods it would take a storm-excited ocean current (typical $f\sim10^{-4}\,$s$^{-1}$ at midlatitudes) to complete even a single orbit — convert the nondimensional period on the hodograph panel to hours. Does this match the day-or-so timescale commonly cited for inertial oscillations in oceanographic textbooks?

Further reading

Vallis, AOFD 2nd ed., §2.5-2.6 (inertial oscillations), §4.1-4.2 (Taylor-Proudman); Pedlosky, Geophysical Fluid Dynamics 2nd ed., §2.3-2.5; Cushman-Roisin & Beckers, Introduction to Geophysical Fluid Dynamics, §7.1-7.3.