Ch 1. Kinematics & the material derivative

Overview

Drop a patch of dye into a fluid near a vortex embedded in a straining current: some of it survives as a coherent blob orbiting the vortex core, the rest is stretched into thin filaments and swept away. This opening chapter builds the two ideas every later chapter leans on without saying so — the material derivative ($D/Dt=\partial_t+\mathbf u\cdot\nabla$, the rate of change following a parcel) and the local strain-vorticity decomposition of a velocity field — using nothing but a prescribed 2D flow and particle trajectories.

The model

Notebook: notebooks/ch01_kinematics.py → exported to /nb/ch01_kinematics.html. Governing equations use the symbols in notation.

gfdlib.kinematics.velocity_field evaluates a steady superposition of uniform strain and a smooth (Lamb-Oseen-like) vortex in closed form; gfdlib.kinematics.okubo_weiss differentiates it with central differences to get the Okubo-Weiss parameter $W$ — the standard diagnostic (Okubo 1970, Weiss 1991) for separating vorticity-dominated cores ($W<0$) from strain-dominated, filament-producing regions ($W>0$). A ring of tracer particles is released around the vortex and advected with gfdlib.timestep.rk4.

Both fluids

• Atmosphere: the Okubo-Weiss decomposition is the standard method for automatically detecting and tracking coherent vortices (cyclones, anticyclones, polar vortex fragments) in reanalysis wind fields — a storm’s core sits in $W<0$ territory, its surrounding frontal shear zones in $W>0$.
• Ocean: mesoscale eddy-detection algorithms applied to satellite altimetry are built almost entirely on this same $W<0$ criterion; the filamentation you watched here is exactly how an eddy’s edge sheds material into the surrounding strain field, the first step in eddy decay.

Exercises

1. (analytic) For pure strain alone ($\Gamma=0$), show by direct differentiation of $u=-\alpha x,\ v=\alpha y$ that $S_n=-2\alpha$, $S_s=0$, $\zeta=0$, hence $W=4\alpha^2$ everywhere — confirm this matches gfdlib.kinematics.okubo_weiss at $\Gamma=0$ (see tests/test_gfdlib.py).
2. (computational) Modify ch01_kinematics.py to release the tracer ring at several different radii at once (e.g. $0.5\sigma$, $1.3\sigma$, $2.5\sigma$) and color each ring differently. Does every particle in the innermost ring stay coherent regardless of $\alpha$, for the range of $\alpha$ the sliders allow?
3. (exploratory) Find, by trial and error with the sliders, the approximate ratio $\Gamma/(\alpha\sigma^2)$ at which the $W=0$ contour just barely encloses a nonzero region. (This is the vortex-survival threshold; compare your empirical value to the literature on vortex stripping by ambient strain, e.g. Legras & Dritschel 1993.)

Further reading

Okubo, A. (1970), Deep-Sea Res. 17, 445–454; Weiss, J. (1991), Physica D 48, 273–294; Legras, B. & Dritschel, D. G. (1993), Fluid Dyn. Res. 10, 159–176 (“Vortex stripping and the generation of high vorticity gradients in two-dimensional flows”); Vallis, AOFD 2nd ed., §2.1-2.3; Salmon, Lectures on Geophysical Fluid Dynamics, ch. 1.