Finite-Size Lyapunov Exponent

Measures how quickly neighbouring water parcels reach a chosen separation, rather than how far apart they get in a chosen time. Where ftle() fixes the clock and measures distance, FSLE fixes the distance and measures the clock:

Usage

fsle(
  env_dat,
  u = "UO",
  v = "VO",
  final_separation = 50,
  initial_separation = NULL,
  max_days = 60,
  direction = c("backward", "forward"),
  step_hours = 6,
  name = NULL
)

Arguments

env_dat: an sf POINT object with one row per location and time step, as datamatch's access functions return, on a regular lon/lat grid, containing eastward and northward velocity columns
u: name of the eastward velocity column, in m/s
v: name of the northward velocity column, in m/s
final_separation: target separation $\delta_f$, in km. This is the scale-selectivity knob: it sets the size of structure resolved.
initial_separation: starting separation $\delta_0$, in km; defaults to roughly one grid cell
max_days: give up after this long. Must exceed the time a typical parcel pair needs, or most of the field returns NA.
direction: "backward" (attracting structures, the default) or "forward" (repelling structures)
step_hours: Runge-Kutta step size, in hours
name: name for the new column

Value

env_dat with an FSLE column added, in 1/day. Parcels that never reach final_separation within max_days are NA.

Details

$$\lambda = \frac{\ln(\delta_f / \delta_0)}{\tau}$$

with $\delta_0$ the starting separation, $\delta_f$ the target, and $\tau$ the time taken to reach it. Units are 1/day.

Why choose this over FTLE

The difference is scale selectivity, and it matters most in a domain whose flow speed varies a lot from place to place.

An FTLE map with a fixed integration time resolves fine structure where the flow is fast and only coarse structure where it is slow. Across a shelf with an energetic break current and a sluggish interior, ridge intensity then partly encodes background current speed rather than frontal activity, and a model cannot tell the two apart.

FSLE asks the same question everywhere — "how long to separate by $\delta_f$?" — so results are comparable between energetic and quiet regions, and $\delta_f$ can be set to a scale that means something biologically: a patch size, a predator's search radius, a survey's resolution.

The cost is that FSLE has no natural place to encode a biologically meaningful timescale. When the relevant duration is known — a retention time, a cohort's accumulation window — ftle() with that integration time is the better tool.

How it is computed

Each grid point is seeded with two companion particles, offset east and north by initial_separation. All three are advected together, and the separation of each pair is checked after every step. $\tau$ is the first time either pair reaches final_separation.

Parcels that never separate that far within max_days return NA: the question "how long to reach this separation" simply has no answer for them, and substituting max_days would report a slow separation rate where there was none at all.

References

d'Ovidio, F., Fernandez, V., Hernandez-Garcia, E., & Lopez, C. (2004). Mixing structures in the Mediterranean Sea from finite-size Lyapunov exponents. Geophysical Research Letters, 31(17).

Examples

if (FALSE) { # \dontrun{
# Structures at the 50 km scale, comparable across the whole shelf
env <- datamatch::accessCopernicus(vars = c("UO", "VO"), ...)
env <- fsle(env, final_separation = 50)       # UO and VO are the defaults

# A scale matched to a predator's search radius
env <- fsle(env, final_separation = 10, max_days = 60)
} # }