Geostrophic Ocean Current Balance

Ocean Setup

Latitude 35°N

Sea-surface height anomaly 1.60

Feature radius 26

Spin-down drag 0.12

Live Balance

Coriolis parameter f

8.34e-5 /s

Inertial period

20.9 h

Geostrophic speed (edge)

0.00 m/s

Mean ageostrophic residual

0.00 m/s

Pressure-gradient force (down-slope)

Coriolis force

Tracer parcel path

How it works

Away from coasts and the surface mixed layer, large-scale ocean flow settles into geostrophic balance: the horizontal pressure-gradient force set up by a sloping sea surface is balanced by the Coriolis deflection, so water flows along contours of sea-surface height rather than down them.

f k × v_g = -g ∇η
v_g = (g/f) k × ∇η
f = 2Ω sin(latitude),  Ω = 7.292×10⁻⁵ rad/s

Each parcel here starts at rest on the slope of a dome (or trough) η(x,z). Two real forces act on it every frame:

dv/dt = -g∇η  -  f (k×v)  -  r·v
x(t+dt) = x(t) + v·dt

-g∇η — the pressure-gradient force, pushing the parcel straight down-slope, toward lower sea-surface height.
-f(k×v) — the Coriolis force, always perpendicular to the parcel's own velocity; it does no work but bends the path.
With no drag the parcel overshoots and performs an inertial oscillation around the geostrophic velocity, tracing loops of period T = 2π/f. A small drag term (r·v, the "spin-down" slider) removes that ageostrophic wobble over a few periods, so the parcel settles onto a closed path parallel to the height contours — exactly the geostrophic current.
Flip the latitude sign to cross the equator: f changes sign, so the same dome now produces the opposite sense of circulation (anticyclonic around a high in the Northern Hemisphere becomes cyclonic in the Southern Hemisphere).

This is the same balance that keeps the Gulf Stream, the Antarctic Circumpolar Current and every ocean gyre's interior flow locked to the basin's mean sea-surface topography, measured today from space by satellite altimetry.

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