Geophysical fluid dynamics studies fluid motion on rotating, stratified planets, including oceans, atmospheres, cores, and planetary interiors.
The Boussinesq approximation treats density as constant except where a small density variation multiplies gravity to produce buoyancy. The velocity remains incompressible.
A rotating fluid is described in a rotating reference frame, where the Coriolis acceleration and centrifugal acceleration supplement the physical forces.
The Coriolis parameter is for planetary rotation rate and latitude . It controls the vertical component of the Coriolis acceleration acting on horizontal flow.
Potential vorticity combines fluid rotation with stretching and stratification. For an inviscid barotropic shallow-water layer,where is relative vorticity and is layer thickness.
Shallow-water potential vorticity is materially conserved by inviscid, unforced shallow water equations. Linearizing about rest with depth gives an anomaly proportional to
The quasi-geostrophic approximation describes slowly evolving, nearly geostrophic flow with small Rossby number. Its dynamics are governed by advection and forcing of potential vorticity.
The quasi-geostrophic streamfunction defines horizontal velocity by and . For a one-layer free-surface model, geostrophic balance gives .
For constant buoyancy frequency on a beta plane, three-dimensional quasi-geostrophic potential vorticity can be written
A quasi-geostrophic vertical mode separates horizontal and vertical dependence of the streamfunction. With rigid horizontal boundaries and constant stratification, the vertical eigenfunctions are .
An isopycnal displacement is the vertical displacement of a constant-density surface. In linear stratified quasi-geostrophic flow,
Geostrophic adjustment is the radiation of inertia-gravity waves from an unbalanced disturbance, leaving a slower flow in geostrophic balance while conserving linearized potential vorticity.
An inertia-gravity wave is restored jointly by buoyancy or free-surface gravity and the Coriolis force.
A coastal Kelvin wave is trapped within a Rossby deformation radius of a boundary. It propagates with the boundary on its right in the Northern Hemisphere and on its left in the Southern Hemisphere, and its linear shallow-water potential-vorticity anomaly vanishes.
Ocean circulation comprises wind-driven, buoyancy-driven, and tidal motions of the ocean over a wide range of spatial and temporal scales.
Wind stress is the tangential traction exerted by the atmosphere on the ocean surface. Its curl injects vorticity into the ocean.
An ocean gyre is a basin-scale rotating circulation driven principally by wind stress and shaped by planetary rotation and boundaries.
Sverdrup balance equates meridional advection of planetary vorticity to wind-stress curl in the weakly frictional ocean interior.
A western boundary current is a narrow, intense return flow that closes a broad wind-driven interior circulation. The beta effect selects the western side for this frictional boundary layer.
The Stommel boundary layer closes a Sverdrup interior using linear drag. Balancing beta-effect advection against drag gives widthfor the normalization in which quasi-geostrophic drag is .
A Rossby wave is a low-frequency wave restored by the spatial variation of planetary or background potential vorticity. On a beta plane its phase commonly propagates westward.
A barotropic Rossby wave has no vertical shear. For horizontal wavenumbers , its quasi-geostrophic frequency is .
A baroclinic Rossby wave has nontrivial vertical structure. Constant stratification and rigid boundaries add a positive vertical-mode contribution to the denominator of its dispersion relation.
An equatorial wave is trapped near the equator by the sign change of the Coriolis parameter. Its meridional structures are Gaussian-weighted Hermite polynomials.
An equatorial Kelvin wave has zero meridional velocity, eastward phase propagation, and Gaussian meridional trapping.
An equatorial Rossby wave propagates westward and has a Gaussian-weighted Hermite-polynomial meridional structure.
Articles by others on the same topic
Geophysical fluid dynamics (GFD) is a branch of fluid dynamics that focuses on the behavior of fluids in the Earth's atmosphere and oceans, as well as in other planetary environments. It combines principles from fluid mechanics, geophysics, and applied mathematics to study the motion of large-scale fluid systems influenced by the Earth's rotation, gravity, and other geophysical forces.