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Under critical balance in a rapidly rotating fluid, define ${\mathbf{u}}_{\perp }^{(0)}=\hat{\mathbf{z}}\times {\nabla }_{\perp }\Phi$ and $D_{\perp }={\partial }_t+\{\Phi ,\cdot \}$. Leading horizontal incompressibility gives zero divergence, but at the next order ${\nabla }_{\perp }\cdot {\mathbf{u}}_{\perp }^{(1)}=-{\partial }_{z}w$. Its Coriolis acceleration contribution survives in the vertical vorticity equation, producing the displayed coupled system. The vertical momentum equation uses the leading pressure $p_0=2\Omega \Phi$. A flow that stays purely horizontal is independent of $z$ in this reduced system; the assertion is not an exact theorem for arbitrary fast solutions of the full rotating Euler equations.

The Rossby number compares inertial acceleration with Coriolis acceleration. For speed $U$, horizontal length $L$, and Coriolis parameter $f$, it is $\mathrm{Ro}=U/(|f|L)$. Small Rossby number supports geostrophic balance and the quasi-geostrophic approximation.

The time derivative of a vector in inertial axes equals its component derivative in rotating axes plus the change of the rotating basis itself. Each basis vector obeys ${\dot{\mathbf{e}}}_j=\mathbf{\Omega }\times {\mathbf{e}}_j$, proving the displayed identity by differentiating $\mathbf{a}={\sum }_j{a}_j{\mathbf{e}}_j$. Applying it twice to position produces the Coriolis acceleration $2\mathbf{\Omega }\times \mathbf{u}$ and the centripetal term $\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{x})$. A time-dependent rotation adds $\dot{\mathbf{\Omega }}\times \mathbf{x}$; constant angular velocity removes that last term.

A rotating fluid is described in a rotating reference frame, where the Coriolis acceleration and centrifugal acceleration supplement the physical forces.

If a frame rotates with angular velocity $\mathbf{\Omega }$ relative to an inertial frame, the derivatives of any vector $\mathbf{a}(t)$ satisfyApplying this formula twice to a position gives the Coriolis acceleration, Euler acceleration, and centrifugal acceleration terms.

The Technische Universität Darmstadt demonstration uses a steel ball launched outward from the centre of a rotating disc. A fixed lecture-room camera records a straight path, while a camera carried above the disc records a curved path. The instructions specify launching toward the fixed camera so that the laboratory motion is clearly visible. The rotating iPod transmits its view wirelessly through Apple TV; two vertical support rods of length $1\mathrm{m}$ are specified for the slow-motion arrangement. The paired video views are the readout: one physical motion is represented in two reference frames.^[ref]

A useful preparation control appears in the Universität Freiburg instructions: with the disc stationary, check that the ball travels radially toward its edge. A camera is mounted on supports attached to the disc; turn the disc by hand as the ball leaves its centre and observe the camera view.^[ref]

The inference is a dependence of the observed trajectory on the rotating axes; a curved camera trace alone does not isolate the Coriolis force. To compare with the constant-angular velocity free particle in rotating coordinates, measure the disc angular velocity, calibrate lengths in the camera image, and compare launches at rest and with opposite rotation directions. These are checks of the ideal model, not reported measurements from the manuals. A rolling ball can exchange forces with the surface through friction, and hand rotation need not have constant angular velocity; an Euler acceleration then enters too. The ideal free-ball drawing deliberately omits those effects. Neither manual supplies a numerical uncertainty or a precision measurement of the Coriolis acceleration.

In an Earth-fixed rotating reference frame, the horizontal displacement of a freely falling body receives contributions from Coriolis acceleration and centrifugal acceleration. A short-fall approximation substitutes the unperturbed downward velocity into the Coriolis term and the Earth's surface position into the centrifugal term. If gravitational acceleration excludes centrifugal acceleration, both contributions must be included explicitly.

For spherical mass density in the anelastic approximation for a rotating barotropic star, the displayed pressure amplitude gives velocity proportional to $(-iz{R}^{m-1},z{R}^{m-1},i{R}^m)$. This velocity is divergence free and tangent to spherical shells, so its mass-weighted divergence vanishes for every spherical mass density profile. It is an inertial r-mode driven by Coriolis acceleration; with phase $e^{i\omega t+im\varphi }$ its inertial frequency is $\omega =-(m-1)(m+2)\Omega /(m+1)$.

For a homogeneous incompressible flow, strict geostrophic balance means that the Coriolis acceleration is exactly balanced by the pressure gradient, after conservative body-force potentials have been absorbed into the pressure per unit density $\pi$. It neglects material acceleration rather than merely assuming that its size is small. With constant vertical rotation, ${\pi }_z=0$ and taking the curl gives ${\mathbf{u}}_z=0$, the Taylor–Proudman theorem. Nonzero cross-contour motion over sloping rigid boundaries requires small departures from this strict limit.

Differential gravitational acceleration, multiplied by the affected mass, across a spatially extended body. In the linear approximation the acceleration difference is $T\Delta \mathbf{r}$, where $T$ is the tidal tensor. Stellar tidal forces limit the domain of a circumplanetary orbit; sufficiently strong ones can produce tidal disruption.

To isolate a tidal force, subtract the gravitational acceleration shared by the body as a whole. Let a reference test particle follow $\ddot{\mathbf{R}}=\mathbf{g}(\mathbf{R})$, and let another have position $\mathbf{R}+\mathbf{x}$. In a translating accelerated reference frame with parallel, nonrotating axes,for a source point mass, when $|\mathbf{x}|/R\ll 1$. Here $\mu =GM$ is its standard gravitational parameter and the last term uses Big O notation. This Taylor expansion removes the common inward pull and keeps the spatial variation. The near side has an excess pull toward the source; the far side has a deficit of inward pull, which is an outward relative acceleration. Both sides can stretch apart while both still accelerate toward the source.^[ref]

For a material element of mass $m$, the residual external force in this translating accelerated reference frame is $m[\mathbf{g}(\mathbf{R}+\mathbf{x})-\mathbf{g}(\mathbf{R})]$. Actual deformation also depends on elasticity, pressure and self-gravity. A rigid body approximation suppresses deformation but can still have internal forces that resist the tidal force. The point-mass tidal stretching and squeezing pattern describes freely falling, noninteracting test particles, rather than promising the same shape for every solid, star or ocean. A rotating reference frame introduces centrifugal acceleration and Coriolis acceleration, which must be included separately.

In a connected inviscid uniformly rotating barotropic fluid, momentum balance makes the displayed sum constant. The pressure term is the barotropic enthalpy function. Perturbations experience Coriolis acceleration, and their rotating-frame frequency depends on the chosen exponential sign convention.

The Rossby number compares advective acceleration with Coriolis acceleration, and the second ratio compares local time dependence with it. Both should be small when pressure and Coriolis terms dominate horizontal momentum. Surface stress, sharp curvature, small scales, rapid evolution and the near-equatorial region can invalidate the approximation. The vertical hydrostatic equilibrium assumption is a separate scale approximation.

Besides the two propagating inertial-acoustic wave branches, a uniform barotropic Keplerian shearing sheet has a stationary branch balancing pressure against Coriolis acceleration. For radial wavenumber $k\ne 0$, setting $\omega =0$ in continuity gives $U_x=0$, and radial momentum gives the displayed azimuthal balance. The full determinant is proportional to $\omega ({\omega }^2-{\Omega }^2-c_s^2{k}^2)$. At $k=0$ the stationary branch is a uniform column-density change; it is not a travelling density wave.