17. Non-Inertial Reference Frames#

A non-inertial reference frame translates, rotates, or does both relative to an inertial frame. The fluid still obeys Newton’s laws, but expressing those laws in moving coordinates introduces apparent accelerations. This appendix develops those accelerations and shows how they enter momentum, pressure balance, buoyancy, and energy.

The discussion assumes a rigid Cartesian frame: its axes remain orthonormal, and the frame may translate and rotate but does not stretch or shear. All moving-frame vectors in the equations below are expressed in the same instantaneous moving basis.

17.1. Rigid-Frame Kinematics#

Let \(I\) denote an inertial frame and \(R\) the moving frame. The position of a fluid parcel is

(17.1.1)#\[\vec{x}_I(t) = \vec{X}_O(t) + \mathbf{Q}(t)\vec{r},\]

where

  • \(\vec{X}_O\) is the inertial-frame position of the moving origin \(O\),

  • \(\vec{r}\) contains the parcel coordinates measured from \(O\) in the moving basis, and

  • \(\mathbf{Q}\) is the proper orthogonal rotation tensor that maps moving-frame components into inertial-frame components.

The origin is simply the reference point used to describe the frame motion. It need not be the center of mass of the vehicle or fluid. The same origin must be used when defining \(\vec{r}\) and the translational motion.

The angular velocity \(\vec{\Omega}\) is defined in moving-frame components by

(17.1.2)#\[\mathbf{Q}^{T}\dot{\mathbf{Q}}\vec{c} = \vec{\Omega}\times\vec{c}\]

for any vector \(\vec{c}\) expressed in the moving basis. Consequently, the derivative of a physical vector satisfies the transport identity

(17.1.3)#\[\frac{d}{dt}\left(\mathbf{Q}\vec{c}\right) = \mathbf{Q} \left[ \left(\frac{d\vec{c}}{dt}\right)_R +\vec{\Omega}\times\vec{c} \right].\]

Equation (17.1.3) is the rotating-frame transport theorem used throughout the derivation [KCD16].

Let

(17.1.4)#\[\vec{u} = \left(\frac{d\vec{r}}{dt}\right)_R\]

be the fluid velocity relative to the moving coordinates. If \(\vec{V}_O=d\vec{X}_O/dt\), differentiating Eq. (17.1.1) gives the inertial velocity

(17.1.5)#\[\vec{U}_I = \vec{V}_O +\mathbf{Q}\left(\vec{u}+\vec{\Omega}\times\vec{r}\right).\]

This relation distinguishes two physically different velocity fields:

  • Fluid fixed relative to the moving frame has \(\vec{u}=\vec{0}\) and moves with the frame.

  • Fluid instantaneously at rest in the inertial frame has

    (17.1.6)#\[\vec{u} = -\mathbf{Q}^{T}\vec{V}_O -\vec{\Omega}\times\vec{r}.\]

Constant translation by itself changes the relative velocity through Eq. (17.1.5), but it does not create an apparent force. This is the Galilean-invariance limit. Translational acceleration and rotation create the additional terms derived next.

17.2. Particle Acceleration#

The material derivative observed in the moving coordinates is

(17.2.1)#\[\frac{D_R}{Dt} = \left.\frac{\partial}{\partial t}\right|_{\vec{r}} +\vec{u}\cdot\nabla_R.\]

Let

(17.2.2)#\[\vec{A}_O = \frac{d^2\vec{X}_O}{dt^2}, \qquad \vec{a}_O = \mathbf{Q}^{T}\vec{A}_O, \qquad \vec{\alpha}=\left(\frac{d\vec{\Omega}}{dt}\right)_R\]

Here \(\vec{A}_O\) is the acceleration of the frame origin in inertial components, \(\vec{a}_O\) is the same acceleration in moving components, and \(\vec{\alpha}\) is angular acceleration in moving components. A second differentiation of Eq. (17.1.1) gives

(17.2.3)#\[\vec{a}_I = \vec{A}_O +\mathbf{Q} \left[ \frac{D_R\vec{u}}{Dt} +2\vec{\Omega}\times\vec{u} +\vec{\alpha}\times\vec{r} +\vec{\Omega}\times\left(\vec{\Omega}\times\vec{r}\right) \right],\]

When Eq. (17.2.3) is solved for the relative acceleration and \(\vec{A}_O\) is expressed in the moving basis, four frame-motion terms appear:

  • \(\vec{a}_O\): translation of the frame origin,

  • \(2\vec{\Omega}\times\vec{u}\): the Coriolis acceleration,

  • \(\vec{\alpha}\times\vec{r}\): the Euler acceleration caused by changing angular velocity, and

  • \(\vec{\Omega}\times(\vec{\Omega}\times\vec{r})\): the inward centripetal acceleration of the rotating kinematics.

These terms appear with the opposite sign when Newton’s second law is written for the relative motion. The last then becomes the outward apparent acceleration commonly called centrifugal acceleration.

17.3. Momentum Equation In Moving Coordinates#

Let \(\vec{g}\) be gravitational acceleration expressed in the moving basis. If gravity is specified in inertial components, its moving-frame representation is

(17.3.1)#\[\vec{g}_R(\vec{r},t) = \mathbf{Q}^{T}(t) \vec{g}_I\!\left(\vec{X}_O+\mathbf{Q}\vec{r},t\right).\]

Thus, even a gravity vector that is constant in the inertial frame generally has time-dependent components in a rotating frame. For brevity, \(\vec{g}\) below means \(\vec{g}_R\).

In the absence of mass addition, continuity in the rigid moving coordinates is

(17.3.2)#\[\left.\frac{\partial\rho}{\partial t}\right|_{\vec{r}} +\nabla_R\cdot(\rho\vec{u}) =0.\]

Because the coordinate transformation is rigid, its volume Jacobian is unity and no dilation source appears. Combining continuity with Eq. (17.2.3) gives the conservative momentum equation

(17.3.3)#\[\left.\frac{\partial(\rho\vec{u})}{\partial t}\right|_{\vec{r}} +\nabla_R\cdot(\rho\vec{u}\otimes\vec{u}) = -\nabla_R p +\nabla_R\cdot\boldsymbol{\tau} +\rho\vec{b} +\vec{f}_{\mathrm{other}},\]

Here \(p\) is physical pressure and is the thermodynamic pressure when an equation of state is used. The tensor \(\boldsymbol{\tau}\) contains the viscous stresses expressed in the moving basis, and \(\vec{f}_{\mathrm{other}}\) contains any remaining force per unit volume. The total body acceleration is

(17.3.4)#\[\vec{b} = \underbrace{\vec{g}-\vec{a}_O}_{\text{gravity and frame translation}} -\underbrace{2\vec{\Omega}\times\vec{u}}_{\text{Coriolis}} -\underbrace{\vec{\alpha}\times\vec{r}}_{\text{Euler}} -\underbrace{\vec{\Omega}\times \left(\vec{\Omega}\times\vec{r}\right)}_{\text{centrifugal}}.\]

Equation (17.3.4) is the central moving-frame result. Every term has units of acceleration, and multiplication by density gives its force per unit volume.

17.3.1. Pure Translation#

When the frame does not rotate, \(\vec{\Omega}=\vec{0}\) and

(17.3.1.1)#\[\vec{b}_{\mathrm{eff}}(t) = \vec{g}(t)-\vec{a}_O(t).\]

Under the usual uniform-gravity approximation, both terms are uniform in space. The effective acceleration may nevertheless change in magnitude or direction with time. This is not a spatial-gravity effect:

  • an upward-accelerating frame increases the apparent downward acceleration,

  • a downward-accelerating frame reduces it, and

  • in free fall, \(\vec{a}_O=\vec{g}\) and \(\vec{b}_{\mathrm{eff}}=\vec{0}\).

Gravity and translational frame acceleration are therefore indistinguishable in the local relative momentum balance except through their physical interpretation and the basis in which their components are expressed.

17.4. Potential And Non-Potential Accelerations#

Representing an acceleration as the gradient of a scalar potential permits it to participate in pressure balance. If pressure alone balances the potential force, taking the curl of \(\nabla_R p=\rho\nabla_R\Psi\) gives

(17.4.1)#\[0 = \nabla_R\times\nabla_R p = \nabla_R\times(\rho\nabla_R\Psi) = \nabla_R\rho\times\nabla_R\Psi.\]

For variable density, a pressure-only equilibrium therefore additionally requires the density surfaces to align with the potential surfaces.

For spatially uniform \(\vec{g}-\vec{a}_O\) and spatially uniform \(\vec{\Omega}(t)\), define

(17.4.2)#\[\Psi(\vec{r},t) = \left(\vec{g}-\vec{a}_O\right)\cdot\vec{r} +\frac{1}{2}\left|\vec{\Omega}\times\vec{r}\right|^2.\]

Its spatial gradient is

(17.4.3)#\[\nabla_R\Psi = \vec{g}-\vec{a}_O -\vec{\Omega}\times \left(\vec{\Omega}\times\vec{r}\right).\]

The gravity, translational, and centrifugal contributions are consequently potential accelerations. They may depend on time while remaining spatial gradients at each instant. The other two rotational terms have different character:

(17.4.4)#\[\vec{u}\cdot\left(-2\vec{\Omega}\times\vec{u}\right)=0, \qquad \nabla_R\times\left(-\vec{\alpha}\times\vec{r}\right) =-2\vec{\alpha}.\]

The Coriolis acceleration depends on velocity and does no pointwise work. The Euler acceleration generally has nonzero curl. Neither can in general be represented by a scalar pressure potential.

For a steady relative equilibrium with \(\vec{u}=\vec{0}\), constant angular velocity, a time-independent \(\Psi\), and no other forces, the momentum equation reduces to

(17.4.5)#\[\nabla_R p = \rho\nabla_R\Psi.\]

This single relation contains ordinary hydrostatics, the free-fall limit, and the radial pressure increase in a steadily rotating fluid. The centrifugal-potential form and the corresponding rotating-fluid equilibrium are standard results [Ped87].

17.5. Reduced Pressure And Buoyancy#

The pressure variable may include or exclude a convenient reference-density part of a potential body force. Let \(\rho_0\) be constant and define the pressure reduced by the complete potential in Eq. (17.4.2) as

(17.5.1)#\[\pi = p-\rho_0\Psi.\]

Then the pressure and potential-force terms satisfy the exact identity

(17.5.2)#\[-\nabla_R p+\rho\nabla_R\Psi = -\nabla_R\pi+(\rho-\rho_0)\nabla_R\Psi.\]

Equation (17.5.2) does not change the physics. It moves the force on the constant reference density into the pressure definition. A formulation may instead retain the full thermodynamic pressure, reduce it by gravity alone, or reduce it by the complete apparent potential. The density multiplying each retained force must follow that pressure definition consistently. The simple identity also requires \(\rho_0\) to be spatially constant. Otherwise, \(\nabla_R(\rho_0\Psi)=\rho_0\nabla_R\Psi+\Psi\nabla_R\rho_0\), and the additional gradient term must be retained.

The equivalence is especially useful for pure translation. Define a gravity-reduced pressure and a pressure reduced by the complete translational potential:

(17.5.3)#\[p_g=p-\rho_0\vec{g}\cdot\vec{r}, \qquad \pi=p_g+\rho_0\vec{a}_O\cdot\vec{r}.\]

The same full-pressure balance can then be written in either reduced form:

(17.5.4)#\[-\nabla_R p_g +(\rho-\rho_0)\vec{g} -\rho\vec{a}_O = -\nabla_R\pi +(\rho-\rho_0)(\vec{g}-\vec{a}_O).\]

Moving translational acceleration into the density-anomaly term is therefore accompanied by a corresponding shift in the reduced pressure. Changing one without the other changes the represented momentum balance whenever \(\rho_0\ne0\).

This pressure decomposition gives buoyancy its natural interpretation. Under the Boussinesq approximation,

(17.5.5)#\[\rho-\rho_0 \approx -\rho_0\beta_T\left(T-T_0\right),\]

where the volumetric thermal-expansion coefficient, evaluated at the reference state, is

(17.5.6)#\[\beta_T = -\frac{1}{\rho_0} \left(\frac{\partial\rho}{\partial T}\right)_{p,Y}.\]

The density-anomaly force associated with the potential acceleration is therefore

(17.5.7)#\[(\rho-\rho_0)\nabla_R\Psi \approx -\rho_0\beta_T\left(T-T_0\right)\nabla_R\Psi.\]

Buoyancy is therefore not tied to the name “gravity.” It is the response of a density variation to the potential acceleration present in the chosen frame. For pure translation, that acceleration is \(\vec{g}-\vec{a}_O\); for steady rotation, it can also include the centrifugal contribution. This is why frame acceleration can change thermal re-stratification even when the physical gravity vector and its spatial uniformity are unchanged.

17.6. Mechanical Work And Energy#

Define total energy and total enthalpy using the relative kinetic energy:

(17.6.1)#\[E_R=e+\frac{1}{2}|\vec{u}|^2, \qquad H_R=h+\frac{1}{2}|\vec{u}|^2, \qquad h=e+\frac{p}{\rho}.\]

If the conserved energy does not include an external potential energy, a body force \(\rho\vec{b}\) contributes the power density

(17.6.2)#\[\dot{w}_b = \rho\vec{b}\cdot\vec{u}.\]

The Coriolis contribution vanishes identically by Eq. (17.4.4). Translation, Euler acceleration, gravity, and centrifugal acceleration can exchange energy with the relative motion.

The potential part may instead be included in the conserved energy. Because the acceleration potential is \(\Psi\), the corresponding potential energy per unit mass is \(-\Psi\). For a time-dependent potential, absorbing this energy removes the spatial potential-force work but leaves the source

(17.6.3)#\[-\rho\left.\frac{\partial\Psi}{\partial t}\right|_{\vec{r}}.\]

Work by any non-potential acceleration remains explicit. Thus a pressure reduction or potential-energy reformulation must be applied consistently to momentum, energy, and mechanical boundary conditions. The reduced pressure \(\pi\) is not thermodynamic pressure; \(p\) must be retained or reconstructed for an equation of state, the definition \(h=e+p/\rho\), and physical normal stresses. For a time-independent potential, the explicit term in Eq. (17.6.3) vanishes.

Key Physical Picture

A rigid translating frame does not make gravity spatially variable. It adds a spatially uniform apparent acceleration \(-\vec{a}_O\), so the fluid responds to \(\vec{g}-\vec{a}_O\). Rotation adds position- and velocity-dependent terms. Potential terms may participate in pressure balance and buoyancy; Coriolis and Euler terms must generally remain explicit.

17.7. References#

[KCD16]

Pijush K. Kundu, Ira M. Cohen, and David R. Dowling. Fluid Mechanics. Academic Press, 6th edition, 2016. ISBN 978-0-12-405935-1. doi:10.1016/C2012-0-00611-4.

[Ped87]

Joseph Pedlosky. Geophysical Fluid Dynamics. Springer-Verlag, 2nd edition, 1987. ISBN 978-0-387-96387-7. doi:10.1007/978-1-4612-4650-3.