Equations of Motion in Rotating Frame Questions and Answers

Answer: A

The equations of motion are Newton's second law expressed for a fluid parcel, with the acceleration measured in an inertial frame rewritten in terms of quantities measured on the rotating Earth. Doing so introduces the Coriolis and centrifugal terms as apparent accelerations moved to the force side of the equation. The first law of thermodynamics, the gas law and continuity are separate members of the governing set and undergo no such transformation, since none of them involves acceleration at all.

Answer: B

Writing the inertial acceleration in terms of rotating frame quantities produces two extra terms, one proportional to the velocity and one proportional to the position vector, which are the Coriolis and centrifugal accelerations. Both are consequences of the frame rather than of any physical interaction. The advective and local terms belong to the material derivative in any frame, and gravity, friction, buoyancy and pressure gradient are all present in an inertial description too.

Answer: C

The cross product of twice the rotation vector with the relative velocity gives the Coriolis acceleration, which is perpendicular to both and hence perpendicular to the motion. The factor of two arises because the transformation is applied twice, once to the position and once to the velocity. The centrifugal term instead involves the rotation vector crossed twice with the position vector, and the advective and local terms are components of the material derivative in any frame at all.

Answer: D

Differentiating the position vector once introduces one rotation term, and differentiating the resulting velocity again introduces a second identical term, so the two combine into a factor of two. The derivation is short and worth performing once, since the factor is otherwise a detail that looks arbitrary. The sidereal and solar day differ by only one part in three hundred and sixty five, and the two components of the rotation vector make quite unequal contributions.

Answer: A

Because the centrifugal acceleration of the Earth's own rotation depends only on position, it can be combined with Newtonian gravitation into a single effective gravity that is the gradient of a modified potential. This is why the centrifugal term rarely appears explicitly in the equations even though the frame is rotating. The Coriolis term depends on velocity and cannot be so absorbed, and the other two are quite different terms. Framing the question as which quantity the change acts upon makes the answer follow without calculation.

Answer: B

Global models are written in spherical coordinates with longitude, latitude and a vertical coordinate, since the Earth is very nearly a sphere and the equations then apply uniformly everywhere. The price is the appearance of metric terms involving the tangent of the latitude, which become singular at the poles. A tangent plane is useful only over a limited region, an inertial Cartesian frame defeats the purpose of the transformation, and cylindrical coordinates fit no natural boundary here.

Answer: C

Expressing the acceleration in spherical coordinates produces additional terms involving the tangent of latitude and the radius, which account for the changing orientation of the unit vectors as a parcel moves. They are small for most purposes but not negligible near the poles or for very fast flows. Coriolis, pressure gradient and friction all appear in any coordinate system, since they represent physics rather than geometry, and confusing the two kinds of term is a common source of error.

Answer: D

All meridians converge at the pole, so the longitudinal grid spacing shrinks to zero and the permissible time step collapses with it. Numerical models handle this with filtering, reduced grids or entirely different meshes such as the cubed sphere or icosahedral grids. The Coriolis parameter reaches a finite maximum of twice the rotation rate, gravity is perfectly well defined, and the pressure gradient is computable given a suitable representation of the field.

Answer: A

Since the atmosphere is only about ten kilometres deep against an Earth radius of six thousand four hundred kilometres, replacing the radial distance by the mean radius introduces an error of well under one per cent. The approximation is applied together with the traditional approximation to preserve conservation of angular momentum and energy. Hydrostatic and Boussinesq approximations concern the vertical momentum equation and the treatment of density respectively.

Answer: B

Applying the shallow atmosphere approximation without the traditional approximation, or the reverse, destroys the conservation of angular momentum and energy in the resulting system. The two must therefore be adopted or rejected as a pair, a result established by Phillips. Hydrostatic and geostrophic approximations are logically independent, and the other pairs listed are alternatives to one another rather than partners. Reading the definition literally, rather than recalling a formula, is the reliable route through questions of this shape.