L2 - Forces on a Rotating Planet

Forces on a Rotating Planet

Lecture Overview

• Describing the atmosphere and ocean in terms of:

  • Velocity: $u = (u, v, w)$

  • Pressure: $P$

  • Density: $\rho$

  • Temperature: $T$

  • Salinity: $S$

• Using a Cartesian frame of reference (east (x), north (y), up (z)).

Governing Equations

• Atmosphere and ocean motions are governed by:

  • Conservation of mass.

  • Conservation of energy.

  • Conservation of momentum (Newton’s laws of motion).

• Newton’s second law of motion:

  • The rate of change of momentum (acceleration) of an object, as measured relative to coordinates fixed in space, equals the sum of all the forces acting.

  • $F = ma$

Fundamental Forces

• Gravitational force

• Pressure gradient force

• Friction

Apparent Forces

• Centrifugal force

• Coriolis force

Gravitational Force

• Newton’s law of gravitation: Two masses attract each other with a force proportional to each of their masses and inversely proportional to the square of the distance between them.

• Gravitational force per unit mass acting on a parcel of air or water at the surface of the Earth (m s-2).

  • Gravitational constant: $G = 6.67 \times 10^{-11} m^3 kg^{-1} s^{-2}$

  • Mass of Earth: $M = 5.97 \times 10^{24} kg$

  • Mean radius of Earth: $a = 6371 km$

• Directed locally downward.

Pressure Gradient Force

• Molecules are continually moving and colliding (Brownian motion).

• Pressure: Force exerted on an imaginary wall per unit area.

• Pressure at any point in a fluid acts equally in all directions.

• A net force requires a pressure gradient (spatial variation of pressure).

• If pressure is uniform, the net force is zero.

• Pressure gradient force = $-\frac{1}{\rho}\frac{\partial P}{\partial x}$

Pressure Gradient Force in 3D

• $-\frac{1}{\rho}(\frac{\partial P}{\partial x},\frac{\partial P}{\partial y},\frac{\partial P}{\partial z})$

Friction

• Force acting on a fluid parcel due to its motion relative to its surroundings (viscosity).

• Frictional drag at the base of the atmosphere.

• Small-scale motions exchange fluid parcels with different momentum, resulting in momentum transfer or shear stress.

• Momentum transfer to the solid Earth acts as a “drag” on the surface flow at the bottom of the atmosphere and ocean.

• Wind stress felt by the ocean.

• Momentum transfer from atmosphere to ocean creates surface ocean currents.

• Often represented by the gradient in a frictional stress $\tau\text{ }(N m^{-2})$.

Non-Inertial Reference Frames

• The Earth is a non-inertial (accelerating) frame of reference due to its rotation.

• Newton’s laws of motion require inclusion of apparent forces.

• Newton’s 1st law of motion: No change in motion unless a resultant force acts on it.

  • An object stationary relative to the stars appears to move when viewed from Earth.

  • An object moving at constant velocity relative to the stars changes direction when viewed from the rotating Earth.

Centrifugal Force

• Centrifugal force per unit mass = $\Omega^2 R$

  • $\Omega$ = rotation rate of the Earth ($7.29 \times 10^{-5} s^{-1}$)

  • $R$ = distance from the axis of rotation (m)

• Pulls objects outwards from the axis of planetary rotation.

Gravity

• Combination of centrifugal force and gravitational force: $g = g^* + \Omega^2 R$

• Except at the poles and the equator, gravity is not directed towards the center of the Earth.

• Surfaces of constant geopotential ($\Phi$) are normal to $g$ and are shaped like oblate spheroids.

• Earth’s equatorial radius is about 21km larger than its polar radius.

Coriolis Force

• Acts at 90 degrees to the right of the motion in the Northern Hemisphere and 90 degrees to the left in the Southern Hemisphere.

• Acts on all moving bodies in a rotating frame of reference.

• Does no work on a fluid parcel (acts at right angles to the velocity).

• In an inertial frame, an object moves in a straight line, but in a rotating frame, the object follows a curved path.

Coriolis Effect Demonstration

• Playing catch on a roundabout illustrates the Coriolis Effect.

• The Coriolis Effect gives hurricanes their spin by deflecting air currents.

• On a merry-go-round spinning counterclockwise, the Coriolis effect makes rolling balls deviate to the right.

Coriolis Acceleration Derivation

• Consider a ball moving radially outward from the center of a rotating table.

• In the inertial frame, the ball moves in a straight line a distance $r = v\Delta t$ in time $\Delta t$.

• In the same time, the table rotates by an angle $\theta = \Omega \Delta t$

• The ball is deflected by a distance $s = r \theta = \Omega v (\Delta t)^2$

• Coriolis acceleration is $a = 2\Omega v$, directed at right angles to the velocity.

Coriolis Parameter

• The component of the Earth’s rotation about the local vertical varies with latitude.

• Coriolis parameter: $f = 2\Omega \sin(\theta)$

• Magnitude = $f$ * velocity

• Direction = 90 degrees to the right in the Northern Hemisphere and 90 degrees to the left in the Southern Hemisphere.

• Strength varies with latitude $\varphi$.

• By convention, $f$ is negative in the Southern Hemisphere.

Summary of Key Points

• Motion of the atmosphere and ocean is governed by mass and energy conservation, Newton’s laws of motion, and gravitation.

• For Newton’s second law to hold in a rotating coordinate system, the centrifugal force and the Coriolis force must be added.

• Dominant forces acting on fluid parcels: gravity, pressure gradient force, Coriolis force, and friction.