L2 - Forces on a Rotating Planet

Notes on Forces Affecting a Rotating Planet

Lecture Overview

• Focus on the dynamics of the atmosphere and ocean, utilizing the following key variables:

  • Velocity (u): Represented as a vector with components (u, v, w) corresponding to the east-west, north-south, and vertical directions.

  • Pressure (P): The force per unit area exerted by the weight of the air above.

  • Density (ρ): Mass per unit volume of air or water, influencing buoyancy and stratification.

  • Temperature (T): Determines thermal energy content and affects gas and liquid behaviors.

  • Salinity (S): The concentration of salts in water, impacting water density and flow characteristics.

• Establishing a Cartesian frame of reference is critical for analysis. The directions are defined as: east (x-axis), up (z-axis), and north (y-axis), facilitating the application of mathematical models to atmospheric and oceanic phenomena.

Governing Principles of Atmosphere and Ocean Motions

• Conservation Laws: These laws are fundamental to understanding fluid dynamics in the atmosphere and oceans, including:

  • Conservation of Mass: Mass cannot be created or destroyed; the total mass of a fluid system remains constant.

  • Conservation of Energy: Energy in a closed system is conserved, influencing temperature and flow patterns.

  • Conservation of Momentum: Described by Newton’s laws, indicating how forces and resulting motion interrelate.

  • Newton’s Second Law: The rate of change of momentum (or acceleration) of an object equals the sum of forces acting on it.

    • Formula: F = m * a (Force equals mass multiplied by acceleration), essential for calculating forces in motion.

Fundamental Forces

• Gravitational Force:

  • Fundamental force that attracts two masses toward each other; its intensity depends on the masses involved and their distance apart.

  • The universal gravitational constant (G) is approximately 6.67 x 10^-11 m³ kg⁻¹ s⁻², essential for calculations in astrophysics and geophysics.

  • The mass of the Earth (M) equals 5.97 x 10²⁴ kg, with a mean radius of about 6371 km, which affects gravitational pull at different altitudes, influencing ocean currents and atmospheric pressure.

  • Gravitational force acts downward at Earth's surface, establishing foundational buoyancy dynamics in atmospheric and oceanic systems.

• Pressure Gradient Force:

  • Results from molecular motion and collisions (related to Brownian motion); significant in fluid dynamics.

  • The pressure gradient drives fluid from areas of high pressure to low pressure, shaping wind patterns and ocean currents.

  • Understanding this force is crucial for weather forecasting and predicting storm tracks.

• Friction:

  • Arises when fluid layers move relative to each other or to solid surfaces, creating shear stress that transfers momentum.

  • Contributions from viscosity of fluids play a pivotal role in the development of surface currents and mixing processes in both the ocean and atmosphere, impacting climate.

Non-Inertial Reference Frames

• Earth's rotation introduces complexities that must be addressed using non-inertial reference frames.

  • Centrifugal Force:

    • Acts outward from the axis of rotation and is dependent on the formula: W²R, where W represents the rotation rate and R the distance from the axis of rotation. This is crucial for understanding the apparent weight variations experienced at different latitudes.

  • The combination of centrifugal force and gravitational force results in effective gravity, represented by:

    • g = gʹ + W²R, where gʹ is gravitational acceleration, influencing water movement and airflow significantly.

Coriolis Force

• The Coriolis effect is essential for understanding fluid dynamics on Earth, acting perpendicular to the direction of motion:

  • In the Northern Hemisphere, it acts 90° to the right of the motion; in the Southern Hemisphere, it acts 90° to the left.

  • The Coriolis effect intricately influences the movement of large weather systems, ocean currents, and the development of cyclones, hurricanes, and tides.

  • Understanding the Influence of Coriolis on Moving Objects:

    • Demonstrated through deflection patterns observed on a rotating table, reinforcing the concept of perceived motion influenced by the Earth’s rotation.

    • Coriolis acceleration can be described by the formula: a = 2Wv, where a is directed at right angles to the direction of motion, an important concept in meteorology and oceanography.

Latitude Dependency of Coriolis Force

• The Coriolis parameter (f) is latitude-dependent:

  • Calculated as f = 2Wsin(θ), where θ is the latitude, impacting motion behavior at different latitudes.

  • The magnitude of the Coriolis force is proportional to f multiplied by velocity, while direction varies between hemispheres.

  • Conventionally, f is considered negative in the Southern Hemisphere, influencing the characteristics of weather systems and oceanic circulation.

Summary of Key Points

• Fundamental principles of atmosphere and ocean motion are based on conservation laws and Newtonian mechanics.

• When applying these principles in a rotating frame, apparent forces such as centrifugal and Coriolis must be factored in due to Earth’s rotation.

• The dominant forces affecting fluid movement—including gravitational force, pressure gradient force, Coriolis force, and friction—play critical roles in shaping weather patterns, ocean currents, and overall climatic conditions.