7: The Equations of Motion
- Page ID
- 30084
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter I consider the response of a fluid to internal and external forces. This leads to a derivation of some of the basic equations describing ocean dynamics. In the next chapter, we will consider the influence of viscosity, and in Chapter 12 we will consider the consequences of vorticity.
Fluid mechanics used in oceanography is based on Newtonian mechanics modified by our evolving understanding of turbulence. Conservation of mass, momentum, angular momentum, and energy lead to particular equations having names that hide their origins (Table \(\PageIndex{1}\)).
| Conservation of Mass: | Leads to Continuity Equation. |
| Conservation of Energy: | Conservation of heat leads to Heat Budgets. Conservation of mechanical energy leads to Wave Equation. |
| Conservation of Momentum: | Leads to Momentum (Navier-Stokes) Eq. |
| Conservation of Angular Momentum: | Leads to Conservation of Vorticity. |
- 7.1: Dominant Forces of Ocean Dynamics
- Discussion of the forces involved in ocean dynamics, with emphasis on the most important three: gravity, friction, and Coriolis force.
- 7.2: Coordinate System
- Brief overview of the Cartesian coordinate system (with introduction to the special cases of the \(f\)-plane and \(\beta\)-plane), and spherical coordinates.
- 7.3: Types of Flow in the Ocean
- Defining the terms commonly used to describe ocean currents and waves.
- 7.4: Conservation of Mass and Salt
- Analyzing flows in the ocean with assumptions of conservation of mass and salt, including a worked example. Introduction to the concept of box models.
- 7.5: The Total Derivative (D/Dt)
- Deriving the total derivative of flow in one direction for a fluid. Expanding the definition of total derivative to three dimensions.
- 7.6: Momentum Equation
- The Momentum Equation in Cartesian coordinates, with discussion of how the pressure, gravity, and Coriolis terms in the equation were derived.
- 7.7: Conservation of Mass: The Continuity Equation
- Derivation for the equation for the conservation of mass (the continuity equation) in a fluid. Simplification of the equation if the Boussinesq Approximation, or the assumption of seawater's incompressibility, is applied.
- 7.8: Solutions to the Equations of Motion
- The continuity equation and the components of the momentum equation are nonlinear partial differential equations, which are almost impossible to solve. The usual assumed boundary conditions of fluid mechanics: no-slip boundaries and lack of flow through boundaries.
- 7.9: Important Concepts
- Summary of important concepts covered in this chapter.