1: General concept- Newton's 2nd law
- Page ID
- 1270
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The wind blowing over your house, the Gulf Stream flowing through the North Atlantic Ocean from Florida to Europe, the jet stream at 10 km altitude... All these different flows in the atmosphere and in the ocean can be understood using ideas from the physics of fluids.
A fluid is any substance that is able to flow which can be a gas, a liquid, or in some cases even a solid. In reality, fluids consist of molecules, particles that interact with each other. However, fluid mechanics deals with very large collections of such particles and it would be impossible to describe all the interactions between them. Therefore, the individual particles and their interactions are not taken into account: fluids are considered as a continuum. Essentially, fluid mechanics is the application of Newton's 2nd law of motion to a flowing continuum. Newton's 2nd law succinctly states:
\[F =m \dfrac{dv}{dt} \label{1.1}\]
Or, the time rate of change in velocity (\(v\)) of a body is proportional to the net force (\(F\)) applied to that body. As a result, the change in the velocity of a small (infinitesimal) parcel of fluid is given by the sum of all the forces that operate on this fluid parcel. The picture on top of this page shows the example of blowing wind, but the same principles apply to ocean flows. All the time, various forces act on the flow; how the flow will develop, depends on all the forces taken together. For flows in the atmosphere and ocean, the collection of forces boils down to:
- Acceleration = pressure gradient force + Coriolis force + friction (horizontal)
- Acceleration = pressure gradient force + gravity + friction (vertical)
Or in mathematical terms:
\[\dfrac{du}{dt}=-\dfrac{\left(\dfrac{dp}{dx}\right)}{\rho}+f * v +K_h \left(\dfrac{d^2u}{dx^2}+\dfrac{d^2u}{dy^2}\right)+K_v\dfrac{d^2u}{dz^2} \label{1.2a}\]
\[\dfrac{dv}{dt}=-\dfrac{\left(\dfrac{dp}{dy}\right)}{\rho}-f * u +K_h \left(\dfrac{d^2v}{dx^2}+\dfrac{d^2v}{dy^2}\right)+K_v\dfrac{d^2v}{dz^2} \label{1.2b}\]
\[\dfrac{dw}{dt}=-\dfrac{\left(\dfrac{dp}{dz}\right)}{\rho}-g+K_h \left(\dfrac{d^2w}{dx^2}+\dfrac{d^2w}{dy^2}\right)+K_v\dfrac{d^2w}{dz^2} \label{1.2c}\]
Here, \(u\) is the velocity in the zonal (West to East) direction, \(v\) is the velocity in the meridional (South to North) direction, and \(w\) is the velocity in the vertical direction. Furthermore, \(p\) is the pressure, \(\rho\) is the density of the fluid, \(f\) the Coriolis parameter, and \(g\) is the gravitational acceleration. \(K_h\) and \(K_v\) are the horizontal and vertical friction coefficients; these are different, because on the large atmospheric and oceanic scales, momentum dispersion is primarily due to turbulence, rather than molecular diffusion. In the following sections, each of these different forces will be discussed in more detail.