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Equations of Motion

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Equations of Motion CHAPTER FIVE Equations of Motion The equations that describe the currents, waves, tides, turbulence, and other forms of fluid motion in the ocean are of a nonlinear nature for which there are no complete, exact, analytical solutions. The best we can do is to work with partial solutions, many of which provide excellent insight into the forces at work. In this chapter, we will examine the indi- vidual terms in the complete equations of motion. In Chapter 6, we will examine a series of simplified equations that demonstrate important aspects of ocean circulation. In Chap- ters 9 and 10, we will use simplified forms of these same equations in our discussion of waves. Any quantitative discussion of forces and motions requires a coordinate system. The system most commonly used in oceanography is the rectilinear, Cartesian system in which the earth is assumed to be flat. A spherical coordinate system would be more realistic, but it is also more complicated. The Cartesian system is adequate for most problems in physi- cal oceanography. The usual convention is to assume a plane in which the x axis points east, the y axis points north, and the z axis is up; more precisely, the z axis is in the direc- tion opposite the gravitational vector. Doing so makes the horizontal xy plane an equal potential surface (see "Gravity: Equal Potential Surfaces" later in this chapter for defini- tion). The corresponding velocity components are u, v, and w. Although meteorologists and oceanographers may agree on the coordinate system in Figure 5.1, they use different conventions for describing winds and currents. A north cur- rent is a current flowing toward the north; a north wind is a wind blowing from the north. The convention is confusing, but there is little likelihood that it will be changed. To mini- mize the confusion, this text refers to northerly winds and northward currents. Newton's second law states that the mass times the acceleration of a particle is pro- portional to the sum of the forces acting on the particle: du_lvF dt Equations of Motion 81 FIVE -x:-u:West x:u:East :ion Figure 5.1 The Cartesian, flat earth coordinate system used in this text. ' fluid exact, In discussing fluid motion, the relationship is usually written which indi- (5.2) series :hap- where it is now understood that the forces are per unit volume, since Eq. (5.2) follows on of from Eq. (5.1): t. The vhich (5.3) – V c, but thysi- As written, Eqs. (5.1) and (5.2) apply to the components of the forces acting in the )oints east–west or x direction Similar equations can be written for the force components acting lirec- along the other two axes: equal efini- —dt 10 2, r x m in dv 1 (5.4) I cur- dt p Y torth. dw i tnini- v F dt p z : pro- There are four important forces acting on a fluid particle in the ocean: gravity, pres- sure gradient, friction, and Coriolis. In a generalized way, Eq. (5.4) may be written (5.1) density x particle acceleration = gravity + pressure gradient + Coriolis + friction (5.5) 82 Equations of Motion The mathematical expression for the forces of gravity, pressure, and Coriolis may be expressed simply. The various forms of the frictional forces are less easy to express in a precise manner, and they are considerably more difficult to measure in the ocean. The two problems are not unrelated. Note that by choosing a coordinate system such that the z axis is along the direction of gravity, there will be no gravitational force in either the x or y direction. Acceleration Before looking at the various force terms, it is necessary to examine the acceleration of fluids. Newton's second law is usually introduced in terms of particle mechanics (a block sliding down an inclined plane or the movement of billiard balls). As written, Eq. (5.4) applies to the motion of a particle. In continuum mechanics (or fluid mechanics), there are two kinds of acceleration for which there are operational definitions, as the following example demonstrates. Consider the water motion in a channel of constant depth but narrowing width (Fig- ure 5.2). The volume of water entering the channel is constant over time, and, in the absence of turbulence, so is the rate of flow along the channel. A current meter suspended A (a) Velocity at B Particle Velocity — — Velocity at A Time (b) Figure 5.2 For steady flow within a channel, the water must flow faster as the channel narrows. Cur- rent meters at points A and B record a constant velocity and zero acceleration. (a) Because the chan- nel narrows, the velocity measured at point B is higher than at point A. (b) However, a particle moving along the channel is accelerated as it moves from A to B. The local acceleration is zero; the average particle acceleration between A and B is not. Equations of Motion 83 at any point within the channel would measure a constant velocity (i.e., zero acceleration). However, if one could tag a water particle (perhaps by using a floating cork) and record its velocity as it traverses the channel, its speed would increase as a constant volume of water is forced to flow through a narrower channel. In this example, the local acceleration is zero; the particle acceleration is not. In many problems, it is desirable to write Eq. (5.4) in terms of the local acceleration rather than the particle acceleration. The two are related in the following way: particle acceleration = local acceleration + field acceleration terms du du du du du dt dt+u—dx+v—dy+w—dz dv dv dv dv dv (5.6) dt dx dy dz dw aw aw dw dw dt dt u-d-x -"Ty +w-Tz or, in general, DE_ d a a d (5.7) Dt dt — Tt +17x +11—±}11dY dz a where we will follow the convention of a number of texts and use D = d Dr dt to emphasize the distinguishing characteristics of acceleration of fluids. The motion of a particle is called Lagrangian motion. The flow past a point is called Eulerian motion. The derivation of Eq. (5.6) is given in Box 5.1. Box 5.1 Acceleration The velocity of a fluid is not only a function of time but also of space: u = f(x,y,z,t) By the chain rule of differentiation, du au duds au dy au dz dt at ax di dy di az di (5.1') du du du du =—at+u—ax+v—dy+W-Tz For emphasis, the total differential is often written Du du du du du Dr dt dt dx dy dz Note that D/Dt is the particle acceleration, and alat is the local acceleration. Thus, Eq. (5.1') can be written 84 Equations of Motion Du du Dt dt = + uux + vu y + wux where we again adopt the notation that du ux —ax Similarly, Dv dv v, + uv x + vvy + wv, Dt dt Dw dw = wt +uwx+vwy+wwx Dt dt In vector notation, Du — = ut +(V V)u Dt Dv — = v, + (V • V)v Dt Dw — = v t + (V V)w Dt or, in general, DV– —+ (V. V)V (5.2') Dt Pressure Gradient Of the various terms in Eq. (5.5), perhaps the pressure gradient is the easiest to visualize. A particle will move from high pressure to low pressure, and the acceleration is simply proportional to the pressure gradient. A mechanical analog is a ball on a frictionless inclined plane. The ball rolls down the plane (from high to low pressure), and the acceler- ation of the ball is proportional to the inclination of the plane (pressure gradient). In math- ematical terms, Eq. (5.5) now becomes (see Box 5.2 for derivation) Du 1 — = ---dP + other forces Dt P ax Dv 1 dp — = — — — + other forces (5.8) Dt P aY Dw 1 — = — — —dP + other forces Dt P az Pressure gradients arise in a variety of ways. One of the simplest is by a sloping water surface. Imagine a container with an ideal fluid (constant density, incompressible, and without viscosity) whose density is pa and that in some manner it is possible to have the water surface slope as in Figure 5.3 without causing any other motion. Remembering Equations of Motion 85 1 2 Pi = Pagz P2 =Pa g (z +ha) Figure 5.3 The slope of the sea surface creates a horizontal pressure gradient throughout the entire fluid. The pressure gradient is proportional to the slope of the sea surface. that the pressure at any point in a motionless fluid is simply the weight of the fluid above—that is, the hydrostatic pressure, Eq. (1.2)— Pi = PagZ (5.9) (5.2') P2 = Pa g( Z + AZ) The resulting pressure gradient term is 1 dp 1 p2 - Pa & Pa AX ualize. AZ ;imply = g — (5.10) AX onless ;celer- = gix math- where ix is the slope of the fluid surface in the x direction. It can be easily shown in a homogeneous fluid that the horizontal pressure gradient is identical everywhere within the fluid; the result is the same regardless of the length Z chosen in Figure 5.3. Thus, if there were no other forces acting, Eq. (5.8) says that the entire fluid in Figure 5.3 would be uniformly accelerated toward the lower pressure. (5.8) Box 5.2 Pressure Gradient Consider a cube of fluid of density p with sides dr, dy, Liz, and let this element of fluid be in a channel where the pressure increases from left to right (i.e.,p 2 > p 1 (Figure 5.1').
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