7
Some 104
Chapter Mathematics: The Equations of Motion
In this chapter we consider the response of a uid 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 inuence of viscosity, and in chapter 12 we will consider the consequences of vorticity. Fluid mechanics used in oceanography is based on Newtonian mechanics modied 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 7.1).
Table 7.1 Conservation Laws Leading to Basic Equations of Fluid Motion Conservation of Mass: Conservation of Energy: Leads to Continuity Equation. Conservation of heat leads to Heat Budgets. Conservation of mechanical energy leads to Wave Equation. Leads to Momentum (Navier-Stokes) Eq. Leads to Conservation of Vorticity.
Conservation of Momentum: Conservation of Angular Momentum:
7.1 Dominant Forces for Ocean Dynamics Only a few forces are important in physical oceanography: gravity, buoyancy due to dierence in density of sea water, and wind stress (table 7.2). Remember that forces are vectors. They have magnitude and direction. 1. Gravity is the dominant force. The weight of the water in the ocean produces pressure, and the varying weight of water in dierent regions of the ocean produces horizontal pressure gradients. Changes in gravity, due to the motion of sun and moon relative to earth produces tides, tidal currents, and tidal mixing in the interior of the ocean. 2. Buoyancy is the upward or downward force acting on a parcel of water that is more or less dense than other water at its level. For example, cold 105
106
CHAPTER 7. THE EQUATIONS OF MOTION air blowing over the sea cools surface waters causing them to be more dense than the water beneath. The dierence in density results in a force that causes the water to sink.
3. Wind blowing across the sea surface transfers horizontal momentum to the sea. The wind drags the water in the direction of the wind, and it creates turbulence that stirs the upper layers of the sea producing the oceanic mixed layer. In addition, wind blowing over ripples on the surface leads to an uneven distribution of pressure over the ripples causing them to grow into waves. 4. Pseudo-forces are apparent forces that arise from motion in curvilinear or rotating coordinate systems. Thus, writing the equations for inertial motion in a rotating coordinate system leads to additional force terms called pseudo forces. For example, Newtons rst law states that there is no change in motion of a body unless a resultant force acts on it. Yet a body moving at constant velocity seems to change direction when viewed from a rotating coordinate system. The change in direction is attributed to a pseudo-force, the Coriolis force. 5. Coriolis Force is the dominant pseudo-force inuencing currents moving in a coordinate system xed to the earth.
Table 7.2 Forces in Geophysical Fluid Dynamics Dominant Forces Gravity Wind Stress Buoyancy Other Forces Atmospheric Pressure Seismic Gives rise to pressure forces and tides. Forces motion at the sea surface. Result from changes in density, leads to convection.
Results in inverted barometer eect. Results in tsunamis driven by earthquakes.
Note that the last two forces are much less important than the rst three.
7.2 Coordinate System Coordinate systems allow us to nd locations in theory and practice. Various systems are used depending on the size of the features to be described or mapped. I will refer to the simplest systems; descriptions of other systems can be found in geography and geodesy books. 1. Cartesian Coordinate System is the one I will use most commonly in the following chapters to keep the discussion as simple as possible. We can describe most processes in Cartesian coordinates without the mathematical complexity of spherical coordinates. The standard convention in geophysical uid mechanics is x is to the east, y is to the north, and z is up. 2. f-Plane is a Cartesian coordinate system in which the Coriolis force is assumed constant. It is useful for describing ow in regions small compared with the radius of the earth and larger than a few tens of kilometers.
7.3. TYPES OF FLOW IN THE OCEAN
107
3. -plane is a Cartesian coordinate system in which the Coriolis force is assumed to vary linearly with latitude. It is useful for describing ow over areas as large as ocean basins. 4. Spherical coordinates are used to describe ows that extend over large distances and in numerical calculations of basin and global scale ows. 7.3 Types of Flow in the ocean Many terms are used for describing the oceans circulation. Here are a few of the more commonly used terms for describing currents and waves. 1. General Circulation is the permanent, time-averaged circulation. 2. Meridional Overturning Circulation also known as the Thermohaline Circulation is the circulation, in the meridional plane, driven by mixing. 3. Wind-Driven Circulation is the circulation in the upper kilometer of the ocean forced by the wind. The circulation can be caused by local winds or by winds in other regions. 4. Gyres are wind-driven cyclonic or anticyclonic currents with dimensions nearly that of ocean basins. 5. Boundary Currents are currents owing parallel to coasts. Two types of boundary currents are important: Western boundary currents on the western edge of the oceans tend to be fast, narrow jets such as the Gulf Stream and Kuroshio. Eastern boundary currents are weak, e.g. the California Current. 6. Squirts or Jets are long narrow currents, with dimensions of a few hundred kilometers, that are nearly perpendicular to west coasts. 7. Mesoscale Eddies are turbulent or spinning ows on scales of a few hundred kilometers. In addition to ow due to currents, there are many types of oscillatory ows due to waves. Normally, when we think of waves in the ocean, we visualize waves breaking on the beach or the surface waves inuencing ships at sea. But many other types of waves occur in the ocean. 1. Planetary Waves depend on the rotation of the earth for a restoring force, and they including Rossby, Kelvin, Equatorial, and Yanai waves. 2. Surface Waves sometimes called gravity waves, are the waves that eventually break on the beach. The restoring force is due to the large density contrast between air and water at the sea surface. 3. Internal Waves are subsea wave similar in some respects to surface waves. The restoring force is due to change in density with depth. 4. Tsunamis are surface waves with periods near 15 minutes generated by earthquakes. 5. Tidal Currents are horizontal currents and currents associated with internal waves driven by the tidal potential. 6. Shelf Waves are waves with periods of a few minutes conned to shallow regions near shore. The amplitude of the waves drops o exponentially with distance from shore.
108
CHAPTER 7. THE EQUATIONS OF MOTION
7.4 Conservation of Mass and Salt Conservation of mass and salt can be used to obtain very useful information about ows in the ocean. For example, suppose we wish to know the net loss of fresh water, evaporation minus precipitation, from the Mediterranean Sea. We could carefully calculate the latent heat ux over the surface, but there are probably too few ship reports for an accurate application of the bulk formula. Or we could carefully measure the mass of water owing in and out of the sea through the Strait of Gibraltar; but the dierence is small and perhaps impossible to measure accurately. We can, however, calculate the net evaporation knowing the salinity of the ow in Si and out So , together with a rough estimate of the volume of water Vi owing in, where Vi is a volume ow in units of m3 /s (gure 7.1).
Precipitation In P Si = 36.3 Atlantic Ocean Sill 330 m
Figure 7.1 Schematic diagram of ow into and out of a basin. From Pickard and Emery, 1990.
Evaporation Out E
Vi Vo
1.75 Sv So = 37.8 Mediterranean
River Flow In R
The mass owing in is, by denition, i Vi . If the volume of the sea does not change, conservation of mass requires: i Vi = o Vo (7.1)
where, i , o are the densities of the water owing in and out. We can usually assume, with little error, that i = o . If there is precipitation P and evaporation E at the surface of the basin and river inow R, conservation of mass becomes: Vi + R + P = Vo + E Solving for (Vo - Vi ): Vo Vi = (R + P ) E (7.3) (7.2)
which states that the net ow of water into the basin must balance precipitation plus river inow minus evaporation when averaged over a suciently long time. Because salt is not deposited or removed from the sea, conservation of salt requires : i Vi Si = o Vo So (7.4)
Where i , Si are the density and salinity of the incoming water, and o , So are density and salinity of the outow. With little error, we can again assume that i = o .
7.5. THE TOTAL DERIVATIVE (D/DT)
109
An Example of Conservation of Mass and Salt Pickard and Emery (1990) in Descriptive Physical oceanography applied the theory to ow into the Mediterranean Sea using values for salinity given in gure 7.1. The incoming volume of water has been estimated to be 1.75 106 m3 /s = 1.75 Sv, where Sv = Sverdrup = 106 m3 /s is the unit of volume transport used in oceanography. Solving Eq. 7.4 for Vo assuming that i = o , and using the estimated value of Vi and the measured salinity gives Vo = 1.68 106 m3 /s. Eq. (7.3) then gives (R + P E) = 7 104 m3 /s. Knowing Vi , we can also calculate a minimum ushing time for replacing water in the sea by incoming water. The minimum ushing time Tm is the volume of the sea divided by the volume of incoming water. The Mediterranean has a volume of around 4 106 km3 . Converting 1.75 106 m3 /s to km3 /yr we obtain 5.5 104 km3 /yr. Then, Tm = 4 106 km3 /5.5 104 km3 /yr = 70 yr. The actual time depends on mixing within the sea. If the waters are well mixed, the ushing time is close to the minimum time, if they are not well mixed, the ushing time is longer. Our example of ow into and out of the Mediterranean Sea is an example of a box model . A box model replaces large systems, such as the Mediterranean Sea, with boxes. Fluids or chemicals or organisms can move between boxes, and conservation equations are used to constrain the interactions within systems.
z,w
qin Particle path q q qout = dt + dx + qin t x
y,v x,u
Figure 7.2 Sketch of ow used for deriving the total derivative.
7.5
The Total Derivative (D/Dt)
If the number of boxes in a system increases to a very large number as the size of each box shrinks, we eventually approach limits used in dierential calculus. For example, if we subdivide the ow of water into boxes a few meters on a side, and if we use conservation of mass, momentum, or other properties within each box, we can derive the dierential equations governing uid ow. Consider the simple example of acceleration of ow in a small box of uid. The resulting equation is called the total derivative. It relates the acceleration of a particle Du/Dt to derivatives of the velocity eld at a xed point in the uid. We will use the equation to derive the equations for uid motion from Newtons Second Law which requires calculating the acceleration of a particles passing a xed point in the uid. We begin by considering the ow of a quantity qin into and qout out of the small box sketched in gure 7.2. If q can change continuously in time and space, the relationship between qin and qout is:
110
CHAPTER 7. THE EQUATIONS OF MOTION
qout = qin +
q q t + x t x
(7.5)
The rate of change of the quantity q within the volume is: Dq qout qin q q x = = + Dt t t x t But x/t is the velocity u; therefore: and Dq q q = +u Dt t x In three dimensions, the total derivative becomes: D = +u +v +w Dt dt x y z D = + u ( ) Dt dt (7.7a) (7.7b) (7.6)
where u is the vector velocity and is the operator del of vector eld theory (See Feynman, Leighton, and Sands 1964: 26). This is an amazing result. The simple transformation of coordinates from one following a particle to one xed in space converts a simple linear derivative into a non-linear partial derivative. Now lets use the equation to calculate the change of momentum of a parcel of uid. 7.6 Momentum Equation Newtons Second Law relates the change of the momentum of a uid mass due to an applied force. The change is: D(mv) =F Dt (7.8)
where F is force, m is mass, and v is velocity; and where we have emphasized the need to use the total derivative because we are calculating the force on a particle. We can assume that the mass is constant, and (7.8) can be written: Dv F = = fm Dt m (7.9)
where fm is force per unit mass. Four forces are important: pressure gradients, Coriolis force, gravity, and friction. Without deriving the form of these forces (the derivations are given in the next section), we can write (7.9) in the following form. Dv 1 = p 2 v + g + Fr Dt (7.10)
7.6. MOMENTUM EQUATION
111
Acceleration equals the negative pressure gradient minus the Coriolis force plus gravity plus other forces. Here g is acceleration of gravity, Fr is friction, and the magnitude of is the Rotation Rate of earth, 2 radians per sidereal day or = 7.292 105 radians/s (7.11)
Momentum Equation in Cartesian coordinates: Expanding the derivative in (7.10) and writing the components in a Cartesian coordinate system gives the Momentum Equation: u u u u 1 p +u +v +w = + 2 v sin + Fx t x y z x v v v v 1 p +u +v +w = 2 u sin + Fy t x y z y w w w w 1 p +u +v +w = + 2 u cos g + Fz t x y z z (7.12a) (7.12b) (7.12c)
where Fi are the components of any frictional force per unit mass, and is latitude. In addition, we have assumed that w << v, so the 2 w cos has been dropped from equation in (7.12a). Equation (7.12) appears under various names. Leonhard Euler (17071783) rst wrote out the general form for uid ow with external forces, and the equation is sometimes called the Euler equation or the acceleration equation. Louis Marie Henri Navier (17851836) added the frictional terms, and so the equation is sometimes called the Navier-Stokes equation. The term 2 u cos in (7.12c) is small compared with g, and it can be ignored in ocean dynamics. It cannot be ignored, however, for gravity surveys made with gravimeters on moving ships.
z
p z y x x y
p + p
Figure 7.3 Sketch of ow used for deriving the pressure term in the momentum equation.
Derivation of Pressure Term Consider the forces acting on the sides of a small cube of uid (gure 7.3). The net force Fx in the x direction is Fx = p y z (p + p) y z Fx = p y z
112 But
CHAPTER 7. THE EQUATIONS OF MOTION
p = and therefore
p x x
p x y z x p Fx = V x Fx = Dividing by the mass of the uid m in the box, the acceleration of the uid in the x direction is: ax = Fx p V = m x m 1 p x
ax =
(7.13)
The pressure forces and the acceleration due to the pressure forces in the y and z directions are derived in the same way. The Coriolis Term in the Momentum Equation The Coriolis term exists because we describe currents in a reference frame xed on earth. The derivation of the Coriolis terms is not simple. Henry Stommel, the noted oceanographer at the Woods Hole Oceanographic Institution even wrote a book on the subject with Dennis Moore (Stommel & Moore, 1989). Usually, we just state that the force per unit mass, the acceleration of a parcel of uid in a rotating system, can be written: af ixed = Dv Dt =
f ixed
Dv Dt
+ (2 v) + ( R)
rotating
(7.14)
where R is the vector distance from the center of earth, is the angular velocity vector of earth, and v is the velocity of the uid parcel in coordinates xed to earth. The term 2 v is the Coriolis force, and the term ( R) is the centrifugal acceleration. The latter term is included in gravity (gure 7.4). The Gravity Term in the Momentum Equation The gravitational attraction of two masses M1 and m is: Fg = G M1 m R2
where R is the distance between the masses, and G is the gravitational constant. The vector force Fg is along the line connecting the two masses. The force per unit mass due to gravity is: Fg G ME = gf = m R2 (7.15)
7.7. CONSERVATION OF MASS: THE CONTINUITY EQUATION
113
Figure 7.4 Downward acceleration g of a body at rest on earths surface is the sum of gravitational acceleration between the body and earths mass gf and the centrifugal acceleration due to earths rotation ( R). The surface of an ocean at rest must be perpendicular to g, and such a surface is close to an ellipsoid of rotation. earths ellipticity is greatly exaggerated here.
where ME is the mass of earth. Adding the centrifugal acceleration to (7.15) gives gravity g (gure 7.4): g = gf ( R) (7.16)
Note that gravity does not point toward earths center of mass. The centrifugal acceleration causes a plumb bob to point at a small angle to the line directed to earths center of mass. As a result, earths surface including the oceans surface is not spherical but it is a prolate ellipsoid. A rotating uid planet has an equatorial bulge.
z u + du r + dr dy dx x
Figure 7.5 Sketch of ow used for deriving the continuity equation.
u, r dz y
7.7 Conservation of Mass: The Continuity Equation Now lets derive the equation for the conservation of mass in a uid. We begin by writing down the ow of mass into and out of a small box (gure 7.5). Mass ow in = u z y Mass ow out = = + u x u+ x z y x x u p p u u + +u + x x y z x x x x
114
CHAPTER 7. THE EQUATIONS OF MOTION
The mass ux into the volume must be (mass ow in) (mass ow out) Mass ux = u p p u +u + x x y z x x x x
The third term inside the parentheses becomes much smaller than the rst two terms as x 0; and Mass ux = In three dimensions: Mass ux = (u) (v) (w) + + x y x x y z (u) x y z x
The mass ux must be balanced by a change of mass inside the volume, which is: x y z t and conservation of mass requires: (u) d(v) (w) + + + =0 t x y z (7.17)
This is the continuity equation for compressible ow, rst derived by Leonhard Euler (17071783). The equation can be put in an alternate form by expanding the derivatives of products and rearranging terms to obtain: u v w +u +v +w + + + =0 t x y z x y z The rst four terms constitute the total derivative of density D/Dt from (7.7), and we can write (7.17) as: 1 D u v w + + + =0 Dt x y z (7.18)
This is the alternate form for the continuity equation for a compressible uid. The Boussinesq Approximation Density is nearly constant in the ocean, and Joseph Boussinesq (18421929) noted that we can safely assume density is constant except when it is multiplied by g in calculations of pressure in the ocean. The assumption greatly simplies the equations of motion. Boussinesqs assumption requires that: 1. ...