Equation Of Motion For A Nonrotating Fluid

The state of the atmosphere or ocean at any time is defined by five key variables:

(six if we include specific humidity in the atmosphere, or salinity in the ocean). Note that by using the equation of state, Eq. 1-1, we can infer p from p and T. To "tie" these variables down we need five independent equations. They are:

1. the laws of motion applied to a fluid parcel, yielding three independent

(x - g-Sx, y --g-Sy, z -g-Sz) (x + g-Sx, y -g-Sy, z --g-Sz)

FIGURE 6.2. An elementary fluid parcel, conveniently chosen to be a cube of sides Sx, Sy, Sz, centered on (x, y, z). The parcel is moving with velocity u.

(x - g-Sx, y --g-Sy, z -g-Sz) (x + g-Sx, y -g-Sy, z --g-Sz)

FIGURE 6.2. An elementary fluid parcel, conveniently chosen to be a cube of sides Sx, Sy, Sz, centered on (x, y, z). The parcel is moving with velocity u.

2Meteorologists like working in pressure coordinates in which p is used as a vertical coordinate rather than z. In this coordinate an equivalent definition of ''vertical velocity'' is:

the rate at which pressure changes as the air parcel moves around. Since pressure varies much more quickly in the vertical than in the horizontal, this is still, for all practical purposes, a measure of vertical velocity, but expressed in units of hPa s-1. Note also that upward motion has negative m.

equations in each of the three orthogonal directions

2. conservation of mass

3. the law of thermodynamics, a statement of the thermodynamic state in which the motion takes place.

These equations, five in all, together with appropriate boundary conditions, are sufficient to determine the evolution of the fluid.

6.2.1. Forces on a fluid parcel

We will now consider the forces on an elementary fluid parcel, of infinitesimal dimensions (Sx, Sy, Sz) in the three coordinate directions, centered on (x, y, z) (see Fig. 6.2).

Since the mass of the parcel is SM = p Sx Sy Sz, then, when subjected to a net force F, Newton's Law of Motion for the parcel is p Sx Sy SzDU = F, (6-2)

where u is the parcel's velocity. As discussed earlier we must apply Eq. 6-2 to the same material mass of fluid, which means we must follow the same parcel around. Therefore, the time derivative in Eq. 6-2 is the total derivative, defined in Eq. 6-1, which in this case is

Gravity

The effect of gravity acting on the parcel in Fig. 6.2 is straightforward: the gravitational force is g SM, and is directed downward,

where z is the unit vector in the upward direction and g is assumed constant.

Pressure gradient

Another force acting on a fluid parcel is the pressure force within the fluid. Consider Fig. 6.3. On each face of our parcel there is a force (directed inward) acting on the parcel equal to the pressure on that face multiplied by the area of the face. On face A, for example, the force is

F(A) = p(x - —, y, z) Sy Sz, directed in the positive x-direction. Note that we have used the value of p at the midpoint of the face, which is valid for small Sy, Sz. On face B, there is an x-directed force

FIGURE 6.3. Pressure gradient forces acting on the fluid parcel. The pressure of the surrounding fluid applies a force to the right on face A and to the left on face B.

FIGURE 6.3. Pressure gradient forces acting on the fluid parcel. The pressure of the surrounding fluid applies a force to the right on face A and to the left on face B.

F(B) = -p(x + —, y, z) Sy Sz, which is negative (toward the left). Since these are the only pressure forces acting in the x-direction, the net x-component of the pressure force is

If we perform a Taylor expansion (see Appendix A.2.1) about the midpoint of the parcel, we have

Sx ^ , , Sx ( dp p(x + —, y, z) = p(x y, z) + y I—

where the pressure gradient is evaluated at the midpoint of the parcel, and where we have neglected the small terms of O(Sx2) and higher. Therefore the x-component of the pressure force is dp

It is straightforward to apply the same procedure to the faces perpendicular to the y- and z-directions, to show that these components are dp

In total, therefore, the net pressure force is given by the vector

Sx Sy Sz

Note that the net force depends only on the gradient of pressure, Vp; clearly, a uniform pressure applied to all faces of the parcel would not introduce any net force.

Friction

For typical atmospheric and oceanic flows, frictional effects are negligible except close to boundaries where the fluid rubs over the Earth's surface. The atmospheric boundary layer—which is typically a few hundred meters to 1 km or so deep—is exceedingly complicated. For one thing, the surface is not smooth; there are mountains, trees, and other irregularities that increase the exchange of momentum between the air and the ground. (This is the main reason why frictional effects are greater over land than over ocean.) For another, the boundary layer is usually turbulent, containing many small-scale and often vigorous eddies; these eddies can act somewhat like mobile molecules and diffuse momentum more effectively than molecular viscosity. The same can be said of oceanic boundary layers, which are subject, for example, to the stirring by turbulence generated by the action of the wind, as will be discussed in Section 10.1. At this stage, we will not attempt to describe such effects quantitatively but instead write the consequent frictional force on a fluid parcel as

where, for convenience, F is the frictional force per unit mass. For the moment we will not need a detailed theory of this term. Explicit forms for F will be discussed and employed in Sections 7.4.2 and 10.1.

6.2.2. The equations of motion

Putting all this together, Eq. 6-2 gives us pSx Sy SzD = ^ + W, + ^

Substituting from Eqs. 6-3, 6-4, and 6-5, and rearranging slightly, we obtain

This is our equation of motion for a fluid parcel.

Note that because of our use of vector notation, Eq. 6-6 seems rather simple. However, when written out in component form, as below, it becomes somewhat intimidating, even in Cartesian coordinates:

— + u— + v— + w— +-— = Fy (b) dt dx dy dz p dy "

dw dw dw dw 1 dp — + u— + v— + w— + - — + g = Fz. (c)

Fortunately we will often be able to make a number of simplifications. One such simplification, for example, is that, as discussed in Section 3.2, large-scale flow in the atmosphere and ocean is almost always close to hydrostatic balance, allowing Eq. 6-7c to be radically simplified as follows.

6.2.3. Hydrostatic balance

From the vertical equation of motion, Eq. 6-7c, we can see that if friction and the vertical acceleration Dw/Dt are negligible, we obtain dp dz

thus recovering the equation of hydrostatic balance, Eq. 3-3. For large-scale atmospheric and oceanic systems in which the vertical motions are weak, the hydrostatic equation is almost always accurate, though it may break down in vigorous systems of smaller horizontal scale such as convection.3

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