Cm

Fig. 11.1 Computation of pressure gradient force

— (1/p){(dp/dx) + (d p/dy) + (dp/dz)} = -(1/p) Vp (11.2.1)

as the force per unit mass due to pressure gradient. Here V denotes the three-dimensional gradient operator with components, d/dx, d/dy, d/dz along the three co-ordinate axes x, y, z respectively, and the minus sign signifies that the force is acting in the direction from high to low values of pressure.

11.2.2 Gravity Force

We have seen in Chap. 1 that the field of earth's gravity at any height may be expressed in terms of the geopotential O at that height. So, if a parcel of air of unit mass is displaced through a vector distance Sr, the work done against gravity, SW, is given by

SW = —VO • Sr where — VO is the gradient of the geopotential.

But, since — SW = F • Sr, where F is the force acting on the parcel, we get

Force due to gravity per unit mass = — VO (11.2.2)

11.2.3 Force of Friction or Viscosity

Like any real fluid, air possesses the property of viscosity by which it resists motion of any part of its medium relative to the other. So, whenever air at a horizontal level in the atmosphere moves relative to a lower level in the atmosphere or the ground below, a vertical gradient of horizontal velocity is created in proportion to the shearing stress between the upper and the lower levels. Since the air at the upper surface is moving faster, there is a net downward transfer of the horizontal momentum by molecular motion and the lower level experiences the same shearing stress as the upper. So, there is no net horizontal force acting on the atmospheric layer between the two levels. In the steady state (see Fig. 11.2), a measure of the shearing stress is given by the expression

Fig. 11.2 Illustrating frictionally-generated steady-state shear flow in the vertical

Fig. 11.3 Computation of the frictional force i / vR&^/az). ôz/2

where Tzx denotes the shearing stress per unit area in the x-direction, u is the x-component of the wind vector at a height z above the lower boundary, and ^ is called the co-efficient of molecular or dynamic viscosity.

However, in a more general case, where the shearing stress varies with height, a horizontal frictional force acts on the layer (see Fig. 11.3).

The magnitude of frictional force may be calculated as follows: Let Tzx be the shearing stress at a horizontal surface at height z at the center of a rectangular volume element with sides Sx, Sy, Sz along the rectangular co-ordinate axes, x, y, z respectively. Then the shearing stress in the x-direction across unit area of the lower boundary surface is Tzx — (dTzx/dz) Sz/2, while that across the upper boundary surface is Tzx + (dxzx/dz) Sz/2. The net frictional force on the volume element acting in the x-direction is then or (dxzx/dz)Sx Sy Sz. On substituting the value of Tzx from (11.2.3) and dividing by the mass, pSxSySz, we get for the x-component of the frictional force per unit mass

For constant Fx may be written as Fx = v d2u/dz2 where v(= ^/p) is called the co-efficient of kinematic viscosity.

Similarly, the y-component of the frictional force per unit mass may be written

[{Tzx + (dTzx/dz) Sz/2} - {Tzx - (dTzx/dz)Sz/2}] Sx Sy,

Fy = vd2v/dz2 Thus, the frictional force per unit mass, F = v d2 V/dz2

11.3 Acceleration of Absolute Motion

Newton's second law of motion applies to absolute motion which is motion measured relative to a fixed frame of reference (fixed relative to space or distant stars). It states that the rate of change of momentum of a body is equal, in magnitude and direction, to the vector sum of the forces acting on the body. If we consider that the forces act on an infinitesimal parcel of air of constant density p, the absolute acceleration of the parcel following motion may be expressed in the vector form p(dVa/dt)a = -Vp - pVOa + pF (11.3.1)

where (d/dt)a denotes differentiation following absolute motion,

Va is the absolute velocity vector, p is pressure,

Oa is geopotential in absolute co-ordinate system, F is the frictional force vector, V is Del operator, t is time, and the subscript 'a' refers to the fixed frame.

In (11.3.1), the first term on the right-hand side gives the pressure force, the second the gravitational force and the third the force of friction.

It is sometimes convenient to consider acceleration per unit mass, instead of per unit volume. In that case, we divide (11.3.1) by p, and obtain

where a is called the specific volume, or volume per unit mass (= 1/p).

Equation (11.3.2) is then the equation in vector form for acceleration of absolute motion per unit mass.

11.4 Acceleration of Relative Motion

In practice, however, it is convenient to measure the velocity or acceleration of a parcel relative to a co-ordinate system which is fixed to the earth and call it relative velocity or acceleration. For this, we allow for the rotation of the earth's surface by adding an extra term VE, which we may call the earth velocity, to the relative velocity. A measure of this extra term is given by the vector product, VE = Q x r, where Q is the angular velocity of the earth and r the radial vector position of a point P on the surface of the spherical earth (see Fig. 11.4).

The absolute velocity Va is then related to the relative velocity Vr by the vector equation

Since r is the position vector of the point P, we may write

Va = (dr/dt)a, and Vr = (d r/dt)r where the subscripts 'a' and 'r' refer to the absolute and the relative motion respectively.

Equation (11.4.1) may then be written in the form

It follows from (11.4.2) that the differential operator

(d/dt)a = (d/dt)r + Q x is quite general and may be applied to any vector.

When applied to the absolute velocity vector Va, we have the equation

Substituting for Va from (11.4.1) on the right-hand side of (11.4.3), we get

(dVa/dt)a = (dVr/dt)r + 2 Q x Vr + Q x(Q x r) (11.4.4)

It can be shown that the triple vector product on the right-hand side of (11.4.4) is equal to —Q2R, where R is the position vector of the point P issuing perpendicularly from the axis of the earth (see Fig. 11.5).

Equation (11.4.4) may, therefore, be written in the final form

Fig. 11.5 Direction of the triple vector product, Q x (Q x r)

Equation (11.4.5) is then the relationship between the accelerations in the two co-ordinate systems.

The equation for relative acceleration in the earth co-ordinate system is obtained by substituting for (d Va/dt)a from (11.4.5) in (11.3.2). This gives

It is immediately clear from (11.4.1) and (11.4.6), that the rotation of the earth introduces two additional forces, viz., —2Q x Vr and Q2 R in the expression for the relative acceleration. These are fictitious or apparent forces and depend upon the rotation of the earth. The first is called the Coriolis force and the second the centrifugal force. Also, the two terms within the bracket on the right-hand side of (11.4.6) constitute g, the earth's effective gravity, which is equal to — VO, vide (1.2.3) (see Chap. 1). Henceforth, we drop the subscript 'r', since, unless otherwise stated, we shall always deal with the relative motion, i.e., motion relative to the earth's surface. The final form of the equation of motion in vector notation is then

11.4.1 Coriolis Force

The Coriolis force per unit mass, — 2Q x V, which is named after the French mathematician, G.G.de Coriolis(1792-1843), who first derived it, acts at right angle to the plane containing the earth's rotational axis and the relative velocity in the horizontal plane. It influences the direction of motion but not the speed of the wind. Since it changes the direction of the wind, it is often called the deviating force due to earth's rotation. The Coriolis force acts to the right of the wind direction in the northern hemisphere (NH) and to the left in the southern hemisphere (SH), as shown in Fig. 11.6.

The term giving the Coriolis force in (11.4.7) is in vector form. It is sometimes convenient to resolve it in a rectangular co-ordinate system with the x-axis pointing eastward, the y-axis pointing northward and the z-axis pointing vertically upward. In

EQUATOR

Fig. 11.6 The direction of the Coriolis force relative to wind direction

Fig. 11.7 Components of the earth's angular velocity vector, Q, at a given latitude, ^

this co-ordinate system, the earth's angular velocity, Q, has the components, Qx = 0; Qy = Qcos Qz = Q sinwhere ^ is the latitude, shown in Fig. 11.7.

If we take u, v, w as the three components of the wind vector V along the x, y, z axes respectively, the resolved components of the Coriolis force, — 2Q x V, along these axes, are respectively,

11.5 The Equations of Motion in a Rectangular Co-ordinate System

The components of the equation of motion (11.4.7) in a rectangular co-ordinate system (x, y, z) are then du/dt = —a dp/dx — 2Q(w cos ^ — v sin ^)+vd2u/dz2

dv/dt = —adp/dy — 2Q u sin ^ + vd2v/dz2 (11.5.1)

In (11.5.1), no vertical component of the frictional force has been added, since it is not regarded as of any consequence for the types of motion we usually consider in the atmosphere, in which motions are largely horizontal.

11.6 A System of Generalized Vertical Co-ordinates

It is sometimes useful and advantageous to use a vertical co-ordinate which is an independent single-valued monotonous function of height. Such a vertical coordinate can be pressure, potential temperature, or any other variable which can be expressed as a function of height. A vertical co-ordinate called sigma a (= p/ps, where ps is surface pressure) has been particularly useful to take care of the variable topography of the earth's surface. Let us take 'c' as the generalized vertical co-ordinate given by c = c(x, y, z, t). The word 'co-ordinate'here simply means a re-labelling of the vertical axis. It does not change the direction when changing from z to p (for example). Our task is then to derive an expression for the gradient of an arbitrary function a along the c = constant surface, when its gradient along a level surface (z = constant) or an isobaric surface (p = constant) is known.

Since a change Sa of a function a in the (x-z) plane is given by

where the first term on the right-hand side gives the x-component of the gradient of a along the z = constant surface, the variation of a in the x-direction along the c-surface (see Fig. 11.8) is given by:

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