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In (12.3.3), dV/dt denotes the tangential acceleration, while V2/R gives the radial acceleration of the moving parcel towards the center of the curve. The latter is also called the centripetal acceleration, since for a unit mass of the parcel it signifies the force with which it is continuously attracted towards the center while it moves along the curve. The curvature effect produces a centrifugal acceleration along the outward normal. Thus, the centripetal and the centrifugal accelerations are equal and opposite of each other.
Since the direction in which the Coriolis force acts is at right angle to the velocity vector, we may write it in the natural co-ordinates, as follows:
Also, the horizontal pressure gradient may be written
Using (12.3.4) and (12.3.5), the horizontal momentum equations (12.2.1) may be resolved into the tangential and radial components as follows:
It follows from (12.3.6) that if the pressure does not vary along the curve, i.e., dp/ds = 0, the tangential wind speed V remains constant along an isobar. The wind blowing along the curved path is called the gradient wind and (12.3.7) is the famous gradient wind equation.
The approximate balance wind that appears to closely conform to the observed wind in the field of a large-scale motion system on a synoptic weather map is the gradient wind given by (12.3.7) which represents a three-way balance between the pressure gradient force, Coriolis force and the centrifugal force in a circular p-field with radius R and is obtained by solving (12.3.7) for V.
Since (12.3.7) is a quadratic equation, its two roots are given by
V = -f R/2± [(f2 R2/4) - (R/p)(dp/dn)]1/2 (12.3.8)
Depending upon the values and signs of R and dp/dn, we may have several possible values of V in (12.3.8), but some of them are clearly not admissible because of the requirement that V be real and non-negative. No solution of (12.3.8) is admissible in which the quantity under the radical is negative. The admissible values of V, however, are found to be better estimates of the observed winds than the geostrophic winds. Fig. 12.2 shows the balance between the forces for the two types of gradient flow commonly observed in the northern hemisphere: (a) a regular low; (b) a regular high.
A point to note here is that the force that deviates the down-the-pressure-gradient movement to a movement along the isobar is the Coriolis force.
In the event of a lack of balance amongst the forces, it is not impossible to have an anomalous flow, such as an anticyclonic flow around a low pressure area, but such anomalies are rare and may occur as a passing phase only when dV/dt is large, i.e., does not obey (12.3.7).
12.3.3 The Geostrophic Wind
In the very restrictive case in which an isobar is straight, i.e., has no curvature, R tends to be infinity and the curvature term V2/R in (12.3.7) vanishes. In that case, (12.3.7) reduces to the balance relation fV = -(1/p) dp/dn Or, Vg = —(1/pf) dp/dn (12.3.9)
Equation (12.3.9) represents a balance between the Coriolis force and the pressure gradient force and is called geostrophic balance. The wind speed Vg is called the geostrophic wind speed and denoted by Vg. The balance is shown schematically in Fig. 12.3.
Fig. 12.3 Geostrophic balance between the pressure gradient force (denoted by PG) and the Coriolis force (denoted by CO) when the isobar has no curvature L denotes Low pressure, H High
12.3.4 Relationship Between the Geostrophic Wind and the Gradient Wind
We have seen from (12.3.9) that in natural co-ordinates, the geostrophic wind is given by the relation
Vg = -(1/pf) dp/dn Substituting this value in the expression for the gradient wind (12.3.7), we obtain
The ratio Vg /V may be evaluated for different types of atmospheric flow. In general, in the northern hemisphere cyclonic flow in which f R is positive, Vg is larger than V, whereas in anticyclonic flow in which f R is negative, Vg is smaller than V. In middle and high latitudes, the difference between Vg and V seldom exceeds 10-20 %. In low latitudes, however, the term V/fR, which is defined as the Rossby number and usually denoted by Ro, is of the order of unity and the difference between V and Vg is large enough to justify use of the gradient wind instead of the geostrophic wind.
When the horizontal pressure field is flat, i.e., there is no pressure gradient in any direction, the balance relation (12.3.7) reduces to the simple form
According to (12.3.6), V is constant in such a case. Further, if we consider a flow at a fixed latitude ^ where f is constant, R becomes constant. This means that the flow will be along a circle (in the clockwise sense) with constant radius R, which we call an inertia circle.
The period of the inertial oscillation, P, is given by
The period P is the time taken by a Foucault pendulum to turn through 180°.
Pure inertial oscillation does not appear to be of any consequence in the atmosphere except for land-sea breeze and the nocturnal jet (see Chap. 14). However, in the ocean where the flow velocities are much slower than in the atmosphere, the radius of the inertial circle is much smaller and significant amount of energy has been detected in currents which oscillate with the inertial period.
If the horizontal scale of the curved flow is very small, the curvature term, V2/R, in (12.3.7) may become much more important than the Coriolis term, fV. In that case, the curvature term may alone balance the pressure gradient term. Thus, where V gives the speed of the cyclostrophic wind, and the radius R is positive in a cyclonic flow, negative in anticyclonic flow.
Figure 12.4 is a schematic showing the balance of forces in a cyclostrophic flow; (a) cyclonic, (b) anticyclonic.
The cyclostrophic flow can be cyclonic or anticyclonic. In either case, the pressure gradient force is always directed towards the center of curvature and the centrifugal force away from it. In (a), R is positive, but dp/dn is negative, whereas in (b) R is negative, but dp/dn is positive.
(V2/R) = —(1/p) dp/dn or, V = {(—R/p) dp/dn}1/2
Fig. 12.4 Balance of forces in cyclostrophic flow: (a) cyclonic, (b) anticyclonic. PG denotes pressure gradient force, and CF the centrifugal force
Fig. 12.4 Balance of forces in cyclostrophic flow: (a) cyclonic, (b) anticyclonic. PG denotes pressure gradient force, and CF the centrifugal force
Cyclostrophic flow usually occurs in fast-rotating small-scale atmospheric motion, such as a tornado or a midget cyclone. On account of very high speed of rotation, the ratio of the centrifugal force to the Coriolis force in such a flow, which is given by V/f R, is very large. We can have an idea of the order of magnitude of Ro in the case of a typical tornado from the following realistic values V, f and R:
These values give Ro — 105. A value of Ro of this magnitude stands in sharp contrast with that for large-scale flows in middle and high latitudes where Ro << 1. This provides a strong justification for neglecting the Coriolis force in comparison with the centrifugal force in cyclostrophic flow, especially in low latitudes. Also, an important point to note here is that when the scale of motion is reduced still further, as in a dust devil or water spout, the sense of rotation can be either cyclonic or anticyclonic.
The reason for this loss of control on the sense of rotation is not difficult to visualize. It is the Coriolis force which forces a parcel of air starting to move down the pressure gradient towards the center of low pressure to deviate from its movement till it attains a balance with the pressure gradient force and moves along the isobar. So when the deviating force is negligible, there is a tug of war between the centrifugal force and the pressure gradient force and the resulting sense of rotation can be somewhat arbitrary. In fact, it can be as often cyclonic as anticyclonic.
It is important to distinguish between two terms frequently used in meteorology in regard to air motion. These are: trajectories and streamlines. A trajectory may be defined as the path traced out by an individual moving parcel of air in a given period of time. In other words, if 5s is a small segment of the path traced out by the parcel in time 5t at wind speed V, the trajectory may be found by integrating the relation, ds/dt = V(x, y, t) (12.4.1)
On the other hand, a streamline is defined as a straight or curved line which is everywhere tangential to the instantaneous wind vector V. The condition for this tangency is V x dr = 0, where dr is a length segment vector along the line. Thus, a streamline is defined by the relation, dy/dx = v(x, y, t0)/u(x, y, to) (12.4.2)
where u, v are the components of the wind vector at location(x,y) at fixed time t0. The streamline may be obtained by integrating (12.4.2).
It is important to note that the streamline refers to the position of a moving parcel of air at a particular instant of time and not to its successive positions in points of time which are given by a trajectory.
Trajectories and streamlines will coincide only in steady-state systems in which there is no local change of velocity with time. To find out a relation between the curvature of a trajectory and that of a streamline in a moving pressure system, we proceed as follows:
Let 5p be the angle between the directions of the wind while it moves over a length 5s of the curve. Then, if Rt and Rs be the radii of the trajectory (t) and the streamline(s) respectively, we may write dp/ds = 1/Rt; and, dp/ds = 1/Rs (12.4.3)
The rate of change of the wind direction following the motion may be written dp/dt = V d p/ds = dp/dt + V dp/ds
Substituting from (12.4.3), we get for the local turning of the wind the relation dp/dt = V (1/Rt — 1/Rs) (12.4.4)
Equation (12.4.4) states that the wind direction remains constant only if the curvatures of the trajectory and the streamline are the same.
It is interesting to see how the local turning of the wind occurs when a pressure system moves relative to some large-scale wind. For simplicity, let us assume that a cyclonic circular isobaric pattern is moving eastward without change of shape with a constant velocity C. Let us further assume that the wind circulating in the pressure system is given by the gradient wind equation (12.3.7). Since the isobars are streamlines, the local turning of the wind is entirely due to the motion of the isobars.
where Z is the angle between the streamlines and the direction of motion of the pressure system.
Substituting from (12.4.3) and (12.4.4) into (12.4.5), we get
We can use (12.4.6) to compute the curvature of the trajectory in any part of a moving pressure system. An example is shown in Fig. 12.5, after Holton (1979), in which a low pressure system with circular isobars moves eastward with a constant speed C relative to the wind speed V.
Figure 12.5 shows the type of trajectories that result for two speeds of the moving system relative to the wind: (a) when C = V/2 and (b) when C = 2V. The curvatures of the trajectories are shown for parcels initially located at the north, east, south and west of the center of the low for both the cases. For simplicity, the geostrophic wind
is used for the computation instead of the gradient wind so that the wind blows along the isobars everywhere and it is assumed that there is no variation of the wind with distance from the center. The curvature of the trajectory is shown at successive times after leaving the initial location.
Streamline-isotach analysis is a common feature of weather maps in low latitudes, while isobaric analysis is preferred in midlatitudes. The distinction is a matter of the most information being in wind or pressure observations. A free-hand streamline-isotach analysis is hardly satisfactory, especially when wind observations on a weather map are few and far between. For this reason, it is recommended that the analysis be based on the use of isogons which are lines joining places which have the same wind direction and on which short line segments may be drawn across each isogon indicating the actual wind direction. The line segments so drawn are then connected by tangent curves. These curves are the desired streamlines. It is usually satisfactory to draw isogons at intervals of 30o, but in regions of small variations of wind direction they may be drawn at intervals of 15 or 10o. This method of drawing streamlines is due to Sandstrom (1910). Petterssen (1956) has described the Sandstrom's method of drawing streamlines and isotachs in great detail. Isotachs are lines drawn through places of equal wind speed. However, if wind observations are scanty, use may be made of certain pressure-wind relationships to obtain additional speed values between observations. For details of this method, the reader may refer to Petterssen's or any other standard book on weather analysis. In Fig. 12.6, we give an example of a streamline-isotach analysis which was prepared by the meteorological group of the International Indian Ocean Expedition (IIOE) (1962-1966) with the data then available.
After introduction of computer analysis of meteorological data as part of data assimilation schemes for numerical models, the traditional streamline-isotach analysis has gradually gone out of use in some of the advanced countries of the world.
12.6 Variation of Wind with Height - The Thermal Wind
The variation of wind with height depends upon the distribution of density which, by the ideal gas law, is a function of temperature and pressure. When the air over a region is cold, the pressure falls off more rapidly with height than when it is warm. In other words, the thickness of an isobaric layer is smaller in cold air than in warm air. This follows from the hydrostatic equation and the ideal gas law and we may write the thickness equation in the form
where pQ, p1 are the pressures at geopotential heights O0 , 01 respectively, and T is the mean temperature of the layer (see Fig. 12.7).
Now, if there are two neighboring regions, one cold and the other warm, a horizontal temperature gradient exists between them, the effect of which upon the horizontal pressure gradient will lead to a vertical variation of the wind between the two regions. If we assume the wind to be largely geostrophic, we can compute the variation of the wind with height as follows:
where Vg is the geostrophic wind vector, f is the Coriolis parameter, and VpO is the horizontal gradient of geopotential O along an isobaric surface. We differentiate (12.6.2) with respect to pressure and obtain d Vg/dp = (1/f) k x Vp (dO/dp) (12.6.3)
Since, by the ideal gas law and the hydrostatic approximation, dO/dp = —RT/p, where T is the mean temperature of the layer, we have, by substitution and rearrangement, dVg/dp = —(R/fp)(k x VpT) (12.6.4)
where VpT is the horizontal temperature gradient in the isobaric layer. Integrating (12.6.4) between pressure surfaces pQ and p1, we get
(Vg)p1 — (Vg)p0 = (R/f)(k x VpT) ln(poM) (12.6.5)
If we denote the vector difference between the geostrophic winds at the two pressure surfaces by VT and call it the thermal wind, then
Fig. 12.7 Thickness of an isobaric layer in warm and cold air
Equation (12.6.6) is the well-known thermal wind equation which controls the vertical variation of the wind with height in a region where there exists a horizontal gradient of temperature.
The components of the thermal wind vector along the x and the y axes may then be written as ut = — (R/f)(dT/dy)p ln(po/p1) vt = (R/f)(dT/dx)p ln (po/p1)
Equation (12.6.7) states that if, in the northern hemisphere, the temperature decreases with latitude, the westerly wind will strengthen with height, while the easterly wind will weaken. However, if the temperature increases with latitude, the westerly wind will weaken, while the easterly will strengthen with height. Examples of the former type are found in the formation of the westerly jetstream at about 250 mb in middle and high latitudes and of the latter type in the formation of the easterly jetstream at about 150 mb over the tropics. Similarly, (12.6.8) states that if the temperature increases (decreases) eastward, the southerly wind will strengthen (weaken) with height in the northern hemisphere. In other words, in the northern hemisphere, the thermal wind will blow so that it keeps cold air to the left and warm air to the right. The direction will reverse in the southern hemisphere. Fig. 12.8 illustrates how a northerly wind will turn with height in the two hemispheres.
Chapter 13
Circulation, Vorticity and Divergence
13.1 Definitions and Concepts - Circulation and Vorticity
Circulation and vorticity are two important parameters of a rotating motion. In the atmosphere, circulation consists of the physical movement of a parcel of air along the closed boundary of a surface area, while vorticity is the tendency of an infinites-imally small area of that surface to turn about an axis normal to it in the same sense as the circulation (see Fig. 13.1).
If V denotes the velocity of a parcel of air and dl a line segment along the boundary of a closed surface, the circulation C along the boundary is measured by the line integral
By convention, C is treated as positive if the integration is carried out in the anti-clockwise direction, and negative if done in the clockwise direction.
Vorticity, on the other hand, is a vector field which gives a microscopic measure of the turning of the air over a unit area of a surface about an axis normal to it in the same sense as the circulation and is measured by the curl of the velocity vector, i.e., by VxV. If we consider a surface area S the boundary of which is given by the length l, then, according to Stokes's theorem?
circulation and vorticity
Fig. 13.1 Illustrating sj
where n is a unit vector normal to the unit area of the surface and the integral on the left-hand side is taken around the whole length l.
It may be noted in Fig. 13.1 that the flows along the common borders of all the unit areas cancel being in opposite directions except those along the outer borders which add up to the circulation along the outer boundary of the surface.
Thus, Stokes's theorem connects circulation and vorticity in solid-body rotation and the relationship is given by
Circulation = vorticity x area (13.1.3)
The meaning of the term 'vorticity' can be further illustrated by taking the case of rotation of a solid circular disk with an angular velocity m. A point at a distance r from the center of the disc has a linear velocity rm, so the circulation at this distance is 2nr2m, where 2nr is the circumference of the circle of radius r. Dividing this by nr2 which is the area of the circle, we obtain by (13.1.3) the vorticity of the disc as 2m.
The problem of atmospheric circulation and vorticity was first studied by Helmholtz about the middle of the nineteenth century. He was followed by V. Bjerknes who in 1898 derived his famous circulation theorem. Here we give a brief review of his theorem. Let us first derive it in the absolute frame of reference by taking the line integral of the Newtonian equations of motion (11.3.2), by neglecting friction. Thus, we start with the equation j) (dVa/dt)a-dl = -j aVpdl -j VOdl (13.2.1)
where the subscript 'a' denotes absolute motion in an absolute frame of reference. The integrand on the left-hand side of (13.2.1) can be written
(dVa/dt)a-dl = d(Va-dl)a/dt - Va ■ (d(dl)/dt)a (13.2.2)
Here, since l is a position vector, (dl/dt)a = Va, and the second term on the right-hand side of (13.2.2) reduces to Va-(dVa). But Va-dVa = (V2)d(Va-Va). Hence, by substitution from (13.2.2), (13.2.1) may be written as
^d(Va • dl)a/dt = -f a dp -f dO + (1/2)^d(Va-Va) (13.2.3)
The line integral of a perfect differential being zero, the second and the third terms on the right-hand side of (13.2.3) disappear, and we are left with dCa/dt = -j> a dp (13.2.4)
where we have written Ca for /(Va-dl)a which constitutes absolute circulation, and the term - a dp gives the number of unit isosteric-isobaric solenoids enclosed by the circulation curve.
In a barotropic atmosphere, where the density or specific volume is a function of pressure only, the isosteric (a = constant) surfaces coincide with the isobaric (p = constant) surfaces and the solenoidal term vanishes and the absolute circulation remains constant. This is the well-known Kelvin circulation theorem for the conservation of absolute circulation in a frictionless barotropic atmosphere. It corresponds to the law of conservation of absolute angular momentum in solid-body rotation in classical mechanics.
The real atmosphere, however, is almost always baroclinic and the solenoidal term plays an important role in atmospheric circulation. In Fig. 13.2(a, b) we show the distinction between the two types of atmospheres, so far as the distribution of the isosteric and isobaric surfaces are concerned. In a barotropic atmosphere, as there are no solenoids, there is no mechanism to change the circulation with time. On the other hand, in a baroclinic atmosphere, the surfaces of pressure (p) and specific volume (a) intersect producing isobaric-isosteric solenoids which bring about a baroclinic circulation, as shown in Fig. 13.2 (b). The direction of the circulation is found by turning the Va vector through an angle (< 180°) so that it coincides with the — Vp vector.
Examples of baroclinic circulation of the type shown in Fig. 13.2 (b) may be found in the generation of land and sea breezes and large-scale monsoons which are driven by differential heating between two parts of the earth's surface, usually between continents and neighbouring oceans.
cold warm
Fig. 13.2 (a) A barotropic atmosphere with no solenoids; (b) A baroclinic atmosphere with isosteric (a)-isobaric (p) solenoids. C - cold, W -warm. The arrow in circle shows the direction of circulation, turning the Va vector towards the - Vp vector cold warm
Fig. 13.2 (a) A barotropic atmosphere with no solenoids; (b) A baroclinic atmosphere with isosteric (a)-isobaric (p) solenoids. C - cold, W -warm. The arrow in circle shows the direction of circulation, turning the Va vector towards the - Vp vector
In meteorology, however, it is more convenient to work with the circulation produced by relative motion which is obtained by subtracting from the absolute circulation the circulation due to the rotation of the earth. Now, the circulation due to earth's rotation, denoted by Ce, may be expressed as / Vedr, where Ve = Qxr, and dr is a line segment along the curved path of the circulation at the position vector r. However, by Stokes's theorem, where dA is a small area of the earth's surface at latitude and the unit vector n is normal to the surface. If we choose the surface to be horizontal, then n is along the local vertical pointing outward. In that case, (Vx Ve)-n = 2Q sin ^ = f , where f is the Coriolis parameter at latitude ^ (see Fig. 13.3).
Equation (13.2.5) then yields for the rate of change of circulation due to earth's rotation, dCe/dt, the expression where Ae (= Asin^) is the projection of the surface A onto the equatorial plane.
If we denote the relative circulation by C, its rate of change, dC/dt, is given by the difference between (13.2.4) and (13.2.6). Thus, we get
This is the well-known Bjerknes circulation theorem. It emphasizes the fact that on the rotating earth, the rate of change of a given circulation around an area is determined by the number of isobaric-isosteric solenoids enclosed by the area and the rate of expansion or contraction of the area with latitude.
Pole
Fig. 13.3 Projection of the area 8A of the earth's surface onto the equatorial plane
13.3 Absolute and Relative Vorticity
The absolute vorticity is defined as the curl of the absolute velocity, VxVa, whereas the relative vorticity is given by the curl of the relative velocity, VxV. The velocity field V being three-dimensional, the vorticity field is also three-dimensional. However, in meteorology, we are largely concerned with horizontal motion and hence the vertical component of the vorticity. If n denotes the vertical component of the absolute vorticity and Z that of the relative vorticity, then we may write
The difference between the absolute and the relative vorticity is the vertical component of the vorticity due to the rotation of the earth, given by k-V xVe,, which, as we have shown earlier, is equal to f, the Coriolis parameter.
Thus, for largely horizontal circulation, in Cartesian co-ordinates,
13.4 Vorticity and Divergence in Natural Co-ordinates
The physical interpretation of vorticity in an atmospheric flow or circulation is often facilitated by expressing it in a natural co-ordinate system. The same remark holds for divergence which may be defined as the rate of increase of an area or volume per unit area or volume. For this, we take a streamline in a rectangular co-ordinate system and assume that at a point P along the streamline, the velocity is V (u, v) and it makes an angle p with the x-axis (see Fig. 13.4).
Then, the components of the speed are: u = Vcos p along the x-axis, and v = Vsin p along an axis at right angles to it.
Differentiating u with respect to y and v with respect to x, we obtain n = kV xVa, Z = k V xV
Fig. 13.4 Vorticity and divergence in natural co-ordinate system
dv/dx = (dV/dx) sin p + (Vcos p)dp/dx du/dy = (dV/dy)cosp - (Vsinp)dp/dy
If we now take a point O along the streamline where the x-axis is tangent to the streamline and measure the distance s along the streamline and n at right angles to it, and note that at O, p = 0, we can write vorticity(Z) and divergence(D) in the form
The interpretation of (13.4.1) is simple. The vorticity Z is positive if the streamline has a cyclonic curvature (anti-clockwise direction) and also if there is a cyclonic wind shear with speed increasing along the outward normal. On a streamline-isotach map, therefore, the maximum positive vorticity between a low and a high pressure area will be found where the streamline has both maximum cyclonic curvature and positive wind shear and the maximum negative vorticity where the streamline has both maximum anticyclonic curvature and negative wind shear. Similarly, it follows from (13.4.2) that dV/ds represents a stretching of the flow downstream and V dp/dn represents the effect of divergence or convergence of the streamlines. Thus, divergence is positive where the wind speed increases downstream and where air tends to stream in isobaric channels in which the speed varies inversely with the width of the channel. This is due to the circumstance that the pressure gradient force is very nearly balanced by the Coriolis force and divergence in the large-scale aircurrents is a very small quantity (^ 10_6s_1).
An example, in the case of a W'ly jetstream (J) in the northern hemisphere, is shown in Fig. 13.5.
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