1. Vorticity and the Maintenance of Rossby WavesWe have seen that
the vorticity associated with airflow has two components: Earth
vorticity and relative vorticity.Earth VorticityEarth vorticity arises
from the planet’s 24-hour rotation. A person sitting on a chair at the
North Pole undergoes one complete rotation each day relative to Earth’s
axis and thus has maximum Earth vorticity, Like the Coriolis force,
Earth vorticity increases with latitude so that it is at its maximum at
either pole and is nonexistent at the equator.Relative VorticityWe
define the second source of rotation, relative vorticity, in terms of
motions of air relative to the surface. Relative vorticity itself has
two sources. The first is the shear that occurs when the speed of a
fluid varies across the direction of flow. Figure 10-2-1a illustrates
this process as water flows through a channel (the same applies to air,
but we use water as an example simply because it is easier to
visualize). The flow is faster along the right bank. If a paddlewheel
was fixed in the middle of this stream, the faster-moving water to the
right would exert a greater force on the wheel than would the
slower-moving flow on the left. This would cause the wheel to rotate
counterclockwise and thereby undergo vorticity due to shear.The second
source of relative vorticity depends on the curvature of the flow, as
shown in Figure 10-2-1b, where the curved channel forces a fluid to turn
counterclockwise. As air flows through a Rossby wave, it undergoes this
type of curvature. As we said, relative vorticity is the sum of
curvature and shear; both are signed quantities. When they have the same
sign, they act in the same direction. When the signs are different,
they offset one another, perhaps completely. In general, shear is much
less important to the occurrence of relative vorticity than
curvature.Absolute Vorticity and Conservation of Angular
MomentumAbsolute vorticity, the overall rotation of a fluid, like
angular momentum, is conserved—that is, in the absence of intervening
forces, it remains constant. Figure 10-2-2 shows a Rossby wave in the
Northern Hemisphere. As the air flows southward, west of the trough
axis, its Earth vorticity decreases (recall that Earth vorticity
decreases toward the equator). But because absolute vorticity is
conserved, an increase in relative vorticity compensates for the
decrease in Earth vorticity, causing the air to turn counterclockwise.
Then, as it starts to flow poleward, the Earth vorticity increases.
Thus, the air turns back to its right and once again exhibits negative
relative vorticity. (For the sake of clarity, we are assuming no shear
exists.)In short, there is a constant trade-off between Earth and
relative vorticity. As the air moves poleward, it assumes a greater
clockwise rotation relative to the surface; as it moves toward the
equator, it turns in a counterclockwise manner. Such reversals in
relative vorticity, along with the conservation of angular momentum,
help maintain Rossby waves.FIGURE 10-2-1 Relative Vorticity.(a) Relative
vorticity by shear, occurs when a fluid moves at a differential speed
across the direction of flow. A paddle wheel fixed across the fluid
rotates counterclockwise as a greater forward stress is exerted to the
right of the direction of flow.(b) Curvature in the direction of flow
can also produce relative voracity....FIGURE 10-2-2 Maintenance of a
Rossby Wave. As the air moves poleward from position 1, the gain in
Earth vorticity is compensated for by a decrease in relative vorticity.
As the parcel approaches position 2, the relative vorticity becomes
negative and the parcel turns back equatorward. At position 3, the
reduction of Earth vorticity causes an increase in relative vorticity,
and the air again turns poleward....What are the two sources of relative
vorticity? Get solution
2. Vorticity and the Maintenance of Rossby WavesWe have seen that the vorticity associated with airflow has two components: Earth vorticity and relative vorticity.Earth VorticityEarth vorticity arises from the planet’s 24-hour rotation. A person sitting on a chair at the North Pole undergoes one complete rotation each day relative to Earth’s axis and thus has maximum Earth vorticity, Like the Coriolis force, Earth vorticity increases with latitude so that it is at its maximum at either pole and is nonexistent at the equator.Relative VorticityWe define the second source of rotation, relative vorticity, in terms of motions of air relative to the surface. Relative vorticity itself has two sources. The first is the shear that occurs when the speed of a fluid varies across the direction of flow. Figure 10-2-1a illustrates this process as water flows through a channel (the same applies to air, but we use water as an example simply because it is easier to visualize). The flow is faster along the right bank. If a paddlewheel was fixed in the middle of this stream, the faster-moving water to the right would exert a greater force on the wheel than would the slower-moving flow on the left. This would cause the wheel to rotate counterclockwise and thereby undergo vorticity due to shear.The second source of relative vorticity depends on the curvature of the flow, as shown in Figure 10-2-1b, where the curved channel forces a fluid to turn counterclockwise. As air flows through a Rossby wave, it undergoes this type of curvature. As we said, relative vorticity is the sum of curvature and shear; both are signed quantities. When they have the same sign, they act in the same direction. When the signs are different, they offset one another, perhaps completely. In general, shear is much less important to the occurrence of relative vorticity than curvature.Absolute Vorticity and Conservation of Angular MomentumAbsolute vorticity, the overall rotation of a fluid, like angular momentum, is conserved—that is, in the absence of intervening forces, it remains constant. Figure 10-2-2 shows a Rossby wave in the Northern Hemisphere. As the air flows southward, west of the trough axis, its Earth vorticity decreases (recall that Earth vorticity decreases toward the equator). But because absolute vorticity is conserved, an increase in relative vorticity compensates for the decrease in Earth vorticity, causing the air to turn counterclockwise. Then, as it starts to flow poleward, the Earth vorticity increases. Thus, the air turns back to its right and once again exhibits negative relative vorticity. (For the sake of clarity, we are assuming no shear exists.)In short, there is a constant trade-off between Earth and relative vorticity. As the air moves poleward, it assumes a greater clockwise rotation relative to the surface; as it moves toward the equator, it turns in a counterclockwise manner. Such reversals in relative vorticity, along with the conservation of angular momentum, help maintain Rossby waves.FIGURE 10-2-1 Relative Vorticity.(a) Relative vorticity by shear, occurs when a fluid moves at a differential speed across the direction of flow. A paddle wheel fixed across the fluid rotates counterclockwise as a greater forward stress is exerted to the right of the direction of flow.(b) Curvature in the direction of flow can also produce relative voracity....FIGURE 10-2-2 Maintenance of a Rossby Wave. As the air moves poleward from position 1, the gain in Earth vorticity is compensated for by a decrease in relative vorticity. As the parcel approaches position 2, the relative vorticity becomes negative and the parcel turns back equatorward. At position 3, the reduction of Earth vorticity causes an increase in relative vorticity, and the air again turns poleward....Explain why there is a trade-off between Earth vorticity and relative vorticity as air flows along a Rossby wave. Get solution
2. Vorticity and the Maintenance of Rossby WavesWe have seen that the vorticity associated with airflow has two components: Earth vorticity and relative vorticity.Earth VorticityEarth vorticity arises from the planet’s 24-hour rotation. A person sitting on a chair at the North Pole undergoes one complete rotation each day relative to Earth’s axis and thus has maximum Earth vorticity, Like the Coriolis force, Earth vorticity increases with latitude so that it is at its maximum at either pole and is nonexistent at the equator.Relative VorticityWe define the second source of rotation, relative vorticity, in terms of motions of air relative to the surface. Relative vorticity itself has two sources. The first is the shear that occurs when the speed of a fluid varies across the direction of flow. Figure 10-2-1a illustrates this process as water flows through a channel (the same applies to air, but we use water as an example simply because it is easier to visualize). The flow is faster along the right bank. If a paddlewheel was fixed in the middle of this stream, the faster-moving water to the right would exert a greater force on the wheel than would the slower-moving flow on the left. This would cause the wheel to rotate counterclockwise and thereby undergo vorticity due to shear.The second source of relative vorticity depends on the curvature of the flow, as shown in Figure 10-2-1b, where the curved channel forces a fluid to turn counterclockwise. As air flows through a Rossby wave, it undergoes this type of curvature. As we said, relative vorticity is the sum of curvature and shear; both are signed quantities. When they have the same sign, they act in the same direction. When the signs are different, they offset one another, perhaps completely. In general, shear is much less important to the occurrence of relative vorticity than curvature.Absolute Vorticity and Conservation of Angular MomentumAbsolute vorticity, the overall rotation of a fluid, like angular momentum, is conserved—that is, in the absence of intervening forces, it remains constant. Figure 10-2-2 shows a Rossby wave in the Northern Hemisphere. As the air flows southward, west of the trough axis, its Earth vorticity decreases (recall that Earth vorticity decreases toward the equator). But because absolute vorticity is conserved, an increase in relative vorticity compensates for the decrease in Earth vorticity, causing the air to turn counterclockwise. Then, as it starts to flow poleward, the Earth vorticity increases. Thus, the air turns back to its right and once again exhibits negative relative vorticity. (For the sake of clarity, we are assuming no shear exists.)In short, there is a constant trade-off between Earth and relative vorticity. As the air moves poleward, it assumes a greater clockwise rotation relative to the surface; as it moves toward the equator, it turns in a counterclockwise manner. Such reversals in relative vorticity, along with the conservation of angular momentum, help maintain Rossby waves.FIGURE 10-2-1 Relative Vorticity.(a) Relative vorticity by shear, occurs when a fluid moves at a differential speed across the direction of flow. A paddle wheel fixed across the fluid rotates counterclockwise as a greater forward stress is exerted to the right of the direction of flow.(b) Curvature in the direction of flow can also produce relative voracity....FIGURE 10-2-2 Maintenance of a Rossby Wave. As the air moves poleward from position 1, the gain in Earth vorticity is compensated for by a decrease in relative vorticity. As the parcel approaches position 2, the relative vorticity becomes negative and the parcel turns back equatorward. At position 3, the reduction of Earth vorticity causes an increase in relative vorticity, and the air again turns poleward....Explain why there is a trade-off between Earth vorticity and relative vorticity as air flows along a Rossby wave. Get solution
