### 5 -- ATMOSPHERIC DYNAMICS

In previous sections, we have explored various aspects of Earth's atmosphere. These have included the atmosphere's composition, its vertical temperature structure and layering, its horizontal wind and temperature fields, and large-scale atmospheric circulation patterns that drive the observed weather and affect the distribution of ozone. We've discussed the wind fields and temperature fields mostly in a qualitative sense. Key concepts like why tight horizontal temperature gradients drive zonal jet streams have been presented but not explained in a dynamical sense. In this section, we explore the area of atmospheric dynamics in order to understand why the general circulation pattern behaves as it does. A good dynamics understanding can unify the previous sections so that they make sense. This includes the general circulation pattern; the observed distribution of temperature and how it varies with latitude, height, and season; and periodic phenomena like QBOs and ENSOs. We can also understand how these dynamical processes drive the observed distribution of ozone, as explored in Chapter 6.

#### 5.1 Geostrophic Balance

All motion in the atmosphere is driven by two key processes: differential heating ultimately provided by the Sun and the spinning of the planet. The first, the differential heating, causes differing temperatures across the surface of Earth. This, in turn, sets up air pressure differences (as explained quantitatively by the equation of state in section 3.2). The second, the spinning of Earth on her axis, provides an acceleration to all objects on the surface. This includes the atmosphere itself, the individual molecules and groupings of molecules considered as air parcels. It is the balance of this horizontal pressure gradient and the acceleration provided by the spinning of the planet that is the dominate force balance in the atmosphere. It is called geostrophic equilibrium or geostrophic balance and it determines the horizontal structure of motion in the atmosphere. The winds that blow as a result of this force balance are called geostrophic winds.

5.1.1 Geostrophic winds -- Newton's Second Law says that the force acting on an object equals the product of its mass and its acceleration. One force acting on an air parcel is the pressure gradient force: that force that causes acceleration from higher pressure to lower pressure. As an analogy, consider the pressure gradient that exists at the nozzle of your vacuum cleaner when it is turned on. A motor inside your vacuum creates a region of relatively low pressure contrasted with the region of higher pressure outside the vacuum cleaner. The dust and air molecules outside the vacuum, being at a higher pressure, feel a force pulling it toward the low pressure inside the vacuum.

In the atmosphere, regions of high and low pressure form ultimately as a result of unequal heating. In moving from a high pressure region to a low pressure region, you are said to be moving along the pressure gradient. The pressure gradient force itself points in the direction of the lower pressure. In the vacuum cleaner example, this is evidenced by the fact that air is sucked into the lower pressure inside. The magnitude of the pressure gradient force is proportional to the magnitude of the pressure gradient itself. The lower the pressure inside the vacuum cleaner, the more powerful the vacuum will be. In the atmosphere, pressure gradient is related to a quantity called geopotential height. The pressure gradient force is proportional to the geopotential height gradient. It is perpendicular to lines of constant geopotential and points towards lower values of geopotential height. The next section discusses geopotential height in detail. It includes some elementary calculus.

5.1.2 Geopotential height -- Because Earth rotates and becaues it is not a perfect sphere, the acceleration due to gravity of any object, including air parcels, will experience two additional components beyond that due to the radial gravitational pull of the planet itself. These two other components are (1) a centrifugal acceleration as experienced in the rotating (non-inertial) frame of reference, and (2) so called "anisotropic" contributions arising from the fact that the planet is not a homogeneous, perfect sphere.

The noninertial frame of reference introduces an apparent centrifugal acceleration that is perpendicular to and directed away from the axis of axis of rotation. The variations in surface terrain and composition, and also the variations due to the slight spheroidal shape of the planet (which possesses a bulge at the equator and hence has the shape of an oblate spheroid) produce minor variations in gravity across the surface. The combination of these terms produce an "effective gravity" in which there is a horizontal component of gravity. This component of gravity effects the hydrostatic balance (see section 3.2.1) and makes determining the balance of forces acting at a given altitude more complicated than necessary. It is for this reason that geopotential coordinates are used, since such coordinates places all the components of gravity into a single vertical coordinate.

A gravitational potential function or geopotential function is defined in which a small change in geopotential equals the amount of work performed against the force of gravity for some small displacement. In formulaic terms:

d = dwg

where wg refers to the work done against gravity. Recall that work is defined as force through a distance (work=force times distance). In this case, the force is gravity (g) and the distance is a slight vertical displacement, dz. The work is done in the opposite sense of gravity, so we need a negative sign. Thus,

d = - g dz

where the vertical elevation z refers to height in place of geometric altitude. That is, z is constant along surfaces of constant geopotential instead of surfaces of constant geometric altitude. These surfaces of constant geopotential allow us to consolidate all the components of gravity into a single vertical coordinate. We can do this, since increases uniformly with height. As for gravity, it can be expressed as a potential function

g = -

By using sea level as z=0 as a reference value of geopotential and integrating from sea level to a height z, we obtain a final expression for geopotential

To simplify things, we assume a constant value for g, in particular a global average of g, which we call go. We now solve for dz, where we denote z as Z for g=go. This gives us our mathematiccal expression for geopotential height

Z = (1/go)(z)

where go is a global average of gravity taken as 9.8 m/s2.

We can now return to our expression for hydrostatic balance, which we expressed previously in non-integral form as

P = dg h

and putting it in differential form,

dP= -dg dz

In terms of geopotential height Z, where g=go , a small change in pressure is

dP= -dgo dZ

What is the usefulness of geopotential height? For one thing, when elevation is represented in terms of geopotential height, the horizontal components of gravity vanish, and we are left with gravitational effects that are solely in the vertical. In addition, when g is represented by go, atmospheric motions are simplified, and we can represent the hydrostatic balance of the atmosphere in terms of gravity. Since the hydrostatic balance takes into account the equation of state, we can relate geopotential height to temperature. As temperature drops, geopotential height drops. As temperature rises, geopotential height rises. We can now restate the conclusion of the last section on the geostrophic wind (section 5.1.1). The pressure gradient force is proportional to the geopotential height gradient and is perpendicular to lines of constant geopotential. In the same way that he pressure gradient force also points from higher to lower values of pressure, it also points from higher to lower values of geopotential height. Air, without taking into account the spinning of Earth, moves from higher to lower pressure and from higher to lower geopotential height.

5.1.3 The Coriolis force -- Observations of motions in the real atmosphere, however, show that air does not blow directly from high to low pressure centers, but instead often flows around the pressure centers. Thus, other forces must be acting. In particular, one major force is acting. It is the Coriolis force. It is a not a real force in the sense that it does any work, but instead it is an "apparent force," arising from the fact that Earth turns on its axis. It is an apparent force that makes sense only because Earth is a noninertial frame of reference. The spinning of Earth creates a constant centrifugal acceleration in which objects appear to curve. If instead of being a spherical, rotating planet, Earth were flat, there would be no Coriolis force. This is because Earth is spherical, with different points on the surface spinning at different speeds. If all points spun at the same speed, as would happen on a flat plane, there would also be no apparent Coriolis force.

The Coriolis force arises from a physical principle known as the conservation of angular momentum. It is really a simple principle that you've witnessed many times in your life.

The classic analogous example to the Coriolis force is the figure ice skater in a spin. She begins spinning with her arms extended away from her body. As she starts to spin, she pulls her arms in towards her body, spinning ever faster all the time. She usually ends up with her arms directly overhead and spinning very, very fast. Why? Her arms act as weights. When extended away from the body, they are far from the axis of rotation (which in a good spin can be found passing through the skater's head, directly down her spine, and through her legs, passing perpendicularly into the ice surface below her). Angular momentum is found to be related to mass, its distance from the axis of rotation, and the rotation speed. When the skater's arms are extended, the mass is far from her body and her rotation speed is correspondingly slow. As the skater pulls her arms in towards her body (towards the axis of rotation), the rotation speed increases. This tells us empirically that the product of mass times distance is inversely related to the rotation speed: the further the mass is from the rotation axis, the slower the rotation; the closer the mass is to the rotation axis, the faster the rotation.

Though the speed of a skater spinning on the ice and the Coriolis force are very different things, both phenomena have in common the concept of distance from an axis of rotation. Think about what happens to a plane flying northward from the Equator. At the Equator, the plane is as far from Earth's rotation axis as it can be. As the planes flies northward, it gets closer and closer to the rotation axis (the axis of rotation of Earth passes perpendicularly through the planet from the North Pole to the South Pole). At the North Pole itself, the plane is directly along Earth's rotation axis. Since the mass of the plane isn't changing in any significant way, its rotation speed must increase. By rotation, though, it is not meant that the plane itself starts rotating. What sort of rotation are we considering?

In the case of rotation, we need to think on a large scale. The plane is flying in the air, which like the solid planet itself, is spinning around the same axis of rotation. Some fixed amount of angular momentum is imparted to the plane when it takes off. It is the same amount that the solid earth has at the take-off location. As the plane flies northward, it gets closer to the axis of rotation--this is analagous to the skater pulling her arms in towards her body. The plane's rotation around Earth's axis must therefore increase. Since Earth rotates from west to east, the plane must accelerate in that direction to make up for the fact that it's closer to the rotation axis. To an observer on the ground, therefore, a plane flying northward from the equator drifts eastward.

It turns out that in the Northern Hemisphere, any moving object "feels" the Coriolis force accelerating it in a direction perpendicular to and towards the right of the direction in which it's moving. In the Southern Hemisphere, the opposite is the case: the acceleration is in a direction perpendicular to and towards the left of the direction in which it's moving. (In both cases, the direction is assigned as viewed from space). The horizontal component of this force has two components of acceleration, one in the zonal (east-west) direction and another in the meridional (north-south) direction. The magnitude of the acceleration always involves the so-called Coriolis parameter f. The Coriolis parameter is given by

f = 2 sin ()

where is Earth's rotation rate and is the latitude. Note that f varies as the sine of the latitude, so f = 0 at the Equator, since = 0. It increases with increasing latitude, reaching a maximum of f = 2 at the North Pole, since = 90°, and f = -2 at the South Pole, since = -90°. The degree of the acceleration is related to the velocity of the object. For an object moving with both zonal and meridional velocity components, denoted u and v, the acceleration in the zonal direction is given by -fv, while the acceleration in the meridional direction is given by fu.

In the case of our northward flying airplane, the degree of acceleration (rightward deflection) increases as the airplane moves northward.

The Coriolis parameter is known as a quantity called planetary vorticity. Vorticity itself refers to the spin or rotation of the air. It is circulation -- like water going down a drain or over a paddle wheel -- but circulation considered about a geometric point. It is the microscopic spin of the air itself. Two types of vorticity exist for our purposes when considering the atmosphere, planetary vorticity and relative vorticity. Planetary vorticity is the spin imparted to the air by the spin of the planet itself. Since f increases from zero at the Equator to a maximum at the poles, air moving from a higher latitude has been imparted an extra bit of planetary vorticity. Relative vorticity refers to the spin of the air associated with weather systems as air moves spirals away from high pressure and into low pressure. It is basically related to the horizontal components of velocity of the air. When this rotation is counterclockwise or cyclonic, as it is around a low pressure system, the relative vorticity is referred to as positive vorticity. Such positive spin is associated with rising motions, as occurs in the center of a low pressure. When the rotation is clockwise or anticyclonic, as it is around a high pressure system, the relative vorticity is referred to as negative vorticity. Such negative spin is associated with sinking motions, as occurs in a region of high pressure (hence the suppression of clouds and precipitation). The positive component of relative vorticity is a central concept in meteorology. Often times it is simply referred to as "vorticity." The two quantities of planetary vorticity and positive (relative) vorticity added together define the quantity of absolute vorticity (see section 5.2.2).

5.1.4 The magnitude of the Coriolis force -- The magnitude of the Coriolis force is rather small. For phenomena with short time scales compared to the period of Earth's rotation (24 hours), such as individual cumulus clouds, the Coriolis force is not important. It is essential, however, for the longer time scale phenomena that are characteristic of much of the stratosphere.

A simple calculation shows how small the Coriolis force is driving a car, and why we can safely ignore it. Consider a car traveling at 100 km/hr (about 60 mph). The magnitude of the Coriolis acceleration is 2 v sin . If we're at a latitude of 30°N, sin 30 is 0.5. The rotation rate of Earth is 360° per day. To calculate the Coriolis force, we need to use a consistent set of units (m, kg, sec), so let's convert 100 km per hour into units of radians per second.

First we convert 100km/hr into meters per second: 100km/hr 105 m/hr = 105 m/ 3600 sec = 27.78 m/sec . Next we determine the rotation rate of Earth in radians per second. Earth spins through 360 degrees or 2 radians per day, so 360° per day = 2 rad/day. There are 86,400 seconds in a day, so 2 rad/ 86,400 sec = 7.27x10-5 rad/sec is the rotation rate of Earth (which we called ). So our final result is (2 x 7.27x10-5 rad/sec) x (27.78 m/sec) * 0.5 = .002 m/sec2.

After about two minutes of driving in a constant direction at 100 km/hr, you would need to make a 1% correction to counteract the Coriolis force. In other words, you're never going to notice it! There is an old tale about toilet bowls flushing the other way in the Southern Hemisphere but it is false. It is false both because the amount of time available is too short and because the Coriolis force can only come into play when the water undergoes a change in latitude sufficient enough to feel the differential rotation speeds of the earth at different latitudes. In neither case is the Coriolis force going to have any effect on which way the water turns. The direction in which the spouts point inside the bowl is far more responsible for the observations than the Coriolis force!

But on a much larger (a thousand kilometers) and longer (days) scale, we can now understand the observed clockwise circulation around a high pressure region and counterclockwise circulation around a low pressure region in the Northern Hemisphere. Just think about the direction of motion caused by the pressure gradient force. The force points away from regions of high pressure. In the Northern Hemisphere, the Coriolis force acts to the right, causing a circulation in the atmosphere around the high pressure region in a clockwise direction. The opposite is true for low pressure regions: the pressure gradient force points inward, while the Coriolis force is still acting to the right, causing a circulation in the atmosphere around the low pressure region in a counterclockwise direction.

To review, the Coriolis force results in an apparent acceleration in a rotating coordinate system. Motions that are in a straight line and at constant speed in an inertial frame of reference appear curved to an observer in a rotating frame (such as Earth). This is because the rotating frame of reference gives rise to a centripetal acceleration. The observer in a rotating frame can account for the acceleration associated with the curved path by posing an apparent force. For an observer in a coordinate system rotating with Earth, this is the Coriolis force.

These apparent forces, not present in an inertial reference frame, do not alter the essential truth of Newton's Second Law, as you might think at first. Such apparent forces are simply additions needed to express that law in a rotating system. If you were to look at an airplane moving northward from the Equator, the eastward deflection arises very naturally without positing the existence of any apparent forces. The plane initially at rest on the ground at the Equator has the same rotational speed as the solid earth itself, moving west-to-east. As the plane traveled due northward, it would fly closer to the axis of rotation. The Coriolis parameter increases towards the North Pole, indicating more rightward deflection. Looking at it another way, consider the rotational speed of the solid earth passing beneath the airplane. Any point on the Equator, the planet and any object on it (including the atmosphere) has to travel nearly 24,000 miles (the circumference of Earth) every 24 hours, while an object at 45° latitude need only travel 12,000 miles in the same period. At the poles, the planet isn't moving at all. The airplane taking off from the Equator thus has quite a bit of built-in rotational velocity from the spin of Earth at the Equator. As the plane moves northward, it passes over places with decreasing amounts of rotational spin. The plane is thus "rotating" to the east faster than the earth beneath it, and hence the apparent acceleration or deflection to the right (east). What holds true for airplanes holds equally true for air parcels. In the northern hemisphere air parcels moving from one latitude to another are deflected toward the right, and in the southern hemisphere air parcels moving from one latitude to another are deflected toward the left.

In Earth's atmosphere, the Coriolis and pressure gradient forces tend to be in balance (consistent with the winds blowing parallel to geopotential height contours). This balance is called geostrophic balance. For example, air blowing around the south polar vortex moves clockwise with the pressure gradient force pointing inward towards the lower pressure inside the vortex, is balanced by the Coriolis force, pointing to the left of the flow, or outward away from the low pressure. This is illustrated in Figure 2.18.

If we assume geostrophic balance, we can solve the resulting equation for the velocity. The wind defined in this way is called the geostrophic wind, and it depends on the latitude and the pressure gradient. The zonal component of the geostrophic wind depends on the latitudinal pressure gradient and the meridional component on the longitudinal pressure gradient. A theory for atmospheric motion has been developed, called quasi-geostrophic theory, which exploits the nearly geostrophic nature of much of the atmosphere's flow patterns. However, in tropical regions, the Coriolis force becomes very small and, therefore, the geostrophic balance is not a very accurate approximation to motions in the atmosphere there.

5.1.5 The thermal wind -- In Section 4 on the horizontal structure of the atmosphere, we discussed the existence of equator-to-pole temperature gradients and zonal winds. In section 4.1.2, it was mentioned that the existence of tight horizontal temperature gradients (warmer towards Equator, colder towards the poles) generate strong westerly jet streams above these gradients. The so-called thermal wind was invoked to explain this.

We are now ready to explore the thermal wind in more detail. The thermal wind balance is a relationship that derives from the balance between horizontal temperature gradients and vertical gradients of the zonal wind. Because the atmosphere is approximately in a state of geostrophic balance above the a certain height, the winds that set up are the geostrophic winds. The existence of a horizontal temperature gradient implies the existence of a zonal jet. The jet stream is a fast moving ribbon of air, which means that there is a vertical gradient in the geostrophic wind from relatively slow winds at the surface to a fast jet of winds at the top of the troposphere and again in the stratosphere.

The thermal wind relationship says that the magnitude of the vertical gradient of the geostrophic (zonal) wind is larger AND positive where the magnitude of the horizontal temperature gradient is larger (positive OR negative). In the northern hemisphere, the temperature gradient is defined as negative (i.e., decreases from Equator to pole). Where the magnitude of the gradient increases, or temperature changes faster, the zonal wind increases with height. In the southern hemisphere, the temperature gradient is defined as positive (i.e., again decreases from Equator to pole.) Again, where the magnitude of the gradient increases, the zonal wind increases with height. In both cases, the zonal wind increases with height. The vector difference between the geostrophic wind at two levels is called the thermal wind. It is a fictitious wind in that it does not refer to air parcels in motion, but rather shows how the wind speed and direction change with height. The zonal (meridional) component of the thermal wind depends on the latitudinal (longitudinal) gradient of the average temperature between the two levels.

The thermal wind has very practical implications for understanding the atmosphere's wind and temperature structure, which the next pair of figures shows. Figures 2.19a and 2.19b are latitude-height cross sections in which a zonal (east-west) average has been taken. The first (2.19a) is of "balanced" zonal wind, meaning that extra momentum terms went into the calculation of the zonal wind in addition to those implied by pure geostrophic balance. (These include curvature effects of Earth and momentum flux quantities that are not discussed here.) The second (2.19b) is of temperature over latitude and altitude.

Because temperature decreases from Equator-to-Pole in the troposphere, the thermal wind relationship indicates that the zonal wind increases with height. The increase is most pronounced where the temperature gradient is largest. Hence, we find the jet stream in the upper troposphere at those latitudes of strongest temperature gradients. In the stratosphere, there is a pole-to-pole temperature gradient between the summer and winter hemispheres, owing to the constant sunlight during the six month "polar day" of the summer hemisphere and the six month "polar night" of the winter hemisphere. Westerly zonal winds are fastest at the location of the polar night terminator, corresponding to the zone of tightest temperature gradient. When the gradient lessens, winds also weaken, though asking which comes first is a chicken-and-egg question. Occasionally, temperatures can warm dramatically in the winter polar stratosphere, as we saw in section 4.2.2. Winds can slacken and even reverse, becoming easterly.

#### 5.2 Conserved quantities

In this section, we explore the concepts of potential temperature and potential vorticity, two central concepts in understanding dynamical meteorological processes. Both these quantities, potential temperature and potential vorticity, are referred to as conserved quantities. Like mixing ratio (see Chapter 3, Section 2), they are quantities that are invariant or fixed for a particular air parcel even as the parcel moves about. Because of its fixed value, it allows the individual air parcel to be traced, and hence the conserved property acts as a tracer (see section 2.2).

5.2.1 Potential temperature -- In section 3.3.2, we introduced the concept of potential temperature. Atmospheric scientists often use this quantity instead of geometric height as a vertical coordinate. As defined, potential temperature is the temperature an air parcel would have if expanded or compressed adiabatically (i.e., without any heat being added or taken away) from its existing pressure to a reference pressure, usually taken to be 1000hPa or 1000mb. It is usually denoted and may be calculated from the formula

= T (Po/P)(R/cp)

where P and T are the parcel's pressure and temperature, Po is the reference pressure, R is the gas constant for air, and cp is the specific heat capacity at constant pressure for air. The First Law of Thermodynamics, when formulated using q, the specific heat, rather than T, states that the time change in is proportional to the diabatic heating.

It can be shown that potential temperature is closely related to a quantity called entropy. The two dimensional surfaces of constant potential temperature in the atmosphere (which are roughly parallel to the land surface) are known as isentropic surfaces (surfaces of constant entropy). Air parcels in which no heat is added or lost move on isentropic surfaces, so that potential temperature is conserved along the air parcel trajectory. It turns out that this assumption of no heat added or lost is a fairly good one for periods of 5-10 days. By using isentropic surfaces, atmospheric scientists are able to reduce the problem of tracking air parcel motion from a three dimensional (latitude, longitude, altitude) problem to a two dimensional (latitude, longitude) problem on an isentropic surface.

Figure 2.20 shows a profile of potential temperature and temperature as a function of altitude. Note that potential temperature is a monotonically increasing function of altitude. Although this is not always the case, it is generally the rule rather than the exception.

The way potential temperature changes with altitude -- the vertical gradient of potential temperature -- determines the stratification of the air. If potential temperature rises with altitude, the air is said to be stably stratified. If potential temperature falls with altitude, the air is said to be negatively stratified. If potential temperature is unchanged, the air is said to be neutrally stratified. In mathematical terms, where potential temperature is denoted by , we have

d/dz > 0 for stable air

d/dz = 0 for neutral air

d/dz < 0 for unstable air

The stratosphere is the layer of the atmosphere where the potential temperature increases with altitude; hence, it is a stably stratified region. The heating, as we shall see in subsequent chapters, is provided by ozone. The static stability of the stratosphere acts as a sort of cap on the weather, which is confined to the troposphere.

5.2.2 Potential vorticity -- A second conserved quantity of fundamental importance is something called Ertel's potential vorticity, named for the German scientist H. Ertel. In section 5.1.3, we explored the concepts of planetary vorticity in the context of the Coriolis parameter and relative vorticity, both positive and negative.

Recall that vorticity is simply the circulation of the air or water (any fluid) considered about a point. Two types of vorticity were identified, planetary vorticity and relative vorticity. The first involves the rotation imparted to the air because of the rotation of the planet itself, and the second involves the degree of spin imparted to the air as it moves from high to low pressure. Though two types of relative vorticity exist, positive and negative, we considered only positive vorticity, since it is associated with the development of low pressure systems. Relative vorticity depends on the horizontal velocity components of the air itself. Planetary plus positive relative vorticity added together give the absolute vorticity.

An intuitive understanding of potential vorticity is more difficult to develop. Potential vorticity is a combination of absolute vorticity and the gradient of potential temperature into a scalar quantity (i.e., a quantity that has a magnitude but no direction) that is conserved under frictionless, adiabatic conditions. Because it contains both dynamic (vorticity) and thermodynamic (potential temperature) properties, the statement of its conservation is quite general.

Potential vorticity, which is the quantity of interest to atmospheric scientists, includes not only the more familiar vorticity already described, but also a component due to vertical changes in potential temperature. It turns out that potential vorticity increases when the stability (given by the vertical gradient of potential temperature) increases, as is the case when you go from the troposphere into the stratosphere. Under appropriate conditions, wind and temperature fields can be derived from potential vorticity. Thus, potential vorticity provides a very powerful dynamic tool.

In addition, potential vorticity is approximately conserved following the motion of an air parcel. That conservation property, along with the natural latitudinal gradient, make potential vorticity an ideal quantity to use as a tracer of atmospheric motion. When the potential vorticity is plotted on an isentropic surface, it can be especially illustrative of an important atmospheric phenomenon known as planetary wave breaking. In this process, long, ever-thinning streamers of air are drawn out of the polar vortex and eventually mixed with mid-latitude air. This is illustrated in Figure 2.21.

#### 5.3 Waves

The most important atmospheric phenomenon for transferring momentum in the atmosphere are waves. Waves allow what goes on in one region to affect what goes on in another region. This process is referred to as "communication," and it allows extra momentum in one area to be transferred to another, such as from the troposphere to the stratosphere, or from the tropics to the extratropics.

The atmosphere exhibits many wave-like motions with a variety of space and time scales ranging from slow moving planetary scale waves to much faster and smaller gravity waves, each playing important roles in the behavior of the stratosphere. Waves are responsible for asymmetries in the polar vortex, stratospheric sudden warmings, mixing of polar vortex air with mid-latitude air, the forcing of the QBO and the control of the midlatitude mean meridional circulation, all of which will be discussed in more detail in Chapter 6.

Wave motions can be categorized according to their restoring mechanism. Just as a block attached to a spring will, when pulled away from its equilibrium position, tend to oscillate under the influence of its restoring mechanism---the tension in the spring---so an air parcel will tend to oscillate around its equilibrium if displaced away from that point. This equilibrium condition usually involves the air parcel's potential temperature and its vorticity. A combination of these effects produces the quantity called potential vorticity (see section 5.2.2). The restoring mechanism for waves can involve these quantities, as well as density and gravity.

5.3.1 Gravity (buoyancy) waves -- Waves whose restoring mechanism is their buoyancy are called gravity waves. These waves can be understood from our previous discussion of static stability. Start out with an atmosphere in which potential temperature increases with altitude (such is found in the stratosphere). If an air parcel is suddenly displaced to a lower altitude, it will be warmer than its surroundings and therefore begin to rise. Like the mass on the spring, as it rises, it picks up momentum, passes through its equilibrium position, and then finds itself surrounded by warmer air. Being cooler than its surroundings, it begins to sink back towards equilibrium. But again, having too much momentum, falls below the equilibrium level to a place where it is again warmer than its surroundings. This oscillation about an equilibrium point, analogous to a pendulum oscillating about an equilbrium point, occur because of the stability parameter. The oscillation occurs with a set frequency. The period or time required for one oscillation for these vertical displacements depends on the stability, but it is typically about 7 minutes. The restoring mechanism in the vertical is the stable stability (stratification) of the wave medium. Such stratification is associated with hydrostatic balance. The term gravity is used because gravity is involved in hydrostatic balance.

Gravity waves are sometimes referred to as internal gravity waves because oscillations occur inside boundaries (i.e. on sharp density changes). An example are gravity waves in water as they travel along a density discontinuity, either within the water or on the surface of the water. In the atmosphere, where density discontinuities are much smaller, gravity ways can travel vertically from the troposphere into the stratosphere and even mesosphere.

5.3.2 Rossby waves -- Waves whose restoring mechanism is the latitudinal (north-south) gradient of potential vorticity are called Rossby waves after the Swedish-American scientist Carl Gustav Rossby, who made so many key contributions to meteorology. To understand why such waves exist, recall our discussion of potential vorticity and the conservation of angular momentum. Assume you have an air parcel moving south. The planetary component of potential vorticity (represented by the Coriolis parameter) will decrease. Since the total potential vorticity remains nearly constant, the local (positive) vorticity of the air must increase. This results in a counterclockwise local rotation of the air mass. But as the air mass begins to turn north again, the planetary component of the potential vorticity increases, resulting in a decrease in the local vorticity. The air mass turns south yet again. The result is a wave pattern in the motion of the air parcel as viewed from above the Earth. Large scale topographical features create undulating Rossby wave patterns around the globe in the northern hemisphere in the 30° to 60°N band. The lack of significant topography in the southern hemisphere leads to a more zonal flow with fewer Rossby waves.

5.3.3 Inertio-gravity waves -- Gravity waves with a sufficiently long periods will begin to feel the rotation of the earth. In such a case, the restoring force becomes a combination of rotation and buoyancy. These are referred to as inertio-gravity waves. Air or water experiencing buoyancy oscillations will also experience a Coriolis deflection. For instance, zonally propagating waves oscillating in the vertical will feel a Coriolis force that imparts a meridional velocity component.

Waves that have both Rossby and inertio-gravity characteristics are referred to as the mixed Rossby-gravity wave. The restoring force is thus the gradient of potential vorticity, static stability, and the Coriolis parameter.

5.3.4 Forced stationary planetary waves -- Rossby waves with very long wavelengths -- upward of 10,000 kilometers -- are referred to as planetary waves. These waves are likely generated by large-scale surface topography like the Rocky Mountains and the Himalaya-Tibet complex (orographically forced) or by land-sea boundaries. These planetary waves do not propagate, but instead are stationary. The fact that they are stationary is related to the fact that the topographical forcing occurs at the same locations. Planetary waves often propagate upward from the troposphere into the stratosphere. Charney and Drazin (1961) developed an important theory about the vertical propagation of these waves. They found that such upward propagation can only occur in an environment of moderate to weak westerly zonal winds. Furthermore, the critical wind speed beyond which propagation cannot occur depends on wavelength in such a way that only the longest waves are likely to propagate into the stratosphere.

5.3.5 Free traveling planetary waves -- Another type of planetary wave are those that propagate with a period of a few days. They are not primarily caused by topographic forcing from below, but instead are free, traveling waves. As an analogy, think of a guitar string: if plucked it will oscillate at a characteristic frequency; it will also respond if that frequency is played by another instrument nearby (sympathetic vibration). Similarly, the atmosphere can also be described by a set of natural frequencies. When excited at those frequencies, it too will respond. The so-called 5-day wave, a westward traveling disturbance in the middle atmosphere with wavelength equal to the distance around a latitude circle, is an example of a free, traveling planetary wave.

5.3.6 Equatorial waves -- The tropical atmosphere supports several types of important wave motions. Eastward and westward moving inertio-gravity waves have high frequencies. Complementing them are westward moving low-frequency equatorial Rossby waves. Bridging the gap between these two is the mixed Rossby-gravity wave which propagates eastward at high frequencies like the inertio gravity waves and westward at low frequencies like the Rossby wave. The change in sign of the Coriolis parameter at the Equator plays a key role in these waves. A final, uniquely equatorial wave type is the Kelvin wave. It propagates eastward like a pure gravity wave (i.e., a gravity wave whose restoring mechanism is buoyancy only). It has no meridional velocity and its zonal velocity is in geostrophic balance with the latitudinal pressure gradient.

The Kelvin wave in the ocean travels along the region of tightest temperature (and hence pressure and density) gradient called the thermocline. The warming and cooling of the waters of the eastern Pacific in the ENSO phenomenon (see section 4.2.4 for review of ENSO) actually occurs as these Kelvin waves propagate along the thermocline, raising and lowering its height. Moving along the Equator where the Coriolis parameter changes sign, these waves are referred to as equatorially- trapped Kelvin waves. When it hits the boundary of the ocean- the South American continent- it reflects in two directions: a Kelvin wave propagates north and south along the coast while a Rossby wave propagates backwards along the Equator into the Pacific. Atmospheric Kelvin and mixed Rossby-gravity waves are involved in the QBO phenomenon (section 4.2.3).

Atmospheric equatorial waves are important because after they are excited in the tropical troposphere, presumably by cumulus convection, they propagate vertically into the middle atmosphere where they can force changes in the zonal mean flow through wave-mean flow interaction which is discussed below.

5.3.7 Midlatitude gravity waves -- Inertio-gravity waves also exist in mid and high latitudes. Just as for the equatorial waves, they can propagate vertically into the middle atmosphere and cause wave-mean flow interaction.

5.3.8 Wave-mean flow interaction -- When doing theoretical or observational studies of wave motions it is often useful to separate the variables (e.g. temperature and winds) into a zonally averaged part and a part that describes deviations from that average. The former define a zonal mean state for the atmosphere, described in section 4. Waves, which are deviations from that average background state, alter the mean flow. This effect is known as the wave-mean flow interaction. The waves are called eddies. They are the departures from the basic state flow. They are frequently assumed to be linear. This means they have a small amplitude relative to their zonally averaged magnitude. This assumption makes their treatment in dynamical theories much simpler mathematically. Eddies play a big role in transporting energy and momentum across latitudes. Transient eddies are departures from the time mean state. There are also standing eddies which arise from the mean field. These standing eddies are found by subtracting out the zonal mean.

Wave-mean flow interaction drives many stratospheric motions. Dissipation of the equatorial Kelvin and mixed Rossby-gravity waves force the westerly and easterly phases, respectively, of the QBO. If wave amplitudes grow, the waves become nonlinear and may "break" in a process quite analogous to ocean waves at the seashore. Above the stratosphere, breaking gravity waves force the semiannual oscillation (SAO). Nonlinear breaking gravity waves also cause a strong forcing of the zonal mean circulation in the upper mesosphere. Finally, the mean meridional circulation is driven by breaking planetary Rossby waves.

#### 5.4 Mean Meridional Circulation

We have so far emphasized the structure of the zonal wind and the various processes that can alter it. But there is also a zonally averaged circulation in the meridional-height plane, i.e., a circulation of rising and poleward motion accompanied by sinking and equatorward flow. The velocities involved are much smaller than the zonal wind, but still important for transport on long time scales. A schematic picture of the circulation is shown in Figure 2.22. The picture should be interpreted as if it represented streamlines of the flow. The circulation is forced, in part, by breaking Rossby waves in the midlatitude middle atmosphere.

The mean meridional circulation is discussed in Chapter 6.