## Atmospheric Forces, Balances, and Weather Systems

### Introduction

The large scale horizontal flow of air in the atmosphere is driven by the imbalance of net radiation over the globe (Fig. 1, more on that next week). This section introduces us to the physical laws governing the horizontal motion of air. It also describes types of motion systems encountered in the atmosphere. These different types of motion are a result of the fact that the balance of forces acting on the atmosphere depends on the scale of motion.

To understand this lecture we should be familiar with:

1. The relationship between forces and motion. The physical law known as Newton's Second Law of motion states: if a net force different than zero is acting on a body, it will accelerate at a rate proportional to the net force. We also need to be aware of the fact that the expression of the Second Law may depend on the frame of reference from which we are watching the motion. Viewing the motion from an accelerating frame of reference introduces apparent forces (or inertial forces). This is the case in the balance describing air flow on Earth as will become clearer later on.
2. Another important principle that controls air flow is that of mass balance or the continuity of the flow. This principle is simply a statement that in a continuous fluid (with no walls or partitions) one can not empty out a region from its mass - when fluid is taken out from one place, surrounding fluid rushes in to take up that space. Viceversa, we cannot accumulate fluid mass in arbitrary locations in space (outside of some possible compressibility). If we try to do that, the fluid will rush out to places where the fluid is less dense. Stated in other ways: convergence and divergence of the flow at any level of the atmosphere and ocean will result in, among other things, vertical motion.

### Take away ideas and understandings

• Know the balance of forces that act on the atmospheric flow.
• Learn about the common types of motion systems in the atmosphere:
• Sea breeze;
• midlatitude cyclones - fronts and weather;

### The concept of horizontal motion.

#### The sea breeze.

Consider the following situation: At the break of day, the radiation from the sun begins to warm up the Earth's surface. Along the coast, surface warming is not horizontally uniform. On the land side of the coast line, the ground warms quickly, but the warming of the sea is slow. (Can you explain why? Consider the heat capacity of soil vs. that of water, the heat expended on evaporation, the depth of the penetration of sun rays on land and water, the albedo of the surface.) Thus a difference in surface temperature between the land and water sides of the coast line quickly develops. As we saw in the previous lecture, the heat absorbed at the surface is transferred to the air column above through conduction and convection. In the late morning hours the lower atmosphere over land is warmer than over the water. The density of the warmer air column over land is lower than in the colder sea column (ideal gas law). The denser the air, the higher its weight, so at the surface the pressure over sea is larger than that over land (hydrostatic balance). However at higher elevations (1-2 km above the surface) the pressure in the warm air is higher than in the cold air . This is because according to the hydrostatic balance, pressure drops more slowly with height in warm air than in cold air. (You can prove this to yourself by using a lower density in Homework problem #2 of Lecture 3 - use a density of 1.125 and recalculate the elevation of the climber above its starting point - it is now 544 meters, higher than in the denser air.) As a fluid, the atmosphere can not sustain pressure imbalances and a flow of air from high to low pressure ensues: at low elevation air flows from sea to land, and at high elevation it flows from land to sea. The cycle is closed by air rising in the warm column over land, and sinking in the cold column over the sea. This is how the sea breeze forms.

This motion continues, fed by the differential heating of land and sea, until the sun's radiation decreases. The land then begins to cool relative to the sea, which stores its heat longer than the land surface (explain why), and the motion dies down. At night, the land surface continues to cool faster than the sea surface, an opposite cycle of events takes place, and the sea breeze reverses its flow.

The motion of the sea breeze is governed by two physical laws:

1. Newton's 2nd law of motion:

F = m x a

If a net force acts on a body (a parcel of air in this case) motion will ensue. Remember also that if the net force is zero, a resting body will stay at rest, while a moving body will move with constant velocity. Here, two forces are acting on the flow: the pressure difference between land and sea is accelerating the day time flow towards the land, and friction, largest near the surface, is trying to slow the motion down.

2. The law of mass continuity (empty spaces are not tolerated in fluids) - is acting to complete the cycle by creating the vertical motion of the air, up in the warm side of the coast, and down in the cold side.

### Atmospheric forces.

The study of atmospheric motion is referred to as Dynamic Meteorology. To handle the physics of motion we need to consider a coordinate system, or a frame of reference. That is because forces and velocities are vectors so both magnitude and direction are important. In meteorology we define an x, y, z coordinate system which has an origin somewhere on the Earth's surface (say at the equator and the Greenwich meridian), and we measure the three directions in the following way:

x: is the zonal (East-West) direction; positive eastward
y: is the meridional (North-South) direction; positive northward
z: is the vertical (up-down) direction; positive upward.

Thus the frame of reference has curved axes and is located on a rotating surface. This system is convenient to us because this is the frame of reference from which we view the atmospheric or oceanic motion. However, because of its peculiar properties it introduces some difficulties. In particular, we need to consider not only fundamental forces like the pressure force, but also apparent, or inertial, forces that result from the fact that the motion is viewed with respect to an accelerating (rotating) frame of reference. We shall see below how these considerations determine the balance of forces acting on the motion of air. Note that aside from within convection cells or clouds, the horizontal atmospheric motion is much more energetic than the vertical one, thus our section deals mainly with the balance of forces governing horizontal motion. The hydrostatic balance equation we studied in the previous lecture is the dominant vertical balance in the types of motion we discuss below.

Pressure, as we saw in previous lecture is the force per unit area exerted by the air molecules on any imaginary surface within the atmosphere. Consider an air parcel suspended in the atmosphere in hydrostatic balance. If pressure on one side of parcel exceeds that on other side, the parcel will experience a net force from high pressure toward low pressure. The force per unit mass acting on the parcel (in Newtons/kg) is given by:

Fpx = - (Δp / Δ x) / ρ

Fpy = - (Δ p / Δ y) / ρ

(It is convenient in dynamic meteorology and oceanography to work with force balances per unit mass since we are not dealing with solids with well defined boundaries.)

The pressure gradient force is thus given by the ratio of pressure difference to the distance over which it acts, divided by density. The pressure gradient force is the active force in the climate system (friction, see below, is passive because it only exists when motion exists). Thus to the weather forecaster or an observer of motion on Earth, the horizontal distribution of pressure is extremely important. Pressure is routinely measured in land stations, on board ships, and from weather balloons, and the reports are disseminated to all weather centers. The data are plotted on maps as isobars (contours of constant pressure) (Fig 2). The direction of the pressure gradient force is perpendicular to isobars, from high to low pressure. Closely spaced isobars indicate a large pressure gradient and strong acceleration of the air parcels.

#### Friction.

Air is not very viscous ("sticky"), so "real" friction (the one that comes from molecular motion) is only important in a very thin layer of atmosphere next to the surface. However, air is very turbulent. This turbulence generates small-scale up and down motion, which mixes slow air from the friction layer with fast air from above, thereby spreading the effect of molecular friction over a layer a few hundred meters thick (turbulence is the reason for wind gusts). This interaction with the surface slows down atmospheric motion.

The physical laws governing atmospheric friction are too complex to be explained here. However, one very simple way of describing the friction in a layer close to the ground is to express it as a force proportional to the velocity of the air and acting to reduce it down. Thus the frictional force per unit mass is:

Ffx = - αu

Ffy = - αv

Where uand v are the zonal and meridional components of the wind (in units of m/sec), and α is a constant equal to about 2 x10-5 1/sec.

#### Frictional balance.

Combining the equations above we find that when an air parcel is subjected to the forces of pressure gradient and friction the equation describing the motion (per unit mass) can be written as:

ax = - (Δ p / Δ x) / ρ - αu

ay = - (Δ p / Δ y) / ρ - αv

Here ax and ay are the acceleration of a unit mass in the west-to-east and south-to-north directions. If a balance is achieved between friction and pressure, the left hand terms in these equations are replaced by 0.

### Apparent (inertial) forces - the large scale flow in the atmosphere.

Apparent or inertial forces are forces resulting from viewing an object in an accelerating frame of reference. When such a situation occurs, the observer has to introduce a "force" into the equation of motion to account for the fact that a force is acting on the frame of reference. However, once the fundamental force in the equation stops acting on the moving body, the apparent force will "disappear" as well. A relatively simple example of an apparent force is the centrifugal force. Another such force, which may be new to many, is the Coriolis force, of utmost importance in meteorology and oceanography.

#### The Centrifugal force - cyclostrophic flow.

When a body is moving in circular motion there must exist a fundamental force pulling it to the center of the circle. For example, we can make a ball move in a circular motion by holding on to it by a string. The fundamental force - called the centripetal force - is exerted on the body by the string and by our holding the string's other end. For an observer "sitting on" the body, there must be another force acting which enables the achievement of the balance, and which keeps the string stretched. This force, called the centrifugal force, acts exactly in opposition to the centripetal force and results from the body's own inertia. If the string is cut, the centrifugal force will cease to exist, and the body will move in a straight line, at a constant velocity, tangent to the circle at the point where the centripetal force stopped acting.

In a similar way, a weather satellite spinning in orbit above the Earth, is held in orbit by a balance between the gravitational force - a fundamental force attracting it towards the center of the Earth, and the centrifugal force pulling away in the opposite direction. There must be an exact balance between the two forces for the circular motion to continue keeping the satellite moving with the same angular velocity, at the same distance from the Earth's surface. An imbalance will result in a change in motion to restore the balance.

In the atmosphere, we sometimes encounter deep circular depressions in pressure with relatively small dimensions (from tens of meters up to a few kilometers) for which the balance of forces can be represented as an equilibrium between the pressure gradient force acting toward the center of the low pressure, and the centrifugal force pulling outwards (the same balance acts on water flowing down a drain). This, for example, is the balance of forces acting in a tornado. In this case we can view the motion of the air parcels from a frame of reference moving circularly with them. In the direction of the motion, there are no forces acting on the air (accept possibly friction, trying to slow the motion down). Perpendicular to the flow there is a balance between pressure gradient and the centrifugal force:

V2 / r = - (Δp/Δr) /ρ

Where:

V is the velocity in the direction tangent to the circle,
r is the radius of the circle,
ρ is the density of air, and
Δp/Δr is the pressure gradient in the direction perpendicular to the motion (along the radius, and directed into the center of the circle).

Motion governed by this balance of forces is called cyclostrophic flow. In cyclostrophic flow the motion can be either clockwise or anti-clockwise around the center of low pressure.

#### The Coriolis force.

On the scales of motion important for weather and climate (anywhere between a few hundred kilometers to the scale of the Earth), motion is governed by the Coriolis force. This force "results from" the fact that we view the movement of air masses on Earth from a point of reference attached to its surface. The Earth is a rotating sphere. As the entire sphere spins around its axis, from west to east, every point on its surface moves in circular motion around the radius connecting it to the Earth's center. This circular motion is largest at the poles where the Earth's angular velocity is equal to one rotation around the Earth's axis in a day or:

Ω = 2π/84600 = 7.27 x 10-5 rad/sec

As we move towards the equator, the rotation of segments of the Earth's surface, along a line connecting them to the Earth's center, decreases until it finally approaches zero on the equator. The rate of rotation of each surface segment around the line connecting them to the center of the Earth is proportional to the sine of the latitude, Φ, passing through that segment:

ω(Φ) = Ω sin Φ

On Earth, latitude angles are measured with respect to the equator, where Φ = 0. North of the equator the latitude angles are positive, and south of it, negative.

Consider an observer standing at the north pole. He might not be aware of it, but he is spinning continuously from west to east (that is, to the left). Now, he throws a ball equatorward, just ahead of him and continues looking ahead. Our observer will soon be facing away from the direction in which he was looking before, and away from the ball. Because he is not aware of his rotation, the observer will conclude that the ball is moving away from him to the right. Since he knows quite well that the force he exerted on the ball sent the ball straight ahead, the observer will conclude that there is another force acting on the ball, pushing it to the right. (If the observer moved to the south pole, the ball would be moving away to the left because the direction of rotation is reversed). This force is the Coriolis force, named after the French engineer, mathematician, and physicist G. G. de Coriolis (1792-1843). The Coriolis force is a force that one has to reckon with everywhere on Earth (accept on the equator) if one throws objects long distances (eg. long-range artillery). Fig 3: merry-go-round movie from University of Illinois WW2010 Project's Forces and Winds online meteorology guide portrays the action of the Coriolis force quite convincingly.

We are watching the motion of air in the atmosphere like the observer watching his ball. From a frame of reference located at the surface, Coriolis force acts everywhere on Earth, deflecting moving air parcels to the right in the Northern Hemisphere, and to the left in the Southern Hemisphere. Only at the equator, where surface segments on the spherical Earth do not exhibit a spinning motion around the Earth's radius, is the deflection zero.

Coriolis force per unit mass of air is expressed as follows:

Fcx = + 2 Ω v sin Φ = + f v

Fcy = - 2 Ω u sin Φ = - f u

Here f is shorthand for the terms depending on the Earth's rotation and latitude. It is known in meteorology and oceanography as the Coriolis factor. Note that this force is only important on large spatial scales and time intervals (distances on the order of hundreds to thousand of kilometers and times of at least close to the Earth's rotation period). Note also that the x-component of the Coriolis force depends on the y-component of the velocity, and vice versa. It thus acts perpendicular to the direction of motion. Depending on the sign of f (that is, depending on the hemisphere).

#### Geostrophic balance.

The balance between the pressure gradient force and the Coriolis force is the most important balance in dynamics of the climate system. Expressed in mathematical terms it is written as follows:

2 Ω v sin Φ= (Δ p / Δ x) / ρ

2 Ω u sin Φ= - (Δ p / Δ y) / ρ

The geostrophic balance gives us the means to calculate wind speed and direction given the pressure gradient. It also tells us that in the large scale atmospheric motion of the Northern Hemisphere, the air flows along the isobars so that the low pressure is to the left of an observer standing with his face in the direction of the wind. In the Southern Hemisphere the low pressure will be to the right of the observer (see Fig 4 for the sequence of steps leading to geostrophic motion, and Fig 5 for a summary of its Northern and Southern Hemisphere manifestations).

Near the surface we do not find good examples of pure geostrophic flow (see Fig 2 above). This is because friction cannot be ignored (see below). But from about one km up it is quite accurate to express the flow as geostrophic. This is quite evident from Fig 6. Deviation from geostrophy occurs when pressure changes suddenly. The air will then accelerate (or decelerate) to reestablish geostrophic balance.

Because of the geostrophic balance, low and high pressure areas in the middle latitudes modulate and affect the temperature along latitude lines. In the Northern Hemisphere, a low pressure area pulls cold air from the north to the west of its center, and warm air from the south to its east (see Fig 6). This motion achieves a distribution of heat from the equator poleward in both hemispheres. In winter, lows and highs move from west to east following one another, causing alternations between relatively warm and relatively cold spells.

### The role of surface friction - Ekman balance.

At the surface, friction must be considered in the balance of forces acting on an air parcel. Both Fig 2 above and Fig 7 are good examples of surface flow where the wind arrows are not perpendicular nor are they parallel to the isobars. To understand why that is so, remember two facts:

1. friction slows down the flow speed, and
2. Coriolis force depends on the flow speed.

This means that a new balance is achieved in which friction and Coriolis forces together counter the pressure gradient force. The effect is known as the Ekman balance (after the German hydrodynamicist V. W. Ekman) and is summarized in Fig. 8.

The Ekman balance is important for forming weather patterns in the atmosphere (see Fig 9). At the Earth's surface, around an area of low pressure, air spirals in toward the center and converges. Because mass can neither be accumulated or depleted (mass continuity) the converging air rises. Adiabatic cooling ensues, which enhances the potential for saturation, cloud formation, and precipitation. In contrast, air spirals outward from the center of a high pressure area and diverges. To accomplish mass continuity, diverging air moves outward and sinks, and adiabatic warming ensues. Warming of the air lowers its relative humidity, and the downward motion (subsidence) suppresses cloud formation and brings fair weather.

### Midlatitude weather systems.

#### Fronts.

From reasons we shall explore in the next lecture, the flow in the middle latitudes is broken into sequences of low and high pressure cells moving from west to east. As we discussed above, these lows and highs affect the local vertical motion through convergence and divergence induced by surface friction. They also sweep air masses of different temperature from north and south bringing alternations in temperature to the regions they pass by in their eastward procession (Fig 10). The moving air masses collide and at their boundaries fronts are created in the low pressure centers (Fig 11).

In their general definition, fronts are the boundary between two air masses of different temperature (and usually also different relative humidities). A passage of a front indicates an imminent change of temperature and also a change in weather. There are two kinds of fronts: Warm fronts and cold fronts (see Fig 2).

Warm fronts: Lie east of the low pressure centers. They bring warm (and often humid) air from the south towards the north to meet with cold air. The warm air, being less dense, rises up and over the cold air mass, causing cloud formation and often precipitation. Most often, the warm air is relatively stable, and the clouds are layered (stratiform) and cover the sky from horizon to horizon in a uniform layer. Because the warm air generally flows over the cold air in a shallow slope, distant warm fronts can be observed hours before they arrive. The winds change from having an easterly component to having a southerly to southwesterly component (the wind direction is the direction from which the wind is blowing). Warm fronts are denoted on weather map by a red line extending out from a low pressure center sometimes marked with half circles pointing to the direction of the fronts motion.

Cold fronts: Lie west and south of the low pressure centers. They bring cold (and often dry) air from north towards the south to meet with warm air. The cold air, being denser, pushes under the warm humid air and lifts it up, creating clouds and often precipitation. The slope of the cold front is usually steep, and the rising air unstable, thus strong convection can occur with cumuliform clouds. Cold fronts usually extend over a narrow zone and their passage is fast. After the frontal passage cold and dry air takes over. At the passage of the cold front the winds veer from a southwesterly direction to a northwesterly one. Cold fronts are denoted on weather map by a blue line extending out from a low with triangles pointing in the direction of the front movement.

Occluded fronts: The cold air behind the cold front tends to race faster around the low pressure center than the warm air does. When weather systems "mature" the cold front can catch up with the warm front. This usually happens closer to the center of the low. The air behind the cold front is sometimes colder than the air in front of the warm front, so the latter gets pushed up. This situation is called a cold-type occlusion. If the opposite happens, and the air behind the cold front is warmer than the air in front of the warm front, the occlusion is called warm-type. In occlusions the weather is a mixed bag of cloud forms, and the signs at the surface are less easy to tell apart.

To see what a midlatitude weather system looks like from space look at Fig 12; Click here to see a movie loop of satellite pictures of this storm and its movement across the US.

For a good summary of midlatitude cyclones, their structure and the weather they bring, look in the colorful book of Moran and Morgan (1995) on the reserve shelves. You can also explore the University of Illinois ww2010 web site for text and illustrations.

#### Tropical cyclones (or hurricanes or typhoons).

Tropical cyclone, also called hurricane and typhoon, is the names given to an intense low pressure region that forms and migrates in the tropical ocean regions and is associated with intense winds and a strong convection activity which brings thunderstorms and large amounts of rainfall (Fig 13).

Tropical Cyclones form over the warm tropical oceans, where the sea surface temperatures are at least 26°C. The energy for the formation of hurricanes comes from the enormous amount of latent heat released when condensation occurs in the massive cloud rings surrounding the hurricane (see Fig 13). The heat forces air to rise, rapidly maintaining the convection, and helps lower the pressure at the surface. This creates a force pulling air into the low pressure at the surface, supplying moisture to the convective system, and the "heat engine" is thus maintained. The Coriolis force is important for the generation of hurricanes, it keeps the convection well organized, and the convergence going on all the time. Thus they can not be found to exist at the equator, but only a few degrees latitude away. The dynamic balance of forces is governed by the pressure gradient force, Coriolis force, centrifugal force, and friction.

Hurricanes form in the "trade wind" belt - the regions just north or south of the equator where the winds blow quite steadily from east to west (easterlies). They form on the eastern side of the ocean basins, initiated by weak pressure perturbations that exist all the time in the tropics. At first they are weak and move west with the local wind direction as they intensify. In the easterly wind belt their motion is steady but not fast (10-20 km/hour). Many of the hurricane seeds (or tropical depressions as they are called) that occur over the ocean never mature into a full hurricane. The massive disturbance that sometimes grows in a time frame of a week or so, needs a lot of favorable conditions to occur. Once it does, it spreads over a radius of a few hundred kilometers. Hurricanes are surrounded by rings of towering thunder clouds spiraling up to a small circle at the center of the storm, with a radius of 30-40 km. Here the winds can reach a speed of 100 km/hour and more, and the most intense rainfall occurs. Inside this ring, lies the eye of the storm, where the air is still and the convection is suppressed by subsidence.

As they move west, the hurricanes drift poleward and when they reach far enough north of the equator in the Northern Hemisphere (or south in the Southern Hemisphere) they enter a region where the prevailing winds are westerlies. These winds turn the hurricane tracks around to go in the opposite direction - eastward. They also speed up their movement, and may reinvigorate in the form of a midlatitude depression.

When strong hurricane winds blow over the ocean they generate very high waves, higher on the side of the storm where the wind moves in the same direction as the storm. The low pressure in the center of the storm also acts to elevate the sea level underneath (similar to a tide) so when the storm hits land it can cause severe flooding on top of heavy wind and rain damage.

For more text and illustrations see the University of Illinois ww2010 web site

Text by Yochanan Kushnir, 2000.