EESC V2100 The Climate System spring 2004 Lecture 4: Laws of Atmospheric Motion and Weather Yochanan Kushnir Lamont Doherty Earth Observatory of Columbia University Palisades, NY 10964, USA kushnir@ldeo.columbia.edu
Horizontal Motion in the Atmosphere - Geostrophic Balance
Horizontal Motion in the Atmosphere The motion of air on Earth is described in terms of a 3-dimensional vector with a zonal (west-to-east) or x component, normally denoted u, a meridional (south-tonorth) or y component denoted v, and a vertical (upward) or z component w. In most cases, the motion is described relative to the rotating Earth. The more prominent component of atmospheric motion are in the horizontal (parallel to the surface of the Earth) dimension. In the vertical dimension, on scales larger than clouds and intense storms such as tornados, the motion is much smaller than in the horizontal and the atmosphere is close to being in hydrostatic balance. The equations governing horizontal motion are based on Newton s Second Law of Motion applied to a fluid system in a spherical reference coordinate system rotating with Earth. The main driving forces in the atmosphere are the pressure gradient produced either through thermal or dynamical effects, and gravity. Retarding or balancing the pressure gradient force is the Coriolis force an apparent force, which results from viewing the motion in reference to the rotating Earth. Close to the surface, friction is also an important force retarding force. Sometimes, when the motion is fast and circular, another apparent force, the centrifugal force comes into play.
The Pressure Gradient Force Just as we saw in the vertical dimension, the pressure gradient forces results from the difference in pressure acting over a distance as described in the following diagram: z - upward y - northward x - eastward p+δp Δz Δx Δy p direction of motion The pressure difference acting over a distance Δx create a force F x = -- ΔpΔyΔz (negative because eastward flow is produced by higher pressure to the west). The eastward acceleration (= force per unit mass) a x, (in m/s) exerted by the pressure gradient differece per unit mass is given by Newton s second law of motion: F x = -- ΔpΔyΔz = (ρδpδyδz)a, where ρ is the density of air. After canceling identical terms on both side of the equation we obtain: Similarly, in the y-direction: a a x = -- (1/ρ)(Δp/Δx). y = -- (1/ρ)(Δp/Δy).
Isobars The mapping of pressure on a horizontal surface, such as the sea level, is the first step in monitoring atmospheric motion Pressure data are collected simultaneously by a network of stations, converted to sea level and reported to weather centers four times daily, to create a synoptic representation of sea level pressure. The station data is interpolated to a regular grid and plotted as isobars - lines of equal pressure.
Pressure Gradients strong gradient weak gradient By viewing the patterns of the isobars, we can determine where the gradients are strong and where they are weak, thus assessing the strength of the horizontal pressure gradient and the direction of the pressure gradient force (PGF). Units of pressure: the MKS unit for pressure is 1 Pascal = 1 Newton/m 2 (the unit is named after the French scientist Blaise Pascal - 1623-1662). The standard sea level pressure in Pascal is 101325 and the order of magnitude of pressure variations is about 100 Pascals. A more workable unit of pressure is a millibar (or one thousands of a bar) which equals 100 Pascals (also called hecto-pascal).
Apparent Forces To study of body movement under force we use a frame of reference. A frame of reference is called inertial if it is at rest or if it moves with constant velocity (that is, constant speed and direction). If the reference frame is moving under acceleration it is non-inertial. Examples for non inertial frames of references are an accelerating car, a rotating platform (even if the angular velocity is constant), and our planet Earth. Non-inertial systems are under the influence of a force acting linearly or by exerting a torque on the system to cause rotation. To use such system as a frame of reference we need to introduce an apparent reaction force equal and opposite to the external one. Consider for example a train moving at a fixed direction and constant speed in the country side. If a passenger on that train decide to get out of her or his seat and move forrard, they will feel and react just as if they are walking in their own back yard. But if the trains decelerated suddenly the passenger would be jolted forward with the same acceleration as that of the braking train. That passenger may conclude that a force acted on her or him pushing them forward while in reality the train they were riding was pushed back. Non Inertial Frame of Reference and The train became a non-inertial frame of reference and in order to explain the effect on the walking passenger in that frame, we will have to introduce and apparent accelerating force equal to the force that slowed the entire train.
The Coriolis Force Next consider an observer, standing in the center of a merry-goround, shaped like a circle with a radius R and rotating in a counterclockwise direction (to the left) with an angular velocity Ω (as in the figure to the right) The observer sets a ball into motion towards the circumference of the circle, in the direction of the radius and with a speed v. Because the observer continues to rotate in the clockwise direction it will appear to him that the ball is turning to the right as it moves away. The faster the observer turns, the faster the ball curves to the right. The closer the ball gets to the circumference of the circle, the faster it will appear to move. To explain the balls behavior, the observer needs to introduce an apparent force which causes the apparent accelerated motion of the ball to the right with respect to the rotating merry-go-round. This force is known as the Coriolis force. d
The Coriolis Force (continued) To determine the magnitude of the Coriolis acceleration consider the following: after the ball reached a distance of R from the observer, we can identify an arc that forms between the point on the circumference of the circle at which the ball was originally aimed (straight ahead of the observer) and the point where it actually crossed the circle. The arc length, d is given by the time t elapsed from the beginning of the motion until the arrival at the circumference of the circle R times the speed at which the circumference is rotating (= angular velocity time the radius), namely: d = ΩRt Under acceleration the distance covered by a moving body is proportional to the square of time. Thus d is also equal: d = a c t 2 / 2 where a c is the acceleration - in this case, the Coriolis acceleration. Equating d from the two equations results in an expression for the Coriolis acceleration: a c = 2 ΩR / t Now, the time t is given by the ratio between the radius and the ball s speed: t = R / v Thus we obtain: a c = 2 Ωv The Coriolis acceleration (or force per unit mass) acting on a particle viewed in a rotating frame of reference is equal to twice the product of the angular velocity and the speed of the particle. The direction of the force and acceleration is perpendicular to the motion, acting in this case to the right when facing in the direction of the motion. R d Ω
R d Ω Ω Φ Ω Coriolis Force on Earth ΩsinΦ How does the situation on Earth resemble the rotating disk? The planet rotates around its axis (with the angular velocity vector pointing to the north (see bottom figure on the left). This is the direction of the vector at every point on the Earth s surface. At the North and South Poles Ω is vertical to the surface and anywhere else it is slanted to the surface with an angle equal to the local colatitude (90 minus the latitude angle, Φ). The local angular velocity component vertical to the surface is thus ΩsinΦ. It describes the rotation of the local surface around the radius connecting it to the center of the Earth with an angular velocity that decreases from the North Pole (Ω) to the equator (0) and then reversing sign in the Southern Hemisphere and continuing to decrease to the South Pole (-Ω). The Coriolis acceleration on Earth is thus a function of the latitude with: a c = 2(ΩsinΦ)v, On Earth: Ω = 2π /84600 = 7.27 x 10-5 rad/sec where v is the velocity of the the moving body with respect to the Earth. Based on this relationship we define the Coriolis factor f = 2(ΩsinΦ).The force is perpendicular to the body s motion acting to the right in the NH and to the left in the SH.
Coriolis Effect on Earth An aircraft heading out from the US West Coast (say San Francisco - latitude ~38 N), flying towards the East Coast (New York - latitude ~40 N) with an almost direct eastward heading, needs to continually correct its course to adjust for the Coriolis force otherwise it will find itself drifting southward to a lower latitude. Without correction, an aircraft flying at a speed of 900 km/hr (= 250 m/s) will drift south at an initial acceleration of: a c = 2Ωsin(38) 250 = 2 7.27 x 10-5 0.62 250 = 0.0224 m/s 2 This implies that for every hour of its flight the aircraft needs to correct its course by heading north a distance of: D = a c (3600 2 )/2 = 145152 m or about 90 miles.
Geostrophic Balance I: The primary horizontal balance of forces in atmospheric motion is the geostrophic balance. When an air parcel begins to move under a pressure gradient force (PGF), in a direction perpendicular to the isobars, and towards the low pressure, the Colriolis force (CF) begins to act, turning the parcel to the right (left in SH). Because CF depends of the particles velocity, this turning intensifies as the particle accelerates until the turning angle is 90 and CF is exactly equal to PGF and they both point perpendicularly to the motion, the pressure gradient force pointing towards the low pressure and the Coriolis force in the opposite direction.
Geostrophic Balance - Derivation To find how to express the geostrophic balance mathematically, examine the diagram on the right. The balance between CF and the PGF is expressed by the equal and 180 apart vectors P and C. In the NH the geostrophic wind blows in a perpendicular direction such that the low pressure is to its left (right in the SH). Note that the CF depends on the wind speed in the direction 90 to its left (right in the SH). The balance must also exist between the x and y components of the forces, separately, related the geostrophic wind components u and v). In the component balance, c x depends on v and c y on u, such that: c x = fv and c y = fu where f is the Coriolis factor. W PGF vector P = (p x,p y ) p x p y c y N S wind vector V = (u,v) v c x u E CF vector C = (c x,c y ) Northern Hemisphere (NH) force diagram Balancing these forces with the PGF results in the geostrophic balance: fv = (1/ρ)(Δp/Δx) fu = (1/ρ)(Δp/Δy) Note that the eqn in the x-direction provides a solution for v and that in the y- direction, for u:
Geostrophic Flow with Friction Friction slows down the wind, causing a weakening in the Coriolis force. A new balance is achieved between the resultant of the Coriolis force (CF) and friction on one hand and the pressure gradient force (PGF) on the other hand.
Friction, Convergence/Divergence, and Mass Continuity Friction leads to the convergence of air into the centers of low pressure and divergence out of the centers of high pressure. An important principle of any fluid motion, mass continuity (or mass balance) implies that there is rising motion in a low pressure system and sinking motion in a high, leading to a reversal of the convergence/divergence patterns aloft. The tendency of air to rise over a low pressure system creates favorable conditions for the formation of rain clouds. In high pressure systems the sinking motion leads to clear and dry conditions.
Mid-latitude Weather Systems
Synoptic Map This segment taken from a synoptic (weather) map of surface pressure shows isobars (contours of equal pressure in mb) and small flags, depicting the wind direction (the flags fly in the direction of the wind) and speed (each full flag bar is 10 knots and half a bar is 5 knots with 1 knot = 1/2 m/s). The flow is very close to geostrophic balance everywhere with a small tendency to flow across isobars towards the low pressure center - a result of the friction effect close to the surface.
Horizontal Motion and Weather The phenomena of weather are linked with the horizontal flow of air in other ways as well. In the midlatitudes chains of lows (cyclones) and highs (anticyclones) migrate steadily eastward mainly in winter. The overall tendency of air to rise over a low is combined with the advection of air by the circulation around it. Northerly winds bring cold air from the north southward to the west of the low-pressure center, and southerly winds bring warm air from the south northward. When cold and warm air masses meet, the warm air tends to move up creating favorable conditions for rain and severe weather. The bands along which air masses meet are called fronts.
Midlatitude Weather Systems
Life Cycle of Midlatitude Cyclones
Life Cycle and Heat Transport This three-dimensional schematic of a midlatitude cyclone life cycle demonstrates how these disturbances reduce the north-south temperature contrast in the midlatitudes by mixing cold air from the north with warm air from the south.
Tropical Weather Systems
Tropical Cyclones Tropical cyclones, also called hurricanes and typhoons, are intense low pressure disturbances that forms and migrates over the tropical ocean regions and are associated with intense winds and a very strong convective activity, which brings thunderstorms and large amounts of rainfall. They have the potential to cause major damage and loss of life when they make landfall.
Hurricane Vertical Cross Section The massive disturbances that sometimes grows in a time frame of a week or so, need specific and favorable conditions to occur, such as high sea surface temperatures (at lease 26 C) and weak vertical wind shears. Once they do, they 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 slow downward motion (subsidence).
Regions of Hurricane Activity Hurricanes are active in the "trade wind" belts - the regions just north or south of the equator where the winds blow quite steadily from east to west (easterlies). Here tropical disturbances generally form, initiated by weak pressure perturbations that exist all the time in the tropics. They move west with the trade winds in a steady, relatively slow motion (10-20 km/hour). During this phase they intensify mainly through the release of latent heat in the surrounding clouds and a small percentage reach full hurricane intensity. Hurricanes tracks curve eastward and they speed up north of ~30 N
Intertropical Convergence Zone (ITCZ) In the tropics, a belt of warmest surface temperatures, surrounds the Earth. Here there is abundant moisture so that small vertical movements of air can lead to spontaneous generation of deep convection. This convection then organizes itself in cells of massive thunderstorms that tend to drift eastward carried in the prevailing winds and in weak wave disturbances somewhat resembling midlatitude disturbances. This region is the ITCZ.