METR 50: Atmospheric Dynamics II Dr. Dave Dempsey Spring 014 Problem #1: The Gradient Wind in Natural Coordinates (Due Friday, Feb. 8; 0 pts total) In natural (s,n,p) coordinates, the synoptic-scaled, horizontal velocity equations (describing how and why horizontal velocity, V, changes), can be written as: s-component equation: n-component equation: DV Dt = Φ s p V = Φ fv R traj For any given geopotential (Φ) pattern, an infinite number of combinations of wind speed (V V ) and trajectory radius of curvature (R traj ) satisfy these equations. That is, no matter what V is, you can find a value of R traj that satisfies the n-component equation. (The s-component equation doesn t depend on V at all.) The gradient wind,, is a particular solution to these equations. is an idealized, mathematically defined, horizontal wind that blows at just the right speed and in just the right direction so that the combined horizontal Coriolis force and pressure-gradient force (other horizontal forces are neglected) accelerate the parcel so that (a) its speed doesn t change, but (b) direction changes at a rate such that its trajectory at the location of the parcel has a radius of curvature, R traj, equal to the radius of curvature of the geopotential height contour,, at that location. That is, if we constrain the infinite number of possible values of R traj to one particular value, then this constrains what the velocity of the parcel can be. Invoking this definition of, the velocity equations for become: s-component equation: n-component equation: D Dt = 0 = Φ f 1
The s-component equation guarantees that the gradient wind speed isn t changing, which occurs only when the gradient wind blows parallel to the geopotential height contour at its location (or when the geopotential gradient = 0). The n-component equation is quadratic in. Rearranged, it reads: + f + Φ = 0 or, more generally: a + b + c = 0 where a = 1, b = f, and c = Φ. This is the gradient wind equation. Quadratic equations such as this one have one or two solutions, or roots. The general solutions to quadratic equations are well known. For the gradient wind equation we can write the solutions as: = b ± b 4ac a = f ± f 4 ( ) / 1 Φ = f ± f Φ Some solutions might be negative or complex (that is, having an imaginary part) and would therefore not be physically meaningful even though they are mathematically possible. To help us isolate the physically meaningful solutions, Table 1 on the next page classifies the roots of the gradient wind equation in the Northern Hemisphere based on the signs of Φ and. With two possible roots for each combination of signs of Φ and, there are eight possible solutions appearing in Table 1, four of which are physically meaningful. Figure 1 on Page 4 illustrates the forces acting in each physically meaningful case.
Table 1: Classification of Roots of the Gradient Wind Equation in the Northern Hemisphere Sign of Sign of Φ > 0 < 0 Φ >0 + root: unphysical + root: antibaric flow (anomalous low) root: unphysical root: unphysical Φ <0 + root: cyclonic flow (regular low) root: unphysical + root: ( > f ) anticyclonic flow (anomalous high) root: ( < f ) anticyclonic flow (regular high) To interpret Table 1, consider the following: (1) Recall that > 0 means that the height contour curves in the direction of positive n in natural coordinates (that is, to the left when viewed looking in the direction of the gradient wind), while < 0 means that the height contour curves in the direction of negative n (that is, to the right of the gradient wind). () Recall also that Φ <0 means that the geopotential (and geopotential height, and pressure) decreases in the direction to the left of the wind (which is the positive n direction). That is, heights (and pressures) are lower to the left of the gradient wind than to the right. Φ >0 means that the geopotential increases in the direction to the left of the wind, so heights and pressures are higher to the left of the gradient wind than on the right. (3) Unless the quantity under the square root sign is zero, then there will be two solutions to the quadratic equation for given values of a, b, and c. + root refers to case in which the + sign precedes the square root symbol, while root refers to the other case, when a sign precedes it. 3
(4) In the cases of both the regular and anomalous highs, the geopotential gradient is limited by the requirement that the quantity under the square root sign be nonnegative; that is, Φ f < 4 This means that if the gradient wind is a good approximation of the real wind (as it is, roughly, on synoptic scales and larger), then the closer to the center of high geopotential or pressure (that is, for decreasing ), the smaller the gradient of geopotential or pressure must be. This means that on synoptic scales, pressure surfaces must become increasingly flatter the closer they are to the center of a high (and contours of geopotential height must become farther apart; the same would be true of isobars on a constant-height surface). It is possible to show that the gradient wind must become correspondingly slower, too. These constraints don t apply to a synoptic-scale low-pressure system; the pressure or geopotential gradients near the center of synoptic-scale lows can be quite large and winds quite fast. It s easy to see this difference between synoptic scale highs and lows on many weather maps; it even seems to be valid near the earth s surface, where friction complicates the picture. Figure 1: Force diagrams in the Northern Hemisphere for the four possible types of gradient wind: (a) regular low; (b) regular high; (c) anomalous low; and (d) anomalous high. L (a) H (b) L (c) H (d) 4
In Figure 1, F Co represents the Coriolis force per unit mass, F PG represents the pressure-gradient force per unit mass, a represents the acceleration arising from the sum of F Co and F PG (that is, a = F PG + F Co ), and is the gradient wind velocity. The tail of each of these vectors is located at the point of interest. The geopotential height contour through the point at which the gradient wind is defined is shown by the thin, curving line. A sense of the geopotential height or pressure distribution is indicated by the contour and by an H or L an L indicates an area where the geopotential height or pressure is lower than it is along the contour shown, while H indicates that it is higher. (As used here, these aren t necessarily local minima or maxima in the height or pressure field.) (1) [15 pts total] The gradient wind. (a) [ pts] Four of the solutions in Table 1 above are classified as unphysical. In each case explain why, both in mathematical terms and (where you can) in physical terms (that is, in terms of the forces acting on the wind, their fundamental properties, and Newton s nd Law). (b) [0.5 pts] In Table 1, one case is classified as an anomalous low. Explain why. (c) [0.5 pts] In Table 1, one of the roots in one case is classified as a regular high while the other is classified as an anomalous high. What is the difference between these two solutions? (d) [8 pts] Construct a table analogous to Table 1 for the Southern Hemisphere, briefly summarizing your reasoning for each entry. (e) [4 pts] For the physically possible cases in your Southern Hemisphere table, construct force-balance diagrams analogous to Figure 1. (Draw wind vectors as fat arrows and force/mass and acceleration vectors as thin arrows, as in Figure 1. The force/mass vectors should sum to equal the acceleration vector.) 5
() [5 pts] "Cyclonic" means turning or rotating in the same sense as the earth does as seen looking down on it from space. (Note that this means that cyclonic motions viewed from above the earth's surface imply turning or rotation in opposite senses in the Southern and Northern Hemispheres.) Explain why fr traj > 0 for cyclonic motions in both hemispheres. [f Ωsinϕ, where ϕ is the latitude, and R traj is the radius of curvature of a parcel trajectory. Note that this applies to any wind, real or not.] 6