Isobaric Coordinates

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Duane Gordon
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1 METR 402: Atmos Dynamics I Fall 20 Isobaric Coordinates Dr. Dave Dempsey Dept. of Geosciences, SFSU In coordinate systems applied to the earth, the vertical coordinate describes position in the vertical direction (that is, parallel to the force of effective gravity. In meteorology, pressure can be a more convenient vertical coordinate than altitude. Why? One reason is that until recently, radiosondes, which are the primary means of gathering observations of weather variables above the earth s surface, measure and reported pressure, temperature, and humidity, but not altitude, as they rise through the atmosphere. (It has become more common to use global positioning systems (GPS to track radiosondes with great enough accuracy to get good altitude reports from them. Another reason is that on scales large enough for the hydrostatic approximation to be valid, the pressure-gradient force/mass term in the equations of motion becomes simpler and density no longer becomes an explicit variable in the tendency equations. There is no free lunch, of course. Pressure surfaces are not generally horiontal, for example, though on sufficiently large scales their slopes are small enough so that we can ignore the fact that isobaric coordinate surfaces are not quite perpendicular to the vertical direction. The Equations of Motion in Isobaric Coordinates In isobaric coordinates, most of the terms in the velocity-component tendency equations (as well as in the other governing equations must be modified, or at least reinterpreted. Here s how. ( By definition the vertical velocity is the rate at which a fluid parcel s vertical position changes with respect to time. If height,, is the vertical coordinate, then the vertical velocity component is w! D Dt. However, in isobaric coordinates the vertical coordinate is pressure, so the vertical velocity component is! Dp Dt. Note that since pressure decreases with increasing altitude, it follows that when w > 0 then! < 0, and vice versa. Not also that the dimensions are quite different: w has dimensions of distance/time, while! has dimensions of pressure/time. Finally, note that horiontal motion in isobaric coordinates is along pressure surfaces, which aren t necessarily flat. (They are not usually very steeply sloped, though, and so are called quasihoriontal. Hence it is possible for ω 0 for true horiontal motion (though of course ω = 0 for motion along a pressure surface. (2 Partial derivatives with respect to time (i.e., local or Eulerian derivatives, also called tendencies are computed with all other independent variables constant. In the case of height coordinates, partial derivatives with respect to time are computed with x, y, and held constant that is, at a fixed location in (x,y, space. However, in isobaric coordinates, local derivatives are computed with x, y, and p held constant that is, at a fixed location in (x,y,p space. The meaning of local therefore changes, because it is possible for a pressure surface to be rising or sinking with time, which means that a fixed location in (x,y,p space is not fixed in (x,y, space and vice-versa. As a result, it is often true that!(!t x,y,!!(!t x,y, p.

2 (3 Each partial derivative with respect to spatial position (such as the components of the gradient operator is computed with other independent variables constant. In the case of height coordinates, partial derivatives with respect to x or y are computed with t and (and y or x constant that is, in a horiontal direction. We might write such partial derivatives using notation such as!(!x y,,t and!(!y x,,t. The horiontal gradient,! H ( or! (, is a vector representing the rate of response of a field variable to changes in horiontal position in the direction in which the field variable increases most rapidly, with t and held constant (that is, in a snapshot of the field on a constant height that is, horiontal surface. However, in isobaric coordinates, horiontal gradients are computed not on constant-height surfaces but on constant-pressure surfaces. That is, as horiontal coordinates (say, x and y vary, t and p are held constant. We would write such a gradient as! p (. Since constant-pressure surfaces aren t generally horiontal, it follows that horiontal gradients in isobaric coordinates aren t truly horiontal. However, if their slopes aren t too large, as is the case on sufficiently large scales, then we can regard them as quasihoriontal. (4 As a consequence of ( and (2, advection terms must be rewritten. For example, terms such as!u #u =!V # H u! wu become!! V # p u!! u p. Note that in general,!! V # p u!! V # H u and!! u p #!wu, although they all have the same dimensions. (Similarly, temperature advection in the temperature tendency equation would be written as!! U #T =!! V # p T!! T p. (5 As a consequence of (2, the horiontal and vertical pressure-gradient force/mass terms must also be rewritten. First, consider the horiontal pressure-gradient force/mass. We want to find the true horiontal pressure-gradient force/mass expressed in isobaric coordinates. (This is what affects the horiontal velocity, V, the definition of which doesn t change when we switch from - to p-coordinates. To figure out how to do this, refer to Figure, below: an isobaric surface p(, 2 = p(x, = p* -- 2 another isobaric surface p(x, x Figure : A vertical cross section through two sloping pressure surfaces 2 -- p(,

3 Figure shows two (sloping constant-pressure surfaces, with x- and - coordinates of three locations and the pressures at those three locations carefully labeled. Note that p( = p(x = p* because the locations ( and (x, lie on the same pressure surface. Our goal here is to represent!( #p #x, the horiontal pressure-gradient force/mass in the x-direction, in terms of a derivative along a constant-pressure surface of a dependent variable in isobaric coordinates. What does this mean? In height coordinates, the independent variables are x, y,, and t and the dependent variables are T, p,, w, ρ, and some moisture variable. (Other dependent variables can be derived from these. In contrast, in isobaric coordinates the independent variables are x, y, p, and t and the dependent variables are T, or Z (i.e., height or geopotential height, respectively,, ω, ρ, and a moisture variable. In other words, we won t be expressing!( #p #x in terms of a spatial derivative of pressure on a constant-height surface because in isobaric coordinates pressure is an independent variable and horiontal derivatives will be computed on a constant-pressure surfaces. Rather, we ll considering derivatives w/r/t x of one or more other dependent variables, on a constant-pressure surface. To make the derivation clearer, start by writing!p!x in terms of its definition in the context of Figure :!p(x!x lim p(x 2 p(x ( ' * # x & x Note that we are going to evaluate!p!x at the point (x, in the diagram. To get where we want to go, focus on the term in brackets and add!p( + p( = 0 to the numerator: p(! p(x = p(! p( + p(! p(x However, since p( = p(x, = p*, the last two terms in the numerator cancel each other out. Hence we can write: p(! p(x = p(! p( (This really just says that from p( on the lower of the two pressure surfaces in Figure we can subtract the pressure from anywhere on the other pressure surface and get the same result, because the pressures everywhere on that other surface are the same. V V 3

4 Now for another algebra trick, this time multiplying by 2 2 =: p(! p(x = p(! p( '( 2 ' # & # 2 & Since a b = b a, we can interchange the denominators of the two factors in brackets: p(! p(x = p(! p( '( 2 ' # 2 & # & Factor! out of the first factor in brackets on the right-hand side: p(! p(x =! p(x,! p(x, '(! 2 ' # 2 & # & Now we re in a position to see what happens as we take the limit as. It s important to recognie that because of the way we ve defined it, 2 will change as changes. That s because 2 represents the height (at of the pressure surface on which the pressure is p(. Similarly, is the height of the pressure surface (at x on which the pressure is p(x, which is the same pressure as p(. That is, we ve defined to be the height of a particular constant-pressure surface. This makes it a function of p and x (and y, and hence it will vary with these independent variables. To emphasie this point, we ll rewrite 2 and in the previous expression so that it reads: p(! p(x =! p(x,! p(x, '( (x, p*! (x, p* 2 ' # 2 & # & where p* represents a particular pressure = p( = p(x. Now, as, we see from the diagram and from the definitions of 2 and that 2 too. Hence, we can write: lim # p( p(x & ( = lim # p( p( & ( lim # (, p* (x, p* & ( x ' 2 2 ' x ' Using conventional short-hand expressions for these derivatives, we can write this more compactly as:!p(x!x # =!p(x, & # (!(x, p* & (! '!x p ' 4

5 and hence the horiontal pressure-gradient force/mass in isobaric coordinates is just:! #p(x #x = #p(x ' & * #(x, p* ' & # ( #x p ( Note that!(x, p*!x p is just the slope of the p* pressure surface at x. That is, the horiontal pressure-gradient force/mass at any point is proportional to the slope of the pressure surface through that point. The expression simplifies further if we can make the hydrostatic approximation, which states (in height coordinates that: 0 =! #p #! g Solving this for (!p and substituting into the previous expression gives:! #p(x #x =!g #(x, p* ( ' * =! #[ g (x, p* ] & #x p #x p +! #,(x, p* #x p Finally, treating the y-component of the horiontal pressure-gradient in the same way, we can write the vector version as follows:! # H p!g# p Z!# p (where Z is the geopotential height. That is, if the hydrostatic approximation is valid, and if we ignore spatial variations in effective gravity (including the relatively small difference between height and geopotential height, then the horiontal pressure-gradient force/mass is proportional to the gradient of geopotential on a pressure surface. It continues to be proportional to the slope of the pressure surface, as well. On typical isobaric maps, which show the pattern of geopotential heights of a pressure surface, the contours of geopotential height tell us visually where the horiontal pressure gradient is relatively large and relatively small, simply by the contour spacing closely packed contours imply that the horiontal pressure gradient is relatively large. 5

Solution to Problems #1. I. Information Given or Otherwise Known. = 28 m/s. Heading of the ultralight aircraft!! h

Solution to Problems #1. I. Information Given or Otherwise Known. = 28 m/s. Heading of the ultralight aircraft!! h METR 520: Atmospheric Dynamics II Dr. Dave Dempsey Dept. of Geosciences, SFSU Spring 2012 Solution to Problems 1 Problem 1 An ultralight aircraft is flying. I. Information Given or Otherwise Known Horizontal