Notes on pressure coordinates Robert Lindsay Korty October 1, 2002

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1 Notes on ressure coordinates Robert Lindsay Korty October 1, 2002 Obviously, it makes no difference whether the quasi-geostrohic equations are hrased in height coordinates (where x, y,, t are the indeendent variables or in ressure coordinates (where x, y,, t are the indeendent variables; the systems of equations exress the same hysical meaning in both cases, and no information may be deduced from the equations hrased in ressure coordinates which could not also have been identified with the equations hrased in height coordinates. So in this sense, either set of variables is useful. The mathematics (i.e., the road taken to solve the equations, however, is not identical because the form that the equations take will differ between the two sets. The ressure coordinate system is referable in most cases in the atmoshere, owing to the simler form of the equations (rincial among these simlifications is the extraction of density [which varies] from the equations of motion. In the ocean, or in any fluid in which an incomressibility assumtion can be made, the equations are generally equally simle in either height or ressure coordinates. A major difference, in this regard, between the ocean and the atmoshere is that density is nearly constant in the ocean, but decays exonentially with altitude in the atmoshere. (As an aside, the ressure and density in the atmoshere dro by an order of magnitude roughly every 16 km; the surface ressure is about 1000 mb [100,000 Pa], at 16 km the ressure is only 100 mb [10,000 Pa], and by 48 km, the ressure is only 1 mb [100 Pa]. If the atmoshere were defined to be 100 km thick, the uer half by volume of a column of air contains only one ercent of the mass of that column. Here, I d like to address some benefits of the ressure coordinate system. First, the continuity equation in ressure coordinates takes a very simle form. Note that in height coordinates, ( ρ + (ρ u + (ρw =0, where is the horiontal gradient oerator in height coordinates ( ( ( + ;the subscrit is a notational reminder that these derivatives are taken holding height constant. In a fluid in which density varies, that fact must be taken into account in this equation. However, in ressure coordinates, the above exression may be rewritten as u + ω =0, ( ( where is the horiontal gradient oerator in ressure coordinates ( +, and ω is the vertical velocity in ressure coordinates. This is a much simler form of the same equation; the simlicity arises from the fact that the coordinate reresents the mass er unit area of the fluid above the corresonding isobaric surface (i.e., since a column of air has a surface ressure of roughly 1000 mb, half of the mass contained in it lies above the

2 500 mb surface. In the ocean, mass varies nearly linearly with deth; in the atmoshere, it decays exonentially with height (but, of course, varies linearly with ressure. From the continuity equation in ressure coordinates, interreting as a vertical coordinate and ω as a vertical velocity shows that in x, y, -sace, air moves as an incomressible fluid. The mathematical treatment of the equations of motion in ressure coordinates is considerably simlified for the atmoshere by using ressure coordinates. Aside from boundary conditions, it is no more difficult to use ressure coordinates in the ocean than it is to use height coordinates, but there is no benefit either. Since the ocean surface has variable ressure, height coordinates generally offer simler boundary conditions in the ocean. Several texts (e.g., Gill, Holton derive the ressure coordinate form of the continuity equation from first rinciles. Attached to this handout, I have shown how the height coordinate form may be mathematically transformed into the ressure coordinate form. In the ressure coordinate form, the equations rove simler than in the height coordinate form for the atmoshere. Some of the equations become formally identical to the equations for a homogeneous and incomressible fluid. Density dros out of the horiontal equations of motion, and the continuity equation is greatly simlified. One trade off of ressure coordinates is that boundary conditions become more comlicated. The surface of the Earth is not one of constant ressure; other coordinates, such as σ coordinates are sometimes used in general circulation models (σ s. But erhas a bigger loss that results from this jum to ressure coordinates is that it is somewhat less intuitive to think in terms of ressure as a vertical coordinate. As noted above, though, the same information deduced in height coordinates may be found in ressure coordinates. As a simle examle, consider the following. φ+δφ φ+δφ +d φ +d φ x x Figure 1: left anel: curves of constant ressure on height surfaces; right anel: the same icture showing how height varies along isobaric surfaces If you were in an airlane confined to fly at a constant height of 5700 meters, your barometer would read somewhere near 500 mb. If you were flying along one of the constant φ lines deicted in the left anel by solid lines (remember, φ is simly g you would observe the

3 ressure to go from high to low to high as you travel from left to right; it may go from 502 to 498 and back to 502 mb. However, if you instead confined your aircraft to fly at an altitude so as to maintain constant ressure (say, 500 mb, you would have to adjust your altitude. Your flight ath is now along the dashed surfaces of constant ressure. You must go from a high altitude, to a lower one, and then rise again to maintain a constant ressure. You can visualie your motion with the left anel; you can see contours of your altitude with the right one. Note that as you travel from left to right, the value of φ at first decreases along your ath, and then starts to rise half way through. Thus, whereas we study ressure erturbations (or, equivalently, ψ in height coordinates, we study height erturbations (or, equivalently, φ in ressure coordinates.

4 Note from the definition of S in equation (1, that S is always ositive. Potential temerature increases with height, so it decreases with ressure (i.e., moving down in the atmoshere toward the surface. Thus, dθ/d is always negative, and the minus sign contained in the definition makes S always ositive. The relationshi between N 2 and S is remarkably simle. N 2 = g dθ θ d = g dθ ( ρg (hydrostatic arox. θ d = ρ2 g 2 dθ ρθ d ( 1 = ρ 2 g 2 dθ π d = ρ 2 g 2 S = g2 S (if you refer secific volume to density α2 So like N 2, S is always ositive (N 2 < 0 is, like S<0, a condition for convective instability. The minus sign contained in the definition of S is simly due to the difference in the direction that the vertical coordinate oints in height and ressure coordinates. Both N 2 and S have the same hysical meaning. The owers of ρg converting S to N 2 are due to the hydrostatic aroximation which relates d to d. I hoe this hels clarify ressure coordinates. Just remember that a rincial simlification comes from absorbing density variations into the vertical coordinate. Terms in the height coordinate form of the definition of q which look daunting when ρ is not constant: 1 ρ become simler in ressure coordinates: ( f 2 o N 2 ρ ψ ( f 2 o S. A nice aer that goes over the transformation from height to ressure coordinates for all of the equations of motion was ublished in 1949 by Eliassen. I have a coy if any of you are interested.

5 Transforming the continuity equation into ressure coordinates Robert Lindsay Korty October 2, 2002 A number of texts cover the derivation of the continuity equation in ressure coordinates from first rinciles. Treatments of this may be found in Wallace and Hobbs ( , Holton ( , or Gill ( Here the derivation is mathematical: the continuity equation is transformed from a system in which height is an indeendent variable (x, y,, t to one in which ressure is the vertical coordinate (x, y,, t. First, note the following coordinate transformations for an arbitrary variable quantity, A: ( ( ( A A A = + ( ( ( A A A = ( A = 1 ( A g. And, with the hydrostatic aroximation, δ = ρgδ 1 δ = ρg δ (in ressure coordinates, = 1 ρ, we may write exressions for the transformation of derivatives: ( A = ( A + ρ Similarly, derivatives with resect to y or t are transformed: ( ( ( A A ( ( ( A A ( A. (2 A (3 A. (4 Before transforming the continuity equation from height to ressure coordinates, make an Eulerian exansion of the material derivative in the definition of w. This will be needed to convert w to ω. From the definition of w, w d dt = ( + u ( + v ( + ω. Using the definition of φ, and the hydrostatic aroximation to relace w= 1 g ( + u g ( + v g, ( ω ρg. (5

6 (As an aside, if numerical accuracy is not imortant, ω may be aroximated by ρgw to within about five ercent. Note that ω and w have oosite signs. Rising motion corresonds to a decrease in ressure, which gives a negative ω. The continuity equation, exressed in height coordinates, may be written: ( ( ( ( ρ ρu ρv ρw =0 (6 x,y,t }{{}}{{}}{{}}{{} A B C D We now must transform the derivatives holding constant into derivatives which hold constant. By equation (2, term B in equation (6 is ( ( ( ρu ρu ρu ( ( [ u ρ = ρ + u + ρ ( ] ρu. But ( ρ =0: ( ( 1 g = 1 g ( ρu = ρ ( =0,sotermB of equation (6 is simly [ ( u + ρ ρu ( ] (7 and from equations (3 and (4 we may write terms C and A from equation (6 as ( ( [ ρv v = ρ + ρ ( ] ρv ( ( ( ρ ρ ρ ( [ = 1 ] [ + ρ ( ] ρ g = 1 ( [ + ρ ( ] ρ g ( [ ρ = ρ ( ] ρ (8 (9 Using the hydrostatic aroximation, we may rewrite term D in equation (6 as ρw ρw = ρg (10

7 Now, lugging (7, (8, (9, and (10 into equation (6 yields: [ ρ ( ( ( ] ( ( u v ρ + ρu + ρv + ρ + ρ ρg ρw =0 And from (5, we note that the term inside the square brackets in the exression above is equal to ω + ρgw. Making this substitution results in the continuity equation transformed into ressure coordinates. ρ [ω + ρgw] + ρ ( u ( v + ρ + ( u ( v ρg ρw + ω u + ω = 0 = 0 = 0 (11 Note that the continuity equation no longer includes any exlicit reference to density. This is a rincial advantage of ressure coordinates. For the ocean, or for any fluid for which an incomressibility assumtion can be made, the continuity equation in height coordinates reduces to u + w = 0, such that height or ressure coordinates are equally simle to use. In the atmoshere, equation (11 is a much simler form than the height coordinate form (equation (6 since an incomressibility assumtion can not be made; in ressure coordinates, the exonential variation of density is subsumed in the coordinate transformation and air moves like an incomressible fluid in x, y, -sace. Mathematically, the ressure coordinate form of the continuity equation for the atmoshere has the same form as the height coordinate form of the continuity equation for the ocean.