2.6 Primitive equations and vertical coordinates

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1 Chater 2. The continuous equations 2.6 Primitive equations and vertical coordinates As Charney (1951) foresaw, most NWP modelers went back to using the rimitive equations, with the hydrostatic aroximation, but without QG filtering, since it is only accurate to the order of the Rossby number = U fl 0.1 in mid latitudes and much larger near the Equator. Quasi-geostrohic models are now reserved for simle roblems where the main motivation is the understanding of atmosheric or ocean dynamics. So far we have used z as the vertical coordinate. When we make the hydrostatic aroximation, as in the rimitive equations, the use of ressure vertical coordinates becomes very advantageous. We can also use any arbitrary variable ( x, y, z, t) as vertical coordinate as long as it is a monotonic function of z (Kasahara, 1974). The most commonly used vertical coordinates are height z, ressure, a normalized ressure σ (Phillis, 1957), otential temerature θ (Eliassen, 1949), and several kinds of hybrid coordinates (e.g., Simmons and Burridge, 1981, Johnson et al, 1993, Purser, ers. comm., Bleck and Benjamin (1993). ch2-6-verticalcoords.doccreated on August 23, :35 PM 1

2 2.6.1 General vertical coordinates When we transform the vertical coordinate, then a variable A( x, y, z, t ) becomes A( x, y, ( x, y, z, t), t). z D z " # B s C s= (x, y, or t) The horizontal coordinates and time remain the same. Let s reresent x, y, or t. Let D be the value of A at oint D. D B "s = C B "s + D C "z # "z "s so that in the limit # " A $ # " A $ # " A $ # " z $ % & = % & + % & % & ' " s ( ' " s ( ' " z ( ' " s ( z s (6.1) and # A& $ % " ' ( s = A z z " A " A " " or = " z " " z. (6.2) ch2-6-verticalcoords.doccreated on August 23, :35 PM 2

3 Relacing (6.2) in (6.1), we get " # $ A% s & ' ( " = A % # $ s & ' z " + A % # $ ( & ' " # $ ( % z & ' " # $ z % s & ' ( (6.3) From this relationshi (for s=x,y) we can get an exression for the horizontal gradient of a scalar A in coordinates: " A " # A = # z A + ( )( )# z " " z (6.4) and for the horizontal divergence of a vector B: $ ".B = z.b + #B ' % & #" ( ). $ #" ' % & #z ( ) z " (6.5) The total derivative of A( x, y,, t) is given by da " A " A = ( ) + v.# A + " t " (6.6) ch2-6-verticalcoords.doccreated on August 23, :35 PM 3

4 The horizontal ressure gradient is therefore 1 " = 1, z " $ & % ). # ' ( %# * + - & ' ( %# ) / %z * + " # z1 (6.7) 0 which becomes, using the hydrostatic equation 1 1 $ z = $ # + $ #" # = $ #" (6.8) In summary the horizontal momentum equations become: dv = $ "% $ %# $ fkxv + F (6.9) the hydrostatic equation $ $ " = %# $ $ # # z = " g $ $ " becomes = # g or $ " $ z (6.10) ch2-6-verticalcoords.doccreated on August 23, :35 PM 4

5 The continuity equation can be derived from the conservation of mass for an infinitesimal arcel: the hydrostatic equation indicates that the mass of a arcel er unit area is roortional to the increase in ressure from the to to the bottom of the arcel (Fig b): M = "xyz = xy g (6.11) Δy g = "z Δx Fig b: Schematic of a arcel of air in a hydrostatic system, where Δ is roortional to the change in mass er unit area. # Now, " = # ", so that taking a logarithmic total derivative, 1 d# x u and noting that ", and the same with the other # x x sace variables, we obtain d (ln " " ) + #. v + H = " " 0 (6.12) ch2-6-verticalcoords.doccreated on August 23, :35 PM 5

6 The thermodynamic equation is as before c T d# dt d = c " # = Q (6.13) The kinematic lower boundary condition is that the surface of the earth is a material surface: the flow can only be arallel to it, not normal. This means that once a arcel touches the surface it is stuck to it. This can be exressed as d( " s ) = 0 at = s or d " s = + v.# s at = s (6.14) " t This kinematic BC is well defined although in ractice it may not be accurate, as for examle, when there is subgrid-scale orograhy. At the to, unfortunately, the BC is not so well defined: As z, 0, but in general there is no satisfactory way to exress this condition for a finite vertical resolution model. Most models assume a simle condition of a rigid to (i.e., making the to surface a material surface) d = 0 at = T (6.15) but this is an artificial boundary condition that introduces surious effects. For examle, Kalnay and Toth (1996) showed that a rigid to introduces artificial uside-down ch2-6-verticalcoords.doccreated on August 23, :35 PM 6

7 baroclinic instabilities in the NCEP global model, and similar observations were made by Hartmann et al (1997) with the ECMWF model. If the to of the model is sufficiently high, and there is enough vertical resolution, the uward moving erturbations get damed in the model (as they do in nature), and the surious interaction with the artificial to may remain small. Alternatively, radiation conditions enforcing the condition that energy can only roagate uwards can be used, but they are not simle to imlement. ch2-6-verticalcoords.doccreated on August 23, :35 PM 7

8 2.6.2 Pressure coordinates This coordinate is a natural choice for a hydrostatic atmoshere (Eliassen, 1949). It greatly simlifies the equations of motion: the horizontal ressure gradient becomes irrotational, and the continuity equation becomes simly zero 3-dimensional divergence, a diagnostic linear equation. As a result the geostrohic wind relationshi is also simler: v g 1 = k x " f For this reason, rawinsonde measurements have been made in ressure coordinates since the early 1950 s. " In ressure coordinates, 1 "#, the total derivative oerator d (6.6) is given by = " + v.# + ", where the vertical " t " d velocity in ressure coordinates is =. The rimitive equations become: dv = "# " fkxv + F (6.16) $ " = # $ (6.17). " # 0 v + = (6.18) " and the thermodynamic equation (6.13) is unchanged. ch2-6-verticalcoords.doccreated on August 23, :35 PM 8

9 The geostrohic and thermal wind relationshis are esecially simle in ressure coordinates: 1 " v g R v g = k x # and = $ k x # T f " f (6.19) On the other hand, the bottom boundary condition is not simle in ressure coordinates because the ressure surfaces intersect the surface: s = " + v.# s at = s (6.20) " t This requires knowing the rate of change of s : " s + v. # s = $ % 0 #. v d (6.21) " t This comlication of the surface boundary condition in ressure coordinates led Phillis (1957) to the invention of sigma coordinates (next subsection). Instead of the horizontal momentum equations, we can use the rognostic equations for the vorticity ζ and divergence δ, obtained by alying the oerators k. x and. to the momentum equations. In ressure coordinates these equations are # #. ( f ) # v + v $ + + " + ( f + ) $. v + k. $ " % = 0 (6.22) # t # # $ $ $ v 2 + %.[ v. % v] + " + %". + %.[ fkxv ] + % # = 0 (6.23) $ t $ $ ch2-6-verticalcoords.doccreated on August 23, :35 PM 9

10 2.6.3 Sigma and eta coordinates Because of the comlication of the bottom BC, Phillis (1957) introduced normalized ressure or sigma coordinates, where = / s and s ( x, y, t) is the surface ressure. These are by far the most widely used vertical coordinates. At the surface, = 1, and at = 0, = 0, so that the to and bottom BC are = 0. More generally, allowing for a to at a finite ressure T = const, # T # T " = = (6.24) S # T with = 0 at = 0, 1 The continuity equation is # #.( ) " = $% v $ # t #" (6.25) The surface ressure tendency equation is (homework, derive (6.25) and (6.26)): &" = & t & S & t = $%.# 1 0 (" v) d (6.26) Substituting back into the continuity equation, one can determine diagnostically from the horizontal wind field v. Desite their oularity, sigma coordinates have a serious disadvantage: the ressure gradient becomes the difference between two terms which over orograhy are large and of oosite sign: ch2-6-verticalcoords.doccreated on August 23, :35 PM 10

11 dv = $ "% $ %# $ fk & v + F (6.27) In order to reduce the noisiness introduced by orograhy due to this effect, Mesinger (1984) introduced a stemountain coordinate denoted eta (used in the Eta model at NCEP, e.g., Mesinger et al, 1988, Janjic, 1990, Black, 1994): O = S 1000mb (6.28) The first factor is the standard sigma coordinate, the second is a scaled standard value of the ressure over orograhy. Mountains are defined as boxes, whose tos have to coincide with a model eta level (Fig ). As a result, the ressure surfaces are almost horizontal, and the ressure gradient is comuted accurately. At NCEP, the Eta model has roven to be very skillful esecially in redicting storms. Recently it has been relaced by the non-hydrostatic Weather and Forecast Model (NCEP version). ch2-6-verticalcoords.doccreated on August 23, :35 PM 11

12 2.6.4 Isentroic coordinates The fact that under adiabatic motion, otential temerature is individually conserved suggested long ago that it could be used as a vertical coordinate. The main advantage, which makes it an almost ideal coordinate, is that vertical motion is aroximately zero in these coordinates (excet for diabatic heating). This reduces finite difference errors in areas such as fronts, where ressure or z-coordinates tend to have large errors associate with oorly resolved vertical motion. Hydrostatic equation: from the definition of otential temerature, and using the hydrostatic and state equations, we get d " dt R d dt d = # = # 1 " T C T C T (6.29) If we define the Exner function R / C = C T /" = C ( ), 0 and the Montgomery otential M = C T + = "# +, we see that " M = (6.30) "# The horizontal ressure gradient becomes very simle, so that the momentum equation is dv = "# M " fk $ v + F (6.31) ch2-6-verticalcoords.doccreated on August 23, :35 PM 12

13 The continuity equation is d " " ln + #. v + = 0 " " (6.32) The otential vorticity is conserved for adiabatic, frictionless flow (Ertel s theorem). This general roerty can be osed in its simlest formulation in isentroic coordinates: dq = 0, (6.33) where " q = ( f + k. # $ v ) ", and integrating between two isentroic surfaces, the otential vorticity is ( f + k. " # v ) q =, (6.34) $ similar to the shallow water equations otential vorticity. Although the isentroic coordinates have many advantages, they have also two main disadvantages: The first is that isentroic surfaces intersect the ground (as do other vertical coordinates excet for sigma-tye coordinates). In ractice this imlies that it is difficult to enforce strict conservation of mass, and this is imortant for long (climate) integrations. For this reason, hybrid sigma-theta coordinates have been used (e.g., Johnson et al, 1993). Other aroaches have been those of Bleck and Benjamin (1993) for the oerational RUC/MAPS model, and that of Arakawa and Konor (1996). ch2-6-verticalcoords.doccreated on August 23, :35 PM 13

14 The second disadvantage is that only statically stable solutions are allowed, since the vertical coordinate has to vary monotonically with height. There are situations, e.g., over hot surfaces, where this is not true even at a grid-scale. Moreover, in regions of low static stability, the vertical resolution of isentroic coordinates can be low. ch2-6-verticalcoords.doccreated on August 23, :35 PM 14