Lecture 10. Vertical coordinates General vertical coordinate

Transcription

1 Lectue 10 Vetical coodinates We have exclusively used height as the vetical coodinate but thee ae altenative vetical coodinates in use in ocean models, most notably the teainfollowing coodinate models and isopycnal models. We ll fist tansfom the equations of motion to a geneal vetical coodinate,, befoe consideing the most common specific vetical coodinates. A good efeence fo the tansfomation is the book by Haltine and Williams, 1980, section 1-9. The book by Haidvogel and Beckmann, 1999, descibes models using all thee vetical coodinates Geneal vetical coodinate Conside a geneal vetical coodinate,, which is assumed to be a monotonic function of height, z. Tansfoming the equations of motion into the coodinates (x, y,, t) makes z = z(x, y,, t) a dependent vaiable. Any dependent vaiable, A = A(x, y, z, t), that can be descibed in the oiginal coodinate can be descibed as a function of the new coodinates: A = A(x, y, z(x, y,, t), t) The vetical patial deivative in the new coodinate is o equivalently A = A A = A 137

2 Atmospheic and Oceanic Modeling, Sping A patial deivative with espect to any othe odinate, s, whee s is one of x, y o t, is obtained using the chain ule: A = A + A s s z s o, substituting in fo z A fom above, A s = A + A s z Hee, the vetical ba with suffix indicates the patial deivative holding constant the suffix. We can now wite down the hoizontal gadient of a scala A in height coodinates expessed in tems of deivatives in coodinates: z A = A A A z Similaly, the hoizontal divegence of a hoizontal vecto, v can be expessed: z v = v v z The vetical velocity in height coodinates, w, can be expessed: w = D t z = + D t x t x + D y t y + D t = + v t z + ṙ Finally, the total deivative of A in z coodinates is the total deivative in coodinates: D t A = A + v t z A + w A z = A ( + v t A + w ) A v t z = A + v t A + ṙ A

3 Atmospheic and Oceanic Modeling, Sping Now we can use the above elations to tansfom the tems in the equations of motion. Hydostatic balance becomes = gρ o moe natually = ρ gz The hoizontal pessue gadient in the momentum equations becomes z p = p z = p + ρ gz whee gz is the geopotential. The incompessible continuity (o non-divegence) equation is: z v + z w = v + w v z The tem w can be found fom the definition of w = D t z above: w = ( + v t z + ṙ ) = ( t z) + v ( z) + ṙ ( z) + v z + ṙ = D t ( z) + v z + ṙ so that continuity in coodinates is D t + ( v + ṙ ) This can be witten equivalently as o D t ln + v + ṙ t + v + ṙ

4 Atmospheic and Oceanic Modeling, Sping Combining all the above elations, the inviscid and adiabatic HPEs can then be witten in coodinates as: D t v + f ˆk v + 1 ρ p + 1 ρ ρ gz + ρ gz D t + ( v + ṙ ) ρ = ρ(θ, S, z) D t θ D t s We ll now chose specific vetical coodinates and see how the above equations simplify o othewise tun out in each case Teain following coodinates The epesentation of bottom topogaphy in z-coodinates has histoically been cude 1. An elegant appoach to incopoate the topogaphy smoothly into models is to nomalize the vetical coodinate by the fluid depth. Thee ae many choices of teain following coodinate but we will demonstate the simplest, known as σ-coodinates (sigma) whee z σ = H(x, y) This maps the bottom at z = H(x, y) to σ = 1 and so the domain is made squae. Using = σ = z/h simplifies the scaling facto = 1 o H = H In the hydostatic equation, the vetical deivative of gz is tivial since z = H = σh. The esulting σ-coodinate equations ae: D t v + f ˆk v + 1 ρ σp 1 ρh σ σz 1 The finite volume method (o patial cell epesentation) has somewhat alleviated this poblem. Howeve, thee ae still advantages to teain-following coodinates such as fixed domain, the ability to use a modal epesentation in the vetical, etc...

5 Atmospheic and Oceanic Modeling, Sping a) 20 b) Figue 10.1: Schematic of sigma-coodinates (a) and s-coodinates (b) in physical x z plane. σ + ρgh σ H v + σ H σ Thee ae two gadient tems in the momentum equations. To a lage degee, these cancel but cannot necessaily be canceled analytically. To see this, imagine a esting fluid of constant density. The pessue is simply a function of height, z. Howeve, wheeve the topogaphy, H, slopes, the coodinate sufaces above must slope too. This means that σ p is non-zeo and indeed is canceled by gρ σ z. At a numeical level, it is impossible to make these two tems balance exactly fo an abitay static initial condition and so leads to spontaneous motion. The imbalance of tems is known as the pessue gadient eo and a lot of effot has been put into educing this eo. With moden schemes, a typical eo will be of the ode of millimetes pe second fo common cicumstances. Ove shallow topogaphy, the effective vetical esolution is inceased. This can be an advantage but cae must be taken to avoid numeical instability due to small gid lengths (CFL). The implicit teatment of vetical tems is theefoe used in models such as SPEM/SCRUM o SEOM. Thee ae many vaiants on the teain following coodinates whee diffeent functions of bottom depth and height ae used. Some also include the

6 Atmospheic and Oceanic Modeling, Sping fee-suface height in the nomalization; the s-coodinate is defined as s = z η H + η so that all though the fee-suface can move the computational domain is fixed ( 1 s 0). The suface vaiations intoduce elatively mino slopes to the coodinate sufaces compaed to those induced by the bottom topogaphy. Teain following coodinates have seveal advantages ove z-coodinate models: Smooth epesentation of bottom topogaphy. Allow concentation of coodinate lines in bounday layes. but have the following dis-advantages: Pessue gadient eo can be significant and/o lead to spontaneous motion. Repesentation of hoizontal o along isopycnal pocesses is awkwad Isopycnal coodinates Whee the fluid is adiabatic, (potential) density is conseved and since, unde statically stable conditions, density is a monotonic function of height it makes a useful vetical coodinate. The eal advantage of isopycnal coodinates is thei Lagangian teatment of vetical motion; explicit advection acts only in the hoizontal. This makes the models models vey adiabatic allowing them to avoid numeical diffusion in the vetical that can be toublesome in z-coodinate and teain-following coodinate models. Stictly speaking, the vetical coodinate is potential density, σ θ but we will use ρ as the vetical coodinate to avoid confusion with the teain following coodinate. Setting = ρ, the continuity equation becomes: t ρ z + ρ z v + ρ ρz ρ

7 Atmospheic and Oceanic Modeling, Sping whee ρ is non-zeo only whee diabatic tems foce flow acoss isopycnals. The continuity equation has become a laye thickness equation fo each density class. The hydostatic equation is ρ + ρ gz = ρ ρ + ρgz gz ρ = ρm gz ρ whee M = p/ρ+gz is the Montgomey potential. The momentum equations become D t v + f ˆk v + 1 ρ ρρm Isopycnal models have seveal advantages ove the height and teain-following coodinates: Ideal fo modeling lateal tansfe pocesses. Adiabatic motions modeled without any spuious diabatic tems. Smooth epesentation of topogaphy. The bottom topogaphy is epesented as piecewise-linea and is included in the model though a vanishing of the laye thickness. Conseves volume of density classes Some dis-advantages ae: Full o non-linea equation of state is difficult. Non-hydostatic effects/dynamics ae not possible. Density is not a natual coodinate fo epesenting mixing pocesses such as the suface BBL (shallow and deep mixed layes). Vetical and hoizontal esolution ae tightly connected in egions whee isopycnals outcop. This can lead to inadequate hoizontal esolution in egions such as the ACC.

8 Atmospheic and Oceanic Modeling, Sping a) a) Figue 10.2: A schematic of (a) step topogaphy in a z-coodinate model and (b) a patial cell epesentation z-coodinate models Note that the z-coodinate equations ae eadily ecoveed. Setting = z means that z = 1, D t z, gz and gz = g. Since we ae so familia with z-coodinates we will efain fom witing them out once moe. One of the poblems with z-coodinates used to be with the cude epesentation of topogaphy as a seies of giant steps (see Fig. 10.2). Now, howeve, we epesent the bottom eithe via a vaiable thickness bottom laye (patial cells) o with a piecewise-linea epesentation. This appoach has been tested and favouably compaed to esults fom a sigma-coodinate model. The thee main z-coodinate models have some fom of patial cells implemented. Some advantages of using z-coodinates ae: Simple to implement and use! The full equation of state can be used. Diabatic pocesses, including the mixed-laye, can be epesented easily. Non-hydostatic and non-boussinesq tems can be included. Some dis-advantages ae: Repesentation of along isopycnal pocesses ae awkwad. Repesenting the bottom bounday laye is awkwad.

9 Atmospheic and Oceanic Modeling, Sping Pimitive equations in pessue coodinates The othe natual choice of vetical coodinate is pessue. Hee, howeve, the non-boussinesq equations ae the best stating point. This leads to equations vey simila to those used in meteoology. They have the advantage of not needing to simplify the equation of state dependence on pessue but have a majo difficulty in epesenting the bottom. Use of the Philip s nomalized pessue coodinate (a sigma-coodinate) endes equations vey simila to the teain-following coodinate outlined above. The compessible, non-boussinesq hydostatic equations in height coodinates ae: D t v + f ˆk v + 1 ρ zp + gρ 1 ρ D tρ + z v + w ρ = ρ(θ, S, p) D t θ D t s Using the same tansfomation ules as befoe, the non-boussinesq equations witten in a geneal coodinate become: D t v + f ˆk v + 1 ρ p + gz + ρ gz D t + ( 1 ρ D tρ + v + ṙ ) ρ = ρ(θ, S, p) D t θ D t s The continuity equation looks unusual. having two total deivatives. A judicious choice of coodinates, = p, will cause these tems to cancel. With

10 Atmospheic and Oceanic Modeling, Sping = p, the hydostaic equation becomes which is nomally witten 1 + ρg α + Φ whee α = 1/ρ is the specific volume and Φ = gz is the geopotential. We can now examine the two total deivatives in the continuity equation: D t + 1 ρ D tρ = D t ( 1 gρ ) 1 gρ D tρ 2 = 1 gρ D tρ 1 2 gρ D tρ 2 The othe majo simplification is that the gadient of pessue on a pessue suface is zeo ( p p ) so that the momentum equation etains only one gadient tem. Making the above substitutions we aive at the pimitive equations in pessue coodinates: D t v + f ˆk v + p Φ α + Φ p v + ω α = α(θ, S, p) D t θ D t s whee ω = ṗ is the coss coodinate flow in pessue coodinates. The equation of state is now witten in tems of specific volume. Examination of the above equations and compaison with the Boussinesq, hydostaic equations in height coodinates eveals a one-to-one coespondance of tems and vaiables; the equations ae isomophic. They take the same stuctual fom and this caies though to the bounday conditions also (de Szoeke and Samelson, 2002; Mashall et al., MWR 2004). Because the

11 Atmospheic and Oceanic Modeling, Sping equations ae isomophic, all of the algoithmic consideations and coodinate tansfomations we discussed can be applied to the pimitive equations in pessue coodinates. This is an appoach used in the MIT GCM to model the atmosphee o non-boussinesq ocean using the same code that is used to model the Boussinesq ocean Atmospheeic equations in pessue coodinates Fo the atmosphee, we simply need to eplace the equation of state with that of the atmosphee (we ll use the ideal gas equation, p = ρrt ) and dop salinity fom the equations. Thee is one simplifying step that can be made due to the ideal gas equation which uses a function called the Exne function, Π: Π = c p ( p p o ) κ Hee, c p is the specific heat at constant pessue, R is the gas constant and κ = R/c p and p o is a efeence pessue used to define the potential tempeaute: θ = c ( ) κ pt p Π = T Note that p Π = c pκ p ( p p o p o ) κ = κ p Π and α = RT p = κc pt = κπθ = θ p Π p p so that the pimitive equations fo an ideal dy atmosphee can be witten: D t v + f ˆk v + p Φ θ + Φ Π p v + ω D t θ Integating the continuity equation vetically ove the ai column and applying appopiate bounday conditions yields the suface pessue equation t p s + p s v

12 Atmospheic and Oceanic Modeling, Sping Hybid coodinates The choice of vetical coodinate is pehaps the single most impotant featue that diffeentiates between models and is still an active aea of eseach. Many goups ae examining how to use hybid coodinates whee the coodinate may be a function of height in the mixed laye, a function of isentopes in the inteio and some function of the teain in the bottom bounday laye.