A simplified model for meteorological now-casting

Transcription

1 A smplfed model for meteorologcal now-castng A. MOLINI a & R. FESTA b a Unversty of Genova, Department of Envronmental Engneerng, 1, Montallegro Genova ITALY,emal: moln@dam.unge.t b Unversty of Genova, Department of Physcs,, Dodecaneso Genova ITALY ABSTRACT A smplfed numercal model for smulaton of the nteracton amongst atmospherc eddes at synoptc scale s presented and dscussed. Snce for all such nteractons the atmosphere can be regarded as a thn layer of non-vscous, ncompressble flud, we started from a revew of the b-dmensonal flows hydrodynamcs n the Hamltonan formulaton to derve the dynamcs for "a vortcty segments system" n moton on a sphercal surface. The extenson to the case of a rotatng sphere becomes possble through the equaton of potental vortcty conservaton d ( Γ+f /h / dt (where h s the effectve thckness of the atmosphere and f = Ω snφ s the Corols factor whch descrbes the nertal effects and the possble consequences of thckness varaton n the atmosphere. Thus, the developed flux model allows to smulate the short range evoluton of cyclonc and antcyclonc systems, wthn affordable computatonal requrements. A few smulatons are presented n order to llustrate the theoretcal developments and to demonstrate the procedure.

2 1 INTRODUCTION In the last decades many numercal models have been devsed and appled n order to forecast the weather dynamcs, both on the large scale (General Crculaton Models, GCM's and on the subcontnental scale (Lmted Area Models, LAM's. Recently some relable algorthm has been checked to forecast the convecton drven weather tme evoluton on a very small spatal scale (note that both the GCM's and the LAM's use the hydrostatc approxmaton to calculate the dynamcs of a stratfed atmosphere. Satellte mages n varous wave ranges are often used, together wth the tradtonal data from the meteorologcal network, to control and almost contnuously tune the weather forecast produced by the models ( although t's stll dffcult to use satellte data from a quanttatve pont of vew. Anyway, both the GCM's and the LAM's requre a sutably dmensoned computng power and take advantage from the knowledge of many expermental data n order to gve ther maxmum performance. On the other sde, snce the tme when the frst satellte mages have been of common experence, the temptaton grew to subectvely predct the poston of clouds and meteorologcal fronts (for a maxmum perod of a day ahead on the bass of the actual and some prevous meteorologcal mages manly n the vsble and nfrared spectrum. In fact, n the case of not too ntrcate meteorologcal patterns, ths subectve now-castng turns out to be same what successful and allow to estmate the cloud system poston wth errors of the order of about 100 Km n the next 1-4 hours. Thus, a smple and fast algorthm has been devsed to numercally perform ths smplfed form of now-castng. The method les on the bass of a twodmensonal vortex flud dynamcs, through the pontng out of deal centers of crculatons whch nteract determnng ther overall moton followng some sutably modfed vortex system dynamc rules. At each nstant the dscrete set of the postons of the vortces determnes the whole dynamc flux, whch n turn determnes the future moton of the vortces. Ths smplfed scheme allows sutable short-tme dynamc forecasts. TWO DIMENSIONAL VORTEX DYNAMICS ON THE PLANE It s well known that a z ndependent (.e. two-dmensonal and plane velocty feld v(x.y,t for an ncompressble, not vscous flow satsfes the dynamc system Γ + v Γ = 0 t v = 0 (1 D v = Γ

3 D x y y and Γ s the only not zero component of x vortcty (along z. Equatons (1 can be solved f the ntal vortcty feld (Γ( x,y,0 and sutable boundary condtons ( n v ( x, y, t = 0 on each boundary pont are gven. The whole problem conssts n an nstantaneous problem (nner brace n eqns.(1, smlar to the magnetostatc problem where (,, (, B = 0 B = µ ( and an evoluton equaton for the votcty feld Γ( x,y,t, whch represent the lagrangan nvarance for the vortcty ( d Γ/ dt = 0,.e. Γ=const along the traectores. Usually one drops out the contnuty equaton by mposng v = Ψ D : then the second nstantaneous equaton n eqns.(1 becomes a Posson equaton Ψ = Γ for the stream functon Ψ( x,y,t..1 Governng equatons If the ntal vortcty feld s gven by a set of parallel straght crculaton lnes normal to the xy plane, wth crculatons values γ and ntersecton ponts R ( 0 = { X ( 0, Y ( 0 } wth the plane (.e. Γ( r, 0 = γ δ( r R ( 0, then one can easly demonstrate that, for t > 0, the ' Bot-Savart ' feld structure holds: where or, explctly, Γ Ψ ( r, t = γ ( r R ( t ( δ 1 γ π ( r, t = lnr R ( t (4 dr dt = dx 1 = dt π dy 1 = dt π 1 π γ γ γ ( R R R R ( X X + ( Y Y ( X X + ( Y Y X Y Y X (5 (6

4 The problem can also be formulated n terms of a Pseudo-Hamltonan system lke the one bellow: dx H γ = dt Y = 1,..., N (7 dy H γ = dt X where H = π 1 γ γ ln R s the pseudo-hamltonan of the system. Thus, almost n the case of unbounded flud, the flud dynamc problem of eqns. (1 can be solved (n R by solvng the dscrete nonlnear dynamc system represented by eqns.(6. f the flud doman s bounded by geometrcally smple boundares, one can try to drectly solve the problem by addng sutably defned crculatons mages to the r.h.s. of eqns.(6. -R (8 Fgure 1: traectores for a system of tree vortces wth the same crculaton ntensty but dfferent spn.

5 Fgure : stream functon for a system of tree vortces wth the same crculaton ntensty but dfferent spn Fgure 4: Stream functon for a system of two vortces wth dfferent crculaton ntensty and spn n a crcularly bounded plane.

6 Fgure 4: Traectores for two vortces wth the same crculaton n ntensty and spn near the crcular boundary. In fgures 1-4 we show some traectores for both unbounded ( R and crcularly bounded crculaton ponts motons and the correspondng nstantaneous stream functons. As we can see from these fgures also apparently smple systems (.e. three vortces at the vertces of a equlateral tranglecan have a complex evoluton. When the flud doman s bounded n the z drecton by two parallel planes, the z translaton nvarance of the vortcty feld breaks down. Gven a system of parallel vortcty segments, all ntally normal to the xy plane, each of these segments tends n general to change ts rectlnear shape nto a curved one. Nevertheless, f the dstance h between the planes s small n comparson wth the dstance among the crculaton segments, the velocty feld s approxmately ndependent of z wthn the flud slab, and one can consder approxmately 'rgd' parallel vortcty segments for the whole (not too long concerned tme nterval. Of course, the Bot - Savart lke nteracton law breaks down too, and must be replaced by the elementary Bot-Savart law. Ths s equvalent to consder vortcty segments nstead of nfnte vortcty lnes. dr 1 ( R R = hγ (9 dt π R R Thus the nteractons among vortex segments vansh wth ther recprocal dstance more rapdly than n the z unbounded case.

7 FLOW INSIDE A SPHERICAL FLUID FILM If we magne the atmosphere as a thn ( h 10 km sphercal flm surroundng the Earth ( mean radus a 678 km,and consder nvscd, ncompressble flows theren on the large synoptc scale, we are compelled to express the equatons of the motons for the 'rgd' vortex elements n term of ther geographcal coordnates ( λ, φ. By calculatng the dstance among vortex segments along the sphercal chords onng them, one obtans wth some algebra that the stream functon for the atmospherc flow s gven by Ψ where ( λ, φ, t h = 4 a dλ h = dt 8 a dφ h = dt 8 a γ γ [ 1 cosφ cosφ ( t cos( λ λ ( t senφ senφ ( t ] senφ cosφ tanφ cos( λ λ [ 1 cosφ cosφ cos( λ λ senφ senφ ] cosφ sn( λ λ [ 1 cosφ cosφ cos( λ λ senφ senφ ] γ 1 (11 (10 The angular zonal and merdonal veloctes of the flow, λ and φ can then be calculated usng the relatons D 1 Ψ λ = cosφ φ 1 Ψ D φ = cosφ λ (1 In the followng fgures (fgs we report (n stereographc proecton some traectores for a few vortcty segments. 4 VORTEX MOTION IN A ROTATING SPHERICAL FLUID FILM In order to represent the nertal effects whch arse from observng the atmospherc flow from a rotatng reference frame fxed wth the Earth, we must modfy the vortex moton equatons consderng the effects of the Earth rotaton both on the velocty components and on the vortces spns.

8 Fgure 5:Traectores for four vortces wth the same crculaton ntensty but dfferent spn on the sphere. Fgure 6: Stream lnes for four vortces wth the same crculaton ntensty an dfferent spn.

9 As well known: va = vrel + Ω r (1 Ths equaton states smply that the absolute velocty of an obect on the rotatng Earth s equal to ts velocty relatve to the Earth plus the velocty owng to the rotaton of the Earth. In terms of the angular zonal and the merdonal components of the velocty feld the eqns.(1 s the same as addng to the zonal component the Earth angular velocty, and so we have that the eqns.(11 smply become: dλ h = dt 8 a dφ h = dt 8 a senφ cosφ tanφ cos( λ λ γ [ 1 cosφ cos cos( sen sen ] φ λ λ φ φ cosφ sn( λ λ γ [ 1 cosφ cosφ cos( λ λ senφ senφ ] +Ω (14 At the same tme, also the absolute vortcty of the eddes s affected by the Earth rotaton accordng to the followng equaton: Γa =Γrel + f (15 where f = Ω snφ s the Corols factor. Then also the vortces crculatons change wth the lattude and must be recalculated at every step. Besdes, we need to estmate vortex spn varaton due to the thckness reducton or the ncrease of the flud slab. Then, n order to estmate the spn varatons we apply to the vortcty segments the potental vortcty conservaton low: d Γ + f = 0 (16 dt h a smulaton s presented n the next page n order to llustrate the Earth rotaton effects on the vortces traectores. 5 THE STRUCTURE OF THE WHOLE NOWCASTING PROCEDURE In the prevous sectons we have brefly descrbed a smplfed theoretcal model whch allow the calculaton of the tme evoluton of an ncompressble, nvscd and almost b-dmensonal flow feld n a thn sphercal flud shell, startng from a dscrete set of well localzed 'crculaton ponts'.in order to apply ths model to the atmosphere one must be able to pontng out a number of ntal crculatons centers from the meteorologcal maps or from the satellte mages. Although the few example reported refer to data deduced from weather

10 Fgure 7:Traectores for four vortces wth the same crculaton ntensty but dfferent spn on the rotatng sphere. charts, programs exst whch are able to guess a velocty feld by examnng a few subsequent satellte mages (usually n the nfrared spectrum followng the dsplacements of cloud systems ; these programs use smple and fast pattern recognton algorthms n order to dentfy n the subsequent mages the same cloud patterns even f they show forms and dmensons (slowly varyng n tme durng ther moton. From the b-dmensonal velocty feld, gven on a sutable defned pxel mesh, a very smple numercal procedure can easly calculate the correspondent vortcty feld and squeeze the maxmum absolute vortcty domans to dentfy a number of deal crculaton centers and the assocated crculaton values. Fnally, the prevously descrbed flud dynamc model operates to produce a guess on the future velocty felds at choose tme ntervals. In the followng fgure we report, for sake of clearness, the analogous guessed dsplacements obtaned startng from meteorologcal maps, comparng the guessed crculaton pont dsplacements wth those observed (and reported on subsequent meteorologcal charts. 6 CONCLUSIONS The whole work has been centered on the developng of a smplfed model descrbng the nteractons between atmospherc structures n terms of vortex dynamcs and able to forecast the moton of these structures at synoptc scale and on short tme scales wthn affordable computatonal requrements.

11 Fgure 8: Smulaton concernng the forecast for 07/07/1994 (forecast nterval =4 h at 500 mb and comparson between the output of the model and real stuaton. Snce n our hypothess the atmosphere can be regarded as a thn layer of nvscd ncompressble flud, we started wth a revew of b-dmensonal flows n the Hamltonan formulaton and we consdered the atmospherc vortces lke localzed pont vortces. Before we analyzed the unbounded case and then we passed, through the hydrodynamcs mages method, to the case of a system of vortces that nteract nsde of a flud doman delmted by a crcular boundary. However, some of the approxmaton n ths frst step of the work turned out not much effectve. In fact, to consder the atmospherc vortces lke nfnte vortcty straght lnes, neglectng n ths way the (geometrcal fnsh of the atmospherc layer, produces nteracton between the vortces much stronger than the real ones. Besdes the dynamcs of a flud ted up to flow on a rotatng sphercal surface s dffculty approxmated wth the one of a plane rotatng flud. Startng from these observatons we have re-expressed the theory of vortces nteractons by the terms of vortcty segments on a sphercal surface. For the rotatng sphere case the conservaton low of potental vortcty (16 descrbes n a effectve and concse way the nertal effects and the possble effects of thckness varaton n the atmospherc slab on the vortex spn. The model, reformed by these terms s able to perform smulatons of wnd feld evoluton on short tmes and at synoptc scale n good agreement wth the real stuaton ponted out from meteorologcal maps.

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