Asymptotic models for the planetary and synoptic scales in the atmosphere

Transcription

1 Asymptotic models for the planetary and synoptic scales in the atmosphere Stamen I. Dolaptchiev Institut für Atmosphäre und Umwelt with Rupert Klein (FU Berlin),,1

2 Outline 1 Planetary scale atmospheric motions and reduced models 2 The Planetary Regime Single scale model Two scale model: interactions with the synoptic scale 3 Anisotropic Planetary Regime 4 Balances on the planetary and synoptic scales in numerical experiments,,2

3 Outline 1 Planetary scale atmospheric motions and reduced models 2 The Planetary Regime Single scale model Two scale model: interactions with the synoptic scale 3 Anisotropic Planetary Regime 4 Balances on the planetary and synoptic scales in numerical experiments,,3

4 Planetary Scale Motions Figure: Time mean geopotential height of the 500 hpa surface for the northern hemisphere, DJF. Based upon ERA40 reanalysis data.,,4

5 Planetary Scale Motions Figure: Power spectrum density of the meridional geostrophic wind at 500 hpa and 50 N (Fraedrich & Böttger, 1978).,,5

6 Planetary Scale Motions atmospheric phenomena: thermally and orographically induced quasi-stationary Rossby waves teleconnection patterns (e.g. NAO, PNA) zonal mean flows (subtropical and polar jets),,6

7 Planetary Scale Motions atmospheric phenomena: thermally and orographically induced quasi-stationary Rossby waves teleconnection patterns (e.g. NAO, PNA) zonal mean flows (subtropical and polar jets) planetary spatial scales: O(6000 km) planetary advective time scale: O(7 days), (u ref 10 m/s),,6

8 Planetary Scale Motions atmospheric phenomena: thermally and orographically induced quasi-stationary Rossby waves teleconnection patterns (e.g. NAO, PNA) zonal mean flows (subtropical and polar jets) planetary spatial scales: O(6000 km) planetary advective time scale: O(7 days), (u ref 10 m/s) interactions with the synoptic eddies synoptic spatial and temporal scales: 1000 km and 1 day.,,6

9 CLIMBER-2 6 Latitude o C Figure 2. Map of the simulated annual-mean surface air temperature change in the year 2100 for a standard scenario of future CO 2 emissions relative to the year (Courtesy of Stefan Rahmstorf, Potsdam Institut für Klimafolgenforschung.) Figure: Annual-mean surface air temperature change in the year 2100 for standard scenario of future CO 2 emissions (Courtesy of S. Rahmstorf). SDM, Atmosphere: 2-D(φ, λ)-ml resolution: 51 longitude, 10 latitude,,7

10 To bridge the gap, Earth System Models of Intermediate Complexity (EMIC s) have been each proposed which can be characterize in the following way. EMIC s describe most of the processes EMIC. It was decided to design a template which will give for each EMIC its scope, the model compo- implicit in comprehensive models, albeit in a more reduced, i.e. a more parameterized form. nents and performance, and selected applications. At They explicitly simulate the interactions among several components of the climate system including Earth biogeochemical System cycles. On the other hand, Models EMIC s are simple enough ofto Intermediate allow for long- the end of August 2000, two EMICs from Belgium Complexity models 4, 8, see Table 1), two from Canada models term climate simulations over several years or even glacial cycles. Similar to those of 6,10), two from Germany models 2, 9), and one each comprehensive models, but in contrast to conceptual models, the degrees of freedom of an EMIC exceed the number of adjustable parameters by several orders of magnitude. Tentatively, from the Netherlands model 3), Russia model 5), we may define an EMIC in terms of a three-dimensional vector: Integration, i.e. number of components of the Earth system explicitly described in the model, number of processes explicitly include socio-econom come closest to acom model in which the an interactive way, rathe prescribed boundary 4.2 Location of EMIC system models described, and detail of description of processes (See Figure 1). Table 1. References to EMICs Integration Conceptual Models Model Short list of references To determine the app spectrum of climate s EMICs Processes 1: Bern 2.5D 2: CLIMBER-2 Stocker et al. 1992), Marchal et al. 1998) Petoukhov et al. 2000), identify the vertical c number of interacting Comprehensive Models Ganopolski et al. 2000) explicitly described in EMICs encompass an 4: EcBilt-CLIO Goosse et al. 2000) 5: IAP RAS Petoukhov et al. 1998), Handorf et al. 1999), module including a 3: EcBilt Opsteegh et al. 1998) Mokhov et al. 2000) modules are labeled 6: MPM Wang and Mysak 2000), balance models, DEM Mysak and Wang 2000) models including so 7: MIT Prinn et al. 1999), Kamenkovich et al. 2000) 8: MoBidiC Cruci x et al. 2000a) 2000b) dynamical models, Q 9: PUMA Fraedrich et al. 1998), GCM: general circula Maier-Reimer et al. 1993) istry is included only i 10: Uvic Weaver et al. 2000) resolution of atmosph Claussen et al.: Earth system models of intermediate complexity 11: IMAGE 2 Alcamo 1994), Alcamo et al. 1996) 583 the case of a spectral Detail of Description Table 2. Interactive components of the climate system being implemented into EMICs for explanation see text) Figure 1: Tentative definition of EMIC s Model Atmosphere Ocean Biosphere Sea ice Inland ice Currently, there are several EMIC s in operation such as 2-dimensional, zonally averaged models (e.g. Gallée et al., 1991), 2.5-dimensional models with a simple energy balance (e.g. Marchal 1 EMBM, 1-D u) 2-D u, z), 3 basins Bo, BT T 2 SDM, 2-D u, k)-ml 2-D u, z) 3 basins Bo, BT, BV TD 3-D, polythermal et. al, 1998; Stocker et al., 1992), or with a statistical-dynamical atmospheric module (e.g. 3 QG, 3-D, T21, L3 3-D, , L12 T Petoukhov et al., 1999), and reduced-form comprehensive models (e.g. Opsteegh et al., 1998). 4 QG, 3-D, T21-L3 3-D, 3 3 BT, BV TD EMIC s have been 5 used for a SDM, number 3-D of palaeostudies, 4.5 6, L8 because they provide SDM, the unique 2-D u, opportunity of transient, long-term ensemble simulations (e.g. Claussen et al., 1999) xed- salinity contrast to k) 4.5 6, L3 T 6 EMBM, 1-D u), land/ocean boxes 2-D u, z), 3 basins TD 2-D u, z), isothermal 7 SDM, 2-D u, z)/atmospheric 3-D, to 3.75, L15 BT T chemistry 8 QG, 2-D u, z)-l2 2-D u, z), 3 basins Bo, BT, BV TD 2-D u, z), isothermal 9 GCM, 3-D, T21, L5 3-D, 5 5, L11 Bo TD 10 DEMBM, 2-D u, k) 3-D, , L 19 TD 3-D, polythermal 11 DEMBM, 2-D u, k)/atmospheric chemistry 2-D u, z), 2 basins Bo, BT, BV T Adapted from Claussen et al. (2002),,8

11 Outline 1 Planetary scale atmospheric motions and reduced models 2 The Planetary Regime Single scale model Two scale model: interactions with the synoptic scale 3 Anisotropic Planetary Regime 4 Balances on the planetary and synoptic scales in numerical experiments,,9

12 Asymptotic regimes synoptic planetary time [hsc/uref] ε 3 ε 2 QG PR PRBF ε 2 length [h sc ] ε 3 Figure: Scale map for the PR and PRBF, the validity range of the quasi-geostrophic (QG) theory is also shown.,,10

13 An unified multiple scales asymptotic approach or the derivation of reduced model equations Klein, 2000; Majda & Klein, 2003; Klein, 2008 ( ) 1 aω Universal small parameter: ε = g 2 Distinguished limit: expression of the characteristic numbers in terms of ε 3 Multiple scales asymptotic ansatz U(t, x, z; ε) = i ε i U (i) ( t ε, t, εt, ε2 t,..., x ε, x, εx, ε2 x,..., z, z,...) ε,,11

14 The Planetary Regime (PR) We consider horizontal velocities of the order of 10 m/s and weak background potential temperature variations, comparable in magnitude to those adopted in the classical QG theory: δθ ε 2. δθ ε 2 : Θ( }{{} ε 3 t, λ P, φ P, z) = 1 + ε 2 Θ (2) ( + O ε 3 ), t P u(t P, λ P, φ P, z) = u (0) + εu (1) + ε 2 u (2) + O ( ε 3).,,12

15 The Planetary Regime Leading order model: planetary geostrophic equations u (0) = 1 fρ (0) e z P p (2), p (2) z ρ = (0) Θ(2), P ρ (0) u (0) + z ρ(0) w (3) = 0, t P Θ (2) + u (0) P Θ (2) + w (3) z Θ(2) = 0, additional boundary condition needed: p (2) at some level or p (2)z. implemented in the atmospheric module of some earth system models of intermediate complexity (EMICs),,13

16 The Planetary Regime Evolution equation for the barotropic component of the pressure p (2)z t P N = ( 1 ỹ P f p y (2) z β P f 2 ) p y (2) z fp (2) z N + β P ỹ P f N = 0, ρ ỹ (0) v (0) u (0) λ P,z ρ (0) v (0) u (0)λ P,z tan φ P P a the barotropic pressure p (2)z is zonally symmetric EMICs use a diagnostic closure! + p (2) λ P,z z p(2). x P ρ (0),,14

17 The Planetary and the Synoptic Scales Interactions between the planetary and the synoptic scales Quasi-geostrophic theory on a sphere: δf f 0 O(1), N const two scales ansatz resolving additionally to the planetary scales the synoptic spatial scales (internal Rossby deformation radius) and the corresponding advective time scale coordinates scaling: Θ(t P, ε 2 t }{{}, λ P, φ P, ε 1 λ }{{ P } t S λ S, ε 1 φ P }{{} φ S, z) = 1 + ε 2 Θ (2) (t P, λ P, φ P, z) + ε 3 Θ (3) (t P, t S, λ P, φ P, λ S, φ S, z) + O(ε 4 ),,15

18 The Planetary and the Synoptic Scales geostrophic balance: u (0) = 1 f e r S π (3) + 1 f e r P π (2), π (i) = p(i) ρ, }{{}}{{} (0) (0) (0) := u S := u P,,16

19 The Planetary and the Synoptic Scales geostrophic balance: u (0) = 1 f e r S π (3) + 1 f e r P π (2), π (i) = p(i) ρ, }{{}}{{} (0) (0) (0) := u S := u P u S (0) = u S (0) (t S, t P, λ P, φ P, λ S, φ S, z) u P (0) = u P (0) (t P, λ P, φ P, z),,16

20 The Planetary and the Synoptic Scales Planetary scale dynamics: ( ) (0) + u P P + w (3) P P V (2) = 0, P V (2) = f Θ (2). t P z ρ (0) z Synoptic scale dynamics: ( ( + u (0) S t S P V (3) = 1 f Sπ (3) + ) ) + u(0) P S P V (3) + βv (0) f ρ (0) z ( ρ (0) π (3) / z Θ (2) / z S + f ρ ) u(0) (0) S P ρ (0) Θ (2) z Θ (2) / z = 0, advection of synoptic scale PV by the planetary scale velocity field advection of PV resulting from the planetary scale gradient of Θ (2) by the synoptic scale velocities,,17

21 The Planetary and the Synoptic Scales Planetary-synoptic interactions in the evolution equation for the planetary scale structure of p (2)z «1 p ỹ P f y (2) z β p P f 2 y (2) z fp (2) z N + β P ỹ P f N = 0, t P N = ρ ỹ (0) P v (0) P u(0) P + p (2) λ,z z p(2). x P ρ (0) + v(0) S u(0) S S,λ P,z ρ (0) v (0) P u(0) P + v(0) S u(0) S S,λ P,z tan φ P a,,18

22 Outline 1 Planetary scale atmospheric motions and reduced models 2 The Planetary Regime Single scale model Two scale model: interactions with the synoptic scale 3 Anisotropic Planetary Regime 4 Balances on the planetary and synoptic scales in numerical experiments,,19

23 Asymptotic regimes synoptic planetary time [hsc/uref] ε 3 ε 2 QG PR PRBF ε 2 length [h sc ] ε 3 Figure: Scale map for the PR and PRBF, the validity range of the quasi-geostrophic (QG) theory is also shown.,,20

24 Motions with planetary zonal & synoptic meridional scales β-plane approximation for a sphere anisotropic scaling Θ( {z} ε 2 t, {z} ε 3 t, λ P, ε 1 λ P, ε 1 φ {z } P, z) =1 + ε 2 Θ (2) (λ P, z, t P ) {z } t S t P λ S φ S + ε 3 Θ (3) (t P, t S, λ P, λ S, φ S, z) + O`ε 4 u(t P, t S, λ P, λ S, φ S, z) =u (0) + εu (1) + ε 2 u (2) + O`ε 3.,,21

25 Motions with planetary zonal & synoptic meridional scales β-plane approximation for a sphere anisotropic scaling Θ( {z} ε 2 t, {z} ε 3 t, λ P, ε 1 λ P, ε 1 φ {z } P, z) =1 + ε 2 Θ (2) (λ P, z, t P ) {z } t S t P λ S φ S + ε 3 Θ (3) (t P, t S, λ P, λ S, φ S, z) + O`ε 4 u(t P, t S, λ P, λ S, φ S, z) =u (0) + εu (1) + ε 2 u (2) + O`ε 3. solvability condition = Θ (2) (z) Leading order model: classical QG on a sphere,,21

26 Motions with planetary zonal & synoptic meridional scales Case: dynamics on a plane: λ P, λ S, φ S X, x, y,,22

27 Motions with planetary zonal & synoptic meridional scales Case: dynamics on a plane: λ P, λ S, φ S X, x, y Next order model: planetary scale structure, next order QG corrections d dt S P V (4) + u (1) S P V (3) = S qg P V (4) = S Φ (4) + 1 f 0 X d P V (3) d dt P dt S x π(3) f 0 2 y2 X v(0), d dt S,P = ( t S,P + u (0) S,P ), = 1 f 0 S + f 0 2 z 2, S qg(π (3) ).,,22

28 Motions with planetary zonal & synoptic meridional scales Case: dynamics on a plane: λ P, λ S, φ S X, x, y Next order model: planetary scale structure, next order QG corrections d dt S P V (4) + u (1) S P V (3) = S qg P V (4) = S Φ (4) + 1 f 0 X d P V (3) d dt P dt S x π(3) f 0 2 y2 X v(0), ( ) d = + u (0) S,P, dt S,P t = 1 2 S + f 0 S,P f 0 z 2, S qg(π (3) ). solvability condition for the planetary-scale dynamics Case: synoptic scales only = QG +1 of Muraki et al. (1999),,22

29 Outline 1 Planetary scale atmospheric motions and reduced models 2 The Planetary Regime Single scale model Two scale model: interactions with the synoptic scale 3 Anisotropic Planetary Regime 4 Balances on the planetary and synoptic scales in numerical experiments,,23

30 The Planetary Regime in numerical experiments Q 2 2 Q x x f u β v time [day] f u β v time [day] f P u (0)S + βv (0)S = 0,,24

31 The Planetary Regime in numerical experiments 2.5 x u ζ v y ζ Q x u ζ v y ζ Q time [day] time [day] S u (0) ζ (0)S = 0,,25

32 Conclusions and outlook,,26

33 Conclusions and outlook behavior of the model in the tropics Majda & Klein 2003; Majda, 2007: X M = ε 2 x, X S = ε 5 2 x, X P = ε 7 2 x = MEW T G, IP ESD,,26

34 Conclusions and outlook behavior of the model in the tropics Majda & Klein 2003; Majda, 2007: X M = ε 2 x, X S = ε 5 2 x, X P = ε 7 2 x = MEW T G, IP ESD numerical implementation of the Planetary Regime, single scale/ two scale,,26

35 Conclusions and outlook behavior of the model in the tropics Majda & Klein 2003; Majda, 2007: X M = ε 2 x, X S = ε 5 2 x, X P = ε 7 2 x = MEW T G, IP ESD numerical implementation of the Planetary Regime, single scale/ two scale MTV strategy for the synoptic scale dynamics,,26

36 Outline 5 An unified multiple scales asymptotic approach,,27

37 An unified multiple scales asymptotic approach or the derivation of reduced model equations 1. Universal parameters: for the rotating earth: a km, Ω s 1, g 10 m s 2 for a variety of atmospheric flow regimes: p ref 1 bar, T ref 290 K, u ref 10 m s 1,,28

38 An unified multiple scales asymptotic approach or the derivation of reduced model equations 2. Characteristic numbers three independent nondimensional numbers π 1 = c ref Ωa 1 2, where c ref = γrt ref. π 2 = u ref 0.03, π 3 = aω , c ref g difficulties with multiple small parameter expansions F = a + bε δ 1 + δ lim lim F = lim lim F = a ε 0 δ 0 δ 0 ε 0 G = lim lim G = a ε 0 δ 0 aε + bδ ε + δ lim lim G = b δ 0 ε 0 II III I ε,,29

39 An unified multiple scales asymptotic approach or the derivation of reduced model equations 3. Distinguished limit Expression of the characteristic numbers in terms of ε: π 1 = c ref Ωa = π 1, π 2 = u ref c ref = ε 2 π 2, π 3 = aω2 = ε 3 π3, g as ε 0 where π 1, π 2 and π 3 are O(1) and ε ,,30

40 An unified multiple scales asymptotic approach or the derivation of reduced model equations 3. Classical dimensionless parameter Mach, Froude and Rossby number are expressed in terms of ε M = F r = u ref γrtref = π 2 ε 2, u ref ghsc = γπ 2 ε 2, Ro hsc = u ref = a π 2 1 2Ωh sc 2h sc ε, where h sc = p ref /(gρ ref ). The Rossby number is small if we use the internal Rossby deformation radius L syn 1000km ε 2 h sc Ro Lsyn = u ref 2ΩL syn ε 2 Ro hsc ε.,,31

41 An unified multiple scales asymptotic approach or the derivation of reduced model equations 4. Multiple scales asymptotic ansatz U(t, x, z; ε) = i ε i U (i) ( t ε, t, εt, ε2 t,..., x ε, x, εx, ε2 x,..., z, z,...) ε Example: quasi-geostrophic theory Characteristic length and time scales, the rescaled coordinates L syn 1000 km ε 2 h sc x S = x = ε2 x = ε 2 x L syn h sc T syn 1 day ε 2 h sc t S = t ε 2 t = = ε 2 t u ref T syn h sc /u ref The asymptotic ansatz for the velocity takes the form u(t, x, z; ε) = i ε i u (i) (x S, y S, t S, z),,32

42 An unified multiple scales asymptotic approach or the derivation of reduced model equations [h sc/u ref] 1/ε 3 planetary geostrophic 1/ε 2 1/ε 1 ε quasigeostrophic incompressible anelastic hydrostatic internal gravity waves acoustics and Lamb waves ε 1 1/ε 1/ε 2 1/ε 3 bulk convective meso synoptic planetary micro [h sc],,33

43 CLASSICAL RESULTS: Coordinate scalings U (i) = U (i) (t, x, z) U (i) = U (i) (t, εx, z) U (i) = U (i) ( t ε, x, z ε ) U (i) = U (i) (ε 2 t, ε 2 x, z) U (i) = U (i) (ε 2 t, ε 2 x, z) U (i) = U (i) (ε 2 t, ε 1 ξ(ε 2 x), z) U (i) = U (i) (ε t, ε 2 x, ε 2 y, z) Simplified model obtained Anelastic & pseudo-incompressible models Linear large scale internal gravity waves Linear small scale internal gravity waves Mid-latitude Quasi-Geostrophic model Equatorial Weak Temperature Gradient models Semi-geostrophic model Equatorial Kelvin, Yanai & Rossby Waves,,34

44 NEW MODELS Intraseasonal Planetary Equatorial Dynamics Majda & Klein (2003); Majda & Biello (2004) Deep Mesoscale Convection Klein & Majda (2006) Hurricanes Mikusky (2007); Mikusky, Klein & Owinoh (in prep.),,35

45 An unified multiple scales asymptotic approach for the derivation of reduced model equations Majda, A. and Klein, R., (2003) Systematic multi-scale models for the tropics, J. Atmos. Sci., 2, Klein, R., (2004) An Applied Mathematical View of Theoretical Meteorology, in: Applied Mathematics Entering the 21st Century, SIAM Proceedings in Applied Mathematics, 116. Klein, R., (2008) An unified approach to meteorological modelling based on multiple-scales asymptotics, Advances in Geosciences, 15, back,,36