Advisor: Prof. Hung Chi Kuo Speaker: Yu Ming Tsai Date: 2008 May 6. Outline

Transcription

1 Filamentation Time Diagnosis Advisor: Prof. Hung Chi Kuo Speaker: Yu Ming Tsai Date: 008 May 6 Outline Two dimensional turbulence Paper Review (through concentric eyewall phenomenon) Introduction to Filamentation Time Okubo Weiss Filamentation Time Hua Klein Filamentation Time Comparison Barotropic model results Isentropic filamentation time as a diagnosis tool for the synoptic scale trough thinning and broadening processes. Summary Future Work 1

2 Some Characteristics of Two Dimensional Flow Material conservation of vorticity or enstrophy. Enstrophy is one half hlfof the mean square vorticity in the flow field. Dilatation in one direction implies contraction in the other direction. Length of material filaments tends to increase, and their width tends to decrease. This increases the vorticity gradients. This is the core of the concept of the enstrophy cascade. Aarnout van Delden (006) (Kuo et al. 008, in press)

3 Taylor Column Montgomery (003) Spiral Bands in Hurricane and Galaxy Airborne-radar reflectivity in Hurricanes Guillermo (1997) (left panels) and Bret (1999) (right panels). Kossin and Schubert (001) 3

4 Electron density redistribution in experimental plasma physics single sign charge axial magnetic field confinement E B drift Coriolis force Core is protected, thin filaments from edges Passive microwave image sequences for four western Pacific typhoons with concentric eyewalls (Kuo et al. 008, in press) (Image courtesy of Naval Research Laboratory, Monterey, California) 4

5 Vortical Hot Towers Tropical Storm Gustav(00) The net effect of the hot towers is to produce strong small scale (10km in diameter on average) lowertropospheric ( below z 5km ) cyclonic PV towers. The strong updrafts in the hot towers converge and stretch existing low level vertical vorticity into intense small scale vortex tubes. Multiple mergers / axisymmetrization of these tubes in the lower troposphere. Hendricks et al. (004) Observations of deformation and mixing of the total ozone field in the Antarctic polar vortex Satellite ozone hole observation Daily TOMS images of total ozone in the Southern Hemisphere for six consecutive days in October Latitude circles are drawn at 40,60, and 80 S. The outermost latitude is 0 S. Bowmen and Mangus (1993) 5

6 Filamentation of Potential Vorticity High Resolution Simulation Aarnout van Delden (006) :46 UTC :46 UTC Guinn and Schubert (1993) 6

7 The formation of concentric eyewall Conserves the angular impulse Melander et al The contraction of the secondary wind maximum by nonlinear advection dynamics (Kuo et al. 008, in press) Lagrangian Parcel Trajectories A A Dilatation in one direction implies contraction in the other direction. A A Guinn and Schubert (1993) 7

8 Vorticity Conserves the angular impulse (km) Melander et al Derivation of Filamentation Time Based on deriving vorticity gradient. From barotropic vorticity equation: t u v x y (1) If neglecting diffusion 0, and computing (1) / x i(1) / y, We could obtain the time dependent equation of vorticity gradient: () u v Sn x y v u Ss x y (stretch deformation) (shear deformation) straining rate ( t) exp( t) 1 1 Q S n S s Rozoff et al. (006) 8

9 1 ( t) exp 1 ( t) exp i S n S S s n S s t t Vorticity gradient will be stretched Vortex is stable In strain dominated region ( S n S s Q > 0 ) straining rate Define 1 ( t) exp filamentation time fil S n fil S S s n S s 1 t (e - folding time) Rozoff et al. (006) r V 0 V r V 0 Vorticity inner shell shielding fil con 30 min Moat formation rapid filamentation zone convection becomes highly distorted and even suppressed fil con Rozoff et al. (006) 9

10 10 Filamentation Time Okubo Weiss type (Weiss, 1991) Hua Klein type considering the acceleration gradient tensor s n fil S S (Hua and Klein, 1998) 1 fil Where : Stretching Deformation. : Shearing Deformation. n S s S f y v x u u t S n y u x v u t S s y u x v u t Vorticity (Hua and Klein, 1998)

11 1 ( ) 0 fil Okubo Weiss Eigenvalue (Hua and Klein, 1998) Hua Klein Eigenvalue 0 1 (Hua and Klein, 1998) 11

12 1 (Hua and Klein, 1998) Scatter plot One dimensional spatial wave number spectra 0 4 k (Hua and Klein, 1998) 1

13 Vorticity OW fila time HK fila time 0 hr 3 hr 1 hr 7.5 min.5 min Rozoff et al. (006, JAS) Rozoff phd dissertation Nondimensional moat width v.s. nondimensional filamentation moat width All cases Cat 5 Cat 4 Cat 3 13

14 D Spectral Barotropic Model ( Near Inviscid ) t (, ) ( x, y ) where : viscosity Double periodic f plane ν = 6.5 m s 1 Fourier pseudo spectral method Time stepping method : 4th order Runge Kutta method Time step : dt = 3 sec Domain : 400 km * 400 km Grids number : 51 * 51 Binary vortex interaction Kuo et al. (008, in press) R1 Vortex radius ratio ( r) R Dimensionless gap ( ) R 1 Vortex strength ratio ( ) Skirt parameter (α) 1 Companion Core An extension of Kuo (004) work. In addition to the original three non dimensional parameter, the skirt parameter is added as a fourth external parameter. 14

15 Authors Hypothesis Summary Relevant Model Results Zipser [1977] Downdrafts from the primary eyewall force a ring of convective updrafts Few downdraft-forced updrafts during this time in the simulations Willoughby [1979] Internal resonance between local inertia period and asymmetric friction due to storm motion No systematic storm motion in the simulated storms Hawkins [1983] Topographic effects No topographic forcing in the simulations Willoughby et al. [1984] Ice microphysics Warm-rain sensitivity case also produces secondary eyewall Molinari, Skubis [1985] Molinari, Vallaro [1989] Montgomery, Kallenbach [1997] Camp, Montgomery [001] Terwey, Montgomery [003] Nong, Emanuel [003] Kuo et al. [004, 007] Synoptic-scale forcings (e.g. low-level wind surges, upper-level momentum fluxes) Internal dynamics axisymmetrization via sheared vortex Rossby wave processes; collection of wave energy near stagnation or critical radii Sustained eddy momentum fluxes and WISHE feedback Axisymmetrization of positive vorticity perturbations around a strong and tight core of vorticity No synoptic-scale forcings in the simulations Possible explanation Possible explanation Possible explanation Terwey and Montgomery (008, in press) Non Dimensional Filamentation Rankine 0.5 (Kuo et al. 008, in press) 15

16 (Kuo et al. 008, in press) Vortex Structure 1) Strong core vortex Radial vorticity profile Radial tangential wind profile α= 0.5 or 1.0 ) Companion vortex: Modified Rankine Vortex 16

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18 The model result of TCM4 (3D fully compressible, nonhydrostatic hurricane model) OW Criterion (Wang et al. 008, in press) The Application of Filamentation Time: The moat formation of typhoon (Rozoff et al., 006) or formation of inner spiral rainbands (Wang et al., 008) Synoptic scale trough thinning or broadening and the formation of cutoff low. Two paradigms of baroclinic wave life cycle behaviour Thorncroft, Hoskins, and McIntyre (1993, QJRMS) 18

19 P. Santurette and C. G. Georgiev 005 The Merits of the θ coordinates 1) Under adiabatic processes, 0 and, therefore, coordinate surfacesareare material surfaces. ) The expression for Ertel s potential vorticity is more straight forward with the θ coordinate. 3) With the θ coordinate, one can obtain a quasi Lagrangian view of the general circulation of the atmosphere. 4) The pressure gradient force in the θ coordinate is M which is irrotational when the curl is taken along an isentropic surface. (Arakawa, 000) 19

20 The demerits of the θ coordinates 1) The coordinate surface intersect the earth s surface even without topography. p Hsu & Arakawa (1990), Fulton & Schubert (199) ) The mass between two coordinate surface can become infinitesimally small. Positive definite methods 3) Coordinate surface may become vertical near the ground due to surface heating. Hybrid coordinate (σ θ coordinate) 4) In nature, unstably stratified layers with 0 may also p exist. In general, not true for large scale atmosphere. (Arakawa, 000) PV theta contour LC1 Trough Thinning LC Trough Broadening Thorncroft et al. (1993) 0

21 mean jet Anti cyclonic shear NE SW tilt trough line, smaller cutoff low mean jet Cyclonic shear NW SE tilt trough line, larger cutoff low Thorncroft et al. (1993) Modeling: LC1 Trough Thinning Anti cyclonic shear NE SW tilt trough line, smaller COL (cutoff low) Thorncroft et al. (1993) 1

22 Modeling: LC Trough Broadening Cyclonic shear NW SE tilt trough line, larger COL Thorncroft et al. (1993) 500 hpa Geopotential Height 1990 Jan 6 00Z ~ 1990 Jan 30 00Z

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31 310 K Isentropic Potential Vorticity 1990 Jan 8 00Z 1990 Jan 9 00Z Shading: 1 to PVU ( dynamic tropopause ) 1990 Jan 8 1Z James,

32 Synoptic Trough Analysis A Event Duration Time: 1990 Jan 6 00Z ~ 1990 Jan 30 00Z Event Features: Both trough thinning and broadening pattern can be seen at the same time on the same weather plot. A cutoff low is formed at 1990 Jan 9 06Z. Analysis Time: 1990 Jan 8 06Z (4hrs before cutoff low formation) Data Source: NCEP Re analysis Resolution:.5 * Jan 8 06Z 3

33 1990 Jan 8 06Z 33

34 1990 Jan 8 06Z 1990 Jan 8 06Z 34

35 Synoptic Trough Analysis B Event Features: A typical and obvious cutoff low formed near Taiwan area during the Mei yu period. A cutoff low is formed at 1998 May 05 1Z. Analysis Time: 1998 May 04 00Z ~ 1998 may 05 1Z (every 1 hrs) Data Source: ECADV Resolution: 0.5 *

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37 Conclusion Filamentation time gives a quantitative diagnosis for straining out process. (the formation of band like structure) Hua Klein criterion produces somewhat larger rapid filamentation zones with smaller values of. The estimates of the size and intensity of rapid filamentation zones given in the figures presented here should be regarded as conservative. Filamentation time could be used as a tool to diagnose synoptic scale trough thinning or broadening and cutoff low processes. From both synoptic examples presented here, we found Okubo Weiss criterion provides good qualitative diagnosis of trough thinning or broadening and cutoff low processes. NCEP reanalysis data with.5 degrees is good enough for qualitative diagnosis. 37

38 Future Work Hua and Klein (1998) showed that the eigenvalues may be written in terms of the pressure field rather than in terms of Lagrangian derivatives. From the nonlinear balance equation: hk Defining pˆ p 0 f 0, we see the eigenvalues, hk, are also equivalent to: We could further use this formula to compute the more accurate (less assumption) filamentation time for both model output and synoptic scale reanalysis field. The End Thank You for Your Attention!! Acknowledgement goes to Prof. H. H. Hsu for his circulation course and discussion. It s very similar between large scale and typhoon phenomenon. 38

39 McWilliams (1984) T=16.5 Q > 0 Q < 0 Q S n S s Straining out regime (Simulation) partial straining - out (PSO) complete straining - out (CSO) Clear gap Adverse shear effect Clear gap The bands are too thin to be called concentric eyewalls. Dritschel and Waugh (199) 39

40 Vortex radius ratio Dimensionless gap Dritschel and Waugh (199) Vorticity (km) 40

41 Okubo Weiss filamentation time v 0 r Hua Klein filamentation time Rozoff et al. (006, JAS) Rozoff phd dissertation Vorticity OW fila time HK fila time 0 hr 9 hr 45 hr 15 min 7.5 min Rozoff et al. (006, JAS) Rozoff phd dissertation 41

42 1) Strong core vortex Vortex Structure 1, if 3 ( x, y) 1 c0 c1r cr c3r, if 1 1 r, 1 if 0 r r r α= 0.5 or 1.0 ) Companion vortex Modified Rankine Vortex exp exp, ( x, y) 1 if r r 0, if r r 1 1 Isentropic Surface (θ constant ) Advantages: 1) As a first approximation the motion is adiabatic (as much as it is quasi geostrophic) and this motion is related to the configuration and origin of air streams. (material conserved) ) Isentropic flow presents a truer picture of the three dimensional air motion than isobaric surfaces and preserves the quasi horizontal behavior of the three dimensional flow. Disadvantages: 1) The atmoshere is not completely adiabatic, especially in the boundary layer and in the vicinity of strong vertical mixing or convection. ) Isnrtropic surfaces may intersect the ground. 3) Isentropic surfaces may extend from low to high levels in the atmosphere and thereby represent no signal quasi horizonal surface. T. N. Carlson (1994) 4

43 James, K IPV maps 500 mb geopotential heights (Hoskins, McIntyre, and Robertson, 1985) 43

44 (Hoskins, McIntyre, and Robertson, 1985) Observation /7 00Z 1/8 1Z 1/7 1Z 1/9 00Z 1/8 00Z on PV= PVU surface 1PVU Km kg 1 1 Isertelic surface (Morgan, 1998) s Thorncroft et al. (1993) 44

45 Future Work Hua and Klein (1998) showed that the eigenvalues may be written in terms of the pressure field rather than in terms of Lagrangian derivatives. From the nonlinear balance equation: hk Defining pˆ p 0 f 0, we see the eigenvalues, hk, are also equivalent to: We could use this formula to compute the more accurate (less assumption) filamentation time for both model output and synoptic scale re analysis field. Nevertheless, Okubo Weiss type used in the present work shows very good qualitative diagnosis result (though under estimated), which encourages us to pursue more accurate quantitative diagnostic analysis. Thorncroft s argument is quite barotropic! m=1~ wave propogates westward m=4~8 wave propogates eastward east west Similar to stationary m=5 m=6 m=7 m=8 (Prof. H. H. Hsu, Course Notes) 45

46 Aarnout van Delden (006) 46