Diapositiva 1 by 7ZRRUmF

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									    Asymptotic expansions for
       geophysical flows
                             Brunello Tirozzi
                 Dipartimento di Fisica, Un. “La Sapienza”

Daniela Bianchi , Department of Physics, Univ. Of Rome “La Sapienza”
Sergey Dobrokhotov, Institute of Problem of Mechanics, Moscow Academy of Sciences
S. Reutskiy , Institute of Magnetohydrodynamic, Kharkov, Ucrain. Ac. of Sciences
Andrei Shafarevich, Faculty of Mechanics and Mathematics, Moscow State University
     Sviluppi Asintotici applicati alla
             fluido dinamica
•   Metodo WKB
•   Operatore canonico di Maslov
•   Creazione e stabilità dei vortici
•   Shallow Water equation
Cauchy Riemann conditions
             and
 stability of perturbations
Perturbed solutions of SW equations   (1/3)
Perturbed solutions of SW equations (2/3)
Perturbed solutions of SW equations   (3/3)
Conserved structure of the solution   (1/2)
Conserved structure of the solution   (2/2)
Stability of the vortex
Non stability of the vortex
Bibliografia
Bibliografia
Nanmadol
Fluid Dynamics with Free
  Boundary Conditions

 Reduction to the Wave equation
Solution with Asymptotic Methods
Wave Equation
Solutions of the Wave equation
• WKB method
• Improvement by means of Maslov
  canonical operator
Solution of linear perturbation in the
            general case
Cauchy Problem
Main problems
First type of solutions
Second type of solutions
Analytic Solution
Front of the waves
Vortical Modes 1
Vortical modes 2
Fast Propagating mode 1
References
Analytical and numerical analysis
  of the wave profiles near the
       fronts appearing in
       Tsunami problems
 Workshop On Renormalization
       Group, Kyoto 2005

    B. Tirozzi, S.Yu. Dobrokhotov,
      S.Ya. Sekerzh-Zenkovich,
          T.Ya. Tudorovskiy
“Gaussian” source of Earthquake



                                                     5




   1                                           2.5
0.75
 0.5
0.25
     0

     -5                                    0


          -2.5



                 0                  -2.5



                     2.5


                               -5
                           5
Level curves of perturbation




                               2
                               1
                               0
                               -1
                               -2
 4




       2




              0




                    -2




                          -4
Wave profiles at the front at time t
      at different angles
                 1.5




                   1




                 0.5




    -4     -2           2     4




                -0.5
3D wave profile for elliptic source



                                                  20




                                             10

 1.5
   1
 0.5
-0.50

   -20                                   0


         -10



               0                   -10



                   10


                             -20
                        20
“Modulated gaussian” source of
         Earthquake



   2




       0                                                    5




                                                      2.5
       -2




            -5                                    0


                 -2.5


                                           -2.5
                        0



                            2.5


                                      -5
                                  5
    Level curves of perturbation




                                        2
                                        1
                                        0
                                        -1
                                        -2
4




          2




                 0




                         -2




                                   -4
Wave profiles at the front at time t
      at different angles
                    1




                 0.75




                  0.5




                 0.25




    -4     -2           2     4


                -0.25




                 -0.5




                -0.75




                   -1
     3D wave profile for elliptic source



                                                      20




2                                                10
 1
 0
-1
-2

-20                                          0


         -10


                                       -10
               0



                     10


                                 -20
                            20
              The ridge near the source of
                      Eathquake

                                      6



                                      4



                                      2

0

                                      0
-1
                                 5

-2


    -3                               -2

                             0

         -5
                                     -4


                0
                                     -6
                        -5



                    5
                                          -6   -4   -2   0   2   4   6
Fronts at different times
                      The set of profiles
                                           20



                                                0.00228282



                                           15




                                           10
                        0.658889

                1.48386



             1.03988
                                            5



            0.0444183

            1.03364


              1.47197
-20   -15                      -10   -5         5            10   15   20



                  0.658554

                                           -5




                                          -10




                                          -15




                                          -20
                        0.658889

                1.48386



             1.03988




            0.0444183

            1.03364


              1.47197
-20   -15                      -10   -5



                  0.658554
              Simulations for Tyrrhenian Sea
              Relief data:
              National Oceanic and Atmospheric Administration (NOAA)
              National Geophysical Data Center (NGDC)
              ETOPO2 2-minute Global Relief
              http://www.ngdc.noaa.gov/mgg/gdas/gd_designagrid.html


 400


                                                   400




 200

                                                   200




   0
                                                     0




-200                                              -200




-400
                                                  -400

       -400       -200     0        200     400

                                                         -400   -200   0   200   400
Rays for imaginary source at
 Stromboli: 38.8 N, 15.2 E
         50   100   150   200   250   300   350   400

   -50


  -100


  -150


  -200


  -250


  -300


  -350


  -400
Amplitudes of wave at different
      points of the coast
         150                  200          250            300              350         400                 450



                                                                0.118962


  -100




                   0.019779


  -150




                                                                                               0.0916952
  -200




  -250

                                                                                             0.00840296
                                                                                            0.129721
               0.00317829

                                                                                 0.155356

  -300
                                    0.306578




                                                    0.0983057
                                                 0.0886875
  -350
3D wave profile
Density plot
Diffraction Days-St. Petersbourg
        29/05/07-01/06/07
Special session on tsunami and typhoons
  Scattering of tsunami on the beach
             S.Dobrokhotov
           Inst. Probl. of Mechanics R. Ac Sc. Moscow

               Petya Zhevandrov
     Escuela de Ciencias Fisico-Matematicas, Morelia, Mexico

                 Brunello Tirozzi
        Department of Physics, Univ. “La Sapienza”, Rome
             ... Workshop on Extreme Events ...
           Max Planck Institut for Complex System
           Dresden, 30 October-2 November 2006


Analysis of the tsunami event in Algeria
                  2003

          D. Bianchi (1),F. Raicich (2), B.
                         Tirozzi (2)


(1) Department of Physics, University “La Sapienza”, Rome, Italy
(2) CNR, Institute of Marine Sciences, Trieste, Italy
                           Barcelona




                Valencia

                                 Ibiza




Earthquake data (USGS): t0: 18:44:19 GMT, 21 may 2003
             USGS: epicenter 36.96°N, 3.63°E, M=6.9, hf=12 km
           Borerro: epicenter 36.90°N, 3.71°E, M=6.8, hf=10 km, strike=56°
          Ellipsoidal deformation

                                             M = 6.9

                                           hf = 12 km
                     a


                     56°
          b


                                    (Pelinovsky et al., 2001)



                                           a = 34 km

                                           b = 12 km

Coordinates: model gridpoints
   Gaussian*cosine deformation


                                      a1= 1.2 10-2 km-1
                                      a2 = 2.7 10-2 km-1
                                      b1 = 0.5 10-2 km-2

                                      b2 = 4.0 10-2 km-2

                                              =0

                                              = 56°




Coordinates: model gridpoints




         (Dobrokhotov et al., 2006)
Linear theory of the scattering of localized waves on a beach of
                         constant slope
     S. Dobrokhotov, Petya Zhevandrov, Brunello Tirozzi
Shape of perturbation
Amplitude of the waves at tsunami front
Average of the too fast oscillating WKB solution
Example: beach of costant slope
Momentum equation
Singular Lagrangian manifold
Dynamic on the lagrangian manifold
Dynamic on the phase space
Solutions before and after the scattering
            of the beach (1/2)
Solutions before and after the scattering
            of the beach (2/2)
Graphics 1
Graphics 2
Graphics 3

								
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