Fluid flow and rotation: a fascinating interplay J urgen Saal - PowerPoint PPT Presentation

Fluid flow and rotation: a fascinating interplay J¨ urgen Saal Mathematics for Nonlinear Phenomena: Analysis and Computation International Conference in honor of Professor Yoshikazu Giga on his 60th birthday J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 1 / 21

About 12 years ago in Sapporo there was ... J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 2 / 21

... some years later ... J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 3 / 21

Contents Rotating fluids 1 The Taylor-Proudman theorem 2 The Ekman boundary layer problem 3 J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 4 / 21

Contents Rotating fluids 1 The Taylor-Proudman theorem 2 The Ekman boundary layer problem 3 J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 5 / 21

The Ekman boundary layer problem (EBLP) Important for: daily weather forecast, tornado and hurricane evolution, aviation, pollen distribution, dew, fog, frost forecasts, air pollution, .... J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 6 / 21

Modeling of the problem Navier-Stokes equations with Coriolis term  ∂ t v − ν ∆ v + ( v , ∇ ) v + Ω e 3 × v = −∇ p    div v = 0 ( EBLP ) U E | ∂ G  v | ∂ G =   v | t =0 = v 0 considered in G = ( R 2 × (0 , d )) × (0 , T ) for d ∈ [ δ, ∞ ]. The Ekman spiral solution � � 1 − e − x 3 /δ cos( x 3 /δ ) , e − x 3 /δ sin( x 3 /δ ) , 0 U E ( x 3 ) = U ∞ p E ( x 2 ) = − Ω U ∞ x 2 . is an exact solution of ( EBLP ) with pressure � 2 ν Here δ = “ layer thickness ”. | Ω | J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 7 / 21

Literature Geophysical: V.W. Ekman, On the influence of the earth’s rotation on ocean currents, Arkiv Matem. Astr. Fysik, 1905. J. Pedlosky, Geophysical Fluid Dynamics, Springer Verlag, 1987. H.P. Greenspan, The Theory of Rotating Fluids, Cambridge Univ. Pr., 1968. Mathematical: D. Lilly ’66, J. Atmospheric Sci. E. Grenier and N. Masmoudi ’97, Commun. Partial. Differ. Equations. B. Desjardins, E. Dormy, and E. Grenier ’99, Nonlinearity. J.-Y. Chemin, B. Desjardin, I. Gallagher, and Grenier E., Ekman boundary layers in rotating fluids ’02, ESAIM Control Optim. Calc. Var. and Rousset, Greenberg, Marletta, Tretter, Giga, Mahalov, Hess, Hieber, Koba, S., ... In infinite energy space ˙ B 0 ∞ , 1 ( R 2 , L p ((0 , d ))): Giga, Inui, Mahalov, Matsui, S. ’07, ARMA. J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 8 / 21

Fluid flow about a rotating obstacle Mathematical model: ∂ t u + ( u · ∇ ) u − ∆ u + Ω e 3 × u − Ω( e 3 × ξ ) ∇ u = ∇ p = 0 div u u | t =0 = u 0 ξ = ( x 1 , x 2 , x 3 ). Literature: T. Hishida ’99, ARMA (iniciated by P.G. Galdi). and Galdi , Farwig, M¨ uller, Shibata, Neustupa, Necasova, Hieber, Geißert, Heck, Dintelmann, Penel, Disser, Maekawa, ... J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 9 / 21

Rotating obstacle vs Ekman ’Rotating obstacle’ situation (e.g. propeller) additional forces: + Ω 2 Ω e 3 × u 2 e 3 × ( e 3 × ξ ) − Ω( e 3 × ξ ) ∇ u � �� � � �� � � �� � Corolis drift centrifugal Ω: twice angular velocity of rotation. ’Ekman’ situation (e.g. Ekman, tornado-hurricane, spin-coating) additional forces: + Ω 2 Ω e 3 × u 2 e 3 × ( e 3 × ξ ) � �� � � �� � Corolis centrifugal J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 10 / 21

Linearized spectrum without rotation: Au = P ∆ u rotating obstacle: A O u = P (∆ u − Ω e 3 × u − Ω( e 3 × ξ ) ∇ u ) A E u = P (∆ u − Ω e 3 × u − ( U E · ∇ ) u − u 3 ∂ 3 U E ) Ekman: J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 11 / 21

Linearized spectrum without rotation: Au = P ∆ u rotating obstacle: A O u = P (∆ u − Ω e 3 × u − Ω( e 3 × ξ ) ∇ u ) A E u = P (∆ u − Ω e 3 × u − ( U E · ∇ ) u − u 3 ∂ 3 U E ) Ekman: Spectra: J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 11 / 21

Linearized spectrum without rotation: Au = P ∆ u rotating obstacle: A O u = P (∆ u − Ω e 3 × u − Ω( e 3 × ξ ) ∇ u ) A E u = P (∆ u − Ω e 3 × u − ( U E · ∇ ) u − u 3 ∂ 3 U E ) Ekman: Spectra: = ⇒ A generates bounded analytic C 0 -semigroup A O generates bounded C 0 -semigroup, which is not analytic A E generates analytic C 0 -semigroup, which is not bounded J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 11 / 21

Contents Rotating fluids 1 The Taylor-Proudman theorem 2 The Ekman boundary layer problem 3 J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 12 / 21

The Taylor-Proudman theorem Theorem (after Taylor ’16 and Proudman ’17) Within a fluid that is steadily rotated at high angular velocity Ω , there is no variation of the velocity field in the direction parallel to the axis of rotation. J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 13 / 21

The Taylor-Proudman theorem Theorem (after Taylor ’16 and Proudman ’17) Within a fluid that is steadily rotated at high angular velocity Ω , there is no variation of the velocity field in the direction parallel to the axis of rotation. � ∂ t u + ( u · ∇ ) u − ∆ u + Ω e 3 × u + ∇ p = 0 ( NSC ) = 0 div u Heuristically: Ω >> 1 ⇒ Ω e 3 × u ≈ −∇ p ⇒ ∇ × ( e 3 × u ) ≈ 0 d ⇒ d z u = e 3 · ∇ u ≈ 0 ⇒ u ( x , y , z ) ≈ u ( x , y ) J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 13 / 21

The Taylor-Proudman theorem Theorem (after Taylor ’16 and Proudman ’17) Within a fluid that is steadily rotated at high angular velocity Ω , there is no variation of the velocity field in the direction parallel to the axis of rotation. � ∂ t u + ( u · ∇ ) u − ∆ u + Ω e 3 × u + ∇ p = 0 ( NSC ) = 0 div u Heuristically: Ω >> 1 ⇒ Ω e 3 × u ≈ −∇ p ⇒ ∇ × ( e 3 × u ) ≈ 0 d ⇒ d z u = e 3 · ∇ u ≈ 0 ⇒ u ( x , y , z ) ≈ u ( x , y ) ⇒ flow is two-dimensional ⇒ global well-posedness ! J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 13 / 21

Literature After more than 80 years first analytical verification: Periodic domains (e.g. torus): ◮ Babin, Mahalov, Nicolaenko ’97, Asymptotic Anal., ’99 + ’01, Indiana Univ. Math. J., ’03 Russian Math. Surveys Crucial pre-condition: uniformness in Ω of local results! J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 14 / 21

Literature After more than 80 years first analytical verification: Periodic domains (e.g. torus): ◮ Babin, Mahalov, Nicolaenko ’97, Asymptotic Anal., ’99 + ’01, Indiana Univ. Math. J., ’03 Russian Math. Surveys Crucial pre-condition: uniformness in Ω of local results! Whole space R n : ◮ J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier ’02, Stud. Math. Appl. ◮ and Koba, Mahalov, Yoneda, Konieczny, Koh, Lee, Takada, ... Crucial ingredients: ◮ exp( − t (∆ + P Ω e 3 × )) = exp( − t ∆) exp( tP Ω e 3 × ), � � � ◮ dispersive estimates: � � R 3 exp( ix · ξ + i Ω ξ 3 / | ξ | ) ψ ( ξ ) d ξ � ≤ C / | Ω | . � � � ( ⇒ Strichartz estimates by Keel-Tao) J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 14 / 21

Literature After more than 80 years first analytical verification: Periodic domains (e.g. torus): ◮ Babin, Mahalov, Nicolaenko ’97, Asymptotic Anal., ’99 + ’01, Indiana Univ. Math. J., ’03 Russian Math. Surveys Crucial pre-condition: uniformness in Ω of local results! Whole space R n : ◮ J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier ’02, Stud. Math. Appl. ◮ and Koba, Mahalov, Yoneda, Konieczny, Koh, Lee, Takada, ... Crucial ingredients: ◮ exp( − t (∆ + P Ω e 3 × )) = exp( − t ∆) exp( tP Ω e 3 × ), � � � ◮ dispersive estimates: � � R 3 exp( ix · ξ + i Ω ξ 3 / | ξ | ) ψ ( ξ ) d ξ � ≤ C / | Ω | . � � � ( ⇒ Strichartz estimates by Keel-Tao) Domains with boundary (e.g. R n + ): ... ??? J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 14 / 21

Contents Rotating fluids 1 The Taylor-Proudman theorem 2 The Ekman boundary layer problem 3 J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 15 / 21