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Gravity Waves , Scale Asymptotics , and the Pseudo- Incompressible Equations

Gravity Waves , Scale Asymptotics , and the Pseudo- Incompressible Equations. Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and Fabian Senf (IAP Kühlungsborn) 13.9.10. Gravity waves in the middle atmosphere. Gravity Waves in the Middle Atmosphere.

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Gravity Waves , Scale Asymptotics , and the Pseudo- Incompressible Equations

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  1. Gravity Waves, ScaleAsymptotics, andthe Pseudo-IncompressibleEquations Ulrich Achatz Goethe-Universität Frankfurt am Main Rupert Klein (FU Berlin) and Fabian Senf (IAP Kühlungsborn) 13.9.10

  2. Gravity waves in the middle atmosphere

  3. Gravity Waves in the Middle Atmosphere Becker und Schmitz (2003)

  4. Gravity Waves in the Middle Atmosphere Becker und Schmitz (2003)

  5. Gravity Waves in the Middle Atmosphere Without GW parameterization Becker und Schmitz (2003)

  6. linear GWs in the Euler equations • no heat sources, friction, … • no rotation

  7. linear GWs in the Euler equations • no heat sources, friction, … • no rotation are equivalent to

  8. linear GWs in the Euler equations • linearization about atmosphere at rest:

  9. linear GWs in the Euler equations • linearization about atmosphere at rest:

  10. linear GWs in the Euler equations • conservation of wave energy: linear equations satisfy

  11. linear GWs in the Euler equations • conservation of wave energy: linear equations satisfy • conservation suggests:

  12. linear GWs in the Euler equations • isothermal atmosphere:

  13. linear GWs in the Euler equations • isothermal atmosphere: gravity waves sound waves

  14. linear GWs in the Euler equations • isothermal atmosphere: gravity waves sound waves GW polarization relations:

  15. GW breaking in the atmosphere Rapp et al. (priv. comm.) Conservation Instabilität in großen Höhen Impulsdeposition …

  16. GW breaking in the atmosphere Rapp et al. (priv. comm.) Conservation Instability at large altitudes Impulsdeposition …

  17. GW breaking in the atmosphere Rapp et al. (priv. comm.) Conservation Instability at large altitudes Turbulence Momentum deposition…

  18. GW Parameterizations • multitude of parameterization approaches(Lindzen 1981, Medvedev und Klaassen 1995, Hines 1997, Alexander and Dunkerton 1999, Warner and McIntyre 2001) • too many free parameters • reasons : • … • insufficent knowledge: conditions of wave breaking • basic paradigm: breaking of a single wave • Stability analyses (NMs, SVs) • Direct numerical simulations (DNS)

  19. GW breaking

  20. GW stability in the Boussinesq theory

  21. Basic GW types Dispersionsrelation: Trägheitsschwerewellen (TSW): Hochfrequente Schwerewellen (HSW):

  22. Basic GW types Coriolis parameter Stability Dispersion relation: Inertia-gravity waves (IGW): High-frequent GWs (HGW):

  23. Basic GW types Coriolis parameter Stability Dispersion relation: Inertia-gravity waves (IGW): High-frequent gravity waves (HGW):

  24. Basic GW types Coriolis parameter Stability Dispersion relation: Inertia-gravity waves (IGW): High-frequent GWs (HGW):

  25. Traditional Instability Concepts • static (convective) instability: amplitude reference (a>1) • dynamic instability • limited applicability to HGW breaking

  26. Traditional Instability Concepts • static (convective) instability: amplitude reference (a>1) • dynamic instability • limited applicability to HGW breaking

  27. Traditional Instability Concepts • static (convective) instability: amplitude reference (a>1) • dynamic instability • BUT: limited applicability to GW breaking

  28. NM analysis, 2.5D-DNS

  29. NMs: Growth Rates (Q = 70°) Achatz (2005, 2007b)

  30. GW amplitude after a perturbation by a NM (DNS): Achatz (2005, 2007b)

  31. Energetics not the gradients in z matter,but those in f! instantaneous growth rate:

  32. Energetics not the gradients in z matter,but those in f! instantaneous growth rate:

  33. Energetics not the gradients in z matter,but those in f! instantaneous growth rate:

  34. Energetics of a Breaking HGW with a0 < 1 Strong growth perturbation energy:buoyancy instability Achatz (2007b)

  35. Singular Vectors: characteristic perturbations in a linear initial value problem: • normal modes: Exponential energy growth • singular vectors:optimal growth over a finite time t(Farrell 1988, Trefethen et al. 1993)

  36. NMs and SVs of an IGW (Ri > ¼!): growth within 5min Achatz (2005) Achatz and Schmitz (2006a,b)

  37. Dissipation rate breaking IGW (a< 1, Ri > ¼): Measured dissipation rates 1…1000 mW/kg (Lübken 1997, Müllemann et al. 2003) Achatz (2007a)

  38. Summary GW breaking • In comparison to assumptions in GW parameterizations: • GW breaking sets in at lower amplitudes, i.e. it sets in earlier (at lower altitudes)

  39. Summary GW breaking • In comparison to assumptions in GW parameterizations: • GW breaking sets in at lower amplitudes, i.e. it sets in earlier (at lower altitudes) • GW dissipation stronger than assumed, i.e. more momentum deposition

  40. Soundproof Modelling and Multi-Scale Asymptotics Achatz, Klein, and Senf (2010)

  41. GW breaking in the middle atmosphere Rapp et al. (priv. comm.) Conservation Instability at large altitudes Momentum deposition…

  42. GW breaking in the middle atmosphere Rapp et al. (priv. comm.) • Conservation • Instability at large altitudes • Momentum deposition… • Competition between wave growth and dissipation: • not in Boussinesq theory

  43. GW breaking in the middle atmosphere Rapp et al. (priv. comm.) • Conservation • Instability at large altitudes • Momentum deposition… • Competition between wave growth and dissipation: • not in Boussinesq theory • Soundproof candidates: • Anelastic (Ogura and Philips 1962, Lipps and Hemler 1982) • Pseudo-incompressible (Durran 1989)

  44. GW breaking in the middle atmosphere Which soundproof model should be used? Rapp et al. (priv. comm.) • Conservation • Instability at large altitudes • Momentum deposition… • Competition between wave growth and dissipation: • not in Boussinesq theory • Soundproof candidates: • Anelastic (Ogura and Philips 1962, Lipps and Hemler 1982) • Pseudo-incompressible (Durran 1989)

  45. Scales: time and space • for simplicity from now on: 2D • non-hydrostatic GWs: same spatial scale in horizontal and vertical characteristic wave number • time scale set by GW frequency characteristic frequency dispersion relation for

  46. Scales: time and space • for simplicity from now on: 2D • non-hydrostatic GWs: same spatial scale in horizontal and vertical characteristic length scale • time scale set by GW frequency characteristic frequency dispersion relation for

  47. Scales: time and space • for simplicity from now on: 2D • non-hydrostatic GWs: same spatial scale in horizontal and vertical characteristic length scale • time scale set by GW frequency characteristic time scale dispersion relation for

  48. Scales: time and space • for simplicity from now on: 2D • non-hydrostatic GWs: same spatial scale in horizontal and vertical characteristic length scale • time scale set by GW frequency characteristic time scale dispersion relation for

  49. Scales: velocities • winds determined by polarization relations: what is ? • most interesting dynamics when GWs are close to breaking, i.e. locally

  50. Scales: velocities • winds determined by polarization relations: what is ? • most interesting dynamics when GWs are close to breaking, i.e. locally

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