Statistical turbulence modelling in the oceanic mixed layer

Transcription

1 Statistical turbulence modelling in the oceanic mixed layer Hans Burchard Baltic Sea Research Institute Warnemünde, Germany Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 1/44

2 Program of presentation NSE Two-equation closure models Two-equation closure models General Ocean Turbulence Model (GOTM) Examples (observations versus simulations) General Estuarine Transport Model (GETM) Three-dimensional model study Conclusions Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 2/44

3 Basic approaches Two approaches to averaged turbulence modelling: Statistical turbulence modelling: Convert NSE to Friedmann-Keller series, cut-off where suitable and parameterise unknown terms. Empirical turbulence modelling: Close equations on lowest order and parameterise relevant processes. Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 3/44

4 NSE Reynolds equation Momentum Equation: t v i + v j j v i ν jj v i + 2ε ijl Ω j v l = ip ρ g i ρ ρ. Reynolds averaging: Reynolds Equation: v i = v i + ṽ i, ρ = ρ + ρ,... t v i + v j j v i j (ν j v i ṽ j ṽ i )+2ε ijl Ω j v l = i p ρ g i ρ ρ. Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 4/44

5 Reynolds Stress Equation ) t ṽ i ṽ j + l ( v l ṽ i ṽ j + ṽ l ṽ i ṽ j ν l ṽ i ṽ j = l v i ṽ l ṽ j l v j ṽ l ṽ i }{{} P ij 2Ω l (ε ilm ṽ j ṽ m + ε jlm ṽ i ṽ m ) }{{} Ω ij 1 ρ {g i ṽ j ρ + g j ṽ i ρ } }{{} B ij 2ν ( l ṽ j )( l ṽ i ). }{{} ε ij 1 ρ ( ṽ i j p + ṽ j i p ) }{{} Π ij Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 5/44

6 Algebraic SMCs The following steps lead to different types of second-moment closures: Empirical closures of pressure-strain correlators. Neglect or simplification of advective and diffusive fluxes of second-moments. Neglect of rotational terms in the second-moment equations. Boundary layer assumption (neglect of horizontal gradients and non-hydrostatic effects).... and many more details... Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 6/44

7 Algebraic SMCs Turbulent Fluxes: ũ w = ν t z ū, w T = ν t z T Eddy Viscosity / Eddy Diffusivity: ν t = c µ (α M, α N ) k2 ε, ν t = c µ(α M, α N ) k2 ε. Shear Number, Buoyancy Number: α M = k2 ε 2 M 2, α N = k2 ε 2 N 2. Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 7/44

8 .22.2 Stability Functions Canuto et al. [21]: c µ c µ α M α M α N α N Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 8/44

9 Exact TKE-Equation ) 1 t k + j ( v j k + ṽ j 2ṽ2 i ν jk + 1 ρ ṽ j p = ṽ j ṽ i i v j }{{} P g ρ ṽ 3 ρ }{{} B ν ( j ṽ i ) 2, }{{} ε This TKE equation will be modelled as it is given above, the only parameterisations needed are for the turbulent f lux terms, for which usually the down-gradient approximation is used. Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 9/44

10 Dissipation equation Exact form (e.g. Wilcox [1998]): ) t ε + j ( v j ε + ṽ j ν( j ṽ i ) 2 ν j ε + 2 ν ρ i ṽ j i p ) = 2ν j v i ( i ṽ k j ṽ k + k ṽ i k ṽ j 2ν jk v i ṽ k j ṽ i }{{} 2ν g ρ j ṽ 3 j ρ }{{} B ε 2ν P ε ( j ṽ i k ṽ i j ṽ k + ν ( ij ṽ k ) 2 ) } {{ } ε ε (1) k-ε model (Launder and Spalding [1972]): ( ) νt t ε z z ε = ε σ ε k (c ε1p + c ε3 B c ε2 ε). Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 1/44

11 Mellor-Yamada model General relation between k, ε and L: L = c 3/4 µ k 3/2 k-kl model (Mellor and Yamada [1982]): t (kl) z (S l z (kl)) = ε L 2 [ E 1 P + E 3 B (1 + E 2 ( LLz ) 2 ) ε ]. Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 11/44

12 Length scale equations (cont d) Other approaches are using equations for ω = ε/k (k-ω model), k/ε, k 2 /ε,..., so why not using the generalised approach of a k n ε m equation? Generic length scale equation (Umlauf and Burchard [23]): t (k n ε m ) z ( νt σ nm k n ε m ) = k n 1 ε m (c nm1 P + c nm3 B c nm2 ε) This works without correction term only for m. Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 12/44

13 Total equilibrium (k-ε) R st i R i = R st i = c µ c µ c2ε c 1ε c 2ε c 3ε..25: Steady-state Richardson number. PSfrag replacements R i st KC RH CA CB c 3ε Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 13/44

14 Kato-Phillips experiment 45 MLD for Canuto et al. version A MLD / m Time / h Empirical Ri_st =.2 Ri_st =.3 Ri_st =.4 Ri_st =.6 Ri_st =.8 Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 14/44

15 Total equilibrium (k-kl) R st i R i = R st i = K M K H 1 E 1 1 E 3..25: Steady-state Richardson number R st i B/ε G H.2.15 PSfrag replacements E 3 Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 15/44

16 GOTM, Challenge Aim The Idea Key features Software Fortran code Test cases Forcing How to run? Information What's New Publications list FAQ User Group Hot Links Who's Who? Guestbook GOTM is a one-dimensional numerical model developed and supported by a core team of ocean modellers. GOTM aims at simulating accurately vertical exchange processes in the marine environment where mixing is known to play a key role. GOTM is freely available under the GPL (Gnu Public License). The interested user can download the source code, a set of test cases (Papa, November, Flex,...) and a comprehensive report. You are warmly invited to join the GOTM mailing list and send any comments/questions to the GOTM team or become a GOTM contributor. The GOTM developers are grateful to their sponsors. Page " maintained by webmaster. Last update: 1/28/ 18:1:2 PPM: 1/26/ 2:48:54 Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 16/44

17 Turbulence under waves How to compare these observations? How to simulate them??? Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 17/44

18 Breaking waves: Simulations eplacements 1 (z z)/hs 1 Terray et al. [1996] Drennan et al. [1996] Anis and Moum et al. [1995] Numerical, z /H s = 2. Numerical, z /H s = 1. Numerical, z /H s =.5 Log-Law No shear production ε / c w u 3 /H s Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 18/44

19 ũ 2 /(w ) 2 ũ 2 /(w ) 2 w 2 Free /(w ) 2 Convection w 2 /(w ) 2 T 2 /(T 2 ) T 2 /(T 2 ) Temperature ε/b KC ε/b RH -.2 CA CB Temperature ( LES -.4 emperature Flux T T max )/T -.6 Variance of w u T /(w T Variance ) of u -.8 Variance of w ũ 2 /(w -1 Variance ) 2 of w Variance of T w /(wvariance ) 2 of T Dissipation Rate Dissipation ( T T max )/T T 2 /(T 2 Rate ) z/dm Temperature Temperature Flux Variance of u Variance of w Variance of T z/dm z/dm Dissipation Rate Temperature Flux -.8 KC RH -1 CA CB LES w T /(w T ) -.8 KC RH -1 CA CB LES ε/b Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 19/44

20 Geographical overview Y Lago Maggiore Northern North Sea Liverpool Bay OWS Papa Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 2/44

21 Lago Maggiore, Italy Observations and simulations of T and ε (Stips et al. [22]) Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 21/44

22 Open Ocean: OWS Papa Temperature Observations Depth [m] Measurements OWS Papa 1961/ Julian Day 1961/62 T [ o C] Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 22/44

23 Open Ocean: OWS Papa Heat budget OWS Papa 1961/62 2e+9 Heat from T-profiles Heat from surface fluxes eplacements Q / J m 2 1.5e+9 1e+9 5e Julian Day 1961/62 Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 23/44

24 Open Ocean: OWS Papa Temperature Simulations: k-ε model with alg. SMC Depth [m] Simulation OWS Papa 1961/ Julian Day 1961/62 T [ o C] Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 24/

25 Open Ocean: OWS Papa Temperature profiles T-profiles, day 12 T-profiles, day 15 T-profiles, day Observed Simulated, KC Simulated, CA Observed Simulated, KC Simulated, CA Observed Simulated, KC Simulated, CA T-profiles, day 21 T-profiles, day 24 T-profiles, day Observed Simulated, KC Simulated, CA Observed Simulated, KC Simulated, CA Observed Simulated, KC Simulated, CA Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 25/44

26 Open Ocean: OWS Papa SST in deg C SST in deg C Sea Surface Temperature OWS Papa 1961/62 Observation Simulation KC Julian Day 1961/62 Observation Simulation CA Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 26/44

27 Northern North Sea Bathymetry and station map ' 45' 1 ' 1 15' 1 3' 1 45' 2 ' 59 45' 59 45' 61 1 Norway ' 59 3' Scotland ' 59 15' ' 59 ' 3' 45' 1 ' 1 15' 1 3' 1 45' 2 ' Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 27/44

28 Northern North Sea Observed temperature Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 28/44

29 Northern North Sea Simulated temperature Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 29/44

30 Northern North Sea Wind and Tides rag replacements Q / (W m 2 ) ( x ζ) 2 + ( y ζ) 2 rag replacements Q / (W m 2 ) ( x ζ) 2 + ( y ζ) 2 τ / (N m 2 ) τ / (N m 2 ) /1 12: ADCP Model Surface stress at station NNS Bed stress at station NNS 22/1 : 22/1 12: Date in /1 : 23/1 12: Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 3/44

31 Northern North Sea Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 31/44

32 NPZD model in GOTM Temperature Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 32/44

33 NPZD model in GOTM Nutrients Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 33/44

34 NPZD model in GOTM Phytoplankton Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 34/44

35 NPZD model in GOTM Zooplankton Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 35/44

36 NPZD model in GOTM Detritus Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 36/44

37 Liverpool Bay Section of Temperature and Salinity Fig 2.a Temperature (Degrees C) 1 6 Depth (m) -2-4 LB Fig 2.b Salinity (PSU) Depth (m) Rippeth, Fisher, Simpson [21] Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 37/44

38 Liverpool Bay Observed and simulated temperature and salinity Simpson, Burchard, Fisher, Rippeth [22] Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 38/44

39 Liverpool Bay Observed and simulated current velocity Simpson, Burchard, Fisher, Rippeth [22] Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 39/44

40 Liverpool Bay Observed and simulated dissipation rates Simpson, Burchard, Fisher, Rippeth [22] Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 4/44

41 GETM General Estuarine Transport Model (Burchard & Bolding [22]) General vertical coordinates Horizontal curvilinear coordinates Uses GOTM as turbulence model Drying & fl ooding Public Domain Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 41/44

42 North Sea simulation PSfrag Sea surface temperature replacements on Aug. 1, PSfrag replacements surface temperature 4 on Aug. 1, 1997 surface T / replacements temperature on Aug. 1, 1997 C 2 3 replacements 2 Aug. 1, T / C Sea surface y / nmsalinity on Aug. 1, x / nm T / C S/psu Temperature section on Aug. 1, 1997 Aug. 1, 1997 T / -1 2 S/psu C Temperature section -2 on Aug. 1, T / C5 y / nm z / m -5-6 y / nm y / nm T / C z / m Sea surface salinity on Aug. 1, x / nm Salinity section on Aug. 1, 1997 S/psu S/psu x / nm x / nm Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 42/44

43 North Sea / Baltic Sea SSS on June 1, 1997, x = y = 3 nm g replacements y / nm S / psu x / nm Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 43/44

44 Conclusions Reynolds decomposition provides a physically sound framework for developing turbulence models. Empirical parameters allow for calibrating such models to the real world. With such models, turbulence is described but not better understood. For mixed layers, two-equation models with algebraic SMCs are sufficient for reproducing observations. These models are economic enough for using them in 3D models for oceanic applications. Seminar at Bjerknes Centre, October 16, 23, Bergen, Norway p. 44/44