Parameterizing Mesoscale Eddy Effects in Large-scale Ocean Models

Transcription

1 Parameterizing Mesoscale Eddy Effects in Large-scale Ocean Models Robert Hallberg with contributions from Malte Jansen, Angélique Melet, Alistair Adcroft, and Matthew Harrison

2 Developments in Parameterizing Eddies Know where to parameterize eddies (and where not to) Use a Resolution Function (Hallberg, Ocean Mod., 013) Along-isopycnal tracer diffusion can coexist with explicit eddies (unlike isopycnal height / G-M diffusion) Make the most of resolution Backscatter energy (Jansen & Held, Ocean Mod., 014) Predict the unresolved eddy intensity dynamically Mesoscale Eddy Kinetic Energy parameterization [Add vertical structure to eddy parameterizations & solve an elliptic equation for eddy transport streamfunction (Danabasoglu & Marshall, 007; Ferrari et al. 010; Abernathey et al. 013; )]

3 Mercator/Tripolar Resolution Required to Admit 1 st Baroclinic Deformation Radius L = c / f Def g ( ) + βc g Coupled Climate Model (CM.5): ~7 years/day on 6,000 Processors at 1/4 res. Cost goes as (Resolution) 3 ; Saturated throughput goes as (Resolution) -1.

4 Phillips model of baroclinic instability -layer stacked shallow water model. (See textbook discussion in Pedlosky, 1987.)! u!! ( ) ( ) " 1!!! + f + kˆ u u = M + u T cd u δ k u M 1 = gη 1/ t h ( ) ( ) 1 x + h1u! gδρ = γ η3/ η3/,ref ( K η M = M1 + η3/ h 3/ ) ρ0 t h ( ) ( ) x + hu! = γ η3/ η3/,ref + ( K η h 3/ ) t Exhibits instabilities with peak growth rates at wavelengths of πλ def. Restoring of zonal-mean interface height reenergizes the flow. Strong bottom drag prevents barotropization. (Arbic & Flierl, 006) η h 1, u! 1/ =η3/ + h1 η3/ = D + h 1 0 km h, u! 40 km Ref: Hallberg, Ocean Mod., 013 λ Def = 1 f gʹ h 1h h + h 1

5 10 km Resolution, K h = 0 Day 1 Day 10 Day 135 Day 730 Lower Layer Upper Layer

6 Upper Layer Flow at Various Resolutions, Day 730, K h = 0.5 km 5 km 10 km 0 km 5 km 33 km 40 km 50 km

7 Upper Layer Relative Vorticity / Coriolis at Various Resolutions, Day 730, K h = 0.5 km 5 km 10 km 0 km 5 km 33 km 40 km 50 km

8 Effects of Resolution on Overturning Transport t V ( y) = v1h1 dx V ( y) = = v y 0 xt 1 h xt 1 wdxdy dx + t vʹ h y 0 1 t 1ʹ h t dx 1 dxdy t y 0 wdxdy t

9 Effects of Resolution on Eddy-Driven Overturning Average Years 5 0 λ Def = 1 f gʹ h 1h h + h km Baroclinic eddy effects are absent or need to be parameterized when the deformation radius is poorly resolved.

10 Effects of Interface Height Diffusion (G-M mixing) at 10 km Resolution, Day 730 K h =0 m s -1 K h =1000 m s -1 K h =3000 m s -1 Relative Vorticity Upper Layer Flow

11 Interface height diffusion is much more effective at suppressing eddies than at parameterizing their effects! Overturning Transport (10 6 m 3 s -1 ) Interface Height Diffusivity (m s -1 ) Interface Height Diffusivity (m s -1 ) V ( y) xt xt t η = + ʹ ʹ + 3/ v1 h1 dx v1h 1 dx Kh dx y t

12 The best solution for the Phillips problem: Abruptly disable the Laplacian interface height diffusivity once the ratio of the grid spacing to the deformation radius reaches R Trans ~ K0 λdef / Δ < R K = ~ 0 λdef / Δ R Trans Trans λ Def = f c + g1 βcg1 ( ) Δx + Δy / ~ This works reasonably for the Phillips problem for any R trans >~! Δ = K/K 0 Step Resolution Function R Trans RH = λ Def ~ / Δ

13 Total Overturning at Various Resolutions, Step Function Parameterized vs. Explicit Eddy Transports, 16 km Res. Overturning Transport (10 6 m 3 s -1 ) ~ K0 λdef / Δ < R K = ~ 0 λdef / Δ R R Trans = Y (km) Strongly unstable case Trans Trans K0 = 8000 m s 1 (Parameterize K 0 later with MEKE.) Y (km) Time- & Zonal-mean Diffusivity

14 Speed with no interface height diffusivity 16 km resolution

15 Speed with step-function Resolution Function applied to interface height diffusivity 16 km resolution

16 Mercator/Tripolar Resolution Required to Admit 1 st Baroclinic Deformation Radius L = c / f Def g ( ) + βc g Coupled Climate Model (CM.5): ~7 years/day on 6,000 Processors at 1/4 res. Cost goes as (Resolution) 3 ; Saturated throughput goes as (Resolution) -1.

17 Annual Mean Step Resolution Function for ¼ Model R Trans =1.41 Marginal instability requires R Trans ~ R Trans can be smaller (explicit eddies over more area) for strong instability.

18 Along-isopycnal Tracer Diffusion vs. Isopycnal Height (or Gent-McWilliams) Diffusion Interface height diffusion suppresses baroclinic eddies; along-isopycnal tracer diffusion does not. Parameterized and resolved tracer mixing can coexist. Eddy mass transport is either explicit or parameterized - not both! Parameterized eddy watermass transport (e.g., G-M) should be disabled abruptly (via a step Resolution Function) when and where eddies are resolved. Eddy tracer diffusivities can (should?) be scaled away smoothly, and can represent mixing by unresolved scales even when some eddy scales are resolved. Use a gradually varying resolution function?

19 Degradation of Mean Flow at Eddy-Permitting Resolution Baroclinically unstable mean flow in a quasigeostrophic model z Warm, light y Cold, dense Topog. Ref: Jansen and Held (Ocean Modelling, 014) Mean Flow (Lower Layer Streamfunction) at Various Resolutions N=56 N=18 N=51 N=64 N=3 m /s

20 Backscatter of Small-Scale Dissipated Energy Ref: Jansen and Held (Ocean Modelling, 014) Kinetic Energy Spectra with Antidissipative Laplacian APE Extraction by Mean Flow KE Dissipation by Bottom Drag Viscous KE Kinetic Energy Spectra Dissipative Biharmonic Viscosity Add anti-viscous Laplacian forcing such t hat 90% of the energy dissipated near the grid scale is returned:! D u! 4! +... = A u A4 u Dt!!! 4! A u u dv = 0. A u u dv ( ) ( ) 9 Kinetic Energy Spectra Add Laplacian Anti-Viscosity 4 51 grid points! 56 grid points 18 grid points 64 grid points 3 grid points wavenumber (k) 51 grid points! 56 grid points 18 grid points 64 grid points 3 grid points wavenumber (k)

21 Mean Flow With and Without Energy Backscatter N=51 Hyperviscosity only Ref: Jansen and Held (Ocean Modelling, 014) N=64 White Noise " y [km] N=64 Hyperviscosity only y [km] 6000" m /s y [km] y [km] N=64 Negative Laplacian Backscatter allows eddy effects to be explicitly modeled at lo wer resolution, and hopefully use smaller values of R Trans

22 MEKE and Resolution Function Scaling Resolution function scaling seems particularly promising when combined with a parameterize to diagnose the unresolved Mesoscale Eddy Kinetic Energy (Cessi, 008; Eden et al., 008; Marshall & Adcroft, 010; Melet et al., 014; Jansen et al., 01X): E t Src = κ c = Src γe d u H bot 1 E + H ( Hκ E) K K ( 1? ) ( 0.001? ) gʹ kκ Int ηk H Int R H k = 1 H k= 1 ( )( κ κ ) κ = F + MEKE Background E h k u k τ ( 0.3? ) min, Δx + Δy, L E MEKE = λd Other Visc Ε : Unresolved Mesoscale Energy/Mass in Column MEKE is self-regulating; the MEKE-parameterized mixing changes the resolved state to reduce MEKE s source of energy.

23 MEKE in Global Models with Step Function F(RH) KH MEKE 1 1/ 4

24 Temperature Bias Reduction in 1/4 Coupled Model No Isopycnal Height Diffusion (Control) With Res. Fn. & Isopycnal Height Diffusion Depth (m) Latitude Change in Bias Latitude Averages over years Same scale as above Atmosphere not retuned Parameterization strength not optimized Vertically uniform diffusivities

25 Changing Energy Fluxes into Lee Waves Ref: Melet et al, 014, J. Climate (in press) Globally, energy input into eddy- and mean-flow driven lee waves decreases with a warming climate due to weaker near-bottom flows But lee-wave energy input to the Southern Ocean increases due to stronger eddies & mean flow Lee Wave Energy Flux RCP 8.5 Preindustrial Control Lee Wave Energy (TW) Globally Integrated Lee Wave Energy Input Historical P.I. Ctrl RCP4.5 RCP6.0 RCP Lee wave energy input RCP8.5 P.I. ctrl. (W m - ) RCP8.5

26 The Gulf Stream in Global Models with Resolution Function Scaling and MEKE 1 1 / 8 With MEKE scaled via a Resolution Function, these global ocean models differ only in their choice of grid-spacing and time-step Backscatter gives plausible explicit eddy effects at lower resolutions Climate projections of heat and carbon update require that eddy parameterizations respond appropriately to an evolving ocean state