Turbulence examined in the frequency-wavenumber domain∗ Andrew J. Morten Department of Physics University of Michigan Supported by funding from: National Science Foundation and Office of Naval Research ∗ Arbic et al. JPO 2012; Arbic et al. in review; Morten et al. papers in preparation Andrew J. Morten LOM 2013, Ann Arbor, MI Collaborators • University of Michigan Brian Arbic, Charlie Doering • MIT Glenn Flierl • University of Brest, and The University of Texas at Austin Robert Scott Andrew J. Morten LOM 2013, Ann Arbor, MI Outline of talk Part I{Motivation (research led by Brian Arbic) • Frequency-wavenumber analysis: {Idealized Quasi-geostrophic (QG) turbulence model. {High-resolution ocean general circulation model (HYCOM)∗: {AVISO gridded satellite altimeter data. ∗We used NLOM in Arbic et al. (2012) Part II-Research led by Andrew Morten • Derivation and interpretation of spectral transfers used above. • Frequency-domain analysis in two-dimensional turbulence. • Theoretical prediction for frequency spectra and spectral transfers due to the effects of \sweeping." {moving beyond a zeroth order approximation. Andrew J. Morten LOM 2013, Ann Arbor, MI Motivation: Intrinsic oceanic variability • Interested in quantifying the contributions of intrinsic nonlinearities in oceanic dynamics to oceanic frequency spectra. • Penduff et al. 2011: Interannual SSH variance in ocean models with interannual atmospheric forcing is comparable to variance in high resolution (eddying) ocean models with no interannual atmospheric forcing. • Might this eddy-driven low-frequency variability be connected to the well-known inverse cascade to low wavenumbers (e.g. Fjortoft 1953)? • A separate motivation is simply that transfers in mixed ! − k space provide a useful diagnostic. Andrew J. Morten LOM 2013, Ann Arbor, MI Oceanic analysis regions • Regions used to analyze AVISO gridded altimeter data and HYCOM output. • Shown against snapshots of SSH (cm) from HYCOM and SSH anomaly (cm) from AVISO. 200 60°N 40°N 100 20°N 0° 0 20°S -100 40°S 60°S -200 0° 40°E 80°E 120°E 160°E 160°W 120°W 80°W 40°W 0° 60°N 50 40°N 20°N 0° 0 20°S 40°S 60°S -50 0° 40°E 80°E 120°E 160°E 160°W 120°W 80°W 40°W 0° Andrew J. Morten LOM 2013, Ann Arbor, MI 2 Frequency spectra of surface streamfunction variance 1 2 and kinetic energy jr 1j (Arbic et al. 2012 JPO) AVISO NLOM (a) ω spectrum of ψ2 (a) ω spectrum of ψ2 1 1 15 15 10 10 13 13 10 10 I Flat spectra at low 11 Midlat SE Pacific 11 Midlat SE Pacific 10 Highlat SE Pacific 10 Highlat SE Pacific frequencies as in Agulhas Agulhas 9 9 10 Malvinas 10 Malvinas Gulf Stream Gulf Stream previous studies 7 Kuroshio 7 Kuroshio 10 10 −3 −2 −1 0 1 −3 −2 −1 0 1 10 10 10 10 10 10 10 10 10 10 (e.g. Richman et ω (days−1) ω (days−1) (b) ω spectrum of |∇ ψ |2 (b) ω spectrum of |∇ ψ |2 al. 1977, Wunsch 1 1 6 6 10 10 2009, 2010) 4 4 10 10 Midlat SE Pacific Midlat SE Pacific Highlat SE Pacific Highlat SE Pacific 2 2 10 Agulhas 10 Agulhas Malvinas Malvinas Gulf Stream Gulf Stream 0 Kuroshio 0 Kuroshio 10 10 −3 −2 −1 0 1 −3 −2 −1 0 1 10 10 10 10 10 10 10 10 10 10 ω (days−1) ω (days−1) Andrew J. Morten LOM 2013, Ann Arbor, MI Spectral transfers of upper layer kinetic energy versus wavenumber at fixed frequency Spectral transfers vs. k QG model Agulhas HYCOM and AVISO (a) QG T (K,ω) vs K at fixed ω KE,1 (a) HYCOM Agulhas T (K,ω) vs K at fixed ω 4 KE,1 3 x 10 ω =0.0012 5 1824 days ω /U))] =0.0105 608 days d 2 ω=0.1001 68 days 0 ω=0.9995 28 days 14 days )(rad/(L ω d =3.1416 1 −5 −4 −3 −2 −1 0 (nW/kg)/[(rad/day)(rad/km)] 10 10 10 10 10 )/[(rad/L d 0 /L ω ω 3 4 (b) AVISO Agulhas T (K, ) vs K at fixed x 10 KE,1 (U −1 5 5880 days −1 0 1 2 10 10 10 10 653 days 68 days 0 28 days (b) QG T(K) integrated over all ω 14 days 600 T −5 KE,1 −4 −3 −2 −1 0 400 T (nW/kg)/[(rad/day)(rad/km)] 10 10 10 10 10 KE,2 )] d T (c) Agulhas T (K), integrated over all ω 200 APE KE,1 T forcing 2000 HYCOM 0 T )/[(rad/L friction AVISO d 1000 /L 3 −200 Residual (U 0 −400 (nW/kg)/[(rad/km)] −1000 −600 −4 −3 −2 −1 0 −1 0 1 2 10 10 10 10 10 10 10 10 10 K (rad/L ) K (rad/km) d Andrew J. Morten LOM 2013, Ann Arbor, MI Spectral transfers of upper layer kinetic energy versus frequency at fixed wavenumber Spectral transfers vs. ! QG model Highlat SE Pac HYCOM & AVISO (a) QG T (K,ω) vs ω at fixed K KE,1 (a) HYCOM Highlat SE Pacific T (K,ω) vs ω at fixed K KE,1 4 1000 K=0.1 2132 km K=0.2 /U))] 3 919 km d K=0.5 500 430 km 2 K=1 117 km K=3 0 56 km )(rad/(L d 1 −500 −4 −2 0 0 (nW/kg)/[(rad/day)(rad/km)] 10 10 10 )/[(rad/L d /L ω ω 3 (b) AVISO Highlat SE Pacific T (K, ) vs at fixed K −1 KE,1 (U −2 2115 km −4 −2 0 2 2000 10 10 10 10 734 km 351 km 1000 114 km (b) QG T(ω) integrated over all K 57 km 15 0 T KE,1 −4 −2 0 10 T (nW/kg)/[(rad/day)(rad/km)] 10 10 10 KE,2 /U))] T (c) Highlat SE Pacific T (ω), integrated over all K d 5 APE KE,1 T forcing 30 HYCOM 0 T friction 20 AVISO )/[(rad/(L d −5 Residual /L 10 3 (U −10 0 (nW/kg)/[(rad/day)] −10 −15 −4 −3 −2 −1 0 1 −4 −2 0 2 10 10 10 10 10 10 10 10 10 10 ω (rad/(L /U)) ω (rad/day) d Andrew J. Morten LOM 2013, Ann Arbor, MI Derivation of spatial transfers • Reminder of derivation of energy and enstrophy spatial transfers @ r2 + J = F + D (1) @t + @ − k2 ~ + J~ = F~ + D~ (2) @t + @ j ~(~k; t)j2 Energy: k2 − Re( ~∗J~) = −Re( ~∗F~) − Re( ~∗D~ ) (3) @t 2 and @ j ~j2 Enstrophy: k4 − Re(k2 ~∗J~) = −Re(k2 ~∗F~) − Re(k2 ~∗D~ ) @t 2 (4) Andrew J. Morten LOM 2013, Ann Arbor, MI Time-frequency analysis needed for temporal transfers Choose either: • Short-time (moving window) Fourier transform: Z τ+T =2 t − τ −i!t f^(τ; !; T ) := σ( )[f (t) − ftrend (t; τ; T )]e dt (5) τ−T =2 T dfc @ =) (τ; !) = i!f^ + f^(τ; !) (6) dt @τ • Wavelet: Z 1 f^(τ; !) := f (t)j!j1=2W ((t − τ)!)dt (7) −∞ dfc @ =) (τ; !) = f^(τ; !) (8) dt @τ Andrew J. Morten LOM 2013, Ann Arbor, MI Derivation of temporal transfers • Derivation of energy and enstrophy temporal transfers: \@ − k2 ~ + J~^ = F~^ + D~^ (9) @t + @ − k2 ~^ + (−i!k2 ~^) +J~^ = F~^ + D~^ (10) @τ | {z } only for STFT + d j ~^(~k; τ; !)j2 Energy: k2 − Re( ~^∗J~^) = −Re( ~^∗F~^) − Re( ~^∗D~^ ) dτ 2 (11) and d j ~^j2 Enstrophy: k4 − Re(k2 ~^∗J~^) = −Re(k2 ~^∗F~^) − Re(k2 ~^∗D~^ ) dτ 2 (12) Andrew J. Morten LOM 2013, Ann Arbor, MI Model choices • 2D Navier-Stokes equation (modified) @ r2 + J( ; r2 ) = F − ν r2 + D + D (13) @t 0 hyper hypo • streamfunction, • jacobian, J(·; ·) • Ekman drag, ν0 = 0, for this talk. • exponential wavenumber filter (hyper-)viscosity, Dhyper • " " filter (hypo-)viscosity, Dhypo • forcing, F , chosen to be localized about wavenumber kF and frequency !F . Andrew J. Morten LOM 2013, Ann Arbor, MI Three different forcing frequencies 15 15 15 x 10 Forcing Spectral Density (k) x 10 Forcing Spectral Density (k) x 10 Forcing Spectral Density (k) 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 Forcing Spectral Density Forcing Spectral Density Forcing Spectral Density 0 0 0 0 1 2 0 1 2 0 1 2 10 10 10 10 10 10 10 10 10 K K K mid kF mid kF mid kF ω ω ω 19 Forcing Spectral Density ( ) 19 Forcing Spectral Density ( ) 18 Forcing Spectral Density ( ) x 10 x 10 x 10 9 7 15 8 6 7 5 6 10 5 4 4 3 3 5 2 2 Forcing Spectral Density Forcing Spectral Density Forcing Spectral Density 1 1 0 0 0 −2 −1 0 −2 −1 0 −2 −1 0 10 10 10 10 10 10 10 10 10 ω ω ω high !F mid !F low !F Andrew J. Morten LOM 2013, Ann Arbor, MI Model choices: spectrally localized forcing In order to allow for statistically homogeneous isotropic turbulence... Start with Maltrud-Vallis (stochastic) forcing, ~ p 2 iφn F~MV (k; tn) = f0 1 − c e + cF~MV (tn−1); (14) Shift the power spectrum peak to !F and −!F ~ ~ i!F tn ~ + ~ −i!F tn ~ − ~ F (k; tn) = e FMV (k; tn) + e FMV (k; tn) (15) Andrew J. Morten LOM 2013, Ann Arbor, MI Typical (~x; t) and r2 snapshots; and forcing Streamfunction snapshot Vorticity snapshot 0.2 100 1000 1000 80 900 0.15 900 60 800 800 0.1 40 700 700 0.05 600 600 20 y 500 0 y 500 0 −20 400 −0.05 400 300 300 −40 −0.1 200 200 −60 −0.15 100 100 −80 −0.2 −100 0 200 400 600 800 1000 0 200 400 600 800 1000 x x Forcing snapshot 1000 0.015 900 0.01 800 700 0.005 600 y 500 0 400 −0.005 300 200 −0.01 100 −0.015 0 200 400 600 800 1000 x Andrew J.