Turbulence Examined in the Frequency-Wavenumber Domain*

Home , Wavenumber

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 ω − 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 ψ1 2 and kinetic energy |∇ψ1| (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. ω 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

∂ ∇2ψ + J = F + D (1) ∂t ⇓ ∂ − k2ψ˜ + J˜ = F˜ + D˜ (2) ∂t ⇓ ∂ |ψ˜(~k, t)|2 Energy: k2 − Re(ψ˜∗J˜) = −Re(ψ˜∗F˜) − Re(ψ˜∗D˜ ) (3) ∂t 2 and ∂ |ψ˜|2 Enstrophy: k4 − Re(k2ψ˜∗J˜) = −Re(k2ψ˜∗F˜) − Re(k2ψ˜∗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 −iωt fˆ(τ, ω; T ) := σ( )[f (t) − ftrend (t, τ; T )]e dt (5) τ−T /2 T

dfc ∂ =⇒ (τ, ω) = iωfˆ + fˆ(τ, ω) (6) dt ∂τ • Wavelet: Z ∞ fˆ(τ, ω) := f (t)|ω|1/2W ((t − τ)ω)dt (7) −∞

dfc ∂ =⇒ (τ, ω) = fˆ(τ, ω) (8) dt ∂τ

Andrew J. Morten LOM 2013, Ann Arbor, MI Derivation of temporal transfers

• Derivation of energy and enstrophy temporal transfers:

\∂ − k2ψ˜ + J˜ˆ = F˜ˆ + D˜ˆ (9) ∂t ⇓ ∂ − k2ψ˜ˆ + (−iωk2ψ˜ˆ) +J˜ˆ = F˜ˆ + D˜ˆ (10) ∂τ | {z } only for STFT ⇓ d |ψ˜ˆ(~k, τ, ω)|2 Energy: k2 − Re(ψ˜ˆ∗J˜ˆ) = −Re(ψ˜ˆ∗F˜ˆ) − Re(ψ˜ˆ∗D˜ˆ ) dτ 2 (11) and d |ψ˜ˆ|2 Enstrophy: k4 − Re(k2ψ˜ˆ∗J˜ˆ) = −Re(k2ψ˜ˆ∗F˜ˆ) − Re(k2ψ˜ˆ∗D˜ˆ ) dτ 2 (12)

Andrew J. Morten LOM 2013, Ann Arbor, MI Model choices

• 2D Navier-Stokes equation (modiﬁed)

∂ ∇2ψ + J(ψ, ∇2ψ) = F − ν ∇2ψ + D + D (13) ∂t 0 hyper hypo • streamfunction, ψ • 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 ωF .

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

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 ωF mid ωF low ωF

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 ωF and −ωF

˜ ~ iωF tn ˜ + ~ −iωF tn ˜ − ~ F (k, tn) = e FMV (k, tn) + e FMV (k, tn) (15)

Andrew J. Morten LOM 2013, Ann Arbor, MI Typical ψ(~x, t) and ∇2ψ 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. Morten LOM 2013, Ann Arbor, MI Energy Fluxes in k-space (three diﬀerent forcing periods)

Energy Fluxes (nonlinear term in red) in k-space

9 9 9 x 10Energy Fluxes in Wavenumber Space x 10Energy Fluxes in Wavenumber Space x 10Energy Fluxes in Wavenumber Space 4 4 4 jacobian jacobian jacobian forcing forcing forcing 3 ekman 3 ekman 3 ekman hyper hyper hyper hypo hypo hypo residual residual residual 2 2 2

1 1 1

0 0 0

−1 −1 −1 Energy Flux Energy Flux Energy Flux

−2 −2 −2

−3 −3 −3

−4 −4 −4 0 1 2 0 1 2 0 1 2 10 10 10 10 10 10 10 10 10 K K K high ωF mid ωF low ωF

• Clear inertial ranges in k-space. • Similar results for all three forcing periods. – slightly more injection of energy for longer forcing periods.

Andrew J. Morten LOM 2013, Ann Arbor, MI Enstrophy Fluxes in k-space (three forcing periods)

Enstrophy Fluxes (nonlinear term in red) in k-space

13 13 13 Enstrophyx 10 Fluxes in Wavenumber Space Enstrophyx 10 Fluxes in Wavenumber Space Enstrophyx 10 Fluxes in Wavenumber Space 1.5 1.5 1.5 jacobian jacobian jacobian forcing forcing forcing ekman ekman ekman hyper 1 1 hyper 1 hyper hypo hypo hypo residual residual residual

0.5 0.5 0.5

0 0 0

−0.5 −0.5 −0.5 Enstrophy Flux Enstrophy Flux Enstrophy Flux

−1 −1 −1

−1.5 −1.5 −1.5 0 1 2 0 1 2 0 1 2 10 10 10 10 10 10 10 10 10 K K K high ωF mid ωF low ωF

• Clear inertial ranges in k-space. • Similar results for all three forcing periods. – slightly more injection of enstrophy for long forcing periods.

Andrew J. Morten LOM 2013, Ann Arbor, MI Energy Fluxes in ω-space (three forcing periods)

Energy Fluxes (nonlinear term in red) in ω-space

17 17 16 x 10 Energy Fluxes in Frequency Space x 10 Energy Fluxes in Frequency Space x 10 Energy Fluxes in Frequency Space 2 1.5 8 jacobian jacobian jacobian forcing forcing forcing 1.5 ekman ekman ekman hyper hyper 6 hyper hypo 1 hypo hypo residual residual residual 1 4

0.5 0.5 2

0 0 −0.5 Energy Flux Energy Flux Energy Flux

−2 −1 −0.5 −4 −1.5

−2 −1 −6 −2 −1 0 −2 −1 0 −2 −1 0 10 10 10 10 10 10 10 10 10 ω ω ω high ωF mid ωF low ωF

• Not really an “inertial range” in ω-space. • Direction of “cascade” depends on forcing frequency. • Probably these transfers are determined by sweeping (by vrms ), – more complicated than a simple Taylor’s hypothesis.

Andrew J. Morten LOM 2013, Ann Arbor, MI Why can the spectral transfers proceed in either direction Local conservation within triads in k-space (Kraichnan 1967)

0 = T (k, p, q) + T (p, q, k) + T (q, k, p), (16) 0 = k2T (k, p, q) + p2T (p, q, k) + q2T (q, k, p),

generalizes to

0 = T (k, p, q; ωk , ωp, ωq) + T (p, q, k; ωp, ωq, ωk ) + T (), (17) 2 2 2 0 = k T (k, p, q; ωk , ωp, ωq) + p T (p, q, k; ωp, ωq, ωk ) + q T (),

which, by integrating over wavenumbers, gives

0 = Q1(ωk , ωp, ωq) + Q1(ωp, ωq, ωk ) + Q1(), (18)

0 = Q2(ωk , ωp, ωq) + Q2(ωp, ωq, ωk ) + Q2(),

where Q1 and Q2 are energy and enstrophy transfers, respectively, within frequency triads.

Andrew J. Morten LOM 2013, Ann Arbor, MI Enstrophy Fluxes in ω-space (three forcing periods)

Enstrophy Fluxes (nonlinear term in red) in ω-space

20 20 20 x Enstrophy10 Fluxes in Frequency Space x Enstrophy10 Fluxes in Frequency Space x Enstrophy10 Fluxes in Frequency Space 6 4 2.5 jacobian jacobian jacobian forcing forcing forcing 2 ekman 3 ekman ekman 4 hyper hyper hyper hypo hypo 1.5 hypo residual residual residual 2 1 2 1 0.5

0 0 0

−0.5 −1 −2 Enstrophy Flux Enstrophy Flux Enstrophy Flux −1 −2 −1.5 −4 −3 −2

−6 −4 −2.5 −2 −1 0 −2 −1 0 −2 −1 0 10 10 10 10 10 10 10 10 10 ω ω ω high ωF mid ωF low ωF

• Not really an “inertial range” in ω-space, except for small ωF . • Direction of “cascade” depends on forcing frequency. • Probably these transfers are determined by sweeping (by vrms ), – more complicated than a simple Taylor’s hypothesis.

Andrew J. Morten LOM 2013, Ann Arbor, MI Energy Spectra in k-space (three forcing periods)

Energy spectra in k-space

Energy Spectrum (k) Energy Spectrum (k) Energy Spectrum (k) 11 10 fit fit fit kolmogorov kolmogorov kolmogorov

−1.57 10 10 10 10 10 10

9 −1.69 9 10 9 10 10 −1.75

8 8 8 10 10 10 Energy(k) Energy(k) Energy(k)

7 7 7 10 10 10

6 6 6 10 10 10 −4.16 −3.9 −3.61

1 2 1 2 1 2 10 10 10 10 10 10 k k k high ωF mid ωF low ωF

• Spectral slopes reasonably close to Kraichnan’s scaling in k-space. • (−5/3 and −3, or more like -4)

Andrew J. Morten LOM 2013, Ann Arbor, MI Enstrophy Spectra in k-space (three forcing periods)

Enstrophy spectra in k-space

Enstrophy Spectrum (k) Enstrophy Spectrum (k) Enstrophy Spectrum (k)

fit fit fit kolmogorov kolmogorov kolmogorov 0.336 0.442

0.328

12 12 10 12 10 10 Enstrophy(k) Enstrophy(k) Enstrophy(k) 11 10 11 11 10 10

−1.57 −2.06 −1.88

1 2 1 2 1 2 10 10 10 10 10 10 k k k high ωF mid ωF low ωF

• Spectral slopes reasonably close to Kraichnan’s scaling in k-space.

Andrew J. Morten LOM 2013, Ann Arbor, MI Taylor’s hypothesis and sweeping

An overview of Taylor’s hypothesis and sweeping. • Taylor’s hypothesis: (strong mean ﬂow) + (spatial structure) =⇒ (Eulerian frequency spectra). • sweeping hypothesis: (zero mean ﬂow) + (strong large-scale vrms ) + (spatial structure) =⇒ (Eulerian frequency spectra).

Andrew J. Morten LOM 2013, Ann Arbor, MI Taylor’s hypothesis and sweeping

• sweeping hypothesis: (zero mean ﬂow) + (strong large-scale vrms ) + (spatial structure) =⇒ (Eulerian frequency spectra).

– sweeping is typically treated similarly to a strong mean ﬂow, but with velocity vrms . (Tennekes 1975) – or, as an ensemble of time-independent sweeping velocities. (Chen et. al. 1989) – but for a general time-dependent sweeping, one needs to be careful. (Morten et al., in preparation)

Andrew J. Morten LOM 2013, Ann Arbor, MI Energy Spectra in ω-space (three forcing periods)

Energy spectra in ω-space

Energy Spectrum (ω) Energy Spectrum (ω) Energy Spectrum (ω)

17 10 fit 17 fit fit naive taylor 10 naive taylor naive taylor 17 10

−1.07 −1.08

16 10 −1.18 16

) 10 ) ) 16 ω 10 ω ω

15 −2.19 Energy(

Energy( −2.25 Energy( −2.16 15 10 10 15 10

14 14 10 10

−3 −2 −1 0 −2 −1 0 −2 −1 0 10 10 10 10 10 10 10 10 10 10 ω ω ω high ωF mid ωF low ωF

• Taylor’s hypothesis and “sweeping” are not quite right; slopes not equal to those in wavenumber spectrum. • Slopes at low frequencies near -1; higher frequencies near -2.

Andrew J. Morten LOM 2013, Ann Arbor, MI Enstrophy Spectra in ω-space (three forcing periods)

Enstrophy spectra in ω-space

Enstrophy Spectrum (ω) Enstrophy Spectrum (ω) Enstrophy Spectrum (ω)

fit fit fit naive taylor naive taylor naive taylor

19 10 19 10 19 10 −0.237 ) ) ) −0.367 −0.231 ω ω ω Enstrophy( Enstrophy( Enstrophy(

18 18 10 10 −1.17 −1.18 −1.11

18 10

−3 −2 −1 0 −3 −2 −1 0 −2 −1 0 10 10 10 10 10 10 10 10 10 10 10 ω ω ω high ωF mid ωF low ωF

• (Naive) Taylor’s hypothesis or “sweeping” give incorrectly signed slope. • Direct cascade enstrophy spectra give best match (−1 slope). • (Correct) Taylor’s hypothesis or “sweeping” give negative slope in ω-spectra for a positive sloping k-spectra (see next two slides).

Andrew J. Morten LOM 2013, Ann Arbor, MI Taylor and Sweeping done right

Summary of some analytical results: • Taylor done right: p (mean velocity v0) and spectra E(k) ∝ k (between kmin and kmax ) =⇒ E(ω) ≈ ωp, for p < 0 p =⇒ E(ω) ≈ ω−p(ω/v0kmax ) ≈ ω−0, for p > 0 –Why? Because in 2D or 3D, v~0 and ~k need not be parallel, so every mode k contributes to arbitrarily low ω in E(ω). • sweeping done right: p (zero mean velocity v0 = 0) and spectra E(k) ∝ k (as above) =⇒ E(ω) ∼ ωp + Ω2ωp−2 + O(ωp−4), for p < 0, as an asymptotic series in the limit ω → ∞, and where

0 0 Ω depends on vrms , h~v (t) · ~v (t)i, kmin and kmax , and note that Ω could be quite large.

Andrew J. Morten LOM 2013, Ann Arbor, MI Summary

• We derived and gave an interpretation for spectral transfers in the wavenumber-frequency domain. • We do see a transfer of energy to lower frequencies due to nonlinearity in some parts of the “ocean” (as measured by AVISO or modelled by HYCOM or a two-layer QG model). • In the case of 2D turbulence, the direction of the transfer is largely determined by the eﬀects of sweeping. • A more rigorous analysis of a time-dependent sweeping is needed to understand slopes of frequency spectra, especially when the cascade is narrow in wavenumber.

Andrew J. Morten LOM 2013, Ann Arbor, MI Summary of part I

• Spectral transfers T and ﬂuxes Π display a tendency for nonlinearities in idealized geostrophic turbulence models to drive energy into low frequencies as well as low wavenumbers.

• Low wavenumber energy associated with low frequencies and vice versa, but not in a simple way.

• Realistic OGCM’s also display a general tendency for nonlinearities to drive energy into lower frequencies, though not as simply or consistently as QG in turbulence models. AVISO gridded altimeter data does only in some of the regions examined; possibly because it is a highly ﬁltered product.

Andrew J. Morten LOM 2013, Ann Arbor, MI