primitive equation simulation results from a 1/48th degree resolution tell us about geostrophic currents? What would high-resolution altimetry

Transcription

1 Scott 2008: Scripps 1 What would high-resolution altimetry tell us about geostrophic currents? results from a 1/48th degree resolution primitive equation simulation Robert B. Scott and Brian K. Arbic The University of Texas at Austin

2 Scott 2008: Scripps 2 A Workshop on Mesoscale and Submesoscale Oceanic Processes: Explorations with Wide-Swath Interferometry Radar Altimetry April 28-30, 2008, at Scripps Institution of Oceanography Organizers: Lee Fu, JPL Raf Ferrari, MIT

3 Scott 2008: Scripps 3 What I ll cover 1. What have we learned from existing altimetry data? and what are the limitations and challenges? Qualitative view of energy cascades, and their relation to instability processes. Need to understand effects of viscosity and spatial resolution. Advantages of measurement redundancy results from Jason calibration phase. 2. What new dynamics can we study with an O(10) km resolution in sea surface height (SSH)? How large an SSH signal would those dynamics produce? What are the desired highest spatial resolution? 1 km? at what precision? 1 cm? Results from idealized PE simulation at run at 2km resolution.

4 Scott 2008: Scripps 4 Altimeters have produced the first synoptic view of the ocean sea surface height: Last week of December

5 Scott 2008: Scripps 5 Geostrophic turbulence of the surface mesoscale flow Kinetic energy wavelength spectra (e.g. LeTraon et al., 1994; Stammer, 1997; Scott and Wang, 2005; Eden, 2007) Kinetic energy cascades

6 Scott 2008: Scripps 6 What is the spectral KE flux? Partition flow field into the large scale flow KE (k<k) (with length scales larger than 1/K, where K = 2π/λ is the horizontal wavenumber, λ is the wavelength), and small scale flow, KE (k>k). Nonlinear advective terms transfer kinetic energy from KE (k<k) to KE (k>k) at a rate Π(K), the spectral KE flux.

7 Scott 2008: Scripps 7 Calculation of Π(K) For geostrophic (quasi 2D) flow: Π(K) = ψ > K J(ψ, 2 ψ) where denotes time and space average, ψ > K is the high-pass filtered streamfunction for the geostrophic flow. The cut-off for the filter is total wavenumber K, and is varied over the range of K of interest, see (Scott and Wang, 2005). Area average is over rectangular box 64x64 grid points on a Mercator grid. ψ estimated from satellite altimetry

8 Scott 2008: Scripps 8 Spectral transfer, T (kx, ky) Relation to Bo s talk T (k) = k= k 2 x +k2 y T (kx, ky)d kx d ky Πk = T (k) dk k

9 Scott 2008: Scripps Figure 1: Boxes used to calculate spectral flux.

10 Scott 2008: Scripps 10 Π(K) vs. K max(π(k)) 0 min(π(k)) upscale downscale Flux divergence = max(π(k))-min(π(k)) = C pe K (m-1 x 10-5) Inf Wavelength (km) Figure 2: Π(K) in the central South Pacific (Scott and Wang, 2005)

11 Scott 2008: Scripps Wavenumber in rad/m 60 o S 50 o S 35 o S 21 o S Figure 3: Normalized Π(K) in the Southern Hemisphere

12 Scott 2008: Scripps o N o N o N Wavenumber in rad/m Figure 4: Normalized Π(K) in the Northern Hemisphere

13 Scott 2008: Scripps S 60S 57S 54S 50S 47S 43S S 35S 30S 26S 21S 16S Wavenumber in rad/m Figure 5: Zonally averaged, normalized Π(K) in the S. Hemisphere

14 Scott 2008: Scripps 14 Zonal Means energy flux energy density deformation radius Inf Wavelength (km) Figure 6: Zonal average of Πk vs k in South Pacific

15 Scott 2008: Scripps 15 Variables Parameters ψ1 U1 H1 = δ H 2 ψ2 U2 H2 Bottom drag Figure 7: 2-layer, QG, flat-bottom, f-plane model (Arbic and Flierl, 2004)

16 Scott 2008: Scripps 16 Nondimensional parameters and friction Nondimensional Laplacian viscosity, ν ν /(U1 U2)Ld, ν = 0.1 for ν = 50m 2 /s (Brown and Owens) (also K. Polzin, submitted), U1 U2 = 1 cm/s, and Ld = 50 km. Nondimensional bottom friction, FL = RLd/(U1 U2) Stratification, δ = H1/H2

17 Figure 8: Effect of viscosity: δ = 0.2, FL = 0.4 k in rad/l d Spectral flux, normalized by maximum value ν=0 ν=0.01 ν=0.1 Scott 2008: Scripps 17

18 Scott 2008: Scripps 18 What we ve learned: Summary Apparent universal shape to Πk. Opens small window to ocean mechanical energy pathways to dissipation: Source of KE from baroclinic instability, Most energy goes to larger scale (bottom drag important?). In-situ viscosity and QG turbulence model forward flux seen by altimetry ν is only nondimensional parameter that led to shift of Πk to lower k. Eddy viscosity of order 0.1 or larger dissipates substantial fraction of energy in model.

19 Scott 2008: Scripps 19 Limitations of altimeter measurements Temporal resolution Spatial resolution Accuracy

20 latitude decorrelation scale in km x dir y dir 350 merged ssh Decorrelation scales used in Aviso Scott 2008: Scripps 20

21 latitude decorrelation scales [L d ] x dir y dir Previous over zonally averaged Ld Scott 2008: Scripps 21

22 Nondimensional spectral flux (c) δ=0.2, F L =0.4, ν= Wavenumber in rad/l d 0.02 FWHM=9.73 L d FWHM=4.87 L d 0 No filter FWHM=3.24 L d (b) δ=0.2, F L =0.4, ν= (a) δ=0.2, F L =0.4, ν=0 0 Effects of spatial resolution on Πk Scott 2008: Scripps 22

23 Scott 2008: Scripps 23 Summary so far Qualitative view of energy cascades, but quantitative aspects might be strongly dependent on spatial resolution. Eddy viscosity and a priori spatial smoothing have similar effects. Related to instability processes: small scale: driving energy sink to geostrophic flow, and large scale: baroclinic instability driving spectral flux divergence.

24 Scott 2008: Scripps 24 To what extent is gridded altimeter selectively removing small scales? Ideally, would compare E(k), the wavenumber kinetic energy spectrum, of gridded data with the true spectrum. Unfortunately, the true spectrum is very difficult to measure. Advantages of measurement redundancy results from Jason calibration phase.

25 Scott 2008: Scripps 25 Problem with second order quantities In general the velocity estimates u e can be decomposed as, u e(t) = u t(t) + n(t) where u t is the true velocity and n is the noise. For sufficiently long temporal average, u e(t) 3 u t(t) 3 Unfortunately the square of the velocity anomaly, u e(t) 2 u t(t) 2 + n(t) 2 The same consideration applies to all even order quantities, including the eddy kinetic energy and its power spectrum.

26 Scott 2008: Scripps 26 Propagation of errors Decompose the cross-track velocity anomalies estimates from TP and J1 altimetry into the true velocity and the noise component, ut (x, t) = u(x, t) + nt (x, t) (1) uj(x, t) = u(x, t) + nj(x, t) (2) where u is the true velocity anomaly and nt and nj are the noise from TP or J1 respectively. The PSD of u is defined as E(k) = û(x, t)û(x, t) where û(x, t) is the discrete Fourier transform in x and denotes expected value, here taken as a time average.

27 Scott 2008: Scripps 27 As a result ET (k) = E(k) + EnT (k) + 2 R(û ˆn T ) (3) EJ(k) = E(k) + EnJ (k) + 2 R(û ˆn J). (4) (5) Key assumption the noise and signal are uncorrelated and so, R(û ˆnT ) 0. Thus we need estimates of the sum of the noise PSDs, obtained from the PSD of δu ut uj = nt nj Eδu = EnT (k) + E nj (k) 2R(ˆn J ˆnT ) (6) The cross term is unavailable, and thus a second key assumption is necessary: for a sufficiently long time average R(ˆnJ ˆnT ) 0.

28 Scott 2008: Scripps 28 Correlated noise due to various factors like aliasing from barotropic signals (including tides) and ionosphere correction are not producing serious contamination of Eδu (Scott and Chambers, 2008). Sea state bias unclear. Combining (3) through (6) and our two assumptions on the noise properties, E(k) = (ET (k) + EJ(k) Eδu(k))/2. (7) Note that the altimeters may have different noise properties, and indeed there is some evidence that J1 is noiser than TP, but (7) still holds.

29 Figure 9: Normalized, zonally averaged spectra P SDu combining Northern and Southern Hemisphere latitude bands. Zonal averaging performed before normalization. Black lines (Extratropical latitudes: 25, 32,40, 47, 55 ) and red lines (tropical: 10, 17 ). Blue line with circles is the average of the black lines. Dashed blue lines are proportional to k 3, k 5/3, and k 1/2. Wavenumber (cycles per m) x Normalized spectral kinetic energy density Scott 2008: Scripps 29

30 Wavenumber (cycles per m) x Normalized spectral kinetic energy density Scott 2008: Scripps 30

31 Figure 10: As in previous Fig. but zonal averaging performed after normalization. Wavenumber (cycles per m) x Normalized spectral kinetic energy density Scott 2008: Scripps 31

32 Figure 11: ST N(k) = P SDu/0.5P SDδu, for various latitude bands: a) 17 N, b) 17 S, c) 32 N, d) 32 S, e) 47 N, f) 47 S. Red lines are the average over all tracks. Light blue lines guide the reader to the length scale of unit signal to noise ratio. Wavenumber (cycles per m) e) f) Normalized spectral kinetic energy density 47 o N 47 o S c) d) a) 17 o N 17 o S b) 32 o N 32 o S Scott 2008: Scripps 32

33 Wavenumber (cycles per m) e) f) Normalized spectral kinetic energy density 47 o N 47 o S c) d) a) 17 o N 17 o S b) 32 o N 32 o S Scott 2008: Scripps 33

34 Scott 2008: Scripps 34 Results from ADCP lines u, v from ship board ADCP, at 8m depth intervals between 16m and 104m depth. every 3km along ship track. four meridional transects in the North Pacific.

35 Figure 12: Remarkably meridional lines Longitude, in degrees 26 Latitude, in degrees North Pacific ADCP transects Scott 2008: Scripps 35

36 Figure 13: Velocity is fairly depth-independent Latitude, degrees North v, m/s u, m/s 0 16m 104m 0.5 Scott 2008: Scripps 36

37 Figure 14: E(k) is fairly depth-independent horizontal wavenumber, k, in rad/m E(k), in m 2 /s 2 /(rad/m) E(k) four transect ave Scott 2008: Scripps 37

38 Scott 2008: Scripps 38 Results from high-res MOM4 simulation Idealized basin Steady forcing: τ x = 0.1 cos((lat 30 )/20 ), relaxing T to linear function of lat Linear equation of state: σ = αθ 40 Vertical levels, and Av = m 2 /s. Horizontal resolution: run 1 is 1/24 th deg. Ah = 0.672m 2 /s, Ab = m 4 /s, Horizontal resolution: run 2 is 1/48 th deg. Ah = 0.168m 2 /s, Ab = m 4 /s 9.5 year spin-up, results from following 4 years.

39 Figure 15: Bottom topography of idealized basin Degrees East 3000 Degrees North Scott 2008: Scripps 39

40 total horizontal wavenumber, k [rad/m] 10 4 k total vel, 7.5m depth total vel, 117m depth geostrophic vel, surface k 5/3 E(k) m 2 /s 2 /(rad/m) /24 th deg. run Scott 2008: Scripps 40

41 total horizontal wavenumber, k [rad/m] 10 4 k 3 E(k) m 2 /s 2 /(rad/m) 10 2 total vel, 7.5m depth total vel, 117m depth geostrophic vel, surface k 5/ /48 th deg. run Scott 2008: Scripps 41

42 Scott 2008: Scripps 42 Physics NOT present in simulation Time variable winds: drive surface waves, inertial oscillations, other internal waves. Tides. Unresolved scales. Non-hydrostatic effects. Nonlinear equation of state, non-boussinesq.

43 total horizontal wavenumber, k [rad/m] 3 2 Pi(k) m 2 /s total vel, 7.5m depth total vel, 117m depth geostrophic vel, surface 2 x /48 th deg. run Scott 2008: Scripps 43

44 Figure 16: Geostrophic velocity E(k) in subdomain only total horizontal wavenumber, k [rad/m] 10 4 k 3 E(k) m 2 /s 2 /(rad/m) 10 2 total vel, 7.5m depth total vel, 117m depth geostrophic vel, surface k 5/ /48 th deg. run Scott 2008: Scripps 44

45 Figure 17: Geostrophic velocity Π(k) in subdomain only total horizontal wavenumber, k [rad/m] Pi(k) m 2 /s total vel, 7.5m depth total vel, 117m depth geostrophic vel, surface 1 x /48 th deg. run Scott 2008: Scripps 45

46 Scott 2008: Scripps 46 Summary Gridded, nadar altimeter data is inadequate for resolving the forward flux of KE. Small-scale processes, which lead to shallow k 5/3 spectrum, might be related to forcing, but need more work to prove this.

47 Scott 2008: Scripps 46-1 References Arbic, B. K., and G. R. Flierl (2004), Baroclinically unstable geostrophic turbulence in the limits of strong and weak bottom Ekman friction: Application to midocean eddies, J. Phys. Oceanogr., 34, Brown, E. D., and W. B. Owens (), Observations of the horizontal interactions between the internal wave field and the mesoscale flow. Eden, C. (2007), Eddy length scales in the North Atlantic Ocean, J. Geophys. Res., 112, C06,004, doi: /2006jc LeTraon, P.-Y., J. Stum, J. Dorandeu, P. Gaspar, and P. Vincent

48 Scott 2008: Scripps 46-2 Scott, R. B., and F. Wang (2005), Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry, J. Phys. Oceanogr., 35, Stammer, D. (1997), Global characteristics of ocean variability estimated from regional TOPEX/Poseidon altimeter measurements, J. Phys. Oceanogr., 27,