Limits of geostrophic dynamics at the ocean surface: Guidance for planning SWOT’s mission and products

Jim McWilliams, Jon Gula, Jeroen Molemaker UCLA

altimetry: ”the most successful ocean experiment of all time” (W. Munk, 2002)

present practice: once all the orbital, gravitational, atmospheric, surface wave, and tidal corrections are made to altimetric SSH measurements, the primary oceanic product is surface geostrophic horizontal velocity,

ug =g

fz⇥rh⌘

with the presumption that it is superimposable with wind-driven Ekman currents. Furthermore, quasi-geostrophic dynamics allows calculation of the vertical velocity wg with some assumptions about upper ocean structure. ug is non-divergent and rotational, and usually its energy transfer is “inverse” from small scales to large.

This framework has served us very well for present altimetry measurements of large-mesoscale eddies with wavelength > 100-200 km.�

With SWOT we will see much smaller scales in . Commonly this regime is called the submesoscale, but it has at least two categories for “balanced” flow:

1. large-submesoscale/small-mesoscale where geostrophic dynamics still reigns.

2. small-submesoscale where geostrophic dynamics is only part of the story.

As I will show, our present estimate for their boundary is around = 30 km, although there are continuous dynamical transitions as deceases.

in addition, of course, to inertia-gravity waves with partly overlapping scales.

⌘

��

Besides theory, the data base for this assessment is realistic simulations using multiply nested grids in the Regional Oceanic Modeling System (ROMS) in a variety of geographical environments. Today’s results come from nesting in basin-scale solutions with small-mesoscale resolution and stepping down to dx = 1.5 km, 0.5 km, and 0.15 km local subdomains:

Gulf Stream after separation from North America Sargasso Sea interiorward of the Gulf Stream California Current System Solomon Sea (strongly topographic and near equatorial)

The relevant behaviors are similar in all places, albeit with quantitative differences.

Gulf Stream surface vorticity/f and sea level snapshots

Surface Spectra vs. Geography

300 100 30 10 3 1 0.3

10−8

10−6

10−4

10−2

E kin (m

2 /s2 ]

λ (km)300 100 30 10 3 1 0.3

10−14

10−12

10−10

10−8

10−6

10−4

10−2

η2 (m

2 )

λ (km)

Azimuthally averaged spectra of uh and from grids with dx = 0.5 km with reference slope lines of kh-2 and kh-4, respectively. Dotting on curves indicates approximate model dissipation range.

The regions are Gulf Stream (blue), California Current (red), and Solomon Sea (black).

⌘

A Helmholtz decomposition of 3D incompressible flow has a horizontal velocity

uh = ur + ud ,

rh · ur = 0

z ·rh ⇥ ud = 0

with a vertical velocity w = �Z

rh · ud dz .

For each velocity component we can compute an azimuthally averaged spectrum E( ), using spatial windowing, and component ratios are the square root of the ratio of spectra, e.g.,

Vd

Vr=

✓Ed

Er

◆1/2

.

�

An analogous ratio, Va/Vg, is calculated for a geostrophic/ageostrophic decomposition, as well as for a rotational flow comparison ratio, Vg/Vr.

Rotational/Divergent Decomposition (I)

Gulf Stream Solomon Sea

Surface kinetic energy spectra showing increasing Vd/Vr as decreases, as well as flattening to the spectrum toward kh-2, either throughout the sub-

mesoscale or just in the small-submesoscale range.

�

[dx = 0.5 km]

Rotational/Divergent Decomposition (II)

300 100 30 10 3 1 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

V d/Vr

λ (km)

Gulf Stream California Current

Solomon Sea Vd/Vr increases as decreases, and it peaks near the onset of dissipation.

Values > ~ 1/4 indicate non-geostrophic dynamics.

Values < 1 indicate mostly balanced flow.

dx = 1.5 (red), 0.5 (blue), and 0.15 (black) km

�

Geostrophic/Ageostrophic Decompositionua = uh � ug

300 100 30 10 3 1 0.30.5

1

1.5

2

2.5

3

V g/Vr

λ (km)

dx = 1.5 (red), 0.5 (blue), and 0.15 (black) km

Gulf Stream

Solomon Sea

California Current

Vg/Vr increases as decreases, and it peaks near the onset of dissipation.

Values slightly > 1 indicate more general “balances” (see later).

Values >> 1 indicate misidentification of as related to balanced flows, with spurious ug = - ua.

The Solomon Sea is very non-geostrophic.

⌘

�

[dx = 0.5 km][dx = 1.5 km]

[dx = 1.5 km]

Surface Divergence/Vertical Velocity Diagnosis

Gulf Stream

Solomon Sea

California Current

The diagnosed wg misses most of the submesoscale w variance in the surface layer. Much of w is a more general “balanced” flow, but some of it is probably IG waves.

The discrepancy is largest in the Solomon Sea, for reasons to be determined.

[dx = 0.5 km]

Kinetic Energy Flux in kh SpaceT( ) = Fourier transform of kinetic energy advection integrated from zero flux at the maximum resolved kh (i.e., ).

T < 0 = inverse transfer to larger scales; T > 0 = forward transfer to smaller scales and dissipation.

red = using ur only blue = using full u

Gulf Stream

All of the small-submesoscale nested grids show a robust forward KE cascade range, with inverse cascade at larger scales, but only if ud is included in the calculation.

California CurrentSargasso Sea

�

300 100 30 10 3 1 0.3−4

−3

−2

−1

0

1

2

3

4 x 10−9

T (m

2 /s3 )

λ [km]300 100 30 10 3 1 0.3

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2 x 10−9

T (m

2 /s3 )

λ [km]300 100 30 10 3 1 0.3

−2

0

2

x 10−9

T (m

2 /s3 )

λ [km]

@tE(kh) = �@khT (kh) + . . .

The implication of these results is that SWOT will have to become more sophisticated in its diagnostic analysis of to deduce the surface velocity field in the small-submesoscale range, with < 30 km or so.

⌘

One can get some distance forward by relying on more general conceptions of surface-layer balances in the submesoscale regime: going from

GB+E = geostrophic + Ekman boundary layer (BL) currents

and proceeding in three combinable directions:

NLB = nonlinear balance (finite Rossby number advection; e.g. gradient-wind balance)

TTW = turbulent thermal wind (coupling of geostrophic and Ekman currents with the turbulence in the BL)

WEC = surface gravity wave effects on currents (and vice versa) through Stokes drift Coriolis and vortex forces and material advection (famous for Langmuir circulations)

All of these have demonstrable relevance to to submesoscale surface flows. Their appli-cation in SWOT will be greatly aided by overlapping measurements of wind stress, SST, and surface waves, where possible.

Beyond these more general diagnostic relations lie more fully prognostic ones, even up to full assimilation OGCMs.

�