Lyapunov, Floquet, and singular vectors for baroclinic waves

The dynamics of the growth of linear disturbances to a chaotic basic state is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave-mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal flow correction. The leading Lyapunov vector is nearly parallel to the leading Floquet vector ø1 of the lowest-order unstable periodic orbit over most of the attractor. Departures of the Lyapunov vector from this orientation are primarily rotations of the vector in an approximate tangent plane to the large-scale attractor structure. Exponential growth and decay rates of the Lyapunov vector during individual Poincaré section returns are an order of magnitude larger than the Lyapunov exponent λ ≈ 0.016. Relatively large deviations of the Lyapunov vector from parallel to ø1 are generally associated with relatively large transient decays. The transient growth and decay of the Lyapunov vector is well described by the transient growth and decay of the leading Floquet vectors of the set of unstable periodic orbits associated with the attractor. Each of these vectors is also nearly parallel to ø1. The dynamical splitting of the complete sets of Floquet vectors for the higher-order cycles follows the previous results on the lowest-order cycle, with the vectors divided into wavedynamical and decaying zonal flow modes. Singular vectors and singular values also generally follow this split. The primary difference between the leading Lyapunov and singular vectors is the contribution of decaying, inviscidly-damped wave-dynamical structures to the singular vectors

Note on a baroclinic analogue of vorticity defects in shear

An approach developed recently to study the dynamics of vorticity defects in homogeneous shear flow extends naturally to the case of baroclinic, quasi-geostrophic flow. It is shown that an inviscid geostrophic flow with uniform vertical shear may be destabilized by introducing a `potential vorticity defect', an arbitrarily small but sufficiently sharp and horizontally uniform change in stratification or vertical shear. The linear baroclinic problem is nearly identical to the linear homogeneous problem, with differences arising only from the boundary conditions. The nonlinear baroclinic problem differs substantially from the nonlinear homogeneous problem, as the leading-order baroclinic nonlinearity is the Jacobian of the `inner' streamfunction and potential vorticity in the horizontal plane aligned with the defect. An example of the linear instability is described

The effects of uncorrelated measurement noise on SWOT estimates of sea-surface height, velocity and vorticity

Author Posting. © American Meteorological Society, 2022. This article is posted here by permission of American Meteorological Society for personal use, not for redistribution. The definitive version was published in Journal of the Atmospheric and Oceanic Technology 39(7), (2022): 1053–1083, https://doi.org/10.1175/jtech-d-21-0167.1.The Ka-band Radar Interferometer (KaRIn) on the Surface Water and Ocean Topography (SWOT) satellite will revolutionize satellite altimetry by measuring sea surface height (SSH) with unprecedented accuracy and resolution across two 50-km swaths separated by a 20-km gap. The original plan to provide an SSH product with a footprint diameter of 1 km has changed to providing two SSH data products with footprint diameters of 0.5 and 2 km. The swath-averaged standard deviations and wavenumber spectra of the uncorrelated measurement errors for these footprints are derived from the SWOT science requirements that are expressed in terms of the wavenumber spectrum of SSH after smoothing with a filter cutoff wavelength of 15 km. The availability of two-dimensional fields of SSH within the measurement swaths will provide the first spaceborne estimates of instantaneous surface velocity and vorticity through the geostrophic equations. The swath-averaged standard deviations of the noise in estimates of velocity and vorticity derived by propagation of the uncorrelated SSH measurement noise through the finite difference approximations of the derivatives are shown to be too large for the SWOT data products to be used directly in most applications, even for the coarsest footprint diameter of 2 km. It is shown from wavenumber spectra and maps constructed from simulated SWOT data that additional smoothing will be required for most applications of SWOT estimates of velocity and vorticity. Equations are presented for the swath-averaged standard deviations and wavenumber spectra of residual noise in SSH and geostrophically computed velocity and vorticity after isotropic two-dimensional smoothing for any user-defined smoother and filter cutoff wavelength of the smoothing.This research was supported by NASA Grant NNX16AH76G

Instability of the Chaotic ENSO: The Growth-Phase Predictability Barrier

The local predictability of the El Nin˜o–Southern Oscillation (ENSO) is examined by the analysis of the evolution of small disturbances to an unstable 4.3-yr ENSO cycle in the Cane–Zebiak model forced by perpetual July conditions. The 4.3-yr cycle represents the dominant near-recurrent behavior in this weakly chaotic regime, so analysis of this single cycle gives useful insights into the dynamics of the irregular oscillation. Growing and neutral time-dependent eigenmodes of the unstable cycle are computed. Disturbance growth analyses based on these eigenmodes, and on singular vectors computed in the unstable-neutral subspace, suggest that there is a predictability barrier associated with the growth phase of El Nin˜o conditions. This barrier arises because the growth mechanism for disturbances to the cycle is nearly the same as the growth mechanism for the El Nin˜o conditions themselves. The local amplification of disturbances during the growth phase is several times greater than the eigenmode amplification associated with time-dependent (Floquet) normal-mode instability of the cycle. It is suggested that the existence of an ENSO predictability barrier tied to the growth phase of El Nin˜o conditions is likely a robust result, independent of the particular model

Towed thermister chain observations of fronts in the subtropical North Pacific

A thermistor chain was towed 1400 km through the eastern North Pacific subtropical frontal zone in January 1980. The observations resolve surface layer temperature features with horizontal wavelengths of 0.2-200 km and vertical scales of 10-70 m. The dominant features, which have horizontal wavelengths of 10-100 km, amplitudes of 0.2°-1.0°C, and random orientation, likely arise from baroclinic instability. Associated with them is a plateau below 0.1 cpkm in the horizontal temperature gradient spectrum. Strong temperature fronts O(1°-2°C/3-10 km) are observed near 33°N, 31°N, and 27°N. Temperature variability is partially density compensated by salinity, with the fraction of compensation increasing northward. There is evidence of vertical mixing during high winds. Temperature at 15-m depth is roughly normally distributed around the climatological surface mean, with a standard deviation of approximately 0.5°C. The standard deviation would correspond to an adiabatic meridional displacement of 80-100 km in the mean gradient. Horizontal temperature gradient at 15-m depth has maximum values in excess of 0.25°C/100 m and kurtosis near 80. In the band 0.10-1 cpkm, the 15-m gradient spectrum is inversely proportional to wave number, consistent with predictions from geostrophic turbulence theory, while the spectrum at 70-m depth has additional variance that is consistent with Garrett-Munk internal wave displacements.Keywords: upper ocean processes, eddies and mesoscale processes, Fronts and jets, Pacific OceanKeywords: upper ocean processes, eddies and mesoscale processes, Fronts and jets, Pacific Ocea

Satellite observations of mesoscale eddy-induced Ekman pumping

Author Posting. © American Meteorological Society, 2015. This article is posted here by permission of American Meteorological Society for personal use, not for redistribution. The definitive version was published in Journal of Physical Oceanography 45 (2015): 104–132, doi:10.1175/JPO-D-14-0032.1.Three mechanisms for self-induced Ekman pumping in the interiors of mesoscale ocean eddies are investigated. The first arises from the surface stress that occurs because of differences between surface wind and ocean velocities, resulting in Ekman upwelling and downwelling in the cores of anticyclones and cyclones, respectively. The second mechanism arises from the interaction of the surface stress with the surface current vorticity gradient, resulting in dipoles of Ekman upwelling and downwelling. The third mechanism arises from eddy-induced spatial variability of sea surface temperature (SST), which generates a curl of the stress and therefore Ekman pumping in regions of crosswind SST gradients. The spatial structures and relative magnitudes of the three contributions to eddy-induced Ekman pumping are investigated by collocating satellite-based measurements of SST, geostrophic velocity, and surface winds to the interiors of eddies identified from their sea surface height signatures. On average, eddy-induced Ekman pumping velocities approach O(10) cm day−1. SST-induced Ekman pumping is usually secondary to the two current-induced mechanisms for Ekman pumping. Notable exceptions are the midlatitude extensions of western boundary currents and the Antarctic Circumpolar Current, where SST gradients are strong and all three mechanisms for eddy-induced Ekman pumping are comparable in magnitude. Because the polarity of current-induced curl of the surface stress opposes that of the eddy, the associated Ekman pumping attenuates the eddies. The decay time scale of this attenuation is proportional to the vertical scale of the eddy and inversely proportional to the wind speed. For typical values of these parameters, the decay time scale is about 1.3 yr.This work was funded by NASA Grants NNX08AI80G, NNX08AR37G, NNX13AD78G, NNX10AE91G, NNX13AE47G, and NNX10AO98G.2015-07-0

Normal-Mode Analysis of a Baroclinic Wave-Mean Oscillation

The stability of a time-periodic baroclinic wave-mean oscillation in a high-dimensional two-layer quasigeostrophic spectral model is examined by computing a full set of time-dependent normal modes (Floquet vectors) for the oscillation. The model has 72 × 62 horizontal resolution and there are 8928 Floquet vectors in the complete set. The Floquet vectors fall into two classes that have direct physical interpretations: wave-dynamical (WD) modes and damped-advective (DA) modes. The WD modes (which include two neutral modes related to continuous symmetries of the underlying system) have large scales and can efficiently exchange energy and vorticity with the basic flow; thus, the dynamics of the WD modes reflects the dynamics of the wave-mean oscillation. These modes are analogous to the normal modes of steady parallel flow. On the other hand, the DA modes have fine scales and dynamics that reduce, to first order, to damped advection of the potential vorticity by the basic flow. While individual WD modes have immediate physical interpretations as discrete normal modes, the DA modes are best viewed, in sum, as a generalized solution to the damped advection problem. The asymptotic stability of the time-periodic basic flow is determined by a small number of discrete WD modes and, thus, the number of independent initial disturbances, which may destabilize the basic flow, is likewise small. Comparison of the Floquet exponent spectrum of the wave-mean oscillation to the Lyapunov exponent spectrum of a nearby aperiodic trajectory suggests that this result will still be obtained when the restriction to time periodicity is relaxed

Singular Vectors and Time-Dependent Normal Modes of a Baroclinic Wave-Mean Oscillation

Linear disturbance growth is studied in a quasigeostrophic baroclinic channel model with several thousand degrees of freedom. Disturbances to an unstable, nonlinear wave-mean oscillation are analyzed, allowing the comparison of singular vectors and time-dependent normal modes (Floquet vectors). Singular vectors characterize the transient growth of linear disturbances in a specified inner product over a specified time interval and, as such, they complement and are related to Lyapunov vectors, which characterize the asymptotic growth of linear disturbances. The relationship between singular vectors and Floquet vectors (the analog of Lyapunov vectors for time-periodic systems) is analyzed in the context of a nonlinear baroclinic wave-mean oscillation. It is found that the singular vectors divide into two dynamical classes that are related to those of the Floquet vectors. Singular vectors in the “wave dynamical” class are found to asymptotically approach constant linear combinations of Floquet vectors. The most rapidly decaying singular vectors project strongly onto the most rapidly decaying Floquet vectors. In contrast, the leading singular vectors project strongly onto the leading adjoint Floquet vectors. Examination of trajectories that are near the basic cycle show that the leading Floquet vectors are geometrically tangent to the local attractor while the leading initial singular vectors point off the local attractor. A method for recovering the leading Floquet vectors from a small number of singular vectors is additionally demonstrated

Coastal Atmospheric Circulation around an Idealized Cape during Wind-Driven Upwelling Studied from a Coupled Ocean–Atmosphere Model

The study analyzes atmospheric circulation around an idealized coastal cape during summertime upwelling-favorable wind conditions simulated by a mesoscale coupled ocean–atmosphere model. The domain resembles an eastern ocean boundary with a single cape protruding into the ocean in the center of a coastline. The model predicts the formation of an orographic wind intensification area on the lee side of the cape, extending a few hundred kilometers downstream and seaward. Imposed initial conditions do not contain a low-level temperature inversion, which nevertheless forms on the lee side of the cape during the simulation, and which is accompanied by high Froude numbers diagnosed in that area, suggesting the presence of the supercritical flow. Formation of such an inversion is likely caused by average easterly winds resulting on the lee side that bring warm air masses originating over land, as well as by air warming during adiabatic descent on the lee side of the topographic obstacle. Mountain leeside dynamics modulated by differential diurnal heating is thus suggested to dominate the wind regime in the studied case. The location of this wind feature and its strong diurnal variations correlate well with the development and evolution of the localized lee side trough over the coastal ocean. The vertical extent of the leeside trough is limited by the subsidence inversion aloft. Diurnal modulations of the ocean sea surface temperatures (SSTs) and surface depth-averaged ocean current on the lee side of the cape are found to strongly correlate with wind stress variations over the same area. Wind-driven coastal upwelling develops during the simulation and extends offshore about 50 km upwind of the cape. It widens twice as much on the lee side of the cape, where the coldest nearshore SSTs are found. The average wind stress–SST coupling in the 100-km coastal zone is strong for the region upwind of the cape, but is notably weaker for the downwind region, estimated from the 10-day-average fields. The study findings demonstrate that orographic and diurnal modulations of the near-surface atmospheric flow on the lee side of the cape notably affect the air–sea coupling on various temporal scales: weaker wind stress–SST coupling results for the long-term averages, while strong correlations are found on the diurnal scale.Keywords: Wind, Coupled models, Coastal flows, UpwellingKeywords: Wind, Coupled models, Coastal flows, Upwellin