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Christopher L. Wolfe
and
Roger M. Samelson

Abstract

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.

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Christopher L. Wolfe
and
Roger M. Samelson

Abstract

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.

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Roger M. Samelson
and
Eli Tziperman

Abstract

The local predictability of the El Niño–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 Niño conditions. This barrier arises because the growth mechanism for disturbances to the cycle is nearly the same as the growth mechanism for the El Niño 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 Niño conditions is likely a robust result, independent of the particular model.

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Roger M. Samelson
and
J. S. Allen

Abstract

Observations of oppositely directed, monthly mean alongshore currents and wind stress over the continental margin off the Pacific coast of North America motivate the theoretical examination of mean flow generation by topographic lee-wave drag. We formulate a barotropic model for wind-forced shelf-slope flow over variable topography. Our central objective is an analytical expression for mean flow generation in a simple case. We specify a linear cross-shelf slope with sinusoidal alongshore variations and use the approximation of Hart, which yields a system with only parametric cross-shelf dependence when the alongshore scales are short compared to the cross-shelf scales. The inviscid unforced equations have two constants of the motion and reduce to a quartic Hamiltonian system similar to that of Duffing's equation. For weak new-resonant time-periodic forcing, we use the method of averaging to obtain evolution equations for the amplitudes of small oscillations. All steady solutions of the averaged equations, which correspond to steadily oscillating small amplitude currents in the model, have mean current in the direction of the observed currents (poleward on an eastern boundary). Multiple equilibria occur. Mean current generation is most efficient for low frequencies, short wavelength topographic variations, and comparable alongshore and cross-shelf topographic slopes. The mean Lagrangian flow is along isobaths. Numerical solutions of the model equations compare well with the averaging analysis predictions. For certain parameter ranges, all steady solutions of the averaged equations are linearly unstable. In these ranges, numerical solutions of the averaged equations yield limit cycles, period doubling sequences, and chaotic behavior, suggesting that the response of slope flow to atmospheric forcing may be irregular.

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Eric D. Skyllingstad
and
Roger M. Samelson

Abstract

A simple, isolated front is modeled using a turbulence resolving, large-eddy simulation (LES) to examine the generation of instabilities and inertial oscillations by surface fluxes. Both surface cooling and surface wind stress are considered. Coherent roll instabilities with 200–300-m horizontal scale form rapidly within the front after the onset of surface forcing. With weak surface cooling and no wind, the roll axis aligns with the front, yielding results that are equivalent to previous constant gradient symmetric instability cases. After ~1 day, the symmetric modes transform into baroclinic mixed modes with an off-axis orientation. Traditional baroclinic instability develops by day 2 and thereafter dominates the overall circulation. Addition of destabilizing wind forcing produces a similar behavior, but with off-axis symmetric-Ekman shear modes at the onset of instability. In all cases, imbalance of the geostrophic shear by vertical mixing leads to an inertial oscillation in the frontal currents. Analysis of the energy budget indicates an exchange between kinetic energy linked to the inertial currents and potential energy associated with restratification as the front oscillates in response to the vertically sheared inertial current. Inertial kinetic energy decreases from enhanced mixed layer turbulence dissipation and vertical propagation of inertial wave energy into the pycnocline.

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Siang Peng Oh
,
Joseph Pedlosky
, and
Roger Samelson

Abstract

The linear and finite-amplitude dissipative dynamics of unstable, zonally localized baroclinic disturbances is investigated in cases where the supercriticality varies in the zonal direction. The zonal confinement occurs due to O(1) variations of the frictional influence on the current's instability. A two-layer f-plane model is used. No meridional shear is present in the basic shear flow.

When the basic current is equal and opposite in the two layers, two zonally localized modes with the same growth rate and opposite symmetries exist for all unstable parameter values. Thus, an infinite family of unstable modes formed from an arbitrary linear combination of these two modes exists. This degeneracy persists in finite amplitude. Hence, the phase of individual crests in the disturbance is a function of initial conditions even for dissipative localized instabilities.

The presence of a mean barotropic flow reduces the growth rates of the localized disturbances and expunges the symmetry properties of the mode and the resulting degeneracy. The disturbance becomes time dependent due to phase translation of crests. Localized modes exist even when the flow in both layers is in the same direction. In finite amplitude there is a weak vacillation in energy level.

A discussion of the appropriate boundary condition for the localized modes suggests that the total geostrophic perturbation streamfunction should vanish on the flow boundaries.

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Roger M. Samelson
and
Roland A. de Szoeke

Abstract

We formulate analytically and solve numerically a semigeostrophic model for wind-driven thermocline upwelling at a coastal boundary. The model has a variable-density entraining mixed layer and two homogeneous interior layers. All variables are uniform alongshore. The wind stress and surface heating are constant. The system is started from rest, with constant layer depths and mixed layer density. A modified Ekman balance is prescribed far offshore, and the normal-to-shore velocity field responds on the scales of the effective local internal deformation radii, which themselves adjust in response to changes in layer depths, interior geostrophic vorticity, and mixed layer density. Sustained upwelling results in a steplike horizontal profile of mixed layer density, as the layer interfaces “surface” and are advected offshore. The upwelled fronts have width O(u */f), as in the two-layer model of de Szoeke and Richman (1984). For fixed initial layer depths, the interior response and the horizontal separation of the upwelled fronts scale with the initial internal deformation radii. Around the fronts, surface layer divergence occurs that is equal in magnitude to the divergence in the upwelling zone adjacent to the coast, but its depth penetration is inhibited by the stratification.

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Roland A. de Szoeke
and
Roger M. Samelson

Abstract

The hydrostatic equations of motion for ocean circulation, written in terms of pressure as the vertical coordinate, and without making the Boussinesq approximation in the continuity equation, correspond very closely with the hydrostatic Boussinesq equations written in terms of depth as the vertical coordinate. Two mathematical equivalences between these non-Boussinesq and Boussinesq equation sets are demonstrated: first, for motions over a level bottom; second, for general motions with a rigid lid. A third non-Boussinesq equation set, for general motions with a free surface, is derived and is shown to possess a similar duality with the Boussinesq set after making due allowance for exchange of the roles of bottom pressure and sea surface height in the boundary conditions, a reversal of the direction of integration of the hydrostatic equation, and substitution of specific volume for density in the hydrostatic equation. The crucial simplification in these equations of motion comes from the hydrostatic approximation, not the Boussinesq approximation. A practical consequence is that numerical ocean circulation models that are based on the Boussinesq equations can, with very minimal rearrangement and reinterpretation, be made free of the strictures of the Boussinesq approximation, especially the ones that follow from its neglect of density dilatation in the conservation of mass.

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Natalie Perlin
,
Eric D. Skyllingstad
, and
Roger M. Samelson

Abstract

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.

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Eric D. Skyllingstad
,
Roger M. Samelson
,
Larry Mahrt
, and
Phil Barbour

Abstract

Numerical simulations of boundary layer evolution in offshore flow of warm air over cool water are conducted and compared with aircraft observations of mean and turbulent fields made at Duck, North Carolina. Two models are used: a two-dimensional, high-resolution mesoscale model with a turbulent kinetic energy closure scheme, and a three-dimensional large-eddy simulation (LES) model that explicitly resolves the largest turbulent scales. Both models simulate general aspects of the decoupling of the weakly convective boundary layer from the surface, as it is advected offshore, and the formation of an internal boundary layer over the cool water. Two sets of experiments are performed, which indicate that complexities in upstream surface conditions play an important role in controlling the observed structure. The first (land–sea) experiments examine the transition from a rough surface having the same temperature as the ambient lower atmosphere, to a smooth ocean surface that is 5°C cooler. In the second (barrier island) experiment, a 4-km strip along the coastline having surface temperature 5°C warmer than the ambient atmosphere is introduced, to represent a narrow, heated barrier island present at the Duck site. In the land–sea case, it is found that the mesoscale model overpredicts turbulent intensity in the upper half of the boundary layer, forcing a deeper boundary layer. Both the mesoscale and LES models produce only a small change in the boundary layer shear and tend to decrease the momentum flux near the surface much more rapidly than the observations. Results from the barrier-island case are more in line with the observed momentum and turbulence structure, but still have a reduced momentum flux in the lower boundary layer in comparison with the observations. The authors find that turbulence in the LES model generated by convection over the heated land surface is stronger than in the mesoscale model, and tends to persist offshore for greater distances because of greater shear in the upper boundary layer winds. Analysis of the mesoscale model results suggests that better estimation of the mixing length could improve the turbulence closure in regions where the surface fluxes are changing rapidly.

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