Glenn R. Ierley

Spurt phenomena of the Johnson-Segalman fluid and related models

Abstract We examine the behavior of viscoelastic fluid models which exhibit local extrema of the steady shear stress. For the Johnson-Segalman and Giesekus models, a variety of steady singular solutions with jumps in shear rate are constructed and their stability to one dimensional disturbances analyzed. It is found that flow-rate versus imposed stress curves in slit-die flow fit experimental observation of the “spurt” phenomenon with some precision. The flow curves involve linearly stable singular solutions, but some assumptions on the dynamics of the spurt process are required. These assumptions are tested by a semi-implicit finite element solution technique which allows solutions to be efficiently integrated over the very long time-scale involved. The Johnson-Segalman model with added Newtonian viscosity is used in the calculations. It is found that the assumptions required to model spurf are satisfied by the dynamic model. The dynamic model also displays a characteristic “latency time” before the spurt ensues and a characteristic “shape memory” hysteresis in load/unload cycles. These as well as other features of the computed solutions should be observable experimentally. We conclude that constitutive equations with shear stress extrema are not necessarily flawed, that their predicted behavior may appear to be arrested “wall slip”, and that such behavior may actually have been observed already.

Symmetry-Breaking Multiple Equilibria in Quasigeostrophic, Wind-Driven Flows

Abstract The classical Munk problem of barotropic flow driven by an antisymmetric wind stress exhibits multiple steady solutions in the range of moderate to high forcing and moderate to low dissipation. Everywhere in the parameter space a perfectly antisymmetric solution exists in which the strength of the cyclonic gyre is equal and opposite to that of the anticyclonic gyre. This kind of solution has been well documented in the literature. In a subset of the parameter a pair of nonsymmetric stationary solutions coexists with the antisymmetric solution. For one member of the pair the amplitude of the cyclonic circulation exceeds that of the anticyclonic flow. The other member of the pair is obtained from the quasigeostrophic symmetry y→&minusy and ψ→−ψ. As a result, the point at which the western boundary current separates from the coast can be either south or north of the latitude at which the antisymmetric Ekman pumping changes sign. This is the first oceanogrphic example of spontaneous breaking of the q...

A Model of the Inertial Recirculation Driven by Potential Vorticity Anomalies

Abstract Some essential features of a recirculating inertial gyre (the “recirculation”) can be analyzed with a very simple, analytically tractable model. In wind-driven eddy-resolving general circulation models the recirculation appears as a strong sub-basin-scale inertial flow with homogeneous potential vorticity. The constant value of potential vorticity decreases with increasing forcing/dissipation ratio while the size and the strength of the recirculating gyre increases. In the subtropical gyre the recirculating gyre might be driven by anomalous values of low potential vorticity carried northward by the western boundary current. We have modeled this process using a barotropic model and prescribing the values of potential vorticity at the edge of the gyre. Our model gyre is contained in a rectangular box in an attempt to simplify the geometry as much as possible and to isolate the processes occurring in the recirculating region. With weak diffusion the prescribed boundary forcing induces a flow with co...

Eigenanalysis of the two-dimensional wind-driven ocean circulation problem

A barotropic model of the wind-driven circulation in the subtropical region of the ocean is considered. A no-slip condition is specified at the coasts and slip at the fluid boundaries. Solutions are governed by two parameters: inertial boundary-layer width; and viscous boundary-layer width. Numerical computations indicate the existence of a wedge-shaped region in this two-dimensional parameter space, where three steady solutions coexist. The structure of the steady solution can be of three types: boundary-layer, recirculation and basin-filling-gyre. Compared to the case with slip conditions (Ierley and Sheremet, 1995) in the no-slip case the wedge-shaped region is displaced to higher Reynolds numbers. Linear stability analysis of solutions reveals several classes of perturbations: basin modes of Rossby waves, modes associated with the recirculation gyre, wall-trapped modes and a resonant mode. For a standard subtropical gyre wind forcing, as the Reynolds number increases, the wall-trapped mode is the first one destabilized. The resonant mode associated with disturbances on the southern side of the recirculation gyre is amplified only at larger Reynolds number, nonetheless this mode ultimately provides a stronger coupling between the mean circulation and Rossby basin modes than do the wall-trapped modes.

Multiple solutions and advection-dominated flows in the wind-driven circulation. Part I: Slip

We consider steady solutions of the barotropic quasigeostrophic vorticity equation for a single subtropical gyre with dissipation in the form of lateral friction. Solutions are governed by two parameters : inertial boundary-layer width ; and viscous boundary-layer width. Numerical computations for slip conditions indicate a wedge-shaped region in this two-dimensional parameter space, where three solutions coexist. One of these is a viscous solution with weak recirculation ; one a solution of intermediate recirculation ; and one a strongly nonlinear recirculation gyre. Parametric scalings based on elementary solutions are numerically corroborated as the first and third of these solutions are continued away from the vicinity of the wedge. The multiplicity of solutions is anticipated by a severely truncated Fourier modal representation paralleling Veronis (1963). The Veronis work was originally applied to predict the possibility of multiple solutions in Stommels (1948) bottom friction model of the circulation. Paradoxically, it appears the solutions are, in that case, unique.

On the convected linear stability of a viscoelastic Oldroyd B fluid heated from below

Abstract We investigate the linear stability problem of convection by the general Oldroyd B fluid and its Maxwell limit in the presence of rigid or free boundaries and fixed temperature or fixed flux. Comparison with recent results by Rosenblat [9] for the analytically accessible case of free boundary conditions shows a qualitative similarity in the shape of the neutral stability curves. But while Newtonian and Jeffreys (general Oldroyd B) fluids are sharply stabilized by the presence of rigid boundaries, the Maxwell fluid is largely unaffected at even moderately large Prandtl number. The reasons for this are discussed. Also, a discrepancy between the earlier works by Vest and Arpaci [3], and Sokolov and Tanner [4], which treat the case of a Maxwell fluid, is found to be due to algebraic error, and not multivaluedness of the stress-strain rate relation as earlier suggested by Eltayeb [6].

A modal α2-dynamo in the limit of asymptotically small viscosity

Abstract A spherical α2-dynamo is presented as an expansion in the free decay modes of the magnetic field. In the limit of vanishing viscosity the momentum equation yields various asymptotic expansions for the flow, depending on the precise form of the dissipation and boundary conditions applied. A new form for the dissipation is introduced that greatly simplifies this asymptotic expansion. When these expansions are substituted back into the induction equation, a set of modal amplitude equations is derived, and solved for various distributions of the α-effect. For all choices of α the solutions approach the Taylor state, but the manner in which this occurs can vary, as previously found by Soward and Jones (1983). Furthermore, as hypothesized by Malkus and Proctor (1975), but not previously demonstrated, the post-Taylor equilibration is indeed independent of the viscosity in the asymptotic limit, and depending on the choice of a may be either steady-state or oscillatory.

Viscous Instabilities in the Western Boundary Layer

Abstract The stability of the western boundary layer is studied by idealizing it as a parallel flow and solving the Orr–Sommerfeld equation, generalized to include the gradient of planetary vorticity. The critical Reynolds number, at which the idealized flow first becomes unstable, is found to be between 20 and 100 depending on the details of the profile. The modes themselves are trapped within the boundary jet because their phase speeds exceed that of the fastest free Rossby wave with the same meridional wavenumber. However, in the important case of a jet with a broad exponential decay, corresponding to a highly inertial flow, we find that the phase speed of the critical mode exceeds that of a free Rossby wave by a very small amount. Consequently, the trapped mode has a very slowly decaying oscillatory tail and so is much wider than the basic state that supports it. The Reynolds stresses in the tail region induce a mean Eulerian flow opposite in direction to the basic state jet. However, Stokes drift is ...

A generalized power-law detection algorithm for humpback whale vocalizations

Conventional detection of humpback vocalizations is often based on frequency summation of band-limited spectrograms under the assumption that energy (square of the Fourier amplitude) is the appropriate metric. Power-law detectors allow for a higher power of the Fourier amplitude, appropriate when the signal occupies a limited but unknown subset of these frequencies. Shipping noise is non-stationary and colored and problematic for many marine mammal detection algorithms. Modifications to the standard power-law form are introduced to minimize the effects of this noise. These same modifications also allow for a fixed detection threshold, applicable to broadly varying ocean acoustic environments. The detection algorithm is general enough to detect all types of humpback vocalizations. Tests presented in this paper show this algorithm matches human detection performance with an acceptably small probability of false alarms (P(FA) < 6%) for even the noisiest environments. The detector outperforms energy detection techniques, providing a probability of detection P(D) = 95% for P(FA) < 5% for three acoustic deployments, compared to P(FA) > 40% for two energy-based techniques. The generalized power-law detector also can be used for basic parameter estimation and can be adapted for other types of transient sounds.

Infinite-Prandtl-number convection. Part 2. A singular limit of upper bound theory

An upper bound on the heat flux for infinite-Prandtl-number convection between two parallel plates is determined for the cases of no-slip and free-slip boundary conditions. For no-slip the large-Rayleigh-number (Ra) scaling for the Nusselt number is consistent with Nu <c Ra 1/3 , as predicted by Chan (1971). However, his commonly accepted picture of an infinite hierarchy of multiple boundary layer solutions smoothly approaching this scaling is incorrect. Instead, we find a novel terminating sequence in which the optimal asymptotic scaling is achieved with a three-boundary-layer solution. In the case of free-slip, we find an asymptotic scaling of Nu <c Ra 5/12 , corroborating the conservative estimate obtained in Plasting & Ierley (2005). Here the infinite hierarchy of multiple-boundary-layer solutions obtains, albeit with anomalous features not previously encountered. Thus for neither boundary condition does the optimal solution conform to the well-established models of finite-Prandtl-number convection (Busse 1969 b), plane Couette flow, and plane or circular Poiseuille flow (Busse 1970). We reconcile these findings with a suitable continuation from no-slip to free-slip, discovering that the key distinction – finite versus geometric saturation – is entirely determined by the singularity, or not, of the initial, single-boundary-layer, solution. It is proposed that this selection principle applies to all upper bound problems.