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Helble, TA, Henderson EE, Ierley GR, Martin SW.  2016.  Swim track kinematics and calling behavior attributed to Bryde's whales on the Navy's Pacific Missile Range Facility. Journal of the Acoustical Society of America. 140:4170-4177.   10.1121/1.4967754   AbstractWebsite

Time difference of arrival methods for acoustically localizing multiple marine mammals have been applied to recorded data from the Navy's Pacific Missile Range Facility in order to localize and track calls attributed to Bryde's whales. Data were recorded during the months of August-October 2014, and 17 individual tracks were identified. Call characteristics were compared to other Bryde's whale vocalizations from the Pacific Ocean, and locations of the recorded signals were compared to published visual sightings of Bryde's whales in the Hawaiian archipelago. Track kinematic information, such as swim speeds, bearing information, track duration, and directivity, was recorded for the species. The intercall interval was also established for most of the tracks, providing cue rate information for this species that may be useful for future acoustic density estimate calculations.

Helble, TA, Ierley GR, D'Spain GL, Martin SW.  2015.  Automated acoustic localization and call association for vocalizing humpback whales on the Navy's Pacific Missile Range Facility. Journal of the Acoustical Society of America. 137:11-21.   10.1121/1.4904505   AbstractWebsite

Time difference of arrival (TDOA) methods for acoustically localizing multiple marine mammals have been applied to recorded data from the Navy's Pacific Missile Range Facility in order to localize and track humpback whales. Modifications to established methods were necessary in order to simultaneously track multiple animals on the range faster than real-time and in a fully automated way, while minimizing the number of incorrect localizations. The resulting algorithms were run with no human intervention at computational speeds faster than the data recording speed on over forty days of acoustic recordings from the range, spanning multiple years. Spatial localizations based on correlating sequences of units originating from within the range produce estimates having a standard deviation typically 10 m or less (due primarily to TDOA measurement errors), and a bias of 20 m or less (due primarily to sound speed mismatch). An automated method for associating units to individual whales is presented, enabling automated humpback song analyses to be performed.

Helble, TA, Ierley GR, D'Spain GL, Roch MA, Hildebrand JA.  2012.  A generalized power-law detection algorithm for humpback whale vocalizations. Journal of the Acoustical Society of America. 131:2682-2699.   10.1121/1.3685790   AbstractWebsite

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. (C) 2012 Acoustical Society of America. [DOI: 10.1121/1.3685790]

Landuyt, W, Ierley G.  2012.  Linear stability analysis of the onset of sublithospheric convection. Geophysical Journal International. 189:19-28.   10.1111/j.1365-246X.2011.05341.x   AbstractWebsite

We consider the onset of small scale convection associated with the cooling of the oceanic lithosphere. This process is investigated by examining the linear stability of a 2-D shear flow that is cooled from the top and obeys an Arrhenius rheology. The perturbation equations are derived assuming a parallel flow, local 1-D analysis. The linear stability results are able to reproduce the structure and growth rate produced by solving the full non-linear equations. In addition, linear stability analysis predicts a similar dependence of onset times on activation energy and Rayleigh number as previous studies when small plate velocities are specified. Our linear stability analysis predicts a significant increase in the onset time with increasing plate velocity. Suggestions are given for the dynamical influence of plate velocity in reducing the convective forcing and delaying onset times.

Livermore, PW, Ierley G, Jackson A.  2011.  The evolution of a magnetic field subject to Taylor's constraint using a projection operator. Geophysical Journal International. 187:690-704.   10.1111/j.1365-246X.2011.05187.x   AbstractWebsite

In the rapidly rotating, low-viscosity limit of the magnetohydrodynamic equations as relevant to the conditions in planetary cores, any generated magnetic field likely evolves while simultaneously satisfying a particular continuous family of invariants, termed Taylor's constraint. It is known that, analytically, any magnetic field will evolve subject to these constraints through the action of a time-dependent coaxially cylindrical geostrophic flow. However, severe numerical problems limit the accuracy of this procedure, leading to rapid violation of the constraints. By judicious choice of a certain truncated Galerkin representation of the magnetic field, Taylor's constraint reduces to a finite set of conditions of size O(N), significantly less than the O(N-3) degrees of freedom, where N denotes the spectral truncation in both solid angle and radius. Each constraint is homogeneous and quadratic in the magnetic field and, taken together, the constraints define the finite-dimensional Taylor manifold whose tangent plane can be evaluated. The key result of this paper is a description of a stable numerical method in which the evolution of a magnetic field in a spherical geometry is constrained to the manifold by projecting its rate of change onto the local tangent hyperplane. The tangent plane is evaluated by contracting the vector of spectral coefficients with the Taylor tensor, a large but very sparse 3-D array that we define. We demonstrate by example the numerical difficulties in finding the geostrophic flow numerically and how the projection method can correct for inaccuracies. Further, we show that, in a simplified system using projection, the normalized measure of Taylorization, tau, may be maintained smaller than O(10(-10)) (where tau = 0 is an exact Taylor state) over 1/10 of a dipole decay time, eight orders of magnitude smaller than analogous measures applied to recent low Ekman-number geodynamo models.

Jackson, A, Livermore PW, Ierley G.  2011.  On Ohmic heating in the Earth's core II: Poloidal magnetic fields obeying Taylor's constraint. Physics of the Earth and Planetary Interiors. 187:322-327.   10.1016/j.pepi.2011.06.003   AbstractWebsite

The extremely small Ekman and magnetic Rossby numbers in the Earth's core make the magnetostrophic limit an attractive approximation to the core's dynamics. This limit leads to the need for the internal magnetic field to satisfy Taylor's constraint, associated with the vanishing of the azimuthal component of Lorentz torques averaged over every cylinder coaxial with the rotation axis. A special class of three dimensional poloidal interior magnetic fields is chosen that satisfies Taylor's constraint identically on every cylinder in a spherical shell exterior to an inner core. This class of fields, which we call small-circle conservative, demonstrates existence of interior fields satisfying Taylor's constraint, regardless of the morphology of the field on the core surface. These poloidal fields are used to examine the Ohmic dissipation present in the Earth's core. To address the question of dissipation, we demand that the 3-D core fields agree with recent observations of the core field structure on the core-mantle boundary. We use these poloidal fields to show that the true lower bound on core dissipation must necessarily lie below a value that we calculate. For 2004 we find that this lower bound must lie below 10(10) W, and when nutation constraints are also considered the bound must lie below 2 x 10(10) W. These numbers are small compared to suggested values of the order of a few Tera Watts. A more restrictive bound may be forthcoming when the time-dependency of the field is considered, using a variational data assimilation technique. (C) 2011 Elsevier B.V. All rights reserved.

Livermore, PW, Ierley GR, Jackson A.  2010.  The construction of exact Taylor states. II: The influence of an inner core. Physics of the Earth and Planetary Interiors. 178:16-26.   10.1016/j.pepi.2009.07.015   AbstractWebsite

The geodynamo mechanism, responsible for sustaining Earth's magnetic field, is believed to be strongly influenced by the solid inner core through its influence on the structure of convection within the tangent cylinder. In the rapidly rotating low-viscosity regime of the geodynamo equations relevant to the Earth's core, the magnetic field must satisfy a continuum of conditions known as Taylor's constraint. Magnetic fields that satisfy this constraint, termed Taylor states, have the property that their axial magnetic torque vanishes when averaged over any geostrophic contour, cylinders of fluid coaxial with the rotational axis. In recent theoretical developments, we proved that when adopting a truncated spherical harmonic expansion, the continuous constraint in space reduced to a finite spectral set of conditions. Furthermore, an expedient choice of regular radial basis presents an under-determined problem when constructing Taylor states in a full-sphere showing the ubiquity of such solutions. A spherical-shell geometry, with a conducting inner core, complicates the formulation of Taylor's constraint due to the partitioning of the geostrophic contours into three distinct regions, ostensibly trebling the stringency of the constraint. This raises questions as to the admissible structures of such Taylor states, and their relation to those in a full-sphere. In this paper, we address these issues in two stages. First, we present a mathematical characterisation of the structure of Taylor's constraint in a spherical-shell. We then enumerate the effective number of conditions that must be satisfied by any magnetic field that is everywhere C(infinity) inside the core, and show that, assuming an equal truncation in radial and solid angle representation, the number of conditions is approximately 5/3 times that for a full-sphere Taylor state. Second, we investigate the influence of the inner core on the structure of admissible Taylor states by constructing a low-degree family of optimally smooth observationally consistent examples in both a spherical-shell and a full-sphere. We show that the introduction of an inner core into a full-sphere increases the minimum magnetic field complexity, simply by virtue of the increased potency of Taylor's constraint, a trait more pronounced in our quasi-axisymmetric models. We speculate that axisymmetric dynamo-generated exact Taylor states, particularly those generated in a spherical-shell, in general have small radial length scales that may be difficult to resolve. (C) 2009 Elsevier B.V. All rights reserved.

Livermore, PW, Ierley GR.  2010.  Quasi-L (p) norm orthogonal Galerkin expansions in sums of Jacobi polynomials. Numerical Algorithms. 54:533-569.   10.1007/s11075-009-9353-5   AbstractWebsite

In the study of differential equations on [ -aEuro parts per thousand 1,1] subject to linear homogeneous boundary conditions of finite order, it is often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight function, w(x) = (1 -aEuro parts per thousand x) (alpha) (1 + x) (beta) . However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials, whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only. That orthogonality of the entire set follows-a property we term "auto-orthogonality"aEuro"is remarkable. Additionally, these basis functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter's useful properties. Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized in an L (p(alpha)) norm given the appropriate choice of alpha = beta. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most of these results in specific cases and certain of the results in the general case. However a proof that such expansions can satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.

Livermore, PW, Ierley GR.  2009.  A new hypergeometric identity linking coefficients of a certain class of homogeneous polynomials motivated from magneto hydrodynamics. Advances in Applied Mathematics. 43:390-393.   10.1016/j.aam.2009.06.001   AbstractWebsite

Considerations of a particular limit of the magnetohydrodynamic equations, appropriate for the generation of magnetic field in planetary interiors, lead to a set of constraints involving a certain class of homogeneous polynomials. This set is significantly degenerate owing to an identity satisfied by the polynomial coefficients, which involves linear combinations of simple (2)F(1) Gauss hypergeometric functions. A generalised version of this new identity is proved by appealing to Wilf-Zeilberger theory. (C) 2009 Elsevier Inc. All rights reserved.

Livermore, PW, Ierley G, Jackson A.  2009.  The construction of exact Taylor states. I: The full sphere. Geophysical Journal International. 179:923-928.   10.1111/j.1365-246X.2009.04340.x   AbstractWebsite

P>The dynamics of the Earth's fluid core are described by the so-called magnetostrophic balance between Coriolis, pressure, buoyancy and Lorentz forces. In this regime the geomagnetic field is subject to a continuum of theoretical conditions, which together comprise Taylor's constraint, placing restrictions on its internal structure. Examples of such fields, so-called Taylor states, have proven difficult to realize except in highly restricted cases. In previous theoretical developments, we showed that it was possible to reduce this infinite class of conditions to a finite number of coupled quadratic homogeneous equations when adopting a certain regular truncated spectral expansion for the magnetic field. In this paper, we illustrate the power of these results by explicitly constructing two families of exact Taylor states in a full sphere that match the same low-degree observationally derived model of the radial field at the core-mantle boundary. We do this by prescribing a smooth purely poloidal field that fits this observational model and adding to it an expediently chosen unconstrained set of interior toroidal harmonics of azimuthal wavenumbers 0 and 1. Formulated in terms of the toroidal coefficients, the resulting system is purely linear and can be readily solved to find Taylor states. By calculating the extremal members of the two families that minimize the Ohmic dissipation, we argue on energetic ground that the toroidal field in the Earth's core is likely to be dominated by low order azimuthal modes, similar to the observed poloidal field. Finally, we comment on the extension of finding Taylor states within a general truncated spectral expansion with arbitrary poloidal and toroidal coefficients.

Livermore, PW, Ierley G, Jackson A.  2008.  The structure of Taylor's constraint in three dimensions. Proceedings of the Royal Society a-Mathematical Physical and Engineering Sciences. 464:3149-3174.   10.1098/rspa.2008.0091   AbstractWebsite

In a 1963 edition of Proc. R. Soc. A, J. B. Taylor (Taylor 1963 Proc. R. Soc. A 9, 274-283) proved a necessary condition for dynamo action in a rapidly rotating electrically conducting. uid in which viscosity and inertia are negligible. He demonstrated that the azimuthal component of the Lorentz force must have zero average over any geostrophic contour (i.e. a fluid cylinder coaxial with the rotation axis). The resulting dynamical balance, termed a Taylor state, is believed to hold in the Earth's core, hence placing constraints on the class of permissible fields in the geodynamo. Such states have proven difficult to realize, apart from highly restricted examples. In particular, it has not yet been shown how to enforce the Taylor condition exactly in a general way, seeming to require an infinite number of constraints. In this work, we derive the analytic form for the averaged azimuthal component of the Lorentz force in three dimensions after expanding the magnetic field in a truncated spherical harmonic basis chosen to be regular at the origin. As the result is proportional to a polynomial of modest degree ( simply related to the order of the spectral expansion), it can be made to vanish identically on every geostrophic contour by simply equating each of its coefficients to zero. We extend the discussion to allow for the presence of an inner core, which partitions the geostrophic contours into three distinct regions.

Ierley, GR, Kerswell RR, Plasting SC.  2006.  Infinite-Prandtl-number convection. Part 2. A singular limit of upper bound theory. Journal of Fluid Mechanics. 560:159-227.   10.1017/s0022112006000450   AbstractWebsite

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 & lerley (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.

Plasting, SC, Ierley GR.  2005.  Infinite-Prandtl-number convection. Part 1. Conservative bounds. Journal of Fluid Mechanics. 542:343-363.   10.1017/s0022112005006555   AbstractWebsite

The methods that have come to be known as the Malkus-Howard-Busse (MHB) and the Constantin-Doering-Hopf (CDH) techniques have, over the past few decades, produced the few rigorous statements available about average properties (e.g. momentum and heat transport) of turbulent flows governed by the Navier-Stokes equation and the heat equation. In this, the first of two papers investigating upper bounds on the heat transport in infinite-Prandtl-number convection, we show that the methods of MHB and CDH yield equivalent optimal bounds: as at a saddle - one from above, and one from below. We also demonstrate that here, in contrast to earlier applications of the CDH method, the simplest possible, one-parameter, 'test function' does not capture the leading-order scaling associated with the fully optimal solution. We explore the consequences of a two-parameter test function in modifying the scaling of the upper bound. In the case of no-slip, the suggestion is that a hierarchy of test functions of increasing complexity is required to yield the correct limiting behaviour.

Balmforth, NJ, Ierley GR, Young WR.  2002.  Tidal conversion by subcritical topography. Journal of Physical Oceanography. 32:2900-2914.   10.1175/1520-0485(2002)032<2900:tcbst>;2   AbstractWebsite

Analytical estimates of the rate at which energy is extracted from the barotropic tide at topography and converted into internal gravity waves are given. The ocean is idealized as an inviscid, vertically unbounded fluid on the f plane. The gravity waves are treated by linear theory and freely escape to z = infinity. Several topographies are investigated: a sinusoidal ripple, a set of Gaussian bumps, and an ensemble of "random topographies.'' In the third case, topographic profiles are generated by randomly selecting the amplitudes of a Fourier superposition so that the power spectral density is similar to that of submarine topography. The authors' focus is the dependence of the conversion rate (watts per square meter of radiated power) on the amplitude of the topography, h(0), and on a nondimensional parameter epsilon(*), defined as the ratio of the slope of an internal tidal ray to the maximum slope of the topography. If epsilon(*) << 1, then Bell's theory indicates that the conversion is proportional to h(0)(2). The results span the interval 0 less than or equal to e(*) <1 and show that the enhancement above Bell's prediction is a smoothly and modestly increasing function of ε(*) : For ε(*) --> 1, the conversion of sinusoidal topography is 56% greater than Bell's epsilon(*) <<1 estimate, while the enhancement is only 14% greater for a Gaussian bump. With random topography, the enhancement at epsilon(*) = 0.95 is typically about 6% greater than Bell's formula. The epsilon(*) << 1 approximation is therefore quantitatively accurate over the range 0 <ε(*)<1, implying that the conversion is roughly proportional to h(0)(2). As epsilon(*) is increased, the radiated waves develop very small spatial scales that are not present in the underlying topography and, when epsilon(*) approaches unity, the associated spatial gradients become so steep that overturns must occur even if the tidal amplitude is very weak. The solutions formally become singular at epsilon(*) = 1, in a breakdown of linear, inviscid theory.

Ierley, G, Miles J.  2001.  On Townsend's rapid-distortion model of the turbulent-wind-wave problem. Journal of Fluid Mechanics. 435:175-189. AbstractWebsite

Townsend's (1980) model of wind-to-wave energy transfer, which is based on a putative interpolation between an inner, viscoelastic approximation and an outer, rapid-distortion approximation and predicts an energy transfer that is substantially larger (by as much as a factor of three) than that predicted by Miles's (1957) quasi-laminar model, is revisited. It is shown that Townsend's interpolation effectively imposes a rapid-distortion approximation throughout the flow, rather than only in the outer domain, and that his asymptotic (far above the surface) solution implicitly omits one of the two admissible, linearly independent solutions of his perturbation equations. These flaws are repaired, and Townsend's dissipation function is modified to render the transport equation for the perturbation energy of the same form as those for the perturbation Reynolds stresses. The resulting wind-to-wave energy transfer is close to that predicted by Townsend's (1972) viscoelastic model and other models that incorporate the perturbation Reynolds stresses, but somewhat smaller than that predicted by the quasi-laminar model. We conclude that Townsend's (1980) predictions, although closer to observation than those of other models, rest on flawed analysis and numerical error.

Ierley, GR, Worthing RA.  2001.  Bound to improve: a variational approach to convective heat transport. Journal of Fluid Mechanics. 441:223-253. AbstractWebsite

To the long established idea of bounding turbulent convective heat transport by a variational method based on energetic constraints, we now add a richer class of 'z-constraints' with the hope of tightening bounds considerably. We establish that only certain moments of the governing equations are effective for this purpose. We explore the initial consequences of groups of such constraints by use of perturbation theory, which clarifies the need that a given set of elements be mutually congruent.

Sheremet, VA, Ierley GR, Kamenkovich VM.  1997.  Eigenanalysis of the two-dimensional wind-driven ocean circulation problem. Journal of Marine Research. 55:57-92.   10.1357/0022240973224463   AbstractWebsite

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 (lerley 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.

Balmforth, NJ, Ierley GR, Worthing R.  1997.  Pulse dynamics in an unstable medium. Siam Journal on Applied Mathematics. 57:205-251. AbstractWebsite

A study is presented of a one-dimensional, nonlinear partial differential equation that describes evolution of dispersive, long wave instability. The solutions, under certain specific conditions, take the form of trains of well-separated purses. The dynamics of such patterns of pulses is investigated using singular perturbation theory and numerical simulation. These tools permit the formulation of a theory of pulse interaction and enable the mapping out of the range of behavior in parameter space. There are regimes in which steady trains form; such states can be studied with the asymptotic, pulse-interaction theory. In other regimes, pulse trains are unstable to global, wavelike modes or radiation. This can precipitate more violent phenomena involving pulse creation or generate oscillating states which may follow Shil'nikov's route to temporal chaos. The asymptotic theory is generalized to take some account of radiative dynamics. In the limit of small dispersion, steady trains largely cease to exist; the system follows various pathways to temporal complexity and typical bifurcation sequences are sketched out. The investigation guides us to a critical appraisal of the asymptotic theory and uncovers the wealth of different types of behavior present in the system.

Ierley, GR, Ruehr OG.  1997.  A combinatorial expression from oceanography. Siam Review. 39:772-774.Website
Ierley, GR, Sheremet VA.  1995.  Multiple Solutions and Advection-Dominated Flows in the Wind-Driven Circulation .1. Slip. Journal of Marine Research. 53:703-737.   10.1357/0022240953213052   AbstractWebsite

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 Stommel's (1948) bottom friction model of the circulation. Paradoxically, it appears the solutions are, in that case, unique.

Cessi, P, Ierley GR.  1995.  Symmetry-Breaking Multiple Equilibria in Quasi-Geostrophic, Wind-Driven Flows. Journal of Physical Oceanography. 25:1196-1205.   10.1175/1520-0485(1995)025<1196:sbmeiq>;2   AbstractWebsite

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 range 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 --> -y and psi --> -psi. 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 oceanographic example of spontaneous breaking of the quasigeostrophic symmetry. Within the region of parameter space where three solutions are found, a second pair of nonsymmetric stationary solutions emerges; bringing the total number of stationary solutions to five. This last pair of nonsymmetric solutions is characterized by basin-filling gyres with amplitudes much above the Sverdrup prediction. Once again, the separation point is displaced from the latitude of vanishing wind stress curl. The existence of nonsymmetric double gyres in an antisymmetrically forced basin shows that there can be no general rule for determining the point of separation of the boundary current in terms of the relative strength of the subtropical and subpolar forcings.

Balmforth, NJ, Ierley GR, Spiegel EA.  1994.  Chaotic Pulse Trains. Siam Journal on Applied Mathematics. 54:1291-1334.   10.1137/s0036139993247422   AbstractWebsite

The authors study a third-order nonlinear ordinary differential equation whose solutions, under certain specific conditions, are individual pulses. These correspond to homoclinic orbits in the phase space of the equation, and the possible pulse types are studied in some detail. Sufficiently close to the conditions under which a homoclinic orbit exists, the solutions take the form of trains of well-separated pulses. A measure of closeness to homoclinic conditions provides a small parameter for the development of an asymptotic solution consisting of superposed, isolated pulses. The solvability condition in the resulting singular perturbation theory is a timing map relating successive pulse spacings. This map of the real line onto itself, together with the known form of the homoclinic orbit, provides a concise and accurate solution of the equation.

Cessi, P, Ierley GR.  1993.  Nonlinear Disturbances of Western Boundary Currents. Journal of Physical Oceanography. 23:1727-1735.   10.1175/1520-0485(1993)023<1727:ndowbc>;2   AbstractWebsite

Viscous shear instability is proposed as a primary mechanism for generating time-dependent eddies at western boundaries. Thus, the authors examine the stability of the Munk flow with constant transport flowing along a straight coast, tilted at an angle with respect to the north-south direction. Various properties of the marginally unstable wave are calculated as a function of the tilting angle, such as the critical Reynolds number and the phase and group velocities. The effects of weak nonlinearity are also examined, and the authors find that the instability is supercritical for the whole range of tilting angles examined. Thus, the marginally unstable mode can equilibrate at a small finite amplitude, and we derive the equation governing its slow evolution. The flow that results after the disturbance has equilibrated to finite amplitude is in agreement with the eddying boundary currents obtained in many wind-driven general circulation models.

Balmforth, NJ, Cvitanovic P, Ierley GR, Spiegel EA, Vattay G.  1993.  Advection of Vector-Fields by Chaotic Flows. Stochastic Processes in Astrophysics. 706( Buchler JR, Kandrup HE, Eds.).:148-160., New York: New York Acad Sciences   10.1111/j.1749-6632.1993.tb24687.x  
Ierley, G, Spencer B, Worthing R.  1992.  Spectral Methods in Time for a Class of Parabolic Partial-Differential Equations. Journal of Computational Physics. 102:88-97.   10.1016/s0021-9991(05)80008-7   Website