Research Article
A multi-scaling analysis of the spin-up problem
- Jean-Pierre St-Maurice, George Veronis
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- 29 March 2006, pp. 417-445
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The response of a contained rotating fluid to a small, abrupt change in the rotation rate is analysed by multi-scaling methods. The procedure makes use of the fact that three different physical processes (inertial oscillations, spin-up response, diffusion) give rise to three different time scales. Since the flow is known to have a boundary-layer character, the variables are derived into interior and boundary-layer parts. The pertinent parameter separating the magnitudes of the amplitudes and the different time scales is the square root of the Ekman number E½, so an expansion in powers of E½ is used. The solution for a homogeneous fluid is derived first and is shown to be consistent with the solution of Green-span & Howard (1963). The results are given in two forms: one is a direct deduction of the expansion method and is valid to O(E); the other is obtained by regrouping the terms to derive a form apparently valid for indefinitely long times. When the fluid is stratified, the physical structure of the system is substantially more complicated, and so is the analysis. Exact results can be obtained for the case where the buoyancy N and the rotational Ω frequencies are the same. For the general case F = N/Ω ≠ 1, results valid for t [Gt ] 1 can be obtained (where t is measured in units of Ω−1). In both cases the exact lowest-order solution for the interior can be derived since it is independent of short time t. For the stratified fluid the elementary spin-up solution of Holton (1965) is part of the solution at O(E½). The remaining part includes the long-time behaviour towards which the system tends as diffusive processes become dominant. The formulation of the long-time problem is complete a t O(E), but parts of it emerge from the analysis at lower order, and it is necessary to treat the lower-order system to obtain a consistent formulation at O(E). In particular, it is possible to show that the thermal boundary condition, which does not affect the elementary spin-up solution, should be satisfied only by the long-time part of the problem. The complete, lowest-order response of the system includes a diffusive part which is quantitatively significant even for times of the order of one spin-up time. It is suggested here that the diffusive contribution may be responsible for parts of the discrepancy between elementary spin-up theory and recent experiments.
The role of dynamic pressure in generating fire wind
- R. K. Smith, B. R. Morton, L. M. Leslie
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- 29 March 2006, pp. 1-19
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Earlier models of fire plumes based on simple entrainment laws and neglecting dynamic pressure have failed to produce the relatively shallow inflow over the fire perimeter known as fire wind. This inflow is of prime importance in fire modelling as it normally provides much of the air required for combustion; for this reason we have carried out a very simple numerical experiment on two-dimensional natural convection above a strip heat source with the intention of simulating those aspects of fire behaviour involved in the generation of fire wind without attempting the formidably difficult task of detailed fire modelling. Our results show clearly that fire wind is driven by the dynamic pressure field which is generated by and intimately related to the region of strong buoyant acceleration close above the ground boundary. Throughout our parametric range there is a concentrated region of large horizontal pressure gradient in a neighbourhood above the perimeter of the fire, and elsewhere the pressure gradients play a lesser role.
We have investigated also the dependence of our solution on the boundary conditions, particularly those at the lateral boundary, where we have imposed as little constraint as possible on flow into and out of the computational region. Considerable effects even of such weak side-boundary constraints persist throughout the solution region at moderate values of the pseudo-Rayleigh number (based on eddy diffusivities), but these can be limited by an appropriate choice of the thermal conditions and kept within acceptable bounds at large pseudo-Rayleigh numbers. Similar effects of boundary conditions are likely to appear in other mesoscale convectively driven atmospheric models, including sea breezes, katabatic winds and locally concentrated convective columns.
On the sedimentation of a sphere in a centrifuge
- Isom H. Herron, Stephen H. Davis, Francis P. Bretherton
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- 29 March 2006, pp. 209-234
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The flow field about a small, slowly sedimenting particle in a centrifuge is examined using matched asymptotic expansions. The near field is dominated by Stokes flow while in the far field a non-axisymmetric cubical conical structure (a viscously modified Taylor column) is found. This far field induces a Coriolis modification in the near field leading to Coriolis corrections to the Stokes drag law. The Coriolis modification of the predicted molecular weight (if the particle were a molecule) of a small particle is calculated. The analysis is applied to an unbounded fluid as well as to a fluid bounded between parallel plates oriented normal to the rotation vector. In the latter case the governing equations for the rotating fluid are posed as a self-adjoint system of partial differential equations and solved using (symmetric) Green's matrices.
On the continuum theory for the large Reynolds number spherical expansion into a near vacuum
- N. C. Freeman, R. S. Johnson, S. Kumar, W. B. Bush
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- 29 March 2006, pp. 625-638
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The steady, spherically symmetric flow of a compressible gas is considered. The gas is both viscous and heat-conducting. In the limit of very high Reynolds number (= α−1, α → 0) and correspondingly low pressure at infinity, the structure of the whole flow field is discussed. The five regions that arise by virtue of the limit α → 0 are briefly considered. Special care is given to the matching across the overlap domains and the first region (close to, but outside, the sonic point) and the fifth (where the pressure adjusts to its ambient value) are carefully examined. It is argued that the application of appropriate matching principles, together with judicious use of numerical solutions, allows an arbitrary pressure and temperature to be assigned to the background gas.
Nonlinear calculation of arbitrarily shaped supercavitating hydrofoils near a free surface
- Okitsugu Furuya
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- 29 March 2006, pp. 21-40
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A nonlinear exact solution to the problem of two-dimensional gravity-free incompressible potential flow around an arbitrarily shaped supercavitating hydrofoil near a free surface is obtained. A combination of Newton's method with a functional iterative procedure is used to solve the nonlinear integral and algebraic equations of this problem. Fast and stable convergence results by starting the iteration with a readily chosen initial solution. Some representative numerical computations are made for practical hydrofoils having both generally shaped camber and leading-edge thickness distributions. The force coefficients, pressure distribution and free-streamline shapes of the cavity are calculated for each case with an execution time on an IBM 370-158 of 200–530 s depending upon the initial trial solution.
Internal waves generated by a translating oscillating body
- R. G. Rehm, H. S. Radt
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- 29 March 2006, pp. 235-258
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The internal-wave system is calculated for a body oscillating transversely, and translating uniformly, through an infinite stratified fluid of constant Brunt-Väisälä frequency. A linearized, time-dependent analysis is used, in which the vertical displacement of a fluid element is the basic dependent variable. Axisym-metric slender-body theory for a homogeneous fluid is used to determine the time-dependent source and dipole distributions required to represent the motion of the body for excitation of the internal waves. The equations are solved by Fourier-transform techniques; and the internal-wave amplitude is evaluated in the far field by the method of stationary phase. The surfaces of constant phase are found to change character as the ratio of the oscillation frequency ω of the body to the Brunt-Väisälä frequency N varies through unity. Along preferred directions, the amplitude of the internal waves is found to decay inversely with distance to the $\frac{5}{6}$ power, whereas, for uniform translation, the amplitude of the internal waves falls off inversely with distance from the body. An asymptotic expression for the amplitude in preferred directions is calculated for several values of the ratio ω/N.
On double-roll convection in a rotating magnetic system
- P. H. Robert, K. Stewartson
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- 29 March 2006, pp. 447-466
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Electrically and thermally conducting inviscid fluid rotating about a vertical axis is confined between two horizontal plates maintained a t different temperatures, the upper plate being the cooler. The fluid is permeated by a horizontal magnetic field that corotates with the fluid. In an earlier paper (Roberts & Stewartson 1974) the fluid is supposed to be in a state of near-marginal instability to convective overturning and the nonlinear evolution of single rolls is discussed. Inertial terms are neglected. However, if q < 2 and λ < 2/3½, where q and λ may be defined by equation (2.3) below, the principle of the exchange of stabilities holds and there is also a degeneracy in the linear stability problem. There are now two distinct unstable rolls equally possible and their nonlinear interaction leads to a violation of the governing equations. This difficulty has already been noted by Taylor (1963) and it is resolved in this paper by adding a geostrophic motion (the Taylor shear) parallel to the magnetic field and by restoring the inertial terms in the governing equations. We consider particularly instabilities in which one roll predominates and find that, if λ is sufficiently small, each of the rolls that can occur is stable with respect to the other, i.e. an initially weak roll of the other type dies out relative to it. This means that we can expect the fluid motion to consist of single rolls at large times. On the other hand when λ is near 2/3½ both rolls are unstable with respect to the other. The Taylor shear does not then die out and the two rolls become comparable in magnitude and modify each other's structure. At intermediate values of λ one of the rolls is stable in this way and the other unstable.
The study is motivated by a desire to understand better the dynamical means by which a large mass of conducting fluid can create its own magnetism. It is argued that these instabilities suggest the existence of a mechanism of self-adjustment preventing λ from either increasing or decreasing indefinitely and noted that, very roughly, λ is of order unity in the earth's core.
Experiments on turbulence in a rotating fluid
- A. Ibbetson, D. J. Tritton
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- 29 March 2006, pp. 639-672
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Experiments have been carried out to investigate the effect of rotation of the whole system on decaying turbulence, generally similar to grid turbulence, generated in air in an annular container on a rotating table. Measurements to determine the structure of the turbulence were made during its decay, mean quantities being determined by a mixture of time and ensemble averaging. Quantities measured (as functions of time after the turbulence generation) were turbulence intensities perpendicular to and parallel to the rotation axis, spectra of these two components with respect to a wavenumber perpendicular to the rotation axis, and some correlation coefficients, selected to detect differences in length scales perpendicular and parallel to the rotation axis. The intensity measurements were made for a wide range of rotation rates; the other measurements were made at a single rotation rate (selected to give a Rossby number varying during the decay from about 1 to small values) and, for comparison, at zero rotation. Subsidiary experiments were carried out to measure the spin-up time of the system, and to determine whether the turbulence produced any mean flow relative to the container.
A principal result is that increasing the rotation rate produces faster decay of the turbulence; the nature of the additional energy sink is an important part of the interpretation. Other features of the results are as follows: the measurements with-outrotation can be satisfactorily related to wind-tunnel measurements; even with rotation, the ratio of the intensities in the two directions remains substantially constant; the normalized spectra for the rotating and the non-rotating cases show surprising similarity but do contain slight systematic differences, consistent with the length scales indicated by the correlations; rotation produces a large increase in the length scale parallel to the rotation axis and a smaller increase in that perpendicular to it; the turbulence produces no measurable mean flow.
A model for the interpretation of the results is developed in terms of the action of inertial waves in carrying energy to the boundaries of the enclosure, where it is dissipated in viscous boundary layers. The model provides satisfactory explanations of the overall decay of the turbulence and of the decay of individual spectral components. Transfer of energy between wavenumbers plays a much less significant role in the dynamics of decay than in a non-rotating fluid. The relationship of the model to the interpretation of the length-scale difference in terms of the Taylor-Proudman theorem is discussed.
The model implies that the overall dimensions of the system enter in an important way into the dynamics. This imposes a serious limitation on the application of the results to the geophysical situations at which experiments of this type are aimed.
The paper includes some discussion of the possibility of energy transfer from the turbulence to a mean motion (the ‘vorticity expulsion’ hypothesis). It is possible, on the basis of the observations, to exclude this process as the additional turbulence energy sink. But this does not provide any evidence either for or against the hypothesis in the conditions for which it has been postulated.
Long-wavelength peristaltic pumping at low Reynolds number
- M. J. Manton
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- 29 March 2006, pp. 467-476
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An asymptotic expansion is found for the low Reynolds number flow induced in an axisymmetric tube by long peristaltic waves of arbitrary shape. Expressions are determined for the relationship between the mean pressure gradient and the volume flux, for the mean rate of working by the wall of the tube and for the shear stress a t the wall. A necessary and sufficient condition for the occurrence of trapping (that is, regions of separated flow near the axis of the tube in a reference frame moving at the wave speed) is obtained. It is shown that reflux (that is, a mean flux in the negative axial direction in a layer of fluid adjacent to the wall when the net mean flux is positive) occurs whenever there is an adverse mean pressure gradient, independently of the shape of the wave. A n estimate of the amount of reflux is derived.
Laminar plume interactions
- Luciano Pera, Benjamin Gebhart
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- 29 March 2006, pp. 259-271
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Interactions between laminar thermal natural convection plumes generated by line and concentrated heat sources were experimentally investigated with a Mach-Zehnder interferometer. Adjacent plane plumes were found to interact more strongly than axisymmetric plumes a t the same spacing. These flows with nominally free boundaries were also found to be affected by nearby surfaces which interfered with the supply of entrainment fluid. Several types of interference with plume entrainment were investigated. The nature of the interacting flows suggests a model which is successful in interpreting the mechanism. An application of the effects of plume interaction is discussed.
The distortion of turbulence by general uniform irrotational strain
- A. J. Reynolds, H. J. Tucker
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- 29 March 2006, pp. 673-693
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This paper describes the measured response of grid turbulence to three limiting types of uniform homogeneous strain: plane straining, axisymmetric elongation and axisymmetric flattening. Straining was achieved by allowing the turbulence to be convected through suitable distorting ducts; the maximum strain ratios were 5·8, 6·0 and 2·3, respectively. An attempt is made, using rapid-distortion theory, to specify an effective strain which accounts for the initial anisotropy of the grid turbulence; in the experiments, this effect was most important for the third species of strain. The maximum effective strain ratios were calculated as 4·05, 7·2 and 2·75, respectively. The rapid-distortion results are able to describe several features of the response of the turbulence with good accuracy: (i) the variation of total turbulence energy through the experimental ducts; (ii) the tendency of one component (that in the direction of the (larger) negative strain) to contain one-half of the turbulence energy after only moderate straining; and (iii) the changes in dimensionless structure parameters composed of ratios of component intensities. The first kind of prediction requires that the concurrent decay be specified in a simple way; (ii) and (iii) require that the initial anisotropy be taken into account. The predictions (iii) are generally less accurate than the others. The degree of success achieved by the rapid-distortion hypo-thesis is rather surprising, since the strain rates in the experiments were not as large as those for which the theory might have been expected to be valid. It is concluded that successful models of turbulence must provide a vorticity amplification essentially like that of rapid-distortion theory. However, the simple distorting flows considered here may not provide severe tests of more refined models, since many features of the response have already been accounted for.
Instability of a periodic boundary layer in a stratified fluid
- R. M. Robinson, A. D. McEwan
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- 29 March 2006, pp. 41-48
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It has been found that the periodic boundary layer formed on a vertically oscil-lating vertical wall bounding a stratified fluid is liable to two distinct modes of wavelike instability. In the first, which arises when the oscillation frequency ω is lower than 0·7 times the buoyancy frequency N, the phase lines are aligned horizontally. The second mode, in which the phase lines are aligned at 45° or more to the horizontal, becomes dominant as ω is increased above 0·9 N.
In distinction from the unstratified periodic Stokes layer, there appears to be, for ω in the vicinity of N, a definite low threshold to the boundary-layer Stokes-Reynolds number (defined as Wo/(2ων)½, where Wo is the maximum vertical wall velocity and v is the kinematic viscosity) above which the instability is sustained a t a detectable level.
The instability of capillary jets
- Arthur M. Sterling, C. A. Sleicher
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- 29 March 2006, pp. 477-495
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At high jet velocity the aerodynamic interaction between a capillary jet and the surrounding medium leads to an enhanced growth rate of axisymmetric disturbances. The available theories which account for this effect fail to agree with experimental observations. The difference is attributed, in part, to the relaxation of the velocity profile in jets formed by fully developed laminar pipe flow. The profile relaxation has a destabilizing effect just as does the aerodynamic interaction. In the absence of velocity-profile relaxation it is shown that the available theories overestimate the aerodynamic effect. A consideration of the viscosity of the ambient fluid yields a semi-empirical modification to the theory which shows good agreement with experimental values.
Wind-induced drift currents
- Jin Wu
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- 29 March 2006, pp. 49-70
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Systematic measurements of drift currents below and of airflows above an air-water interface have been made under various wind conditions. The current near but not immediately below the water surface is found to follow a Kármán-Prandtl (logarithmic) velocity distribution. The current immediately below the water surface varies linearly with depth. The transitions of the current boundary layer to various regimes appear to lag behind, or to occur a t a higher wind velocity than, those of the airflow. The fraction of the wind stress supported by the wave drag seems to vary with the wind and wave conditions: a large fraction is obtained at low wind velocities with shorter waves and a small fraction is obtained a t high wind velocities with longer waves. At the air-water interface, the wind-induced current is found to be proportional to the friction velocity of the wind. The Stokes mass transport, related to wave characteristics, is only a small component of the surface drift in laboratory tanks. However, in terms of the fraction of the wind velocity, the mass transport increases, while the wind drift decreases, as the fetch increases. The ratio between the total surface drift and the wind velocity decreases gradually as the fetch increases and approaches a constant value of about 3·5% at very long fetches.
On the prediction of intermittent turbulent flows
- Paul A. Libby
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- 29 March 2006, pp. 273-295
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In this paper we present a theoretical model which permits the conservation equations of fluid dynamics to be conditioned in a fashion analogous to the experimentalist's technique of ‘conditioned sampling’. The detailed analysis refers to the best-known sampling condition, outer-edge intermittency; but the model equation may be applicable to other flow situations, wherein condition- ing exposes details of the physical phenomena. The analysis results in predic- tions of the flow variables within the turbulent flow and of the intermittency. Comparison is made with two sets of experimental results for the two-dimensional mixing layer and with a boundary layer.
Modal equations for cellular convection
- D. O. Gough, E. A. Spiegel, Juri Toomre
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- 29 March 2006, pp. 695-719
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We expand the fluctuating flow variables of Boussinesq convection in the planform functions of linear theory. Our proposal is to consider a drastic truncation of this expansion as a possibly useful approximation scheme for studying cellular convection. With just one term included, we obtain a fairly simple set of equations which reproduces some of the qualitative properties of cellular convection and whose steady-state form has already been derived by Roberts (1966). This set of ‘modal equations’ is analysed at slightly supercritical and at very high Rayleigh numbers. In the latter regime the Nusselt number varies with Rayleigh number just as in the mean-field approximation with one horizontal scale when the boundaries are rigid. However, the Nusselt number now depends also on the Prandtl number in a way that seems compatible with experiment. The chief difficulty with the approach is the absence of a deductive scheme for deciding which planforms should be retained in the truncated expansion.
Inertial Taylor columns on a beta plane
- Michael S. McCartney
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- 29 March 2006, pp. 71-95
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The effect of variable Coriolis parameter on the formation of inertial Taylor columns is determined for the case of a two-layer fluid with moderate stratification. Analytic solutions of the inertial, quasi-geostrophic, β-plane equations are obtained. As a special case, sohtions corresponding to a single-layer, homo-geneous fluid are also obtained. When both layer velocities are retrograde (westward), the effect of β is to limit the horizontal extent of the disturbance due to the bump. When both layer velocities are prograde (eastward), an extensive meandering wake is found downstream of the bump. Associated with this wake can be large stationary cyclonic and anti-cyclonic eddies. The meander amplitudes in the two layers are typically nearly the same. I n both the retrograde and prograde cases, the strength of the disturbance to the flow above the bump is less in the upper layer compared with the lower, indicating an attenuation in the vertical due to stratification. For a counter-flow situation, the solutions are complicated by the possibility of a stationary baroclinic wave, one that would exist even for β = 0. In all the situations in which a meandering wake is formed, there is a wave-drag force on the bump. Some laboratory experiments corres- ponding to the single-layer solutions are described.
Kelvin–Helmholtz wave growth on cylindrical sheets
- G. D. Crapper, N. Dombrowski, G. A. D. Pyott
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- 29 March 2006, pp. 497-502
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A first-order analysis of Kelvin-Helmholtz wave growth on cylindrical sheets is carried out. It is demonstrated that the growth rate of both symmetric and antisymmetric waves increases significantly with reduction of the radius of the core.
Entrainment rates in turbulent shear flows
- S. T. Paizis, W. H. Schwarz
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- 29 March 2006, pp. 297-308
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The rate of entrainment of ambient fluid across a turbulent interface has been defined as the mean rate of increase of turbulent fluid in the flow direction. Experiments to measure this quantity by conditional sampling in a two-dimen- sional wall jet are described. Further, estimates of this entrainment rate were made for the turbulent boundary layer, two-dimensional wake, two-dimensional jet and round jet and the results are discussed.
Dissipative heating in convective flows
- J. M. Hewitt, D. P. Mckenzie, N. O. Weiss
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- 29 March 2006, pp. 721-738
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Dissipative heating is produced by irreversible processes, such as viscous or ohmic heating, in a convecting fluid; its importance depends on the ratio d/HT of the depth of the convecting region to the temperature scale height. Integrating the entropy equation for steady flow yields an upper bound to the total rate of dissipative heating in a convecting layer. For liquids there is a regime in which the ratio of dissipative heating to the convected heat flux is approximately equal to c(d/HT), where the constant c is independent of the Rayleigh number. This result is confirmed by numerical experiments using the Boussinesq approximation, which is valid only if d/HT is small. For deep layers the dissipative heating rate may be much greater than the convected heat flux. If the earth's magnetic field is maintained by a convectively driven dynamo, ohmic losses are limited to 5% of the convected flux emerging from the core. In the earth's mantle viscous heating may be important locally beneath ridges and behind island arcs.