5 results
Balanced and unbalanced routes to dissipation in an equilibrated Eady flow
- M. JEROEN MOLEMAKER, JAMES C. MCWILLIAMS, XAVIER CAPET
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- Journal:
- Journal of Fluid Mechanics / Volume 654 / 10 July 2010
- Published online by Cambridge University Press:
- 17 June 2010, pp. 35-63
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The oceanic general circulation is forced at large scales and is unstable to mesoscale eddies. Large-scale currents and eddy flows are approximately in geostrophic balance. Geostrophic dynamics is characterized by an inverse energy cascade except for dissipation near the boundaries. In this paper, we confront the dilemma of how the general circulation may achieve dynamical equilibrium in the presence of continuous large-scale forcing and the absence of boundary dissipation. We do this with a forced horizontal flow with spatially uniform rotation, vertical stratification and vertical shear in a horizontally periodic domain, i.e. a version of Eady's flow carried to turbulent equilibrium. A direct route to interior dissipation is presented that is essentially non-geostrophic in its dynamics, with significant submesoscale frontogenesis, frontal instability and breakdown, and forward kinetic energy cascade to dissipation. To support this conclusion, a series of simulations is made with both quasigeostrophic and Boussinesq models. The quasigeostrophic model is shown as increasingly inefficient in achieving equilibration through viscous dissipation at increasingly higher numerical resolution (hence Reynolds number), whereas the non-geostrophic Boussinesq model equilibrates with only weak dependence on resolution and Rossby number.
Local balance and cross-scale flux of available potential energy
- M. JEROEN MOLEMAKER, JAMES C. McWILLIAMS
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- Journal:
- Journal of Fluid Mechanics / Volume 645 / 25 February 2010
- Published online by Cambridge University Press:
- 08 February 2010, pp. 295-314
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Gravitational available potential energy is a central concept in an energy analysis of flows in which buoyancy effects are dynamically important. These include, but are not limited to, most geophysical flows with persistently stable density stratification. The volume-integrated available potential energy
ap is defined as the difference between the gravitational potential energy of the system and the potential energy of a reference state with the lowest potential energy that can be reached by adiabatic material rearrangement; ap determines how much energy is available for conservative dynamical exchange with kinetic energy k. In this paper we introduce new techniques for computing the local available potential energy density Eap in numerical simulations that allow for a more accurate and complete analysis of the available potential energy and its dynamical balances as part of the complete energy cycle of a flow. In particular, the definition of Eap and an associated gravitation disturbance field permit us to make a spectral decomposition of its dynamical balance and examine the cross-scale energy flux. Several examples illustrate the spatial structure of Eap and its evolutionary influences. The greatest attention is given to an analysis of a turbulent-equilibrium simulation Eady-like vertical-shear flow with rotation and stable stratification. In this regime Eap exhibits a vigorous forward energy cascade from the mesoscale through the submesoscale range – first in a scale range dominated by frontogenesis and positive buoyancy-flux conversion from ap to k and then, after strong frontal instability and frontogenetic arrest, in a coupled kinetic-potential energy inertial-cascade range with negative buoyancy-flux conversion – en route to fine-scale dissipation of both energy components.
The formation and evolution of a diffusive interface
- M. JEROEN MOLEMAKER, HENK A. DIJKSTRA
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- Journal:
- Journal of Fluid Mechanics / Volume 331 / 25 January 1997
- Published online by Cambridge University Press:
- 21 May 2009, pp. 199-229
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The formation and evolution of a diffusive interface in a stable salt-stratified layer cooled from above is studied in a two-dimensional geometry by direct numerical simulation. For a typical example with realistic parameters, the evolution of the flow is computed up to the moment where three layers can be distinguished. Focus is on the development of the first mixed layer. The convective velocity scaling as proposed by Hunt (1984) and previously proposed expressions for the interfacial heat flux (Huppert 1971; Fernando 1989a) are shown to correspond well with the results of the simulation. The evolution of the first layer can be well described by an entrainment relation based on a local balance between kinetic and potential energy with mixing efficiency γ. The new entrainment relation is shown to fit the numerical results well and an interpretation of γ in terms of the overall energy balances of the flow is given.
Previously, two rival mechanisms have been proposed that determine the final thickness of the first layer (Turner 1968; Fernando 1987). One of the distinguishing features of both mechanisms is whether a transition in entrainment regime – as the first layer develops – is a necessary condition for the mixed layer to stop growing. Another is the presence of a buoyancy jump over the interface before substantial convection in the second layer occurs. From the numerical results, we find a significant buoyancy jump even before the thermal boundary layer ahead of the first layer becomes unstable. Moreover, the convective activity in the second layer is too small to be able to stop the growth of the first layer. We therefore favour the view proposed by Fernando (1987) that a transition in entrainment regime determines the thickness of the first layer. Following this, a new one-dimensional model of layer formation is proposed. Important expressions within this model are verified using the results of the numerical simulation. The model contains two constants which are determined from the numerical results. The results of the new model fit experimental results quite well and the parameter dependence of the thickness of the first layer is not sensitive to the values of the two constants.
Symmetry breaking and overturning oscillations in thermohaline-driven flows
- HENK A. DIJKSTRA, M. JEROEN MOLEMAKER
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- Journal:
- Journal of Fluid Mechanics / Volume 331 / 25 January 1997
- Published online by Cambridge University Press:
- 21 May 2009, pp. 169-198
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The bifurcation structure of thermohaline-driven flows is studied within one of the simplest zonally averaged models which captures thermohaline transport: a Boussinesq model of surface-forced thermohaline flow in a two-dimensional rectangular basin. Under mixed boundary conditions, i.e. prescribed surface temperature and fresh-water flux, it is shown that symmetry breaking originates from a codimension-two singularity which arises through the intersection of the paths of two symmetry-breaking pitchfork bifurcations. The physical mechanism of symmetry breaking of both the thermally and salinity dominated symmetric solution is described in detail from the perturbation structures near bifurcation. Limit cycles with an oscillation period in the order of the overturning time scale arise through Hopf bifurcations on the branches of asymmetric steady solutions. The physical mechanism of oscillation is described in terms of the most unstable mode just at the Hopf bifurcation. The occurrence of these oscillations is quite sensitive to the shape of the prescribed fresh-water flux. Symmetry breaking still occurs when, instead of a fixed temperature, a Newtonian cooling condition is prescribed at the surface. There is only quantitative sensitivity, i.e. the positions of the bifurcation points shift with the surface heat transfer coefficient. There are no qualitative changes in the bifurcation diagram except in the limit where both the surface heat flux and fresh-water flux are prescribed. The bifurcation structure at large aspect ratio is shown to converge to that obtained by asymptotic theory. The complete structure of symmetric and asymmetric multiple equilibria is shown to originate from a codimension-three bifurcation, which arises through the intersection of a cusp and the codimension-two singularity responsible for symmetry breaking.
Non-axisymmetric instability of centrifugally stable stratified Taylor–Couette flow
- IRAD YAVNEH, JAMES C. MCWILLIAMS, M. JEROEN MOLEMAKER
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- Journal:
- Journal of Fluid Mechanics / Volume 448 / 10 December 2001
- Published online by Cambridge University Press:
- 26 November 2001, pp. 1-21
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The stability is investigated of the swirling flow between two concentric cylinders in the presence of stable axial linear density stratification, for flows not satisfying the well-known Rayleigh criterion for inviscid centrifugal instability, d(Vr)2/dr < 0. We show by a linear stability analysis that a sufficient condition for non-axisymmetric instability is, in fact, d(V/r)2/dr < 0, which implies a far wider range of instability than previously identified. The most unstable modes are radially smooth and occur for a narrow range of vertical wavenumbers. The growth rate is nearly independent of the stratification when the latter is strong, but it is proportional to it when it is weak, implying stability for an unstratified flow. The instability depends strongly on a non-dimensional parameter, S, which represents the ratio between the strain rate and twice the angular velocity of the flow. The instabilities occur for anti-cyclonic flow (S < 0). The optimal growth rate of the fastest-growing mode, which is non-oscillatory in time, decays exponentially fast as S (which can also be considered a Rossby number) tends to 0. The mechanism of the instability is an arrest and phase-locking of Kelvin waves along the boundaries by the mean shear flow. Additionally, we identify a family of (probably infinitely many) unstable modes with more oscillatory radial structure and slower growth rates than the primary instability. We determine numerically that the instabilities persist for finite viscosity, and the unstable modes remain similar to the inviscid modes outside boundary layers along the cylinder walls. Furthermore, the nonlinear dynamics of the anti-cyclonic flow are dominated by the linear instability for a substantial range of Reynolds numbers.