Dissipation of shear-free turbulence near boundaries

The rapid-distortion model of Hunt & Graham (1978) for the initial distortion of turbulence by a flat boundary is extended to account fully for viscous processes. Two types of boundary are considered: a solid wall and a free surface. The model is shown to be formally valid provided two conditions are satisfied. The first condition is that time is short compared with the decorrelation time of the energy-containing eddies, so that nonlinear processes can be neglected. The second condition is that the viscous layer near the boundary, where tangential motions adjust to the boundary condition, is thin compared with the scales of the smallest eddies. The viscous layer can then be treated using thin-boundary-layer methods. Given these conditions, the distorted turbulence near the boundary is related to the undistorted turbulence, and thence profiles of turbulence dissipation rate near the two types of boundary are calculated and shown to agree extremely well with profiles obtained by Perot & Moin (1993) by direct numerical simulation. The dissipation rates are higher near a solid wall than in the bulk of the flow because the no-slip boundary condition leads to large velocity gradients across the viscous layer. In contrast, the weaker constraint of no stress at a free surface leads to the dissipation rate close to a free surface actually being smaller than in the bulk of the flow. This explains why tangential velocity fluctuations parallel to a free surface are so large. In addition we show that it is the adjustment of the large energy-containing eddies across the viscous layer that controls the dissipation rate, which explains why rapid-distortion theory can give quantitatively accurate values for the dissipation rate. We also find that the dissipation rate obtained from the model evaluated at the time when the model is expected to fail actually yields useful estimates of the dissipation obtained from the direct numerical simulation at times when the nonlinear processes are significant. We conclude that the main role of nonlinear processes is to arrest growth by linear processes of the viscous layer after about one large-eddy turnover time.

Liquid-vapour homogenisation of fluid inclusions in stalagmites: Evaluation of a new thermometer for paleoclimate research

We present a new approach to determine palaeotemperatures (mean annual surface temperatures) based on measurements of the liquid–vapour homogenisation temperature of fluid inclusions in stalagmites. The aim of this study is to explore the potential and the limitations of this new palaeothermometer and to develop a reliable methodology for routine applications in palaeoclimate research. Therefore, we have investigated recent fluid inclusions from the top part of actively growing stalagmites that have formed at temperatures close to the present-day cave air temperature. A precondition for measuring homogenisation temperatures of originally monophase inclusions is the nucleation of a vapour bubble by means of single ultra-short laser pulses. Based on the observed homogenisation temperatures (Th(obs)) and measurements of the vapour bubble diameter at a known temperature, we calculated stalagmite formation temperatures (Tf) by applying a thermodynamic model that takes into account the effect of surface tension on liquid–vapour homogenisation. Results from recent stalagmite samples demonstrate that calculated stalagmite formation temperatures match the present-day cave air temperature within ± 0.2 °C. To avoid artificially induced changes of the fluid density we defined specific demands on the selection, handling and preparation of the stalagmite samples. Application of the method is restricted to stalagmites that formed at cave temperatures greater than ~ 9–11 °C.

Determination of Holocene cave temperatures from Kr and Xe concentrations in stalagmite fluid inclusions

From the concentrations of dissolved atmospheric noble gases in water, a so-called “noble gas temperature” (NGT) can be determined that corresponds to the temperature of the water when it was last in contact with the atmosphere. Here we demonstrate that the NGT concept is applicable to water inclusions in cave stalagmites, and yields NGTs that are in good agreement with the ambient air temperatures in the caves. We analysed samples from two Holocene and one undated stalagmite. The three stalagmites originate from three caves located in different climatic regions having modern mean annual air temperatures of 27 °C, 12 °C and 8 °C, respectively. In about half of the samples analysed Kr and Xe concentrations originated entirely from the two well-defined noble gas components air-saturated water and atmospheric air, which allowed NGTs to be determined successfully from Kr and Xe concentrations. One stalagmite seems to be particularly suitable for NGT determination, as almost all of its samples yielded the modern cave temperature. Notably, this stalagmite contains a high proportion of primary water inclusions, which seem to preserve the temperature-dependent signature well in their Kr and Xe concentrations. In future work on stalagmites detailed microscopic inspection of the fluid inclusions prior to noble gas analysis is therefore likely to be crucial in increasing the number of successful NGT determinations.

A unified theory of balance in the extratropics

Many physical systems exhibit dynamics with vastly different time scales. Often the different motions interact only weakly and the slow dynamics is naturally constrained to a subspace of phase space, in the vicinity of a slow manifold. In geophysical fluid dynamics this reduction in phase space is called balance. Classically, balance is understood by way of the Rossby number R or the Froude number F; either R ≪ 1 or F ≪ 1. We examined the shallow-water equations and Boussinesq equations on an f -plane and determined a dimensionless parameter _, small values of which imply a time-scale separation. In terms of R and F, ∈= RF/√(R^2+R^2 ) We then developed a unified theory of (extratropical) balance based on _ that includes all cases of small R and/or small F. The leading-order systems are ensured to be Hamiltonian and turn out to be governed by the quasi-geostrophic potential-vorticity equation. However, the height field is not necessarily in geostrophic balance, so the leading-order dynamics are more general than in quasi-geostrophy. Thus the quasi-geostrophic potential-vorticity equation (as distinct from the quasi-geostrophic dynamics) is valid more generally than its traditional derivation would suggest. In the case of the Boussinesq equations, we have found that balanced dynamics generally implies hydrostatic balance without any assumption on the aspect ratio; only when the Froude number is not small and it is the Rossby number that guarantees a timescale separation must we impose the requirement of a small aspect ratio to ensure hydrostatic balance.

An ‘ideal’ form of decaying two-dimensional turbulence

In decaying two-dimensional Navier-Stokes turbulence, Batchelor's similarity hypothesis fails due to the existence of coherent vortices. However, it is shown that decaying two-dimensional turbulence governed by the Harney-Hasegawa-Mima (CHM) equation ∂/∂t (V^2 φ-λ^2 φ)+J(φ,∇^2 φ)=D where D is a damping, is described well by Batchelor's similarity hypothesis for wave numbers k ≪ λ (the so-called AM regime). It is argued that CHM turbulence in the AM regime is a more `ideal' form of two-dimensional turbulence than is Navier-Stokes turbulence itself.

Nonlinear symmetric stability of planetary atmospheres

The energy–Casimir method is applied to the problem of symmetric stability in the context of a compressible, hydrostatic planetary atmosphere with a general equation of state. Formal stability criteria for symmetric disturbances to a zonally symmetric baroclinic flow are obtained. In the special case of a perfect gas the results of Stevens (1983) are recovered. Finite-amplitude stability conditions are also obtained that provide an upper bound on a certain positive-definite measure of disturbance amplitude.

Nonlinear stability of continuously stratified quasi-geostrophic flow

Nonlinear stability theorems analogous to Arnol'd's second stability theorem are established for continuously stratified quasi-geostrophic flow with general nonlinear boundary conditions in a vertically and horizontally confined domain. Both the standard quasi-geostrophic model and the modified quasi-geostrophic model (incorporating effects of hydrostatic compressibility) are treated. The results establish explicit upper bounds on the disturbance energy, the disturbance potential enstrophy, and the disturbance available potential energy on the horizontal boundaries, in terms of the initial disturbance fields. Nonlinear stability in the sense of Liapunov is also established.

Nonlinear stability of multilayer quasi-geostrophic flow

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.

Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows

A novel method is presented for obtaining rigorous upper bounds on the finite-amplitude growth of instabilities to parallel shear flows on the beta-plane. The method relies on the existence of finite-amplitude Liapunov (normed) stability theorems, due to Arnol'd, which are nonlinear generalizations of the classical stability theorems of Rayleigh and Fjørtoft. Briefly, the idea is to use the finite-amplitude stability theorems to constrain the evolution of unstable flows in terms of their proximity to a stable flow. Two classes of general bounds are derived, and various examples are considered. It is also shown that, for a certain kind of forced-dissipative problem with dissipation proportional to vorticity, the finite-amplitude stability theorems (which were originally derived for inviscid, unforced flow) remain valid (though they are no longer strictly Liapunov); the saturation bounds therefore continue to hold under these conditions.

The stability of a two-dimensional vorticity filament under uniform strain

The quantitative effects of uniform strain and background rotation on the stability of a strip of constant vorticity (a simple shear layer) are examined. The thickness of the strip decreases in time under the strain, so it is necessary to formulate the linear stability analysis for a time-dependent basic flow. The results show that even a strain rate γ (scaled with the vorticity of the strip) as small as 0.25 suppresses the conventional Rayleigh shear instability mechanism, in the sense that the r.m.s. wave steepness cannot amplify by more than a certain factor, and must eventually decay. For γ < 0.25 the amplification factor increases as γ decreases; however, it is only 3 when γ e 0.065. Numerical simulations confirm the predictions of linear theory at small steepness and predict a threshold value necessary for the formation of coherent vortices. The results help to explain the impression from numerous simulations of two-dimensional turbulence reported in the literature that filaments of vorticity infrequently roll up into vortices. The stabilization effect may be expected to extend to two- and three-dimensional quasi-geostrophic flows.

A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids

In addition to the Hamiltonian functional itself, non-canonical Hamiltonian dynamical systems generally possess integral invariants known as ‘Casimir functionals’. In the case of the Euler equations for a perfect fluid, the Casimir functionals correspond to the vortex topology, whose invariance derives from the particle-relabelling symmetry of the underlying Lagrangian equations of motion. In a recent paper, Vallis, Carnevale & Young (1989) have presented algorithms for finding steady states of the Euler equations that represent extrema of energy subject to given vortex topology, and are therefore stable. The purpose of this note is to point out a very general method for modifying any Hamiltonian dynamical system into an algorithm that is analogous to those of Vallis etal. in that it will systematically increase or decrease the energy of the system while preserving all of the Casimir invariants. By incorporating momentum into the extremization procedure, the algorithm is able to find steadily-translating as well as steady stable states. The method is applied to a variety of perfect-fluid systems, including Euler flow as well as compressible and incompressible stratified flow.

Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere

It is shown that, for a sufficiently large value of β, two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate Ω. This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Liapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute-vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity is that the wave steepness ε < 1, where ε = 2[surd radical]2Z/βU in the planar case and $\epsilon = 2^{\frac{1}{4}} a^{\frac{5}{2}}Z^{\frac{7}{4}}/\Omega U^{\frac{5}{2}}$ in the spherical case, and where Z is the enstrophy, U the r.m.s. velocity, and a the radius of the sphere. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which ε decreases in time) seem to evolve into a non-ergodic regime at large scales.

Modelling the rheology of sea ice as a collection of diamond-shaped floes

In polar oceans, seawater freezes to form a layer of sea ice of several metres thickness that can cover up to 8% of the Earth’s surface. The modelled sea ice cover state is described by thickness and orientational distribution of interlocking, anisotropic diamond-shaped ice floes delineated by slip lines, as supported by observation. The purpose of this study is to develop a set of equations describing the mean-field sea ice stresses that result from interactions between the ice floes and the evolution of the ice floe orientation, which are simple enough to be incorporated into a climate model. The sea ice stress caused by a deformation of the ice cover is determined by employing an existing kinematic model of ice floe motion, which enables us to calculate the forces acting on the ice floes due to crushing into and sliding past each other, and then by averaging over all possible floe orientations. We describe the orientational floe distribution with a structure tensor and propose an evolution equation for this tensor that accounts for rigid body rotation of the floes, their apparent re-orientation due to new slip line formation, and change of shape of the floes due to freezing and melting. The form of the evolution equation proposed is motivated by laboratory observations of sea ice failure under controlled conditions. Finally, we present simulations of the evolution of sea ice stress and floe orientation for several imposed flow types. Although evidence to test the simulations against is lacking, the simulations seem physically reasonable.