Perturbation Kinetic Energy Research Articles

Madelung transformation of the quantum bouncer problem

The “quantum bouncer” problem of a quantum particle in a gravitational field, bouncing elastically from a horizontal surface, is mapped via the Madelung transformation into the dynamics of a 1D, compressible, inviscid fluid. Under this transformation the energy states of the quantum bouncer, given in terms of the Airy function, are hydrostatic equilibria states where gravity is balanced by the corresponding vertical pressure gradient force exerted by the fluid. The stability of the hydrostatic states, with respect to small perturbations, is expressed in terms of the positive definite pseudoenergy integral derived here. Pseudoenergy, which is a constant of motion of the linearized perturbation dynamics, is in this case the sum of the perturbation kinetic and available potential energies. The latter is found to be proportional to the square of the vertical derivative of the buoyancy perturbation rather than the square of the buoyancy itself, as is generally the case in classical fluids. For sinusoidal perturbations with vertical wavenumber k, the effective frequency of the buoyancy oscillation is found to be , which is twice the ratio between the gravitational force and the momentum of a free quantum particle.

Variations in atmospheric perturbation potential energy associated with the South China Sea summer monsoon

This study investigates the energy conversion processes and their relation to convection (circulation) during the South China Sea summer monsoon (SCSSM) years from the viewpoint of atmospheric perturbation potential energy (PPE). An atmospheric PPE dipole pattern associated with the SCSSM develops over the western North Pacific (WNP) and southern Maritime Continent (SMC) in the boreal summer, serving as a link between the SCSSM and diabatic heating. Actually, the conversion between the energy variations and the convection over the WNP is distinctly different with that over the SMC. The precipitation leads the PPE over the WNP, while similar situation is reversed over the SMC. During strong SCSSM years, the higher PPE over the WNP, controlled primarily by the latent heat released from condensation related to surplus precipitation, is corresponding to the negative energy conversion ( text{C}_{k} ) over there. This indicates that more PPE is converted to perturbation kinetic energy and further intensifying ascending motion over the WNP. Consequently, the descending movement reduces the PPE and is corresponding to positive text{C}_{k} over the SMC, suggesting that the less PPE converts into the perturbation kinetic energy and in turn favors the descending movement and deficit precipitation there. The enhanced southwesterly induced by this SCSSM Hadley circulation, superimposed on the mean southwesterly wind, further favors the intensification of the SCSSM, implying that the SCSSM can maintain development through the positive convection–PPE–circulation feedback.

Modal and non-modal linear stability of Poiseuille flow through a channel with a porous substrate

Abstract We present modal and non-modal linear stability analyses of Poiseuille flow through a plane channel with a porous substrate modeled using the Volume Averaged Navier–Stokes (VANS) equations. Modal stability analysis shows the destabilization of the flow with increasing porosity of the layer. The instability mode originates from the homogeneous fluid region of the channel for all the values of porosity considered but the governing mechanism changes. Perturbation kinetic energy analysis reveals the importance of viscous dissipation at low porosity values while dissipation in the porous substrate becomes significant at higher porosity. Scaling analysis highlights the invariance of the critical wavenumber with changing porosity. On the other hand, the critical Reynolds number remains invariant at low porosity and scales as R e c ∼ ( H ∕ δ ) 1 . 4 at high porosity where δ is the typical thickness of the vorticity layer at the fluid–porous interface. This reveals the existence of a Tollmien–Schlichting-like viscous instability mechanism at low porosity values, and Rayleigh analysis shows the presence of an inviscid instability mechanism at high porosity. For the whole range of porosities considered, the non-modal analysis shows that the optimal mechanism responsible for transient energy amplification is the lift-up effect, giving rise to streaky structure as in single-phase plane Poiseuille flow. The present results strongly suggest that the transition to turbulence follows the same path as that of classical Poiseuille flow at low porosity values, while it is dictated by the modal instability for high porosity values.

The effect of magnetic field on perturbation evolution in homogeneously sheared flows

The goal of this paper is to identify and characterize the fundamental mechanisms that contribute toward flow stabilization in sheared plasma flows. Toward that end, we investigate the evolution of velocity and magnetic field perturbations in homogeneously sheared magnetohydrodynamic (MHD) flows subjected to an imposed streamwise magnetic field. The influences of magnetic field strength ($B_{0}$) and perturbation wavevector orientation ($\unicode[STIX]{x1D6FD}$) are characterized using linear analysis and direct numerical simulations. The linear analysis of ideal MHD indicates that the perturbation evolution is governed by four processes: pressure redistribution, kinetic energy production, kinetic–magnetic energy exchange and magnetic energy production due to magnetic stretching. The interplay between these processes can be characterized by the ratio of shear-to-magnetic timescales ($R_{A}\equiv V_{A}\unicode[STIX]{x1D705}/S$) and $\unicode[STIX]{x1D6FD}$, where $V_{A}$, $\unicode[STIX]{x1D705}$ and $S$ are Alfvén wave speed, initial wavenumber and mean flow shear, respectively. For cases with low values of $\boldsymbol{R}_{\boldsymbol{A}}~(\ll \mathbf{1})$, a three-stage perturbation evolution is seen. At the first stage, pressure redistribution and production dominate leading to hydrodynamic-type behaviour. In the second stage, the onset of magnetic stretching process leads to an increase in magnetic energy. At late stages, production subsides and the dynamics is dominated by harmonic exchange between velocity and magnetic fields. For cases of $\boldsymbol{R}_{\boldsymbol{A}}\approx \mathbf{1}$, the magnetic field reacts rapidly enough that hydrodynamic and magnetic production stages occur simultaneously followed by harmonic exchange. In the case of $\boldsymbol{R}_{\boldsymbol{A}}\gg 1$, all three stages occur simultaneously leading to harmonic exchange between kinetic and magnetic energies without any perturbation growth. For all cases considered, the late stage harmonic exchange results in equipartition between perturbation magnetic and kinetic energies. For a given $R_{A}$, the effect of increasing $\unicode[STIX]{x1D6FD}$ is to reduce the intensity of coupling and progressively slow down the three stages of evolution. For spanwise wavevector perturbations, the velocity–magnetic field interaction mechanism vanishes and there is no effect of pressure or magnetic field on individual velocity components.

Atmospheric energetics over the tropical Indian Ocean during Indian Ocean dipole events

The evolution of atmospheric perturbation potential energy (PPE) over the tropical Indian Ocean is analyzed during a composite positive IOD event using reanalysis datasets for the period 1948–2015. The IOD modulates the variation in PPE, which then affects perturbation kinetic energy (PKE) through energy conversion. The PPE anomalies in the lower troposphere (1000–850 hPa) as the dominant layer of the PPE in the whole troposphere (1000–150 hPa) present a dipole pattern corresponding to the anomalous variation in SST during the IOD event. When cold SST anomalies (SSTAs) first appear in the eastern Indian Ocean (IOD-E), they reduce the atmospheric PPE in the lower troposphere rapidly. The negative PPE anomalies lead to less energy conversion to PKE, restraining the surface wind convergence over the IOD-E and weakening the climatological Walker circulation. Meanwhile, the surface easterly wind anomalies strengthen, which depresses the thermocline to the west and gives warmer SSTAs in the western Indian Ocean (IOD-W). The PPE anomalies and energy conversion (CK) over the IOD-W are opposite to those over the IOD-E, the anomalous easterly wind continues to develop, and the positive SSTAs in the IOD-W reach a peak. Thus, the response of the Walker circulation over the Indian Ocean provides a positive feedback during the IOD event and explains the delayed effect of IOD-E SSTAs on IOD-W SSTAs.

The scale dependence of initial‐condition sensitivities in simulations of convective systems over the southeastern United States

The sensitivity of ensemble simulations of deep convective events in the southeastern United States to initial‐condition (IC) errors is examined by imposing idealized moisture perturbations at small and large scales. Four severe weather events are considered, ranging from a springtime frontal system to convection driven almost exclusively by daytime heating. Events with strong synoptic‐scale forcing were insensitive to the scale of IC errors, but weakly forced events exhibited greater sensitivity to small‐scale than large‐scale IC errors. Additional ensemble simulations of idealized convective systems suggest that the greater sensitivity to small‐scale IC errors of the weakly forced cases arises from their higher sensitivity to the strength and location of the first convective elements. Ensemble spread and predictability are characterized by two measures: the ratio of the perturbation kinetic energy (KE) about the ensemble mean to the background KE and the neighborhood‐based fractions skill score (FSS) of hourly precipitation with respect to that in an unperturbed reference simulation. For simulations of both observed events and idealized convective systems, the FSS appears to be a more discriminating indicator of differences in predictability between different convective events.

Ensemble Spread Grows More Rapidly in Higher-Resolution Simulations of Deep Convection

Idealized ensemble simulations of mesoscale convective systems (MCSs) with horizontal grid spacings of 1, 1.4, and 2 km are used to analyze the influence of numerical resolution on the rate of growth of ensemble spread in convection-resolving numerical models. The ensembles are initialized with random phases of 91-km-wavelength moisture perturbations that are captured with essentially identical accuracy at all resolutions. The rate of growth of ensemble variance is shown to systematically increase at higher resolution. The largest horizontal wavelength at which the perturbation kinetic energy (KE′) grows to at least 50% of the background kinetic energy spectrum is also shown to grow more rapidly at higher resolution. The mechanism by which the presence of smaller scales accelerates the upscale growth of KE′ is clear-cut in the smooth-saturation Lorenz–Rotunno–Snyder (ssLRS) model of homogeneous surface quasigeostrophic turbulence. Comparing the growth of KE′ from the MCS ensemble simulations to that in the ssLRS model suggests interactions between perturbations at small scales, where KE′ is not yet completely saturated, and somewhat larger scales, where KE′ is clearly unsaturated, are responsible for the faster growth rate of ensemble variance at finer resolution. These results provide some empirical justification for the use of deep-convection-related stochastic parameterization schemes to reduce the problem of underdispersion in coarser-resolution ensemble prediction systems.

Abrupt Climate Shift in the Mature Rainy Season of the Philippines in the Mid-1990s

A robust climate shift around 1993/1994 from early August to early September, which corresponds to the mature rainy season of the Philippines, was identified in stations located over the western coast of the country. The convection in the mature rainy season during 1994–2008 (E2) was suppressed compared with 1979–1993 (E1). The possible role of the changes in the large-scale conditions and tropical cyclone (TC) activity were analyzed. The results show that the western North Pacific Subtropical High has extended further westward in E2 leading to an enhanced lower-level divergence and less moisture transport over the Philippines. The changes in the large-scale conditions, which featured a mid-tropospheric descent, a decrease in low-level relative humidity, an enhanced vertical zonal wind shear, and a decrease in the perturbation kinetic energy, also inhibited the synoptic-scale disturbances in the vicinity of the Philippines. In particular, fewer TCs developed and made landfall over the Philippines in E2. We also found inconsistent climate shifts in May, June, July, and September between the rainfall data from the stations and the Climate Prediction Center Merged Analysis of Precipitation, which highlights the importance of sub-seasonal analysis in decadal-to-interdecadal climate change studies.

Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Péclet and Richardson numbers

We consider the nonlinear optimisation of irreversible mixing induced by an initial finite amplitude perturbation of a statically stable density-stratified fluid with kinematic viscosity $\unicode[STIX]{x1D708}$ and density diffusivity $\unicode[STIX]{x1D705}$. The initial diffusive error function density distribution varies continuously so that $\unicode[STIX]{x1D70C}\in [\bar{\unicode[STIX]{x1D70C}}-\unicode[STIX]{x1D70C}_{0}/2,\bar{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D70C}_{0}/2]$. A constant pressure gradient is imposed in a plane two-dimensional channel of depth $2h$. We consider flows with a finite Péclet number $Pe=U_{m}h/\unicode[STIX]{x1D705}=500$ and Prandtl number $Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}=1$, and a range of bulk Richardson numbers $Ri_{b}=g\unicode[STIX]{x1D70C}_{0}h/(\bar{\unicode[STIX]{x1D70C}}U^{2})\in [0,1]$ where $U_{m}$ is the maximum flow speed of the laminar parallel flow, and $g$ is the gravitational acceleration. We use the constrained variational direct-adjoint-looping (DAL) method to solve two optimisation problems, extending the optimal mixing results of Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241–277) to stratified flows, where the irreversible mixing of the active scalar density leads to a conversion of kinetic energy into potential energy. We identify initial perturbations of fixed finite kinetic energy which maximise the time-averaged perturbation kinetic energy developed over a finite time interval, and initial perturbations that minimise the value (at a target time, chosen to be $T=10$) of a ‘mix-norm’ as first introduced by Mathew et al. (Physica D, vol. 211, 2005, pp. 23–46), further discussed by Thiffeault (Nonlinearity, vol. 25, 2012, pp. 1–44) and shown by Foures et al. (2014) to be a computationally efficient and robust proxy for identifying perturbations that minimise the long-time variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the time-averaged kinetic-energy maximising perturbations are significantly suboptimal at mixing compared to the mix-norm minimising perturbations, and also that minimising the mix-norm remains (for density-stratified flows) a good proxy for identifying perturbations which minimise the variance at long times. Although increasing stratification reduces the mixing in general, mix-norm minimising optimal perturbations can still trigger substantial mixing for $Ri_{b}\lesssim 0.3$. By considering the time evolution of the kinetic energy and potential energy reservoirs, we find that such perturbations lead to a flow which, through Taylor dispersion, very effectively converts perturbation kinetic energy into ‘available potential energy’, which in turn leads rapidly and irreversibly to thorough and efficient mixing, with little energy returned to the kinetic-energy reservoirs.

Characteristics of the leading Lyapunov vector in a turbulent channel flow

The values of the highest Lyapunov exponent (HLE)$\unicode[STIX]{x1D706}_{1}$for turbulent flow in a plane channel at Reynolds numbers up to$Re_{\unicode[STIX]{x1D70F}}=586$are determined. The instantaneous and statistical properties of the corresponding leading Lyapunov vector (LLV) are investigated. The LLV is calculated by numerical solution of the Navier–Stokes equations linearized about the non-stationary base solution corresponding to the developed turbulent flow. The base turbulent flow is calculated in parallel with the calculation of the evolution of the perturbations. For arbitrary initial conditions, the regime of exponential growth${\sim}\exp (\unicode[STIX]{x1D706}_{1}t)$which corresponds to the approaching of the perturbation to the LLV is achieved already at$t^{+}<50$. It is found that the HLE increases with increasing Reynolds number from$\unicode[STIX]{x1D706}_{1}^{+}\approx 0.021$at$Re_{\unicode[STIX]{x1D70F}}=180$to$\unicode[STIX]{x1D706}_{1}^{+}\approx 0.026$at$Re_{\unicode[STIX]{x1D70F}}=586$. The LLV structures are concentrated mainly in a region of the buffer layer and are manifested in the form of spots of increased fluctuation intensity localized both in time and space. The root-mean-square (r.m.s.) profiles of the velocity and vorticity intensities in the LLV are qualitatively close to the corresponding profiles in the base flow with artificially removed near-wall streaks. The difference is the larger concentration of LLV perturbations in the vicinity of the buffer layer and a relatively larger (by approximately 80 %) amplitude of the vorticity pulsations. Based on the energy spectra of velocity and vorticity pulsations, the integral spatial scales of the LLV structures are determined. It is found that LLV structures are on average twice narrower and twice shorter than the corresponding structures of the base flow. The contribution of each of the terms entering into the expression for the production of the perturbation kinetic energy is determined. It is shown that the process of perturbation development is essentially dictated by the inhomogeneity of the base flow, as well as by the presence of transversal motion in it. Neglecting of these factors leads to a significant underestimation of the perturbation growth rate. The presence of near-wall streaks in the base flow, on the contrary, does not play a significant role in the development of the LLV perturbations. Artificial removal of streaks from the base flow does not change the character of the perturbation growth.

Modulation of tropical cyclone tracks over the western North Pacific by intra-seasonal Indo-western Pacific convection oscillation during the boreal extended summer

This study investigates the effect of the intra-seasonal Indo-western Pacific convection oscillation (IPCO) on tropical cyclone (TC) tracks over the western North Pacific (WNP) during the boreal extended summer (May−October). The number of west- and northwest-moving TC tracks is found to sharply increase over the WNP in the positive intra-seasonal IPCO phase. Recurving tracks have greater weight in the negative intra-seasonal IPCO phase. Possible physical mechanisms are further examined in terms of steering flow, energy conversion, and energy propagation. When the intra-seasonal IPCO phase is positive, the first-order moment term of perturbation potential energy (PPE1) converts into perturbation kinetic energy (PKE) at lower latitudes. The pressure trough spreads farther to the east. Meanwhile, Rossby waves emanating from the convective centers of the intra-seasonal IPCO over the WNP (WaveWNP) and EEIO (WaveEEIO) travel into the trough region, thereby deepening the trough. These features enhance the westward and northwestward steering flow between 20°N and about 30°N, sharply increasing the number of straight west- and northwest-moving TC tracks over the WNP. When the intra-seasonal IPCO is in a negative phase, conversion from PPE1 to PKE at lower latitudes is suppressed and the trough weakens. More PPE1 converts to PKE in the climatological western Pacific subtropical high (WPSH) region and the WPSH is intensified. Moreover, WaveEEIO intensifies the north–south ridge of the WPSH over the southern Indian Peninsula. Meanwhile, part WaveWNP propagate northeastward. These features favor northeastward motion of TCs over the WNP.

The Impact of Dimensionality on Barotropic Processes during KWAJEX

Abstract In this study, the 2D and 3D cloud-resolving model simulations of the Tropical Rainfall Measuring Mission (TRMM) Kwajalein Experiment (KWAJEX) are compared to study the impact of dimensionality on barotropic processes during tropical convective development. Barotropic conversion of perturbation kinetic energy is associated with vertical transport of horizontal momentum under vertical shear of background horizontal winds. The similarities in both 2D and 3D model simulations show that 1) vertical wind shear is a necessary condition for barotropic conversion, but it does not control the barotropic conversion; 2) the evolution of barotropic conversion is related to that of the vertical transport of horizontal momentum; and 3) the tendency of vertical transport of horizontal momentum is mainly determined by the covariance between horizontal wind and the cloud hydrometeor component of buoyancy. The differences between the 2D and 3D model simulations reveal that 1) the barotropic conversion has shorter time scales and a larger contribution in the 2D model simulation than in the 3D model simulation and 2) kinetic energy is generally converted from the mean circulations to perturbation circulations in the 3D model simulation. In contrast, more kinetic energy is transferred from perturbation circulations to the mean circulations in the 2D model simulation. The same large-scale vertical velocity may account for the similarities, whereas the inclusion of meridional winds in the 3D model simulation may be responsible for the differences in barotropic conversion between the 2D and 3D model simulations.

Internal Intraseasonal Variability of the West African Monsoon in WRF

AbstractThe West African monsoon (WAM) and its landmark features, which include African easterly waves (AEWs) and the African easterly jet (AEJ), exhibit significant intraseasonal variability in boreal summer. However, the degree to which this variability is modulated by external large-scale phenomena, such as the Madden–Julian oscillation (MJO), remains unclear. The Weather Research and Forecasting (WRF) Model is employed to diagnose the importance of the MJO and other external influences for the intraseasonal variability of the WAM and associated AEW energetics by removing 30–90-day signals from initial and lateral boundary conditions in sensitivity tests. The WAM produces similar intraseasonal variability in the absence of external influences, indicating that the MJO is not critical to produce WAM variability. In control and sensitivity experiments, AEW precursor signals are similar near the AEJ entrance in East Africa. For example, an eastward extension of the AEJ increases barotropic and baroclinic energy conversions in East Africa prior to a 30–90-day maximum of perturbation kinetic energy in West Africa. The WAM appears to prefer a faster oscillation when MJO forcing is removed, suggesting that the MJO may serve as a pacemaker for intraseasonal oscillations in the WAM. WRF results show that eastward propagating intraseasonal signals (e.g., Kelvin wave fronts) are responsible for this pacing, while the role of westward propagating intraseasonal signals (e.g., MJO-induced Rossby waves) appears to be limited. Mean state biases across the simulations complicate the interpretation of results.

Linear stability of thermocapillary liquid layers of a shear-thinning fluid

The linear stability analysis has been performed for the thermocapillary liquid layers of a shear-thinning fluid. The Carreau fluid model is applied to describe the rheological property. The critical parameters are determined as a function of Prandtl number (Pr), degree of shear-thinning, and gravity level. For linear flow, the shear-thinning effect is destabilizing for small and moderate Pr but increases the stability slightly for large Pr. For return flow, the perturbation kinetic energy concentrates near the surface, and the flow is stabilized when the surface viscosity is used. The streamwise wave is excited at large Pr, and a new mechanism is found at moderate Pr, where the hot spots appear at the bottom of the layer. In the presence of gravity, the viscosity stratification is enhanced and more kinds of different modes are excited. The preferred mode changes to downstream at large Pr while the gravity becomes an important perturbation energy source at small Pr. The shear-thinning effect for the instability mechanism is discussed and the comparisons are made with channel flows.

The energy transfer mechanism of a perturbed solid-body rotation flow in a rotating pipe

Three-dimensional direct numerical simulations of a solid-body rotation superposed on a uniform axial flow entering a rotating constant-area pipe of finite length are presented. Steady in time profiles of the radial, axial, and circumferential velocities are imposed at the pipe inlet. Convective boundary conditions are imposed at the pipe outlet. The Wang and Rusak (Phys. Fluids 8:1007–1016, 1996. doi:10.1063/1.86882) axisymmetric instability mechanism is retrieved at certain operational conditions in terms of incoming flow swirl levels and the Reynolds number. However, at other operational conditions there exists a dominant, three-dimensional spiral type of instability mode that is consistent with the linear stability theory of Wang et al. (J. Fluid Mech. 797: 284–321, 2016). The growth of this mode leads to a spiral type of flow roll-up that subsequently nonlinearly saturates on a large amplitude rotating spiral wave. The energy transfer mechanism between the bulk of the flow and the perturbations is studied by the Reynolds-Orr equation. The production or loss of the perturbation kinetic energy is combined of three components: the viscous loss, the convective loss at the pipe outlet, and the gain of energy at the outlet through the work done by the pressure perturbation. The energy transfer in the nonlinear stage is shown to be a natural extension of the linear stage with a nonlinear saturated process.

Turbulent mixing due to the Holmboe wave instability at high Reynolds number

We consider numerically the transition to turbulence and associated mixing in stratified shear flows with initial velocity distribution $\overline{U}(z,0)\,\boldsymbol{e}_{x}=U_{0}\,\boldsymbol{e}_{x}\tanh (z/d)$ and initial density distribution $\overline{\unicode[STIX]{x1D70C}}(z,0)=\unicode[STIX]{x1D70C}_{0}[1-\tanh (z/\unicode[STIX]{x1D6FF})]$ away from a hydrostatic reference state $\unicode[STIX]{x1D70C}_{r}\gg \unicode[STIX]{x1D70C}_{0}$. When the ratio $R=d/\unicode[STIX]{x1D6FF}$ of the characteristic length scales over which the velocity and density vary is equal to one, this flow is primarily susceptible to the classic well-known Kelvin–Helmholtz instability (KHI). This instability, which typically manifests at finite amplitude as an array of elliptical vortices, strongly ‘overturns’ the density interface of strong initial gradient, which nevertheless is the location of minimum initial gradient Richardson number $Ri_{g}(0)=Ri_{b}=g\unicode[STIX]{x1D70C}_{0}d/\unicode[STIX]{x1D70C}_{r}U_{0}^{2}$, where $Ri_{g}(z)=-([g/\unicode[STIX]{x1D70C}_{r}]\,\text{d}\overline{\unicode[STIX]{x1D70C}}/\text{d}z)/(\text{d}\overline{U}/\text{d}z)^{2}$ and $Ri_{b}$ is a bulk Richardson number. As is well known, at sufficiently high Reynolds numbers ($Re$), the primary KHI induces a vigorous but inherently transient burst of turbulence and associated irreversible mixing localised in the vicinity of the density interface, leading to a relatively well-mixed region bounded by stronger density gradients above and below. We explore the qualitatively different behaviour that arises when $R\gg 1$, and so the density interface is sharp, with $Ri_{g}(z)$ now being maximum at the density interface $Ri_{g}(0)=RRi_{b}$. This flow is primarily susceptible to Holmboe wave instability (HWI) (Holmboe, Geophys. Publ., vol. 24, 1962, pp. 67–113), which manifests at finite amplitude in this symmetric flow as counter-propagating trains of elliptical vortices above and below the density interface, thus perturbing the interface so as to exhibit characteristically cusped interfacial waves which thereby ‘scour’ the density interface. Unlike previous lower-$Re$ experimental and numerical studies, when $Re$ is sufficiently high the primary HWI becomes increasingly more three-dimensional due to the emergence of shear-aligned secondary convective instabilities. As $Re$ increases, (i) the growth rate of secondary instabilities appears to saturate and (ii) the perturbation kinetic energy exhibits a $k^{-5/3}$ spectrum for sufficiently large length scales that are influenced by anisotropic buoyancy effects. Therefore, at sufficiently high $Re$, vigorous turbulence is triggered that also significantly ‘scours’ the primary density interface, leading to substantial irreversible mixing and vertical transport of mass above and below the (robust) primary density interface. Furthermore, HWI produces a markedly more long-lived turbulence event compared to KHI at a similarly high $Re$. Despite their vastly different mechanics (i.e. ‘overturning’ versus ‘scouring’) and localisation, the mixing induced by KHI and HWI is comparable in both absolute terms and relative efficiency. Our results establish that, provided the flow Reynolds number is sufficiently high, shear layers with sharp density interfaces and associated locally high values of the gradient Richardson number may still be sites of substantial and efficient irreversible mixing.

Barotropic and baroclinic processes associated with convective development in the tropical deep convective regime

The budget of perturbation kinetic energy is calculated to study barotropic and baroclinic processes associated with convective development in the tropical deep convective regime. The barotropic kinetic-energy conversion is further examined through the derivation and analysis of budget of vertical transport of zonal momentum. The two-dimensional cloud-resolving model simulation data during TOGA COARE are analyzed in this budget study. The analysis of the energy budget shows that while baroclinic conversion from perturbation available potential energy is a major source for the growth of perturbation kinetic energy, barotropic conversion from the mean kinetic energy has significant modification on the variation of perturbation kinetic energy. The general evolution of barotropic conversion is associated with that of the vertical wind shear of the mean circulations, but the significant change in barotropic conversion corresponds to the change in the vertical transport of zonal momentum. Zonal transport of cloud hydrometeor is responsible for the variation of vertical transport of zonal momentum.

Multiscale Temporal Mean Features of Perturbation Kinetic Energy and Its Budget in the Tropics: Review and Computation

Abstract The authors examined the maintenance mechanisms of perturbation kinetic energy (PKE) in the tropical regions for multiple time scales by computing and analyzing its budget equation. The emphasis has been placed on the mean features of synoptic and subseasonal variabilities using a 33-yr dataset. From analysis of the contributions from u-wind and υ-wind components, the PKE maximum in the Indian Ocean is attributed less to synoptic variability and more to intraseasonal variability in which the Madden–Julian oscillation (MJO) dominates; however, there is strong evidence of seasonal variability affiliated with the Asian monsoon systems. The ones in the eastern Pacific and Atlantic Oceans are closely related to both intraseasonal and synoptic variability that result from the strong MJO and the relatively large amplitude of equatorial waves. The maintenance of the PKE budget mainly depends on the structure of time mean horizontal flows, the location of convection, and the transport of PKE from the extratropics. In the regions with strong convective activities, such as the eastern Indian Ocean to the western Pacific, the production of PKE occurs between 700 and 200 hPa at the expense of perturbation available potential energy (PAPE), which is generated by convective heating. This gain in PKE is largely offset by divergence of the geopotential component of vertical energy flux; that is, it is redistributed to the upper- and lower-atmospheric layers by the pressure field. Strong PKE generation through the horizontal convergence of the extratropical energy flux takes place in the upper troposphere over the eastern Pacific and Atlantic Ocean, and is largely balanced by a PKE loss due to barotropic conversion, which is determined solely by the sign of longitudinal stretching deformation. However, over the Indian Ocean, there is a net PKE loss due to divergence of energy flux, which is compensated by PKE gain through the shear generation.

Linear stability and energetics of rotating radial horizontal convection

The effect of rotation on horizontal convection in a cylindrical enclosure is investigated numerically. The thermal forcing is applied radially on the bottom boundary from the coincident axes of rotation and geometric symmetry of the enclosure. First, a spectral element method is used to obtain axisymmetric basic flow solutions to the time-dependent incompressible Navier–Stokes equations coupled via a Boussinesq approximation to a thermal transport equation for temperature. Solutions are obtained primarily at Rayleigh number $\mathit{Ra}=10^{9}$ and rotation parameters up to $Q=60$ (where $Q$ is a non-dimensional ratio between thermal boundary layer thickness and Ekman layer depth) at a fixed Prandtl number $\mathit{Pr}=6.14$ representative of water and enclosure height-to-radius ratio $H/R=0.4$. The axisymmetric solutions are consistently steady state at these parameters, and transition from a regime unaffected by rotation to an intermediate regime occurs at $Q\approx 1$ in which variation in thermal boundary layer thickness and Nusselt number are shown to be governed by a scaling proposed by Stern (1975, Ocean Circulation Physics. Academic). In this regime an increase in $Q$ sees the flow accumulate available potential energy and more strongly satisfy an inviscid change in potential energy criterion for baroclinic instability. At the strongest $Q$ the flow is dominated by rotation, accumulation of available potential energy ceases and horizontal convection is suppressed. A linear stability analysis reveals several instability mode branches, with dominant wavenumbers typically scaling with $Q$. Analysis of contributing terms of an azimuthally averaged perturbation kinetic energy equation applied to instability eigenmodes reveals that energy production by shear in the axisymmetric mean flow is negligible relative to that produced by conversion of available potential energy from the mean flow. An evolution equation for the quantity that facilitates this exchange, the vertical advective buoyancy flux, reveals that a baroclinic instability mechanism dominates over $5\lesssim Q\lesssim 30$, whereas stronger and weaker rotations are destabilised by vertical thermal gradients in the mean flow.

Nonlinear Control of Axisymmetric Swirling Flows in a Long Finite-Length Pipe

Feedback stabilization of inviscid and high Reynolds number, axisymmetric, swirling flows in a long finite-length circular pipe using active variations of pipe geometry as a function of the evolving inlet radial velocity is studied. The complicated dynamics of the natural flow requires that any theoretical model that attempts to control vortex stability must include the essential nonlinear dynamics of the perturbation modes. In addition, the control methodology must establish a stable desired state with a wide basin of attraction. The present approach is built on a weakly nonlinear model problem for the analysis of perturbation dynamics on near-critical swirling flows in a slightly area-varying, long, circular pipe with unsteady changes of wall geometry. In the natural case with no control, flows with incoming swirl ratio above a critical level are unstable and rapidly evolve to either vortex breakdown states or accelerated flow states. Following an integration of the model equation, a perturbation kinetic-energy identity is derived, and an active feedback control methodology to suppress perturbations from a desired columnar state is proposed. The stabilization of both inviscid and high-Re flows is demonstrated for a wide range of swirl ratios above the critical swirl for vortex breakdown and for large-amplitude initial perturbations. The control gain for the fastest decay of perturbations is found to be a function of the swirl level. Large gain values are required at near-critical swirl ratios while lower gains provide a successful control at swirl levels away from critical. This feedback control technique cuts the feed-forward mechanism between the inlet radial velocity and the growth of perturbation's kinetic energy in the bulk and thereby enforces the decay of perturbations and eliminates the natural explosive evolution of the vortex breakdown process. The application of this proposed robust active feedback control method establishes a branch of columnar states with a wide basin of attraction for swirl ratios up to at least 50% above the critical swirl. This study provides guidelines for future flow control simulations and experiments. However, the present methodology is limited to the control of high-Reynolds number (nearly inviscid), axisymmetric, weakly nonparallel flows in long pipes.