scholarly journals Step-size effect in the time-transformed leapfrog integrator on elliptic and hyperbolic orbits

2020 ◽  
Vol 497 (4) ◽  
pp. 4384-4389
Author(s):  
Long Wang ◽  
Keigo Nitadori
Keyword(s):  
Phase Transition ◽  
Kinetic Energy ◽  
Potential Energy ◽  
Size Effect ◽  
Mathematical Explanation ◽  
Symplectic Integrator ◽  
Step Size ◽  
Hyperbolic Orbit ◽  
The Mean ◽  
Integration Errors

ABSTRACT A drift-kick-drift (DKD) type leapfrog symplectic integrator applied for a time-transformed separable Hamiltonian (or time-transformed symplectic integrator; TSI) has been known to conserve the Kepler orbit exactly. We find that for an elliptic orbit, such feature appears for an arbitrary step size. But it is not the case for a hyperbolic orbit: When the half step size is larger than the conjugate momenta of the mean anomaly, a phase transition happens and the new position jumps to the non-physical counterpart of the hyperbolic trajectory. Once it happens, the energy conservation is broken. Instead, the kinetic energy minus the potential energy becomes a new conserved quantity. We provide a mathematical explanation for such phenomenon. Our result provides a deeper understanding of the TSI method, and a useful constraint of the step size when the TSI method is used to solve the hyperbolic encounters. This is particular important when an (Bulirsch–Stoer) extrapolation integrator is used together, which requires the convergence of integration errors.

Study of the Mechanisms of Vortex Variability in the Lofoten Basin Based on the Energy Analysis

2021 ◽  
Vol 37 (3) ◽  
Author(s):  
V. S. Travkin ◽  
◽  
T. V. Belonenko ◽  
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Baroclinic Instability ◽  
Positive Trend ◽  
Barotropic Instability ◽  
Available Potential Energy ◽  
Conversion Rates ◽  
Order Of Magnitude ◽  
The Mean ◽  
Mean Kinetic Energy

Purpose. The Lofoten Basin is one of the most energetic zones of the World Ocean characterized by high activity of mesoscale eddies. The study is aimed at analyzing different components of general energy in the basin, namely the mean kinetic and vortex kinetic energy calculated using the integral of the volume of available potential and kinetic energy of the Lofoten Vortex, as well as variability of these characteristics. Methods and Results. GLORYS12V1 reanalysis data for the period 2010–2018 were used. The mean kinetic energy and the eddy kinetic one were analyzed; and as for the Lofoten Vortex, its volume available potential and kinetic energy were studied. The mesoscale activity of eddies in winter is higher than in summer. Evolution of the available potential energy and kinetic energy of the Lofoten Vortex up to the 1000 m horizon was studied. It is shown that the vortex available potential energy exceeds the kinetic one by an order of magnitude, and there is a positive trend with the coefficient 0,23⋅1015 J/year. It was found that in the Lofoten Basin, the intermediate layer from 600 to 900 m made the largest contribution to the potential energy, whereas the 0–400 m layer – to kinetic energy. The conversion rates of the mean kinetic energy into the vortex kinetic one and the mean available potential energy into the vortex available potential one (barotropic and baroclinic instability) were analyzed. It is shown that the first type of transformation dominates in summer, while the second one is characterized by its increase in winter. Conclusions. The vertical profile shows that the kinetic energy of eddies in winter is higher than in summer. The available potential energy of a vortex is by an order of magnitude greater than the kinetic energy. An increase in the available potential energy is confirmed by a significant positive trend and by a decrease in the vortex Burger number. The graphs of the barotropic instability conversion rate demonstrate the multidirectional flows in the vortex zone with the dipole structure observed in a winter period, and the tripole one – in summer. The barotropic instability highest intensity is observed in summer. The baroclinic instability is characterized by intensification of the regime in winter that is associated with weakening of stratification in this period owing to winter convection.

scholarly journals Ageostrophic instability in rotating, stratified interior vertical shear flows

2014 ◽  
Vol 755 ◽  
pp. 397-428 ◽  
Author(s):  
Peng Wang ◽  
James C. McWilliams ◽  
Claire Ménesguen
Keyword(s):  
Energy Source ◽  
Kinetic Energy ◽  
Potential Energy ◽  
Gravity Waves ◽  
Shear Flows ◽  
Vertical Shear ◽  
Mean Flow ◽  
Efficient Energy Transfer ◽  
The Mean ◽  
Inertia Gravity Waves

AbstractThe linear instability of several rotating, stably stratified, interior vertical shear flows $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\overline{U}(z)$ is calculated in Boussinesq equations. Two types of baroclinic, ageostrophic instability, AI1 and AI2, are found in odd-symmetric $\overline{U}(z)$ for intermediate Rossby number ($\mathit{Ro}$). AI1 has zero frequency; it appears in a continuous transformation of the unstable mode properties between classic baroclinic instability (BCI) and centrifugal instability (CI). It begins to occur at intermediate $\mathit{Ro}$ values and horizontal wavenumbers ($k,l$) that are far from $l= 0$ or $k = 0$, where the growth rate of BCI or CI is the strongest. AI1 grows by drawing kinetic energy from the mean flow, and the perturbation converts kinetic energy to potential energy. The instability AI2 has inertia critical layers (ICL); hence it is associated with inertia-gravity waves. For an unstable AI2 mode, the coupling is either between an interior balanced shear wave and an inertia-gravity wave (BG), or between two inertia-gravity waves (GG). The main energy source for an unstable BG mode is the mean kinetic energy, while the main energy source for an unstable GG mode is the mean available potential energy. AI1 and BG type AI2 occur in the neighbourhood of $A-S= 0$ (a sign change in the difference between absolute vertical vorticity and horizontal strain rate in isentropic coordinates; see McWilliams et al., Phys. Fluids, vol. 10, 1998, pp. 3178–3184), while GG type AI2 arises beyond this condition. Both AI1 and AI2 are unbalanced instabilities; they serve as an initiation of a possible local route for the loss of balance in 3D interior flows, leading to an efficient energy transfer to small scales.

scholarly journals Scaling Laws and Regime Transitions of Macroturbulence in Dry Atmospheres

2008 ◽  
Vol 65 (7) ◽  
pp. 2153-2173 ◽  
Author(s):  
Tapio Schneider ◽  
Christopher C. Walker
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Surface Potential ◽  
Scaling Laws ◽  
Potential Temperature ◽  
Eddy Kinetic Energy ◽  
Eddy Flux ◽  
Available Potential Energy ◽  
Mean Fields ◽  
The Mean

Abstract In simulations of a wide range of circulations with an idealized general circulation model, clear scaling laws of dry atmospheric macroturbulence emerge that are consistent with nonlinear eddy–eddy interactions being weak. The simulations span several decades of eddy energies and include Earth-like circulations and circulations with multiple jets and belts of surface westerlies in each hemisphere. In the simulations, the eddy available potential energy and the barotropic and baroclinic eddy kinetic energy scale linearly with each other, with the ratio of the baroclinic eddy kinetic energy to the barotropic eddy kinetic energy and eddy available potential energy decreasing with increasing planetary radius and rotation rate. Mean values of the meridional eddy flux of surface potential temperature and of the vertically integrated convergence of the meridional eddy flux of zonal momentum generally scale with functions of the eddy energies and the energy-containing eddy length scale, with a few exceptions in simulations with statically near-neutral or neutral extratropical thermal stratifications. Eddy energies scale with the mean available potential energy and with a function of the supercriticality, a measure of the near-surface slope of isentropes. Strongly baroclinic circulations form an extended regime in which eddy energies scale linearly with the mean available potential energy. Mean values of the eddy flux of surface potential temperature and of the vertically integrated eddy momentum flux convergence scale similarly with the mean available potential energy and other mean fields. The scaling laws for the dependence of eddy fields on mean fields exhibit a regime transition between a regime in which the extratropical thermal stratification and tropopause height are controlled by radiation and convection and a regime in which baroclinic entropy fluxes modify the extratropical thermal stratification and tropopause height. At the regime transition, for example, the dependence of the eddy flux of surface potential temperature and the dependence of the vertically integrated eddy momentum flux convergence on mean fields changes—a result with implications for climate stability and for the general circulation of an atmosphere, including its tropical Hadley circulation.

scholarly journals Biases in Thorpe-Scale Estimates of Turbulence Dissipation. Part II: Energetics Arguments and Turbulence Simulations

2015 ◽  
Vol 45 (10) ◽  
pp. 2522-2543 ◽  
Author(s):  
Alberto Scotti
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Turbulent Flows ◽  
Companion Paper ◽  
Mean Flow ◽  
Available Potential Energy ◽  
Ozmidov Scale ◽  
Thorpe Scale ◽  
The Mean ◽  
Shear Driven Flows

AbstractThis paper uses the energetics framework developed by Scotti and White to provide a critical assessment of the widely used Thorpe-scale method, which is used to estimate dissipation and mixing rates in stratified turbulent flows from density measurements along vertical profiles. This study shows that the relevant displacement scale in general is not the rms value of the Thorpe displacement. Rather, the displacement field must be Reynolds decomposed to separate the mean from the turbulent component, and it is the turbulent component that ought to be used to diagnose mixing and dissipation. In general, the energetics of mixing in an overall stably stratified flow involves potentially complex exchanges among the available potential energy and kinetic energy associated with the mean and turbulent components of the flow. The author considers two limiting cases: shear-driven mixing, where mixing comes at the expense of the mean kinetic energy of the flow, and convective-driven mixing, which taps the available potential energy of the mean flow to drive mixing. In shear-driven flows, the rms of the Thorpe displacement, known as the Thorpe scale is shown to be equivalent to the turbulent component of the displacement. In this case, the Thorpe scale approximates the Ozmidov scale, or, which is the same, the Thorpe scale is the appropriate scale to diagnose mixing and dissipation. However, when mixing is driven by the available potential energy of the mean flow (convective-driven mixing), this study shows that the Thorpe scale is (much) larger than the Ozmidov scale. Using the rms of the Thorpe displacement overestimates dissipation and mixing, since the amount of turbulent available potential energy (measured by the turbulent displacement) is only a fraction of the total available potential energy (measured by the Thorpe scale). Corrective measures are discussed that can be used to diagnose mixing from knowledge of the Thorpe displacement. In a companion paper, Mater et al. analyze field data and show that the Thorpe scale can indeed be much larger than the Ozmidov scale.

Energetics and mixing in buoyancy-driven near-bottom stratified flow

2019 ◽  
Vol 869 ◽  
pp. 214-237
Author(s):  
Pranav Puthan ◽  
Masoud Jalali ◽  
Vamsi K. Chalamalla ◽  
Sutanu Sarkar
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Energy Loss ◽  
Convective Instability ◽  
Mixing Efficiency ◽  
Time Interval ◽  
Mean Flow ◽  
Available Potential Energy ◽  
Long Time ◽  
The Mean

Turbulence and mixing in a near-bottom convectively driven flow are examined by numerical simulations of a model problem: a statically unstable disturbance at a slope with inclination $\unicode[STIX]{x1D6FD}$ in a stable background with buoyancy frequency $N$ . The influence of slope angle and initial disturbance amplitude are quantified in a parametric study. The flow evolution involves energy exchange between four energy reservoirs, namely the mean and turbulent components of kinetic energy (KE) and available potential energy (APE). In contrast to the zero-slope case where the mean flow is negligible, the presence of a slope leads to a current that oscillates with $\unicode[STIX]{x1D714}=N\sin \unicode[STIX]{x1D6FD}$ and qualitatively changes the subsequent evolution of the initial density disturbance. The frequency, $N\sin \unicode[STIX]{x1D6FD}$ , and the initial speed of the current are predicted using linear theory. The energy transfer in the sloping cases is dominated by an oscillatory exchange between mean APE and mean KE with a transfer to turbulence at specific phases. In all simulated cases, the positive buoyancy flux during episodes of convective instability at the zero-velocity phase is the dominant contributor to turbulent kinetic energy (TKE) although the shear production becomes increasingly important with increasing  $\unicode[STIX]{x1D6FD}$ . Energy that initially resides wholly in mean available potential energy is lost through conversion to turbulence and the subsequent dissipation of TKE and turbulent available potential energy. A key result is that, in contrast to the explosive loss of energy during the initial convective instability in the non-sloping case, the sloping cases exhibit a more gradual energy loss that is sustained over a long time interval. The slope-parallel oscillation introduces a new flow time scale $T=2\unicode[STIX]{x03C0}/(N\sin \unicode[STIX]{x1D6FD})$ and, consequently, the fraction of initial APE that is converted to turbulence during convective instability progressively decreases with increasing $\unicode[STIX]{x1D6FD}$ . For moderate slopes with $\unicode[STIX]{x1D6FD}<10^{\circ }$ , most of the net energy loss takes place during an initial, short ( $Nt\approx 20$ ) interval with periodic convective overturns. For steeper slopes, most of the energy loss takes place during a later, long ( $Nt>100$ ) interval when both shear and convective instability occur, and the energy loss rate is approximately constant. The mixing efficiency during the initial period dominated by convectively driven turbulence is found to be substantially higher (exceeds 0.5) than the widely used value of 0.2. The mixing efficiency at long time in the present problem of a convective overturn at a boundary varies between 0.24 and 0.3.

scholarly journals Energetics of Eddy–Mean Flow Interactions in the Gulf Stream Region

2015 ◽  
Vol 45 (4) ◽  
pp. 1103-1120 ◽  
Author(s):  
Dujuan Kang ◽  
Enrique N. Curchitser
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Gulf Stream ◽  
Mean Flow ◽  
Available Potential Energy ◽  
Flow Energy ◽  
Flow Interaction ◽  
The Mean ◽  
Eddy Field ◽  
Coast Region

AbstractA detailed energetics analysis of the Gulf Stream (GS) and associated eddies is performed using a high-resolution multidecadal regional ocean model simulation. The energy equations for the time-mean and time-varying flows are derived as a theoretical framework for the analysis. The eddy–mean flow energy components and their conversions show complex spatial distributions. In the along-coast region, the cross-stream and cross-bump variations are seen in the eddy–mean flow energy conversions, whereas in the off-coast region, a mixed positive–negative conversion pattern is observed. The local variations of the eddy–mean flow interaction are influenced by the varying bottom topography. When considering the domain-averaged energetics, the eddy–mean flow interaction shows significant along-stream variability. Upstream of Cape Hatteras, the energy is mainly transferred from the mean flow to the eddy field through barotropic and baroclinic instabilities. Upon separating from the coast, the GS becomes highly unstable and both energy conversions intensify. When the GS flows into the off-coast region, an inverse conversion from the eddy field to the mean flow dominates the power transfer. For the entire GS region, the mean current is intrinsically unstable and transfers 28.26 GW of kinetic energy and 26.80 GW of available potential energy to the eddy field. The mesoscale eddy kinetic energy is generated by mixed barotropic and baroclinic instabilities, contributing 28.26 and 9.15 GW, respectively. Beyond directly supplying the barotropic pathway, mean kinetic energy also provides 11.55 GW of power to mean available potential energy and subsequently facilitates the baroclinic instability pathway.

scholarly journals The storm tracks and the energy cycle of the Southern Hemisphere: sensitivity to sea-ice boundary conditions

1999 ◽  
Vol 17 (11) ◽  
pp. 1478-1492 ◽  
Author(s):  
C. G. Menéndez ◽  
V. Serafini ◽  
H. Le Treut
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Southern Ocean ◽  
Sea Ice ◽  
Sea Level ◽  
Zonal Flow ◽  
Sea Level Pressure ◽  
Level Pressure ◽  
The Mean ◽  
Ice Edge

Abstract. The effect of sea-ice on various aspects of the Southern Hemisphere (SH) extratropical climate is examined. Two simulations using the LMD GCM are performed: a control run with the observed sea-ice distribution and an anomaly run in which all SH sea-ice is replaced by open ocean. When sea-ice is removed, the mean sea level pressure displays anomalies predominantly negatives near the Antarctic coast. In general, the meridional temperature gradient is reduced over most of the Southern Ocean, the polar jet is weaker and the sea level pressure rises equatorward of the control ice edge. The high frequency filtered standard deviation of both the sea level pressure and the 300-hPa geopotential height decreases over the southern Pacific and southwestern Atlantic oceans, especially to the north of the ice edge (as prescribed in the control). In contrast, over the Indian Ocean the perturbed simulation exhibits less variability equatorward of about 50°S and increased variability to the south. The zonal averages of the zonal and eddy potential and kinetic energies were evaluated. The effect of removing sea-ice is to diminish the available potential energy of the mean zonal flow, the available potential energy of the perturbations, the kinetic energy of the growing disturbances and the kinetic energy of the mean zonal flow over most of the Southern Ocean. The zonally averaged intensity of the subpolar trough and the rate of the baroclinic energy conversions are also weaker.Key words. Air-sea interactions · Meteorology and atmospheric dynamics (climatology; ocean · atmosphere interactions)

Analysis of mean and eddy energy of the Black Sea circulation derived under account the climatic and real atmospheric forcing

2021 ◽  
Author(s):  
Olga Dymova ◽  
Sergey Demyshev ◽  
Dmitry Alekseev
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Black Sea ◽  
Baroclinic Instability ◽  
Mesoscale Eddies ◽  
The Black Sea ◽  
Barotropic Instability ◽  
Mesoscale Variability ◽  
The Mean ◽  
Mean Current

&lt;p&gt;The aim of the work is to study the mechanisms of the Black Sea mesoscale variability based on an analysis of Lorenz energy cycles calculated from the density and currents velocity obtained by the results of three numerical experiments. An eddy-resolving z-model with a horizontal resolution of 1.6 km was used. Three experiments were carried out with different atmospheric forcing: 1) - climatic data; 2) - SKIRON data for 2011; 3) &amp;#8211; SKIRON data for 2016. The mean current kinetic energy MKE, the eddy kinetic energy EKE, the mean available potential energy MPE, the eddy available potential energy EPE and the rates of energy conversion, generation and dissipation were considered in detail.&lt;/p&gt;&lt;p&gt;For all experiments the generation and dissipation rates of the MKE and EKE are close to each other, so the kinetic energy from wind dissipated inside the sea. A buoyancy work (described by the conversion between the MPE and MKE) increase the MKE. The EKE was increasing due to the energy transport from the mean current into eddies and the transport from the EPE to the EKE for all experiments. It is shown that these two energy fluxes were comparable in the experiment 1, while the ratio between of them has changed almost six times in the experiments 2 and 3. The c(MKE, EKE) prevailed in 2011, but the c(EPE, EKE) dominated in 2016.&lt;/p&gt;&lt;p&gt;The maps analysis of the EKE spatial distribution showed that its maximum in the climatic field was located above a continental slope and in areas of the biggest mesoscale eddies. The mesoscale variability of the climatic circulation was due to the influence of both baroclinic and barotropic instability. The zones of the EKE maximum were located in the abyssal part of the sea in the experiments 2 and 3. EKE was increasing in 2011 mainly due to the inflow from the mean current through barotropic instability. The growth of EKE in 2016 was due to conversion of EPE induced by baroclinic instability.&lt;/p&gt;&lt;p&gt;The difference in the EKE variability by the results of climatic and real forcing experiments is associated with the wind forcing. The contribution of the wind stress work to MKE was decreased for the experiments 2 and 3, so as a result, it was observed weakening in the mean current, intensive stream meandering and generation of mesoscale eddies not only in the coastal zones but also in the abyssal part of the sea. Thus, the Black Sea mesoscale variability is determined by barotropic instability or by the combined contribution of barotropic and baroclinic instability processes under intense wind action. The mesoscale variability is due to baroclinic instability under weak wind action.&lt;/p&gt;&lt;p&gt;The reported study was funded by RFBR and Government of the Sevastopol according to the research project No 18-45-920019 and the state task No. 0555-2021-0004.&lt;/p&gt;

scholarly journals Using Kinetic Energy with Potential Energy When Determining Power During the Stair Climbing Test

2020 ◽  
Author(s):  
James Roush ◽  
John Heick ◽  
Joseph Genovese ◽  
Kyle Kurashima ◽  
Dallin Yarrington
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Stair Climbing ◽  
Total Power ◽  
Accurate Assessment ◽  
Acceleration Phase ◽  
Deceleration Phase ◽  
Vital Component ◽  
The Mean ◽  
Functional Task

ABSTRACT Stair climbing is an important functional task that indicates independence, and generating power to climb stairs is a vital component of this task. Power during stair climbing is traditionally calculated using potential energy (PE), but it may be important to determine power expended using kinetic energy (KE). Purpose: The current study assessed power output for stair climbing with and without the inclusion of KE. Methods: Sixty participants (21-35 years) climbed a 12-step stairway with a 2-meter acceleration phase before the first step and a 2-meter deceleration phase after the last step. Participants completed 3 trials, and average time was used for calculating energy expended and power. Results: The mean difference between power from PE and total power was 6.16 W (SD = 2.50, t29 = 13.49, p < 0.001) for males and 64.76 W (SD = 2.90, t29 = 8.99, p < 0.001) for females. Agreement between power calculated from PE and total power was 0.99 (95% confidence interval = 0.98-1.0). Conclusion: Power calculated using PE and KE was significantly different from using PE alone, which may be clinically important. When conducting stair-climbing tests, both PE and KE may be necessary for the most accurate assessment of power.

scholarly journals Study of the Mechanisms of Vortex Variability in the Lofoten Basin Based on Energy Analysis

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
V. S. Travkin ◽  
T. V. Belonenko ◽  
◽  
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
World Ocean ◽  
Positive Trend ◽  
Barotropic Instability ◽  
Available Potential Energy ◽  
Conversion Rates ◽  
Order Of Magnitude ◽  
The Mean ◽  
Mean Kinetic Energy

Purpose. The Lofoten Basin is one of the most energetic zones of the World Ocean characterized by high activity of mesoscale eddies. The study is aimed at analyzing different components of general energy in the basin, namely the mean kinetic and vortex kinetic energy calculated using the integral of the volume of available potential and kinetic energy of the Lofoten Vortex, as well as variability of these characteristics. Methods and Results. GLORYS12V1 reanalysis data for the period 2010–2018 were used. The mean kinetic energy and the eddy kinetic one were analyzed; and as for the Lofoten Vortex, its volume available potential and kinetic energy was studied. Mesoscale activity of eddies in winter is higher than in summer. Evolution of the available potential energy and kinetic energy of the Lofoten Vortex up to the 1000 m horizon was studied. It is shown that the vortex available potential energy exceeds the kinetic one by an order of magnitude, and there is a positive trend with the coefficient 0,23·1015 J/year. It was found that in the Lofoten Basin, the intermediate layer from 600 to 900 m made the largest contribution to the potential energy, whereas the 0–400 m layer – to kinetic energy. The conversion rates of the mean kinetic energy into the vortex kinetic one, and the mean available potential energy into the vortex available potential one (baroclinic and barotropic instability) were analyzed. It is shown that the first type of transformation dominates in summer, while the second one is characterized by its increase in winter. Conclusions. The vertical profile shows that kinetic energy of eddies in winter is higher than in summer. The available potential energy of a vortex is by an order of magnitude greater than the kinetic energy. Increase in the available potential energy is confirmed by a significant positive trend and by decrease of the vortex Burger number. The graphs of the barotropic instability conversion rate demonstrate the multidirectional flows in the vortex zone with the dipole structure observed in a winter period, and the tripole one – in summer. The barotropic instability highest intensity is observed in summer. The baroclinic instability is characterized by intensification of the regime in winter that is associated with weakening of stratification in this period owing to winter convection.

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