Emergent behaviors of the Cucker–Smale ensemble under attractive–repulsive couplings and Rayleigh frictions

In this paper, we revisit an interaction problem of two homogeneous Cucker–Smale (in short CS) ensembles with attractive–repulsive couplings, possibly under the effect of Rayleigh friction, and study three sufficient frameworks leading to bi-cluster flocking in which two sub-ensembles evolve to two clusters departing from each other. In the previous literature, the interaction problem has been studied in the context of attractive couplings. In our interaction problem, inter-ensemble and intra-ensemble couplings are assumed to be repulsive and attractive, respectively. When the Rayleigh frictional forces are turned on, we show that the total kinetic energy is uniformly bounded so that spatially mixed initial configurations evolve toward the bi-cluster configuration asymptotically fast under some suitable conditions on system parameters, communication weight functions and initial configurations. In contrast, when Rayleigh frictional forces are turned off, the flocking analysis is more delicate mainly due to the possibility of an exponential growth of the kinetic energy. In this case, we employ two mutually disjoint frameworks with constant inter-ensemble communication function and exponentially localized inter-ensemble communication functions, respectively, and prove the bi-clustering phenomenon in both cases. This work extends the previous work on the interaction problem of CS ensembles. We also conduct several numerical experiments and compare them with our theoretical results.

Super- and sub-rotating equatorial jets in shallow water models of Jovian atmospheres: Newtonian cooling versus Rayleigh friction

Numerical simulations of the shallow water equations on rotating spheres produce mixtures of robust vortices and alternating zonal jets, as seen in the atmospheres of the gas giant planets. However, simulations that include Rayleigh friction invariably produce a sub-rotating (retrograde) equatorial jet for Jovian parameter regimes, whilst observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted over several decades. Super-rotating equatorial jets have recently been obtained in shallow water simulations that include a Newtonian relaxation of perturbations to the layer thickness to model radiative cooling to space, and in simulations of the thermal shallow water equations that include a similar relaxation term in their temperature equation. Simulations of global quasigeostrophic forms of these different models produce equatorial jets in the same directions as the parent models, suggesting that the mechanism responsible for setting the direction lies within quasigeostrophic theory. We provide such a mechanism by calculating the effective force acting on the thickness-weighted zonal mean flow due to the decay of an equatorially trapped Rossby wave. Decay due to Newtonian cooling creates an eastward zonal mean flow at the equator, consistent with the formation of a super-rotating equatorial jet, while decay due to Rayleigh friction leads to a westward zonal mean flow at the equator, consistent with the formation of a sub-rotating equatorial jet. In both cases the meridionally integrated zonal mean of the absolute zonal momentum is westward, consistent with the standard result that Rossby waves carry westward pseudomomentum, but this does not preclude the zonal mean flow being eastward on and close to the equator.

On the Dynamical Control of the Mesosphere–Lower Thermosphere by the Lower and Middle Atmosphere

The NCAR Whole Atmosphere Community Climate Model (WACCM) is used to investigate the dynamical influence of the lower and middle atmosphere on the upper mesosphere and lower thermosphere. In simulations using a methodology adapted from the “specified dynamics” (nudged) version of the model, horizontal winds and temperature over part of the vertical range of the atmosphere are relaxed toward results from a previous simulation that serves as the true simulation, equivalent to meteorological analysis. In the upper mesosphere, the magnitude of the divergence of the constrained simulations from the true simulation depends on the vertical extent and frequency of the data used for nudging the model and grows with altitude. The simulations quantify the error growth of the model dynamical fields when data and forcing terms are known exactly and there are no model biases. The error growth rate and the ultimate discrepancy between the nudged and true fields depend strongly on the method used for representing gravity wave drag. The largest error growth occurs when the gravity wave parameterization uses interactive wave sources that depend on convective activity or fronts. Errors are reduced when the same parameterization is used with smoothly varying specified wave sources. The smallest errors are seen when the parameterized gravity wave drag is replaced by linear Rayleigh friction damping on the wind speed. These comparisons demonstrate the role of gravity waves in transporting the variability of the troposphere into the mesosphere and lower thermosphere.

On Adding Thermodynamic Damping Mechanisms to Refine Two Classical Models of Katabatic Winds

The Prandtl and layer-averaged models of katabatic winds contain some nonphysical singularities in the analytical solutions, which give unbounded steady flow anomalies at zero slope angles or adiabatic lapse rates. This study presents some simple refinements of these two classical models, in which the aforementioned singularities are removed when Newtonian cooling and Rayleigh friction are included in the system. It is pointed out that, in the limit of zero slope angles or adiabatic lapse rates, the along-slope buoyancy force and the adiabatic heating caused by air descending approach zero. Under such circumstances, a bounded steady solution for the katabatic winds is impossible unless some damping mechanisms are included to retard the anomalies induced by the radiative cooling effect in the boundary layer. Newtonian cooling and Rayleigh friction are the two simplest thermodynamic damping mechanisms that can be included to balance the effects of eddy viscosity and eddy thermal conductivity in the katabatic-flow model. Physically speaking, the Newtonian cooling term represents a small partition of the radiative effect and the Rayleigh friction term represents an approximation of the bottom drag effect in a turbulent boundary layer.

Steady-State Dynamics of a Density Current in an f-Plane Nonlinear Shallow-Water Model

The authors study the nonlinear dynamics of a density current generated by a diabatic source in a rotating and a nonrotating system, both in the presence and in the absence of frictional losses, using a steady-state hydrostatic shallow-water model and producing solutions as a function of the Coriolis parameter and of the Rayleigh friction coefficient. Results are presented in the range of the parameter values that are relevant for shallow atmospheric flows as sea–land breezes and as cold pool outflows. In the shallow-water approximation, single-layer flows and two-layer flows with a lid have three degrees of freedom, and their steady-state dynamics are governed by three ordinary differential equations (ODEs), whereas two-layer flows bounded by a free surface have six degrees of freedom, and their dynamics are governed by six ODEs. It is shown that in the limit case of frictionless flow, the problem has an explicit analytical solution, and in the presence of friction, the system for a one-layer flow and for a two-layer flow bounded by a lid can be reduced to two algebraic equations, plus one second-order ordinary differential equation, which can be integrated numerically. Results show that the maximum runout length of the current occurs when the Rayleigh friction coefficient in the lower layer is on the order of the Coriolis parameter. This length is larger when the upper layer is deeper than the lower layer, but it shortens when the friction coefficient of the upper layer is smaller than that in the lower layer. In addition, the relative error of the solution to the linearized equations is computed. This error, which is enhanced when the width of the forcing is smaller than the Rossby radius, is sizable when the friction coefficient is smaller than the Coriolis parameter. In addition, by comparing the nonlinear solution with a lid (three degrees of freedom) to the nonlinear solution with a free surface as an upper boundary (six degrees of freedom), it is shown that the solution with the lid overestimates the geopotential for low values of the friction coefficient and it underestimates the geopotential for large values of this coefficient. The error, sizable when the two layers have a comparable depth, rapidly decreases when the upper layer becomes deeper than the lower layer; accordingly, a rigid lid can be safely adopted only when the depth of the upper layer is twice the depth of the lower layer, or deeper.

Summertime Low-Level Jets over the High-Latitude Arctic Ocean

The application of a simple analytic boundary layer model developed by Thorpe and Guymer did not produce good agreement with observational data for oceanic low-level jet observations even though this model has worked well for the predictions of low-level jets over continental surfaces. This failure to properly predict the boundary layer wind maxima was very puzzling because more detailed numerical boundary layer models have properly predicted these low-level oceanic wind maxima. To understand the reasons for its failure to explain the ocean observations, the authors modified the frictional terms in the horizontal linear momentum equations of Thorpe and Guymer, using a standard eddy viscosity closure technique instead of the Rayleigh friction parameterization originally used. This improvement in the modeling of the dissipation terms, which has resulted in the use of an enhanced Rayleigh friction parameterization in the horizontal momentum equations, modified the boundary layer winds such that the continental predictions remained nearly identical to those predicted previously using the Thorpe and Guymer model while the oceanic predictions have now become more representative of the measured wind speed from recent Arctic expeditions.

Sensitivity of the Freie Universität Berlin Climate Middle Atmosphere Model (FUB-CMAM) to different gravity-wave drag parameterisations

We report the sensitivity of the Berlin Climate Middle Atmosphere Model (CMAM) to different gravity-wave (GW) parameterisations. We perform five perpetual January experiments: 1) Rayleigh friction (RF) (control), 2) non-orographic GWs, 3) orographic GWs, 4) orographic and non-orographic GWs with no background stress, and 5) as for 4) but with background stress. We also repeat experiment 4) but for July conditions. Our main aim is to improve the model climatology by introducing orographic and non-orographic parameterisations and to investigate the individual effect of these schemes in the Berlin CMAM. We compare with an RF control to determine the improvement upon a previously-published model version employing RF. Results are broadly similar to previously-published works. The runs having both orographic and non-orographic GWs produce a statistically-significant warming of 4-8K in the wintertime polar lower stratosphere. These runs also feature a cooling of the warm summer pole in the mesosphere by 10-15K, more in line with observations. This is associated with the non-orographic GW scheme. This scheme is also associated with a heating feature in the winter polar upper stratosphere directly below the peak GW-breaking region. The runs with both orographic and non-orographic GWs feature a statistically-significant deceleration in the polar night jet (PNJ) of 10-20ms-1 in the lower stratosphere. Both orographic and non-orographic GWs individually produce some latitudinal tilting of the polar jet with height, although the main effect comes from the non-orographic waves. The resulting degree of tilt, although improved, is nevertheless still weaker than that observed. Accordingly, wintertime variability in the zonal mean wind, which peaks at the edge of the vortex, tends to maximise too far polewards in the model compared with observations. Gravity-planetary wave interaction leads to a decrease in the amplitudes of stationary planetary waves 1 and 2 by up to 50% in the upper stratosphere and mesosphere, more in line with observations. Comparing modelled and observed Eliassen-Palm fluxes suggests that planetary wave (PW) breaking occurs too far polewards in the model. The wind and temperature changes are consistent with changes in the Brewer-Dobson (BD) circulation. Results suggest that the effect of enforcing a minimum background wave stress in the McFarlane scheme could be potentially important. In the Southern Hemisphere (SH) in July, the GW schemes had only a small impact on the high-latitude lower stratosphere but there featured strong warming near 0.1hPa.