Two-layer Quasi-geostrophic Model Research Articles

The Parameter Dependence of Eddy Heat Flux in a Homogeneous Quasigeostrophic Two-Layer Model on a β Plane with Quadratic Friction

Abstract This study investigates the parameter dependence of eddy heat flux in a homogeneous quasigeostrophic two-layer model on a β plane with imposed environmental vertical wind shear and quadratic frictional drag. We examine the extent to which the results can be explained by a recently proposed diffusivity theory for passive tracers in two-dimensional turbulence. To account for the differences between two-layer and two-dimensional models, we modify the two-dimensional theory according to our two-layer f-plane analyses reported in an earlier study. Specifically, we replace the classic Kolmogorovian spectral slope, −5/3, assumed to predict eddy kinetic energy spectrum in the former with a larger slope, −7/3, suggested by a heuristic argument and fit to the model results in the latter. It is found that the modified theory provides a reasonable estimate within the regime where both and the strength of the frictional drag, , are much smaller than unity (here, c D is the nondimensional drag coefficient divided by the depth of the layer, k d is the wavenumber of deformation radius, and U is the imposed background vertical wind shear). For values of and that are closer to one, the theory works only if the full spectrum shape of the eddy kinetic energy is given. Despite the qualitative, fitting nature of this approach and its failure to explain the full parameter range, we believe its documentation here remains useful as a reference for the future attempt in pursuing a better theory.

N-symmetric interaction of N hetons. I. Analysis of the case N = 2

We examine the motion of N symmetric hetons (oppositely signed vertical dipoles) in a two-layer quasi-geostrophic model. We consider the special case of N-fold symmetry in which the original system of 4N ordinary differential equations reduces to just two equations for the so-called “equivalent” heton. We perform a qualitative analysis to classify the possible types of vortex motions for the case N = 2. We identify the regions of the parameter space corresponding to unbounded motion and to different types of bounded, or localized, motions. We focus on the properties of localized, in particular periodic, motion. We identify classes of absolute and relative “choreographies” first introduced by Simó [“New families of solutions to the N-body problems,” in Proceedings of the European 3rd Congress of Mathematics, Progress in Mathematics Vol. 201, edited by C. Casacuberta, R. M. Miró-Roig, J. Verdera, and S. Xambó-Descamps (Birkhäuser, Basel, Barcelona, 2000), pp. 101–115]. We also study the forms of vortex trajectories occurring for unbounded motion, which are of practical interest due to the associated transport of heat and mass over large distances.

Solitary waves of nonlinear barotropic–baroclinic coherent structures

This study describes the evolutionary mechanisms of nonlinear barotropic–baroclinic interactions, especially, on the excitations, propagations, and decreases of nonlinear coherent structures. Starting from the classical two-layer quasi-geostrophic potential vorticity conservation model equations, the barotropic and baroclinic model equations are derived from the classical work of Pedlosky and Thomson [J. Fluid Mech. 490, 189–215 (2003)]. By considering the effects of bottom topography and beta-plane approximation, the coupled nonlinear Korteweg–de Vries model equations for the evolutions of barotropic and baroclinic coherent structures are obtained by using the methods of multiple scales and perturbation expansions, respectively. Solitary wave solutions are given according to the method of elliptic function expansions, and the physical mechanisms for the evolutions of the nonlinear barotropic–baroclinic interactive coherent structures are analyzed based on the obtained solitary wave solutions. It will be potentially useful for further theoretical investigations on atmospheric blocking phenomena or wave–flow interactions.

Baroclinic Modes over Rough Bathymetry and the Surface Deformation Radius

AbstractThe deformation radius is widely used as an indication of the eddy length scale at different latitudes. The radius is usually calculated assuming a flat ocean bottom. However, bathymetry alters the baroclinic modes and hence their deformation radii. In a linear quasigeostrophic two-layer model with realistic parameters, the deep flow for a 100-km wave approaches zero with a bottom ridge roughly 10 m high, leaving a baroclinic mode that is mostly surface trapped. This is in line with published current meter studies showing a primary EOF that is surface intensified and has nearly zero flow at the bottom. The deformation radius associated with this “surface mode” is significantly larger than that of the flat bottom baroclinic mode. Using World Ocean Atlas data, the surface radius is found to be 20%–50% larger over much of the globe, and 100% larger in some regions. This in turn alters the long Rossby wave speed, which is shown to be 1.5–2 times faster than over a flat bottom. In addition, the larger deformation radius is easier to resolve in ocean models.

Assimilating Observations with Spatially Correlated Errors Using a Serial Ensemble Filter with a Multiscale Approach

AbstractThe serial ensemble square root filter (EnSRF) typically assumes observation errors to be uncorrelated when assimilating the observations one at a time. This assumption causes the filter solution to be suboptimal when the observation errors are spatially correlated. Using the Lorenz-96 model, this study evaluates the suboptimality due to mischaracterization of observation error spatial correlations. Neglecting spatial correlations in observation errors results in mismatches between the specified and true observation error variances in spectral space, which cannot be resolved by inflating the overall observation error variance. As a remedy, a multiscale observation (MSO) method is proposed to decompose the observations into multiple scale components and assimilate each component with separately adjusted spectral error variance. Experimental results using the Lorenz-96 model show that the serial EnSRF, with the help from the MSO method, can produce solutions that approach the solution from the EnSRF with correctly specified observation error correlations as the number of scale components increases. The MSO method is further tested in a two-layer quasigeostrophic (QG) model framework. In this case, the MSO method is combined with the multiscale localization (MSL) method to allow the use of different localization radii when updating the model state at different scales. The combined method (MSOL) improves the serial EnSRF performance when assimilating observations with spatially correlated errors. With adjusted observation error spectral variances and localization radii, the combined MSOL method provides the best solution in terms of analysis accuracy and filter consistency. Prospects and challenges are also discussed for the implementation of the MSO method for more complex models and observing networks.

Single-Precision in the Tangent-Linear and Adjoint Models of Incremental 4D-Var

Abstract The use of single-precision arithmetic in ECMWF’s forecasting model gave a 40% reduction in wall-clock time over double-precision, with no decrease in forecast quality. However, using reduced-precision in 4D-Var data assimilation is relatively unexplored and there are potential issues with using single-precision in the tangent-linear and adjoint models. Here, we present the results of reducing numerical precision in an incremental 4D-Var data assimilation scheme, with an underlying two-layer quasigeostrophic model. The minimizer used is the conjugate gradient method. We show how reducing precision increases the asymmetry between the tangent-linear and adjoint models. For ill-conditioned problems, this leads to a loss of orthogonality among the residuals of the conjugate gradient algorithm, which slows the convergence of the minimization procedure. However, we also show that a standard technique, reorthogonalization, eliminates these issues and therefore could allow the use of single-precision arithmetic. This work is carried out within ECMWF’s data assimilation framework, the Object Oriented Prediction System.

Data Assimilation for a Quasi-Geostrophic Model with Circulation-Preserving Stochastic Transport Noise

This paper contains the latest installment of the authors’ project on developing ensemble based data assimilation methodology for high dimensional fluid dynamics models. The algorithm presented here is a particle filter that combines model reduction, tempering, jittering, and nudging. The methodology is tested on a two-layer quasi-geostrophic model for a beta -plane channel flow with O(10^6) degrees of freedom out of which only a minute fraction are noisily observed. The model is reduced by following the stochastic variational approach for geophysical fluid dynamics introduced in (Holm in Proc R Soc A 41:20140963, 2015) as a framework for deriving stochastic parameterisations for unresolved scales. The reduction is substantial: the computations are done only for O(10^4) degrees of freedom. We introduce a stochastic time-stepping scheme for the two-layer model and prove its consistency in time. Then, we analyze the effect of the different procedures (tempering combined with jittering and nudging) on the performance of the data assimilation procedure using the reduced model, as well as how the dimension of the observational data (the number of “weather stations”) and the data assimilation step affect the accuracy and uncertainty of the results.

Overtransmission of Rossby Waves at a Lower-Layer Critical Latitude in the Two-Layer Model

Abstract Rossby waves, propagating from the midlatitudes toward the tropics, are typically absorbed by critical latitudes (CLs) in the upper troposphere. However, these waves typically encounter CLs in the lower troposphere first. We study a two-layer linear scattering problem to examine the effects of lower CLs on these waves. We begin with a review of the simpler barotropic case to orient the reader. We then progress to the baroclinic case using a two-layer quasigeostrophic model in which there is vertical shear in the mean flow on which the waves propagate, and in which the incident wave is assumed to be an external-mode Rossby wave. We use linearized equations and add small damping to remove the critical-latitude singularities. We consider cases in which either there is only one CL, in the lower layer, or there are CLs in both layers, with the lower-layer CL encountered first. If there is only a CL in the lower layer, the wave’s response depends on the sign of the mean potential vorticity gradient at this lower-layer CL: if the PV gradient is positive, then the CL partially absorbs the wave, as in the barotropic case, while for a negative PV gradient, the CL is a wave emitter, and can potentially produce overreflection and/or overtransmission. Our numerical results indicate that overtransmission is by far the dominant response in these cases. When an upper-layer absorbing CL is encountered, following the lower-layer encounter, one can still see the signature of overtransmission at the lower-layer CL.

The role of the asymmetric Ekman dissipation term on the energetics of the two-layer quasi-geostrophic model

In the two-layer quasi-geostrophic model, the friction between the flow at the lower layer and the surface boundary layer, placed beneath the lower layer, is modeled by the Ekman term, which is a linear dissipation term with respect to the horizontal velocity at the lower layer. The Ekman term appears in the governing equations asymmetrically; it is placed at the lower layer, but does not appear at the upper layer. A variation, proposed by Phillips and Salmon, uses extrapolation to place the Ekman term between the lower layer and the surface boundary layer, or at the surface boundary layer. We present theoretical results that show that in either the standard or the extrapolated configurations, the Ekman term dissipates energy at large scales, but does not dissipate potential enstrophy. It also creates an approximately symmetric stable distribution of potential enstrophy between the two layers. The behavior of the Ekman term changes fundamentally at small scales. Under the standard formulation, the Ekman term will unconditionally dissipate energy and also dissipate, under very minor conditions, potential enstrophy at small scales. However, under the extrapolated formulation, there exist small “negative regions”, which are defined over a two-dimensional phase space, capturing the distribution of energy per wavenumber between baroclinic energy and barotropic energy, and the distribution of potential enstrophy per wavenumber between the upper layer and the lower layer, where the Ekman term may inject energy or potential enstrophy.

Solitary waves of two-layer quasi-geostrophic flow and analytical solutions with scalar nonlinearity

In this paper we investigate the instability of two-layer quasi-geostrophic model with scalar nonlinearity, which is a geophysical fluid dynamics model, and give a nonlinear stability criterion. Furthermore, the analytical solutions of the two-layer quasi-geostrophic model are obtained via the Jacobi elliptic periodic function transform method and symbolic computation. It is efficient in solving a range classes of nonlinear partial differential equations for this method. The result shows that the structure of Rossby waves is influenced by many factors such as base flow and scalar nonlinearity.

Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model

The stochastic variational approach for geophysical fluid dynamics was introduced by Holm (Proc Roy Soc A, 2015) as a framework for deriving stochastic parameterisations for unresolved scales. This paper applies the variational stochastic parameterisation in a two-layer quasi-geostrophic model for a $ \beta $-plane channel flow configuration. We present a new method for estimating the stochastic forcing (used in the parameterisation) to approximate unresolved components using data from the high resolution deterministic simulation, and describe a procedure for computing physically-consistent initial conditions for the stochastic model. We also quantify uncertainty of coarse grid simulations relative to the fine grid ones in homogeneous (teamed with small-scale vortices) and heterogeneous (featuring horizontally elongated large-scale jets) flows, and analyse how the spread of stochastic solutions depends on different parameters of the model. The parameterisation is tested by comparing it with the true eddy-resolving solution that has reached some statistical equilibrium and the deterministic solution modelled on a low-resolution grid. The results show that the proposed parameterisation significantly depends on the resolution of the stochastic model and gives good ensemble performance for both homogeneous and heterogeneous flows, and the parameterisation lays solid foundations for data assimilation.

Role of Surface-Layer Coherent Eddies in Potential Vorticity Transport in Quasigeostrophic Turbulence Driven by Eastward Shear

The transport by materially coherent surface-layer eddies was studied in a two-layer quasigeostrophic model driven by eastward mean shear. The coherent eddies were identified by closed contours of the Lagrangian-averaged vorticity deviation obtained from Lagrangian particles advected by the flow. Attention was restricted to eastward mean flows, but a wide range of flow regimes with different bottom friction strengths, layer thickness ratios, and background potential vorticity (PV) gradients were otherwise considered. It was found that coherent eddies become more prevalent and longer-lasting as the strength of bottom drag increases and the stratification becomes more surface-intensified. The number of coherent eddies is minimal when the shear-induced PV gradient is 10–20 times the planetary PV gradient and increases for both larger and smaller values of the planetary PV gradient. These coherent eddies, with an average core radius close to the deformation radius, propagate meridionally with a preference for cyclones to propagate poleward and anticyclones to propagate equatorward. The meridional propagation preference of the coherent eddies gives rise to a systematic upgradient PV transport, which is in the opposite direction as the background PV transport and not captured by standard Lagrangian diffusivity estimates. The upgradient PV transport by coherent eddy cores is less than 15% of the total PV transport, but the PV transport by the periphery flow induced by the PV inside coherent eddies is significant and downgradient. These results clarify the distinct roles of the trapping and stirring effect of coherent eddies in PV transport in geophysical turbulence.

A Multiscale Alignment Method for Ensemble Filtering with Displacement Errors

AbstractHigh-resolution models nowadays simulate phenomena across many scales and pose challenges to the design of efficient data assimilation methods that reduce errors at all scales. Smaller-scale features experience rapid nonlinear error growth that gives rise to displacement errors, which cause suboptimal ensemble filter performance. Previous studies have started exploring methods that can reduce displacement errors. In this study, a multiscale alignment (MSA) method is proposed for ensemble filtering. The MSA method iteratively processes the model state from the largest to the smallest scales. At each scale, an ensemble filter is applied to update the state with observations, and the analysis increments are utilized to derive displacement vectors for each member that align the ensemble at smaller scales before the next iteration. The nonlinearity in smaller-scale priors is reduced by removing larger-scale displacement errors. Because the displacement vectors are derived from analysis increments in the state space rather than the nonlinear observation-space cost function formulated in previous studies, this method provides a less costly and more robust way to solve for the displacement vectors. Observing system simulation experiments using a two-layer quasigeostrophic model were conducted to provide a proof of concept of the MSA method. Results show that the MSA method significantly improves the accuracy of posteriors compared to the existing ensemble filter methods with or without multiscale localization. Advantage of the MSA method are more evident when the ensemble size is relatively small and the cycling period is comparable to the average eddy turnover time of the dynamical system.

Role of Residual Overturning for the Sensitivity of Southern Ocean Isopycnal Slopes to Changes in Wind Forcing

AbstractThe Antarctic Circumpolar Current plays a central role in the ventilation of heat and carbon in the global ocean. In particular, the isopycnal slopes determine where each water mass outcrops and thus how the ocean interacts with the atmosphere. The region-integrated isopycnal slopes have been suggested to be eddy saturated, that is, stay relatively constant as the wind forcing changes, but whether or not the flow is saturated in realistic present day and future parameter regimes is unknown. This study analyzes an idealized two-layer quasigeostrophic channel model forced by a wind stress and a residual overturning generated by a mass flux across the interface between the two layers, with and without a blocking ridge. The sign and strength of the residual overturning set which way the isopycnal slopes change with the wind forcing, leading to an increase in slope with an increase in wind forcing for a positive overturning and a decrease in slope for a negative overturning, following the usual conventions; this behavior is caused by the dominant standing meander weakening as the wind stress weakens causing the isopycnal slopes to become more sensitive to changes in the wind stress and converge with the slopes of a flat-bottomed simulation. Eddy saturation only appears once the wind forcing passes a critical level. These results show that theories for saturation must have both topography and residual overturning in order to be complete and provide a framework for understanding how the isopycnal slopes in the Southern Ocean may change in response to future changes in wind forcing.

Solitary waves, breathers, and rogue waves modulated by long waves for a model of a baroclinic shear flow.

Investigated in this paper is a quasigeostrophic two-layer model for the wave packets in a marginally stable or unstable baroclinic shear flow. We find that the wave packets can be modulated by certain long waves, resulting in different behaviors from those in the existing literature. Via the bilinear method, we construct the modulated Nth-order (N=1,2,...) solitary waves, breathers, and rogue waves for the wave-packet equations. Based on the modulation effects of the long waves, the solitary waves are classified into three types, i.e., Type-I, Type-II, and Type-III solitary waves. Type-I solitary waves, without the modulations, are the bell shaped and propagate with constant velocities; Type-II solitary waves, with the weak modulations, are shape changing within a short time and subsequently return to the bell-shaped state; and Type-III solitary waves, with the strong modulations, show not only the variations of shapes but also the appearances, splits, combinations, and disappearances of certain bulges in the evolution. For the interaction between the two unmodulated solitary waves, two Type-I solitary waves can bring about the oscillations in the interaction zone when they possess different velocities, and bring into being the bound-state, oscillation-state, and bi-oscillation-state solitary waves when they possess the same velocity. For the two interactive modulated solitary waves, bound-state, oscillation-state, and bi-oscillation-state solitary waves with the short-time variations of shapes or appearances of bulges can occur. Due to the modulations of the long waves, breathers and rogue waves are distorted and stretched, mainly in two aspects: one is the evolution trajectories for the breathers; the other is the shape variations for each element of the breathers and rogue waves. Numbers of the peaks and valleys for the rogue waves are adjustable via the modulations. In addition, modulated breathers and rogue waves can degenerate into the M- or W-shaped or multipeak solitary waves under certain conditions.

Tilted drifting jets over a zonally sloped topography: effects of vanishing eddy viscosity

Oceanic multiple jets are seen to possess spatio-temporal variability imposed by varying bottom topography resulting in jets that can drift and merge. The dynamics of multiple jets over a topographic zonal slope is studied in a two-layer quasi-geostrophic model. The jets tilt from the zonal direction and drift meridionally. In addition to the tilted jets, other large-scale spatial patterns are observed, which are extracted using the principal component analysis. The variances of these patterns are strongly influenced by the values of eddy viscosity and bottom friction parameters. The contribution of the tilted jets to the full flow field decreases with decreasing friction and viscosity parameters, and purely zonal large-scale modes, propagating in the meridional direction, populate the flow field. Linear stability analysis and two-dimensional kinetic-energy spectrum analysis suggest that the zonal modes gain energy from ambient eddies as well as from the tilted jets through nonlinear interactions. However, viscous dissipation and bottom friction tend to suppress the nonlinear interactions, which results in the inhibition of the upscale energy transfer from eddies to the zonal modes. These simulations suggest that, in the presence of topography, alternating jet patterns may be sustained through interactions among various large-scale modes. This is different from the classical zonal jet formation arguments, in which direct eddy forcing maintains the jets.

The Control of Surface Friction on the Scales of Baroclinic Eddies in a Homogeneous Quasigeostrophic Two-Layer Model

AbstractIn idealized models of the extratropical troposphere, both β and surface friction can control the equilibrated scales of baroclinic eddies by stopping the inverse cascade. A scaling theory on how surface friction alone sets these scales was proposed by Held in 1999 in the case of a quadratic drag law. However, the theory breaks down when friction is modeled by linear damping, and there are other reasons to suspect that it is oversimplified. An ideal system to test the theory is the homogeneous two-layer quasigeostrophic model in the β = 0 limit with quadratic damping. This study investigates some numerical simulations of the model to analyze two causes of the theory’s breakdown. They are 1) the asymmetry between two layers due to confinement of friction to the lower layer and 2) deviation from a spectrally local inverse energy cascade due to the spread of wavenumbers over which energy is input into the barotropic mode. The former is studied by comparing the simulations with drag appearing asymmetrically or symmetrically between the two layers. The latter is addressed with a heuristic modification of the theory. A regime where eddies equilibrate without an inverse cascade is also examined. A comparison is then made between quadratic and linear drag simulations. The connection to a competing theory based on the dynamics of equivalent barotropic vortices with thermal signatures is further discussed. Finally, we present an example of an inhomogeneous statistically steady state to argue that the diffusivity obtained from the homogeneous model has relevance to more realistic configurations.

On Stability of the Thomson’s Vortex N-gon in the Geostrophic Model of the Point Vortices in Two-layer Fluid

A two-layer quasigeostrophic model is considered. The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a circle of radius R in one of the layers is presented. The vortices have identical intensity and length scale is $$\gamma ^{-1}>0$$ . The problem has three parameters: N, $$\gamma R$$ and $$\beta $$ , where $$\beta $$ is the ratio of the fluid layer thicknesses. The stability of the stationary rotation is interpreted as orbital stability. The instability of the stationary rotation is instability of system reduced equilibrium. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The parameter space $$(N,\gamma R,\beta )$$ is divided on three parts: $$\varvec{A}$$ is the domain of stability in an exact nonlinear setting, $$\varvec{B}$$ is the linear stability domain, where the stability problem requires the nonlinear analysis, and $$\varvec{C}$$ is the instability domain. The case $$\varvec{A}$$ takes place for $$N=2,3,4$$ for all possible values of parameters $$\gamma R$$ and $$\beta $$ . In the case of $$N=5$$ , we have two domains: $$\varvec{A}$$ and $$\varvec{B}$$ . In the case $$N=6$$ , part $$\varvec{B}$$ is curve, which divides the space of parameters $$(\gamma R, \beta )$$ into the domains: $$\varvec{A}$$ and $$\varvec{C}$$ . In the case of $$N=7$$ , there are all three domains: $$\varvec{A}$$ , $$\varvec{B}$$ and $$\varvec{C}$$ . The instability domain $$\varvec{C}$$ takes place always if $$N=2n\geqslant 8$$ . In the case of $$N=2\ell +1\geqslant 9$$ , there are two domains: $$\varvec{B}$$ and $$\varvec{C}$$ . The results of research are presented in two versions: for parameter $$\beta $$ and parameter $$\alpha $$ , where $$\alpha $$ is the difference between layer thicknesses. A number of statements about the stability of the Thomson N-gon is obtained for the systems of interacting particles with the general Hamiltonian depending only on distances between the particles. The results of theoretical analysis are confirmed by numerical calculations of the vortex trajectories.

Multiple Equilibria in a Land–Atmosphere Coupled System

Many low-order modeling studies indicate that there may be multiple equilibria in the atmosphere induced by thermal and topographic forcings. However, most work uses uncoupled atmospheric model and just focuses on the multiple equilibria with distinct wave amplitude, i.e., the high- and low-index equilibria. Here, a low-order coupled land–atmosphere model is used to study the multiple equilibria with both distinct wave phase and wave amplitude. The model combines a two-layer quasi-geostrophic channel model and an energy balance model. Highly truncated spectral expansions are used and the results show that there may be two stable equilibria with distinct wave phase relative to the topography: one (the other) has a lower layer streamfunction that is nearly in (out of) phase with the topography, i.e., the lower layer ridges (troughs) are over the mountains, called ridge-type (trough-type) equilibria. The wave phase of equilibrium state depends on the direction of lower layer zonal wind and horizontal scale of the topography. The multiple wave phase equilibria associated with ridge- and trough-types originate from the orographic instability of the Hadley circulation, which is a pitch-fork bifurcation. Compared with the uncoupled model, the land–atmosphere coupled system produces more stable atmospheric flow and more ridge-type equilibrium states, particularly, these effects are primarily attributed to the longwave radiation fluxes. The upper layer streamfunctions of both ridge- and trough-type equilibria are also characterized by either a high- or low-index flow pattern. However, the multiple wave phase equilibria associated with ridge- and trough-types are more prominent than multiple wave amplitude equilibria associated with high- and low-index types in this study.

Role of Eddies in the Maintenance of Multiple Jets Embedded in Eastward and Westward Baroclinic Shears

Multiple zonal jets observed in many parts of the global ocean are often embedded in large-scale eastward and westward vertically sheared background flows. Properties of the jets and ambient eddies, as well as their dynamic interactions, are found to be different between eastward and westward shears. However, the impact of these differences on overall eddy dynamics remains poorly understood and is the main subject of this study. The roles of eddy relative vorticity and buoyancy fluxes in the maintenance of oceanic zonal jets are studied in a two-layer quasigeostrophic model. Both eastward and westward uniform, zonal vertically sheared cases are considered in the study. It is shown that, despite the differences in eddy structure and local characteristics, the fundamental dynamics are essentially the same in both cases: the relative-vorticity fluxes force the jets in the entire fluid column, and the eddy-buoyancy fluxes transfer momentum from the top to the bottom layer, where it is balanced by bottom friction. It is also observed that the jets gain more energy via Reynolds stress work in the layer having a positive gradient in the background potential vorticity, and this is qualitatively explained by a simple reasoning based on Rossby wave group velocity.