Annual Rossby Waves Research Articles

Wind stress curl as a driving force of annual waves in the upper ocean for interpreting energetics at all latitudes

The dynamics of waves and eddies in the upper ocean plays an important role in the climate variation of tropical and subtropical regions. Previous diagnoses for annual Rossby waves in oceanic model outputs manifested zonally alternating signals (ZASs) in the time-averaged distributions of wind input as well as pressure-flux divergence terms in the budget equation of wave energy. This is the case when the annual mean of the wind input is estimated as the inner product of simulated velocity vector and wind stress vector in previous studies. The present study proposes a new mathematical expression for estimating the wind input that is analogous to one derived from the quasi-geostrophic potential vorticity equation. Namely, the wind input is estimated as the negative of the product of pseudo-streamfunction and wind stress curl, the latter of which is associated with the horizontal divergence of Ekman velocity. This can be interpreted as replacement of kinetic energy input with gravitational potential energy input. Pseudo-streamfunction in the present study is inverted from Ertel’s potential vorticity anomaly and is seamlessly available at all latitudes. This contrasts with the quasi-geostrophic streamfunction which is singular at the equator. The new expression enables reducing ZASs in the horizontal distributions of both wind input and pressure-flux divergence terms, without harming the qualitative advantage of energy flux vectors to indicate the group velocity of waves at all latitudes.

Modulation of Annual Rossby Waves on the Formation of Basin‐Wide Salinity Fronts

AbstractThis study reveals the formation mechanisms of a group of salinity fronts across the Arabian Sea (AS) and explores their associations with annual Rossby waves, utilizing Soil Moisture Active Passive (SMAP) satellite data and other relevant data sets. As low‐salinity water is transported westward and northward at varying rates between 6°N ∼ 20°N, substantial basin‐wide northeast‐southwest oriented salinity fronts emerge at its leading edge from March to May. Further analysis indicates a strong correlation between the transport of low‐salinity water and the westward‐propagating Rossby waves. The salinity fronts are primarily influenced by the geostrophic circulation, which significantly contributes to the transport of low‐salinity water. As a result, the low‐salinity water closely aligns with positive sea level anomaly (SLA), representing westward propagating downwelling Rossby waves. A continuously stratified linear ocean model (LOM) analysis suggests second mode baroclinic Rossby waves dominate seasonal sea level changes, surpassing the influence of the first mode. The spatial mean standard deviation of the first mode within the region of 56 ∼ 75°E, 9 ∼ 11°N is approximately 60% that of the second mode. Positive SLA is mainly caused by the annual downwelling Rossby waves radiating from the eastern boundary. Due to the latitudinal decrease in the phase speed of the annual Rossby waves, the sea level fronts manifest the northeast‐southwest direction, resembling the location of the sea surface salinity fronts. The salinity fronts highlight the crucial role of Rossby waves in shaping upper ocean water mass distribution in the AS, potentially impacting the broader Indian Ocean climate.

Interior Route and Seasonal Dynamics of the Meridional Current in the Eastern Indian Ocean Tropical Gyre

AbstractThe eastern boundary current of the tropical Indian Ocean (IO) connects the equatorial/south tropical IO, contributing to the meridional heat and salinity exchange. As a part of the IO tropical gyre, the meridional current in the eastern IO tropical gyre (East Sumatra Current, ESC) flows southward in the mean state. However, the route of this current varies with months. Drifter trajectories display an exceptional route that turns southeastward before reaching Sumatra in the austral winter and spring, considered the interior ESC. This study uses observations and reanalysis data sets to investigate the seasonal structure and forcing mechanisms behind the interior ESC. The velocity is decomposed into geostrophic and wind‐driven components. The results suggest that the geostrophic component plays a dominant role in the ESC recirculating early from April to June. The southward geostrophic currents are mainly caused by positive zonal temperature gradients, which are induced by both the cold anomaly of the Chagos Dome and the warm water carried by the Indonesian Throughflow, particularly in April and May. Meanwhile, the annual Rossby wave contributes to the temperature anomaly of the Chagos Dome. During the monsoon that prevails after July, the wind‐driven component increases and becomes the primary forcing, accompanying the slowdown of geostrophic velocity. Therefore, the local wind stress, wave propagation, and water mass transport account for the formation of the seasonal interior ESC.

Possible Mechanisms of Interannual Variations in Surface Mixed Layer Temperatures off Somalia in Boreal Summer

Abstract The key processes responsible for the interannual variation of the surface mixed layer temperature off Somalia in the western Arabian Sea in boreal summer are investigated by the use of a regional ocean model. Our focus is on influences of remotely forced annual Rossby waves as well as local southwesterly monsoonal winds in the years with anomalously warm or cold mixed layer temperature conditions. Composite and heat budget analyses of the simulated results indicate that the interannual mixed layer temperature variations in the region off the coast of Somalia are generated by the combined effects of local upwelling, Rossby wave intrusion, and horizontal advection. In particular, interannual modulation of the annual Rossby wave before the onset of the southwest monsoon causes a subsurface temperature anomaly below the mixed layer along the coast of Somalia. It is also shown that this subsurface temperature anomaly is upwelled into the mixed layer by the upward flow associated with the seasonally evolving coastal upwelling and that the mixed layer temperature anomaly in the coastal region is advected into the offshore region by seasonal developments of the Somali Current. Results from sensitivity experiments with different wind forcing scenarios demonstrate that the contribution of this remotely forced Rossby wave to the interannual variations of the mixed layer temperature anomaly off Somalia is comparable to that of the local wind stress anomalies alone. Significance Statement The purpose of this study is to investigate the key processes responsible for the interannual variation of the temperature within the ocean surface layer off Somalia in boreal summer. Because this temperature variation is shown to be related to the Indian summer monsoon rainfall, this study can lead to improvement in the predictability of the Indian summer monsoon. Previous studies have demonstrated that interannual modulation of the local monsoonal winds along the coast of Somalia in spring and summer generate the temperature variation. We find for the first time that the large-scale ocean waves excited remotely in previous winter play an important role, and its contribution is comparable to that of the effect of the wind influences.

Full Vorticity Budget of the Arabian Sea from a 0.1° Ocean Model: Sverdrup Dynamics, Rossby Waves, and Nonlinear Eddy Effects

Abstract The Arabian Sea, influenced by the Indian monsoon, has many unique features, including its basin-scale seasonally reversing surface circulation and the Great Whirl, a seasonal anticyclonic system appearing during the southwest monsoon close to the western boundary. To establish a comprehensive dynamical picture of the Arabian Sea, we utilize numerical model output and design a full vorticity budget that includes a fully decomposed nonlinear term. The ocean general circulation model has 0.1° resolution and is mesoscale eddy-resolving in the region. In the western boundary current system, we highlight the role of nonlinear eddies in the life cycle of the Great Whirl. The nonlinear eddy term is of leading-order importance in this feature’s vorticity balance. Specifically, it contributes to the Great Whirl’s persistence in boreal fall after the weakening of the southwesterly winds. In the open ocean, Sverdrup dynamics and annual Rossby waves are found to dominate the vorticity balance; the latter is considered as a key factor in the formation of the Great Whirl and the seasonal reversal of the western boundary current. In addition, we discuss different forms of vertically integrated vorticity equations in the model and argue that the bottom pressure torque term can be interpreted analogously as friction in the western boundary and vortex stretching in the open ocean.

On the low western Pacific sea levels observed prior to strong East Pacific El Niños

Abstract. Research, based on both observations and ocean models, has indicated that strong East Pacific El Niños are preceded by the development of unusually low sea levels at the western end of the North Equatorial Trough early in the year. This results in an increase in the strength of the North Equatorial Counter Current which, aided by low sea levels due to the annual Rossby wave, then transports West Pacific Warm Pool water to the central and eastern Pacific. Here, an ocean model is used to investigate the factors affecting sea level prior to the 1982–1983 East Pacific El Niño. The results indicate that during this period the reduction in sea level was caused by Ekman pumping, due to the local winds, acting over a period of many months. The north–south distances involved mean that such upwelling can be connected to the westerly wind phase of Madden–Julian Oscillations. Since the 1980s, the quality and quantity of global wind measurement has steadily improved. So, if the Ekman pumping hypothesis is correct, the better-quality wind data available prior to the 1997–1998 and 2015–2016 East Pacific El Niños should also show unusually large Ekman pumping in the same region, over a similar long period. This is tested and found to be correct. However, a study of the integrated Ekman pumping for the period 1976 to 2015 indicates that in some years similar events did not develop despite a comparable amount of pumping during the first part of the year. The results imply that significant Ekman pumping early in the year is a necessary but not sufficient requirement for the development of a strong East Pacific El Niño.

Baroclinic Characteristics and Energetics of Annual Rossby Waves in the Southern Tropical Indian Ocean

AbstractThe first baroclinic mode Rossby wave is known to be of critical importance to the annual sea level variability in the southern tropical Indian Ocean (STIO; 0°–20°S, 50°–115°E). In this study, an analysis of continuously stratified linear ocean model reveals that the second baroclinic mode also has significant contribution to the annual sea level variability (as high as 81% of the first baroclinic mode). The contributions of residual high-order modes (3 ≤ n ≤ 25) are much less. The superposition of low-order (first and second) baroclinic Rossby waves (BRWs) primarily contribute to the high energy center of sea level variability at ~10°S in the STIO and the vertical energy penetration below the seasonal thermocline. We have found that 1) the low-order BRWs, having longer zonal wavelengths and weaker damping, can couple more efficiently to the local large-scale wind forcing than the high-order modes and 2) the zonal coherency of the Ekman pumping results in the latitudinal energy maximum of low-order BRWs. Overall, this study extends the traditional analysis to suggest the characteristics of the second baroclinic mode need to be taken into account in interpreting the annual variability in the STIO.

Revisit of seasonal variability of subsurface temperature in the tropical Pacific with Argo data

Abstract Seasonal variability of temperature in the tropical Pacific is studied with Argo data from 2001 to 2017. It is found that the strongest seasonal variation is in the subsurface layer corresponding to the thermocline, which is located in three zonal bands centering at 12°N, 5°N and 5°S across the entire tropical Pacific. Further analysis indicates that seasonal variation of subsurface temperature is caused by the vertical movement of the thermocline, and the signal is generated in the eastern Pacific and propagates westward to the western Pacific. A linear wind-driven first-mode baroclinic Rossby wave model is then employed to investigate the seasonal fluctuation of thermocline. The seasonal shoaling and deepening of thermocline and associated temperature variations are well captured by this Rossby wave model. Basin-scale seasonal anomalies of wind stress curl in the tropical Pacific produce significant thermocline fluctuations, which propagate westward in the form of annual Rossby wave, and cause the seasonal variation of subsurface temperature. Thermocline fluctuations are revealed to be tightly associated with the meridional shift of equatorial currents such as North Equatorial Current, North Equatorial Countercurrent and South Equatorial Current.

Equatorial Pacific 1,000‐dbar Velocity and Isotherm Displacements From Argo Data: Beyond the Mean and Seasonal Cycle

AbstractEquatorial Pacific zonal velocity at 1,000 dbar and vertical isotherm displacements from 0–2,000 dbar are analyzed using Argo data through February 2018. In agreement with previous studies, the mean 1,000‐dbar zonal velocity is characterized by alternating eastward and westward jets of amplitude 5–10 cm/s. Similarly, the seasonal cycle is dominated by an annual Rossby wave that is apparent in annual harmonic fits to both the isotherm displacement and velocity fields. Residual (mean, annual, and semiannual harmonics removed) zonal velocity and isotherm displacements are analyzed using Complex Empirical Orthogonal Functions (CEOFs) and their Principal Components. For the analysis at 1,000 dbar, the first velocity CEOF accounts for 28% of the variance and the first isotherm displacement CEOF 24%. Spatial patterns of the lead CEOFs are broadly consistent with Rossby waves, but only isotherm displacement CEOF1 can be clearly linked to known physical processes in the equatorial Pacific. This is true of the isotherm displacement CEOF1s across pressure surfaces, despite being estimated independently on each one. The Principal Component of isotherm displacement CEOF1 at 1,000 dbar is well correlated (r = –0.65) with the Niño3.4 index at a 12‐month lag, suggesting that El Niño signals propagate deep into the equatorial Pacific Ocean. The 3‐D structure of the isotherm displacement CEOF1s suggests that the deep isotherm displacement signals are consistent with a vertically propagating Rossby wave of vertical mode 3–4.

Dynamics on Seasonal Variability of EKE Associated with TIWs in the Eastern Equatorial Pacific Ocean

AbstractEnergetic mesoscale eddies (vortices) associated with tropical instability waves (TIWs) exist in the eastern equatorial Pacific Ocean between 0° and 8°N. This study examines the seasonal variations in eddy kinetic energy (EKE) of TIWs using in situ and satellite observations and elucidates the underlying dynamical mechanisms. The results reveal that the cross-equatorial southerly winds are key to sustaining the high-level EKE (up to ~600 cm2 s−2) from boreal summer to winter in 0°–6°N and 155°–110°W. Because of the β effect and the surface wind divergence, the southerly winds generate anticyclonic wind curls north of the equator that intensify the sea surface temperature (SST) fronts and force the downwelling annual Rossby waves. The resultant sea surface height ridge induces strong horizontal current shears between 0° and 5°N. The intensified current shears and SST fronts generate EKE via barotropic and baroclinic instabilities, respectively. To the extent that the seasonal migration of a northward-displaced intertropical convergence zone intensifies the southerly winds north of, but not south of, the equator, our study suggests that the climatic asymmetry is important for the oceanic eddy generations in the eastern equatorial Pacific Ocean—a result with important implications for coupled climate simulation/prediction.

Annual Rossby Waves Below the Pycnocline in the Indian Ocean

AbstractAnnual variability in pressure and horizontal velocity below the pycnocline of the Indian Ocean is examined using in situ observations obtained from Argo floats and conductivity‐temperature‐depth sensors. Pressure and velocity at 1,000‐m depth show significant annual variability in the Arabian Sea and between 10°S and 20°S in the South Indian Ocean. Wavenumbers were estimated by fitting a straight line to annual harmonic phase by amplitude‐weighted least squares fit. Results showed that vertical wavenumber tends to decrease poleward in both hemispheres, and vertical wavelength is more than tripled from about 5,000 m at 5°N/S to 16,000 m at 20°N/S. The inverse radius of deformation squared (defined as f2m2/N2) does not show clear dependence on latitude, because poleward increase in the magnitude of the Coriolis parameter (f) compensates for poleward decrease in vertical wavenumber (m), and Brunt‐Väisälä frequency (N) varies little meridionally in the analysis domain. According to the dispersion relation of quasi‐geostrophic Rossby waves, these meridional structures in m and deformation radius result in steeper angle of ray trajectory at a higher latitude in the longitude‐depth plane. Energy estimated from in situ observations shows a consistent pattern with this expectation. The line of constant phase of annual pressure harmonic is steeper in angle at higher latitudes, which is also expected from the uniform deformation radius and constant N. This result indicates that energy penetrates deeper at higher latitudes.

Seasonal Cycle and Annual Reversal of the Somali Current in an Eddy‐Resolving Global Ocean Model

AbstractThe annual cycle of the Somali Current is studied using a forced strongly eddy‐active global ocean general circulation model. The annual cycle and timing of the reversal of the Somali Current differ depending on the location along the coast of northeastern Africa. A momentum budget analysis calculated from the model output shows that both annual Rossby waves and the life cycle of the Great Whirl impact the northern part of the Somali Current (roughly 5°N to 10°N). In particular, the nonlinear term is of leading order importance when the Great Whirl is present. This is in contrast to the leading‐order geostrophic balance between 2°N and 5°N. South of 2°N, equatorial dynamics prevail, and the nonlinear term is again important, and its reversal is strongly affected by the northward flowing East African Coastal Current. Alongshore wind forcing is influential at all latitudes, but only the transition in flow direction during boreal spring between 2°N and 5°N directly follows that of the wind.

On the role of the North Equatorial Counter Current during a strong El Niño

Abstract. An analysis of archived data from the NEMO 1∕12th degree global ocean model shows the importance of the North Equatorial Counter Current (NECC) in the development of the strong 1982–1983 and 1997–1998 El Niños. The model results indicate that in a normal year the core of warm water in the NECC is diluted by the surface Ekman transport, by geostrophic inflow and by tropical instability waves. During the development of the 1982–1983 and 1997–1998 El Niños, these processes had reduced effect at the longitudes of warmest equatorial temperatures and to the west. During the autumns of 1982 and 1997, the speed of the NECC was also increased by a stronger-than-normal annual Rossby wave. The increased transport of warm water by the NECC due to these changes resulted in warm water reaching the far eastern Pacific and appears to have been a major factor in moving the centre of deep atmospheric convection eastwards across the Pacific.

Water Masses and Circulation in the Tropical Pacific off Central Mexico and Surrounding Areas

AbstractThe seasonal variations and the interactions of the water masses in the tropical Pacific off central Mexico (TPCM) and four surrounding areas were examined based on an extensive new hydrographic database. The regional water masses were redefined in terms of absolute salinity (SA) and conservative temperature (Θ) according to the Thermodynamic Equation of Seawater 2010 (TEOS-10). Hydrographic data and the evaporation minus (precipitation + runoff) balance were used to investigate the origin and seasonality of two salinity minima in the area. The shallow (50–100 m) salinity minimum originates with the California Current System and becomes saltier as it extends southeastward and mixes with tropical subsurface waters while the surface salinity minimum extends farther north in the TPCM in summer and fall because of the northward advection of tropical surface waters. The interactions between water masses allow a characterization of the seasonal pattern of circulation of the Mexican Coastal Current (MCC), the tropical branch of the California Current, and the flows through the entrance of the Gulf of California. The seasonality of the MCC inferred from the distribution of the water masses largely coincides with the geostrophic circulation forced by an annual Rossby wave.

Vertical Structure of Upper-Ocean Seasonality: Extratropical Spiral versus Tropical Phase Lock

AbstractSeasonality is a fundamental feature of the coupled ocean–atmosphere system. The hemispherically opposite general pattern at the air–sea interface is well known, but its penetrative behavior below the sea surface is poorly understood. Over 10 years of Argo data reveal for the first time the spiral-like structure of vertical seasonality in the upper extratropical ocean: the amplitude (strength of annual sea temperature variability) decreases rapidly with depth while the phase (peaking time) rotates from August (February) at the sea surface to December (June) at the bottom of the mixed layer in the Northern (Southern) Hemisphere under a clockwise representation. It is found that, analogous to the Ekman current spiral, the oceanic seasonality is almost reversed at approximately 500-m depth with about 5% of the surface intensity left. In contrast, the seasonality of subtropical oceans below the thermocline exhibits a phase-lock pattern around May and October to the north and south of the equator, respectively. Meanwhile, a systematic westward progression of annual phase corresponding to the warmest month is observed in the equatorial regions between 10°S and 10°N. It is suggested that the seasonality spiral of extratropical oceans occurs as a result of the vertical decay of solar penetration in tandem with delayed annual maximum mixing in the context of buoyancy and turbulent induced convections, while the month of full summer appears to be “constant” May (October) in the subtropical oceans of the Northern (Southern) Hemisphere under the stratified subsurface layer due to a 9-month trapping of penetrating solar energy from May (October) to next January (June). The vertically locked westward annual phase progression in the equatorial regions is likely to be a consequence of the first baroclinic mode of β-refracted annual Rossby waves. The spiral and phase-lock behaviors of the upper oceans are of critical significance to the understanding of mixed layer and thermocline dynamics.

Vertical Structure of Upper-Ocean Seasonality: Annual and Semiannual Cycles with Oceanographic Implications

Abstract A decade of newly available Argo float data for the period 2004–13 are used to investigate the three-dimensional structures of upper-ocean seasonality with emphasis on the vertical aspects of annual and semiannual cycles, yielding three main findings with oceanographic implications. First, the vertical evolution of the horizontal pattern of annual and semiannual amplitudes appears to be highly “nonlinear,” suggesting that the thermodynamic causes are depth dependent. The global ocean seasonality exhibits a vertically varying pattern in space, including midlatitude maxima in the near-surface layer due to solar forcing, zonal “strips” in the subsurface layer due to the equatorial current system, and systematic westward phase propagation in the intermediate layer due to annual Rossby waves. Second, a zone of 500 ± 300-m depths along with a 6-month periodicity are chosen as appropriate space–time “windows” for detecting eddy signatures via Argo-derived temperature amplitude and phase, respectively. It is revealed that the eddy-induced “blobby” pattern observed previously by satellite altimeter appears in the Agro result as “woodsy” bulks, which can be well illustrated in the semiannual amplitude and phase maps at window depths. Meanwhile, six eddy deserts paired in each ocean basin have also been identified. Third, the existence of a dozen vertical quasi-annual amphidromes is first reported, with cophase lines that may radiate toward the ~2000-m lower limit of Argo measurement. The well-known global meridional overturning circulation and the pseudozonal overturning currents in the equatorial Pacific, Atlantic, and Indian Oceans may possibly contribute to the observed vertical amphidromes.

Equatorward shift of annual Rossby waves in the Equatorial Pacific Ocean

Annual Rossby wave is a key component of the ENSO phenomenon in the equatorial Pacific Ocean. Due to the paucity and seasonal bias in historical hydrographic data, previous studies on equatorial Rossby waves only gave qualitative description. The accumulation of Argo measurements in recent years has greatly alleviated the data problem. In this study, seasonal variation of the equatorial Pacific Ocean is examined with annual harmonic analysis of Argo gridded data. Results show that strong seasonal signal is present in the western equatorial Pacific and explains more than 50% of the thermal variance below 500 m. Lag-correlation tracing further shows that this sub-thermocline seasonal signal originates from the eastern equatorial Pacific via downward and southwestward propagation of annual Rossby waves. Possible mechanisms for the equatorward shift of Rossby wave path are also discussed.

Bottom‐pressure signature of annual baroclinic Rossby waves in the northeast tropical Pacific Ocean

AbstractThe annual cycle in bottom pressure (pb) in the northeast tropical Pacific Ocean (NTPO) is studied. Focus is on a zonal section along 12°N between 105°W and 145°W that is characterized by a strong annual cycle in sea level from satellite altimetry. Estimates of pb from the Gravity Recovery and Climate Experiment (GRACE), a state estimate produced by the Estimating the Circulation and Climate of the Ocean (ECCO) consortium, and a linear Rossby wave model (LRWM) are used. The GRACE NTPO pb annual cycle shows amplitudes as large as 1 cm water equivalent. The GRACE data also evidence westward propagation that is consistent with the behavior of long mode‐1 Rossby waves at this latitude, with phase increasing from east to west at a rate of ∼0.34 m s−1. The ECCO and LRWM pb estimates corroborate the notion that GRACE reveals the pb signature of annual Rossby waves driven by interior wind stress curl and possibly damped by frictional processes. Results have implications for attempts to constrain global ocean mass using a single point mooring.

Evidence of equatorial Rossby wave propagation obtained by deep mooring observations in the western Pacific Ocean

We investigated long-term zonal flow (U) oscillations in the deep layer in the western Pacific Ocean by comparing measurements obtained from deep mooring observations near the equator with a high-resolution model. For the comparison, we used mooring data at 2,000 m and below north and south of the equator obtained from preceding studies. In our model results, vertically propagating equatorial annual Rossby waves were detected in the velocity and pressure anomaly (P a) fields. P a had a multi-cell meridional structure symmetric about the equator. The corresponding meridional distribution of U, averaged along the wave propagation, was the same as that previously obtained from Argo float data, suggesting the existence of the multi-cell structure in P a in the real ocean. A harmonic analysis of the simulated U indicated that the equatorial annual Rossby waves excited over the central and eastern equatorial Pacific propagated down to below 2,000 m in the western equatorial Pacific. Decomposition of the annual U harmonics indicated that the multi-cell structure in pressure was not a manifestation of a single meridional mode but a composite of several lowest meridional modes, suggesting that the cell widths may change according to relative mode intensities. The mooring data north and south of the equator was consistent with the simulated multi-cell structure in P a of the equatorial Rossby waves, although some discrepancies remained. We consider that the observed variation in U is a manifestation of the Rossby wave propagation in the real ocean. The present study is the first to recognize the existence of equatorial annual Rossby waves below 2,000 m in the western Pacific Ocean from direct current observations.

On quartet interactions in the California Current system

Abstract. Sea surface height (SSH) altimetry observations for 1992 to 2009 off California are used to show that observed quasi-zonal jets were likely driven by near-resonance interactions between different scales of the flow. Quartet (modulational) instability dominated and caused non-local transfer of energy from waves and eddies to biannual oscillations and quasi-zonal jets. Two types of quartets were identified: those composed of scales corresponding to (a) quasi-zonal jets, annual and semiannual Rossby waves and mesoscale eddies, and (b) biannual oscillations, semiannual Rossby waves and mesoscale eddies. The spectral centroid regularly shifted into the domain of low-order modes. However, the spectrum of SSHs does not demonstrate a power behavior. This says that the classical inverse cascade is absent. For a case with bottom friction, quartet instability required the existence of a certain level of dissipativity in the flow.