On unravelling mechanism of interplay between cloud and large scale circulation: a grey area in climate science

Abstract

The interaction between cloud and large scale circulation is much less explored area in climate science. Unfolding the mechanism of coupling between these two parameters is imperative for improved simulation of Indian summer monsoon (ISM) and to reduce imprecision in climate sensitivity of global climate model. This work has made an effort to explore this mechanism with CFSv2 climate model experiments whose cloud has been modified by changing the critical relative humidity (CRH) profile of model during ISM. Study reveals that the variable CRH in CFSv2 has improved the nonlinear interactions between high and low frequency oscillations in wind field (revealed as internal dynamics of monsoon) and modulates realistically the spatial distribution of interactions over Indian landmass during the contrasting monsoon season compared to the existing CRH profile of CFSv2. The lower tropospheric wind error energy in the variable CRH simulation of CFSv2 appears to be minimum due to the reduced nonlinear convergence of error to the planetary scale range from long and synoptic scales (another facet of internal dynamics) compared to as observed from other CRH experiments in normal and deficient monsoons. Hence, the interplay between cloud and large scale circulation through CRH may be manifested as a change in internal dynamics of ISM revealed from scale interactive quasi-linear and nonlinear kinetic energy exchanges in frequency as well as in wavenumber domain during the monsoon period that eventually modify the internal variance of CFSv2 model. Conversely, the reduced wind bias and proper modulation of spatial distribution of scale interaction between the synoptic and low frequency oscillations improve the eastward and northward extent of water vapour flux over Indian landmass that in turn give feedback to the realistic simulation of cloud condensates attributing improved ISM rainfall in CFSv2.

1 Introduction

The agriculture, drinking water, health, power and the gross domestic product of a country in monsoon region are strongly affected by the monsoon rainfall variability and its sensible prediction. After the initiation of Monsoon Mission Project (http://www.tropmet.res.in/monsoon/files/about_us.php) there is a continuous and concerted effort and trend to improve the dynamical prediction of Indian summer monsoon(ISM) either by modifying different physical processes (Hazra et al. 2016; De et al. 2016) or by incorporating as well as modifying different parameterization techniques (Goswami et al. 2015, 2016; Saha et al. 2017) or by changing the ensemble technique of perturbed fields with the bias corrected forecast (Abhilash et al. 2013, 2014) in the dynamical model Climate Forecast System version 2 (CFSv2) of National Centre for Environmental Prediction (NCEP) (hereafter referred as NCEP CFSv2 or simply CFSv2). One such initiative was the nonlinear relation between the cloud modification and the large scale dynamics. As the cloud was the harbinger of the turbulent weather and hence, the rainfall, previous efforts were made by the cloud physicists to relate the cloud and the convective processes with the rainfall without considering how the clouds affected nonlinearly on the large scale dynamical processes. On the contrary, the monsoon meteorologists were trying to establish the physical and dynamical relationship between the large scale process and the rainfall without acknowledging the attribution of the nonlinear aspect of global circulation on the cloud modification and its role on rainfall change. Then, how do the cloud, circulation and climate interact among themselves?—a fundamental puzzle, raised by Stevens and Bony (2013) and Bony et al. (2015). The inadequate computer power and the sparse observational networks compelled the cloud physicists to investigate within the cloud system as it was impossible to look into the global phenomena through the scale in which the cloud processes were examined. With the advent of the powerful computer and enhanced conceptual understanding developed through the exhaustive visualization obtained from high resolution observational networks (both in-situ and remote sensing), the scientists are now capable to explore the cloud processes with the resolution in which the large scale systems can be evaluated simultaneously.

Theoretically, tropical circulation is thermodynamically controlled by diabetic heating which is dependent on convective flux of heat and moisture parameterized in climate model (Held and Hoskins 1985; Stevens and Bony 2013; Shepherd 2014). The convective fluxes as well as the large scale advection of heat and moisture determine the realistic cloud condensate in the climate model. Therefore the relationship between cloud and the tropical circulation is highly nonlinear in nature. So, an advanced understanding utilising a state-of-the-art supercomputers and enormous observing facility is imperative to improve the numerical representation of how the clouds interplay nonlinearly with the general circulation through the heat and moisture convection. Moreover, an exploration of coupling between cloud and circulation is prerequisite for reducing the uncertainty in the climate sensitivity and to increase the precision in climate projection (Stevens and Bony 2013). One such recent endeavour was that how the modification of one of the cloud parameters viz. critical relative humidity (CRH) (that controls the cloud condensation mixing ratio by adjusting the area of condensation and evaporation of model convection scheme) changed the Indian summer monsoon rainfall (ISMR) and the large scale flow pattern during the contrasting monsoon season (De et al. 2016). It was beyond the scope of the paper De et al. (2016) to pinpoint the underlying processes of the model responsible for linking cloud and the large scale circulation to resolve the puzzle raised by Bony et al. (2015).

The interannual variability of seasonal ISMR can be explained by the analysis of variance (Rowell 1998) which may be bifurcated by its external and internal component. The external variance is controlled by the slowly varying boundary condition like sea surface temperature (SST) anomaly, El Nino Southern Oscillation (ENSO) forcing etc. (Palmer 1994; Kang et al. 2004; Kang and Shukla 2006) whereas the internal variance arises from the interactions among different scales of motions like the interaction of seasonal mean with intraseasonal oscillations (ISOs) and synoptic scale disturbances, interaction between ISOs and synoptic scale (Kang and Shukla 2006; Suhas et al. 2012). Moreover, the interaction between large scale circulation and orography and many more small scale instability processes contribute to the internal variance (Kang et al. 2004; Suhas et al. 2012). The potential predictability attributing to the external variance is large as the external variance is controlled by the slowly varying boundary conditions whereas the predictability due to the internal variance is inherently small as it is associated with the random variability of internal dynamics invoking nonlinearity of the system (Rowell 1998; Kang et al. 2004). The spatial distribution of external and internal variances delineated that the extratropics is basically controlled by the internal variance showing less potential seasonal predictability (Fig. 15.2 in Kang and Shukla 2006). Unlike the extratropics, tropics was dominated by the external variance and therefore, showed the large potential predictability. But the Indian monsoon region is an exception with at least two models (Korean Meteorological Agency and NCEP) in Fig. 15.2 of Kang and Shukla (2006) exhibit large internal variance and least signal to noise ratio. Moreover, Suhas et al. (2012) showed that the internal dynamics evolved from the nonlinear kinetic energy (KE) exchanges between the seasonal mean and the ISOs could explain 20% of the interannaul variability and about 50% of the interdecadal variability of monsoon. This implies that the seasonal prediction of ISM rainfall (ISMR) is a challenging task due to the inhibition of the predictability attributing the internal dynamics of the monsoon system which is generated from the nonlinear aspect of monsoon flow. As a part of the interaction between cloud and the large scale monsoon flow, the paper De et al. (2016) has already underscored the seminal role of the cloud modification through CRH on the nonlinear error energetics of ISM. This entails a more insightful investigation of how the cloud in terms of the CRH modification modulates the internal dynamics of ISM arising from the nonlinear scale interactions between different prominent scales like ISOs and seasonal mean; synoptic scale and seasonal mean; synoptic scale and ISOs during monsoon period. Another aspect of internal dynamics is the inherent uncertainty arises from the inadequate physical and dynamical processes and the insufficient mathematical formula governing the dynamical processes of the model (Kanamitsu 1985; Jung 2005). As this work has been concentrated to a particular physical process i.e. cloud modification through one of its parameter CRH in coupled model, uncertainty in wind field developed due to the different CRH profiles of the model has been examined in the wavenumber domain through the comprehensive investigation of possible answers of the following questions. How does the CRH affect the spectral behaviour of error variance and its growth rate? How do the different inertial terms like error flux, generation terms in error growth rate budget (Boer 1984) substantiate the error and its growth rate spectra in terms of the nonlinear scale interactions among the planetary/ultra-long waves (wavenumber band 1–4), long waves (wavenumber range 5–10) and the synoptic and small scale waves (wavenumbers > 10) for different CRH experiments of CFSv2 during the contrasting monsoon seasons. This study is an extension of the work done so far in the paper De et al. (2016).

Two types of scale interactions are explored in this study. One is in frequency domain i.e. the time domain and other is in wavenumber domain, obtained from spatial domain. In frequency domain three prominent oscillations 30–60, 10–20 and 3–5 day along with the seasonal mean have been selected for scale interaction study. The 30–60 day oscillation is responsible for northward propagation of monsoon trough/tropical convergence zone (Dakshinamurti and Keshavamurti, 1976; Sikka and Gadgil 1980; Krishnamurti and Subrahmanyam 1982) during monsoon period. The westward propagation of monsoon low pressure systems, generating from the Bay of Bengal and controlling the active and break phase of Indian monsoon typically follows the 10–20 day oscillations (Krishnamurti and Ardanuy 1980; Kripalani et al. 2004). The low pressure systems like lows, depressions, deep depressions, cyclonic storms during the monsoon period which are the rain bearing systems of Indian monsoon have a life time of 3–5 days (Mooley and Shukla 1987; Sikka 1980). In wavenumber domain, the spectral distribution of error energy and the growth rate spectra of error have been computed to explore the role of CRH on the spectral error energetics study using the lower tropospheric (850 hPa) observed and model output winds of CFSv2 for three different vertical profiles of CRH. In addition to this, the nonlinear inertial terms like the convergence or divergence of error flux and the generation of error attributing error growth are computed in terms of the wave–wave exchange of error energy through nonlinear triad interactions. The nonlinear kinetic energy exchanges among the dominant mode of oscillations having different time period in frequency domain and the spectral error energetics study in wavenumber domain during the contrasting monsoon period entail the internal dynamics of Indian monsoon. The second section describes the CFSv2 model and design of experiment very briefly. Methodology for computation of nonlinear KE exchange in frequency domain and the spectral error energy and its growth rate budget in wavenumber domain are delineated in Sect. 3. Results are described comprehensively in Sect. 4 and Sect. 5 comprises of the conclusion in summarised form.

2 CFSv2 model and experimental design

The CFSv2 model is a fully coupled ocean–atmosphere model developed by NCEP with spectral triangular truncation T126 (horizontal resolution 100 km.) and 64 sigma-pressure hybrid vertical layers (Saha et al. 2013). The ocean model is Geophysical fluid dynamics Laboratory modular ocean model at 0.25°–0.5° grid spacing with 40 vertical layers (Griffies et al. 2004). Pattnaik et al. (2013) investigated the influence of Simplified Arakawa Schubert (SAS) and Relax Arakawa Schubert (RAS) cumulus parameterization schemes in Coupled Forecast System (CFS) model. The key point was that rainfall forecast skills over the Indian landmass are consistently higher in SAS as compared to RAS convective parameterization scheme. Therefore, in this study we have used simplified Arakawa–Schubert convection scheme with momentum mixing (Pan and Wu 1995; Hong and Pan 1998) along with Zhao and Carr (1997) large-scale microphysical scheme. The effect of sub-grid scale variations in moisture on the large scale condensation is accounted by critical relative humidity (CRH) as suggested by Zhao and Carr (1997) and the tendency of relative humidity was calculated by Sundqvist et al. (1989). It is assumed that all the cloud condensate to be in the cloudy region in order to calculate cloud optical thickness. The Rapid Radiative transfer model for shortwave radiation with maximum random cloud overlap (Iacono et al. 2000; Clough et al. 2005) is used as radiation scheme along with a four-layer Noah land surface model (Ek et al. 2003) and a two-layer sea ice model (Wu et al. 2005). The details are already available in De et al. (2016).

In a relative humidity (RH) based cloud scheme, cloud is not formed in general circulation model (GCM) until RH attains some critical value (i.e. CRH), above which the process of condensation initiates. Previous literatures (Walcek et al. 1990; Slingo et al. 1987) advocated a strong relationship between the CRH and the cloud coverage in GCMs assuming the magnitude of the CRH within the range of 60–90%. The CRH is a tuneable parameter based on observation showing different value for different GCM indicating its model dependency (Quass 2012; Williamson et al. 1987). The CFSv2 model has constant CRH of the value of 85% at the model generated vertical levels, that leaves a space to modify the vertical profile of CRH in the perspective of better simulation of ISM. Based on the vertical profile of RH (De et al. 2016) obtained from the radio-sonde data of nine stations on the monsoon trough zone and from the Modern-Era Retrospective Analysis for Research and Application Analysis (MERRA) daily RH product over the core monsoon zone (75°E–85°E, 15°N–25°N), it was observed that the RH increased initially upto the maximum value 90% for cloudy days and then decreased. The data was taken during June, July, August and September period of the years 2003 and 2009. The year 2003 was the normal monsoon year with the seasonal departure of rainfall 2.3% of long period average of ISMR whereas the severe deficient monsoon was occurred in the year 2009 with the seasonal shortfall of rainfall − 22% of long period average as per the end of season reports of India Meteorological Department (IMD) (http://www.imd.gov.in; De et al. 2016 and the references therein). The choice of 2003 as normal monsoon year was purely arbitrary but the year 2009 was chosen as it was the most severe drought year in recent past (after the year 2000) and could not be predicted either by the operational empirical models at IMD or by the dynamical models at national and international centres (Gadgil et al. 2005; Niranjan Kumar et al. 2013). On the basis of observation and reanalysis data, the design of experiment is given in tabular form shown in Table-I. The model generated medium level cloud has been defined by the boundary of 650 and 400 hPa at low latitude and 750–500 hPa at mid to high latitude (Moorthi et al. 2010; Sun et al. 2010). The high and low level clouds in CFSv2 model are at above and below the above mentioned boundary, respectively. The model has been executed for four initial conditions (ICs) on 5th day of February, march, April and May during the year 2003 and 2009 initialized with the climate forecast system reanalysis datasets (CFSR, Saha et al. 2010) and has been integrated for the period of monsoon upto the end of September following De et al. (2016). All the CSFv2 model runs for different CRH versions are performed at the high power computing system deployed at Indian Institute of Tropical Meteorology, Pune, India.

3 Data and computation

The reanalysis wind data at lower tropospheric level (850 hPa) of CFSR (Saha et al. 2010) and the European Centre for Medium range weather forecast reanalysis (ERA) are utilized over the Indian monsoon region (60°E–100°E; 0°–30°N) in addition to the CFSv2 model output of 850 hPa level wind field generated from the total 24 runs at four initial conditions during the contrasting years for three different CRH experiments. All the data are daily with one degree resolution. The period of study is June, July, August and September (JJAS) during the years 2003 and 2009.

3.1 Methodology for computation of nonlinear kinetic energy exchanges in frequency domain

Hayashi (1980) showed that the total nonlinear KE exchange denoted by 〈K · K(n)〉 can be partitioned into the two parts i.e. 〈K · K(n)〉 = 〈L(n)〉 + 〈K(0) · K(n)〉; where 〈K(0) · K(n)〉 is the KE exchanges between the time mean flow and the time transient flow, represented by mean–wave interactions. In this study it implies the amount of KE exchanges between seasonal mean and the oscillation of any frequency no. n except the time mean. Here, only the three dominant oscillations during the monsoon are considered i.e. the frequency numbers n correspond to the time period 30–60, 10–20 and 3–5 days oscillations. The term 〈L(n)〉 in above equation represents the nonlinear KE exchanges among different modes associated with frequency no. n excluding the seasonal mean and is denoted by wave–wave interactions. Three different modes of oscillations of above mentioned time period are considered here to evaluate the wave–wave interactions during monsoon period. In 〈L(n)〉, three waves of different frequency no. n, r and s interact among themselves in terms of individual triad interactions denoted by L(n,r,s) where n, r and s are inter-related with one of the following selection rules n = r + s, n = r − s and n = s − r (Saltzman 1957). The nonlinear KE exchange related with frequency no. n, may be evaluated by summing up all possible individual triad interaction L(n,r,s) following the above mathematical rule and is represented by ΣL(n,r,s). Hence, ΣL(n,r,s) = 〈L(n)〉. The detail mathematical equation is given in Agarwal et al. (2016). The mean–wave interaction 〈K(0) · K(n)〉 has been computed based on the formulation following Hayashi (1980).

3.2 Formulation of error energy and its growth rate budget in wavenumber domain

The spectral errors in seasonal simulation of lower tropospheric wind field of CFSv2 for three different CRH experiments (rhctl, rh90 and rhvar) have been evaluated in the energy/variance form following Boer (1984). The error in simulation of seasonal wind can be measured as

$${V_e}={\text{ }}{V_m}-{\text{ }}{V_a};\;{u_e}={\text{ }}{u_m}-{\text{ }}{u_a}\;{\text{and}}\;{v_e}={\text{ }}{v_m}-{\text{ }}{v_a},$$

(1)

where V, u and v are the total, zonal and meridional winds, respectively. X_e, X_m and X_a are the error, model output and the analysis/observed part (here CFSR data) associated with parameter X, respectively. Now, the error energy in seasonal simulation may be formulated as,

$$K\left( t \right) = ~\left\langle {\overline {\frac{1}{2}~{V_e}~\cdot~~{V_e}} } \right\rangle ~.$$

(2)

The overbar represents the spatial average over the domain of study (here the global tropical belt i.e. 30°S–30°N; 0°–360°E) and the symbol 〈〉 indicates the ensemble average over the whole season for a fixed initial condition at which the CFSv2 model has been executed. The associated growth rate equation taken from Boer (1984) may be written as,

$$\frac{{\partial K(t)}}{{\partial t}}=~ - \nabla \cdot \left( {\overline {{\frac{{{V_e} \cdot {V_e}}}{2}~{V_m}}} } \right) - \left( {\overline {{{u_e}{V_e}~ \cdot \nabla {u_a}+{v_e}{V_e}~ \cdot \nabla {v_a}}} } \right)+\overline {{{V_e} \cdot R}} .$$

(3)

The above equation is a barotropic equation applied at 850 hPa wind field. The error growth rate at any point, computed by the left hand side term is controlled by nonlinear convergence or divergence of error by simulated flow shown by the first term within first bracket in right hand side of Eq. (3), barotropic generation of error due to the nonlinear interaction between erroneous simulated flow and the observed/analysed flow represented by the 2nd term within the first bracket in r.h.s of same equation following Boer (1984). Errors due to all other processes like computational error, dissipation etc. except the above inertial processes are linked with the source/sink term shown by the last term in the r.h.s of above equation. Applying the Fourier transform in Eq. (2), the spectral form of error energy in wavenumber domain may be written as,

$${K_n}\left( t \right) = \left\langle {\overline {\frac{1}{2}({u_e}C_n^2 + {u_e}S_n^2 + {v_e}C_n^2 + {v_e}S_n^2)} } \right\rangle .$$

(4)

The subscript n in above equation represents wavenumber. The term u_eC and u_eS indicate the cosine and sine component of zonal wind error, respectively and the same are for meridional wind v also. As each term of error growth rate Eq. (3) is a triple product term, the equation can be expressed in the form of nonlinear error energy exchanges in terms of individual triad interaction associated with a wave of wavenumber n interacting with waves of other wavenumbers r and s following the selection rule for the permissible transfer of energy, already discussed in Sect. 3.2. Therefore, the spectral form of the error growth rate budget equation in seasonal simulation of wind field associated with three different CRH experiments of CFSv2 is given in Eq. (5) of Appendix.

4 Results

Before describing the results, it is necessary to delineate the gist of the conclusions made by De et al. (2016) as a prequel to maintain the continuity of the study. Among the three CRH experiments control (ctl), rh90 and rhvar (variable CRH) in CFSv2 model (see Table 1 for experimental design), the rhvar experiment showed more realistic variability of rainfall and precipitable water vapour with proper modulation of tropospheric temperature gradient compared to the ctl and rh90 studies during the contrasting monsoon years 2003 and 2009. This was substantiated with the modulation of convection through outgoing long wave radiation and the realistic variability of high cloud fraction, observed best in rhvar CRH experiment compared to other CRH runs of CFSv2 during good and bad monsoons. One of the reasons of dry rainfall bias in CFSv2 was the weakening of the eastward water vapour flux (WVF) and the large systematic error in the cross equatorial flow region over Arabian sea at the ctl run in normal monsoon. This dry bias was reduced due to the stronger eastward WVF and the lowering of the wind bias for rhvar experiment in respect of other CRHs. The reduced northward extent of WVF over the Indian landmass (another factor for dry bias in control experiment) was also enhanced in variable CRH simulation shown by the length of the rainy season resulting good seasonal ISMR in CFSv2 during 2003. From the whole study the following questions may be raised.

Table 1 Tabular form of experimental design for different value of critical relative humidity(CRH) in CFSv2 model

• 1. What is the exact mechanism that may account for the improvement of large scale dynamics in terms of the moisture flux and the easterly vertical shear of mean zonal wind (measure of length of the rainy season) by modulating clouds through CRH?

• 2. What are the inherent underlying processes in the GCM that may attribute to improve the systematic biases in wind field by modifying the cloud condensate only?

The mechanistic exploration of the coupling between cloud and large scale circulation in terms of multi-scale interactions (revealed as internal dynamics of monsoon) may possibly answer the above questions.

The modulation of the internal dynamics of Indian monsoon due to the different CRH experiments on CFSv2 is to be observed through scale interaction mechanism in terms of nonlinear KE exchanges among different prominent scales during monsoon in frequency as well as wavenumber domain. From the perspective of frequency domain it is to be seen how the CRH influences the mean–wave interaction between seasonal mean and the prominent low and high frequency oscillations mentioned in Sect. 1 and the wave–wave interactions among those waves excluding seasonal mean. All the interactions are to be validated with the observed KE obtained from CFSR and ERA wind data. From the point of view of wavenumber domain, the influence of CRH on the spectral error energy and its growth rate is to be examined that may help to evaluate the uncertainty in climate projection and the inaccuracy in climate sensitivity studies. The role of nonlinear inertial terms like error flux and generation terms to error growth in CFSv2 for different CRH experiments is to be evaluated through wave–wave interactions in terms of different triads among the planetary, large and small scale waves to explain the spectral error growth in the seasonal simulation of the 850 hPa wind field generated from different CRH version of CFSv2 model for exploring the underlying processes of a climate model that lead to the improvement of biases in large scale circulation due to the cloud modification.

4.1 Frequency domain

The total kinetic energy of the seasonal mean and different low and high frequency oscillations are calculated before evaluating the KE exchanges among those modes. The total KE associated with either seasonal mean or any low and high frequency oscillation has been computed as the summation of all KE exchanges while the mean or any wave interacts with all possible waves. The objective is to examine whether any quasilinear and nonlinear KE exchanges is responsible either due to the total KE exchanges or due to the change in internal dynamics of the CFSv2 model. Figure 1 describes the total KE associated with the seasonal mean, 30–60, 10–20 and 3–5 day synoptic scale waves during the normal (2003) and deficient (2009) monsoon year. The kinetic energy has shown for lower tropospheric (850 hPa) wind of CFSv2 model run at February, March, April and May initial conditions(IC) and at the average of all ICs denoted by ‘avg’ shown by hatched bars in above figure. Each IC comprises of three bars of different colours which represent three different CRH experiments ctl, rh90 and rhvar of CFSv2. The last two single bars indicate the observed KE computed from reanalysis wind field of CFSR and ERA data. The unit of KE is J/kg. The figures reveal the followings.

• 1. The seasonal mean and the 3–5 day synoptic disturbances are appeared to be having maximum and minimum KE, respectively showing a hierarchy of energy containing capacity of different modes from longer to shorter time period in the contrasting years of 2003 and 2009.

• 2. Comparing each mode of the year 2003 with that of 2009, it has been observed that there is not much variability of total KE associated with each mode for most of the cases except the KE of 30–60 day mode in Fig. 1c vs. 1d shown in avg.

• 3. The total KE of each mode associated with ctl, rh90 and rhvar experiments of CFSv2 is found to be approximately similar to the total KE of seasonal mean, 30–60, 10–20 and 3–5 day oscillations obtained from the reanalysis wind field of CFSR and ERA data in most of the cases.

Fig. 1

Total kinetic energy (KE) associated with seasonal mean, 30–60, 10–20 and 3–5 day oscillations during the normal (2003; a, c, e, g) and deficient (2009; b, d, f, h) monsoon year. KE is shown at each initial condition (Feb, Mar, Apr and May) and at their ensemble average (avg) along with the observed (here reanalysis) KE [CFSR (pink) and ERA (purple)]. The KE associated with averaged initial condition are represented by hatched bar with black border

Following Lorenz (1969), the scale of any wave is proportional to its energy containing capacity and that is appeared in Fig. 1 where the longer time period mode is having more KE compared to the KE of shorter one. From the above analysis it has been understood that the monsoon variability during the normal and deficient year has no role on the total kinetic energy of each mode as the energy associated with each mode of oscillations has been almost unaltered during the contrasting seasons. The variability in KE averaged over all the ICs (shown in avg) is appeared to be not significant for the three different CRH experiments of CFSv2 for most of the cases and this implies that the cloud modification through CRH may not modulate the total KE of each mode. Hence, it is imperative to explore the KE exchanges among the different modes of oscillations instead of total KE of these modes during the monsoon period.

4.1.1 Mean–wave interactions

The KE exchanges between the mean flow and the different mode of oscillations have been computed through mean–wave interactions following Hayashi (1980). It is a quasi-linear interaction with a positive value implying the seasonal mean losing energy to different low and high frequency oscillations. The negative value of interaction represents the gain of KE by the seasonal mean from other waves. Figure 2 describes the mean–wave interaction for the three different CRH experiments of CFSv2 during the JJAS period of the contrasting monsoon years of 2003 and 2009. The mean–wave interactions in 850 hPa wind field generated from CFSv2 run at February, March, April and May IC are shown in X-axis of all the Fig. 2a–f. The interactions averaged over all the ICs are denoted by avg in Fig. 2a–f. The observed mean–wave interactions are computed from lower tropospheric (850 hPa) CFSR and ERA wind data and are represented by single bar in Fig. 2. The followings are revealed from the figure.

Fig. 2

Mean–wave interactions between seasonal mean and 30–60, 10–20, 3–5 day oscillations for all CRH experiments during the contrasting monsoon years 2003 and 2009. KE is shown at each initial condition (Feb, Mar, Apr and May) and at their ensemble average (avg) along with the observed (here reanalysis) KE [CFSR (pink) and ERA (purple)]. The KE associated with averaged initial condition are represented by hatched bar with black border

The magnitude of interaction of seasonal mean with low and high frequency oscillations is proportional to the scale of oscillations interacting with seasonal mean during 2003 and 2009. This implies that the mean–wave interactions for 30–60 day mode are shown maximum particularly in KE exchanges averaged over all ICs (avg) compared to the interactions associated with 10–20 and 3–5 day modes. Similarly, the interaction between seasonal mean and 10–20 day wave is appeared to be larger than the mean–wave interactions of 3–5 day synoptic scale waves. Following the Lorenz (1969) concept, the energy associated with a wave is proportional to the scale of the wave. Hence, the amount of energy exchange between mean and any wave in mean–wave interaction is also proportional to the scale of participating wave.

All the CRH experiments of CFSv2 at all the ICs as well as the observations exhibit positive interactions implying transfer of KE from seasonal mean to other waves irrespective of normal and deficient monsoon. This indicates that the seasonal mean is the primary source of KE for all low and high frequency oscillations during the contrasting years, obtained from both model and observations (detail observational results was found utilizing the reanalysis dataset in Agarwal et al. 2016).

The 30–60 day mode is drawing more energy from seasonal mean during 2009 (Fig. 2b) compared to the good monsoon year 2003 (Fig. 2a) shown in KE averaged over all ICs (avg) for three CRH experiments of CFSv2. As far as 10–20 day oscillation is concerned, this low frequency mode is gaining more energy in 2009 (Fig. 2d) compared to that during 2003 (Fig. 2c) shown in avg column except the variable CRH experiment (rhvar) of CFSv2 model where 10–20 day mode is receiving slightly more energy during 2003 compared to 2009. This is because the maximum energy exchange from the seasonal mean to 10–20 day is occurred in February and May IC for rhvar experiment in respect of other experiments during 2003 (Fig. 2c). The 3–5 day synoptic scale wave is taking almost equal amount of energy from seasonal mean for rh90 and rhvar experiments during the contrasting seasons observed in avg comparing Fig. 2e, f whereas the control (ctl) experiment shows less KE exchange during 2003 in respect of 2009. The variability in KE exchanges at different initial conditions are appeared to be more during 2003 compared to in 2009 for all 30–60, 10–20 and 3–5 day oscillations.

The observed mean–wave interactions evaluated from CFSR and ERA 850 hPa wind field are exhibited as similar in nature with nearly equal in magnitude except the interactions associated with 3–5 day during both the years (Fig. 2e, f) where the underestimation of CFSR in respect of ERA is more in the year 2003 (Fig. 2e) compared to the year 2009 (Fig. 2f). Analysing the mean–wave interactions between model and observation, the model overestimated the interaction compared to the observed interaction in most of the cases except the mean–wave interactions of 30–60 day (Fig. 2a) and that of 3–5 day (Fig. 2e) during the good monsoon year 2003. As far as 30–60 day oscillation is concerned, the interaction derived from model wind field is comparable with the observed interactions in good monsoon year (Fig. 2a). The interaction of model generated synoptic scale with seasonal mean (Fig. 2e) is comparable with the observed interactions obtained from ERA data during normal monsoon year whereas the CFSR shows the underestimation.

It is understood from the mean–wave interactions that the overestimation of total KE derived from CFSv2 model compared to the observed KE in Fig. 1 is mainly because of overestimation in mean–wave interactions of the model in respect of observation shown in Fig. 2. Sikka (1980) and Kripalani et al. (2004) documented that the 30–60 low frequency mode was stronger (weaker) than 10–20 day mode during deficient (good) monsoon year. Comparing the mean–wave interactions of 30–60 day mode between the normal (Fig. 2a) and weak (Fig. 2b) years, it has been revealed that the low frequency mode is drawing more (less) energy from seasonal mean during 2009 (2003) for all CRH versions of CFSv2 model as well as observations. Hence, the possible inherent cause for stronger 30–60 day wave during weak monsoon is that the mode is receiving more KE from seasonal mean compared to that during normal monsoon year substantiated by both model and reanalysis data. The rhvar experiment among all the CRH studies exhibits the most realistic variability in mean–wave interactions showing the less gain of KE by 30–60 day from seasonal mean in 2003 in respect of energy gain during 2009 comparing the averaged ICs in Fig. 2a, b. As far as 10–20 day mode is concerned, this low frequency mode in Fig. 2c is gaining more energy from seasonal mean for the year 2003 compared to that in the year 2009 (Fig. 2d) associated with the rhvar CRH experiment among other experiments in avg of CFSv2 model indicating that the variable CRH experiment delineates the realistic variability in mean–wave interaction of 10–20 day oscillations following Sikka (1980) and Kripalani et al. (2004). The synoptic scale disturbances are generally drawing more energy from seasonal mean due to its larger activity during good monsoon year compared to drought year as revealed from reanalysed wind field (Agarwal et al. 2016) and that is reflected in ERA also comparing Fig. 2e, f. Model experiments for averaged initial condition describe comparable mean–wave interactions of 3–5 day wave for rh90 and rhvar CRH experiments between 2003 and 2009, however, the control run of CFSv2 describes unrealistic variability showing weaker interaction in good monsoon year compared to the interaction in bad monsoon. Therefore, the change in the vertical profile of CRH in CFSv2 in respect of the existing CRH profile modulates the internal dynamics measured through the realistic variability of the quasi-linear KE exchanges in mean–wave interactions during the normal and deficient Indian monsoon. The interaction between seasonal mean and different low and high frequency oscillations are discussed above comprehensively for different CRH experiments of CFSv2 compared with observations during the contrasting years. Now, the nature of nonlinear interaction between the low and high frequency oscillations excluding the seasonal mean will be examined through the wave–wave exchanges of KE in CFSv2 model experiments compared with observations to explore nonlinear part of the internal dynamics in frequency domain.

4.1.2 Wave–wave interactions

In wave–wave interactions any wave of frequency number (hereafter referred as frequency) n while interacting with the waves of frequency r and s will lose (gain) KE to (from) other waves according to the sign of the magnitude of the interactions. The positive (negative) value of interaction represents the wave of frequency n is losing (gaining) energy to (from) other frequencies r and s. All the frequencies are converted into the corresponding time period in Fig. 3. The nonlinear interactions are shown between two low and high frequency waves having time period X and Y described in Fig. 3a, f. The positive (negative) bar represents the KE flows from X to Y (Y to X) in Fig. 3a, f. The nonlinear KE exchanges are delineated at each IC February, March, April and May in which CFSv2 model has been run for all three CRH experiments, along with the wave–wave interactions obtained from CFSR and ERA reanalysis data. To compare the interactions of the two waves between the contrasting years, y-axis associated with the two modes during 2003 is kept same as the y-axis of those modes in 2009 applicable for all the Fig. 3a, f. The followings have been revealed from Fig. 3.

• (A) The maximum nonlinear KE exchanges are observed when the two low frequency oscillations 10–20 and 30–60 day oscillations are interacting among each other during both the years 2003 and 2009 shown in Fig. 3e, f. The energy exchanges are found to be minimum between 3–5 and 10–20 day as revealed from Fig. 3a, b. This implies that the magnitude of energy between the two oscillations depends upon the length of the time period of oscillations. The larger the time period of participating waves represent the more KE exchanges shown in Fig. 3 irrespective of deficient and normal monsoon years.

• (B) The magnitude of wave–wave interactions (whether positive or negative) are appeared to be stronger during 2009 compared to 2003 in most of the cases shown in Fig. 3 and the variability of interactions at different IC of CFSv2 model are found to be more in deficient monsoon year compared to good monsoon.

• (C) The wave–wave interactions between synoptic scale wave 3–5 day and the low frequency oscillations 30–60 day show a very weak interactions in normal monsoon year (Fig. 3a) observed in the averaged ICs (avg) of CFSv2 model in comparison with the interactions (shown in avg) occurred in deficient monsoon (Fig. 3b) for all the CRH experiments. The figures indicate that the 3–5 day mode receives insignificant amount of energy from 30 to 60 day mode during 2003 compared to the energy gain by the same mode in 2009 for ctl and rhvar experiments whereas the rh90 shows the opposite interactions.

• (D) Unlike the 3–5 and 30–60 day interactions, the synoptic scale mode is drawing more energy from 10 to 20 day mode during the normal monsoon year (Fig. 3c) compared to deficient year (Fig. 3d) shown in averaged ICs of  CFSv2 model. Out of the three CRH experiments, both the control and rh90 experiments in avg exhibits the larger gain of KE by the synoptic scale from 10–20 day low frequency mode in normal monsoon in respect of weak monsoon, however, rhvar delineates the less gain of KE in 2003 compared to 2009 (comparing Fig. 3c with 3d). In weak monsoon, 3–5 day mode is losing KE to 10–20 day as observed in control experiment (Fig. 3d).

• (E) The wave–wave interactions between two low frequency oscillations reveal that the 10–20 day mode is receiving KE from 30 to 60 day for the ctl experiment irrespective of the contrasting years (Fig. 3e, f). The energy has been transferred from (to) 30–60 day period to (from) 10–20 day during 2003 (2009) in rh90 CRH experiment whereas the rhvar shows the opposite flow of energy between the two low frequency waves observed in Fig. 3e, f.

• (F) As far as the comparison of wave–wave interactions derived from CFSv2 wind with the interactions obtained from the observed wind field are concerned, the CRH experiments (ctl, rh90 and rhvar) exhibit qualitative similarity with the observed interactions computed from CFSR anf ERA data in most of the cases except Fig. 3b, c. In Fig. 3b, the interaction in control and rhvar experiments shown in avg are found to be opposite interaction in respect of both CFSR and ERA and the interactions between 3–5 and 10–20 day for all experiments of CFSv2 are in disagreement with the observed interactions as revealed from Fig. 3c. Among all the experiments the wave–wave interaction of wind field derived from variable CRH experiment rhvar exhibits the realistic variability qualitatively for most of the cases in Fig. 3 (in avg) substantiated by either CFSR or ERA or both observed interactions.

Fig. 3

Wave–wave interactions among 30–60, 10–20 and 3–5 day oscillations for all CRH experiments during the contrasting monsoon years 2003 and 2009. KE is shown at each initial condition (Feb, Mar, Apr and May) and at their ensemble average (avg) along with the observed (here reanalysis) KE [CFSR (pink) and ERA (purple)]. The KE associated with averaged initial condition are represented by hatched bar with black border

Like mean–wave interactions, wave–wave interactions are in agreement with Lorenz (1969) theory. This implies that the KE transfer between the waves is proportional to the scale (here time period) of the interacting waves. As the low frequency oscillations 30–60 day is generally less strong during the normal monsoon year compared to the 30–60 day mode in weak monsoon (Sikka 1980; Kripalani et al. 2004), there is a weak interaction observed between the synoptic scale system and 30–60 day mode in Fig. 3a compared to the interaction revealed in avg during the year 2009 (Fig. 3b). Similarly, the other low frequency oscillations 10–20 day oscillations regulate the westward propagation of monsoon low pressure systems by controlling active and break phases of Indian monsoon (Krishnamurti and Ardanuy 1980; Sikka 2006) and hence, it is observed strong (weak) during normal (deficient) monsoon seasons (Kulkarni et al. 2011; Preethi et al. 2011). Substantiating with above findings, the CFSv2 model also exhibits that the synoptic scale mode is receiving more nonlinear KE from 10 to 20 day during 2003 compared to in 2009 shown in averaged ICs of control and rh90 CRH experiments, however, the CFSR and ERA interactions between 3–5 and 10–20 day are found to be opposite to the interactions derived from model (Fig. 3c, d). Here, the rhvar experiment has failed to represent the realistic variability in wave–wave interaction comparing Fig. 3c, d. It is beyond the scope of the present study to explore the variability in nonlinearity between model and observation as revealed from Fig. 3c. From Fig. 3 it is not understood where the interactions are taking place over the Indian landmass. Does the CRH influence the geographical distribution as well as the magnitude of interactions at different locations over Indian region? The nonlinear interactions at spatial domain may give some explanations of the above questions.

Fig. 4

Geographical distribution of interaction between 3–5 and 30–60 day oscillations for different CRH experiments (ctl, rh90 and rhvar) of CFSv2 model and for ERA and CFSR reanalysis data during the contrasting years. The interactions using the model outputs for each experiment are shown after averaging all initial conditions at which model is executed

Fig. 5

Geographical distribution of interaction between 3–5 and 10–20 day oscillations for different CRH experiments (ctl, rh90 and rhvar) of CFSv2 model and for ERA and CFSR reanalysis data during the contrasting years. The interactions using the model outputs for each experiment are shown after averaging all the initial conditions at which model is executed

Figures 4 and 5 delineate the spatial distribution of nonlinear interactions of high frequency mode 3–5 day with the low frequency modes 30–60 and 10–20 day, respectively over the Indian monsoon region. The positive (negative) value contour of interactions represent the KE flows from X to Y (Y to X) while the nonlinear interactions are taking place between the time period X and Y in Figs. 4 and 5. The first and third row represent the interactions for three different CRH experiments (ctl, rh90 and rhvar) of CFSv2 model during 2003 and 2009, respectively, whereas the second and fourth row show the observed interactions obtained from ERA and CFSR data during the same years, respectively in above figures. Most of the interactions between the synoptic scale disturbances and the 30–60 day mode are taking place over the core monsoon zone/monsoon trough area (18°N–29°N, 69°E–96°E) as revealed from Fig. 4a, e during the normal monsoon year. Observation exhibits more interactions over land than that over ocean and it is quite understandable because of more variability of lower tropospheric wind over Indian landmass due to the topography compared to the wind over oceanic region (Fig. 4d, e). Moreover, the synoptic scale systems interact with 30–60 day starting from Bay of Bengal and adjoining ocean and land region extending upto Pakistan region causing the organisation of the monsoon transients over the core monsoon zone (Mooley and Shukla 1987; Sikka 2006; Agarwal et al. 2016) during the good monsoon year. As far as the nonlinear interactions derived from different CRH experiments of CFSv2 model are concerned, the control experiment depicts strong negative interactions at Myanmar region (17°N, 95°E) which is not supported by observations (Fig. 4d, e). In addition to this, there is no interaction in westward direction over the Indian landmass as observed in ctl and rh90 experiments (as the contours are not seen beyond 74°E longitudinal line in westward direction over land region) (Fig. 4a, b). On the contrary, in rhvar experiment (Fig. 4c), the monsoon transients (3–5 day scale) have interacted with low frequency oscillations in northwest ward direction extending upto Pakistan region. This implies that the KE transfer between the monsoon systems and low frequency oscillations has been restricted in westward direction and is not reaching upto Gujarat region for both ctl and rh90 CRH experiments of CFSv2 model. Unlike these two experiments, variable CRH experiment (Fig. 4c) shows the distribution of interactions substantiated with observations (Fig. 4d, e) with more number of positive and negative value contours which extend beyond the Gujarat region in westward direction compared to ctl and rh90 experiments. This may be one of the inherent causes for less rainfall in control and rh90 version of CFSv2 compared to rhvar CRH experiment as the dry rainfall bias has been reduced in rhvar during the normal monsoon year (De et al. 2016). All the CFSv2 model experiments and the observation depict more negative value contours compared to the positive value contours resulting net negative interactions which imply that the synoptic scale systems are gaining KE from 30 to 60 day during good monsoon year. Relatively weak interactions are observed in all the three versions of CFSv2 model (Fig. 4f–h) and observations (Fig. 4i, j) during the deficient monsoon 2009. The dispersive nature of spatial distribution of interactions during 2009 indicates that the interactions do not confine within the core monsoon zone which is nicely reflected in all CRH versions of CFSv2 model (Fig. 4f–h). That is why the monsoon transients have less affinity to organise themselves along the monsoon trough region causing the trough to be less cyclogenesis during the weak monsoon in respect of the normal monsoon year (Agarwal et al. 2016). Among all the model experiments rh90 (Fig. 4g) and rhvar (Fig. 4h) exhibit the less interactions over India and adjoining region compared to the interactions observed in ctl experiment (Fig. 4f). Hence, rh90 and rhvar experiments show more realistic distribution of interactions supported by observations compared to control experiment. One of the causes for the reduction of wet bias in rh90 and rhvar simulations compared to the control experiment in CFSv2 during the drought year 2009 (De et al. 2016) may be due to the improved nonlinear interaction between synoptic scale and 30–60 day mode revealed as internal dynamics of monsoon.

Similar to Fig. 4, the nonlinear interactions between synoptic scale and 10–20 day low frequency oscillations underscore large density of interactions over the monsoon trough region found in model experiments (Fig. 5a–c) and observed data (Fig. 5d, e) as well during 2003. The contours are appeared at head Bay of Bengal and adjoining region and extend upto Pakistan region over the landmass as revealed from rh90 and rhvar experiments but not from ctl experiment. Moreover, the northern limit of the contour has reached upto 30°N in rh90 and rhvar CRH experiments whereas the ctl experiment shows the limitation of interactions beyond 26°N latitudinal belt. The westward (upto Pakistan region) and the northward (upto 30°N) extent of spatial distribution of interactions are depicted in both ERA (Fig. 5d) and CFSR (Fig. 5e) reanalysis data. The negative value contours are more than the positive value that result a net negative value of interactions between synoptic scale and 10–20 day mode as observed in both model experiments and observation during 2003. This implies the gain of the nonlinear KE by the monsoon transients from the 10–20 day oscillations during the normal monsoon. Weak interactions are delineated during the deficient year (Fig. 5f–j). There is hardly any interactions between 3 and 5 and 10–20 day observed in westward (less than 65°E longitudinal line) and northward (beyond 27°N latitudinal line) direction as revealed from all model experiments (Fig. 5f–h). Hence, the westward and northward propagation of synoptic scale disturbances have been restricted as because the monsoon transients are not getting adequate KE from low frequency to move further causing suppress cyclogenesis of monsoon trough during the deficient monsoon (Agarwal et al. 2016). In ERA (Fig. 5i) and CFSR (Fig. 5j), although large number of contour are appeared but due to the comparable amount of positive and negative value contours, the gross interaction is insignificant after nullifying each other. Hence, the energy exchanges between monsoon transients and low frequency oscillations are small that actually limit the propagation of monsoon systems in 2009. The contrasting variability in the geographical distribution of interactions are clearly depicted between the normal and deficient monsoon year in all the CRH experiments of CFSv2 comparing Fig. 5a–c with f–h, respectively. So, in most of the cases the CRH modification in CFSv2 dictates not only the magnitude but the geographical distribution of nonlinear interaction of synoptic scale with low frequency oscillations also during the contrasting monsoon season over the Indian landmass. Hence, the interplay between the cloud and circulation through CRH may be manifested as a change in internal dynamics of Indian monsoon revealed from the mean–wave as well as the wave–wave interactions among the low and high frequency oscillations. One of the reasons for the improved seasonal simulation of monsoon in the variable CRH experiment of CFSv2 may be the realistic modulation of internal dynamics of monsoon. Other facet of internal dynamics of Indian monsoon is to study the error KE in wavenumber domain for different CRH experiments of CFSv2 model with respect to CFSR data to explore the uncertainty in climate sensitivity to the cloud modification. Therefore, the following section deals with the spectral evolution of the error in simulation of seasonal wind field and how the different nonlinear inertial terms vary for the prescribed relative humidity profiles of CFSv2 in wavenumber domain. How do the inter-scale error propagation characteristics differ with the different CRH structures in CFSv2 model?

4.2 Wavenumber domain

The spectral form of error growth rate equation shown in Eq. (5) of “Appendix” is formulated in terms of the wave–wave exchanges through triad interactions. In a triad (n, r, s) three waves of different wavenumbers n, r and s interact among themselves and exchange energy following some permissible selection rule n = r + s or n = r ± s (same as in Sect. 3.1). Here, n is the wavenumber whose triad interactions are to be calculated when the wavenumber n interacts with the waves of wavenumbers r and s. As each term in r.h.s. of Eq. (5) is a triple product term, the double Fourier transform may be applied in each term to formulate in triad interactive form following the previous studies by Krishnamurti et al. (2003) and De (2010a). The spectral analysis of any term in Eq. (5) can be evaluated by computing the spectrum at any wavenumber n of that term after summing all individual triad associated with the wavenumber n following the methodology (Sect. 3.1) discussed in frequency domain. Thus the nonlinear convergence or divergence of error flux and the generation of error in Eq. (3) have been transformed into the scale interactive form through the participating triad interactions shown in the r.h.s of Eq. (5). The positive (negative) value of interactions represent the gain (loss) of error energy of the wave of wavenumber n from (to) other waves of wavenumber r and s in triads (n, r, s). The error spectra have been evaluated from Eq. (4) and the error growth rate spectra have been computed from the left hand side of Eq. (5). The error energy, its growth rate and the different inertial terms in error growth rate budget equation have been calculated averaging over all the initial conditions of different CRH runs of the CFSv2 model.

De et al. (2016) depicts that the variable CRH profile in CFSv2 model has improved the seasonal simulation of wind field by showing the reduced error in spatial distribution over India and adjoining region during the contrasting monsoon years 2003 and 2009. But it is not understood whether the internal dynamics has some role in lowering the error due to the change in relative humidity structure in the CFSv2 model. This can be measured by exploring the spectral error energy (Lorenz 1969), its growth rate and how the different nonlinear inertial terms respond to the error growth in terms of the scale interactions through individual triad. Figure 6 depicts the spatial distribution of the error and its growth rates for three different CRH experiments ctl, rh90 and rhvar of CFSv2 model during 2003 (Fig. 6a–f) and 2009 (Fig. 6g–l). The scale of the error remains same for both years and similarly for error growth rate for easy comparison. There is a systematic build-up of error kinetic energy at wavenumber 4 as observed in all CRH experiments (Fig. 6a, b, c) with the maximum and minimum error shown in ctl (Fig. 6a) and rhvar (Fig. 6c) experiment, respectively. The secondary maximum is observed at wavenumber 1 but the magnitude of error associated with wavenumber 1 exhibits insignificant variations in different experiments of CFSv2 comparing among the Fig. 6a–c. Hence, the decrease in the geographical distribution of error of lower tropospheric (850 hPa) seasonal wind appeared in rhvar experiment revealed from De et al. (2016) is mainly because of the lowest error observed at wavenumber 4 in that experiment compared to the error associated with the same wavenumber in other CRH experiments during 2003. Previous studies by Kanamitsu (1985), Kanamitsu et al. (1995) and De (2010a, b) were documented that the error was dominated in the spectral band of ultra-long/planetary scale waves (wavenumber band 1–4) in global atmospheric models like NCEP Global forecast system (NCEP GFS) (previously called NCEP MRF model) and the National Centre for Medium Range Weather Forecast model. In this study the coupled model delineates the error maxima associated with the same spectral range in which the error has also been observed maximum at atmospheric GCM models which implies that the inclusion of ocean model with atmospheric GCM does not change the spectral range of error maxima. Hence, it is speculated that the uncertainty may be governed by the internal variance of the model. Here, the internal variance of the model is computed by the spectral evolution of error energy/variance, its growth rate and the different inertial terms responsible for error growth due to the cloud modification by changing different CRH profiles of CFSv2. The spectral error growth rate curves exhibit maximum growth rate at wavenumber 4 during 2003 shown in Fig. 6d–f. Among all the CRH experiments smallest error growth rate is appeared at rhvar experiment (Fig. 6f) that causes the smallest error (Fig. 6c) in good monsoon year. During the deficient monsoon year, the error has developed at wavenumber 1 for all CRH experiments (Fig. 6g–i), out of which the rhvar depicts the least error (Fig. 6i) with respect to other CRH experiments. Hence, the lowest spatial error variance observed in seasonal simulation of the lower tropospheric (850 hPa) wind obtained from rhvar CFSv2 model run revealed from De et al. (2016) is due to the minimum error accumulation at wavenumber 1 of rhvar run compared to other CRH version of CFSv2 during the weak monsoon. The error growth rate curves are showing peaks at wavenumber 1 for all CRH runs (Fig. 6j–l) with the smallest error growth rate is observed in rhvar experiment (Fig. 6l). Now, the questions arise—which nonlinear term is responsible for spectral error growth during the contrasting monsoon seasons? From which scale has the error been propagated to the planetary scales? The comprehensive triad interaction analysis of wavenumber 4 and 1 of the dominant inertial term in error growth rate equation (5) during the year 2003 and 2009, respectively may give some answers of the above questions by evaluating the scale interactive aspects of error growth for different experiments of CFSv2.

Fig. 6

Spectral error energy and its growth rate for ctl (cntrl.), rh90 and rhvar CRH experiments of CFSv2 during the years 2003 and 2009. The error is calculated based on CFSR reanalysis data. The graphs shown, are averaged over all initial conditions. The unit of the error is m²/s² and the error growth rate is m²/s³

In order to examine the relative importance of each term with respect to residual in error growth rate Eq. (5), all the terms are analysed and it is understood that the nonlinear convergence or divergence of error flux is the most dominating dynamical term as was inferred from Boer (1984), De (2010a, b). The positive (negative) triad interaction in error flux term represents the nonlinear convergence (divergence) of error to (from) the wavenumber n. It has been observed from Fig. 6 that the error and its growth rate are insignificant beyond the wavenumber 20. Hence, the analysis of triad interactions in Figs. 7 and 8 has been restricted to wavenumber 20 only. The spectra of error energy at any wave (here wavenumber 4 and 1 in Figs. 7, 8, respectively) are associated with all possible triads following n = r + s and n =|r − s| comprising of positive and negative value triad interactions and are computed from net interaction (whether positive or negative) after combining all the positive and negative value triads. The Y-axis of Figs. 7 and 8 show the plethora of individual triads in the form of (n, r, s) upto wavenumber 20 (taken for s) and the X-axis depicts the magnitude of the nonlinear error KE exchange for each triad. The triad spectra for n = r − s (n = s − r) are shown in Fig. 7a, c, e (Fig. 7b, d, f) for ctl, rh90 and rhvar experiments of CFSv2, respectively. The net interaction at any wavenumber can be obtained by combing the spectra for n = r − s and n = s − r of the respective CRH experiment. Here, the spectra of wavenumber 4 for n = r + s [i.e. the triads (4, 3, 1), (4, 2, 2) and (4, 1, 3)] are found to be negligibly small compared to the spectra for other selection rules for all the CFSv2 experiments and hence, are not included in Fig. 7. Figure 7 depicts the lowest interactions in rhvar CRH run (Fig. 7e, f) compared to the ctl and rh90 experiments shown in their respective figures associated with wavenumber 4 that may cause the lowest error and its growth rate, appeared at wavenumber 4 in rhvar run of CFSv2 shown in Fig. 6 during the good monsoon year. The more number of positive value triads with respect to its negative value may lead to the net positive interaction by summing up all the triads of Fig. 7a, b which implies the net convergence of error at wavenumber 4 in ctl experiment. Further analysis of triad spectra depicts that the wavenumber 4 is drawing error KE from wavenumber 5, 8, 9, 12 and 13 through the triads (4, 4, 8), (4, 5, 9), (4, 8, 12) and (4, 9, 13) dominantly (Fig. 7b) whereas there is a loss of error KE of wavenumber 4 in terms of the triad (4, 13, 9) shown in Fig. 7a. In rh90 experiment (Fig. 7c, d), the number of positive value triads are more than the triads of negative value implying the convergence of error at wavenumber 4. But no significant peaks are observed except the triad (4, 17, 13) in Fig. 7c where the error has been transferred from planetary wave of wavenumber 4 to small scales (wavenumber 13 and 17). The number of negative value triads is found to be more in the variable CRH experiment (Fig. 7e, f) that may lead to lowering the convergence of error at wavenumber 4 (as the net error flux may be less positive by adding up the triads for n = r − s and n = s − r) compared to other CFSv2 experiments in Fig. 7. Individual triad interaction analysis shows that the wavenumber 4 is gaining energy from ultra-long and long waves of spectral range 2–10 through the triads (4, 6, 2) to (4, 10, 6) in n = r − s (Fig. 7e) whereas the error has been propagated to the small scales significantly in terms of the triads (4, 17, 13), (4, 21, 17) and (4, 8, 12), (4, 17, 21) in Fig. 7e, f, respectively. The wavenumber 4 loses energy also to the large scale through the triad (4, 2, 6) in n = s − r.

Fig. 7

Individual nonlinear triad interaction spectra for error energy flux term at wavenumber N = 4 interacting with waves of other wavenumbers R and S for N = R − S and N = S − R associated with different CRH experiments in CFSv2 during the normal monsoon year 2003

The deficient monsoon year also shows the lowest scale interactions in rhvar run (Fig. 8e, f) compared to other CRH study of CFSv2 model like the good monsoon year. The decrease in error and its growth rate in variable CRH profile at wavenumber 1 shown in Fig. 6 is due to the less error flux observed compared to the ctl (Fig. 8a, b) and rh90 (Fig. 8c, d) profile in climate model. The larger number and magnitude of the positive value triads compared to the triads of negative value are depicted in control experiment while individual triads are analysed, leading to the net convergence of error at wavenumber 1 (Fig. 8a, b). Dominant error energy exchanges are taking place at smaller scales during the drought year compared to the scale interactive energy exchanges in normal monsoon. The error has been transferred significantly from the small and synoptic scale waves of wavenumbers 11, 12, 13, 15 and 16 to the planetary scale of wavenumber 1 showing in triads (1, 12, 11), (1, 13, 12) and (1, 16,15) for n = r − s (Fig. 8a) and (1, 12, 13) and (1, 15, 16) for n = s − r (Fig. 8b). There is a prominent divergence of error from planetary scale to long waves (wavenumbers 6–8) and short waves having wavenumber range 14–17 delineated in triads (1, 7, 6), (1, 8, 7) and (1, 15, 14), (1, 17, 16), respectively. The increased relative humidity at all vertical levels of the model in rh90 experiment than that in control experiment (Table 1) enhances the scale interactions maximum in rh90 run for n = r − s and n = s − r spectra with more number of positive value triads than the triads of negative value (Fig. 8c, d). Like the ctl experiment, energy exchanges in rh90 are significant in the small scale range i.e. the wavenumber range 10–12 for n = r − s and the wavenumber band 11–13 for n = s − r. The wavenumber 1 is gaining energy from the spectral band 10–12 through the respective triads [(1, 11, 10), (1, 12, 11) and (1, 11, 12)] in Fig. 8c, d and loses energy to wavenumbers 12 and 13 observed in triads (1, 13, 12) and (1, 12, 13) for n = r − s and n = s − r, respectively. As far as the variable relative humidity experiment is concerned (Fig. 8e, f), the density of the interactions are more in wavenumber band > 10 and there is net convergence of error at wavenumber 1 due to the large number and magnitude of positive value triads with respect to its negative value. The wavenumber 1 is receiving energy significantly from small scale waves of wavenumbers 14, 15 and 18–20 though the triads (1, 15, 14) and (1, 19, 18), (1, 20, 19), respectively for n = r − s, whereas there is only one triad (1, 8, 7) in which large scale is losing energy to wavenumber 1. Similarly, there is a divergence of error from ultra-long wave to small scales 12, 13, 15 and 16 through the triads (1, 13, 12) and (1, 16, 15). For n = s − r, only one triad (1, 14, 15) is significant where the largest wave is losing energy to small scales of wavenumbers 14 and 15.

Fig. 8

Individual nonlinear triad interaction spectra for error energy flux term at wavenumber N = 1 interacting with other waves of wavenumber R and S for N = R − S and N = S − R associated with different CRH experiments in CFSv2 during the weak monsoon year 2009

The comprehensive spectral analysis of error, its growth rate and the nonlinear error flux in triad interactive form at the dominant wavenumbers reveal that the error has been accumulated at the planetary scale range irrespective of normal and weak monsoon years depicting the error associated with wavenumber 4 (1) during the good (bad) year at all CRH studies of CFSv2 model. Both the atmospheric general circulation model (NCEP GFS) and the coupled model (CFSv2) have shown the dominant spectral mode of error variance within the wavenumber band 1–4 which imply that the coupling of ocean with atmosphere in climate model does not alter the error accumulation at the predominant ultra-long scale range. The rhvar CRH experiment exhibits the lowest error and its growth rate at the respective wavenumber during the normal and deficient monsoon season among all the CFSv2 experiments. The normal monsoon year is dominated by the error energy peak at the planetary scale of wavenumber 4 showing the convergence of error at this wavenumber from the planetary and long waves of spectral range 2–10 whereas the weak monsoon year exhibits the maximum error energy at wavenumber 1 in which the error has been transported from the synoptic and small scale range of wavenumbers > 10. The bias reduction in lower tropospheric wind due to the less nonlinear convergence of error to the planetary scales in wavenumber domain and the realistic modulation of spatial distribution of interaction of synoptic scale with low frequency oscillations in frequency domain observed in rhvar CRH simulation improves the eastward WVF entering the west coast of Indian from Arabian Sea and the northward extent of WVF over Indian landmass that eventually give feedback to the realistic modulation of cloud condensates attributing improved seasonal ISMR.

5 Conclusions

Recognising the fact that the Indian monsoon is controlled by the slowly varying boundary conditions (Charney and Shukla 1981; Palmer 1994) the predictability of Indian monsoon is largely inhibited by the internal variance even if the GCM model is perfect having accurate boundary and initial conditions (Kang and Shukla 2006). The internal variance is highly model dependent and is not only regulated by the internal dynamics of the model but by different physical processes also in the model. The cloud is one such important physical process in global model for the following reasons.

• (a) There is always cause and effect relationship between the cloud and ISM rainfall. But, the proper understanding of coupling between the cloud and the circulation in the climate model is missing (Stevens and Bony 2013). Hence, it is imperative to explore the relationship between the cloud and large scale circulation by evaluating the inaccuracy in climate sensitivity through the cloud modification in global model for the improved simulation of ISMR and for better future projection by the climate model.

• (b) Although examining the effect of cloud modification on the large scale dynamics is prerequisite to evaluate the interplay between cloud and circulation (Bony et al. 2015), it is still a grey area in climate science.

In view of above, a study was already executed by De et al. (2016) where the simulation of contrasting Indian monsoon had been evaluated through the large scale aspects of monsoon modifying the cloud in CFSv2 coupled model by changing the vertical profile of CRH. The realistic cloud modification by the variable CRH experiment improved the rainfall and precipitable water vapour biases substantiated by realistic variability in tropospheric temperature gradient, high cloud fraction and the convection. As far as the large scale dynamics is concerned, the eastward WVF entering the Indian landmass and the northward extent of WVF were appeared to be stronger in rhvar CRH experiment that reduced the dry rainfall bias of normal monsoon. One of the reasons of improved WVF was the reduction of systematic error energy in wind field of CFSv2 with variable CRH profile during the contrasting monsoon season. But, from this study it remains elusive about the relationship between the cloud modification and the improvement of nonlinear systematic error energy in wind field. Multi-scale interactions are to be computed to evaluate the internal dynamics of monsoon that ultimately governs the internal variance of CFSv2 model. So, for the first time, to explore the interactions between the cloud and the large scale circulation for unfolding the mechanism of coupling between these two parameters, the internal dynamics of monsoon has been computed in terms of the nonlinear scale interactions among the dominant scales in frequency as well as in wavenumber domain for the different CRH experiments of CFSv2 model as a sequel of the above study.

In frequency domain, the synoptic scale system 3–5 day as high frequency mode and the 10–20, 30–60 day waves as low frequency mode are considered here along with the seasonal mean for scale interaction study. Two types of scale interaction—mean–wave and wave–wave interactions are computed for three different CRH experiments of CFSv2 model. In mean–wave interaction the most realistic variation in KE exchanges between the seasonal mean and the low frequency oscillations are observed in the variable CRH experiment rhvar compared to other CRH studies ctl and rh90 during the normal and weak monsoon year. Similarly, the synoptic scale system is drawing more (less) KE from seasonal mean during 2003 (2009) revealed from modified CRH runs rh90 and rhvar substantiated by observation, whereas the existing CRH of CFSv2 (ctl) shows the opposite transfer of KE unrealistically during the contrasting monsoon year. In wave–wave interactions the stronger nonlinear interaction between the 3–5 and 30–60 day mode is depicted during the deficient year compared to the same in normal monsoon obtained from rhvar simulation similar to the interactions derived from reanalysis wind field. On the contrary, the other CRH experiments are incapable to show this modulation. As far as the geographical distribution of wave–wave interaction is concerned, the rhvar shows the spread of the interaction within the core monsoon zone starting from the Head Bay and extending north westward upto the eastern side of Pakistan region during the normal monsoon, however, this spread has been restricted both in northward as well as westward direction in weak monsoon following the observed interaction. The existing CRH and rh90 profiles of CFSv2 do not exhibit the modulation of the spatial distribution of interactions during the normal and deficient monsoon period for all the interactions between the high and low frequency mode. One of the reasons for the improved simulation of Indian monsoon by the variable CRH in De et al. (2016) may be due to the proper representation of internal dynamics revealed from quasi-linear and nonlinear KE exchanges through the mean–wave and wave–wave interactions, respectively in frequency domain.

In wavenumber domain, the CFSv2 coupled model shows the error accumulation at the planetary scale range in which the atmospheric general circulation model also exhibits the peak. This implies that the coupling of ocean with atmosphere in dynamical model does not alter the error development at the predominant ultra-long scale and hence, may be governed by the internal variance of the model. The lowest bias in seasonal simulation of lower tropospheric wind appeared in rhvar experiment of CSFv2 (De et al. 2016) can be explained in terms of spectral error energy, its growth rate and the triad spectra of the inertial term responsible for error growth. The smallest error and its growth rate are observed for rhvar CRH run compared to other CRH profiles of CFSv2 in the planetary scale of wavenumber 4 (1) during the good (bad) monsoon year. Analysing the individual triad interactions, it is revealed that the error is accumulated to wavenumber 4 from the planetary and long scale of spectral band 2–10 during the normal monsoon, whereas the weak monsoon delineates the convergence of error from the small and synoptic scale range of wavenumber > 10 to the planetary scale of wavenumber 1. So, the lowering of the 850 hPa wind error energy in rhvar simulation of CFSv2 model may possibly be due to the reduced nonlinear convergence of error to the planetary scale from the shorter scales compared to other CRH simulations of CFSv2. Therefore, the cloud modification through CRH may lead to change the large scale circulation system by modulating the internal dynamics of Indian monsoon revealed from the quasi-linear and nonlinear KE exchanges among different scales through the process of scale interactions in frequency as well as in wavenumber domain that in turn modify the internal variance of CFSv2 model. On the contrary, the bias reduction in lower tropospheric wind due to the reduced error convergence and the realistic modulation of spatial distribution of interaction of synoptic scale with low frequency oscillations observed in rhvar CRH simulation improves the eastward WVF entering India through west coast from Arabian Sea and the northward extent of WVF over Indian landmass that eventually give feedback to the realistic simulation of cloud condensates resulting improved ISMR. Thus, the mechanism of interplay between the cloud and circulation has been unfolded. This work has paved the way for linking cloud with large scale circulation which is missing in climate model.

The interaction between the cloud and large scale circulation is a subject of large spectrum. Many aspects like diabetic heating, cumulus schemes, microphysical processes, triggering mechanism etc. are responsible for cloud modification and hence, the convection. For each case of cloud modification how it shows an impact on large scale circulation or conversely, the large scale dynamical feedback to the cloud scale for those parameters is a subject of individual study in any GCM. Further studies may be pursued for each factor responsible for cloud modification mentioned above showing the interaction of cloud on large scale dynamics and vice versa.

Appendix

Mathematical equation for error energy growth rate budget:

$$\begin{aligned} &\mathop { \sum }\limits_{n} \left[ {\frac{\partial }{{\partial t}}\left( {\frac{1}{2}{u_e}C_{n}^{2}} \right)+\frac{\partial }{{\partial t}}\left( {\frac{1}{2}{u_e}S_{n}^{2}} \right)+\frac{\partial }{{\partial t}}\left( {\frac{1}{2}{v_e}C_{n}^{2}} \right)+\frac{\partial }{{\partial t}}\left( {\frac{1}{2}{v_e}S_{n}^{2}} \right)} \right]= - \frac{1}{2}\left( {\mathop \sum \limits_{{n=r+s}}^{{}} +\mathop \sum \limits_{{n=r - s}} +\mathop \sum \limits_{{n=s - r}} } \right), \\ & \quad \times \left\{ {\begin{array}{*{20}{c}} {{u_m}{C_n} \cdot {u_e}{C_r}\frac{{\partial {u_e}{C_s}}}{{\partial x}}+{u_m}{C_n} \cdot {v_e}{C_r}\frac{{\partial {v_e}{C_s}}}{{\partial x}}+\frac{1}{2}\frac{{\partial {u_m}{C_n}}}{{\partial x}}{u_e}{C_r} \cdot {u_e}{C_s}+\frac{1}{2}\frac{{\partial {u_m}{C_n}}}{{\partial x}}{v_e}{C_r} \cdot {v_e}{C_s}} \\ {+{v_m}{C_n} \cdot {u_e}{C_r}\frac{{\partial {u_e}{C_s}}}{{\partial y}}+{v_m}{C_n} \cdot {v_e}{C_r}\frac{{\partial {v_e}{C_s}}}{{\partial y}}+\frac{1}{2}\frac{{\partial {v_m}{C_n}}}{{\partial y}}{u_e}{C_r} \cdot {u_e}{C_s}+\frac{1}{2}\frac{{\partial {v_m}{C_n}}}{{\partial y}}{v_e}{C_r} \cdot {v_e}{C_s}} \end{array}} \right\}F, \\ & \quad - \frac{1}{2}\left( { - \mathop \sum \limits_{{n=r+s}} +\mathop \sum \limits_{{n=r - s}} +\mathop \sum \limits_{{n=s - r}} } \right)\left\{ {\begin{array}{*{20}{c}} {{u_m}{C_n} \cdot {u_e}{S_r}\frac{{\partial {u_e}{S_s}}}{{\partial x}}+{u_m}{C_n} \cdot {v_e}{S_r}\frac{{\partial {v_e}{S_s}}}{{\partial x}}+\frac{1}{2}\frac{{\partial {u_m}{C_n}}}{{\partial x}}{u_e}{S_r} \cdot {u_e}{S_s}} \\ {+\frac{1}{2}\frac{{\partial {u_m}{C_n}}}{{\partial x}}{v_e}{S_r} \cdot {v_e}{S_s}+{v_m}{C_n} \cdot {u_e}{S_r}\frac{{\partial {u_e}{S_s}}}{{\partial y}}+{v_m}{C_n} \cdot {v_e}{S_r}\frac{{\partial {v_e}{S_s}}}{{\partial y}}} \\ {+\frac{1}{2}\frac{{\partial {v_m}{C_n}}}{{\partial y}}{u_e}{S_r} \cdot {u_e}{S_s}+\frac{1}{2}\frac{{\partial {v_m}{C_n}}}{{\partial y}}{v_e}{S_r} \cdot {v_e}{S_s}} \end{array}} \right\}L, \\ & \quad - \frac{1}{2}\left( {\mathop \sum \limits_{{n=r+s}} - \mathop \sum \limits_{{n=r - s}} +\mathop \sum \limits_{{n=s - r}} } \right)\left\{ {\begin{array}{*{20}{c}} {{u_m}{S_n} \cdot {u_e}{C_r}\frac{{\partial {u_e}{S_s}}}{{\partial x}}+{u_m}{S_n} \cdot {v_e}{C_r}\frac{{\partial {v_e}{S_s}}}{{\partial x}}+\frac{1}{2}\frac{{\partial {u_m}{S_n}}}{{\partial x}}{u_e}{C_r} \cdot {u_e}{S_s}} \\ {+\frac{1}{2}\frac{{\partial {u_m}{S_n}}}{{\partial x}}{v_e}{C_r} \cdot {v_e}{S_s}+{v_m}{S_n} \cdot {u_e}{C_r}\frac{{\partial {u_e}{S_s}}}{{\partial y}}+{v_m}{S_n} \cdot {v_e}{C_r}\frac{{\partial {v_e}{S_s}}}{{\partial y}}} \\ {+\frac{1}{2}\frac{{\partial {v_m}{S_n}}}{{\partial y}}{u_e}{C_r} \cdot {u_e}{S_s}+\frac{1}{2}\frac{{\partial {v_m}{S_n}}}{{\partial y}}{v_e}{C_r} \cdot {v_e}{S_s}} \end{array}} \right\}U, \\ & \quad - \frac{1}{2}\left( {\mathop \sum \limits_{{n=r+s}} +\mathop \sum \limits_{{n=r - s}} - \mathop \sum \limits_{{n=s - r}} } \right)\left\{ {\begin{array}{*{20}{c}} {{u_m}{S_n} \cdot {u_e}{S_r}\frac{{\partial {u_e}{C_s}}}{{\partial x}}+{u_m}{S_n} \cdot {v_e}{S_r}\frac{{\partial {v_e}{C_s}}}{{\partial x}}+\frac{1}{2}\frac{{\partial {u_m}{S_n}}}{{\partial x}}{u_e}{S_r} \cdot {u_e}{C_s}} \\ {+\frac{1}{2}\frac{{\partial {u_m}{S_n}}}{{\partial x}}{v_e}{S_r} \cdot {v_e}{C_s}+{v_m}{S_n} \cdot {u_e}{S_r}\frac{{\partial {u_e}{C_s}}}{{\partial y}}+{v_m}{S_n} \cdot {v_e}{S_r}\frac{{\partial {v_e}{C_s}}}{{\partial y}}} \\ {+\frac{1}{2}\frac{{\partial {v_m}{S_n}}}{{\partial y}}{u_e}{S_r} \cdot {u_e}{C_s}+\frac{1}{2}\frac{{\partial {v_m}{S_n}}}{{\partial y}}{v_e}{S_r} \cdot {v_e}{C_s}} \end{array}} \right\}X, \\ & \quad - \frac{1}{2}\left( {\mathop \sum \limits_{{n=r+s}} +\mathop \sum \limits_{{n=r - s}} +\mathop \sum \limits_{{n=s - r}} } \right)\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial {u_a}{C_n}}}{{\partial x}}{u_e}{C_r} \cdot {u_e}{C_s}+\frac{{\partial {u_a}{C_n}}}{{\partial y}}{u_e}{C_r} \cdot {v_e}{C_s}+\frac{{\partial {v_a}{C_n}}}{{\partial x}}{u_e}{C_r} \cdot {v_e}{C_s}} \\ {+\frac{{\partial {v_a}{C_n}}}{{\partial y}}{v_e}{C_r} \cdot {v_e}{C_s}} \end{array}} \right\}GE, \\ & \quad - \frac{1}{2}\left( { - \mathop \sum \limits_{{n=r+s}} +\mathop \sum \limits_{{n=r - s}} +\mathop \sum \limits_{{n=s - r}} } \right)\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial {u_a}{C_n}}}{{\partial x}}{u_e}{S_r} \cdot {u_e}{S_s}+\frac{{\partial {u_a}{C_n}}}{{\partial y}}{u_e}{S_r} \cdot {v_e}{S_s}+\frac{{\partial {v_a}{C_n}}}{{\partial x}}{u_e}{S_r} \cdot {v_e}{S_s}} \\ {+\frac{{\partial {v_a}{c_n}}}{{\partial y}}{v_e}{S_r} \cdot {v_e}{S_s}} \end{array}} \right\}NE, \\ & \quad - \frac{1}{2}\left( {\mathop \sum \limits_{{n=r+s}} - \mathop \sum \limits_{{n=r - s}} +\mathop \sum \limits_{{n=s - r}} } \right)\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial {u_a}{S_n}}}{{\partial x}}{u_e}{C_r} \cdot {u_e}{S_s}+\frac{{\partial {u_a}{S_n}}}{{\partial y}}{u_e}{C_r} \cdot {v_e}{S_s}+\frac{{\partial {v_a}{S_n}}}{{\partial x}}{u_e}{C_r} \cdot {v_e}{S_s}} \\ {+\frac{{\partial {v_a}{S_n}}}{{\partial y}}{v_e}{C_r} \cdot {v_e}{S_s}} \end{array}} \right\}RA, \\ & \quad - \frac{1}{2}\left( {\mathop \sum \limits_{{n=r+s}} +\mathop \sum \limits_{{n=r - s}} - \mathop \sum \limits_{{n=s - r}} } \right)\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial {u_a}{S_n}}}{{\partial x}}{u_e}{S_r} \cdot {u_e}{C_s}+\frac{{\partial {u_a}{S_n}}}{{\partial y}}{u_e}{S_r} \cdot {v_e}{C_s}+\frac{{\partial {v_a}{S_n}}}{{\partial x}}{u_e}{S_r} \cdot {v_e}{C_s}} \\ {+\frac{{\partial {v_a}{S_n}}}{{\partial y}}{v_e}{S_r} \cdot {v_e}{C_s}} \end{array}} \right\}TION, \\ & \quad +{\text{ }}{V_e} \cdot R{\text{ }}\left( {RESIDUAL} \right). \\ \end{aligned}$$

(5)

In the above equation, u_mC, v_mC and u_mS, v_mS are the cosine and sine component of zonal and meridional wind field of model output, respectively. Similarly, u_aC, v_aC and u_aS, v_aS are the cosine and sine component of zonal and meridional analysis/observed wind field, respectively. The subscript n, r and s represent three wavenumber of participating waves in triad interactions. The left hand side of the equation evaluates the spectral error growth rate term. The extent of flux, generation and residual term are shown by indicating outside the second bracket in the right hand side of above equation.