The basis of the modelling framework proposed in this study is the statistical model developed in Part 1. Only a fraction of the model is needed in Part 2, which primarily considers the variance of nonharmonic internal tides. This section introduces relevant relationships from Part 1 for independent waves and then extends them to correlated waves.

The statistical model in Part 1 considers internal tides with a single vertical-mode structure in a narrow frequency band observed at a fixed observation location and approximates them as a sinusoidal time series that has the deterministic angular frequency ω, the deterministic mean phase lag φ, a random amplitude A, and a random phase-lag deviation Θ. Furthermore, it is assumed that this signal results from the superposition of independent and non-identically distributed N sinusoidal wave components, each of which has the deterministic mean phase lag φj, a random amplitude Aj, and a random phase-lag deviation Θj. Then, the signal can be expressed as 1Ae-i(φ+Θ)eiωt=∑j=1NAje-iφj+Θjeiωt, where t is time. Unlike Part 1, the mean phase lags are subtracted from the total phase lags to make Θ and Θj random variables with zero mean, and only deterministic amplitudes Aj=aj are hereafter considered for individual wave components. The phase PDF is assumed to be the wrapped normal (or Gaussian) distribution as in Part 1: 2fΘjθj=12πσj∑k=-∞∞exp⁡-θj+2πk22σj2, where σj is the standard deviation of the phase. The wrapped normal distribution is a circular analogue of the Gaussian distribution and defined for any one period of 2π. It approaches the Gaussian distribution in the limit σj→0 but approaches the uniform distribution in the limit σj→∞. Since harmonic analysis determines harmonic amplitudes and phase lags using the method of least squares, the complex-valued amplitudes (i.e. their magnitudes represent wave amplitudes and their angles represent wave phases as the coefficients of a complex Fourier series) are further decomposed into the expected values and deviations from them: 3Ae-i(φ+Θ)=re-iφ+A′e-i(φ+Θ′)=∑j=1Nrje-iφj+Aj′e-iφj+Θj′. Here, rj is the magnitude of the expected complex-valued amplitude on the complex plane, and Aj′ and Θj′ are the amplitude and phase lag of the deviation, respectively (see Fig. ). Note that (r, φ) and (A′, φ+Θ′) correspond to harmonic and nonharmonic internal tides, respectively. Note also that Θ′ and Θj′ are random variables with zero mean unlike Part 1 and that A, A′, and Aj′ are random variables even though aj is deterministic (see Fig.  and Part 1). Assuming tentatively that σj in Eq. () is known and that all the wave components are independent, the expectation and variance of the complex-valued random amplitudes aje-i(φj+Θj) are (see Part 1) 4aEaje-iφj+Θj=rje-iφj=ajμje-iφj,4bVaraje-iφj+Θj=EAj′2=aj2ςj2,4cμj=e-σj2/2,4dςj2=1-e-σj2. Hereafter, E(⋅) and Var(⋅) denote the expectation and variance, respectively. For complex-valued variables, the variance is defined as Var(X)=E((X-E(X))(X-E(X))*). Hereafter, the superscript * denotes complex conjugate. Then, because of the independence of individual wave components, E(A′2) is given by (see Part 1 for justification) 5EA′2=∑j=1NEAj′2. Note that E(A′2) is the variance of the envelope amplitude of nonharmonic internal tides and is twice the nonharmonic internal-tide variance because the sinusoidal “carrier” wave (i.e. eiωt in Eq. ) has the variance of 1/2.

The above argument assumes the independence of individual wave components; however, the horizontal correlation of phase modulation along the propagation paths introduces the correlation of wave components arriving from individual sources. To consider the horizontal correlation, we remove the assumption of independent wave components in Eqs. () and () and calculate the covariance of the ith and jth wave components. Using Eqs. (), (), and ()–(), we get 6CovAi′e-iφie-iΘi′,Aj′e-iφje-iΘj′=siςiRijςjsj*, where sj=aje-iφj represents complex-valued pre-modulation wave amplitudes from individual sources (hereafter referred to as “sources”), Rij is the correlation coefficient of e-iΘi and e-iΘj, and the covariance is defined as Cov(X,Y)=E((X-E(X))(Y-E(Y))*). Note that Θj′ does not follow the wrapped normal distribution in Eq. (), but Aj′e-iΘj′ can be expressed in terms of aje-iΘj, μj, and ςj using Eqs. (), (), and ()–(). This yields 7Rij=1ςiςjexp⁡-12EΔΘ2-μiμj, where ΔΘ=Θi-Θj. Since the difference of correlated wrapped normal variables is a wrapped normal variable, the expectation in the above equation is obtained using 8Ee-iΔΘ=exp⁡-12EΔΘ2, which is the same relationship as for the normally distributed phase . Note that this relationship makes the correlation coefficients Rij real-valued, although the original variable, e-iΔΘ, is complex-valued. To derive this convenient relationship, the definition of Θj is changed from Part 1 to have zero mean. To proceed, and assumed the correlation functions of Θi and Θj, but we aim to express E(ΔΘ2) as a function of the variance and correlation length of phase speed. This is done by stochastic modelling, as described in Sect. .

The correlation coefficients in Eq. () can be used to convert correlated sources (e.g. from hydrodynamic modelling) to effectively independent sources that can be used in the statistical model. To do so, we write the complex-valued amplitude of nonharmonic internal tides A′e-i(φ+Θ′) in Eq. () in two ways. On the one hand, we assume that the waves from individual sources sj=aje-iφj are later modulated by horizontally correlated random phase shifts, yielding 9A′e-i(φ+Θ′)=sphysTΣncorr. Here, s is the vector containing sj, and Σ is a diagonal matrix whose diagonal components are ςj defined in Eq. (). Hereafter, the superscript T denotes transpose. The above form is chosen so that the vector n, with its components ςj-1(e-iΘj-E(e-iΘj)), is a vector containing random variables with zero mean and unit variance (but not Gaussian) on the complex plane. The subscript “phys” emphasizes that the variable is calculated based on physics (in this study, by the adjoint frequency response analysis introduced in Sect. ), and the subscript “corr” emphasizes horizontally correlated random variables. The statistical model, on the other hand, requires independent random variables: 10A′e-i(φ+Θ′)=sstatTΣn, where the vector sstat contains the amplitudes of independent sources. Now, we may assume that two random vectors are related as ncorr=R1/2n, where R=R1/2RT/2 is the horizontal correlation coefficient matrix whose components are given by Eq. (). Note that n is complex-valued, but R is real-valued because of Eq. (). Assuming tentatively that R1/2 is known, the comparison of the above two equations shows 11sstat=Σ-1RT/2Σsphys. We use this relationship to convert horizontally correlated sources calculated based on physics to effectively independent sources that can be used in the statistical model. Then, considering Eqs. () and () in a matrix form and E(n*nT)=I, we get 12EA′2=sstatHΣ2sstat=RT/2ΣsphysHRT/2Σsphys. Hereafter, the superscript H denotes conjugate transpose. Note that the (i, j) component in the summation corresponds to Eq. (). Appendix  provides detailed points regarding the above treatment of horizontal correlation using R1/2.

The continuous version of Eq. () is useful in this study. The equation divided by 2 can be written as 1312EA′2=∫snh(x→)dx→=12∫∫R1/2(x→′,x→)ς(x→′)s(x→′)dx→′*×∫R1/2(x→′,x→)ς(x→′)s(x→′)dx→′dx→. The variables s(x→) and snh(x→) are hereafter referred to as the “source function” and “nonharmonic variance source function” (more correctly source density function), and s(x→), ς(x→), and R1/2(x→′,x→) are the continuous versions of sphys, Σ, and RT/2, respectively. (Note that Σ is a diagonal matrix.) The factor 1/2 is multiplied in the above equation so that the integral of snh corresponds to the variance of a nonharmonic internal-tide time series from observations or numerical modelling, rather than the variance of the envelope amplitude. The above expression shows that, because the horizontal integral of snh yields the total nonharmonic internal-tide variance, snh can be mapped to identify their important source regions. Also, a regional integral of snh yields the contribution of that region to the total variance. Although not shown in this paper, snh can also be used to calculate PDFs using the theory in Part 1. (However, note that snh is nonunique within the correlation length of phase modulation because RT/2 in Eq. () or R1/2(x→′,x→) in Eq. () is nonunique, as explained in Appendix .)

In order to calculate the deterministic sources of internal tides for a fixed observation location, we use a combination of adjoint sensitivity modelling and the frequency response analysis from Fourier theory, referred to as “adjoint frequency response analysis” in this study. A brief summary and the major output of the method are described below. Appendix  provides an overview of the adjoint method, which is often used in inverse problems, and the details of the adjoint frequency response analysis.

The basic idea of the adjoint frequency response analysis is as follows. Since internal tides are linear waves and their major generation forces are deterministic as a first approximation, the forcing and so-called impulse response function can be used to obtain spatially and temporally varying internal waves excited by forcing at a particular location and time. A problem converse to this yields spatially and temporally varying sources of internal waves at a particular location and time (including both harmonic and nonharmonic components) by considering the forcing and the so-called adjoint sensitivity (or the Green's function; e.g. ). These methods can be extended to sinusoidal internal tides using the Fourier transform.

The application of the adjoint frequency response analysis to internal tides under realistic stratification and bathymetry requires a linear numerical hydrodynamic model and its adjoint. In this paper, we use a linear hydrodynamic model based on vertical-mode decomposition in and . The formulation employs horizontally varying vertical modes that are calculated using local water depths and stratification in order to include the effects of steep slopes (for approximately linear waves). More details are described in Appendix . An advantage of this formulation is that it yields the evolutionary equations analogous to the shallow water equations with explicit forcing functions from barotropic tides to individual baroclinic modes. As an example, the forcing function from the barotropic to VM1 M₂ tide is shown in Fig. .

If only one baroclinic mode is considered in the hydrodynamic model, the adjoint frequency response analysis allows us to write the complex-valued internal-tide amplitude as 14ae-iφ=∫s(x→)dx→=∫2λ̃*(x→)f̃(x→)dx→, where a and φ are the pre-modulation amplitude and phase of isopycnal displacement due to the baroclinic mode of interest at the location of interest, respectively. The variable f̃ is the forcing function from the barotropic tidal currents to the baroclinic mode, and λ̃ is the adjoint frequency response function of ae-iφ to f̃ at other locations, calculated by the adjoint of the linear hydrodynamic model. These variables are defined in Appendix . The function s(x→) is the source function appearing in Eq. (). The middle expression shows that, because the horizontal integral of the source function yields the complex-valued amplitude ae-iφ, s can be mapped to identify important source regions. The right expression shows that the adjoint frequency response function λ̃ acts as a transfer function from the forcing function f̃, which provides forcing in a global sense, to the source function s; this provides forcing relevant to the location of interest. The maps of f̃ and λ̃ can be used to identify regions where forcing and dynamic response are large. The important advantage of the source function in this study is that it provides horizontally distributed sources of internal tides observed at a fixed location so that different phase statistics can be assigned to different sources.

It is also convenient to write Eq. () in a discretized form. The equation can be written as 15ae-iφ=∑j=1Nsj=∑j=1Naje-iφj, where sj represents the discretized version of s(x→). The variables sj=aje-iφj are sought-after wave sources corresponding to sphys in Eq. ().

To develop stochastic models of phase statistics, we consider waves with a constant frequency that arrive at an observation location after travelling through regions of random phase-speed variability. Following and the analysis in Appendix , the random phase deviation along the wave propagation path between a source located at x→j and the observation location (say, jth path) can be calculated considering the variation of the total wave phase d(phase)=ω(dt-cj-1dξj)-d(φj+θj) and that of the phase speed cj, where ξj is the coordinate along the path. (Note that θj is the stochastic version of Θj in Eq. .) Some examples of wave propagation paths are shown in Fig. a. To introduce random components in the phase lag and phase speed, we write cj=c‾j+cj′ and assume that φj and c‾j are the respective mean components, and θj and cj′ are the respective stochastic components with zero mean. Assuming |cj′|≪c‾j and following the constant mean phase (dξj=c‾jdt), the deviation of total phase due to cj′ or θj is given by cj′c‾j-2ωdξj and -dθj, respectively. This yields (see Appendix  for alternative derivation) 16dθj=-ωcj′c‾jdt, where time t is used as the independent variable because it is a convenient common coordinate variable for multiple paths.

Since θj and cj′ are stochastic variables, Eq. () is a stochastic differential equation. Stochastic differential equations are commonly forced by white Gaussian noise, but it is undesirable to assume cj′ is white noise because cj′ certainly has spatial correlation. A common “trick” used to deal with correlated noise is to introduce an additional stochastic equation driven by white noise, which yields the desired correlation function see e.g.chap. 12.3. For example, we may assume that cj′ follows 17dcj′=-c‾jLCcj′dt+c‾jLCdbj, where LC is the e-folding correlation length of cj′. The variable bj is a random variable called Brownian motion see e.g.chap. 4.1. Intuitively, the above equation can be formally divided by dt and dbj/dt regarded as white noise, although this view is mathematically incorrect in general.

The stochastic phase models used in this study are developed by considering covariance equations associated with the above two stochastic differential equations. The details of the derivation are given in Appendix , and only the final equations used for the modelling are provided in the following two sections. Note that we need to integrate only ordinary differential equations in this study because the covariance equations are ordinary differential equations, although the formulation is based on stochastic differential equations.

To model the phase variance σj2 in Eqs. ()–(), we consider Eqs. () and () along a single wave propagation path. The evolutionary equations of the covariance between cj′ and θj, Pcθ, and that between θj and θj, Pθθ, are given by 18adPcθdt=-c‾LCPcθ-ωc‾σC2,18bdPθθdt=-2ωc‾Pcθ, where σC2 is the phase-speed variance, and the subscript j is suppressed for brevity. If c‾ and LC remain constant, the solution under the initial condition Pcθ=Pθθ=0 at t=0 is 19Pθθ=2σC2c‾2ωLCc‾2c‾tLC-1+e-c‾t/LC. This agrees with Eq. (12) in if the correlation function of c′ is assumed to be exponential (Eq. ). Note that it is essential to consider the phase-speed correlation length LC because a small correlation length makes phase-speed variability less efficient in inducing phase variance.

The straightforward approach for solving Eqs. ()–() is to integrate the equations from a source location to the observation location; however, this approach is computationally inefficient because it needs separate (forward) integration from each source location along the same path. Alternatively, we can exploit the adjoint method described in Appendix . The adjoint sensitivity of Pθθ at the observation location to [PcθPθθ]T at other locations can be calculated by integrating the equations adjoint to Eqs. ()–() once, backwards in time from the observation location. Then, Pθθ can be calculated as the convolution of the adjoint sensitivity and the forcing (i.e. the σC2 term in Eq. ) along the path. The resultant phase variance Pθθ, which grows with distance from the observation location, is used as the phase variance σj2 in the statistical model. Note that Pθθ can grow without a limit, but this does not cause any problem because the wrapped normal distribution in Eq. () can be used with arbitrarily large phase spread σj.

We now consider the calculation of the variance of the phase difference E(ΔΘ2) in Eq. (). Note that full evaluation of E(ΔΘ2) is difficult for relatively large problems because E(ΔΘ2) depends on pairs of two source locations, which vary over the area considered (e.g. model domain). For this reason, a number of approximations are introduced in the theory in this section and in numerical methods later in Sect. . To simplify the calculation of phase difference ΔΘ, we consider ΔΘ only in the cross-path direction in this section.

The modelling of cross-path phase difference ΔΘ is done by considering Eqs. () and () along two wave propagation paths passing through the same observation location and by calculating the phase difference Δθ=θi-θj (Δθ is the stochastic version of ΔΘ in Eq. ). In this section and Appendix , the subscripts i and j indicate variables along the ith and jth paths, respectively. We take into account the variability of the mean phase speed c‾ and the phase-speed correlation length LC along the propagation paths but neglect their cross-path variability. The evolutionary equations of the covariance between ci′ and Δθ, PciΔθ, that of cj′ and Δθ, PcjΔθ, and that of Δθ and Δθ, PΔθΔθ, are given by 20adPciΔθdt=-c‾LCPciΔθ-ωc‾σC21-Rη|Δη|l,20bdPcjΔθdt=-c‾LCPcjΔθ+ωc‾σC21-Rη|Δη|l,20cdPΔθΔθdt=-2ωc‾PciΔθ-PcjΔθ, where 21Rη|Δη|l=F(|Δη|/l)1+F2(|Δη|/l) is the cross-path correlation function of phase speed, Δη is the cross-path distance, 22aF(|Δη|/l)=e-|Δη|/l,22bl=2π-1LC, and l is the cross-path correlation length. Generally, PΔθΔθ needs to be calculated numerically. However, if c‾, LC, and |Δη|/l remain constant, the comparison of Eqs. ()–() and ()–() leads to the explicit solution 23PΔθΔθ=2Pθθ1-Rη(|Δη|/l), where Pθθ is given by Eq. (). (It may appear odd to assume constant |Δη|/l because |Δη| certainly varies; however, an empirical relationship is introduced later in Sect.  to account for the variation.) This shows that the cross-path correlation length of Δθ depends on the phase-speed correlation length LC through Eqs. ()–(). This is important because LC can be estimated from observations or hydrodynamic modelling more easily than the correlation length of phase difference Δθ.

Similar to Eqs. ()–(), Eqs. ()–() can be solved using the adjoint method explained in Appendix , and the resultant variance PΔθΔθ corresponds to E(ΔΘ2) in Eq. (). However, note that the analysis has been simplified substantially by the assumptions introduced above. In particular, note that PΔθΔθ=0 at Δη=0, which implies Rij=1 in Eq. () because Eqs. ()–() neglect along-path correlation. To take into account the effects of along-path correlation, an empirical adjustment is introduced later in Sect. .

To illustrate application of the proposed model suite, we took vertical-mode-one (VM1) semidiurnal internal tides at the PIL200 mooring site (115.915° E, 19.435° S, ≈200 m deep) of the Australian Integrated Marine Observing System on the Australian North West Shelf (Figs.  and ) as an example. Part 1 analysed the nonharmonic VM1 to vertical-mode-four (VM4) diurnal, semidiurnal, and quarter-diurnal internal tides in the observations.

In the model suite, we included the four major semidiurnal tidal constituents (M₂, S₂, K₂, and N₂) and four lowest baroclinic modes (VM1–VM4). Figure  shows a flowchart for the application of the proposed model suite to multiple tidal constituents and vertical modes. Forcings from the major constituents were considered separately, assuming that the nonharmonic internal-tide variance (and the associated statistics) is calculated for a sufficiently long time series. Since it was impractical to separate nonharmonic internal tides into constituents in the PIL200 observations (Part 1), the resultant variance, E(A′2)/2 in Eq. (), and the nonharmonic variance source functions from individual constituents were summed to obtain the total for semidiurnal internal tides. It may sound confusing to include multiple baroclinic modes for modelling VM1 internal tides at the PIL200 location. This is required because barotropic forcing excites not only VM1 but also higher modes, which can be converted to VM1 by topographic interaction before arriving at the PIL200 location (see Fig. ). To distinguish overall barotropic forcing to VM1 internal tides at the PIL200 location from barotropic forcing to individual baroclinic modes in the intermediate process, the latter is hereafter referred to as, for example, “barotropic-to-VM2” or “VM0-to-VM2” forcing.

In the hydrodynamic modelling, we considered linear hydrostatic internal tides under climatological stratification without background currents. Note that mesoscale oceanic variability is intentionally omitted because its effects are represented by random phase-speed variability in the stochastic models (see Appendix  for the justification of this treatment). A sinusoidal periodic motion was assumed (as in Eq. ) in the governing equations (Eqs. – in without the nonlinear terms) so that the hydrodynamic model directly calculates the adjoint frequency response function (λ̃ in Eq. ). The frequency response function was calculated for complex-valued VM1 isopycnal displacement amplitude at the PIL200 location (i.e. ae-iφ in Eq. ), whose magnitude is scaled to have the value of extreme (maximum or minimum) displacement within the water column.

Details of the hydrodynamic model set-up are as follows. The model grid encompass most of the Australian North West Shelf and part of the Lesser Sunda Islands in Indonesia (Fig. a). The horizontal coordinates are oriented in the cross-shelf (NNW–SSE) and along-shelf (SSW–NNE) directions at the PIL200 location. The horizontal grid size is 0.01°. The model extent and grid resolution are not ideal but were limited by available computational resources. The four lowest baroclinic modes (VM1–VM4) are included in the calculation. Vertical modes are calculated using the 2019 version of GEBCO bathymetry and stratification from the 2018 version of World Ocean Atlas annual climatology over the 2005–2017 period . TEOS-10 is used to calculate density. The model includes horizontally varying linear bottom friction, which is calculated using the (nondimensional) quadratic bottom drag coefficient of 10⁻³ and the barotropic tidal current speed from the TPXO9-atlas version 5 (updated from ). Since the grid resolution is not sufficiently high to resolve internal tides in regions with shallow water depths or weak stratification, we exclude regions where the celerity of each (nth) vertical mode cn is less than 0.1 m s⁻¹, which roughly corresponds to four grid points per wavelength for semidiurnal tides. (In this study, the term “celerity” is deliberately used for the propagation speed of non-rotating, long, linear gravity waves with one of the vertical-mode structures, which differs from the phase speed of internal tides.) The Flather open boundary condition (; see also ) is applied to individual vertical modes at the open boundaries. The adjoint frequency response function was calculated separately for the M₂, S₂, K₂, and N₂ tidal frequencies.

The source function (s(x→) in Eq. ) was calculated from the adjoint frequency response function for the four lowest baroclinic modes and barotropic currents from the TPXO9-atlas for the four major semidiurnal constituents. This provided 16 source functions in total.

The phase variance Pθθ was calculated based on Eqs. ()–(), but it required finding wave propagation paths from the PIL200 location. We took the simplest approach and calculated the propagation paths by standard ray theory e.g.chap. 4.5, but applying it backwards in time. The initial location is the PIL200 location and the initial angles are in 0.1 and 1° intervals for rays propagating towards offshore and onshore, respectively. Additional rays are used to ensure that some rays propagate into the southern part of the major straits in the Lesser Sunda Islands, such as the Lombok Strait. Figure a shows about 1/30 of the calculated ray paths as examples.

The standard ray equations and the equations adjoint to Eqs. ()–() were integrated backwards in time using the fourth-order Runge–Kutta method for VM1 to VM4 semidiurnal internal tides. The time steps are 300, 450, 600, and 900 s for VM1, VM2, VM3, and VM4, respectively. In the calculation, the along-path variability of water depth, phase speed, and Coriolis parameter are taken into account. Since the results were insensitive to small frequency differences among the major semidiurnal constituents, the M₂ frequency was used in the modelling.

The phase-speed variance σC2 in the model was chosen based on the PIL200 observations, which yielded σC,PIL2002≈12, 9.5, 8.2, and 8.2×10-3 m² s⁻² for VM1, VM2, VM3, and VM4 semidiurnal internal tides, respectively (Appendix ). Although the observations were made on the continental shelf at ≈ 200 m water depth, the phase-speed variance of VM1 is not unreasonable for deep ocean. For example, previous numerical modelling suggests σC/c‾ = 1 %–3 % in deep ocean for VM1 semidiurnal internal tides. Since these values include only low-frequency components, they are likely to be underestimates for σC2, which needs to include all frequency components as explained in Appendix . So, σC2≈1.2×10-2 m² s⁻², which yields σC/c‾≈3.6 % assuming c‾=3 m s⁻¹, appears to be roughly the upper limit of the current estimate of σC2 for deep ocean. For higher modes, phase-speed variance appeared to be unavailable except those from Appendix . These facts suggest that horizontally constant phase-speed variance is not a bad assumption, so we chose σC2 by scaling σC,PIL2002 as 24σC2=αCσC,PIL2002, where αC is a model parameter. This choice is also a simple and convenient way to show the dependence of the results on σC2. We used αC varying between 0.4 and 1.0. As already explained, αC=1.0 is the estimate for the PIL200 location and appears to be roughly the current upper limit for deep ocean. The choice αC=0.4 (σC/c‾≈2.3 %) is about the middle range of the current estimate for deep ocean, but it would be a substantial underestimate for shallow water. We chose the middle of these likely upper and lower limits, αC=0.7, as a reference value.

Regarding the correlation length of phase speed LC, we assumed LC to be proportional to the Rossby radius of deformation Rd=c‾1/f: 25LC=αLRd, where f is the Coriolis parameter, and αL is a model parameter. This choice was made for two reasons. First, Rd is a common length scale used for mesoscale oceanic variability. Second, LC is expected to vary substantially between continental shelves and deep ocean, and the mean VM1 celerity c‾1 in the expression of Rd conveniently reflects at least some part of this variability. Note that the same c‾1 is used to calculate LC for all the higher modes, considering that the phase-speed modulations of all vertical modes are caused by the same oceanic variability. The phase-speed correlation length appears to be rarely evaluated, but showed that the correlation length was about 3 times Rd around Hawaii. This value might be affected by the smoothing scale of the reanalysis product used in their study and is larger than the typical radius of mesoscale eddies for the latitude e.g.. However, phase-speed correlation could be affected by processes that have a length scale larger than eddies e.g.. Since the typical eddy radius is roughly Rd for the latitude range of the model domain e.g., the realistic parameter range is αL≳1. We chose the middle-ground value of αL=2 as a reference value. Note that the wavelength of VM1 semidiurnal internal tides is about 1–2 times Rd in the modelled region.

After the ray-based calculation, the travel time and phase variance Pθθ along the ray paths were horizontally interpolated to obtain gridded results using a Gaussian kernel. This interpolated Pθθ was used as σj2 in the statistical model.

The horizontal correlation coefficient matrix R was implemented as a diffusion operator following , which is a numerical technique commonly used in data assimilation see e.g.chap. 3.1.6. This is because, although R could be calculated in principle using Eqs. () and ()–(), it was prohibitive to store the whole R on computer memory in practice. The method approximates the correlation function as Gaussian and requires the correlation lengths at individual grid points, which are equivalent to the standard deviation of the Gaussian function (i.e. impulse response solution to the diffusion equation). Since Eqs. ()–() calculate the variance of the cross-path phase difference for different cross-path distance |Δη|, Eqs. () and ()–() yield only the cross-path correlation length ση, and the along-path correlation length σξ is still missing. In this study, an empirical relationship between ση and σξ was introduced, and equivalent isotropic diffusion was assumed for simplicity. Then, the phase correlation modelling requires the equivalent isotropic correlation length of phase modulation at each grid point σr calculated from PΔθΔθ in Eqs. ()–().

To determine σr, we assume 26|Δη|=αr-1Δr in Eqs. ()–(), where Δr is the distance between the sources, and αr is an empirical parameter whose meaning is explained shortly. This assumption has the advantage that PΔθΔθ could be integrated (backwards in time) for various values of Δr together with the ray tracing and integration of Pθθ, and the results can be gridded in the same way. Substituting the resultant PΔθΔθ into E(ΔΘ2) in Eq. () yields the horizontal correlation function at each grid point R(Δr). (In Eq. , μi=μj and ςi=ςj are assumed.) Then, by approximating the first peak of R(Δr) as Gaussian, we get 27R(Δr)≈exp⁡-Δr22αr2ση2. The empirical factor αr represents two effects: anisotropy of the horizontal correlation of phase modulation and the along-path variation of cross-path distance. Typical values of αr for these effects are considered in the following.

To estimate αr for anisotropic phase correlation, we tentatively regard Rij in Eq. () as the correlation function R(Δξ,Δη) (Δξ is a lag distance in the along-path direction) and compare its integral scale with that of the equivalent isotropic correlation function R(Δr). Assuming that the correlation functions are Gaussian and equating the integrals, we get 28∫-∞∞∫-∞∞exp⁡-Δξ22σξ2-Δη22ση2dΔξdΔη≈2π∫0∞Δrexp⁡-Δr22σr2dΔr, where σξ is the unknown standard deviation in the along-path direction. This yields the relationship of the integral scales, 29σξση≈σr2. The comparison of Eqs. ()–() shows that αr=σξ/ση, and αr=1 for the isotropic correlation function (σξ=ση). Note the relatively weak dependence of αr on σξ. For example, the correlation function is highly anisotropic for σξ=9ση, but it yields αr=3.

To estimate αr for the along-path variation of cross-path distance, we consider the linear variation of cross-path distance |Δη| between the observation location and source locations. Since the distance between the sources is Δr, an intuitive value for average |Δη| over the paths is αr=2. However, note that ray tracing suggests large along-path variability of |Δη| (Fig. a).

Based on the above consideration, the equivalent isotropic correlation length of phase modulation σr was calculated from Eqs. () and ()–(), and () as follows. Considering both the anisotropy of the phase correlation and the along-path variation of cross-path distance, αr between 1 and 5 appears to be reasonable. We chose the middle of these likely upper and lower limits, αr=3, as a reference value. Using Eq. () with a chosen αr, PΔθΔθ from Eqs. ()–() was substituted into Eq. () to calculate the correlation coefficient for a different source distance Δr. This yielded the isotropic correlation function R(Δr). Since σr is required for the diffusion operator method, the Gaussian shape was fitted to the first peak of the correlation function where R(Δr)>0.5 by the least-squares method, and the resultant standard deviation is used as σr in the diffusion operator method.

In addition to σr, the diffusion operator method also requires normalization factors that impose Rii≈1 after applying the diffusion operator (i.e. the matrix Λ in ). The normalization factors are calculated by the ensemble method explained in . We used 200 ensemble members, which correspond to the standard error of 5 % in the normalization of R.

As in the ray tracing and phase spread modelling, σr and the normalization factors were calculated separately for the four lowest baroclinic modes using the M₂ frequency. The frequency differences among semidiurnal constituents were neglected.

The nonharmonic variance source function was calculated for each constituent from Eq. () using sphys from the source function, Σ calculated from the phase variance Pθθ=σj2, and R implemented as a diffusion operator with the equivalent isotropic correlation length of phase modulation σr; however, it required one more assumption because it was not obvious which phase spread and phase correlation should be applied to each source function. For example, if higher modes are directly excited by barotropic forcing and converted to VM1 near the sources, and then the VM1 internal tides propagate to the observation location (follow VM0-to-VM2 forcing, left-hand-side “Topographic interaction”, and then VM1 propagation in Fig. ), the phase spread and correlation lengths for VM1 should be applied to the source functions for higher modes because the phases are modulated as VM1 internal tides. However, if higher modes are directly excited by barotropic forcing, propagate as higher modes, and are then converted to VM1 near the observation location (follow VM0-to-VM2 forcing, VM2 propagation, right-hand-side “Topographic interaction”, and then VM1 propagation in Fig. ), the phase spread and correlation lengths for higher modes should be applied to the source functions of respective higher modes. The latter scenario is assumed in this study because the continental slope near the PIL200 location induces strong topographic interaction between VM1 and higher modes, as shown later.

The adjoint frequency response function (λ̃ in Eq. ) of VM1-induced isopycnal displacement at the PIL200 location to the barotropic (VM0)-to-VM1 forcing qualitatively shows a pattern of internal waves spreading from a point source but affected by topography-induced variation of the propagation speed (Fig. a). For internal-wave signals propagating offshore, wave spreading gradually reduces the magnitudes. By the time the signals reach the Indonesian archipelago, the magnitudes are reduced by a factor of more than 10. For internal-wave signals propagating towards the Australian coast, the wavelengths decrease rapidly because shallower water depths and weaker stratification reduce the propagation speed. The signals disappear on the shelf shallower than 100 m, partly because of bottom friction and partly because the grid resolution gradually becomes insufficient to adequately resolve internal tides there. This numerical dissipation does not change the overall results of this study because the shallow shelf has mild slopes and hence no important sources of internal tides at the PIL200 location.

The adjoint frequency response function to the VM0-to-VM2 forcing also shows a pattern of internal waves spreading from a point source (Fig. b). The magnitudes are smaller than the VM1 signals because the VM2 (and other higher-mode) signals result from the topographic conversion of VM1 signals on the continental slope. The shorter wavelength shows that the signals are propagating as a free VM2 internal-wave signal, at least as a first approximation. These features justify our choice of applying the phase spread and horizontal phase correlation for VM2 to the VM2 source function (Sect. ). This observation is significant because the spatial pattern would be very different if the topographic conversion occurred near the sources or if VM2 signals resulted from a directly forced response rather than a free-wave response. Additionally, these different scenarios affect which phase spread and horizontal phase correlation should be applied to the VM2 (and higher-mode) source function.

The source function (s(x→) in Eq. ) was calculated simply by multiplying the forcing function (Fig. ) and the complex conjugate of the adjoint frequency response function (Fig. ). Figure  shows the source function of the VM1 M₂ internal tide at the PIL200 location as an example. It shows alternating signs at the wavelength of the VM1 M₂ internal tide. Physically, it means, for example, that the internal tides generated at half a wavelength away from the PIL200 location and then propagated there have the opposite phase from those locally and currently generated at the location. So, these waves tend to cancel each other, and the opposite signs in the source function reflect this wave cancelling. Although the adjoint frequency response function decays with distance (Fig. a), remote locations with strong barotropic tides and/or steep bottom slopes can be sources as strong as those near the observation location. For example, the magnitudes of the source function in the straits of the Indonesian archipelago, which are well-known source regions of internal tides, are comparable to those on the Australian shelf.

VM1 internal tides from most of the model domain except the Australian shelf are only partially random (Fig. b). The travel time τ for VM1 semidiurnal internal tides calculated by ray theory increases roughly radially from the PIL200 location (Fig. a), which agrees with the adjoint sensitivity (Fig. a). A clear exception is the Australian shelf where τ grows quickly because of small group velocity. The phase variance Pθθ=σj2 also increases roughly radially, but the rate of increase is faster on the shelf because the phase-speed variance σC2 relative to the squared mean phase speed c‾2 is much larger there (Fig. b). Note that σj>1 is a convenient threshold for random sources (see Eq. ; also Fig. 2d in Part 1 for illustration).

Unlike VM1, VM2 internal tides are mostly random (Fig. e). This is partly because the phase-speed variance σC2 relative to the squared mean phase speed c‾2 is larger for VM2 than VM1, so the rate of increase of phase variance is higher. Another reason is that VM2 internal tides have about twice the travel time compare to VM1, and hence VM2 has more time to be affected by random oceanic variability (Fig. d).

Since the diffusion operator method by was used to represent the horizontal correlation of phase modulation, the equivalent isotropic phase correlation length σr characterizes the horizontal correlation. It shows an order-of-magnitude variability between the deep ocean and continental shelf for VM1 (Fig. c) and tends to have a magnitude comparable to but smaller than αrLC over a large part of the model domain. The reason for this can be seen by considering Eqs. () and () in the limit of small Pθθ=σj2, which suggests the length scale 2π-1αrLC. For example, the gradual increase in σr towards north reflects the latitudinal variation of the Rossby radius of deformation, which is assumed to be proportional to LC. The small σr on the continental shelf results from small celerity (and hence small Rossby radius of deformation). The modelled equivalent isotropic correlation functions at three contrasting locations are shown in Fig. a. The correlation function generally has a broader tail than the Gaussian function. The modelled and fitted correlation functions agree around the correlation value of 0.6, which corresponds to σr (standard deviation of the Gaussian function).

The equivalent isotropic correlation length σr for VM2 is substantially smaller than VM1 (Fig. f) and does not have the rough relationship with LC, although the same LC is used for VM1 and VM2. This is because the phase variance Pθθ=σj2 is much larger for VM2 than VM1 (Fig. b and e), which makes the gradient of PΔθΔθ around |Δη|/l≪1 larger (see Eq. ) and the decay of the exponential function in Eq. () faster. As a result, the latitudinal variation does not exist for VM2, but the order-of-magnitude variability between the deep ocean and continental shelf remains. Figure b shows that the modelled and fitted correlation functions agree well for correlation values larger than 0.6 for VM2.

The results of the model suite provide the contributions of different source regions, vertical modes, and tidal constituents to the modelled nonharmonic internal tides and their dependence on the model parameters. We look at different contributions using the reference case (αC, αL, αr) = (0.7, 2, 3) as an example in this section and then the parameter dependence in the next section.

The total modelled nonharmonic VM1 semidiurnal internal-tide variance is 38 m² in the reference case compared to the observed variance of 45±12 m² (confidence interval based on twice the standard error). As explained in Sect. , the variance is calculated based on VM1-induced extreme (maximum or minimum) isopycnal displacements within the water column. The modelled variance can be converted to vertically integrated potential energies in J m⁻² by multiplying by 7.6 and the variance of surface displacements in m² by multiplying by 7.1×10-7 (without seasonal variation).

The contributions of different regions are shown in Fig.  as a map of nonharmonic variance source function and in Table  as regionally integrated contributions. The following regions are arbitrarily chosen for illustration purposes. The LOC region is the local region near the PIL200 location on the Australian North West Shelf shallower than 1500 m, and the NWS region is the Australian shelf region excluding the LOC region. The LAS and SS regions cover the Lombok and Alas straits and Sape Strait, respectively. The IND region is the rest of the model domain, mostly the deep Indian Ocean. These regions are indicated by dashed blue lines in Fig. . Figure  shows that important source regions are the Australian shelf and the straits in the Indonesian archipelago. The nonharmonic variance source function appears much smoother than the source function in Fig.  because the diffusion operator that approximates the correlation coefficient matrix R is applied, and the phase correlation lengths are relatively large (Fig. c and f). The horizontal scale of the nonharmonic variance source function is smaller than the correlation length for VM1 (Fig. c). This is partly because higher modes have smaller correlation lengths (Fig. f) and partly because the diffusion operator averages the opposing contributions from the source function (e.g. red and blue patches in Fig. ) when the correlation length is comparable to or larger than the wavelength. However, note that the locations of sources in the nonharmonic variance source function are uncertain within the phase correlation length in the current approach, as explained in Appendix . This is why contributions from relatively large regions are compared in Table .

Table  shows that remote regions are more important sources of the nonharmonic internal tides than local sources. For example, the contributions of the Australian shelf are smaller than those of the Indonesian straits, and the local contribution on the Australian shelf is smaller than the rest of the shelf. This is because remote sources can be as strong as local sources before phase modulation (Fig. ), and it takes time for random phase-speed variability to make internal tides nonharmonic (Fig. b and e). Although the magnitude of the nonharmonic variance source function in the deep ocean (IND region) is nearly 2 orders of magnitude smaller than the peak values in the major sources (Fig. ), Table  shows that the overall contribution is substantial because it occupies a much larger area than the other regions. Figure  also suggests that, although we used a relatively large model domain for available computational resources, the current modelling is likely to have missed remote sources. It is likely that at least a few m² of variance is missing from the deep Indian Ocean to the west of the model domain.

Table  shows the contributions of different vertical modes and tidal constituents to the modelled variance. The tabular entry for VM2 and M₂ represents, for example, the contribution of the VM2 internal tide that is excited by the M₂ barotropic forcing and then converted to VM1 before arriving at the PIL200 location. Regarding the contributions of different vertical modes, the model results show that VM1 contributes about 3/4 of the total variance, and the contributions decrease with increasing mode number. Regarding the contributions of different tidal constituents, M₂ and S₂ forcings contribute roughly 3/4 and 1/4 of the total variance, respectively. The contributions of K₂ and N₂ are small (1.8 m²). The VM1 directly forced by M₂ alone contributes roughly half of the total variance. So, VM1 and M₂ are dominant, but focusing only on VM1 and M₂ would cause substantial underestimation of the nonharmonic semidiurnal internal-tide variance in this case.

The results shown in the previous section are based on the reference model parameters, but the parameters have relatively large uncertainty. In this section, we investigate the dependence of the results on the model parameters and compare the results with observations at the PIL200 location. The model parameters are varied beyond the realistic range for process understanding.

The results show that the modelled nonharmonic internal-tide variance strongly depends on the variance (αC or σC2) and correlation length (αL or LC) of phase speed (Fig. a). These parameters affect the nonharmonic internal-tide variance in two ways. First, they determine the partitioning of the variance into harmonic and nonharmonic components through the phase variance σj2 (see Eqs.  and ). Second, they affect the phase correlation length σr through μj and ςj in Eq. (), as well as the variance of horizontal phase difference PΔθΔθ in Eqs. ()–(). The dependence on αL shows that it is essential to consider the phase-speed correlation length (see the small variance at αL=0 in Fig. a) because phase-speed variability with a small correlation length is inefficient in producing phase variance (see Eq. ). The dependence on αL gradually decreases with increasing αL for a few reasons. First, the ratio of the variance partitioned to nonharmonic component (ςj2 in Eq. ) increases with the phase variance σj2, but the rate of increase becomes much slower for σj2>1 (see also Fig. 2d in Part 1 for illustration). Second, the horizontal phase correlation tends to increase nonharmonic internal-tide variance as explained in Appendix , but the increase ceases when the equivalent isotropic phase correlation length σr becomes comparable to the internal-tide wavelength. This is because regions separated by half a wavelength tend to have opposing contributions to internal-tide amplitude (see blue and red patches in Fig. ), and the opposing contributions are averaged in Eq. () when the correlation length is larger than half the wavelength.

The nonharmonic internal-tide variance also strongly depends on αr (Fig. b). The dependence illustrates the aforementioned roles played by the phase correlation length σr and internal-tide wavelength more clearly because σr is roughly proportional to αr. The phase correlation increases the nonharmonic internal-tide variance when αr is small. Although αr≪1 (negligible horizontal correlation) is unrealistic, small variance in this limit shows that it is essential to consider horizontal phase correlation for gridded sources, as explained in Appendix . When αr becomes larger, the nonharmonic internal-tide variance decreases gradually with increasing αr by the averaging of sources with opposite phases. The peak of the variance should occur when σr is around a quarter of the wavelength. Considering that the internal-tide wavelength is 1–2 times the Rossby radius of deformation in the modelled region and σr tends to be comparable to but smaller than αrLC for VM1, this suggests αLαr is roughly 1/2 at the peak. Figure b shows the peak around αr≈1/2 for αL=2. This shows that anisotropy of the horizontal correlation of phase is an important controlling parameter for a realistic parameter range (αL≳1,αr≳1), especially if αL≈1. More generally, the result shows that the ratio of the phase correlation length and internal-tide wavelength is important for nonharmonic internal-tide variance.

The comparison of the model results and the PIL200 observations shows that the model results are not inconsistent with the observations for a realistic parameter range (αL≳1, αr≳1), although the modelled variance tends to be smaller than the observed mean. The larger phase-speed variance case (αC=1.0) used phase-speed variance from the PIL200 location on the continental shelf, which provides phase-speed variance that appeared to be roughly the upper limit of previous estimates for deep ocean. In this case, the model results are around the observed mean for αL≥1. The smaller phase-speed variance case (αC=0.4) used phase-speed variance that is about the middle of previous estimates for deep ocean but is an underestimate for shallow water. So, it is reasonable that the modelled variance is around or below the approximate 95 % confidence interval for αL≥1. In the reference case for phase-speed variance (αC=0.7), the model results are between the observed mean and the lower bound of the approximate 95 % confidence interval for αL≥1. Considering the number of assumptions and simplifications used in the model suite, the results are encouraging. This demonstrates the feasibility of the proposed modelling framework and model suite.