a. The radiative energy balance

We consider internal gravity waves propagating in a stratified ocean with stability frequency N under the influence of a vertically sheared mean flow U. For simplicity, we ignore any horizontal and time dependency of U(z) and N(z). The spectrum of the waves is described by the wave energy spectral density $E\left(\mathbf{x},z,t,k,\varphi ,m\right)$, which is a function of time t and horizontal and vertical coordinates x and z, respectively, but also of wavenumber coordinates, m, k, and ϕ. The vertical wavenumber is denoted by m and the horizontal wavenumber k is expressed here as k = kn, with the horizontal unit vector n = (cosϕ, sinϕ), horizontal wave angle ϕ, and amplitude k = |k|. The dispersion relation of the gravity waves is given by ω_enc = ω + k ⋅ U, where ω_enc denotes the frequency of encounter which is measured at a fixed point and where ω denotes the intrinsic internal gravity wave frequency $\omega =\pm \sqrt{\left(f^2{m}^2+N^2{k}^2\right)/\left(m^2+k^2\right)}$.

b. Lee wave spectrum

The flow U₀ = U|_z=−h over topography with a spectrum F_top(k, ϕ) will generate stationary lee waves with vanishing frequency of encounter ω_enc = 0 and so with intrinsic frequency ω_lee = −kU_n with U_n = n ⋅ U₀, and vertical wavenumber $\left|m_{\text{lee}}\right|=k\sqrt{N^2-{\omega }_{\text{lee}}^2}/\sqrt{{\omega }_{\text{lee}}^2-f^2}$ (Bell 1975). To calculate the mean wave parameter c^± and Λ^± in Eq. (2), we assume that the shape of the lee wave energy spectrum in the interior stays close to the spectrum generated at the bottom and use the energy balance Eq. (2) to simulate the balance between forcing and dissipation of the lee waves. This assumption will be assessed below with a direct simulation of Eq. (1).

Note that Eq. (7) is an assumption since the lee waves might change their frequency and wavenumbers while propagating through the water column, they might also exchange energy with other waves by nonlinear effects, and/or they might dissipate by convective or shear instability. It is assumed that all such processes remain small in the sense that they do not affect much the values of the mean group velocity c^± and mean flow exchange parameter Λ^± in Eq. (2). All these processes will be included in Eq. (2), only their effect on c^± and Λ^± is neglected. Note also that by using Eq. (7) we do not exclude other wave energy besides the lee waves. In fact, below we will include a coexisting wave field described by the closure of Olbers and Eden (2013), which interacts with the lee wave part of the spectrum and thereby leads to the dissipation of the lee wave energy. Using Eq. (7) in Eq. (3) acts as a bandwidth filter to isolate the lee wave energy compartment and in Eqs. (4) and (5) to obtain their bulk vertical propagation and mean flow interaction properties.

c. Choosing the lee wave spectral form

Following Eq. (11), the spectral shape function A_lee in Eq. (7) is defined by

$A_{\text{lee}}\left(k,\varphi \right)=E_{\text{bell}}\left(k,\varphi \right){\left[{{\displaystyle \int }}_0^{\infty }{E}_{\text{bell}}\left(k,\varphi \right)dk\right]}^{-1}.$(12)

To calculate A_lee and further the mean parameter Λ^± and c^± from Eq. (9), it is necessary to specify the topographic spectrum F_top. Goff (2010) and Goff and Arbic (2010) provide a statistical description of F_top, given by

$F_{\text{top}}\left(k,\varphi \right)={\displaystyle \frac{h_{\text{rms}}^2\nu }{\pi k_s{k}_n}}\hspace{0.17em}k{\left[1+{\cos }^2\left(\varphi -{\varphi }_s\right)k^2/k_s^2+{\sin }^2\left(\varphi -{\varphi }_s\right)k^2/k_n^2\right]}^{-\left(\nu +1\right)}$(13)

with the Hurst parameter ν and the root-mean-square topographic height $h_{\text{rms}}^2={\displaystyle \int }{F}_{\text{top}}dk\hspace{0.17em}d\varphi$. In Eq. (13), k_s and k_n are characteristic wavenumbers in strike and normal direction, respectively, with k_s ≤ k_n, and θ_s = π/2 − ϕ_s is the angle of the wavenumber vector measured clockwise with respect to the true northward direction. An isotropic form of Eq. (13) is obtained setting k_n = k_s,

$F_{\text{top}}\left(k,\varphi \right)={\displaystyle \frac{h_{\text{rms}}^2\nu }{\pi k_s^2}}\hspace{0.17em}k{\left(1+k^2/k_s^2\right)}^{-\left(\nu +1\right)}.$(14)

Goff (2010) and Goff and Arbic (2010) provide global maps of k_s, k_n, θ_s, and h_rms for fixed Hurst parameter ν = 0.8. It turns out, however, that the numerical integration of F_bell (and the related bottom stress) in k and ϕ for a comprehensive ocean model experiment is numerically very expensive and a further simplification is still necessary which allows for a semianalytical treatment. We follow Nikurashin and Ferrari (2011) and simplify the isotropic topography spectrum Eq. (14) for k ≫ k_s as

$F_{\text{top}}\left(k,\varphi \right)={\displaystyle \frac{h_{\text{rms}}^2\nu }{\pi }}\hspace{0.17em}{k}_s^{2\nu }{k}^{-2\nu -1}.$(15)

The shape function A_lee(k, ϕ) then becomes

$A_{\text{lee}}={\left({\displaystyle \frac{\left|U_n\right|}{N}}\right)}^{-2\nu }{J}^{-1}\hspace{0.17em}{\displaystyle \frac{{\left(k{U}_n\right)}^2}{{\left(k{U}_n\right)}^2-f^2+r^2}}\hspace{0.17em}{k}^{-2\nu -1},J={{\displaystyle \int }}_{\left|f\right|/N}^1\hspace{0.17em}{\displaystyle \frac{t^{-2\nu +1}}{t^2-{\left(f/N\right)}^2+{\left(r/N\right)}^2}}\hspace{0.17em}dt,$(16)

with a dimensionless integral J independent of U_n (and ϕ). The integral ${{\displaystyle \int }}_0^{\infty }{A}_{\text{lee}}\left|\dot{z}\right|\hspace{0.17em}dk$, relevant to compute Λ^± and c^±, then becomes

${{\displaystyle \int }}_0^{\infty }{A}_{\text{lee}}\left|\dot{z}\right|\hspace{0.17em}dk={\left({\displaystyle \frac{\left|U_n\right|}{N}}\right)}^{-2\nu }{J}^{-1}\hspace{0.17em}{\displaystyle \frac{\left|U_n\right|}{N^2-f^2}}{{\displaystyle \int }}_{f/\left|U_n\right|}^{N/\left|U_n\right|}{k}^{-2\nu -1}{\left(k^2{U}_n^2-f^2\right)}^{1/2}{\left(N^2-k^2{U}_n^2\right)}^{1/2}dk.$(17)

With the transformation t = |U_n|k/N the last integral on the right-hand side can be converted to one that is independent of U_n (and ϕ),

$I={{\displaystyle \int }}_{\left|f\right|/N}^1{t}^{-2\nu -1}\hspace{0.17em}\sqrt{1-t^2}\sqrt{t^2-f^2/N^2}\hspace{0.17em}dt,$(18)

such that

${{\displaystyle \int }}_0^{\infty }{A}_{\text{lee}}\left|\dot{z}\right|\hspace{0.17em}dk={\displaystyle \frac{\left|U_n\right|N^2}{N^2-f^2}}{\displaystyle \frac{I}{J}}\approx \left|U_n\right|{\displaystyle \frac{I}{J}},$(19)

where the latter assumption is for N² ≫ f². The integrals I and J are monotonically increasing with N/f and can be expressed in terms of the hypergeometric function ₂F₁ as shown in appendix A. This is, however, of little practical value because an evaluation of I and J would be by a numerical integration for ₂F₁ anyhow. It is shown in appendix A that reasonable approximations for I and J are

$I\approx 0.65{\left({\displaystyle \frac{N}{f}}\right)}^{\nu },J\approx {\left({\displaystyle \frac{N}{f}}\right)}^{2\nu }\log \left(f/r\right)$(20)

for 10 < N/|f| < 100 and N² ≫ f² ≫ r². Given the uncertainty in the spectral shape, we believe that such approximations are sufficient, and also set ν = 1 for simplicity to calculate Λ^± and c^± in the practical applications below. We then find

${{\displaystyle \int }}_0^{\infty }{A}_{\text{lee}}\left|\dot{z}\right|\hspace{0.17em}dk=\left|U_n\right|{\text{Λ}}_0,{\text{Λ}}_0=I/J\hspace{0.17em}\approx \hspace{0.17em}0.65\left(f/N\right)/\log \left(f/r\right).$(21)

In any case, even without these approximations, the parameter Λ₀ is independent of wave angle ϕ and the parameters Λ^± and c^± become

${\text{Λ}}^{\pm }\left(\varphi \right)=\mp {\sigma }_U{\text{Λ}}_0,c^{\pm }\left(\varphi \right)=\pm \left|U_n\right|{\text{Λ}}_0.$(22)

Both Λ^± and c^± are then weakly dependent on the frictional parameter r. Figure 1 shows the approximated Λ₀ as function of N/f and r/f and the approximated form of Λ₀ is compared to the exact one in Fig. 12. We will determine the appropriate choice of r for the closure using a direct numerical simulation of Eq. (1) below.

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The (approximated) dimensionless parameter log₁₀Λ₀ is shown from Eq. (21) as function of N/f and r/f.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0230.1

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d. Pseudomomentum fluxes

The effect of the waves on the mean flow is given by the divergence of the wave induced vertical momentum flux $\overline{{\mathbf{u}}^{\prime }{w}^{\prime }}$ in the mean momentum equation and the buoyancy flux $\overline{b^{\prime }{w}^{\prime }}$ in the mean buoyancy equation. It turns out that interpreting the mean momentum as the residual momentum instead of the Eulerian mean momentum, one can combine both effects into the pseudomomentum flux¹ $\overline{{\mathbf{u}}^{\prime }{w}^{\prime }}+f\overline{\underset{\neg }{{\mathbf{u}}^{\prime }}{b}^{\prime }}/N^2$, which then figures only in the mean (residual) momentum equation. It was shown in Eden and Olbers (2017) how these wave-driven fluxes can be calculated from the wave polarization relations and the wave energy density $E$, and that the resulting energy transfer terms in the mean residual energy equation U²/2 match exactly (but of opposite sign) the corresponding energy transfer in Eq. (1) (first term on the right-hand side).

e. Unidirectional model

The role of the vertical dependency of N and U for the parameters Λ₀ and c_lee may need some clarification. In the radiative transfer equation Eq. (1), the local mean flow velocity U determines the energy exchange with the mean flow and the wave drag on the mean flow, and thus the term ∂_zU in Eq. (30) is taken from the local mean flow also. However, the bottom flow U₀ determines the lee wave bottom energy flux, and thus also the spectral shape A_lee of the lee waves, which we therefore use to determine the parameter c_lee. For the parameter Λ₀ = I/J from Eq. (21) we use the local N for both integrals I and J, which we find to fit reasonably well a direct simulation of Eq. (1) as shown below, where we also show the impact of varying U on the parameters.

f. Multidirectional model and sink terms

The terms with ${\tau }_s^{-1}$ introduce linear damping with a time scale τ_s of the energy of the difference of up- and downward propagating wave, $\text{Δ}{E}_{\text{lee}}=E_{\text{lee}}^{+}-E_{\text{lee}}^{-}$, but has no effect on total energy $E_{\text{lee}}=E_{\text{lee}}^{+}+E_{\text{lee}}^{-}$. The decay of a generic GM spectrum but with vertically asymmetric distribution is attributed by McComas (1977) to elastic scattering and τ_s is estimated as a few days. The same may be expected to occur in a vertically asymmetric lee wave spectrum but the determination of the exact value needs further investigation. Here we use as in Olbers and Eden (2013) τ_s = 3 days.

The lee waves are assumed to be embedded in a background internal wave field with total energy E_gm, which is governed by the closure described in Olbers and Eden (2013). The terms with α_ww remove total lee wave energy E_lee, but drop for the asymmetry spectrum ΔE_lee. The energy sink α_wwE_gmE_lee is added as a source in the corresponding equation for E_gm of the background waves. It is quadratic in energy akin to the scattering integral of wave–wave interaction. The closure and the parameter α_ww are taken from Olbers and Eden (2013). The quadratic form of the closure is also the basis of the so-called finestructure parameterization (e.g., Gregg 1989) and was recently confirmed by direct numerical calculations of the scattering integral of wave–wave interactions by Eden et al. (2019). Here, we use the same value for α_ww as in Eden et al. (2019) [with revised parameter according to the findings of Pollmann et al. (2017)], but it is clear that further investigation for the dissipation of a lee wave spectrum in the presence of a background GM field is necessary.

a. Model validation

To validate some of the assumptions necessary to develop the model given by Eq. (30), we first directly integrate the radiative transfer equation Eq. (1) numerically for a given lee wave generation, stratification N, and mean flow U. Different to the model given by Eq. (30), this model is fully resolved in wavenumber (k, m) space, and also in vertical dimension z and time t, while we assume horizontal homogeneity as before. The term S on the right-hand side of Eq. (1) contains wave–wave interactions among the lee wave field, but also with the generic background wave field, and scattering at mesoscale flow and the bottom. All these effects need subsequent investigation, however, here we specify S simply as Rayleigh damping with $S=-2rE$ as in the assumed lee wave spectral shape Eq. (12).

Figures 2a–c show the wave energy $E$ using f = 10⁻⁴ s⁻¹, an exponentially increasing N(z) with maximal amplitude of 30f and a decay scale of 800 m (shown in Fig. 5b), r = 0.01f, and vertically constant U = (0.1, 0) m s⁻¹. Due to the Rayleigh damping, there is an asymmetry between up- and downward propagating waves, with more energy in the upward propagating waves at negative m. Since $\dot{m}=-k^2/\left(k^2+m^2\right)\left(N/\omega \right)dN/dz$, the increase in N toward the surface leads to an increase (decrease) of the vertical wavenumber magnitude of the upward (downward) propagating waves. Since the magnitude of $\dot{z}$ decreases with larger vertical wavenumber magnitude, wave energy $E$ accumulates toward the surface, such that the total wave energy ${\displaystyle \int }Ed\mathbf{k}\hspace{0.17em}dm$ (not shown) also increases toward the surface. The corresponding integration for negative intrinsic frequency looks identical but mirrored in each wavenumber axis.

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(a)–(c) Wave energy $E$ in a direct simulation of Eq. (1) for the case of vanishing mean flow shear, shown as a function of depth and integrated over different wavenumber vector components. (d),(e) Spectral shapes A(k_x) from the theoretically predicted [E_bell(k_x, k_y = 0); solid] and of the simulation of Eq. (1) [${\displaystyle \int }E\left(k_x,k_y=0\right)\hspace{0.17em}dm$; dashed] at two different depths. The spectra are normalized by the area under the curves. (f) The parameter ${\displaystyle \int }A\left|\dot{z}\right|\hspace{0.17em}dk$ (times 1000) calculated from the spectral shapes (dashed for the simulated spectrum) at all depth levels. (g) The parameter ${\displaystyle \int }A\left|\dot{z}\right|\hspace{0.17em}dk$ (times 1000) for different wave angles at the bottom. The simulated spectrum (dashed) is shown for −π/2 < ϕ < π/2 for a model with negative intrinsic frequencies, otherwise for positive ones. The dotted line denotes the approximation Eq. (21), i.e., |U_n|Λ₀.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0230.1

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For the quality of the approximations made to derive the lee wave closure Eq. (30), the assumed lee wave spectral shape A and in particular the parameter ${\displaystyle \int }A\left|\dot{z}\right|\hspace{0.17em}dk$ are important. We compare the theoretically predicted spectrum E_bell with the simulated spectrum at k_y = 0 m⁻¹ in Figs. 2d and 2e. Both spectra are normalized, i.e., are shown as shape functions A. Close to the bottom, the simulated spectral shape is indeed very similar to the theoretically predicted one, although the resonance peak where ω_lee → f is smaller in the simulated spectral shape, and higher up in the water column the resonance peak gets smaller and broader in the simulated spectral shape. Figure 2f shows the resulting parameter ${\displaystyle \int }A\left|\dot{z}\right|\hspace{0.17em}dk$ calculated from both spectral shapes. Since the simulated spectral shape A is in general larger for larger k than the theoretically predicted one, and since $\dot{z}$ gets larger for larger k, the parameter ${\displaystyle \int }A\left|\dot{z}\right|dk$ from the simulated spectral shapes is larger than the theoretically predicted one. The value of the parameter also depends on the resonant peak close to ω_lee → f. Although we use a rather large number of grid points in the spectral domain, this peak is still not particularly well resolved in the numerical model, such that the simulated parameter is larger than the theoretically predicted one. However, the depth dependency of the parameter is very similar.

Since the dependency on wave angle of both spectra are nearly identical (not shown), the wave angle dependency of the parameter calculated from both spectra is also very similar, as shown in Fig. 2g. The approximated parameter |U_n|Λ₀ from Eq. (21) is slightly smaller than the theoretically predicted one (as expected from Fig. 12, but the dependency on wave angle is also very similar to the others.

Figure 3 shows corresponding results including vertical shear: a jet centered at z = −750 m of double exponential profile with a decay scale of 200 m superimposed on the eastward velocity of 0.1 m s⁻¹ (also shown in Fig. 5a). Total wave energy (not shown) is still maximum at the surface, but has now also a secondary local maximum around 1000 m depth at the lower flank of the jet. Figure 3 shows that close to the bottom and at the surface, the simulated spectral shape is again close to the theoretically predicted one, only at the depth range of the mean flow interaction there is a slight distortion.

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As in Fig. 2, but for a case with mean flow shear.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0230.1

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The other important approximation necessary for our new lee wave closure is Eq. (28), which is needed to account for the unknown wave angle dependency of the energy compartment. For the wave energy $E$ from the simulation of Eq. (1), the difference between left- and right-hand side of Eq. (28) stays within 10%–15% both for the case with and without mean flow shear (not shown), and between 5% and 10% for the theoretically predicted spectrum. While this justifies the use of Eq. (28) for the unidirectional model, we made no comparisons with more complicated mean flow configuration.

Overall, we find reasonable agreement between the assumed spectral shape in the new closure and the simulation of the radiative transfer equation, although the model does not simulate the pronounced resonance peak close to ω_lee → f, which may be due to numerical reasons. A vertically varying N changes the parameter ${\displaystyle \int }A\left|\dot{z}\right|dk$ significantly, but its vertical dependency is fitted well by the approximation. Mean flow interaction does not effect the parameter much. The effect on the parameter ${\displaystyle \int }A\left|\dot{z}\right|\hspace{0.17em}dk$ is overall an increase by a factor of 2, thus we believe that choosing r/f = 0.1—which generates this factor of 2 in the resulting parameter ${\displaystyle \int }A\left|\dot{z}\right|\hspace{0.17em}dk$—is currently optimal to calculate the parameter in the following model experiments.

b. Idealized model simulations

The model variables of the unidirectional model Eq. (30) are discretised on an equidistant vertical grid and Eq. (30) is integrated in time. The discretization uses a second-order advection with superbee limiter for the flux terms. A simplified mean flow equation ∂_tU = −∂_zτ is also integrated in time to demonstrate the effect of the wave drag. We use 200 grid points in the vertical, 2000 m depth, and a time step of 360 s. Energy is conserved to numerical precision between the kinetic energy of the mean flow and the total lee wave energy $E_{\text{lee}}^{+}+E_{\text{lee}}^{-}$. The vertical boundary conditions for the closure are detailed in appendix B. They include the lee wave bottom energy and momentum flux, for which the calculation follows closely the previous studies (e.g., Scott et al. 2011; Nikurashin and Ferrari 2011; Naveira Garabato et al. 2013).

Figure 4 shows a simulation with initially zero wave energy, U = (0.1, 0) m s⁻¹, f = 10⁻⁴ s⁻¹, and constant N = 30 f as environmental conditions, h_rms = 10 m, k_s = 1/10 km⁻¹, ν = 0.8, and L = 1 for the bottom boundary conditions (see appendix B), and r = 0.1 f, ${\tau }_s^{-1}=0\hspace{0.17em}{\text{s}}^{-1}$ and α_ww = 0 s m⁻² for the model parameter. During the first 50 days of the integration a lee wave energy flux into $E_{\text{lee}}^{+}$ of about 10⁻³ W m⁻² constantly force upward propagating waves of the same magnitude. Since there is initially a vanishing compartment $E_{\text{lee}}^{-}$, the magnitude of the stress $\tau =-\left(2/\pi \right)\mathrm{sign}\left(U_0\right){\text{Λ}}_0\left(E_{\text{lee}}^{+}-E_{\text{lee}}^{-}\right)$ is largest where the upward propagating waves have reached, and zero above that vertical location. That divergent momentum flux generates a strong wave drag leading to decreasing mean flow amplitudes at those vertical locations. After about 50 days, the $E_{\text{lee}}^{+}$ compartment is reflected at the surface into the $E_{\text{lee}}^{-}$ compartment with identical magnitude, which leads to a vanishing momentum flux τ and a large wave drag where the downward propagating waves have reached. This again generates decreasing mean flow amplitudes. After bottom reflection the cycle is repeated, but with decreasing wave drag amplitudes since the energy flux into the wave field ceases. This also generates a smaller propagation speed c_lee. After 200 days most of the energy of the mean flow is taken up by the waves.

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(a) Energy compartment of upward propagating lee waves $\rho E_{\text{lee}}^{+}$ in an idealized model simulation with initially U = const and N = const. (b) same as (a) but for the energy compartment of downward propagating lee waves $\rho E_{\text{lee}}^{-}$. (c) Wave-induced vertical momentum flux ρτ. (d) Zonal mean flow U.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0230.1

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