Hydraulics and Mixing of the Deep Overflow in the Lifamatola Passage of the Indonesian Seas

a. Observed transport estimates

The mean ADCP velocity profile combined with the high-resolution (1′ × 1′) topography from ETOPO1 (dataset available on https://www.ngdc.noaa.gov/mgg/global/; Amante and Eakins 2009), are used to estimate the Lifamatola Passage volume transports in the along-thalweg direction, which is aligned northwest to southeast at a compass angle of 150.5° (Fig. 1b). Instead of characterizing the overflow and water mass by its potential density, we define the overflow in terms of Conservative Temperature (Θ) since the density is mostly controlled by temperature in the deep Lifamatola Passage. Conservative Temperature Θ, which is proportional to potential enthalpy ($h°$), obeys a simple conservation law (see appendix C) and leads to a precise measure of heat flux (McDougall 2003). Conservative Temperature Θ and $h°$ are derived from the in situ temperature and salinity using the new equation of state of seawater TEOS-10 (McDougall and Barker 2011). The mean overflow transport is obtained from interpolating the mean velocities onto a 10-m grid and integrating from the seafloor to a selected Conservative Temperature ($\Theta$) surface.

The Conservative Temperature section along the thalweg of the Lifamatola Passage displayed in Fig. 2 is constructed from nine CTD casts made in 2017 and from the time-mean mooring measurements. The deep flow that spills down across the sill is largely confined to waters colder than Θ = 3.6°C. Moreover, of the surfaces that participate in the spilling, those in the range 3.0° < Θ ≤ 3.6°C tend to experience no further descent as they enter the Seram Sea, whereas surfaces with Θ ≤ 3.0°C continue to descend, suggesting a possible role in ventilation of the deep basins downstream. In the following, we estimate mean volume transport for waters bounded above by the Θ = 3.0° and 3.6°C surfaces as well as several others.

Also required for transport estimates is information regarding the lateral (cross-passage) extent and the structure of the flow. The cross-section chosen to estimate the transport below Θ = 3.6°C is indicated by the dark line in the inset map of Fig. 1b, with the solid dots at each end marking the lateral boundaries. The northeastern boundary (1°44.21′S, 127°00.61′E) is chosen as the isobath that matches the elevation of the Θ = 3.6°C surface at the mooring. Since to the southwest the topography shallows to a crest at about 1700-m depth and then deepens further to the west (Fig. 1c), we define the topographic crest as the position of our western boundary, which places the boundary at a location (1°54.40′S, 126°54.85′E) very close to that defined by van Aken et al. (2009). For transport below Θ = 3.0°C, the flow is confined by topography on either side and we define the boundaries accordingly. In their transport estimate of 2.5 Sv for flow below 1250 m, van Aken et al. (2009) assumed that the velocity was laterally uniform. With the same assumption of lateral uniformity and with our edge definitions, our estimate of the transport below Θ = 3.6°, 3.4°, 3.2°, and 3.0°C are 2.8, 2.4, 2.0, and 1.5 Sv, respectively. However, uncertainties exist due to assumptions of laterally uniform flow and of the location of the southwest boundary.

Flows in sea straits are often concentrated in a jet (e.g., Siddall et al. 2004) or a boundary current (Gill 1977). We rule out the possibility of the overflow being a boundary current since the local deformation radius $\sqrt{g^{\prime }D}/\left|f\right|~124\hspace{0.17em}\mathrm{km}$ (where $g^{\prime }~6.5\times {10}^{-4}\hspace{0.17em}\mathrm{kg}\hspace{0.17em}{\mathrm{m}}^{-3}$ is the reduced gravity, discussed further in section 3b, D ~ 500 m is the typical layer thickness, and f = −4.6 × 10⁻⁶ s⁻¹ is the Coriolis parameter) is much larger than the ~21-km width occupied by the deep layer. For a jet-like flow, assuming the mean velocity varies linearly from its maximum at the mooring site to zero at the edges, the transports are then exactly one-half of the values of the uniform flows: for example, 1.4 Sv below Θ = 3.6°C and 0.75 Sv below Θ = 3.0°C.

b. Hydraulic criticality

The spilling and descent of dense fluid across the sill and into the Seram Sea suggests that the deep transport in the Lifamatola Passage may be hydraulically controlled, which implies some degrees of regulation of the overflow transport. Hydraulic control also suggests that the deep transport may be estimated based only on the topography and stratification instead of velocity. This could potentially be used as a parameterization in a model that is unable to resolve the topographic details of the Lifamatola Passage. It also suggests blocking of certain types of horizontally propagating waves that attempt to travel from the Seram Sea upstream into the Maluku Sea. In hydraulics, one is primarily interested in nondispersive long waves (wave lengths much greater than fluid depth) since these are most efficient at altering the upstream flow. Since we only have velocity measurements at the mooring location, we will evaluate the hydraulic state of flow there (slightly downstream of the sill crest) by calculating Froude numbers based on different simplifications of the mean flow.

By fitting the observed velocity and stratification to a 1.5-layer model configuration, one can make a crude estimate of the hydraulic state described by the common Froude number $F=V/\sqrt{g^{\prime }D}$, a ratio of the advection speed to the phase speed of long internal gravity waves. The 1.5-layer model assumes an active lower layer with uniform velocity V and thickness D and a quiescent upper layer, with a density jump represented by the reduced gravity g′ at the interface. Earth’s rotation is neglected due to the very large ratio of internal deformation radius to the channel width. The flow is hydraulically subcritical, critical, or supercritical according to F < 1, F = 1, or F > 1, respectively. The transition of subcritical-to-supercritical overflow is expected to take place at the critical or control section, which generally coincides with the crest of the sill, and can be shifted elsewhere, usually downstream, as a result of bottom drag or entrainment (Pratt 1986; Gerdes et al. 2002).

The estimation of reduced gravity $g^{\prime }=g\left(\mathrm{\Delta }\rho /{\rho }_0\right)$, which is proportional to the density jump Δρ across the interface, can be problematic. As shown in Figs. 3a and 3b, no distinct density jump is evident in the time-average density profile, only a band of weakened stratification between Θ = 3.4° and 3.6°C. If we choose the Θ = 3.4°C surface as the interface, and estimate the upper-layer density as that averaged between Θ = 3.4° and Θ = 3.6°C, and the lower-layer density and velocity as the vertical average below Θ = 3.4°C, then $g^{\prime }~6.5\times {10}^{-4}\hspace{0.17em}\mathrm{kg}\hspace{0.17em}{\mathrm{m}}^{\mathrm{-3}}$. The estimated Froude number of the mean flow is about 0.8, which would be lower had we chosen the upper-layer density to be a vertical average between Θ = 3.4°C and the sea surface.

A more sophisticated version of the Froude number was derived by Nielsen et al. (2004), who accounted for the shear and density variations in the lower layer using “shape” coefficients α and β (see appendix A for definitions). The resulting Froude number $F={\left(\alpha /\beta \right)}^{1/2}V/\sqrt{g^{\prime }D}$ tends to be larger than the 1.5-layer estimate (Nielsen et al. 2004; Girton et al. 2006). For the present case, evaluation of this Froude number using the same value of g′ as given above leads to a value of 1.1.

A third estimate, also based on an idealization of the flow, is to calculate the wave propagation speed of the first baroclinic mode for a flow with vertically uniform buoyancy frequency N and velocity V in the lower layer, and with quiescent and homogenous fluid above. This approach assumes that the weakly stratified layer between Θ = 3.4° and 3.6°C decouples the stratified underlying flow from shallower regions and avoids estimation of g′. In the long-wave limit, the waves of the first baroclinic mode are analogous to waves in the 1.5-layer model. The propagation speed of the first mode is c₋₁ = V − 2ND/π, with the negative sign of the subscript representing upstream-propagating waves relative to the mean flow (see section 6.2 in Baines 1995). The corresponding Froude number yields F₁ = πV/(2ND), which compares the flow speed with the phase speed of the first baroclinic mode for a quiescent fluid. Here we take the lower layer N and V as the vertical averages of the measured values below Θ = 3.4°C, we find a wave propagation speed of −0.03 m s⁻¹ (i.e., very slow upstream propagation) and F₁ = 0.9.

All three estimates are quite close to unity, suggesting that the flow is nearly critical at the mooring location. Although the lack of velocity measurements upstream and downstream prevents evaluation of the hydraulic states there, the near-criticality of the flow near the sill crest is consistent with the inspected subcritical-to-supercritical hydraulic transition from the descending feature of isotherms observed both in the hydrographic profiles from van Aken et al. (1988) and Fig. 2.

c. Hydraulic transport estimates

There are two types of transport (or “weir”) formulas that relate the volume transport Q to the observed stratification and topography. The first type requires knowledge of the stratification far upstream of the control section, and assumes the flow there is nearly quiescent, as in a large reservoir. Since our knowledge of the upstream conditions is limited, we employ a second class of formulas that depend only on the measured stratification at the mooring location, where the overflow is presumed to be critical. The results of the previous section suggest that this is approximately so. The transports for rectangular, triangular, and parabolic cross sections are given in the context of the 1.5-layer model described above (Pratt and Whitehead 2008; Chow 1959):

For a rectangular cross section, W and D stand for the uniform channel width and layer thickness, respectively. For the other two cross sections, D represents the layer thickness at the deepest point of the idealized geometry. Parameters γ_L and γ_R are the slopes of the left-hand side and right-hand side walls (facing downstream) of the triangular cross section (γ_L = W_L/D, γ_R = W_R/D, with W_L and W_R the widths of the flow interface to the left and right of the deepest point), and δ represents the curvature of the channel bottom at the sill for parabolic cross section ($\delta =4D/W_{\mathrm{1}}^{\mathrm{2}}$, where W₁ is the width of the interface).

Assuming that the overflow is critical at the mooring location and the flow interface coincides with the Θ = 3.4°C isotherm, one can determine the best fit γ_L, γ_R, δ, and D to the cross-section topography using a least squares fit (see the triangle and parabola in Fig. 1c). For the rectangular weir formula, W and D in (1) are determined by the mean channel width below the level of the interface ${\displaystyle {\int }_h^{h+D}w\left(z\right)\hspace{0.17em}dz}/D$, where h is the bottom depth and the distance between the deepest depth and the interface, respectively. The rectangular, triangular, and parabolic weir formulas arrive at transports of 2.2, 2.6, and 2.8 Sv, respectively. We note that these hydraulic estimates of the volume flux are subject to uncertainty regarding flow criticality at the mooring station as well as uncertainty due to the estimate of g′ and that the direct mooring measurements are also subject to a high degree of uncertainty due to the unknown spatial structure of the flow. In this regard, all three weir formulas estimates lie reasonably close to the mooring measurement of 2.4 Sv, although the rectangular and triangular weir formulas give results that are closest to the observed value.

d. Wave blockage and upstream propagation: Wave modes for continuously stratified shear flow

A more general approach to assess the flow criticality, preferable where the flow lacks a clear layer interface, is to consider the wave modes of the continuously varying density and velocity profiles over the full water column, though this analysis does not easily lead to a weir formula. Here we summarize a computation of the wave modes at the mooring site, not only to assess the proximity of the flow at the mooring to hydraulic criticality but also to determine whether there are waves that are able to propagate upstream, from the Indian Ocean–Banda–Seram Sea regions, into the Maluku Sea and regions farther north.

We limit discussions to two-dimensional, infinitely-long waves since they tend to propagate more rapidly than waves with the same modal structure but finite wavelengths and are generally the key to upstream influence. The “upstream influence” refers to the ability of a sill or width contraction to regulate certain features of the flow far upstream and is generally communicated by long waves. These properties are discussed more fully in Pratt and Whitehead (2008) and Baines (1977, 1995).

The effects of Earth’s rotation are ignored. The discrete vertical wave modes and corresponding propagation speeds are the eigenfunctions and eigenvalues of the Taylor–Goldstein (hereafter TG) equation:

with $\widehat{w}\left(0\right)=\widehat{w}\left(D\right)=0$. Here $\widehat{w}\left(z\right)$ gives the vertical structure of vertical velocity due to the wave, c is the horizontal propagation speed of a vertical mode, N²(z) is the squared buoyancy frequency, V(z) is the horizontal fluid velocity (positive in the direction of the Lifamatola outflow), V″(z) represents the vorticity gradient due to the vertical shear, and the primes denote differentiation with respect to z. The vertical structure of the along-channel velocity of perturbation is given by $\widehat{\upsilon }\left(z\right)=-\left(i/k\right){\widehat{w}}^{\prime }\left(z\right)$. For stationary (c = 0), long waves (k = 0), l² is called the Scorer parameter (Scorer 1949). We will refer to l² with arbitrary c and k as the generalized Scorer parameter. Where the function l² is positive, the solutions for $\widehat{w}$ oscillate in the z direction. Sufficiently large V″ can cause l² to become negative, in which case the restoring mechanism associated with stratification has been vanquished by shear (Hogg et al. 2001). We use a code developed by Smyth et al. (2011) to solve (4), and we set k = 10⁻⁶ m⁻¹ in all cases in order to approximate long waves since the nondispersive character of these waves is key to hydraulic control. The profile of the generalized Scorer parameter l² for each mode is calculated to assess the nature of wave modes given k ~ 0 and corresponding wave propagation speed $c$ (usually nonzero).

Mean profiles of V and N² from the mooring measurements only cover ranges of 300–2000 m and 850–2000 m, respectively. In order to compute wave modes for the full-depth profiles, we extend the velocity profile above 300 m by filling data voids with zeros and construct a full-depth N² profile by linearly interpolating the potential density profiles between CTD stations 4 and 5, which lie immediately upstream and downstream of the mooring. The potential density profile is then processed with a fourth-order Butterworth low-pass filter with a cutoff distance of 50 m to eliminate small-scale inversions due to intermittent overturns. The profile of N² is computed from the smoothed density profile and linearly interpolated onto a 10-m grid. The constructed V and N² profiles ranging over 0–2000 m (Fig. 4a) are then used to solve the TG equations. Although the true bottom depth is 2007 m, we treat the deepest data point at 2000 m as the lower boundary where a free-slip boundary condition (nonzero velocity) is implied. This choice of boundary condition for the background flow is consistent with the inviscid TG equation for disturbances. The Richardson number is greater than 1/4 at nearly all z for this profile, with just a few values lying slightly below 1/4. Were it the case that the Richardson number was greater than 1/4 at all $z$, it is known that an infinite number of discrete vertical modes ${\widehat{w}}_j\left(z\right)$ and corresponding eigenvalues c_j exists (Drazin and Reid 1981; Baines 1995). Each mode comes in a pair [denoted ${\widehat{w}}_j\left(z\right)$ and ${\widehat{w}}_{-j}\left(z\right)$], corresponding to waves that attempt to propagate downstream and upstream relative to the mean flow. Figures 4b and 4c show that the gravest vertical modes of the full-depth profiles are largely determined from the strongly stratified subsurface layer (~100 m) and correspond to waves propagating on a shallow waveguide. They are insensitive to the velocity and stratification within the deep overflow. Modes j = −1, −2, and −3 have negative propagation speeds in the range −0.82 to −2.48 m s⁻¹, suggesting that they are able to propagate from the Seram Sea to the Maluku Sea unimpeded. The generalized Scorer parameter l² (Figs. 5a–c) shows the largest positive values dominated by the stratification term in the upper part of the water column at around 100 m, suggesting that the large near-surface stratification provides the primary restoring tendency to the corresponding waves. It should be noted that the velocity profile above 300 m is arbitrarily filled with zeros, but the upstream propagation tendency would not be affected unless the shallow velocities were positive and approached the propagation speeds listed above.

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Modal structure of selected normalized eigenvectors $\widehat{w}\left(z\right)$ and $\widehat{\upsilon }\left(z\right)$ as determined by the numerical solution of the Taylor–Goldstein equation. The vertical modes correspond to upstream propagating waves (j = −1, −2, −3). (a) The full-depth squared buoyancy frequency profile N² (s⁻²) from CTD measurements (red) and the velocity profile V(m s⁻¹) from mooring measurements (black), while the dashed line indicates that the velocity profile above 300 m is filled with zeros. (b),(c) Vertical structure of the three modes in terms of $\widehat{w}\left(z\right)$ and $\widehat{\upsilon }\left(z\right)$, respectively, based on the profiles in (a). (d) The modified stratification and velocity profiles. The stratification profile (red) assumes a homogenous upper layer. The solid and dash–dotted velocity profiles (black) assume an inactive layer placed above a moving layer with depth-dependent and uniform velocities, respectively. (e),(f) The gravest modes (j = −1) of the profiles in (d). The solid and dash–dotted modes in (e) and (f) are based on the solid and the dash–dotted velocity profiles in (d), respectively. Phase speeds are color-coded and listed with units of m s⁻¹, with the negative sign representing upstream propagation. The dashed and solid lines in (d)–(f) indicate the locations of the Θ = 3.6° and 3.4°C surfaces, respectively.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0326.1

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(a)–(c) Color-coded generalized Scorer parameter l² = N²/(V − c)² − V″/(V − c) (unit of m⁻²) and its first and second constituents for modes (j = −1, −2, −3) based on the full-depth stratification and velocity. (d)–(f) As in (a)–(c), but for the mode (j = −1) obtained from the modified stratification and velocity profiles indicated by the solid curves in Fig. 4d. Note that the maxima of the generalized Scorer parameters shown in (d)–(f) can be an order of magnitude smaller than those in (a)–(c), and plots in (d)–(f) are limited to depths below 1400 m.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0326.1

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Identification of the waves whose upstream propagation is arrested by the overflow, and that are therefore hydraulically relevant, is more problematic. There are many modes with large numbers of zero-crossings that have near-zero phase speed and are arrested, or nearly so, by the deep flow, and therefore are difficult to interpret (e.g., Pratt et al. 2000; Hogg et al. 2001). However, the situation may be simplified by invoking an argument that has its origins in literature pertaining to mountain waves and downslope winds (Smith and Sun 1987): the top of the descending flow is often subject to shear instabilities and overturns that may lead to a homogenized layer of fluid (e.g., Peltier and Clark 1979; Pawlak and Armi 1997; Farmer and Armi 1999), and the base of this layer tends to act as a perfect reflector of internal waves, implying that the spilling layer is insulated from properties of the flow above and may be treated in isolation. The wave modes relevant to the spilling layer may then be found by treating the fluid within and above the mixed layer as homogeneous. Figures 3a and 3b suggest the presence of a weakly stratified (though not completely mixed) layer between Θ = 3.6° and 3.4°C. We will treat the base of this layer as the interface bounding the spilling flow and will solve the TG equation in the lower portion of the water column using a pliant boundary condition at that interface. A pliant surface is a material interface separating the stratified fluid from the homogenous layer above, and across which the pressure is continuous (see section 1.2 in Baines 1995). This procedure is tantamount to solving the TG equation over the full depth, but replacing the fluid above the bottom of the mixed layer with a homogeneous, inactive layer. We use the time-mean profiles of N² and V from mooring measurements, with estimated Richardson numbers mostly larger than 1/4 within the overflow (Fig. 3d).

Sensitivity tests were made by choosing different levels between Θ = 3.6° and 3.4°C as the interface. Profiles of V and N² for one such choice along with profiles of the corresponding gravest mode are given by solid curves in Figs. 4d–f. Each choice of interface yields a gravest mode (j = −1) with maximum amplitude of ${\widehat{w}}_{-1}$ and zero-crossing of ${\widehat{\upsilon }}_{-1}$ located at the same level around the interface of the overflow (within the range of Θ = 3.4°–3.6°C). The waves so obtained all propagate slowly upstream, with speeds c₋₁ ranging from −0.08 to −0.17 m s⁻¹. To put these values into context, an equivalent Froude number V/(V − c₋₁) (a ratio of the flow speed and the phase speed of the first baroclinic mode for a quiescent fluid), where V = 0.45 m s⁻¹ is the layer-average velocity, would lie in the range 0.73–0.85, which is only slightly below the range of estimates for layer-model Froude numbers given earlier. For comparison, we also show the solution of the TG equation with uniform overflow velocities (black dash–dotted velocity profile and vertical modes in Figs. 4d–f). The wave propagation speed of mode j = −1 is found to be very close to zero (c₋₁ = −0.01 m s⁻¹), which is consistent with the estimate of −0.03 m s⁻¹ assuming uniform N² and V (see section 3b). Interestingly, we find that the effects of shear on the structure of deep modes are particularly pronounced near the bottom. As shown in Figs. 5d–f, the generalized Scorer parameter l² of the shear flow is positive and dominated by the velocity curvature term, suggesting that the vorticity gradient provides the main restoring tendency for the corresponding waves.

a. Estimates of diapycnal mixing coefficients

We estimated the interfacial mixing of the Lifamatola Passage overflow by evaluating Thorpe-scale overturning lengths from CTD casts. Diapycnal diffusivity estimates have also been made from the mass and heat budgets integrated over control volumes. The former are local estimates, and the later provide a spatial integral view of the mixing.

1) Thorpe-scale-based estimates

One widely used estimate of the turbulent dissipation rates is based on Thorpe scales (Thorpe 1977), which assumes that the density inversions are due to the turbulent mixing. The turbulent dissipation rates ε and corresponding diapycnal diffusivity coefficients K of individual overturns, shown in Fig. 6a, were estimated at all nine CTD stations using Thorpe scales estimated from Conservative Temperature profiles and Osborn’s model for the relationship between ε and K (details are described in appendix B). Identified overturns with large ε and K are found concentrated downstream of the crest of the sill, while fewer overturns and weaker mixing are found upstream of the crest. At CTD sites 5 and 6, regions of large overturns are separated by the Θ = 3.4°C isotherm, which is associated with the two mixed wedges described above (also seen in Fig. 2). High mixing levels are also found within the descending flow downstream of the CTD site 6, but weaker values are seen at the final section (CTD site 9), where the flow is laterally unconfined by bathymetry.

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(a) Estimates of ε (blue bars) and K (orange bars) for each overturn at CTD sites 1–9, with black lines indicating the levels of the Θ = 3.4°C isotherm. (b) The mean overflow dissipation rates across the sill. The depth averages were calculated using a zero and nonzero background dissipation coefficient. (c) Diapycnal diffusivity coefficients from overturns averaged into isothermal layers (squares) and over all the CTD sites (black line) and CTD sites 5–8 (red line), respectively. The gray and black dashed lines indicate the diffusivity coefficients for unresolved overturns and estimated in the deep Banda Sea by van Aken et al. (2009), respectively.

Citation: Journal of Physical Oceanography 50, 9; 10.1175/JPO-D-19-0326.1

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To assess the spatial variation of dissipation rates across the sill, we calculated the depth average of the dissipation rate within the overflow as

$\overline{\mathit{\epsilon }}={\displaystyle \frac{1}{H}}{\displaystyle {\sum }_i{\mathit{\epsilon }}_i{h}_i},$(5)

where H is the thickness of the overflow, ε_i and h_i are the dissipation rate and thickness of an individual overturn within the overflow layer. The upper boundary of the overflow was chosen to be the Θ = 3.4°C isotherm since the time-mean overflow velocities at the mooring location are found mostly confined below this isotherm and it is a convenient choice for doing the budget calculation detailed in sections 4b and 4c. In regions where no overturns were detected, we alternatively assume a background ε of zero and 0.2 × 10⁻⁷ W kg⁻¹, and we compute the corresponding lower and upper bounds of $\overline{\mathit{\epsilon }}$. The value 0.2 × 10⁻⁷ W kg⁻¹ is different from a typical oceanic background level of O(10⁻¹⁰) W kg⁻¹ [observed in the abyssal Brazil basin by St. Laurent et al. (2001)]: instead, it indicates the upper limit for unresolved overturns (with Thorpe scales smaller than 1 m) and is calculated using the maximum background buoyancy frequency of N = 3 × 10⁻³ s⁻¹ determined from the resolved overturning regions within the overflow. A corresponding estimate of the upper limit of K for unresolved overturns is 0.4 × 10⁻³ m² s⁻¹. As shown in Fig. 6b, the highest values of $\overline{\mathit{\epsilon }}$ (0.6–1.1 × 10⁻⁷ W kg⁻¹) are found immediately downstream the crest of the sill (CTD sites 5–7), where the mixed wedge is located and hydraulic jumps are suggested, while the regions upstream and further downstream have smaller $\overline{\mathit{\epsilon }}$. The differences between upper and lower bounds of $\overline{\mathit{\epsilon }}$ at these locations are less than 0.1 × 10⁻⁷ W kg⁻¹.

The spatial pattern seen in the mean overflow dissipation rates is also found in the diapycnal diffusivity coefficient averaged over isothermal layers $\overline{K}$ within the overflow (Fig. 6c). The layer average is calculated as

$\overline{K}={\displaystyle \frac{1}{\mathrm{\Delta }\Theta }}{\displaystyle {\sum }_i{K}_i\delta \hspace{0.17em}{\Theta }_{\mathit{\iota }}},$(6)

where ΔΘ = 0.2°C is the temperature spacing, and K_i and δΘ_i are the diffusivity coefficient of an overturn and the change of the sorted Conservative Temperature values across the overturning region, respectively. The value of $\overline{K}$ was calculated within layers centered at isotherms ranging from 2.6° to 4.0°C. The values of $\overline{K}$ averaged over CTD sites 5–8 are 2.0–4.6 times of those averaged over all profiles within layers below the overflow interface (the 3.4°C isotherm), suggesting an enhancement in tracer diffusion immediately downstream of the mooring, although the averages are subject to large uncertainties due to the very small set of data (e.g., only four overturns are identified within the layer 2.6° ± 0.1°C). In fact, following the standard error propagation methods (Bevington and Robinson 1992), the standard error of the mean is estimated to be larger than the region averages. However, the means still lie well above the maximum value of diffusivity for unresolved overturns (gray dashed line in Fig. 6c), and using this background K only increase the results by at most 1 × 10⁻³ m² s⁻¹. The diffusivity coefficients averaged over CTD sites 5–8 range from 2.3 × 10⁻³ to 10.1 × 10⁻³ m² s⁻¹ within the overflow layer, and these values are 1–5 times of the bulk estimate in the deep Banda Sea, 2.2 × 10⁻³ m² s⁻¹ given by van Aken et al. (2009). This suggests that a mixing hotspot is located downstream of the hydraulically controlled sill in the Lifamatola Passage. We caution that the correlations between temperature gradients and local diffusivity coefficients can significantly affect the actual region-averaged dispersion. Using an approach presented by Wagner et al. (2019), we estimate that the spatial variation of temperature gradients can modulate the diffusivity coefficient averaged across the sill by a factor ranging from 0.7 to 1.6 for layers within the overflow. Our data do not have the resolution in time and space required to assess whether and in which layer this factor would be statistically significantly different from one.

b. Entrainment estimate

An important consequence of mixing around the interfacial region is the downward entrainment of mass into the overflow and the consequential increasing transport. For the 1.5-layer model configuration, the entrainment process can be represented by a velocity component across the interface separating the layers (Gerdes et al. 2002). For a continuously stratified overflow, identification of the boundary across which entrainment is measured is less clear. Here we choose the Θ = 3.4°C surface as a reasonable interface and define a lower-layer control volume that is bounded above by this surface and extends from the mooring section downstream to CTD site 7 or 8. This definition distinguishes the water that sinks to great depth in the Banda Sea from water that moves towards the Banda Sea but does not sink appreciably. Other choices of the interface are certainly possible. As detailed in appendix C, the downward entrainment across this interface can be estimated by again combining budgets for mass and potential enthalpy. The lack of direct velocity data at the downstream section requires an approximation, namely, that the potential enthalpy flux there is equal to the depth-average potential enthalpy, which we know, times the vertical integral of the horizontal velocity, which is not known. This simplification allows us to eliminate the unknown velocity integral and close the problem.

Use of the above procedure, with the depth-average potential enthalpy at the downstream section estimated from the vertical arithmetic mean and area-weighted mean, yields volume entrainment rates of 0.6–0.7 and 1.0–1.2 Sv, respectively [Eq. (C4)]. The gap between the two estimates indicates the uncertainty of the estimated entrainment due to lack of knowledge of the downstream velocity profile. The results are relatively insensitive to the diapycnal diffusivity along the interface, the choices of downstream sections, and the dissipation rate within the control volume. Summing the overflow transport (2.4 Sv) at the mooring station and the downstream entrainment yields a total volume flux of 3.0–3.6 Sv below Θ = 3.4°C (~1500 m at mooring location). The updated overflow transport is about 20%–40% larger than the 2.5 Sv below 1250 m estimated by van Aken et al. (2009), which includes some water above Θ = 3.4°C that may not be affected by entrainment as it moves downstream.

c. Ventilation, mixing, and turbulent heat flux in the Banda Sea

The cold deep branch of the ITF passes through the Lifamatola Passage, descends into the deep Banda Sea, and is thought to account for the deep ventilation in that basin (Wyrtki 1961; Gordon et al. 2003b; van Aken et al. 1988, 2009). A mean upwelling velocity of 50 m yr⁻¹, responsible for the deep-water renewal, was estimated by van Aken et al. (1988) using a deep transport estimate of 1 Sv below 1500 m divided by the surface area of the Banda Sea. This transport was later updated to 2.5 Sv by van Aken et al. (2009) leading to a proportionally larger upwelling of 125 m yr⁻¹. From the 72 WOCE profiles, the average depth of the Θ = 3.4°C surface in the Banda Sea lies at 1555 $\pm$ 84 m (with uncertainties represented by one standard derivation) and the corresponding area is (6.1 ± 0.1) × 10¹¹ m². Substituting the interface area at the average depth of the Θ = 3.4°C surface, our estimated mooring transport of 2.4 Sv, plus the entrainment of 0.6–1.2 Sv into Eq. (C5) yields an upwelling rate of ~160–190 m yr⁻¹, 25%–50% larger than the value without entrainment.

Van Aken et al. (1988, 2009) estimated the diapycnal diffusivity coefficient in the Banda Sea using the approach of Munk (1966) in which an advection–diffusion length scale is estimated from data. Their early estimate 9 × 10⁻⁴ m² s⁻¹, valid for an inflow of 1 Sv, was later raised to 2.2 × 10⁻³ m² s⁻¹ in accordance with their revised transport estimate of 2.5 Sv. Here we present a slightly different estimate based on the mass and thermodynamic budgets for a control volume that is bounded upstream by CTD site 8 and above by the Θ = 3.4°C surface, which is topographically confined within the Banda Sea. Note that any downward entrainment downstream of mooring station would eventually upwell across the same (Θ = 3.4°C) surface and would therefore not alter the thermodynamic budget. The control volume is shown schematically in Fig. C1 and the calculation is detailed in appendix B. The estimated diapycnal diffusivity coefficient over the Θ = 3.4°C surface is 1.6 × 10⁻³ ± 5 × 10⁻⁴ m² s⁻¹, with uncertainties (one standard error) mostly due to the unknown spatial dependence of the vertical gradient of the potential enthalpy at the upper boundary of the control volume. The contribution from the uncertainty of area is on the order of 10⁻⁵ m² s⁻¹. The potential enthalpy profiles calculated from the WOCE measurements were subsampled onto a 100-m grid. The vertical gradients were computed using first differences at midpoints of bins and then interpolated to desired levels.

Based on the same control volume, we estimate a downward turbulent heat flux of 9 W m⁻² at the Θ = 3.4°C surface [Eq. (C8)]. Van Aken et al. (2009) proposed that a downward heat flux of 28 W m⁻² is required to support upwelling of the cold deep branch of ITF from below 1250 m (~3.2°C) to about 1000 m (~5.0°C) in the Banda Sea, from whence it can enter the Indian Ocean. Their estimate assumes that no flow passes through the Lifamatola Passage between the two depths, although the same authors reported a ~1.4-Sv northward flux above 1250 m. Their estimate would be lower if the advective heat flux above the overflow were included (Gordon et al. 2010). Our estimate of turbulent heat flux avoids the assumption of zero advective heat fluxes above the overflow layer and instead provides an estimate of the heat flux required to maintain the stratification at the level of the Θ = 3.4°C isothermal surface in the Banda Sea.