Interpreting Patterns of Concentric Rings Within Small Buoyant River Plumes

remote sensing

Communication Interpreting Patterns of Concentric Rings within Small Buoyant River Plumes

George Marmorino * and Thomas Evans

Remote Sensing Division, U.S. Naval Research Laboratory, Washington, DC 20375, USA; [email protected] * Correspondence: [email protected]

Abstract: High-resolution imagery of small buoyant plumes often reveals an extensive pattern of concentric rings spreading outward from near the discharge point. Recent remote sensing studies of plumes from rivers ﬂowing into the Black Sea propose that such rings are internal waves, which form near a river mouth through an abrupt deceleration of the current, or hydraulic jump. The present study, using numerical simulations, presents an alternative viewpoint in which no hydraulic jump occurs and the rings are not internal waves, but derive instead through shear instability. These two differing dynamical views point to a clear need for additional ﬁeld studies that combine in-water measurements and time-sequential remote sensing imagery.

Keywords: high-resolution satellite imagery; buoyant plume; turbulent mixing; Mzymta River (Black Sea)

1. Introduction Rivers flowing into the sea add momentum and buoyancy that influence coastal Citation: Marmorino, G.; Evans, T. dynamics; and they carry sediment, nutrients, and contaminants, which impact the coastal Interpreting Patterns of Concentric ecosystem. Where laterally unconfined, the freshwater discharge spreads freely as a plume Rings within Small Buoyant River over the ambient ocean water. An important objective of studies of river plumes, as well Plumes. Remote Sens. 2021, 13, 1361. as their laboratory analogs, has been to quantify the mixing of the buoyant layer with the https://doi.org/10.3390/rs13071361 ambient fluid [1,2]. An intriguing feature often found in buoyant plumes from small rivers is a distinctive Academic Editor: Daniel F. Carlson pattern of nearly concentric bands, or “rings”, which appear to emanate from near the river mouth [3–5]. Figure1 shows an example for the Mzymta River, which is representative of Received: 21 January 2021 Accepted: 27 March 2021 several other small rivers discharging into the Black Sea. Plumes from these rivers show a Published: 2 April 2021 spacing between consecutive rings in the range of 30 to 60 m [4]. Similar ring-like patterns have been found to occur in a near-surface discharge of heating water from an electric

Publisher’s Note: MDPI stays neutral power-generating plant [6]. In the literature, such features have been referred to as internal with regard to jurisdictional claims in bands, fronts, and bores (e.g., reference [2]). published maps and institutional affil- Recent remote sensing studies of plumes like the Mzymta’s claim such features within iations. a plume are high-frequency internal waves that propagate away from a source region near the river mouth [4,5,7,8]. Note that this is distinct from internal waves that can form ahead of the plume, if the ambient water is stratified [9–11]. The explanation given for the origin of the internal waves is that as the river flows into the coastal sea it abruptly decelerates through a quasi-stationary hydraulic jump, and that the internal waves are generated Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. through an assumed oscillation of this jump. No explicit theoretical formulation, however, This article is an open access article is provided for this mechanism. It is also assumed these internal waves significantly distributed under the terms and increase mixing of the plume with underlying ambient sea water; but there is, as yet, no conditions of the Creative Commons evidence to confirm this. It is fair to say, therefore, that, while the phenomenon may be Attribution (CC BY) license (https:// widespread [4], the physical mechanisms and potential environmental consequences are creativecommons.org/licenses/by/ not well understood. 4.0/).

Remote Sens. 2021, 13, 1361. https://doi.org/10.3390/rs13071361 https://www.mdpi.com/journal/remotesensing Remote Sens. 2021, 13, 1361 2 of 13

Remote Sens. 2020, 11, x FOR PEER REVIEW 2 of 13

FigureFigure 1. WorldView-3 1. WorldView-3 image image of theof the Mzymta Mzymta River River plumeplume on 4 April April 2017, 2017, illustrating illustrating the thenear nearlyly concentric concentric bands, bands, or or “rings”, that are the focus of this study. Shown are band 3 data (518 to 586 nm wavelengths), geo-referenced to a UTM “rings”,map that projection are the (zone focus 37N; of this WGS-84), study. and Shown using aresquare band 1.4-m 3 data pixels. (518 An to independent 586 nm wavelengths), color-composite geo-referenced version appears to in a UTM map projectionreference [4]. (zone WorldView-3 37N; WGS-84), is a commercial and using Earth square observation 1.4-m pixels. satellite An owned independent by DigitalGlobe, color-composite Inc., Westminster, versionappears CO, in referenceUSA. [4 ]. WorldView-3 is a commercial Earth observation satellite owned by DigitalGlobe, Inc., Westminster, CO, USA.

TheThe objective objective of of thethe presentpresent report report is is to to demonstrate, demonstrate, using using a nu americal numerical model, model, that that ring-likering-like features features resembling resembling those those in in Figure Figure 1 1can can be be simulated, simulated, but butthat thatsuch such rings ringsare are notnot internal internal waves, waves, nornor dodo they derive derive from from an an abrupt abrupt transition transition in the in plume the plume velocity velocity ﬁeld.field. The The results results areare thusthus incompatible with with the the hypothesis hypothesis stated stated above, above, and this and has this has implicationsimplications for for interpreting interpreting not not only only remotelyremotely sensedsensed imagery of of the the river river plume, plume, but but also also measurements of water-column hydrography. These differing dynamical views measurements of water-column hydrography. These differing dynamical views point to a point to a clear need for additional studies that combine in-water measurements and cleartime-sequential need for additional remote sensing studies imagery. that combine in-water measurements and time-sequential remote sensing imagery.

Remote Sens. 2021, 13, 1361 3 of 13

2. Materials and Methods 2.1. Background The prototype for this study is the plume formed by the Mzymta River. The Mzymta River, which flows through the western edge of the Caucasus mountains, discharges into the eastern Black Sea about 25 km southeast of the city of Sochi, Russia. There the river is characterized by relatively high speed, but small vertical extent and volume. The Mzymta discharge rate is, however, highly variable with monthly means ranging from 20 to 120 m3 s−1 [5]. During periods of significant discharge, river speed (either measured directly or inferred from discharge gauge data) is in the range of about 1 to 2 m s−1 [5,12]; water depth at the river mouth is nominally 1 to 1.5 m; and river width near the mouth averages about 40 m. The small length scale but large velocity of the Mzymta discharge gives rise to a near-field supercritical flow [4], meaning that the plume water velocity exceeds the speed of a linear internal wave phase speed. Within the near field, the plume is dominated by the inertia of the discharge and behaves much like a turbulent buoyant jet. Previous studies suggest that a small river plume will transition from supercritical near field to subcritical far field gradually and over a significant distance (on the order of a kilometer) from the river mouth [1]. For the Mzymta, the near field can extend as far as 1 to 2 km from the river mouth, beyond which the dynamics are dominated by wind stress and the effect of Earth’s rotation [12]. The Mzymta carries out to sea large amounts of silt and organic suspended matter, resulting in a turbid plume that is easily detectible in ocean color satellite imagery. The WorldView-3 satellite image shown in Figure1 may be the best highest-resolution data of the plume that exists currently. It was acquired in early spring (4 April 2017), during a discharge of approximately 60 m3 s−1 (Figure 4 in reference [4]). Shown are data from band 3 (green: 518 to 586 nm wavelengths), as it was found to have the greatest dynamic range, or signal contrast, and thus best highlights key features of the plume. The discharge can be seen to have an initial width of about 25 m at the river mouth. The plume extends 780 m from the mouth (in the direction normal to shore), and the signal level decays with distance with an e-folding scale of about 300 m. A gradual turn of the plume toward the north may indicate a transition to far-field dynamics. The seaward-most extent of the plume is marked by a distinct front and narrow scum line. The plume front displays a lobe-cleft structure that is characteristic of gravity current-like geophysical flows [13,14]. Lobes protrude seaward from the front, while clefts are indentations in the frontal edge. In Figure1, successive lobes and clefts are spaced most commonly in the range of 40 to 80 m. On the plume side of the front, displaced by about 15 m, lies an approximately parallel, or “companion”, dark band. This band may indicate an upwelling of relatively clear water in the rear of the frontal head, where the plume protrudes downward behind the surface front (for example, reference [15]). Rings in Figure1 appear relatively bright, suggesting they are regions where the turbid plume water is locally deeper than the water between rings. Individual rings can extend coherently over a wide azimuth; some appear to either split or merge with others. Over the bulk of the plume, the ring spacing averages 39 m (±9 m). Rings are more finely spaced, however, in the plume area northwest of the river mouth. The rings appear to emanate from a source region located 100 to 200 m from the river mouth and extending 25 to 100 m alongshore [4]. There is, as yet, no WorldView stereo imagery of the Mzymta plume from which one might deduce currents (for example, [16]). But pairs of images from Sentinel-2 and Landsat 8 satellite overpasses that were closely spaced in time have been used by Osadchiev [4] to deduce information about the ring features. Under the assumption that the features represent internal waves, he reports wavelengths in the range of 30 to 60 m, phase speeds of 0.45–0.65 m s−1, and wave periods (that is, wavelength divided by phase speed) of 65 to 90 s [4,17]. We posit that what was actually measured was the advection speed of the features; hence the period should be interpreted instead as the time interval between generation of each successive ring (c.f., Section 3.2). Remote Sens. 2021, 13, 1361 4 of 13

In-water data for the Mzymta plume are sparse. The most relevant hydrographic measurements were made by Osadchiev [4], on a day (29 May 2014) when a LANDSAT-8 pass revealed plume structure similar to that of Figure1. The measurements are comprised of eight “tow-yo” profiles made in water depths of 7 to 8 m and over a horizontal distance of 50 m, beginning about 150 m from the river mouth; hence, the data were collected within the apparent source region of the ring features. While six of the profiles showed the expected structure of a plume confined to the upper meter or so of the water column, profiles 1 and 3 were anomalous in that they showed low-salinity plume water penetrating downward over nearly the entire depth of the water column. These two localized, downward penetrations of plume water were interpreted as evidence of a hydraulic jump.

2.2. Model The plume is modeled in three-dimensions using non-hydrostatic and non-rotational dynamics. We use the OpenFOAM solver TwoLiquidMixingFoam, run as a large-eddy simulation. OpenFOAM is a free, open-source library of source code for solvers and utilities [18]. A large-eddy simulation resolves spatial scales from the domain size down to a filter size ∆; effects of smaller scales must be modeled. In this study, turbulent effects on scales smaller than ∆ are modeled using a k-equation eddy viscosity model. The model solves a turbulence kinetic energy equation, yielding the sub-grid-scale turbulent kinetic 1/2 energy, ksgs. A sub-grid-scale eddy viscosity, υsgs, is then computed as υsgs = Ck (ksgs) ∆, where Ck is a model constant set to a value of 0.094. The dissipation term in the turbulence 3/2 kinetic energy equation is given by e = CE k /∆, where CE = 1.048 is another model constant. The quantity e thus provides an estimate of the dissipation rate of turbulent kinetic energy; hence, of vertical mixing. As the focus of the model is the near field of the river plume, neglect of Earth’s rotation is justified. Model geometry is shown in Figure2. A notch at y = 0, x < 0 represents the river inflow, which has a width W0 = 26 m and depth H0 =1.6 m. For 0 < x < 137 m, the bottom depth is specified as z = −1.6 − 1.5 × 10−6 (x + 25)3; for x > 137 m the bottom is fixed at 8-m depth; there is no depth variation in the alongshore (y) direction. In reality, the ocean floor continues to deepen offshore, reaching 100 m within 1 to 2 km of the river mouth. The bottom boundary is modeled as no slip, using a wall function to adjust the bottom-layer eddy viscosity. The sea surface is assumed to be a rigid lid. Boundary conditions of slip, zero-gradient are used on the two across-shore boundaries, and a zero-gradient condition is applied at the offshore boundary. Grid resolution varies both horizontally and vertically (Figure2). Near the inflow the horizontal resolution (∆x, ∆y) is 1 m; it becomes progressively coarser with increasing x. In the vertical direction, z, resolution is greatest near the sea surface, where 12 layers cover 1.6 m; another 24 layers, each successively 5.57% larger, cover the remaining 6.4 m; the thickest layer is 46 cm. The filter length scale ∆ is taken as proportional to the cube root 1/3 of the cell volume: ∆ = C∆ (∆x ∆y ∆z) , where C∆ is a constant. In this work, we used C∆ = 0.2, as much larger values were found to damp the formation of rings, while smaller values resulted in an incoherent flow field. The model time step ∆t was varied to limit the Courant number at 0.4; the mean value of ∆t was 0.14 s. A model run typically ends when the plume intersects an across-shore boundary. −3 The river, or inflow, water has a density ρ1 = 1000 kg m , and the ambient sea water a −3 density ρ2 = 1017 kg m [17,19,20]; thus, the inflow reduced gravity g00 = g (ρ2 − ρ1)/ρ2 = 0.1640 m s−2, where g = 9.81 m s−2 is the gravitational acceleration constant. In portraying

the modeled density field in the results section, we use a normalized density ρ* = (ρ − ρ1)/(ρ2 − ρ1), so that a value of zero corresponds to river water, and a value of one to sea −1 water. The base-case simulation uses a constant inflow velocity U0 = 1.5 m s . This value was chosen as it falls in the middle of the range of Mzymta River flow speeds (Section 2.1). 3 −1 The inflow freshwater discharge rate is thus U0 W0 H0 = 62.4 m s , which is comparable to the discharge at the time of Figure1. The inflow conditions can be characterized in terms Remote Sens. 2021, 13, 1361 5 of 13

of an inﬂow Froude number, Fri = U0/sqrt (g00 H0) = 2.93. A value of Fri > 1 indicates Remote Sens. 2020, 11, x FOR PEER REVIEW supercritical inﬂow. 5 of 13

Figure 2. ModelFigure geometry. 2. Model (a) geometry. Plan (x-y) view,(a) Plan showing (x-y) view, spatial showing variation spatial in horizontalvariation in resolution horizontal (∆ resolutionx, ∆y). A 25 × 26-m notch in the shoreline(∆x, ∆y). represents A 25 × 26-m the notch river inﬂow.in the shorelin (b) Verticale represents (x-z) section, the river showing inflow. offshore (b) Vertical bathymetry; (x-z) section, red lines indicate vertical resolution,showing which offshore varies bathymetry; from 0.13 m nearred lines the surfaceindicate to ve 0.46rtical m resolution, at the bottom. which varies from 0.13 m near the surface to 0.46 m at the bottom.

3. Results Grid resolution3.1. Spatial varies Structure both horizontally and vertically (Figure 2). Near the inflow the horizontal resolutionRepresentative (∆x, ∆y) model is 1 m; results it becomes are shown progressively in Figure 3coarser, corresponding with increasing to a time of 1630 s x. In the vertical(27.2 direction, min) after thez, resolution start of the is simulation. greatest near By the this sea time surface, the plume where has 12 expanded layers to a size of cover 1.6 m; 710another by 750 24 mlayers, in the each across- successively and along-shore 5.57% larger, directions. cover the Ring-like remaining features 6.4 m; appear in the the thickest surfacelayer is density46 cm. ﬁeldThe filter as narrow length bands scale of ∆ relativelyis taken as buoyant proportional ﬂuid (Figure to the3 a);cube in the surface root of the cellvelocity volume: as bands∆ = C∆ of(∆x locally ∆y ∆z) higher1/3, where speed C∆ is (Figure a constant.3b); and In this as bandswork, we of increased used plume C∆ = 0.2, as muchdepth largerh (Figure values3c). were The modelfound to rings damp resemble the formation those in of satellite rings, while imagery smaller of the Mzymta values resultedplume. in an Rings incoherent appear flow over field. nearly The allmodel azimuth time step directions, ∆t was and,varied as to will limit be the demonstrated Courant numberbelow, at move0.4; the continuously mean value of offshore ∆t was 0.14 and s. toward A model the run plume typically front. ends The when rings appear to the plume intersectsemanate an from across-shore a source region,boundary. approximately 75 to 175 m from the model river mouth The river,and or having inflow, an water alongshore has a density extent 𝜌 of1= 1000 about kg 80 m m;−3, and this the is comparable ambient sea to water estimates a derived density 𝜌2= 1017from kg satellite m−3 [17,19,20]; imagery (Sectionthus, the 2.1 inflow). The reduced spacing gravity between g0 successive′ = g (𝜌2 - 𝜌 rings1)/𝜌2 = in the model 0.1640 m s−2,is where about g 37 = m9.81 (± m9 m),s−2 is representing the gravitational an average acceleration over a constant. central, 20 In◦-azimuth portraying sector. This is

the modeledcomparable density field to thein the mean results ring spacingsection,of we 39 use m in a thenormalized satellite image density in Figure𝜌⁎ = (𝜌1. - Model rings 𝜌1)/(𝜌2 - 𝜌1), areso that more a closelyvalue of spaced zero corresponds well off the plumeto river centerline, water, and similar a value to Figureof one 1to. sea water. The base-case simulation uses a constant inflow velocity U0 = 1.5 m s−1. This value was chosen as it falls in the middle of the range of Mzymta River flow speeds (Section 2.1). The inflow freshwater discharge rate is thus U0 W0 H0 = 62.4 m3 s−1, which is comparable to the discharge at the time of Figure 1. The inflow conditions can be characterized in terms of an inflow Froude number, Fri = U0/sqrt (g0′ H0) = 2.93. A value of Fri > 1 indicates supercritical inflow.

Remote Sens. 2020, 11, x FOR PEER REVIEW 6 of 13

3. Results 3.1. Spatial Structure Representative model results are shown in Figure 3, corresponding to a time of 1630 s (27.2 min) after the start of the simulation. By this time the plume has expanded to a size of 710 by 750 m in the across- and along-shore directions. Ring-like features appear in the surface density field as narrow bands of relatively buoyant fluid (Figure 3a); in the surface velocity as bands of locally higher speed (Figure 3b); and as bands of increased plume depth h (Figure 3c). The model rings resemble those in satellite imagery of the Mzymta plume. Rings appear over nearly all azimuth directions, and, as will be demonstrated below, move continuously offshore and toward the plume front. The rings appear to emanate from a source region, approximately 75 to 175 m from the model river mouth and having an alongshore extent of about 80 m; this is comparable to estimates derived from satellite imagery (Section 2.1). The spacing between successive rings in the model is Remote Sens. 2021, 13, 1361 about 37 m (± 9 m), representing an average over a central, 20°-azimuth sector. This6 ofis 13 comparable to the mean ring spacing of 39 m in the satellite image in Figure 1. Model rings are more closely spaced well off the plume centerline, similar to Figure 1.

FigureFigure 3. 3. RepresentativeRepresentative model model results, results, shown shown at att =t 1630= 1630 s. (a s.) surface (a) surface normalized normalized density, density, 𝜌⁎; (b) ρsurface*;(b) surface velocity velocity mag- magnitude,nitude, |U ||;U (|;c) plume (c) plume depth depth h, definedh, deﬁned as where as where the the normalized normalized density density 𝜌⁎ ρ=* 0.9;= 0.9; (d) ( dFroude) Froude number, number, Fr; Fr; (e) ( eRichardson) Richardson number,number, Ri; Ri; (f ()f kinetic) kinetic energy energy dissipation dissipation raterate e𝜖,, averagedaveraged over plume plume depth depth hh; ;discontinuities discontinuities in in 𝜖 eoccuroccur at at jumps jumps inin the the modelmodel grid grid spacing. spacing.

OtherOther characteristicscharacteristics of of the plume plume are are also also in in reasonable reasonable agreement. agreement. The The modeled modeled fieldsfields ofof surface surface density density and and speed speed decay decay approximately approximately exponentially exponentially with with radial radial distance dis- fromtance mouth,from mouth, with anwith e-folding an e-folding scale scale of about of about 325 m;325 this m; isthis comparable is comparable to the to the turbidity tur- decaybidity scaledecay of scale the Mzymtaof the Mzymta plume (Sectionplume (S 2.1ection). The 2.1). model The plumemodel frontplume has front a lobe-cleft has a structure (dominant spacing of 40 to 70 m) that resembles that in Figure1. The model result for plume depth (Figure3c) shows the expected frontal head structure: a narrow, leading band where with the plume is deepest; and a narrow, trailing area that is (persistently) shallow and resembles the companion dark band in the satellite data (Figure1). One apparent difference between the simulation and imagery is that the model inflow begins spreading laterally almost immediately, while the imagery shows little initial spreading. This is in large part because the density and velocity plots in Figure3 show surface fields, while the imagery is showing depth-integrated backscattered light. When model results are integrated over a plausible optical penetration depth of 1 m, the spreading is much reduced so that the shape of the discharge resembles that of the depth field in Figure3c. The dynamical character of the plume can be revealed by examining two parameters: the Froude number Fr = U/(g’ h)1/2, where and U and h are local values of plume depth and velocity, respectively, g’ is the local reduced gravity; and the Richardson number, Ri = −(g/ρ) ∂ρ/∂z/(∂U/∂z)2, evaluated at the depth where |∂ρ/∂z| is maximum. The value of Fr can be seen (Figure3d) to exceed one almost everywhere, indicating that at this evolutionary stage the entire plume is supercritical and can thus be considered a near-field Remote Sens. 2021, 13, 1361 7 of 13

buoyant flow. In particular, there is no evidence of an abrupt transition to sub-critical flow close to the river mouth; this is distinctly different from the view of references [4,5,7,8]. And because the plume is supercritical nothing that occurs downstream can affect the creation of rings that is occurring upstream; hence, modeled rings continue to form and evolve in a similar way throughout a simulation. In the plot of Richardson number (Figure3e ), shades of blue correspond to values of Ri less than the critical value of 0.25. Values of Ri < 0.25 indicate regions of the flow field that are susceptible to Kelvin–Helmholtz shear instability (KHI). Ri falls significantly below its critical value over the innermost part of the plume (see later Section 3.3). The spatial distribution of kinetic energy dissipation rate e (Figure3f) provides an indication of where turbulent mixing is occurring within the plume. Mixing is thus predicted to be highest (nearly 10−3 m2 s−3) in the source region of the rings, where values of Ri are smallest, and to decay strongly with increasing radial distance. This is consistent with mixing in river plumes being dominated by stratified shear-flow instabilities [1]. Rings have higher dissipation rates than do plume areas between rings. This provides some evidence that ring features enhance vertical mixing between the plume and the underlying sea water. Figure3c shows that, away from the source region, the plume thickens with radial distance, the rate of increase being approximately constant over radial distances of 200 to 550 m. This thickening is evidence of turbulent entrainment of lower-layer fluid into the plume.

3.2. Evolving Nature of the Plume Rings produced by the model are not internal waves. This can be demonstrated by tracking virtual surface particles chosen to lie within a particular ring at some initial time, as shown in Figure4a. Subsequent particle positions were determined by computing their Lagrangian displacements, using the surface velocity field at each model time step. An animation of the evolving density field (Figure S1) shows the particle positions over a time period of 460 s, the final time being shown in Figure4b. Over time, the chosen particles advect radially and, as a group, stretch-out azimuthally. But, throughout, the particles remain within the same ring features. If those features were internal waves, they would propagate through the plume and leave the particles behind them. That does not occur; thus, the rings are material features. Rings move continuously offshore and toward the plume front. This is illustrated through a Hovmöller diagram (Figure5) generated from the centerline surface density. The diagram shows a pattern of blue-white streaks (that is, locally more buoyant water) inclined from the lower-left toward the upper-right, and which represents the offshore movement of the rings. Rings move at 1 m s−1 near their source region (at x ~ 100 m), but slow with increasing offshore distance until reaching the plume front, which itself advances into ambient water at a steady radial speed of 0.27 m s−1. The rings thus advance faster than the plume front; but they never move beyond the plume front, which is consistent with Figure1 and other satellite imagery [4,5]. The range of predicted ring speeds is consistent with that deduced from satellite imagery by Osadchiev (Section 2.1). Inshore of x~300 m, the period between successive rings in Figure5 is about 60 s. While this value is comparable to internal wave periods reported for the Mzymta plume [4,17], our interpretation is as an estimate of the time interval at which rings are generated within the source region. Note that the plume translation speed is often described in terms of the long-wave internal wave phase speed c = (g’ h)1/2, which can be calculated using the Fr and U data in Figure3. Near the front, c = 0.28 ± 0.02 m s−1, which is not significantly different from the speed of frontal advance derived from Figure5. The continuous offshore advance of the plume would, of course, in the real world be arrested by far-field dynamics (Section 2.1). Remote Sens. 2021, 13, 1361 8 of 13

Remote Sens. 2020, 11, x FOR PEER REVIEW 8 of 13

Figure 4.FigureAdvection 4. Advection of surface of surface water water particles particles (black (black dots) dots) chosen chosen to to lie lie initially initially within a a ring ring feature. feature. (a) (Initiala) Initial density density field, field, t = 1280 s; (b) 7.7-min later, t = 1740 s. The ten initial particle positions were determined from local density minima t = 1280 s;within (b) 7.7-min a specified later, subsett = 1740area; s.subsequent The ten initialparticle particle position positionss were determined were determined by computing from their local Lagrangian density displace- minima within a specifiedments, subset using area; the subsequent evolving surface particle velocity positions field at were each determinedmodel time step by ( computing∆t ~ 0.14 s). For their clarity, Lagrangian the particles displacements, are shown using the evolvinglarger surface than the velocity underlying field model at each cell dimension. model time An step animation (∆t ~ 0.14(Figure s). S1) For shows clarity, particle the particlespositions every are shown 10 s over larger the than the time period of (a) to (b). underlying model cell dimension. An animation (Figure S1) shows particle positions every 10 s over the time period of (a) to (b).Remote Sens. 2020, 11, x FOR PEER REVIEWRings move continuously offshore and toward the plume front. This is illustrated9 of 13 through a Hovmöller diagram (Figure 5) generated from the centerline surface density. The diagram shows a pattern of blue-white streaks (that is, locally more buoyant water) inclined from the lower-left toward the upper-right, and which represents the offshore movement of the rings. Rings move at 1 m s−1 near their source region (at x ~ 100 m), but slow with increasing offshore distance until reaching the plume front, which itself advances into ambient water at a steady radial speed of 0.27 m s−1. The rings thus advance faster than the plume front; but they never move beyond the plume front, which is consistent with Figure 1 and other satellite imagery [4,5]. The range of predicted ring speeds is consistent with that deduced from satellite imagery by Osadchiev (Section 2.1). Inshore of x~300 m, the period between successive rings in Figure 5 is about 60 s. While this value is comparable to internal wave periods reported for the Mzymta plume [4,17], our interpretation is as an estimate of the time interval at which rings are generated within the source region. Note that the plume translation speed is often described in terms of the long-wave internal wave phase speed c = (g’ h) 1/2, which can be calculated using the Fr and U data in Figure 3. Near the front, c = 0.28 ± 0.02 m s−1, which is not significantly different from the speed of frontal advance derived from Figure 5. The continuous offshore advance of the plume would, of course, in the real world be arrested by far-field dynamics (Section 2.1).

Figure 5. A Hovmöller (x-t) diagram showing the evolving centerline surface density. Rings, indicated by blue-white Figure 5. A Hovmöller (x-t) diagram showing the evolving centerline surface density. Rings, indicated by blue-white streaks, move at 1 m s−1 near their source region, but slow with increasing offshore distance. The plume front propagates −1 streaks, moveinto ambient at 1 m water s near (red theirarea) at source a steady region, radial speed but slow of 0.27 with m s− increasing1. The time period offshore shown distance. matches Thethat in plume Figure front S1. propagates into ambient water (red area) at a steady radial speed of 0.27 m s−1. The time period shown matches that in Figure S1.

Viewed in a centerline vertical plane (Figure 6), the plume can be seen to lift off the bottomViewed at x~40 in a m, centerline then to develop vertical shear plane instabilities, (Figure as6 expected), the plume from Figure can be3e. The seen in- to lift off the bottomstabilities, at which x~40 have m, horizontal then to developwavelengths shear in the instabilities, range of 10 to as12 m, expected result in vertical from Figure 3e. oscillations of the plume interface that are highly variable in both space and time (c.f., The instabilities, which have horizontal wavelengths in the range of 10 to 12 m, result Figure 3c). In particular, localized penetrations of plume water to very near the seabed occur (Figure 6c, arrow) that are qualitatively similar to Osadchiev’s tow-yo data (Section 2.1). Beyond this region, from x~150 m, there emerge periodically localized downward bulges in the plume layer (Figure 6, features 1–3); these are incipient ring features.

Remote Sens. 2021, 13, 1361 9 of 13

in vertical oscillations of the plume interface that are highly variable in both space and time (c.f., Figure3c). In particular, localized penetrations of plume water to very near the seabed occur (Figure6c, arrow) that are qualitatively similar to Osadchiev’s tow-yo data (Section 2.1). Beyond this region, from x~150 m, there emerge periodically localized Remote Sens. 2020, 11, x FOR PEER REVIEWdownward bulges in the plume layer (Figure6, features 1–3); these are incipient10 of ring 13 features.

FigureFigure 6. 6.Centerline Centerline ( x(x-z- )z section) section of of normalized normalized density,density, shownshown atat 10-s10-s intervals: ( (aa)) tt=1610= 1610 s; s;(b) ( bt=1620) t = 1620 s; (c) s; t=1630 (c) t = s; 1630 (d) s; (dt)=1640t = 1640 s; (e s;) (te=1650) t = 1650s; (f) s;t=1660 (f) t = s. 1660 Note s. that Note only that the only plume’s the plume’s inner portion inner portion (x < 300 (x m)< 300 is being m) is shown. being shown. Features Features 1–3 cor- 1–3 correspondrespond to to evolving evolving ring ring feat features.ures. Arrow Arrow in panel in panel c points c points to a to localize a localized,d, near-bottom near-bottom penetration penetration of plume of plume water. water.

3.3. Sensitivity to Inflow Speed and Geometry 3.3. Sensitivity to Inflow Speed and Geometry Sensitivity of the ring features to the model inflow velocity U0 is illustrated in Figure Sensitivity of the ring features to the model inflow velocity U is illustrated in Figure7 , 7, for U0 values of 1, 1.5, and 2 m s−1. Compared with the base case0 (middle panel), rings −1 forappearU0 values more disorganized of 1, 1.5, and for 2 m larger s . inflow Compared (right); with but the for base smaller case inflow (middle (left) panel), ring fea- rings appeartures emerge more disorganized from the source for larger region inflow having (right); an increased but for smaller level of inflow azimuthal (left) ringcoherence, features emergeand thus from more the closely source resembling region having the anreal increased plume (Figure level of 1). azimuthal This behavior coherence, is likely and the thus more closely resembling the real plume (Figure1). This behavior is likely the consequence consequence of higher levels of shear-induced turbulence with increasing U0; and is con- ofsistent higher with levels the ofincreasingly shear-induced larger turbulence areas of es withpecially increasing low RiU values0; and (< is consistent0.1), as the withlower the increasinglythe value of Ri larger is below areas its of critical especially value low of 0.25, Ri values the higher (<0.1), is asthe the growth lower rate the of value KHI. ofAlso Ri is belowincreasing its critical with inflow value ofspeed 0.25, are the the higher spacing is the of growthrings and rate their ofKHI. period Also (that increasing is, time be- with inflowtween creation speed are of thesuccessive spacing rings): of rings the and ring their spacings period are (that 29±6 is, m, time 37±9 between m, 39±9 creationm, while of successivetheir periods rings): are about the ring 45, spacings60, and 70 are s. An 29 ±explanation6 m, 37 ± for9 m, those 39 ± values9 m, while eludes their us. Addi- periods aretional about cases 45, (not 60, shown) and 70 were s. An done explanation for still lower for those and valueshigher eludesinflow speeds. us. Additional For U0 = cases0.5 −1 (notm s−1 shown), or a discharge were done rate forof only still 21 lower m3 s− and1, there higher are no inflow rings speeds.at all; for For U0 =U 2.50 = m 0.5 s−1 m, which s , or 3 −1 −1 aat discharge 104 m3 s−1 rate is near ofonly the upper 21 m limits , of there the areMzymta no rings discharge at all; rate, for U the0 = plume 2.5 m is s further, which elongated, more generally turbulent, and without coherent rings. The model thus produces rings only over a middle range of discharge rates. This is qualitatively consistent with the real Mzymta River plume, as ring-like patterns are not apparent if the discharge rate is too low (less than about 50 m3 s−1), and often absent when the discharge rate is too high [4].

Remote Sens. 2021, 13, 1361 10 of 13

at 104 m3 s−1 is near the upper limit of the Mzymta discharge rate, the plume is further elongated, more generally turbulent, and without coherent rings. The model thus produces rings only over a middle range of discharge rates. This is qualitatively consistent with the real Mzymta River plume, as ring-like patterns are not apparent if the discharge rate is too 3 −1 Remote Sens. 2020, 11, x FOR PEER REVIEWlow (less than about 50 m s ), and often absent when the discharge rate is too high11 of 13 [4 ].

Figure 7. Variation of plume morphology with inflow velocity U0. (a) U0 = 1.0 m −s−11; (b) U0 = 1.5 m s−1;1 (c) U0 = 2.0 m s−1.− 1 Figure 7. Variation of plume morphology with inflow velocity U0.(a) U0 = 1.0 m s ;(b) U0 = 1.5 m s ;(c) U0 = 2.0 m s . Corresponding values of river discharge rate are 42, 62, and 83 m3 s−1; values of inlet Froude number are 2.0, 2.9, and 3.9. Corresponding values of river discharge rate are 42, 62, and 83 m3 s−1; values of inlet Froude number are 2.0, 2.9, and 3.9. Curves indicate values of Richardson number of 0.1 and 0.25; panel (a) has no 0.1 or lower values. Model time is 1630 s for Curveseach case. indicate values of Richardson number of 0.1 and 0.25; panel (a) has no 0.1 or lower values. Model time is 1630 s for each case. Runs were also done using different eddy-viscosity models, different model grids, and Runsdifferent were river also geometries done using and different nearshore eddy-viscosity bathymetry. All models, these runs different produced model rings grids, and different river geometries and nearshore bathymetry. All these runs produced rings for for U0 = 1.5 m s−1. An example, in which a flat ocean bottom was used, is shown in Figure −1 U8a.0 = In 1.5 this m case, s . Anthe lift-off example, of the in whichplume aoccu flatrs ocean uniformly bottom across was used,the width is shown of the ininflow,Figure at8 a . In this case, the lift-off of the plume occurs uniformly across the width of the inflow, at x = x=0; similar to a spreading, supercritical (Fri = 2.1) plume used in laboratory studies [2]. 0;As similar compared to a spreading,with the sloping supercritical bathymetry (Fri = 2.1)result plume (Figure used 8b), in the laboratory area with studies very [low2]. As comparedvalues of Richardson with the sloping number bathymetry (Ri < 0.1) resultis reduced, (Figure indicating8b), the areaa reduction with very in the low initial values oflevel Richardson of turbulence, number and(Ri ring < features 0.1) is reduced, emerge from indicating the source a reduction region having in the an initial increased level of turbulence,level of azimuthal and ring coherence. features emerge from the source region having an increased level of azimuthalOverall coherence. model results suggest the physical mechanism responsible for creating rings is linkedOverall to spatially model results developing suggest shear the physicalinstabilities, mechanism arising after responsible the point for of creatinglift-off. De- rings isvelopment linked to spatiallyof KHI in developingan area of the shear plume instabilities, where Ri ~ arising 0.1 appears after theto be point consistent of lift-off. with Devel- the opmentlaboratory of KHIstudy in by an Yuan area ofand the Horner-Devine plume where [2]. Ri ~They 0.1appears found that to bethe consistent vorticity associ- with the laboratoryated with KHI study becomes by Yuan aggregated and Horner-Devine into larger-scale [2]. They structures, found that which the stretch vorticity and associated main- withtain coherent KHI becomes shapes aggregated as the plume into spreads larger-scale laterally; structures, but different which from stretch the model and results, maintain coherentthe laboratory shapes structures as the plume were spreads linear in laterally; shape, advanced but different more from slowly the modelthan the results, plume the laboratoryfront, and structureswere observed were linearonly far in behind shape, advancedthe plume more front. slowly Just how than coherent the plume structures front, and wereemerge observed from a onlyturbulent far behind low-Ri the source plume region front. in Justthe model how coherent plume is structures not clear; emergebut a close from aexamination turbulent low-Ri of model source animations region in thedoes model reveal plume behaviors is not clear;that are but not a close reported examination in the of modellow-Reynolds-number animations does laboratory reveal behaviors study. thatThese are include not reported pairing in of the KH low-Reynolds-number instabilities, as well laboratoryas a joining study. up, Theseor merging, include of pairing initial ofring KH fragments, instabilities, both as effects well as tending a joining to up, increase or merging,length of initial scales ring with fragments, increasing both radial effects distance tending. Such to effects, increase including length scales their withsensitivity increasing to radialthe model distance. parameterization Such effects, of including turbulence, their will sensitivity need to be to investigated the model parameterizationfurther. of turbulence, will need to be investigated further.

Remote Sens. 2021, 13, 1361 11 of 13

Remote Sens. 2020, 11, x FOR PEER REVIEW 12 of 13

FigureFigure 8. 8.Variation Variation of plumeof plume morphology morphology with with nearshore nearshore bathymetry: bathymetry: (a) flat (a ocean) flat ocean bottom; bottom; and (b )and base (b case) base with case nearshore with nearshore slope. Curves indicate values of Richardson number of 0.1. Model time is 1630 s for each case. slope. Curves indicate values of Richardson number of 0.1. Model time is 1630 s for each case. 4. Conclusions 4. ConclusionsThis study has attempted to achieve a clearer understanding of patterns of concentric ringsThis studyoccurring has attemptedwithin small to achievebuoyant a river clearer plumes, understanding as has been of patternsrevealed ofrecently concentric in ringsocean occurring color satellite within imagery. small buoyant The Mzymta river plumes, River is as used has beenas a revealedprototype recently for numerical in ocean colorsimulations satellite of imagery. the plume The dynamics. Mzymta RiverThe focus is used is on as athe prototype near field for of numerical the plume, simulations which is ofcharacterized the plume dynamics. by supercritical The focus flow is onand the enhanced near field mixing of the plume,of the plume which iswith characterized ambient bywaters. supercritical The main flow result and is enhancedthat the model mixing is able of the to plumeproduce with ring-like ambient features waters. resembling The main resultthose isin that the theMzymta model plume. is able toThe produce rings are ring-like material features features resembling that, as compared those in the with Mzymta the plume.bulk plume The ringswater, are are material deeper, featuresmore buoyant, that, as and compared have more with forward the bulk momentum. plume water, In the are deeper,model, morethe origin buoyant, of the and rings have appears more forward linked to momentum. shear instability In the occurring model, the close origin to ofthe the ringsinflow appears source, linked which to is shearas suggested instability by occurringsome previous close towork the [2]. inflow This source, is distinctly which dif- is as suggestedferent from by the some internal previous wave work hypothesis [2]. This proposed is distinctly in references different from[4,5,7,8]. the Determining internal wave hypothesiswhich dynamical proposed viewpoint in references is closer [4 ,to5,7 the,8]. truth Determining will require which additional dynamical study. viewpoint A mod- is closereling issue to the is truth the sensitivity will require of the additional ring featur study.es to Athe modeling mixing scheme issue is parameterization. the sensitivity of In the ringthe field, features an experiment to the mixing to test scheme whether parameterization. a ring is a material In the feature field, or an a experimentpropagating to one test whethermight be a ringdone is by a materialusing a featureboat, or or perhaps a propagating an aerial one drone, might to be inject done a by fluorescent using a boat, wa- or perhapster-tracing an dye aerial into drone, part toof injecta ring, a thus fluorescent creating water-tracing an artificial ocean dye into color part signal of a [21]. ring, The thus creatingsubsequent an artificial spatial and ocean temporal color signal evolution [21]. Theof the subsequent dye can then spatial be andcaptured temporal in an evolution aerial ofsurvey the dye [8]. can If the then dye be capturedand (turbid) in an ring aerial mo surveyve together—similar [8]. If the dye to and our (turbid) Figure ring4—then move together—similarrings are material tofeatures; our Figure if the4—then ring propagates rings are material away from features; the dye, if the then ring rings propagates are in- awayternal from waves. the Use dye, might then ringsalso be are made internal of a waves. horizontal Use acoustic might also Doppler be made current of a horizontalprofiler, acousticdeployed Doppler either from current a boat profiler, [22] to deployedmeasure the either radial from motion a boat of [turbid22] to measurerings through the radial the motionplume, ofor turbidfrom a ringsnear-shore through station the plume,[23] to examine or from velocity a near-shore structure station within [23] the to examinelift-off velocityand source structure regions. within the lift-off and source regions.

Supplementary Materials: The following are available online at www.mdpi.com/xxx/s1, Figure S1 Supplementary Materials: The following are available online at https://www.mdpi.com/article/10 (Video S1): Animation of surface density field, including particle tracking as in Figure 4. .3390/rs13071361/s1, Figure S1 (Video S1): Animation of surface density ﬁeld, including particle tracking as in Figure4. Author Contributions: G.M. conceived the idea for this study and drafted the manuscript. T.E. did all the numerical work. All authors have read and agreed to the published version of the manuscript.

Remote Sens. 2021, 13, 1361 12 of 13

Author Contributions: G.M. conceived the idea for this study and drafted the manuscript. T.E. did all the numerical work. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the Office of Naval Research under Naval Research Laboratory (NRL) Project 72-1R25-Z-0-5. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: No new data were created or analyzed in this study. Data sharing is not applicable to this article. Acknowledgments: This is contribution NRL/JA/7230-20-0767. Conflicts of Interest: The authors declare no conflict of interest.

References 1. Horner-Devine, A.R.; Hetland, R.D.; MacDonald, D.G. Mixing and transport in coastal river plumes. Ann. Rev. Fluid Mech. 2015, 47, 569–594. [CrossRef] 2. Yuan, Y.; Horner-Devine, A.R. Experimental investigation of large-scale vortices in a freely spreading gravity current. Phys. Fluids 2017, 29, 106603. [CrossRef] 3. Garvine, R.W. Radial spreading of buoyant, surface plumes in coastal waters. J. Geophys. Res. Oceans. 1984, 89, 1989–1996. [CrossRef] 4. Osadchiev, A.A. Small mountainous rivers generate high-frequency internal waves in coastal ocean. Sci. Rep. 2018, 8, 16609. [CrossRef][PubMed] 5. Osadchiev, A.A.; Zavialov, P.O. Structure and Dynamics of Plumes Generated by Small Rivers. In Estuaries and Coastal Zones— Dynamics and Response to Environmental Changes; Pan, J., Ed.; IntechOpen: London, UK, 2019. 6. Marmorino, G.; Savelyev, I.; Smith, G.B. Surface thermal structure in a shallow-water, vertical discharge from a coastal power plant. Env. Fluid Mech. 2015, 15, 207–229. [CrossRef] 7. Marchevsky, I.K.; Osadchiev, A.A.; Popov, A.Y. Numerical Modelling of High-frequency Internal Waves Generated by River Discharge in Coastal Ocean. In GISTAM; 2019; pp. 384–387. Available online: https://www.scitepress.org/Papers/2019/78402/7 8402.pdf (accessed on 1 December 2020). 8. Osadchiev, A.; Barymova, A.; Sedakov, R.; Zhiba, R.; Dbar, R. Spatial Structure, Short-temporal Variability, and Dynamical Features of Small River Plumes as Observed by Aerial Drones: Case Study of the Kodor and Bzyp River Plumes. Remote Sens. 2020, 12, 3079. [CrossRef] 9. Kilcher, L.F.; Nash, J.D. Structure and dynamics of the Columbia River tidal plume front. J. Geophys. Res. 2010, 115, C05S90. [CrossRef] 10. Wang, C.; Wang, X.; Da Silva, J.C. Studies of internal waves in the strait of Georgia based on remote sensing images. Remote Sens. 2019, 11, 96. [CrossRef] 11. Mendes, R.; da Silva, J.C.B.; Magalhaes, J.M.; St-Denis, B.; Bourgault, D.; Pinto, J.; Dias, J.M. On the generation of internal waves by river plumes in subcritical initial conditions. Sci. Rep. 2021, 11, 1–12. [CrossRef][PubMed] 12. Osadchiev, A.A.; Sedakov, R.O. Spreading dynamics of small river plumes on the northeastern coast of the Black Sea observed by Landsat 8 and Sentinel-2. Remote Sens. Environ. 2019, 221, 522–533. [CrossRef] 13. Horner-Devine, A.R.; Chickadel, C.C. Lobe-cleft instability in the buoyant gravity current generated by estuarine outflow. Geophys. Res. Lett. 2017, 44, 5001–5007. [CrossRef] 14. Horner-Devine, A.; Chickadel, C.C.; MacDonald, D. Coherent Structures and Mixing at a River Plume Front. In Coherent Flow Structures in Geophysical Flows at the Earth’s Surface; Venditti, J., Best, J.L., Church, M., Hardy, R.J., Eds.; Wiley: Chichester, UK, 2013; pp. 359–369. 15. Marmorino, G.O.; Trump, C.L. Gravity current structure of the Chesapeake Bay outflow plume. J. Geophys. Res. 2000, 105, 28847–28861. [CrossRef] 16. Delandmeter, P.; Lambrechts, J.; Marmorino, G.O.; Legat, V.; Wolanski, E.; Remacle, J.F.; Chen, W.; Deleersnijder, E. Submesoscale tidal eddies in the wake of coral islands and reefs: Satellite data and numerical modelling. Ocean Dyn. 2017, 67, 897–913. [CrossRef] 17. Osadchiev, A.A.; Zavialov, P.O. Lagrangian model of a surface-advected river plume. Cont. Shelf Res. 2013, 58, 96–106. [CrossRef] 18. OpenFOAM and The OpenFOAM Foundation. Available online: https://openfoam.org/ (accessed on 10 December 2020). 19. Osadchiev, A. A method for quantifying freshwater discharge rates from satellite observations and Lagrangian numerical modeling of river plumes. Environ. Res. Lett. 2015, 10, 085009. [CrossRef] 20. Osadchiev, A.; Korshenko, E. Small river plumes off the northeastern coast of the Black Sea under average climatic and flooding discharge conditions. Ocean Sci. 2017, 13, 465–482. [CrossRef] 21. Savelyev, I.; Miller, W.D.; Sletten, M.; Smith, G.B.; Savidge, D.K.; Frick, G.; Menk, S.; Moore, T.; De Paolo, T.; Terrill, E.J.; et al. Airborne remote sensing of the upper ocean turbulence during CASPER-East. Remote Sens. 2018, 10, 1224. [CrossRef] Remote Sens. 2021, 13, 1361 13 of 13

22. Trump, C.L.; Allan, N.; Marmorino, G.O. Side-looking ADCP and Doppler radar measurements across a coastal front. IEEE J. Ocean Eng. 2000, 25, 423–429. [CrossRef] 23. Marmorino, G.O.; Trump, C.L. Shore-based acoustic Doppler measurement of near-surface currents across a small embayment. J. Coast. Res. 2000, 16, 864–869.