Suppression of Baroclinic Instabilities in Buoyancy-Driven Flow over Sloping Bathymetry

Robert D. Hetland Department of Oceanography, Texas A&M University, College Station, Texas

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Abstract

Baroclinic instabilities are ubiquitous in many types of geostrophic flow; however, they are seldom observed in river plumes despite strong lateral density gradients within the plume front. Supported by results from a realistic numerical simulation of the Mississippi–Atchafalaya River plume, idealized numerical simulations of buoyancy-driven flow are used to investigate baroclinic instabilities in buoyancy-driven flow over a sloping bottom. The parameter space is defined by the slope Burger number S = Nf−1α, where N is the buoyancy frequency, f is the Coriolis parameter, and α is the bottom slope, and the Richardson number Ri = N2f2M−4, where M2 = |∇Hb| is the magnitude of the lateral buoyancy gradients. Instabilities only form in a subset of the simulations, with the criterion that SHSRi−1/2 = Uf−1W−1 = M2f−2α 0.2, where U is a horizontal velocity scale and SH is a new parameter named the horizontal slope Burger number. Suppression of instability formation for certain flow conditions contrasts linear stability theory, which predicts that all flow configurations will be subject to instabilities. The instability growth rate estimated in the nonlinear 3D model is proportional to ωImaxS−1/2, where ωImax is the dimensional growth rate predicted by linear instability theory, indicating that bottom slope inhibits instability growth beyond that predicted by linear theory. The constraint SH 0.2 implies a relationship between the inertial radius Li = Uf−1 and the plume width W. Instabilities may not form when 5Li > W; that is, the plume is too narrow for the eddies to fit.

Denotes Open Access content.

Corresponding author e-mail: Robert D. Hetland, hetland@tamu.edu

Abstract

Baroclinic instabilities are ubiquitous in many types of geostrophic flow; however, they are seldom observed in river plumes despite strong lateral density gradients within the plume front. Supported by results from a realistic numerical simulation of the Mississippi–Atchafalaya River plume, idealized numerical simulations of buoyancy-driven flow are used to investigate baroclinic instabilities in buoyancy-driven flow over a sloping bottom. The parameter space is defined by the slope Burger number S = Nf−1α, where N is the buoyancy frequency, f is the Coriolis parameter, and α is the bottom slope, and the Richardson number Ri = N2f2M−4, where M2 = |∇Hb| is the magnitude of the lateral buoyancy gradients. Instabilities only form in a subset of the simulations, with the criterion that SHSRi−1/2 = Uf−1W−1 = M2f−2α 0.2, where U is a horizontal velocity scale and SH is a new parameter named the horizontal slope Burger number. Suppression of instability formation for certain flow conditions contrasts linear stability theory, which predicts that all flow configurations will be subject to instabilities. The instability growth rate estimated in the nonlinear 3D model is proportional to ωImaxS−1/2, where ωImax is the dimensional growth rate predicted by linear instability theory, indicating that bottom slope inhibits instability growth beyond that predicted by linear theory. The constraint SH 0.2 implies a relationship between the inertial radius Li = Uf−1 and the plume width W. Instabilities may not form when 5Li > W; that is, the plume is too narrow for the eddies to fit.

Denotes Open Access content.

Corresponding author e-mail: Robert D. Hetland, hetland@tamu.edu

1. Introduction

Baroclinic instabilities are a ubiquitous feature of oceanic flows. They occur at large scales within western boundary currents like the Gulf Stream and Kuroshio (e.g., Tulloch et al. 2011) and at very small scales within the oceanic surface mixed layer, referred to as the submesoscale (e.g., Boccaletti et al. 2007; Capet et al. 2008). The primary difference between the very large- and small-scale instabilities is in the Rossby number Ro = U(fL)−1 of the flow, where U is a horizontal velocity scale, f is the Coriolis parameter, and L is a horizontal length scale; at large scales Ro ≪ 1, while at small scales Ro ~ 1. Instabilities form at approximately the first baroclinic deformation radius (Stone 1966), such that Ro2Ri ~ 1; the Richardson number Ri is defined such that a thermal wind balance specifies the vertical shear , where M2 = |∇Hb| is the magnitude of the lateral buoyancy gradients, H is a vertical length scale, and N is the buoyancy frequency.

Buoyancy-driven flow in the coastal ocean, however, is generally not associated with a large, unstable eddy field. There are exceptions, of course; evidence from satellite sea surface temperature imagery described by Mork (1981) suggests that the Norwegian Coastal Current undergoes baroclinic instability. Meanders also form in the mid-Atlantic shelfbreak front (e.g., Zhang and Gawarkiewicz 2015); however, these meanders do not grow into a fully developed eddy field (Garvine et al. 1988, 1989). There is evidence from numerical simulations that suggests tidal mixing fronts may occasionally experience baroclinic instabilities (Brink 2013). Though forced by wind stress, not buoyancy inputs, instabilities also form along upwelling fronts along the U.S. West Coast (e.g., Barth 1994; Marchesiello et al. 2003; Rogers-Cotrone et al. 2008). Thus, there is ample evidence that baroclinic instabilities do occasionally occur in coastal ocean settings.

However, curiously, there is very little evidence for baroclinic instabilities in river plumes (Horner-Devine et al. 2015). Plumes may form in complex shapes that may appear to have formed from instabilities, but these patterns have been shown to be caused by changes in river discharge (Yankovsky et al. 2001) or changes in wind direction (Hetland 2005; MacCready et al. 2009). Idealized numerical simulations by Isobe (2005) suggest that plumes may be susceptible to inertial instabilities, but there is little evidence for such a phenomenon; many numerical and laboratory studies of rotating plumes with no wind forcing show that the plume is stable over many rotational periods (Fong and Geyer 2002; Lentz and Helfrich 2002; Hetland 2005; Horner-Devine et al. 2006) and does not develop instabilities along the plume front. Rong and Li (2012) use a hydrodynamic simulation of the Changjiang plume, and show that instabilities form along the plume front in a modified case that excludes tides; the real Changjiang plume is in a region with significant tidal energy and does not exhibit such instabilities. Unlike other river plumes, the Mississippi–Atchafalaya River plume displays large instabilities over the Texas–Louisiana (TXLA) shelf in summer. The instabilities are seen clearly in the surface salinity field from a realistic numerical simulation, and an example is shown in Fig. 1. This system will be the prototype for an unstable plume in the present study. During summer, winds are weakly onshore and slightly upcoast,1 causing freshwater to pool south of Atchafalaya Bay and be pulled offshore (Zhang et al. 2012a).

Fig. 1.
Fig. 1.

An example of eddies seen in the surface salinity field along the Mississippi–Atchafalaya plume front from a realistic hydrodynamic model. The area surrounded by the magenta line indicates the region over which water column properties shown in Fig. 5 were calculated. Isobath contours (white) are at 10, 20, 50, 100, 200, 500, 1000, 1500, 2000, and 3000 m.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

Instabilities are seen along the Mississippi–Atchafalaya River plume front in satellite images of surface chlorophyll or true color images where the plume boundary is defined by a color change caused by the sediment load in the river water; some example images are shown in Fig. 2. Unobscured satellite images are difficult to obtain due to cloudiness during summer, so 3-day composite images are used. Even on clear days summer sea surface temperature is very uniform across the entire Gulf of Mexico, such that visual band images are the only reasonable alternative to examine coastal features. The MODIS K490 band, shown in Fig. 2, shows the diffuse attenuation coefficient for 490-nm wavelength light and is associated with turbidity within the river plume. Although it is difficult to attribute features seen in the images directly to dynamical features, the features in the images show structures that look similar to the instabilities shown in Fig. 1.

Fig. 2.
Fig. 2.

MODIS satellite images showing 3-day composites of the log of the K490 band, which measures the penetration of 490-nm light into the water column. The color scale shows a range of light penetration between 0.1 (dark green) to 100 m.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

There are a number of horizontal scales that are apparent in the satellite images and numerical model results. DiMarco et al. (2010) suggested that there is a “wave” in water column processes that appears to be phase locked to three shoals south of Atchafalaya Bay. However, Marta-Almeida et al. (2013) demonstrate that the flow farther offshore is chaotic and that small changes in the forcing will result in differences in simulated salinity on the same order as the seasonal variability in the region on the shelf where these large eddies are seen.

This paper attempts to address some of the dynamical reasons that baroclinic instabilities may be suppressed in most river plumes and why they form in the Mississippi–Atchafalaya plume region. This question will be explored using a series of idealized numerical simulations in the context of existing theory on baroclinic instabilities over a sloping bottom. There are a number of processes thought to inhibit the formation of instabilities, such as frictional processes due to tides or wind. For example, Capet et al. (2008) find reduced submesoscale activity inshore of 50 m on the Argentinian shelf and attribute this to bottom friction. A novel constraint put forward in this paper is that narrow flows do not allow instability formation because the dominant unstable wavelength is large compared to the width of the current; in other words, the eddies do not fit.

2. Theory

A theory describing baroclinic instabilities over a sloping bottom was first put forward by Blumsack and Gierasch (1972) in a study examining the Martian atmospheric circulation; the derivation is reproduced in appendix A, using a notation more conventional to shelf circulation studies. The derivation involves a perturbation expansion assuming a small Rossby number Ro ≪ 1 and then using the resulting dispersion relation to find the largest imaginary component of the frequency as a function of the alongshore wavenumber and slope parameter. The results are shown in Fig. 3.

Fig. 3.
Fig. 3.

Normalized growth rate of the unstable modes as a function of normalized wavenumber and the shelf slope parameter δ based on Eq. (A34).

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

Many studies of mixed layer instabilities do not make the assumption of a small Rossby number. Instead the equations of motion, including the nonlinear terms, may be nondimensionalized using a single parameter, the Richardson number Ri = N2H2U−2 = Bu2Ro−2 (e.g., Stone 1966; Boccaletti et al. 2007), where Bu = (NH)/(fL). Although the initial perturbation expansion is done in terms of Ro, the analysis presented in this paper will instead be done using the Richardson number Ri and the slope Burger number S = Nf−1α, where is the dimensional bottom slope, as the nondimensional parameter space. The two variables representing the nondimensional wavenumber and frequency may be rewritten dimensionally as , where Rd = (NH)/f is the local deformation radius, and . The slope parameter δ that represents the relationship between the shelf slope and the stratification then becomes δ = hyBu2 = (U−1H−1fLα)(NHf−1L−1)2 = SRi1/2.

As Blumsack and Gierasch (1972) discuss, δ is a ratio of the bathymetric and isopycnal slopes. To see this, note
e1
Thus, δ = 1 will be associated with an isopycnal slope equal and opposite to the bottom slope, with buoyant water in the shallower regions; δ = 0 is a flat-bottomed case similar to the classic Eady baroclinic instability solution.

Subsequent studies extended this analysis and examined the possibility of a sloping upper boundary as well (Mechoso 1980) and nonuniform vertical stratification (Isachsen 2011). Blumsack and Gierasch’s solution is similar to the Eady model (Eady 1949) and is based on quasigeostrophic theory in that both the density gradients and bottom slope are considered to be weak, such that the Rossby number of the flow is small, where Ro ≪ 1; it is not a priori clear how well this theory will describe flows that are more energetic, where Ro ~ 1. Also, the linear theory only describes the initial growth of the instabilities, the period when the growth rate is exponential, but the amplitude is still small. This means that the most unstable wavenumbers predicted by the linear theory may not directly correspond to finite-amplitude flow features, such as eddy size.

The results of Blumsack and Gierasch’s linear stability analysis are shown in Fig. 3. For a given value of the slope parameter δ, there is a limited range of unstable wavenumbers. The instabilities are generated through the same mechanism as the Eady model, that is, with interacting vortices trapped to the surface and bottom boundaries (e.g., Cushman-Roisin and Beckers 2011). The primary finding relevant to the present study is that for all positive values of the slope parameter δ, there is always a narrow band of unstable wavenumbers, with the fastest growing instabilities ωImax, associated with length scales a few times smaller than the local deformation radius Rd = NHf−1. For δ = 1, for example, linear theory predicts the most unstable wavenumber will be kRd ~ 3, with the most unstable modes growing at about half the rate compared to a flat-bottomed case. In this manuscript, the convention is chosen that positive values of δ are associated with buoyant water near shore and downcoast geostrophic flow, typical for most buoyancy-driven coastal flows in the absence of strong wind forcing.

There are three relevant nondimensional numbers describing the parameter space of the numerical solutions: the Richardson number Ri = N2f2M−4, the slope Burger number S = Nf−1α, and the slope parameter δ = N2M−2α. The three parameters are related through the relation δ = Ri1/2S, so that specification of two of these parameters determines the third. Approximate parameters for the Texas–Louisiana shelf in summer are S ~ 0.3 and Ri ~ 10, leading to δ ~ 1.

A new parameter is presented in this manuscript, the horizontal slope Burger number:
e2
which is shown in the results section to be the primary parameter controlling the suppression of instabilities in the idealized simulations presented. The horizontal slope Burger number may be written in terms of the nondimensional numbers Ri, S, and δ; it also defines the initial partitioning of kinetic to available potential energy, as derived in appendix B.

The term “submesoscale” has often been used to describe small-scale baroclinic instabilities found in the mixed layer (e.g., Haine and Marshall 1998; Boccaletti et al. 2007; Thomas and Ferrari 2008), which are much smaller than the “mesoscale” flow features such as warm-core rings shed from the Gulf Stream. However, while these flow features are small, they are still well characterized by baroclinic instability theory (e.g., Boccaletti et al. 2007) and still have the most unstable growth rates near the deformation radius (Stone 1966). The primary difference between mesoscale instabilities and submesoscale instabilities is the energy of the flow, as characterized by a high Rossby number or low Richardson number. Thus, while the eddies seen on the Texas–Louisiana shelf resemble mesoscale eddies in that they span the entire water column so that the deformation radius is defined by the stratification through the whole water column as opposed to only the surface mixed layer, they are energetic, with Ro ~ 0.3 even after the instabilities have fully formed, so they have more dynamic similarity to submesoscale instabilities.

3. Numerical methods

a. Realistic model

Results from the realistic numerical model of the northern Gulf of Mexico are based on the TXLA shelf model (Zhang et al. 2012b). The TXLA shelf model is based on the Regional Ocean Modeling System (ROMS; Shchepetkin and McWilliams 2005; Haidvogel et al. 2008) and has been used as the basis of previous studies of buoyancy- and wind-driven flow over the Texas–Louisiana shelf (Zhang et al. 2012a; Thyng et al. 2013). Briefly, the model uses realistic atmospheric forcing based on the North American Regional Reanalysis (NARR) and river inputs from the Mississippi–Atchafalaya and seven other regional rivers. The model has 30 vertical layers and a horizontal resolution of about 1 km in the region of interest, the continental shelf south of Louisiana, between 10- and 75-m depth. The first-mode baroclinic deformation radius in this region is around 5 to 20 km, so flow features on the scale of the local deformation radius are well resolved. The model is nested within the Gulf of Mexico HYCOM model, which provides tracer, flow, and sea level boundary information. The model has demonstrated skill at reproducing the salinity field; the errors are unbiased, and the standard deviation of the model error is about half that of the standard deviation in the observed salinity field relative to climatology. These unbiased errors in salinity have been attributed to chaotic, small-scale flow features over the Texas–Louisiana shelf (Marta-Almeida et al. 2013).

b. Idealized model

ROMS is configured in an idealized, reentrant domain with a uniformly sloping bottom and an initial density field that is varied to examine the parameter dependence of eddy formation. The grid consists of 256 × 128 grid points with 1-km uniform along- and cross-shore resolution, such that the domain is 256 km in the alongshore direction and 128 km in the cross-shore direction. The model has 30 vertical layers, so that the vertical resolution is about a meter in the area of interest. The boundary conditions are periodic alongshore, closed at the coast, and open along the offshore boundary. The bottom slope (10−3) is uniform across all simulations, with 5-m depth at the coast and 133 m offshore. The bathymetry has added noise, with a standard deviation equal to 1% of the total depth, to provide some explicit, small, stochastic forcing to provide small disturbances for instability formation.

The model is run as an initial-value problem, with the initial state evolving unforced. There are no surface momentum or buoyancy fluxes. The bottom boundary condition is specified using a logarithmic velocity profile in the bottom boundary layer, with the bottom stress defined by with the von Kármán constant κ = 0.41 and bottom roughness z0 = 0.003 m. This specification has an advantage over a quadratic drag law specification of bottom stress in that it is less sensitive to vertical resolution in the bottom layer of the model grid. Typical values for z0 in the ROMS test case suite vary between 0.02 and 0.003. Although z0 is on the low side of this range, sensitivity experiments showed that the results are not particularly sensitive to the value of z0 unless it is perturbed by many orders of magnitude.

The initial configuration of the tracer fields is based on observations and simulations of the Mississippi–Atchafalaya plume over the Louisiana shelf. Inshore of 50-m depth, there is a constant cross-shore horizontal density gradient determined by salinity. This laterally stratified region associated with fresher water near the coast defines the plume width W. Everywhere in the domain, there is a constant, horizontally uniform vertical stratification controlled by temperature. Density is determined by a linear equation of state:
e3
so that the horizontal and vertical density gradients may be explicitly formed through the combination of temperature and salinity. The flow is initialized with a geostrophic vertical shear, with no flow at the seafloor, a configuration common to idealized models of shelf flow (e.g., Yankovsky and Chapman 1997) as well as realistic simulations of the Texas–Louisiana shelf (Zhang et al. 2014). An example of these fields is shown for the base case in Fig. 4.
Fig. 4.
Fig. 4.

Cross sections of temperature, salinity, and density are shown for the base case (Ri = 1.0, δ = 0.1, and S = 0.1). The plan view of surface density at the lower right includes information about the lateral boundary conditions.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

The offshore open boundary condition needs some special consideration. The tracers and three-dimensional velocities use a simple no-gradient condition. The plume never crosses the entire domain in the range of parameters considered here, so the solution is not very sensitive to the details of the tracer open boundary condition. The two-dimensional variables use a Chapman–Flather combination for the free surface and depth-averaged currents (Flather 1976; Chapman 1985). However, trial simulations developed a barotropic wave that propagated along the open boundary that interacted with the developing instabilities in the buoyant plume region. To ensure that no waves develop along the open boundary, a sponge layer is added along the five points adjacent to the open boundary. Alongshore depth-averaged flow is relaxed to zero with a time scale ranging from 10 s at the boundary to 1000 s five points in, linearly interpolated at the points in between. This strong nudging was required because the open boundary edge waves travel at about 36 m s−1, meaning that a wave characteristic traverses a single grid cell in about 28 s; this time scale was identified as the most relevant for eliminating the formation of these spurious open boundary waves.

Table 1 lists the range of parameters used in the parameter space simulations. A broader range of simulations beyond those listed in the table were performed to examine model sensitivity and robustness. The simulations presented are a good representation of the range of different simulated conditions.

Table 1.

Parameter space of simulations. The upper group is shown in Fig. 9. All simulations are shown in Figs. 12, 10, and 11. All simulations were run with N2 = 1.00 × 10−4 and α = 1.00 × 10−3. The instability and frictional time scales Ti and Tf are defined in Eqs. (11) and (10).

Table 1.

4. Results

a. Conditions over the Texas–Louisiana shelf

A snapshot of predicted sea surface salinity from the realistic model is shown in Fig. 1. The distribution of salinity shows a general increase in salinity away from the two river sources: the Mississippi and Atchafalaya River mouths. There are also large eddies apparent in the salinity field along the river plume front, bringing freshwater offshore and salty water onshore. The spatial scales of these eddies are from a few tens of kilometers to nearly a 100 km, covering the entire cross-shore extent of the continental shelf. Marta-Almeida et al. (2013) show that this region is associated with strong variability in sea surface salinity during summer based on the spread in an ensemble of simulations with perturbed forcing conditions. In the region where these large eddies are observed, they calculated the variability between these ensemble runs to be as large as the variability in the ensemble mean, with relatively smaller variability elsewhere. As the relative ensemble spread in the properties over the shelf was much larger than the relative variance of the wind or river forcing perturbations, the processes responsible for the ensemble spread are nonlinear and chaotic. Thus, the variability in the ensemble sea surface salinity was attributed to energetic, nonlinear eddies along the Mississippi–Atchafalaya plume front.

To estimate the mean dynamical properties associated with these eddies, characteristic values of vertical and horizontal density gradients were estimated from the realistic simulation in the region where large eddies are typically seen, defined operationally as between 10- and 50-m water depth and between 94.0° and 90.5°W (the region marked in Fig. 1). Vertical stratification is defined by N2 = bz, and horizontal density gradients are defined by , where the buoyancy is defined as . Histograms of these values, weighted by the volume of each grid cell, calculated every 3 days for 6 yr are shown in Fig. 5. The distribution of stratification seen in the histograms indicates that both the vertical and horizontal density gradients have distinct ranges, modulated by a seasonal cycle, with stronger horizontal and vertical stratification in spring and summer. Interestingly, the characteristic vertical and horizontal density gradients are similar from summer to summer, despite changes in discharge (Fig. 5, bottom panel) and wind (not shown). Wind can have a strong influence on the Mississippi–Atchafalaya plume position (Zhang et al. 2012a) and consequently also the shelf ecosystem (Forrest et al. 2011; Thyng et al. 2013), but wind forcing is not discussed further in this paper.

Fig. 5.
Fig. 5.

Water properties within the region delimited in Fig. 1 are shown as volume-weighted histograms (shading) as a function of time. The bottom panel shows the corresponding time series of discharge from the Mississippi River.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

Three additional parameters were calculated from the estimates of bulk stratification over the shelf: the Richardson number Ri, the slope Burger number S, and the slope parameter δ. Each of these parameters was calculated within each cell; the volume-weighted histograms are shown in the third and fourth panels of Fig. 5. Because of the small-scale bathymetric features, the local bottom slope is variable. However, for the case here, the relevant value of the bottom slope is an average across a flow feature. Because of this, the approximate mean bottom slope over the whole shelf α = 10−3 was used for the bottom slope in calculating the slope Burger number and slope parameter.

The most likely values for the slope Burger number during summer are between 0.25 and 0.35, with all values falling within a fairly narrow range. This indicates the slope is a factor in controlling the flow evolution on the shelf. Most typical values for the Richardson number range from about 10 to 30 in summer, with much more spread in the histograms, which range from about 1 to 100. This means the flow field is relatively energetic, as equivalent Rossby numbers are in the range of 0.1 to 0.3, indicating likely nonlinear flow evolution. Finally, the slope parameter δ is in the range of about 1 to 5 during summer, which suggests that the bottom slope is a factor in determining the evolution of any baroclinic instabilities that form.

b. The idealized model base case

Three parameters are required to define the idealized simulation, given the geometric configuration of a constant bottom slope α = 10−3 and a 50-km-wide region of constant cross-shore density gradient. All of the idealized cases presented in this paper have a vertical stratification of N2 = 10−4 s−2, with M2 and f determined by specifying two nondimensional parameters; two of Ri, δ, and S as the three are related through δ = SRi1/2. Based on the water mass analysis from the realistic model, a set of parameters defining the base case were selected, Ri = 1.0 and S = 0.1, with associated derived parameters M2 = 10−6 s−2, f = 10−4 s−1, and δ = 0.1. Note that these parameters represent conditions slightly more energetic than those seen in the realistic model; however, this configuration evolves to a final state that resembles the realistic simulation parameters.

A time series of sea surface salinity and surface vorticity snapshots from the base case simulation (Fig. 6) shows the formation and evolution of instabilities. The eddy field is first visible as an organized disturbance on day 4, evolving into a more chaotic field over the next few days. From day 10 onward, the eddy field does not grow, and the eddy characteristics are roughly constant. As the isopycnals relax, the reach of the buoyant water extends from the initial condition, roughly 50 km offshore, to about 70 km offshore on day 10. The maximum (dimensional) rate for instability growth for δ = 0.1, based on Eq. (A34) in appendix A (Fig. 3), is = 2.50 days−1.

Fig. 6.
Fig. 6.

The evolution of the surface salinity and normalized vorticity fields are shown for the base case (Ri = 1.0, δ = 0.1, S = 0.1); see Fig. 4 for initial conditions. Instabilities form around day 4 and are mature, with little evolution of the eddy structure or offshore motion of the buoyancy front beyond day 10.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

A histogram of stratification parameters and associated nondimensional numbers in the base case simulation is shown in Fig. 7. Although the base case parameters are slightly more energetic (lower Ri, S, and δ) than observed in the realistic simulation, the base case rapidly evolves due to the effect of the instability growth such that Ri and S increase to values similar to those found on the Texas–Louisiana shelf in summer: Ri ~ 10, S ~ 0.3, and δ ~ 1.0. The state evolves over a period of about 6 days, from day 3 to day 9, the same period that eddies in sea surface salinity are seen to grow. The horizontal density gradient M2 remains roughly constant, whereas the vertical stratification N2 is increased as the isopycnals slump to N2 ~ 10−3 s−1, similar to the values seen in the realistic model. The alongshore mean isopycnal motion in a cross-shore section may be compared to rotating venetian blinds, where the spacing between the blinds remains constant along the suspending cord, analogous to the constant horizontal density gradients, but the distance between adjacent blinds is reduced as they rotate closed, analogous to the increasing vertical stratification as the isopycnals relax.

Fig. 7.
Fig. 7.

The upper five panels show volume-weighted histograms of various stratification metrics and nondimensional numbers, described further in the text, similar to those shown in Fig. 5 for the base case (Ri = 1.0, δ = 0.1, and S = 0.1). Properties are calculated only within the initially stratified region, inshore of the 50-m isobath. The initial condition for each parameter is marked with a horizontal blue line. The bottom panel shows a time series of the mean (blue), eddy (red), and total (black) kinetic energy all normalized by the initial mean kinetic energy. The mean bottom stress (gray), averaged over the area of the plume, is shown with the scale on the right.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

A Reynolds decomposition, u = U + u′ and b = B + b′, is used to divide the flow and buoyancy into a mean and perturbed state, with the alongshore mean flow and alongshore mean buoyancy defined as
e4
e5
such that the perturbations u′ and b′ are defined as deviations from the instantaneous alongshore mean. Note that because the domain is periodic in the x direction, Ux = Bx = 0 by definition.

The kinetic energy is shown in the bottom panel of Fig. 7. The mean kinetic energy is defined using the alongshore-averaged velocity ½U2, and the eddy kinetic energy is defined using the total flow minus the alongshore-averaged velocity ½. All kinetic energy values are normalized such that the initial total energy is one. As the isopycnals relax, there is a corresponding increase in the eddy kinetic energy and decrease in mean kinetic energy. The total kinetic energy increases slightly, indicating some energy taken from the potential energy field. The domain average magnitude of the bottom stress is initially weak, as the flow is in geostrophic balance with near-zero bottom flow but increases dramatically as eddies form. The initially weak bottom stress suggests that bottom friction is unimportant in the formation of instabilities but becomes important during the formation of the eddy field, as indicated by the decreasing total kinetic energy.

As the eddy field develops, eddies grow in size, seen qualitatively in Fig. 6. This is consistent with two-dimensional geostrophic turbulence, an approximation for mesoscale baroclinic instabilities, where energy in the flow field at smaller scales shifted toward larger-scale motions (Salmon 1998; Vallis 2006), referred to as an inverse energy cascade. This leads to an unanswered question about how energy is transferred from mesoscale eddies to the smallest scales, where molecular dissipation can occur. In the numerical simulations presented here, the increased bottom stress associated with the formation of eddies creates a clear pathway for a forward energy cascade from the large scale of the eddies to small-scale dissipation in the bottom boundary layer.

The conversion from the mean state to eddy kinetic energy is driven primarily by baroclinic instabilities. Following Barth (1994), this was tested by comparing the dominant terms in the eddy kinetic energy budget:
e6
with the three terms on the right-hand side representing, respectively, the horizontal Reynolds stress, associated with lateral shear instabilities; the vertical Reynolds stress, associated with vertical shear instabilities; and the vertical buoyancy flux, associated with baroclinic instabilities.

The calculated time series of the three terms in the eddy kinetic energy budget [Eq. (6)] is shown in Fig. 8. The energy conversion is dominated by vertical buoyancy flux , indicating that the instabilities are primarily driven by baroclinic instabilities and only weakly by shear instabilities.

Fig. 8.
Fig. 8.

The terms in the eddy kinetic energy budget for the base case, as described in the text.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

Barotropic instabilities, were they to appear at all, would form on the seaward side of the buoyant flow where the lateral shear is strongest. The uniform cross-shore density gradients that define the initial condition of the plume must merge with the offshore region where density varies only vertically. The model configuration is set up such that the cross-shore density gradients to decay exponentially away from the plume edge with a characteristic scale of 5 km. The cross-shore density gradients within the plume will determine the vertical shear and thus the maximum alongshore current at the plume edge. This in turn will determine the lateral shear in the region just offshore of the plume, which will also have a characteristic decay scale of 5 km. The maximum lateral shear will be a function of only M2, since the geometry of the buoyant flow does not vary between cases. As the base case, shown in Fig. 8, has the largest value for M2 of all the cases considered, it also has the largest lateral shear and is correspondingly the most favorable conditions for barotropic instabilities to occur. Therefore, since barotropic shear instabilities are unimportant in the base case, baroclinic instabilities are the dominant instability mechanism in all of the simulations presented.

c. Suppression of instabilities

To examine how the characteristics of baroclinic instability formation change under different conditions, two parameters used to define the base case, Ri and S, were modified to cover a range of values defining horizontal density gradients and rotation. The parameter range is not extremely large but does cover a range of realistic values for the shelf as well as span three dynamical regimes: unstable, stable, and frictional. Table 1 shows a complete list of the simulations presented.

Figure 9 shows sea surface salinity at a snapshot in time after instabilities are expected to form for a subset of the simulations, as a function of Ri and S. Generally, cases where S < 0.3 undergo some sort of instability, cases with S > 0.3 do not necessarily; exceptions are typically caused by frictional effects. Even in the cases where the flow is unstable, the structure and magnitude of the instabilities varies over the simulations.

Fig. 9.
Fig. 9.

Surface salinity for simulations covering a range of Ri and S are shown at a time corresponding to t′ = 50, except the Ri = 10 and S = 0.5 case, which would be shown at 716.8 days but is shown at 365 days instead. The deformation radius Rd, inertial length scale Li, and the time of the snapshot T are shown along with the parameters associated with the particular run.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

The time at which each case in Fig. 9 is plotted is based on the normalized time
e7
where is based on the nondimensional time scale of fastest instability growth ωImax [Eq. (A34); Fig. 3] for the particular value of the slope parameter δ, corresponding to a simulation. The additional factor S−1/2 increases the time of expected instability generation relative to that expected by alone in every case, so that the criterion for instability formation defined in Eq. (7) is more stringent than that expected by linear theory. The nondimensional time t′ = 50 corresponds roughly to 6 days in the base case.

Normalized eddy kinetic energy (EKE) is used to quantify the magnitude of instabilities in the different cases. All of the simulations listed in Table 1 are included in this analysis. Figure 10 shows a time series of domain-integrated, normalized eddy kinetic energy, plotted as a function of normalized time. Two normalizations are used: the left panel shows time normalized by the instability growth rate predicted by linear theory , and the right panel shows time normalized by . The collapse of the curves in the right panel indicates that bottom slope inhibits instability formation beyond that predicted by linear theory.

Fig. 10.
Fig. 10.

Normalized eddy kinetic energy of the surface flow is shown as a function of time normalized in two ways: (left) Imax and (right) ImaxS−1/2 for all the cases presented in Table 1. The gray lines are the simulations that have been identified to be strongly influenced by bottom friction; the Tf < 20Ti cases shown in Fig. 12. The thick black line is the base case Ri = 1.0 and S = 0.1.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

The normalized eddy kinetic energy is defined by subtracting the alongshore mean of the alongshore flow (the cross-shore mean flow is assumed to always be zero) from the flow field and calculating the domain-average kinetic energy of the residual; the eddy kinetic energy is then normalized by the initial integrated mean kinetic energy MKEInitial:
e8
where U is the alongshore mean flow, defined in Eq. (4). The choice for eddy kinetic energy normalization is not obvious; the initial available potential energy APEInitial is another reasonable alternative. These two choices are related:
e9
using the equation for the initial energy ratio derived in appendix B, where SH = SRi−1/2 is a new parameter called the horizontal slope Burger number.

Normalization of the eddy kinetic energy based on the initial mean kinetic energy was chosen because of the relationship between the eddy kinetic energy and the horizontal slope Burger number. The left panel of Fig. 11 shows a strong, significant relationship (r = −0.94, p = 10−14) between the eddy kinetic energy, normalized by the initial mean kinetic energy, and the logarithm of the horizontal slope Burger number (excluding five cases influenced by friction, defined below). As this relationship is not obvious, other choices for the controlling parameter were investigated. The right panel shows the correlation between the normalized eddy kinetic energy to SRin as a function of n. The range of n considered covers the slope Burger number S = SRi0, the horizontal Slope Burger number SH = SRi−1/2, the slope parameter δ = SRi1/2, as well as other likely combinations of these parameters. The highest correlation is for n = −0.6, statistically indistinguishable from n = −½ but statistically distinct from both n = 0 and n = ½. Correlations for the eddy kinetic energy normalized by the initial available potential energy were significantly lower than eddy kinetic energy normalized by the initial mean kinetic energy, as seen in the right panel of Fig. 11.

Fig. 11.
Fig. 11.

(left) The normalized eddy kinetic energy at t′ = 50 is plotted as a function of the horizontal slope Burger number SH. The five open circles are simulations that have been identified to be strongly influenced by bottom friction; the Tf < 20Ti cases are shown in Fig. 12. The star marks the base case. The dashed line shows the best fit to the remaining (filled) points (r2 = −0.88, p = 10−14). (right) The coefficient of determination r2 using different powers n of Ri in the regression. The black line shows r2 using eddy kinetic energy normalized by the initial mean kinetic energy; the gray line shows the same except normalized by initial available potential energy.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

Friction could be a potential factor in inhibiting instabilities (e.g., Capet et al. 2008). To examine the effects of friction, the frictional spindown time is compared to the time scale of instability formation. Analysis of the evolution of the domain average magnitude of the bottom stress (not shown) indicates that bottom friction is relatively small in the beginning of each simulation, increasing greatly during instability growth. Friction is expected to be small in the initial configuration because the flow is geostrophically balanced, with linearly sheared currents decreasing to near zero at the seafloor, similar in configuration to a slippery bottom boundary layer (MacCready and Rhines 1993) or a shelfbreak front at its trapping depth (Yankovsky and Chapman 1997). Instabilities increase the variance in the flow field and disrupt the large-scale balance corresponding to weak bottom flow. As the eddies interact with each other and move on- and offshore, they create localized regions of increased bottom velocity, thereby increasing the associated bottom stress. The frictional time scale Tf is based on the average bottom stress from the model diagnostics output during the second day of simulation—to avoid any spurious currents associated with initialization shock—and is defined as
e10
where is the depth-averaged flow at the edge of the buoyant plume y = W, where H = 50 m and is the alongshore kinematic bottom stress. The time scale of instability formation is
e11
based on the analysis shown in Fig. 10. Figure 12 shows a comparison of these two time scales for all the cases listed in Table 1. Cases where Tf < 20Ti were identified as those where friction played a significant role in the evolution of the simulation. The factor of 20 is somewhat arbitrary, and the results are not sensitive to small changes; factors from 10 to 50 also serve to roughly partition the cases where friction is important. The factor may be taken to mean slightly more than an order of magnitude. Some cases, where Ti < 0.15 days, had a relative decrease in the frictional time-scale time because instabilities had already formed by the second simulation day when the bottom stress was calculated; the bottom friction is elevated, and the frictional time scale is reduced when eddies are present.
Fig. 12.
Fig. 12.

The time scale of instability formation, proportional to Ti [Eq. (11)] is compared to the frictional time scale based on mean bottom stress Tf [Eq. (10)]. See text for details.

Citation: Journal of Physical Oceanography 47, 1; 10.1175/JPO-D-15-0240.1

5. Discussion

a. Interpretations of and physical explanations of instability suppression

The exploration of parameter dependence shows that there is a range of parameters where instabilities are inhibited or even suppressed. This is in contrast with linear theory that predicts instabilities occur for all values of a positive shelf slope, though at progressively higher wavenumbers. Also, the analysis of the parameter dependence shows that the horizontal slope Burger number
e12
is the primary parameter that controls the reduction of instability growth. Instabilities are suppressed for SH 0.2, and instability intensity, as measured by eddy kinetic energy, increases linearly as log(SH) decreases. This is in contrast to linear theory that predicts that the slope parameter δ is the primary variable determining instability formation over a sloping shelf.

One interpretation of SH is that it is the slope parameter δ divided by the Richardson number, such that the relative (local) proportions of energy in the flow Ri ~ PE/KE modify the dependence on the slope parameter. Flows with relatively more kinetic energy are comparatively stabilized. One reason for this might be because strong flows with weak or no stratification are constrained to roughly follow isobaths in rotating systems. Similarly, increased stratification decreases the influence of bathymetry on the flow, similar to the transition from barotropic to baroclinic mode coastal-trapped waves with increasing stratification [see, e.g., Brink (1991) and references therein].

The initial partitioning of (global) kinetic to potential energy is proportional to SH; here again, increased kinetic energy is a stabilizing influence. It may seem counterintuitive then that SH is also proportional to M2, but horizontal density gradients have a stronger influence on kinetic energy through the thermal wind balance (MKEInitialM4) than potential energy (APEInitialM2).

Likewise, SH can also be written as a ratio of the inertial length scale Li = Uf−1 to the plume width W. Instabilities cannot occur in relatively narrow plumes, where W 5Li, thus strong flows that increase the inertial length scale stabilize the plume. The importance of the inertial length scale can be seen in Fig. 9. For example, all the cases where S = 0.1 all have the same deformation radius, but Li decreases with increasing Ri, and the structure of the eddies clearly shows smaller spatial scales with increasing Ri.

The factor of S−1/2 in the estimate of the eddy growth rate [Eq. (7)] demonstrates that increasing slope increases the characteristic time scale of eddy growth. This may be because linear theory is based on very small slopes, small enough that the fluid depth may be considered constant over the domain. In the idealized cases presented here, and in the real Mississippi–Atchafalaya River plume, eddies span a wide range of depths. For example, an eddy with a length scale of 15 km (roughly 3Rd or 3Li in the base case) would experience a 15-m change in depth across the eddy. This is a significant fraction of the total depth: 30% if the eddies are in the deepest part of the plume, more if shallower.

The simulations do not span a very wide range of Ri, and Ri1/2 varies only by a factor of about 3 over the parameter space explored. If this weak dependence on Ri is ignored, the inhibition of instabilities can also be attributed to the dominant dependence of SRi−1/2 ~ S = RdW−1. In this case, eddies are suppressed when the deformation radius is too large to fit within the plume with or W 3Rd, based on a regression of normalized eddy kinetic energy to log(S) (not shown).

There are a few exceptions to the general rule that instabilities are suppressed for SH 0.2 apparent in Fig. 9. These cases are generally associated with either Ri = 1 or long integration times (T 100 days). In both cases, the reason for the instability is that the state of the density field and associated flow evolves before the instabilities form.

In the case in which Ri = 1, the state is on the boundary of symmetric instabilities, such that symmetric instabilities form as the state oscillates slightly about Ri = 1. When the state is shifted to Ri < 1, these instabilities bring the buoyancy-driven current rapidly back to Ri = 1, through a rotational convective adjustment (symmetric instability). This mixing creates conditions where the Richardson number is slightly higher than one, different from the initial conditions. The instabilities that form for Ri = 1 and S ≥ 0.3 are quite weak, however, and they do not grow into a fully developed eddy field.

The cases where the integration time is long, over 100 days in dimensional time, are affected by friction, as shown in Fig. 12. Small, background vertical mixing erodes the vertical stratification such that it decreases. More importantly, the lateral density gradients also decrease, by up to a factor of 2, due to both salinity near the coast increasing and the plume moving offshore, increasing the plume width. For example, in the S = 0.5, Ri = 5 case in Fig. 9, the lateral density gradients decrease such that M2 is reduced by about one half by day 245, decreasing SH by half to about 0.25, near the critical level of 0.2. One interpretation is that as the plume width increases and the buoyancy-driven current slows due to the decrease in lateral density gradients, the eddies have a smaller inertial length scale (as well as a smaller deformation radius due to the decreases in vertical stratification), eddies are better able to fit within the waveguide of the plume, and they are allowed to form and grow.

b. Comparison to real plumes and other idealized studies

Baroclinic instabilities do not form in idealized simulations of river plumes nor is there much evidence for baroclinic instabilities in real plumes (Horner-Devine et al. 2015). The classic unforced river plume, with a bulge and coastal current, does not go unstable in idealized numerical simulations with only buoyancy forcing (e.g., Yankovsky and Chapman 1997; Fong and Geyer 2002; Hetland 2005) or under wind forcing (Fong and Geyer 2001; García Berdeal et al. 2002; Hetland 2005; Jurisa and Chant 2013). While there is no evidence for baroclinic instabilities in rotating tank laboratory experiments specifically designed to examine river plume structure and evolution (Horner-Devine et al. 2006; Avicola and Huq 2003b,a), there is evidence for instability formation in a wide range of buoyancy-driven flows in the laboratory [e.g., Griffiths (1986) and references therein]. Griffiths and Linden (1981) note a transition from stable to unstable flow conditions in a series of experiments with a continuous buoyancy flux from a line source as the width of the current grew past a critical value and attributed the stable configuration to frictional effects. Thus, there is some evidence that narrow currents may be more stable and that the processes through which the current is formed are an important factor in determining stability. However, it is not clear if frictional processes that may be responsible for stability of laboratory-scale flows act similarly at large, geophysical scales.

Pennel et al. (2012) consider how topography affects the stability of a buoyant coastal current, but they consider a very different configuration than presented in this paper; they consider a surface-trapped plume where the buoyant water is only found in the upper 15% of the water column such that the plume never directly interacts with the sloping topography below. However, Pennel et al. still find that topography has a profound influence on the formation of instabilities within the plume, with the slope parameter δ as the relevant parameter to describe topographic influences. They also find that the maximum growth rate is greatly reduced for strong bathymetric slopes, although instabilities were always present in the portion of parameter space they explored and that increasing bottom slope delays the formation of large eddies. Isachsen (2011) also finds that the highest equivalent lateral tracer diffusivities associated with baroclinic instabilities occurs with a flat bottom δ = 0; a nonzero bottom slope always decreases the equivalent diffusivity despite the prediction from linear theory that the maximum should occur around δ = −0.5.

It should be noted that the “bottom-advected” experiments performed by Yankovsky and Chapman (1997) develop a meandering current, but the flow does not go unstable. An estimate of the flow conditions (Yankovsky and Chapman 1997, their Fig. 6) indicate that the frontal width of the bottom-advected plume front is about 15 km wide, with a deformation radius of about 4 km, indicating that the front is on the edge of instability: S ~ RdW−1 = 0.26. Note that in this case, the region of horizontal stratification is confined to the front—water on either side of the front is homogeneous—so the frontal width is the appropriate length scale. For the realistic Mississippi–Atchafalaya plume simulations and the idealized simulations presented here, the lateral density gradients that define the plume extend from the coastline to the plume front, so that the bottom slope and water depth at the plume front can be used to determine the plume width. The meanders in the plume appear to have the same scale as baroclinic instabilities (Yankovsky and Chapman 1997, their Figs. 7 and 8), roughly 3Rd. An estimate of SH in the plume front (based on W = 15 km, U = 0.3 m s−1, and f = 10−4 s−1) is SH = 0.2, the point where instabilities are predicted to be suppressed. Estimates based on M2 are less accurate, as the lateral density gradients vary across the plume front. The conditions downstream, near where the meanders begin, are much different than those near the mouth. For example, the Rossby number at the source is Ro = 0.067, whereas downstream, Ro ~ 0.3. Thus, the evolution of the plume front is important in determining if instabilities can form.

Jia and Yankovsky (2012) present a study that is an exception to the general observation that baroclinic instabilities do not occur in idealized river plume simulations by demonstrating that the presence of background temperature stratification can make a plume go unstable. They used a configuration with a source of buoyancy along a coastal wall, releasing freshwater into a uniformly sloping shelf. When the plume was released into homogeneous receiving waters, no instabilities formed, in agreement with previous studies. However, the addition of a uniform temperature stratification in the receiving waters of N2 = 1.2 × 10−4 s−1 (beneath a 15-m-thick mixed layer) caused the plume to go unstable. An important difference between this study and Jia and Yankovsky (2012) is that in their study, the plume was allowed to evolve and the structure of the plume at the point where it goes unstable is determined by the plume history. It seems likely that the addition of vertical stratification allowed the plume to evolve to a wider state than it would have otherwise because the plume pushes the thermocline downward and that displaced thermocline increases the de facto width of the plume offshore as the total region with significant horizontal density gradients is increased compared to the homogeneous receiving water case. In the cases presented here, what would be equivalent to the downstream plume structure was specified as an initial condition; the flow does not evolve as a coastal current from a point source. However, even when the buoyancy-driven flow structure is specified, gradual evolution of the flow can cause instabilities, as in the cases influenced by friction.

Although it is plausible that simulations like those described by Yankovsky and Chapman (1997) and Jia and Yankovsky (2012) will also have some aspects of instability formation that do not conform to linear theory, it is not yet clear if the results presented in this paper—the importance of SH in inhibiting instabilities and the S−1/2 modification to instability growth rates—can be directly applied to other plume configurations. For example, the relative importance of bottom friction may be greater in plumes that occur in shallower water. There may also be some other geometric aspects or forcing conditions that enhance or inhibit instability formation.

The results of these previous studies (e.g., Griffiths and Linden 1981; Yankovsky and Chapman 1997; Jia and Yankovsky 2012) imply that one of the reasons that the Mississippi–Atchafalaya plume may go unstable is due to the plume history. The residence time of freshwater from the Mississippi and Atchafalaya Rivers on the Texas–Louisiana shelf is between 3 months to a year (Zhang et al. 2012a). Thus, a significant fraction of the freshwater on the shelf during calm summer experienced the more stormy conditions of the previous spring. The mixing associated with these spring storms may result in a broader plume front. Also, winds during summer are weak and upwelling favorable, which tends to pull older plume water eastward to the region south of Louisiana as well as pulling the plume slightly offshore (Cochrane and Kelly 1986; Cho et al. 1998). This may cause the plume front to be farther offshore and the plume to be broader than it would otherwise have been. Both of these processes would tend to broaden the plume and reduce the vertical stratification, both factors enhancing the possibility of baroclinic instability formation.

6. Conclusions

The primary finding of this study is that baroclinic instabilities do not form in a coastally trapped, buoyancy-driven flow under all conditions; when the inertial length scale is larger than about a fifth of the plume width, instabilities do not form. Exceptions in the cases considered are due to modification of the flow by mixing—background mixing or due to symmetric instabilities—prior to the formation of baroclinic instabilities.

Comparison with previous studies suggests that the evolution of a river plume is critical in determining if instabilities will occur or not. It seems that the classic idealized plume problem of buoyant water introduced into a homogeneous ocean evolves in a way that suppresses instabilities. The coastal current that is formed is too narrow, relative to both the local deformation radius and inertial length scale, to allow eddies to form and grow. The constraint that eddies must fit within the buoyancy-driven flow may be the reason that baroclinic instabilities are seldom observed in both real and idealized river plume studies.

In real plumes, some forcing agent—either winds or tides—must be present to mix the plume and increase its width. However, these forcing agents also appear to inhibit instabilities through mixing. Instabilities form in the Mississippi–Atchafalaya plume because there is a relaxation in the forcing during summer. The plume is mixed in nonsummer to increase its width but is subject to very weak forcing by winds and tides in the summer, when instabilities are strongest.

Acknowledgments

This work was funded by the Texas General Land Office (Contract Number 10-096-000-3927) and the Gulf of Mexico Research Initiative through the Gulf Integrated Spill Research consortium (Contract Number SA12-09/GoMRI-006). Data for the realistic model results are publicly available through the Gulf of Mexico Research Initiative Information and Data Cooperative (GRIIDC; at https://data.gulfresearchinitiative.org; doi:10.7266/N7VH5KSK). I thank Ken Brink, Xavier Capet, Baylor Fox-Kemper, Gordon Zhang, and two anonymous reviewers for very helpful comments when preparing this manuscript.

APPENDIX A

Linear Stability Analysis

The following derivation closely follows Blumsack and Gierasch (1972) but written in notation more familiar to coastal oceanographers. Start with the inviscid, Boussinesq equations of motion
ea1
ea2
ea3
ea4
ea5
where , so that ; other notation is standard, and stared variables are dimensional. These equations are subject to the boundary conditions
ea6
ea7
These equations are normalized using a horizontal length scale L, a vertical length scale H, a time scale L/U, and a horizontal velocity scale U, such that, for example, the nondimensional x coordinate is defined by . Other variables are then normalized as
ea8
ea9
ea10
ea11
The normalized equations (A1) through (A5) become
ea12
ea13
ea14
ea15
ea16
with boundary conditions
ea17
ea18
where the Rossby number is Ro = U/(fL), and the Burger number is Bu = (NH)/(fL). Note that the Burger number defines H and L by the length scale of the flow features; below the slope Burger number S = (N/f)α defines these length scales by the bottom slope. We will assume that Bu ~ 1 and Ro ≪ 1 and do a perturbation expansion about Ro. The Ro = 0 solution is
ea19
ea20
ea21
ea22
In other words, the basic state is geostrophic with no vertical velocities. Substitute these geostrophic relations into the small O(Ro) terms in Eqs. (A12) through (A16) to obtain a quasigeostrophic potential vorticity conservation equation
ea23
where
ea24
The pressure p0 is chosen to be consistent with a geostrophically balanced, constant vertical shear plus a small perturbation Φ:
ea25
where Φ ≪ 1. The unperturbed part of the base state corresponds to by = −1; the y coordinate is oriented positive offshore with buoyancy increasing shoreward, and the associated geostrophic currents are directed alongshore. The geostrophic flow is such that the flow is zero at the bottom, increasing linearly in the positive z direction, that is, the geostrophic flow is downcoast in the Kelvin wave direction. This configuration is very similar to an arrested bottom boundary layer flow (e.g., MacCready and Rhines 1993) or a bottom-advected, buoyancy-driven current on a shelf (Chapman and Lentz 1997; Yankovsky and Chapman 1997). The alongshore flow dominates, so that Eq. (A25) substituted into Eq. (A23) is approximately, ignoring terms ≤ O2),
ea26
The boundary conditions are w = 0 at the surface and w = −Roυhy at the bottom; hy > 0 is the normal case with deeper, denser water offshore. If the vertical coordinate is aligned so that the bottom is at z = 0 and the surface is at z = 1, the boundary conditions, cast in terms of pressure, are combining Eqs. (A14) and (A15) and using the assumed form of the pressure in Eq. (A25):
ea27
ea28
which, ignoring terms ≤ O2), reduces to
ea29
ea30
where δ = Bu2hy is the slope parameter. Note that the bottom slope was defined differently as compared to Blumsack and Gierasch (1972), so that δ has the opposite sign.
Assume that the solution is wavelike in the horizontal direction and oriented alongshore (∂/∂y = 0), so that
ea31
Plugging this form into Eq. (A26), subject to the bottom boundary condition (A29), gives a general structure function for the z dependence of the vertical modes:
ea32
where the amplitude A is an arbitrary constant. Now, applying the surface boundary condition (A30) gives a quadratic equation for the frequency
ea33
The imaginary part of the solution to this equation represents the growth of the wave Φ riding on the mean state
ea34
and is plotted in Fig. 3 {equivalent to Blumsack and Gierasch [1972, their Eq. (3.11) and Fig. 2]}.

APPENDIX B

Energetics

The kinetic and available potential energy of the initial state may be calculated analytically. Over a continental shelf with a uniform slope α, the initial state is defined by a region of uniform horizontal buoyancy gradient, defined by M2 within a distance W from the coast. There is also a uniform vertical gradient in buoyancy in this laterally stratified region that continues out into the deep ocean, such that the vertical buoyancy gradient is everywhere N2. The available potential energy is defined using a buoyancy anomaly that assumes no lateral gradients in buoyancy and that the deep ocean is large enough such that the buoyancy anomaly is completely dispersed without significantly altering the reference ground state. The velocity υ is defined using a thermal wind balance, with zero velocity at the seafloor. The sea surface is defined to be zero at offshore edge of the plume region, with sea surface gradients defined using a geostrophic balance with the surface flow. Thus,
eb1
eb2
eb3
eb4
Using these values the kinetic energy and available potential energy can be written in terms of the parameters defining the configuration:
eb5
eb6
eb7
where the available potential energy has been split into two components: APEρ, associated with lateral density gradients (assuming η = 0), and APEη, associated with the energy contained in the free-surface anomalies (assuming constant density).
The majority of the available potential energy is contained within the stratification, as can be seen in the ratio
eb8
For the base case (M2 = 1 × 10−6 s−1; W = 50 km), this ratio is approximately 0.057, indicating only about 5% of the energy is contained in the sea level anomalies.
The ratio of kinetic energy to available potential energy associated with the lateral density gradients is
eb9
This parameter looks somewhat like a “horizontal” slope Burger number, proportional to lateral gradients in buoyancy rather than vertical gradients. Lateral density gradients have also been shown to be important in governing estuarine circulation [see Geyer and MacCready (2014) and references therein], in particular in relation to the Simpson number Si, similar to a Richardson number but proportional to horizontal gradients in density instead of vertical. Given that this parameter has a number of physical interpretations, none of which are obviously the primary interpretation, it will be referred to the horizontal slope Burger number SH.

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1

Downcoast is defined as alongshore and in the direction of Kelvin wave propagation, upcoast is the opposite. On the Texas–Louisiana shelf, upcoast is eastward, from Texas toward Louisiana.

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    • Search Google Scholar
    • Export Citation
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    • Export Citation
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    • Export Citation
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    • Export Citation
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    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Export Citation
  • Hetland, R. D., 2005: Relating river plume structure to vertical mixing. J. Phys. Oceanogr., 35, 16671688, doi:10.1175/JPO2774.1.

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    • Search Google Scholar
    • Export Citation
  • Horner-Devine, A. R., R. D. Hetland, and D. G. MacDonald, 2015: Mixing and transport in coastal river plumes. Annu. Rev. Fluid Mech., 47, 569594, doi:10.1146/annurev-fluid-010313-141408.

    • Search Google Scholar
    • Export Citation
  • Isachsen, P. E., 2011: Baroclinic instability and eddy tracer transport across sloping bottom topography: How well does a modified Eady model do in primitive equation simulations? Ocean Modell., 39, 183199, doi:10.1016/j.ocemod.2010.09.007.

    • Search Google Scholar
    • Export Citation
  • Isobe, A., 2005: Ballooning of river-plume bulge and its stabilization by tidal currents. J. Phys. Oceanogr., 35, 23372351, doi:10.1175/JPO2837.1.

    • Search Google Scholar
    • Export Citation
  • Jia, Y., and A. Yankovsky, 2012: The impact of ambient stratification on freshwater transport in a river plume. J. Mar. Res., 70, 6992, doi:10.1357/002224012800502408.

    • Search Google Scholar
    • Export Citation
  • Jurisa, J. T., and R. J. Chant, 2013: Impact of offshore winds on a buoyant river plume system. J. Phys. Oceanogr., 43, 25712587, doi:10.1175/JPO-D-12-0118.1.

    • Search Google Scholar
    • Export Citation
  • Lentz, S. J., and K. R. Helfrich, 2002: Buoyant gravity currents along a sloping bottom in a rotating fluid. J. Fluid Mech., 464, 251278, doi:10.1017/S0022112002008868.

    • Search Google Scholar
    • Export Citation
  • MacCready