## 1. Introduction

Baroclinic instabilities are ubiquitous throughout the global ocean. They release potential energy stored in horizontal density gradients, creating unsteady and evolving motions in the flow at approximately the first baroclinic deformation radius. As such, the Burger number (Bu) associated with baroclinicly unstable flow is Bu = *R*_{d}*L*^{−1} = RoRi^{1/2} ~ 1 (Eady 1949; Stone 1966, 1970). Beyond this, there are two general categories of baroclinic instabilities associated with the Rossby number (Ro) and the Richardson number (Ri) of the flow: mesoscale instabilities within the large-scale, geostrophic fronts with Ro ≪ 1 and Ri ≫ 1, and submesoscale instabilities with Ro ~ 1 and Ri ~ 1 (Boccaletti et al. 2007). Examples of the submesoscale instabilities include instabilities within the ocean mixed layers (Boccaletti et al. 2007; Fox-Kemper et al. 2008), along the fronts of mesoscale eddies (Callies et al. 2015; Brannigan et al. 2017), in the bottom boundary layers (Wenegrat et al. 2018), and, rarely, in certain coastal buoyancy-driven flows (Hetland 2017); they are potentially more nonlinear and energetic in nature than can be accurately described by quasigeostrophic (QG) theory.

The dynamics of QG baroclinic instabilities over a flat bottom has been well studied since the seminal papers by Eady (1949) and Phillips (1954), few studies have investigated the nongeostrophic scenario with a flat bottom (Stone 1966, 1970) and the QG scenario with a sloping bottom (Blumsack and Gierasch 1972; Mechoso 1980), but fewer studies have considered both the sloping-bathymetry effect and nongeostrophic effect. The submesoscale baroclinic instabilities in bottom boundary layers and coastal buoyancy-driven flows are influenced by these two effect (Wenegrat et al. 2018; Hetland 2017), and are less understood than the submesoscale instabilities in the surface mixed layers and the mesoscale instabilities, which are not influenced by the effects at the same time. One unique feature is the instability suppression—the submesoscale instabilities over sloping bathymetry exhibit weaker growth compared to the QG theories (Blumsack and Gierasch 1972; Mechoso 1980). Based on linear stability analysis, Wenegrat et al. (2018) shows that the growth of the instabilities in bottom boundary layers decreases as the regime shifts from QG to nongeostrophic. Also, although coastal buoyancy-driven flows are often associated with stronger lateral density gradients than open ocean fronts, they are seldom observed to be associated with instabilities; in particular, baroclinic instabilities are seldom observed in river plumes, even though lateral buoyancy gradients within the fronts are strong (Horner-Devine et al. 2015). External forcing agents, such as winds and tides, could suppress baroclinic instabilities through mixing processes. However, it has been demonstrated that a rotating plume without external forcing can be very stable over many rotational periods (Fong and Geyer 2002; Lentz and Helfrich 2002; Hetland 2005; Horner-Devine et al. 2006; Hetland 2017); this implies that the suppression can be due to the intrinsic inhibiting effects of the front. Baroclinic instabilities can enhance dispersion of water borne particles (Thyng and Hetland 2018), and decrease predictability in numerical simulations (Marta-Almeida et al. 2013). Better understanding the instability growth would improve our understanding on numerical predictability and be helpful for further investigating other submesoscale processes (e.g., symmetric instabilities and frontogenesis), material (e.g., nutrients and sediments) transport, mixing, and turbulence in the bottom boundary layers and coastal buoyancy-driven flows.

This paper explores nongeostrophic baroclinic instability theories adapted to the scenario with a sloping bathymetry. The two-layer model is adapted from Sakai (1989), and the continuously stratified model is an existing model adapted from Stone (1971) by Wenegrat et al. (2018). Both models are used to investigate the suppression of nongeostrophic baroclinic instabilities over sloping bathymetry, which is not revealed in the classical QG theories (Blumsack and Gierasch 1972; Mechoso 1980). In particular, this paper attempts to seek a nondimensional parameter for indicating the instability suppression and understand the underlying mechanism controlling the suppression in the buoyant flow regime.

## 2. Theory

### a. Layered model of nongeostrophic baroclinic instability

Phillips (1954) transformed the continuously stratified fluid to a two-layer system and constructed the layered model of QG baroclinic instabilities. Sakai (1989) investigated the ageostrophic instabilities using an ageostrophic version of the Phillips model. We adapt the Sakai model to the scenario with sloping bottom and surface (hereinafter referred to as the adapted Sakai model). The schematics of the Sakai model and the adapted Sakai model are shown in Fig. 1. The adapted Sakai model is a rotating two-layer channel with the sloping topography and background flow in the thermal wind balance. Considering the time scale as *f*^{−1}, the horizontal length scale as the Rossby deformation radius *H*_{0} (see the primitive equations and scale analysis in appendix A), the dimensionless form of the equations governing the perturbations is

subject to

where the subscript 1 denotes the variables in the upper layer, subscript 2 for the lower layer, *u* is the along-slope velocity, *υ* is the across-slope velocity, *p* is the pressure, ±*Y*_{max} are the across-slope boundaries, *g*′ = (Δ*ρ*/*ρ*_{0})*g* is the reduced gravity, *H*_{0} is the dimensional average thickness, and *U*_{0} is the dimensional background flow (*U*_{0} in the upper layer and −*U*_{0} in the lower layer). The term *δ* ≡ *α*/*α*_{p} is the slope parameter, the ratio of the bottom slope *α* and the isopycnal slope *α*_{p} = 2*U*_{0}*f*/*g*′; both *α* and *α*_{p} are taken to be positive in this study. Here, we use the sign convention of *δ* as Hetland (2017) and Wenegrat et al. (2018), the opposite sign convention compared to the original one in the study of Blumsack and Gierasch (1972), such that positive *δ* represents the common case of buoyancy-driven flow over sloping bathymetry, where the isopycnal and bathymetric slopes are opposite. In this study, we will only focus on the scenarios with positive *δ*, which is referred to as the buoyant flow regime. The across-slope boundary *H* [Δ*H* = 0.5 is set as in Sakai (1989)].

Substituting a wave-like solution *σ*_{i} = Imag[*σ*] is obtained numerically through linear stability analysis. We use the software package DEDALUS for the stability analysis; DEDALUS is an open-source PDE solver written in Python that uses spectral methods (Burns et al. 2016). Figure 1 shows the normalized growth rate in the flat-bottom case (*δ* = 0) and the sloping-bottom case (*δ* = 0.5); see section 3a for more details.

### b. Continuously stratified model of nongeostrophic baroclinic instability

Eady (1949) developed a continuously stratified QG framework to study baroclinic instabilities. Blumsack and Gierasch (1972) adapted the Eady model to the scenario with a sloping bottom (hereinafter referred to as the BG model). Mechoso (1980) extended the BG model to the scenario with a sloping surface. Stone (1966, 1970, 1971) extended the Eady model to the nongeostrophic limit and constructed the continuously stratified model of nongeostrophic baroclinic instabilities. We use the modified model (Wenegrat et al. 2018), which adapts the Stone model to the scenario with sloping bottom and surface (hereinafter referred to as the adapted Stone model). The schematic of the BG, Mechoso, and adapted Stone models are shown in Fig. 2. In the adapted Stone model, the coordinates are rotated to align with the sloping bottom. The background buoyancy has a constant vertical gradient *N*^{2} and a constant horizontal gradient *M*^{2}, and the background flow is constrained by the thermal wind relation. Considering the time scale as *f*^{−1}, the horizontal length scale as *U*/*f*, and the vertical length scale as *H* (see the primitive equations and scale analysis in appendix B), the dimensionless form of the equations governing the perturbations in the rotated coordinates is

subject to

where *u* is the along-slope velocity, *υ* is the across-slope velocity, and *w* is the slope-normal velocity, *p* is the pressure, *b* is the buoyancy, *θ* is the bottom slope angle, *z* = 0 (*z* = 1) is the sloping bottom (surface). Ri = *N*^{2}*f*^{2}*M*^{−4} is the Richardson number, and *δ* ≡ *α*/*α*_{p} = *αN*^{2}*M*^{−2} is the slope parameter, the ratio of the bottom slope *α* and the isopycnal slope *α*_{p} = *M*^{2}/*N*^{2}. The term *ε* = *f*^{2}*M*^{−2} is the nonhydrostatic parameter (Stone 1971).

Noting that the surface is also assumed to be tilted, the adapted Stone model is intrinsically suitable for baroclinic instabilities in a tilted bottom boundary layer, but seems not directly applicable to the situation with a flat surface that is not parallel with the tilted bottom (e.g., coastal buoyancy-driven flow). One way is to adapt the Stone model to the scenario with a flat surface, but this presents two challenges for theoretical approaches: first, making the uniform depth assumption like Eq. (4) is theoretically invalid; second and subsequently, assumed solutions with a wave form in the across-slope direction become theoretically invalid. Consequently, instead of adapting the Stone model to the flat-surface scenario, we will address the feasibility of the adapted Stone model (Wenegrat et al. 2018) in the flat-surface cases, for example, baroclinic instabilities in a coastal buoyancy-driven flow over a continental shelf.

Assuming a wave-like solution *σ*_{i} = Imag[*σ*]. Figure 2 shows the normalized growth rate based on the BG model (Blumsack and Gierasch 1972), the Mechoso model (Mechoso 1980), and the adapted Stone model (Wenegrat et al. 2018); See section 3b for more details. To keep consistent with the normalization in the QG models (the BG and Mechoso models), the dimensionless variables *σ*_{i} and *k* in the adapted Stone model are converted to the normalized variables *R*_{d} = *NH*/*f*; that is, according to the scaling relations,

The normalized growth rate *R*_{d}/*U*,

With regard to the normalization in the adapted Sakai model, the wavenumber *R*_{d}, so *k* in the adapted Sakai model, while the dimensionless

## 3. Results

In this section, we will address the suppression of nongeostrophic baroclinic instabilities over sloping bathymetry from the temporal perspective—the growth rate of instabilities—and discuss the suppression mechanisms in the layered and continuously stratified models. The suppression problem will be addressed in the dimensionless space; the dimensionless parameter space is Ri–*δ*, where Ri represents the nongeostrophic effect and *δ* represents the effect of sloping bathymetry.

### a. Suppression of instabilities in the layered model

Based on the adapted Sakai model, baroclinic instabilities start to be suppressed (meaning the growth rate of the instabilities is reduced), when the bottom slope increases (slope parameter *δ* increases from 0 to 0.5, see Fig. 1). Also, instabilities are found to be suppressed with decreasing bulk Richardson number Ri_{b}. The suppression of instabilities has opposite dependencies on Ri_{b} and *δ*. To demonstrate this, the maximum growth rate as a function of Ri_{b} and *δ* is calculated. The maximum normalized growth rate _{b} and *δ*. The left panel of Fig. 3 shows the _{b} and *δ*. The maximum normalized growth rate _{b} and *δ*, with the suppression as Ri_{b} decreases and *δ* increases. Baroclinic instabilities are sufficiently suppressed at low Ri_{b} and high *δ*. To understand the suppression of baroclinic instabilities, we will identify the primary parameter linked with the suppression and explore the controlling mechanism.

A new parameter, the slope-relative Burger number *S*_{r} ≡ (*M*^{2}/*f*^{2})(*α* + *α*_{p}), is considered as the coefficient controlling the suppression of instabilities in the nongeostrophic case. Compared to the horizontal slope Burger number *S*_{H} ≡ (*M*^{2}/*f*^{2})*α* (Hetland 2017), the slope-relative Burger number *S*_{r} uses a bottom slope relative to the isopycnal slope *α*_{p}. Thus, if the slopes are aligned, with the bottom parallel to the isopycnals, then *S*_{r} = 0. The slope-relative Burger number uses the layer thicknesses in the adapted Sakai model; *M*^{2}/*f*^{2})(*H*/*L*) = (*M*^{2}/*f*^{2})(*α* + *α*_{p}). The slope-relative Burger number can be written in terms of the Richardson number Ri and the slope parameter *δ* as

where *δ*_{r} is the slope-relative parameter,

with an interpretation similar to the slope-relative Burger number *S*_{r}. For clarity, hereafter 1 + *δ* will be written as *δ*_{r}.

The bulk form of the slope-relative Burger number is

The distribution of *S*_{r} in Ri_{b}–*δ* space (shown in the center panel of Fig. 3) exhibits a similar pattern as the maximum normalized growth rate *S*_{r} (see the right panel of Fig. 3); *S*_{r} contains the opposite dependencies of _{b} and *δ*. In addition, instabilities are sufficiently suppressed for *S*_{r} and _{b} ~ *O*(100)].

A strength of the Sakai model is to interpret instabilities in the framework of wave resonance based on the physical wave coordinates (Sakai 1989). For instance, Kelvin–Helmholtz instabilities are interpreted as the resonance of interacting gravity waves. In this study, we will only focus on the interactions between Rossby waves to interpret the suppression of baroclinic instabilities. The physical wave coordinates, then, only consist of the Rossby waves; since the Rossby wave coordinates are the Fourier basis (see appendix C), the projection onto the physical wave coordinates is equivalent to conducting the Fourier transform (Pedlosky 2013). The details of the Rossby wave resonance in the adapted Sakai model can be found in appendix C. The resonance rate

where *n* = 1, 2, 3, …) is the dimensional across-slope wavenumber. If *δ*_{r} = 1 (the flat-bottom case), Eq. (10) is reduced to *n* = 1) and using the *S*_{r}, and they are highly correlated (*r*^{2} = 0.999, *p* = 0.0); it indicates that *S*_{r} controls the growth of instabilities by influencing the Rossby wave resonance.

The underlying mechanism is that *S*_{r} influences the wave resonance by modifying the properties of the Rossby waves. According to appendix C, the normalized Doppler-shifted phase speed of the Rossby waves in both layers is

where *k* is the normalized along-slope wavenumber, and *n* = 1, 2, 3, …) is the normalized across-slope wavenumber. The slope-relative parameter *δ*_{r} could be considered as the normalized topographic beta, which is *fα*/*H*_{0}. Noticing that *S*_{r}; higher *S*_{r} would indicate larger difference of the Doppler-shifted phase speed *S*_{r} (using the lowest mode and the *k* corresponding to the maximum growth rate). As the schematics of Fig. 5 show, low *S*_{r} represents the scenario where *S*_{r} represents the scenario where *S*_{r} influences the Rossby wave resonance (thus the instability growth) by altering the Doppler-shifted phase speed of the Rossby waves, thereby controlling the growth of instabilities.

### b. Suppression of instabilities in the continuously stratified model

The suppression of instabilities through the reduction of instability growth rate is also found in the adapted Stone model (Wenegrat et al. 2018). Particularly, the suppression is significant when the regime shifts from QG to nongeostrophic. To demonstrate the suppression, the QG models, that is, the BG model (Blumsack and Gierasch 1972) and the Mechoso model (Mechoso 1980), are briefly reviewed here. In the BG model, the background state consists of a density field with constant vertical and horizontal buoyancy gradients and a velocity field constrained by the thermal wind relation. By assuming that the Rossby number Ro = *U*/(*fL*) ≪ 1 and Burger number Bu = (*NH*)/(*fL*) ~ 1 (leading to the Richardson number Ri = Bu^{2}Ro^{−2} ≫ 1), the equations governing perturbations can be reduced to a single PDE about the QG potential vorticity. Then, the growth rate of the instabilities is analytically obtained by solving the associated eigenvalue problem; the normalized growth rate of the BG model is then

where *δ* is the slope parameter. On the other hand, the Mechoso model relaxes the flat-surface assumption of the BG model and takes into account a surface that could be tilted as the bottom would be. Here, we only consider the scenario where the surface is parallel with the sloping bottom; the normalized growth rate of the Mechoso model is, then,

Figure 2 shows

Maximum growth rates predicted by the QG and adapted Stone models have different dependencies on the Richardson number Ri. The maximum normalized growth rate *δ*. The Ri–*δ* space is spanned within 1 ≤ Ri ≤ 5 and 0 ≤ *δ* ≤ 0.6, which follows Hetland (2017). This parameter space covers most scenarios of energetic coastal fronts (Hetland 2017) and a large variety of energetic deep ocean BBLs (Wenegrat et al. 2018). The maximum normalized growth rate *δ*. Here we implicitly associate QG theories to high Ri conditions, because it is unknown how well QG theories describe flow at low Ri conditions. The growth rates *δ*, but do not vary with Ri, since the QG normalized growth rates are independent of Ri [see Eqs. (12) and (13)]. In contrast,

A dimensionless number is sought for indicating the suppression of instabilities in the adapted Stone model. One requirement of the number is that it can reduce to the QG limit (Blumsack and Gierasch 1972; Mechoso 1980) and the nongeostrophic limit with flat bathymetry (Stone 1970). The following discussion reviews the dimensionless numbers ^{−1})^{−1/2}, which controls the instability growth in the QG limit and nongeostrophic limit with flat bathymetry, respectively. First, as demonstrated in Pedlosky (2016), the slope-relative parameter *δ*_{r} is the only dimensionless number appearing at the bottom boundary condition of the Eady problem [Pedlosky 2016, Eq. (5b)]; it involves the vertical shear of the background flow that is supplemented by the topographic production of vertical vorticity by the perturbed across-slope flow. Figure 7 (left) shows *r*^{2} = 0.999, *p* = 0.0) and *r*^{2} = 0.993, *p* = 0.0)—^{−1})^{−1/2},

which is obtained based on the asymptotic analysis of Stone (1970). Figure 7 (right) shows the comparison between the numerical ^{−1})^{−1/2}; (1 + Ri^{−1})^{−1/2} has a robust relation with *r*^{2} = 0.999, *p* = 0.0). Equation (14) is also shown in Fig. 7 (right); the offset between ^{−1})^{−1/2} is the controlling number in the nongeostrophic limit with flat bathymetry.

Noticing that *S*_{r}|_{Flat} = Ri^{−1}, the controlling number (1 + Ri^{−1})^{−1/2} might be the reduction of (1 + *S*_{r})^{−1/2} in the flat-bottom case. The multiplication of *S*_{r})^{−1/2} (as the nongeostrophic modification) would be a physically intuitive form of the dimensionless number in the adapted Stone model, because it can simply reduce to ^{−1})^{−1/2} as *δ* → 0 (the nongeostrophic limit with flat bathymetry; Stone 1970). The dimensionless number *δ* space is shown in Fig. 8a, which has a similar distribution as the *r*^{2} = 0.991 and *p* = 0.0). Given that the QG growth is controlled by *S*_{r})^{−1/2} on

The fact that *δ*_{r} (or *δ*) and *S*_{r}. Energetics is used to understand these dependencies. According to appendix D (following Wenegrat et al. 2018), the eddy potential energy (EPE) and eddy kinetic energy (EKE) equations (the primed variables are the dimensional perturbations) are

and

where _{c} and HBF_{n} are the driving energy sources [HBF_{n} ≪ HBF_{c} because (HBF_{n}/HBF_{c}) ~ *ε* tan*θ*], which convert the mean available potential energy (APE; stored in the sloping isopycnals) to EPE, as the classic nonrotated energetics would suggest. However, the two VBF terms exhibit opposite roles in terms of transferring energy. VBF_{n} represents the stratifying effect of the instabilities; it is the energy transfer, induced by the slope-normal motion, from EPE to EKE and hence is an energy source of EKE and a sink of EPE. VBF_{c} represents the EPE gained by overcoming the cross-slope gravity during the stratifying process; so it is an energy sink of EKE and a source of EPE. Based on the scaling relation in appendix B and the nonhydrostatic parameter *ε* ≪ 1,

which is the ratio of two different types of EPE sources, that is, the EKE conversion and APE conversion to EPE; larger *δ* indicates larger EKE conversion back to EPE and hence stronger suppression. On the other hand,

larger *S*_{r} indicates smaller EPE conversion to EKE and hence weaker instability growth.

It has been suggested by previous studies that the horizontal slope Burger number *S*_{H} = *δ*/Ri is relevant to the instability suppression (Hetland 2017; Wenegrat et al. 2018). Wenegrat et al. (2018) demonstrate that VBF_{c}/VBF_{n} can be scaled as *S*_{H}; larger *S*_{H} indicates larger reduction of VBF and hence weaker instabilities. The difference of scaling is that *S*_{H} scales the ratio between the VBF components rather than the components as Eqs. (17) and (18) do. Figure 8b shows *S*_{H} in the Ri − *δ* space, and Fig. 8d shows the comparison between *S*_{H} and *S*_{H} but in a complex way. Although both *S*_{H} and _{c} (EKE sink) and VBF_{n} (EKE source), the later formula seems to be more accurate in term of indicating growth rate.

## 4. Discussion

### a. Application to coastal buoyancy-driven flow

The adapted theories are directly applicable to baroclinic instabilities in a bottom boundary layer (Wenegrat et al. 2018), where both the bottom and surface can be assumed to be tilted and parallel. However, these theories may not seem as directly applicable to surface-intensified baroclinic instabilities that are usually associated with tilted bottoms but flat surfaces (e.g., instabilities in coastal fronts). In this section, we will address, by scale analysis, the applicability of the adapted Stone model to the baroclinic instabilities in coastal buoyancy-driven flow over sloping bathymetry.

The following scale analysis demonstrates that the adapted Stone model can be used as an approximation of the flat surface case so long as the slope Burger number *H* is the depth at the center of the front *α* is a constant bottom slope. Considering the motions within the length scale of baroclinic instabilities that are on the order of the Rossby deformation radius *R*_{d} = *NH*/*f* (Eady 1949; Stone 1966, 1970), the dimensional flat surface can be scaled as

where *S* ≡ (*N*/*f*)*α* = *δ*Ri^{−1/2} is the slope Burger number. For the situations characterized by *δ* ~ *O*(10^{−1}) and Ri ~ *O*(1) (which are the cases in this study), the slope Burger number *S* is *O*(10^{−1}) so that the uniform fluid depth *z* = 1 will be a reasonable assumption. In other words, if *S* ~ *O*(10^{−1}), motions with lateral displacements *O*(Rd) will span a depth range of *H* ± 0.1*H*, and hence the uniform fluid depth will be a first-order approximation.

With the uniform fluid depth approximation, the rigid-lid boundary condition at the flat surface in the rotated coordinates is

where *S*_{H} ≡ *αM*^{2}*f*^{−2}= *δ*Ri^{−1} is the horizontal slope Burger number (Hetland 2017). For the situations characterized by *δ* ~ *O*(10^{−1}) and Ri ~ *O*(1) (which are the cases in this study), the horizontal slope Burger number *S*_{H} is *O*(10^{−1}) so that the no-flow boundary condition

### b. Numerical simulations of instabilities in coastal buoyancy-driven flow

A series of existing idealized numerical simulations, first analyzed by Hetland (2017), are used to examine the feasibility of the adapted Stone model. The idealized model domain is a 260 km (along-slope) × 128 km (across-slope) continental shelf with the uniform bathymetric slope *α* = 10^{−3} across all simulations. The depth increases from 5 m onshore to 133 m offshore. The model grid has 1-km uniform horizontal resolution and 30 layers in the vertical direction. The boundary conditions are periodic along-slope, open (with a sponge layer) offshore, and closed (no-slip) at the coast. The *k*–*ε* turbulence closure scheme is used to calculate the vertical mixing, and bottom friction is defined using a specified bottom roughness and a log-layer approximation. The model is unforced and run as an initial-value problem. The initial density field is a coastal buoyant front with a constant vertical stratification *N*^{2} over the whole shelf and a constant lateral buoyancy gradient *M*^{2} within the offshore distance of *W* = 50 km. The initial current field is configured in the thermal wind balance with the density field. The initial density and current fields of the base case (Ri = 2.0 and *δ* = 0.1) are shown in Fig. 9. Initial fields are varied among simulations to cover a range of situations of instability formation.

The idealized simulations were configured in the parameter space of the Richardson number Ri and the slope parameter *δ*, and all the simulations used in this study are listed in Table 1. All the simulations were run with same stratification *N*^{2} and same bottom slope *α*, but with different lateral buoyancy gradients *M*^{2} and Coriolis parameters *f* that are determined by each combination of Ri and *δ*. Note that the simulations listed in Table 1 are part of the simulations conducted in Hetland (2017); simulations with the Richardson number Ri = 1 or Ri = 10 or the slope parameter *δ* > 0.5 or the horizontal slope Burger number *S*_{H} > 0.2 were excluded. Ri = 1 is around the boundary between baroclinic instabilities and symmetric instabilities (Haine and Marshall 1998; Boccaletti et al. 2007), so the simulations with Ri = 1 are excluded to ensure only baroclinic instabilities can form. The simulations with Ri = 10 are excluded to ensure the nongeostrophic regime, and to minimize the influence of bottom friction in these simulations with long instability growth rates. The instability formations in the simulations with *δ* > 0.5 are excluded because they are generally weak and also strongly influenced by bottom friction. The simulations with *S*_{H} > 0.2 are excluded to ensure

Simulations in the parameter space of Ri and *δ*. All simulations were run with *N*^{2} = 1.00 × 10^{−4} s^{−2} and *α* = 1.00 × 10^{−3}. Other parameters are determined by varing Ri and *δ*. The slope Burger number is determined by *S* = *δ*Ri^{−1/2}. The horizontal slope Burger number is determined by *S*_{H} = *δ*Ri^{−1}. The Coriolis parameter is determined by *f* = *Nα*/*S*. The horizontal buoyancy gradient is determined by *M*^{2} = *Nf*Ri^{−1/2}. The terms ^{−1}, based on the adapted Stone model, the BG model, and the flat-bottom Stone model, respectively. The term ^{−1}, and *T*_{reg} is the time scale to truncate a EKE series for the regression.

The growth rate of instabilities is estimated in each simulation and then compared to the QG model and the adapted Stone model. Note that the spatial scale of the instabilities increases in the offshore direction (see Fig. 10) because the deformation radius, Rd = *NH*/*f*, increases with increasing depth offshore. We only focus on the growth of the largest instabilities at ~50 km offshore with the depth *H* ~ 50 m, because they are the most energetic and dispersive components (Thyng and Hetland 2017, 2018).

EKE is used to quantify the growth rates of instabilities. Given that the domain is periodic in the along-slope direction, the velocity field associated with the instability eddies is calculated by subtracting the along-slope background velocity from the original velocity field. So, EKE is dominated by the largest eddies. Then, the EKE can be determined by integrating the kinetic energy of the eddy flow field over the whole domain. Last, the EKE is normalized by the initial domain-integrated mean kinetic energy MKE_{Initial}. Figure 11 (top) shows the normalized EKE, EKE/MKE_{Initial}, of all the simulations listed in Table 1.

The EKE in each case appears to increase exponentially from the start (Fig. 11, top), but eventually the rapid increase is retarded by friction and finite amplitude effects. To isolate our results from these frictional effects and compare our results more directly with the theories that do not consider the influence of friction, we truncate the EKE time series where it reaches half of the maximum, removing the later part that is potentially influenced by friction. The truncated time scale for each simulation is listed in Table 1, and the truncated EKE time series are shown in Fig. 11 (middle). We take the base case (Ri = 2.0, *δ* = 0.1) as an example to show the comparison between the simulated growth rate and the theoretical predictions. The truncated EKE time series of the base run is shown in Fig. 11 (bottom), and the best exponential function to fit it has a growth rate of 1.73 day^{−1} (*r*^{2} = 0.996). On the other hand, the maximum dimensional growth rate for the base case is estimated based on the adapted Stone model (Wenegrat et al. 2018), the flat-bottom Stone model (Stone 1971), and the QG model (Blumsack and Gierasch 1972); they are

A regressed estimate of simulated EKE growth rate, *t* tests are conducted to see if the regressed and predicted growth rates are statistically equivalent. The two-tailed *p* value for the adapted Stone theory comparison is *p* = 0.42, for the QG theory *p* = 0.04, and for the flat-bottom Stone theory *p* = 0.03. We cannot reject the null hypothesis for the adapted Stone theory if a *p* value threshold of 5% is used, indicating the regressed and predicted growth rate distributions are indistinguishable. However, we can reject the null hypothesis in the tests for the QG theory and the flat-bottom Stone theory; the regressed and predicted distributions are distinct, as apparent from the offset from the 1:1 line in Figs. 12b and 12c. Moreover, the lower panels of Fig. 12 show the distribution of growth rate errors in the Ri–*δ* space. Compared to the adapted Stone model, the QG model and the flat-bottom Stone model have higher growth rate errors, particularly in the low Richardson number cases (Ri = 2.0 and 3.0); this implies that these models are not able to accurately describe the development of the submesoscale baroclinic instability eddies under energetic flow situations. However, under the conditions of

Both the nongeostrophic effect and sloping bathymetry are important for indicating the instability growth. The overestimation of growth rates in QG theory is presumably due to the missing of the nongeostrophic suppressing effect. Applying the flat-bottom Stone theory misses the suppressing effect of the sloping bathymetry. From the perspective of energetics, assuming flat-bottom (*δ* = 0) will overestimate the total VBF (by underestimating the EKE sink VBF_{c} and overestimating the EKE source VBF_{n}) and hence underestimate the suppression. From the perspective of wave interactions, assuming the flat-bottom will facilitate/overestimate the wave interaction and hence underestimate the suppression as well.

Furthermore, the adapted Stone theory might overestimate the suppression in the coastal case by assuming a sloping surface, as the comparison between the BG and Mechoso models suggests (see Fig. 2). But the nonlinear simulations indicate that the adapted Stone theory is most accurate among these existing theories, which suggests that the overestimation is not prohibitively large. On the other hand, there is little improvement when moving from the QG theory to the flat-bottom Stone theory; the underestimation of the suppression by assuming a flat-bottom seems not negligibly small.

### c. Linking S_{r} to potential vorticity

Above, the parameter *S*_{r} is linked to instability growth rate. Here we show how this parameter is related to normalized potential vorticity. The reference potential vorticity in this case is *fN*^{2}, the case with no horizontal density gradients and thereby no associated balanced flow. In the adapted Stone model, the normalized Ertel potential vorticity (Thomas et al. 2016) can be written as

For the adapted Stone model, the potential vorticity is uniform throughout the domain, but in the adapted Sakai model, the sloping interface creates a gradient in potential vorticity. In this case, we consider the change in potential vorticity, ΔPV, across an inertial radius *L*_{i} = *U*_{0}/*f*

where *β*_{T} = [(*α* + *α*_{p})*f*]/*H*_{0} is the topographic beta supplemented by the isopycnal slope. Thus, if we consider a parcel going from high to low potential vorticity, the normalized potential vorticity will be _{0} = *f*/*H*_{0} is the reference potential vorticity starting location on the high potential vorticity side of the inertial circle. Thus the two interpretations are somewhat similar.

## 5. Conclusions

Layered and continuously stratified models of nongeostrophic baroclinic instability over sloping topography are explored in the buoyant flow regime to study the suppression mechanism. The layered model (i.e., the adapted Sakai model) reveals a new parameter, the slope-relative Burger number *S*_{r} ≡ (*M*^{2}/*f*^{2})(*α* + *α*_{p}) {the bulk form is *S*_{r} = [*U*_{0}/(*H*_{0}*f*)](*α* + *α*_{p})}, which is a dimensionless parameter that controls the suppression of instabilities in the nongeostrophic limit. The continuously stratified model (i.e., the adapted Stone model) shows that, when the regime shifts from QG to nongeostrophic, the growth of instabilities is inhibited with increasing *S*_{r}.

In the adapted Sakai model, the instability growth rate linearly decreases with increasing *S*_{r}. The physical mechanism behind *S*_{r} is explored based on the wave resonance theory (Sakai 1989). In the physical wave coordinates consisting of the Rossby waves, baroclinic instabilities are interpreted as the Rossby wave resonance; supported by the adapted Sakai model, the maximum normalized growth rate of instabilities is found to be nearly 1:1 to the resonance rate of the Rossby waves. The slope-relative Burger number *S*_{r} modifies the Doppler-shifted phase speed of the Rossby waves, alters the wave resonance, and hence influences the growth of instabilities.

In the adapted Stone model, the instability growth rate linearly decreases with decreasing *δ*_{r} (or *δ*) and *S*_{r}. Supported by the energetics, larger *δ*_{r} represents larger VBF_{c}, an EKE sink, and hence stronger suppression; large *S*_{r} represents smaller VBF_{n}, an EKE source, and hence weaker instability growth. Given that *S*_{r})^{−1/2} represents the instability suppression when the regime shifts from QG to nongeostrophic.

The adapted models are intrinsically applicable to baroclinic instabilities in a bottom boundary layer but has one limitation when applying to coastal buoyancy-driven flow—the sloping-surface assumption. Supported by scale analysis, idealized numerical simulations of coastal buoyancy-driven flow are used to test the feasibility of the adapted Stone model in flat-surface situations. The comparison of the numerical results and theoretical predictions indicates that the limitation is not prohibitive if the slope Burger number

## Acknowledgments

This work was funded by NSF (Grant OCE-1851470), Texas General Land Office (Contract 18-132-000-A673), and Gulf of Mexico Research Initiative through the CSOMIO consortium (Contract R01984). Lixin Qu was supported by a graduate research fund from Texas Sea Grant and a scholarship from China Scholarship Council. We thank Jacob Wenegrat, Leif Thomas, and two anonymous reviewers for very helpful suggestions and codes when preparing this manuscript.

## APPENDIX A

### Adapted Sakai Model

The following derivation follows Sakai (1989) but is modified to account for the presence of sloping bottom and surface. Considering a rotating two-layer channel with sloping bottom and top and currents in the thermal wind balance, the linearized equations of the perturbations are

subject to

where *g*′ = (Δ*ρ*/*ρ*_{0})*g* is the reduced gravity, *f* is the Coriolis parameter, *U*_{0} and −*U*_{0} for simplicity (Sakai 1989). Also, *α* is the bottom slope and *α*_{p} = 2*U*_{0}*f*/*g*′ is the isopycnal slope.

Considering the time scale as 1/*f*, the horizontal length scale as the Rossby deformation radius *H*_{0}, the scaling relations about the variables in Eq. (A1) are

where *δ*_{r} ≡ (*α* + *α*_{p})/*α*_{p} = *δ* + 1 is the slope-relative parameter [*δ* ≡ *α*/*α*_{p} = (*αg*′)/(2*U*_{0}*f*) is the slope parameter]. The dimensionless form of Eq. (A1) is, then,

subject to

Assuming an ansatz of the form

subject to

## APPENDIX B

### Adapted Stone Model

The following derivation follows Stone (1966, 1970, 1971) but is modified to account for the presence of sloping bottom and surface. The coordinates are rotated to align with the sloping topography as Wenegrat et al. (2018), and the derivation is essentially equivalent to Wenegrat et al. (2018) but with a different coordinate orientation. Dimensionally, the equations describing a rotational fluid field with the Boussinesq approximation in the rotated coordinates are

subject to

where *θ* is the slope angle, *ρ* and *ρ*_{0} are the seawater density and the reference, respectively). Rigid-lid boundary conditions are applied at the surface (

Considering the horizontal velocity scale as *U*, the time scale as *f*^{−1}, the horizontal length scale as *U*/*f*, and the vertical length scale as *H*, the scaling relations about the variables in Eq. (B1) are, then,

Thus, Eqs. (B1) and (B2) have the dimensionless form

subject to

where Ri = *N*^{2}*H*^{2}*U*^{−2} = *N*^{2}*f*^{2}*M*^{−4} is the Richardson number, *δ* = *αN*^{2}*M*^{−2} is the slope parameter, and *ε* = *fHU*^{−1} = *f*^{2}*M*^{−2} is the nonhydrostatic parameter.

Considering a background current flowing only in the along-slope direction, constrained by the thermal wind relation, the background state can be described as

where *M*^{2}/*f* is the thermal wind shear in the nonrotated coordinates and

where *S*_{r} = *δ*_{r}Ri^{−1} is the slope-relative Burger number. Assuming *u*′, *υ*′, *w*′, *b*′, and *p*′ are the small perturbations from the background state, the equations governing the perturbed state can be linearized by neglecting the product of small terms as (dropping the primes for clarity and neglecting the viscosity and diffusion terms)

Assuming an ansatz of the form

subject to

## APPENDIX C

### Rossby Wave Interactions in the Adapted Sakai Model

The following derivation follows Sakai (1989) but is modified to account for the presence of sloping bottom and top. The details about the interaction theory and associated derivation can be found in section 4 and appendix A of Sakai (1989). We will only focus on the interactions between Rossby waves. In the adapted Sakai model, the physical wave coordinates consisting of the Rossby waves in the upper and lower layers are, then,

where *n* = 1, 2, 3, …) is the dimensional across-slope wavenumber, *δ*_{r} = 1 + [(*αg*′)/(2*U*_{0}*f*)] is the slope-relative parameter. Assuming an ansatz of the form

where *A*_{n} and *B*_{n} are the magnitudes in the physical wave coordinates, **E**_{1n} is the complex conjugate of the adjoint vector *d*_{2n}. The interactions between Rossby waves can be described by

where *A*_{n} and *B*_{n} in Eq. (C3):

which can be reduced to the flat-bottom case, *δ*_{r} = 1.

## APPENDIX D

### Energetics in the Rotated Coordinates

The following energetics closely follows Wenegrat et al. (2018) but with a different coordinate orientation. The coordinates are rotated to align with the sloping topography. The relation between the nonrotated and rotated coordinates is

where the tildes denote the nonrotated coordinates. The background buoyancy has a constant vertical gradient

So, the background buoyancy in the rotated coordinates could be set as

By linearizing the dimensional governing equations, Eq. (B1), we can get the dimensional equations describing the perturbations as below (dropping asterisks for clarity)

where the bars denote the background variables, the primes denote the perturbation variables, and

By multiplying *b*′/*N*^{2} to the buoyancy equation in Eq. (D3) and taking the Reynolds averaging, the eddy potential energy (EPE) equation can be obtained as

where HBF_{c} is the horizontal buoyancy flux (HBF) contributed by cross-slope motion, HBF_{n} is the HBF contributed by slope-normal motion, VBF_{c} is the vertical buoyancy flux (VBF) contributed by cross-slope motion, VBF_{n} is the VBF contributed by slope-normal motion, DPE is the dissipation of EPE, and RPE is the redistribution of EPE. Baroclinic instabilities get energy from the available potential energy of the background front. The energy transfer from mean available potential energy to EPE is through the HBF terms, which requires _{c} ≫ HBF_{n}.

By multiplying

where SP is the shear production, DKE is the dissipation of EKE, RKE is the redistribution of EKE, and _{n} is an EKE source. Since _{c} is an EKE sink; it represents the energy transfer back to EPE due to overcoming the cross-slope gravity.

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