1. Introduction
An important development in our understanding of the upper ocean, occurring largely over the last decade, has been the recognition that submesoscale (horizontal scales of approximately 0.1–10 km) fronts, eddies, and instabilities are a prominent component of the dynamics of the ocean surface mixed layer (Boccaletti et al. 2007; Callies et al. 2015). These submesoscale processes alter the classic one-dimensional picture of boundary layer dynamics, giving rise to a host of new physical processes that have been shown to affect both large-scale ocean processes (Lévy et al. 2010; Wenegrat et al. 2018) and the turbulence properties of the boundary layer itself (Taylor and Ferrari 2010; Thomas and Taylor 2010; Taylor 2016). However, despite the large body of literature that has developed on submesoscale processes in the surface boundary layer [as reviewed in Thomas et al. (2008) and McWilliams (2016)], these types of processes have received much less attention in the bottom boundary layer (BBL),1 despite BBLs over sloping topography exhibiting key similarities with surface mixed layers at a front, particularly the existence of available potential energy in the form of a horizontal buoyancy gradient (Fig. 1).
(a) Schematic of a BBL over a linear topographic slope, with solid contours indicating buoyancy surfaces and the dashed line indicating the top of a weakly stratified BBL. The rotated coordinate system is indicated in the lower left. (b) Simplified domain used in the basic parameter space exploration (section 2c). Throughout, the background velocity is assumed to be a function only of the slope-normal coordinate z. The relationships between the slope-normal derivative and derivatives in the nonrotated frame are also indicated.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
(a) Schematic of a BBL over a linear topographic slope, with solid contours indicating buoyancy surfaces and the dashed line indicating the top of a weakly stratified BBL. The rotated coordinate system is indicated in the lower left. (b) Simplified domain used in the basic parameter space exploration (section 2c). Throughout, the background velocity is assumed to be a function only of the slope-normal coordinate z. The relationships between the slope-normal derivative and derivatives in the nonrotated frame are also indicated.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
(a) Schematic of a BBL over a linear topographic slope, with solid contours indicating buoyancy surfaces and the dashed line indicating the top of a weakly stratified BBL. The rotated coordinate system is indicated in the lower left. (b) Simplified domain used in the basic parameter space exploration (section 2c). Throughout, the background velocity is assumed to be a function only of the slope-normal coordinate z. The relationships between the slope-normal derivative and derivatives in the nonrotated frame are also indicated.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Recently, however, observations and numerical modeling have begun to suggest that the BBL does indeed support an active submesoscale turbulence field, with dynamical implications potentially as broad as for submesoscale processes in the surface boundary layer. Submesoscale processes in BBLs along topography have been shown to enhance cross-shelf exchanges (Gula et al. 2015), contribute to interior water mass transformation (Ruan et al. 2017), generate long-lived submesoscale coherent vortices (Molemaker et al. 2015), and provide another route for cascading energy from the large-scale flow into unbalanced motions and eventual dissipation (Gula et al. 2016). Particular focus has been given to centrifugal (CI) and symmetric instabilities (SI) in the BBL, both of which are associated with negative potential vorticity (Hoskins 1974; Haine and Marshall 1998), which can occur in BBLs over steep topography (Dewar et al. 2015) or in the presence of interior flows driving downslope Ekman transport in the boundary layer (Allen and Newberger 1998). The role of baroclinic instability in the BBL is currently less clear, despite the fact that surface mixed layer baroclinic instabilities are one of the more thoroughly studied aspects of submesoscale dynamics, with demonstrated effects on boundary layer restratification (Boccaletti et al. 2007; Fox-Kemper et al. 2008), surface potential vorticity fluxes (Wenegrat et al. 2018), boundary layer turbulence (Taylor 2016), biological productivity (Mahadevan et al. 2012), and the mesoscale eddy field (Sasaki et al. 2014). Understanding the conditions under which baroclinic instability can be expected to be active in the BBL is therefore a key step toward understanding submesoscale turbulence in the BBL.
In this article, we extend a variety of earlier works on baroclinic instability over topography (e.g., Blumsack and Gierasch 1972; Mechoso 1980; Pedlosky 2016; Solodoch et al. 2016) to the nongeostrophic regime appropriate for the low Richardson numbers typical of the BBL. Using a linear stability analysis, we explore the parameter dependence of baroclinic instabilities in a weakly stratified lower layer over sloping topography, which can be considered as a bottom counterpart to the surface mixed layer instability (Boccaletti et al. 2007). The submesoscale BBL mode is shown to be more robust to topographic slope than mesoscale baroclinic instability, and we demonstrate that the solutions to two classic one-dimensional theories of the BBL structure are unstable to BBL baroclinic instability. An example idealized nonlinear simulation further suggests that at finite amplitude the BBL baroclinic instability mode can generate vertical buoyancy fluxes, and vertical velocities, similar in magnitude to those associated with surface mixed layer instabilities. It is therefore expected that submesoscale baroclinic instability, along with the symmetric and centrifugal modes (Allen and Newberger 1998; Molemaker et al. 2015; Dewar et al. 2015; Gula et al. 2016), plays a leading-order role in the dynamics of both coastal (Brink 2016; Brink and Seo 2016; Hetland 2017), and deep ocean BBLs, with implications for closing the upwelling limb of the abyssal circulation (Ferrari et al. 2016; Callies 2018).
The article is organized as follows. In section 2 the equations governing linear perturbations in the BBL are developed, relevant quasigeostrophic results are briefly reviewed, and the parameter dependence of the baroclinic mode in the nongeostrophic limit is explored numerically. In section 3 a steady one-dimensional solution for a BBL in the presence of bottom-intensified turbulence, with parameters similar to those observed along the mid-Atlantic ridge, is shown to support growing baroclinic instability in a weakly stratified outer layer on the order of hundreds of meters thick. In section 4 numerical solutions of the time-dependent, one-dimensional, Ekman adjustment problem for flow along a slope are likewise shown to be unstable to submesoscale baroclinic instability in a well-mixed bottom boundary layer that is on the order of tens of meters thick. An example nonlinear simulation is presented in section 5, allowing us to comment briefly on the finite-amplitude behavior, and in section 6 the baroclinic mode is discussed in relation to symmetric and centrifugal instability, both of which can also be present in BBLs over sloping topography. Major findings and broader implications are summarized in section 7.
2. Baroclinic instability in the bottom boundary layer
a. Theory
For simplicity, we will only consider background flows that are invariant in the rotated along- and across-slope direction, which excludes slope-normal background flows (i.e., W = 0). These restrictions are consistent with the conceptual model of a BBL of uniform thickness along a linear slope (sections 3 and 4) and still allow for a vertically and horizontally sheared background flow in the nonrotated coordinate system (where the horizontal shear is a consequence of the sloping topography, as shown schematically in Fig. 1b). We will also retain only the slope-normal component of the turbulent diffusion terms, which for an isotropic turbulent diffusivity can be understood as an assumption of small aspect ratio, H2/L2 ≪ 1, where H and L scale the slope-normal and rotated horizontal dimensions, respectively. This choice does not affect the conclusions of this article; however, we emphasize that the properties of diapycnal and isopycnal turbulent mixing in the BBL are currently poorly constrained from available observations, and it is possible that under some circumstances—for example, along very steep topography—the full three-dimensional diffusion operator may be important.
To motivate the following sections, we first consider the baroclinic growth rates for an idealized profile of stratification (N2) and shear (Λ), with varying slope angle. Figure 2 shows profiles of background velocity and buoyancy, where we assume there is an inviscid along-slope flow, V(z) = Λz secθ, and a stratified interior overlying a weakly stratified BBL. Note that these profiles are held fixed in the nonrotated frame, hence the isopycnal slope in the rotated frame changes as a function of the slope angle. The inviscid problem only requires boundary conditions on the slope-normal velocity, which are given by w = 0 at z = 0 and w = −u tanθ at z = 1000 m, representing a rigid horizontal upper boundary 1000 m from the bottom. Growth rates of the baroclinic mode (k = 0) are calculated numerically and are shown as a function of increasing slope angle in Fig. 3. The full-depth baroclinic mode can be seen at wavelengths of ~30 km. This mode is strongly modulated by slope angle, with maximum growth rates reduced, and shifted to higher wavenumber, as the slope angle increases, consistent with the expectation from quasigeostrophic (QG) theory (discussed further below). The fastest-growing instability, however, is found at wavelengths near 1500 m, with perturbation structure and energetics shown in Fig. 4. This is a bottom-enhanced submesoscale baroclinic instability, resulting from the interaction of counterpropagating Rossby waves along the lower-boundary and on the interface of changing stratification between the boundary layer and the interior. These ocean bottom modes are thus similar to baroclinic instabilities in the atmospheric boundary layer (Nakamura 1988) and mirror the structure of surface mixed layer instabilities (Boccaletti et al. 2007; Callies et al. 2016). As shown in Fig. 3, the growth rates of the submesoscale mode are relatively insensitive to increasing slope angle, suggesting that submesoscale baroclinic instability may be a robust feature of the bottom boundary layer. To understand this behavior, we will first briefly review relevant results from the QG limit and then explore parameter space in the non-QG limit relevant for submesoscale instabilities.
Example idealized vertical structure of a stratified interior overlying a weakly stratified BBL.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Example idealized vertical structure of a stratified interior overlying a weakly stratified BBL.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Example idealized vertical structure of a stratified interior overlying a weakly stratified BBL.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Growth rates, normalized by f −1, for the idealized profile shown in Fig. 2 for various slope angles (legend). The low-wavenumber, full-depth baroclinic modes are indicated by the dashed box. Growth rates of the low-wavenumber instability are highly modulated by the slope angle, whereas the high wavenumber features are robust to changing slope angle.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Growth rates, normalized by f −1, for the idealized profile shown in Fig. 2 for various slope angles (legend). The low-wavenumber, full-depth baroclinic modes are indicated by the dashed box. Growth rates of the low-wavenumber instability are highly modulated by the slope angle, whereas the high wavenumber features are robust to changing slope angle.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Growth rates, normalized by f −1, for the idealized profile shown in Fig. 2 for various slope angles (legend). The low-wavenumber, full-depth baroclinic modes are indicated by the dashed box. Growth rates of the low-wavenumber instability are highly modulated by the slope angle, whereas the high wavenumber features are robust to changing slope angle.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Structure of the fastest-growing perturbation (l ≈ 6 × 10−4 m−1) for θ = 5 × 10−3. Velocities are normalized relative to the maximum across-slope velocity and buoyancy relative to the maximum buoyancy perturbation. Kinetic energy tendency terms are defined in appendix A, with buoyancy production given by VBP and shear production by the sum of the lateral and vertical shear production terms (LSP + VSP).
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Structure of the fastest-growing perturbation (l ≈ 6 × 10−4 m−1) for θ = 5 × 10−3. Velocities are normalized relative to the maximum across-slope velocity and buoyancy relative to the maximum buoyancy perturbation. Kinetic energy tendency terms are defined in appendix A, with buoyancy production given by VBP and shear production by the sum of the lateral and vertical shear production terms (LSP + VSP).
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Structure of the fastest-growing perturbation (l ≈ 6 × 10−4 m−1) for θ = 5 × 10−3. Velocities are normalized relative to the maximum across-slope velocity and buoyancy relative to the maximum buoyancy perturbation. Kinetic energy tendency terms are defined in appendix A, with buoyancy production given by VBP and shear production by the sum of the lateral and vertical shear production terms (LSP + VSP).
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
b. Quasigeostrophic baroclinic instability over sloping topography
These growth rates are shown as a function of α in Fig. 5 for the case of a horizontal upper boundary (left) and for a sloping upper boundary, αub = α (right). In both cases there is a region of instability for any α > −1, with increasing positive values of α associated with reduced growth rates, and an increasingly narrow region of instability at high wavenumbers, as seen for the low-wavenumber modes in Fig. 3. While linear theory predicts finite growth rates at arbitrarily high α, in reality nonlinear (Pedlosky 2016), viscous, and non-QG (section 2c) effects may provide an upper bound on physically relevant wavenumbers. With an upper boundary parallel to the topography, there is no longer a barotropic potential vorticity (PV) gradient, and hence there is no longer a long-wave cutoff for α < 0 (Fig. 5, right). The sloping boundaries also lead to a destabilization of low wavenumbers for α < 0, with maximal growth rates for α = −0.5. This is a consequence of parcel trajectories being forced by the boundary slope to cross isopycnals in a manner that achieves maximal extraction of available potential energy (Mechoso 1980). The increase of growth rates as 
Baroclinic instability growth rates from (18) as a function of slope parameter α and wavenumber normalized by the deformation radius NH/f 
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Baroclinic instability growth rates from (18) as a function of slope parameter α and wavenumber normalized by the deformation radius NH/f
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Baroclinic instability growth rates from (18) as a function of slope parameter α and wavenumber normalized by the deformation radius NH/f
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
c. Nongeostrophic baroclinic instabilities over sloping topography
To explore parameter space in the non-QG limit, using (11)–(15) in the rotated frame, we continue to assume the flow is inviscid, with uniform background vertical stratification, N2, and along-slope flow with uniform slope-normal shear. This allows the background velocity field to be written in the rotated frame as V(z) = Λz secθ, and in the nonrotated frame as 
In the nondimensionalization used here, l acts as a Rossby number, quantifying advection, vis-à-vis Doppler shifting of the perturbations by the mean flow, relative to the Coriolis term. Non-QG effects will therefore become important when 
Growth rates for Ri = (left) 10 and (right) 1, as a function of slope parameter α and wavenumber normalized by the deformation radius NH/f 
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Growth rates for Ri = (left) 10 and (right) 1, as a function of slope parameter α and wavenumber normalized by the deformation radius NH/f
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Growth rates for Ri = (left) 10 and (right) 1, as a function of slope parameter α and wavenumber normalized by the deformation radius NH/f
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
An alternate, physically intuitive, way to consider the parameter dependence of the instabilities is to hold the slope angle and horizontal buoyancy gradient fixed (Fig. 7). Varying the stratification thus varies the Richardson number and α, approximating interior stratification encountering deep topography, and adjusting within a turbulent boundary layer (sections 3 and 4). At low Richardson numbers the growth rates increase, and higher wavenumbers become unstable, consistent with the decreasing stratification allowing greater vertical penetration and coupling of the boundary waves. Assuming values typical of interior flows (V ~ 0.1 m s−1, f = 10−4 s−1) the most unstable modes for Ri < 10 have a horizontal scale of O(5) km and growth rates on the order of 0.5–3 inertial periods.
Growth rates for a fixed topographic slope angle (θ = 5 × 10−3) and vertical shear Λ = 10−3 s−1, with varying N2. Wavenumbers are normalized by ΛH/f, growth rates by f−1.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Growth rates for a fixed topographic slope angle (θ = 5 × 10−3) and vertical shear Λ = 10−3 s−1, with varying N2. Wavenumbers are normalized by ΛH/f, growth rates by f−1.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Growth rates for a fixed topographic slope angle (θ = 5 × 10−3) and vertical shear Λ = 10−3 s−1, with varying N2. Wavenumbers are normalized by ΛH/f, growth rates by f−1.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
It is important to note that for isopycnals that intersect the topography at right angles, a reasonable first approximation to the BBL structure, the slope parameter is given by α = tan2θ, which for small slope angle can be approximated as α ≈ θ2. It is estimated that approximately 95% of the world’s ocean has 
Up until this point we have considered inviscid flows with greatly simplified vertical structures for the background buoyancy and shear fields, as an aid to interpreting the problem parameter dependencies. In the following sections, we will consider the stability characteristics of the bottom boundary layer structure predicted by two somewhat more realistic dynamical theories of the ocean bottom boundary layer. The first results from bottom intensified turbulent mixing, the second from Ekman adjustment of the bottom boundary layer to an imposed interior flow.
3. Bottom-intensified turbulent mixing
Available potential energy near the bottom boundary can be generated by turbulent mixing associated with processes such as the breaking of internal waves over sloping topography. Observations show that turbulence over rough topography is strongly bottom enhanced, suggesting there should be a dipole of vertical velocities along topography, with downwelling in an outer layer a few hundred meters thick, where diapycnal buoyancy fluxes are divergent, and upwelling in an inner layer adjacent to the bottom, where diapycnal buoyancy fluxes are convergent (e.g., Polzin 1997; Ledwell et al. 2000; St. Laurent et al. 2001; Ferrari et al. 2016; McDougall and Ferrari 2017). However, 1D boundary layer solutions, with turbulent viscosities based on observations, imply outer-layer stratification much weaker than observed, and hence weak vertical velocity dipoles (Callies 2018). Baroclinic instabilities in the mixing layer would alter the 1D buoyancy budget, with important implications for closing the upwelling branch of the abyssal circulation, and hence here we consider the stability characteristics of 1D boundary layer solutions with turbulent diffusivities based on observations.
The solution for the background fields is shown in Fig. 8. There is an inner layer of thickness 
Solution to (19)–(21) with parameters as given in (25) and (26). In the upper-left panel the along-slope velocity V is shown in orange and the across-slope velocity U is shown in blue.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Solution to (19)–(21) with parameters as given in (25) and (26). In the upper-left panel the along-slope velocity V is shown in orange and the across-slope velocity U is shown in blue.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Solution to (19)–(21) with parameters as given in (25) and (26). In the upper-left panel the along-slope velocity V is shown in orange and the across-slope velocity U is shown in blue.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Growth rate of the baroclinic mode for the base state shown in Fig. 8.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Growth rate of the baroclinic mode for the base state shown in Fig. 8.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Growth rate of the baroclinic mode for the base state shown in Fig. 8.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
The perturbation structure and kinetic energy tendency of the fastest-growing mode are shown in Fig. 10. Phase lines are inclined into the mean shear (note f < 0), and the instabilities grow by releasing available potential energy in the outer layer, with maximum vertical buoyancy fluxes near z = 250 m. Perturbations quantities are enhanced near the bottom boundary, except for in the thin inner layer where dissipation dominates the energetics. At finite amplitude the eddy vertical buoyancy fluxes will provide a restratifying tendency, altering the simple 1D balance given by (19)–(21), and potentially modifying the structure of the vertical velocity dipole in the boundary layer. The net secondary circulation along sloping topography may therefore involve both eddy and Eulerian components, with implications for the dynamics of the abyssal circulation (Callies 2018).
Structure of the fastest-growing baroclinic mode for the base state shown in Fig. 8, discussed in section 3. Velocities are normalized relative to the maximum across-slope velocity and buoyancy relative to the maximum buoyancy perturbation. Kinetic energy tendency terms are defined in appendix A, with buoyancy production given by VBP, shear production by the sum of the lateral and vertical shear production terms (LSP + VSP), and dissipation by DKE.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Structure of the fastest-growing baroclinic mode for the base state shown in Fig. 8, discussed in section 3. Velocities are normalized relative to the maximum across-slope velocity and buoyancy relative to the maximum buoyancy perturbation. Kinetic energy tendency terms are defined in appendix A, with buoyancy production given by VBP, shear production by the sum of the lateral and vertical shear production terms (LSP + VSP), and dissipation by DKE.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Structure of the fastest-growing baroclinic mode for the base state shown in Fig. 8, discussed in section 3. Velocities are normalized relative to the maximum across-slope velocity and buoyancy relative to the maximum buoyancy perturbation. Kinetic energy tendency terms are defined in appendix A, with buoyancy production given by VBP, shear production by the sum of the lateral and vertical shear production terms (LSP + VSP), and dissipation by DKE.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
4. Stability of the bottom Ekman layer
The classic analysis of the laminar bottom boundary layer holds that, in the presence of an interior flow along the slope, a bottom Ekman layer must also develop to satisfy the bottom boundary condition on the momentum (Pedlosky 1979). Interior flow thus gives rise to across-slope flow in the Ekman layer that advects the across-slope buoyancy gradient, modifying the depth and stratification of the boundary layer (MacCready and Rhines 1991, 1993). For interior flow in the direction of topographic wave propagation (topography sloping upward to the right of the direction of flow in the Northern Hemisphere) the resulting Ekman flow is downslope, advecting light water under dense, causing convective mixing and deepening of the BBL (Garrett et al. 1993; Moum et al. 2004), an analog to the surface “Ekman buoyancy flux” associated with downfront winds at a surface mixed layer front (Thomas 2005). For interior flow in the opposite direction, the Ekman flow is upslope, increasing boundary layer stability by advecting dense water under light. In each case this adjustment process generates available potential energy in the BBL (Umlauf et al. 2015), and we demonstrate here that this available potential energy can allow for growing baroclinic instability in the BBL.
a. Downwelling favorable interior flow
The numerical solution for the case of downwelling-favorable interior flow with magnitude typical of a deep western boundary current (Toole et al. 2011), 
BBL response to a downwelling-favorable interior flow, 
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
BBL response to a downwelling-favorable interior flow,
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
BBL response to a downwelling-favorable interior flow,
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Instability growth rates (Fig. 12) are calculated using snapshots of the numerical solutions for U, V, B, κ, and ν as the background fields in the eigenvalue problem, (6)–(10). The baroclinic mode emerges after approximately five inertial periods, with growth rates increasing and wavenumbers decreasing as the boundary layer deepens in time. Growth of the baroclinic mode becomes faster than the Ekman adjustment time scale after only t ≈ 0.5τEk, emphasizing that these instabilities grow rapidly relative to the shutdown process. High-frequency variability is associated with weak inertial oscillations modifying the boundary layer structure. The structure of the fastest-growing mode at t = 10 inertial periods is shown in Fig. 13. Vertical buoyancy production of kinetic energy dominates the instability growth, with dissipation reducing the kinetic energy tendency by a factor of approximately 1/2. Shear production is a small contribution to the vertically integrated kinetic energy tendency and is dominated by the lateral shear production, although there are locally significant contributions from both vertical and lateral shear production terms at the top of the boundary layer.
Hovmöller plot of the baroclinic growth rates as a function of wavenumber and time, normalized by the inertial period, for the case of downwelling favorable interior flow (Fig. 11). Growth rates are normalized by f−1.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Hovmöller plot of the baroclinic growth rates as a function of wavenumber and time, normalized by the inertial period, for the case of downwelling favorable interior flow (Fig. 11). Growth rates are normalized by f−1.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Hovmöller plot of the baroclinic growth rates as a function of wavenumber and time, normalized by the inertial period, for the case of downwelling favorable interior flow (Fig. 11). Growth rates are normalized by f−1.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Structure of the fastest-growing baroclinic mode for the downwelling-favorable Ekman layer (Fig. 11) at t = 10 inertial periods, discussed in section 4. Velocities are normalized relative to the maximum across-slope velocity, buoyancy relative to the maximum buoyancy perturbation. Kinetic energy tendency terms are defined in appendix A, with buoyancy production given by VBP, shear production by the sum of the lateral and vertical shear production terms (LSP + VSP), and dissipation by DKE.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Structure of the fastest-growing baroclinic mode for the downwelling-favorable Ekman layer (Fig. 11) at t = 10 inertial periods, discussed in section 4. Velocities are normalized relative to the maximum across-slope velocity, buoyancy relative to the maximum buoyancy perturbation. Kinetic energy tendency terms are defined in appendix A, with buoyancy production given by VBP, shear production by the sum of the lateral and vertical shear production terms (LSP + VSP), and dissipation by DKE.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Structure of the fastest-growing baroclinic mode for the downwelling-favorable Ekman layer (Fig. 11) at t = 10 inertial periods, discussed in section 4. Velocities are normalized relative to the maximum across-slope velocity, buoyancy relative to the maximum buoyancy perturbation. Kinetic energy tendency terms are defined in appendix A, with buoyancy production given by VBP, shear production by the sum of the lateral and vertical shear production terms (LSP + VSP), and dissipation by DKE.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
It is important to emphasize that, for downwelling-favorable interior flow, the BBL can become unstable to symmetric or centrifugal instability, with growth rates that exceed the baroclinic mode (Allen and Newberger 1998). Future work will therefore consider the full nonlinear Ekman adjustment process; however, the findings presented here suggest that fast-growing submesoscale baroclinic instability likely plays an important role in the 3D adjustment of the BBL to interior flows, emerging after symmetric/centrifugal instability restratifies the boundary layer to a state of marginal stability (discussed further in section 6).
b. Upwelling favorable interior flow
When the interior flow is in the opposite direction, 
As in Fig. 11, but for upwelling favorable interior flow, 
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
As in Fig. 11, but for upwelling favorable interior flow,
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
As in Fig. 11, but for upwelling favorable interior flow,
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
As in Fig. 12, but for the upwelling favorable case shown in Fig. 14.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
As in Fig. 12, but for the upwelling favorable case shown in Fig. 14.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
As in Fig. 12, but for the upwelling favorable case shown in Fig. 14.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
5. Nonlinear simulation
In this section we present the results of an example of idealized nonlinear simulation. Solutions are found using Dedalus, in a computational domain that is doubly periodic in the rotated horizontal coordinate system. This is achieved by imposing a fixed across-slope buoyancy gradient, 
The model is initialized with a uniformly stratified interior (NI = 3.4 × 10−3 s−1) and a barotropic along-slope interior flow of 0.1 m s−1. The BBL is initialized in thermal wind balance, with a thickness of 100 m, such that Ri = 1.5 in the BBL (Fig. 16). The slope angle is θ = 10−2, giving a slope parameter in the BBL of α ≈ 0.15. For simplicity we consider the case of weak slope-normal viscosity and diffusivity, ν = κ = 10−5 m2 s−1, and, for numerical stability, a biharmonic diffusivity in the rotated horizontal (νh = 2 × 105 m4 s−1) (appendix B). A more complete exploration of parameter space using nonlinear simulations, including the use of physically realistic turbulence closures, will be the subject of future work.
Initial conditions for along-slope velocity (color) and buoyancy (gray contours) for the nonlinear simulation discussed in section 5 and appendix B.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Initial conditions for along-slope velocity (color) and buoyancy (gray contours) for the nonlinear simulation discussed in section 5 and appendix B.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Initial conditions for along-slope velocity (color) and buoyancy (gray contours) for the nonlinear simulation discussed in section 5 and appendix B.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Snapshots of the instability evolution are shown in Fig. 17. By day 10 the baroclinic mode has reached finite amplitude and is clearly evident in both the buoyancy and slope-normal velocity fields. The along-slope wavelength of the dominant instability, λ ≈ 6.4 km, is similar to that predicted by linear stability analysis of the domain-averaged buoyancy and velocity fields at day 1.5, λ ≈ 5.5 km. Importantly, the slope-normal velocities exceed 250 m day−1 (Fig. 17, bottom row), comparable to values associated with baroclinic instability in the surface boundary layer. These large-magnitude vertical velocities suggest that eddy fluxes in the BBL have the potential to affect a wide range of BBL processes, including, for example, nutrient fluxes between the BBL and interior, sediment transport, and coastal hypoxia.
Snapshots of (top) buoyancy and (bottom) slope-normal velocity (w) evaluated at 50 m above the bottom.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Snapshots of (top) buoyancy and (bottom) slope-normal velocity (w) evaluated at 50 m above the bottom.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Snapshots of (top) buoyancy and (bottom) slope-normal velocity (w) evaluated at 50 m above the bottom.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
The energetics of the nonlinear simulation are shown in Fig. 18. In the top panel the domain-integrated eddy kinetic energy (EKE) is shown, defined as 
(top) Domain-integrated EKE compared to the prediction of linear theory (calculated as discussed in section 5) and (bottom) domain-averaged EKE tendency (bottom) with components as indicated in the legends.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
(top) Domain-integrated EKE compared to the prediction of linear theory (calculated as discussed in section 5) and (bottom) domain-averaged EKE tendency (bottom) with components as indicated in the legends.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
(top) Domain-integrated EKE compared to the prediction of linear theory (calculated as discussed in section 5) and (bottom) domain-averaged EKE tendency (bottom) with components as indicated in the legends.
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
In this simulation, domain-averaged vertical buoyancy fluxes in the BBL reach 
6. Instability regimes
In addition to the baroclinic mode, (6)–(10) also admit symmetric and centrifugal instabilities (Haine and Marshall 1998), each of which may be active in the ocean BBL under certain conditions (Allen and Newberger 1998; Brink 2012; Molemaker et al. 2015; Gula et al. 2016). While the focus of this work is primarily on baroclinic instability, it is useful to also briefly summarize where in parameter space each of these modes is anticipated to be dominant.
An example regime diagram is shown in Fig. 19, as a function of the slope Burger number and the Richardson number. The q′ = 0 line divides the domain (dashed red line), with the entire area to the right of the line linearly unstable to baroclinic instability, although instabilities at small α are likely the most physically relevant (section 2c). For q′ < 0 the domain is further divided by the energetics, with LSP/VSP = 1 shown from theory [(33); dashed blue line in Fig. 19], assuming δ/Ri1/2 = f/N = 0.1. Below this line the symmetric mode grows by extracting mean kinetic energy through the vertical shear production. Above this line the energetics are dominated by the lateral shear production, either in a pure centrifugal instability mode, to the left of the green line indicating zero absolute vertical vorticity (1 − α/Ri = 0), or in a mixed symmetric/centrifugal type mode, for positive absolute vertical vorticity to the right of the green line. Numerical solutions of (6)–(10) [with l = 0, k = 20f/(NH)] were also used to calculate 
Example regime diagram for baroclinic instability (BI), symmetric instability (SI), centrifugal instability (CI), and mixed instability (SI/CI) in the ocean BBL. The red line indicates q = 0, the green line indicates 
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Example regime diagram for baroclinic instability (BI), symmetric instability (SI), centrifugal instability (CI), and mixed instability (SI/CI) in the ocean BBL. The red line indicates q = 0, the green line indicates
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
Example regime diagram for baroclinic instability (BI), symmetric instability (SI), centrifugal instability (CI), and mixed instability (SI/CI) in the ocean BBL. The red line indicates q = 0, the green line indicates
Citation: Journal of Physical Oceanography 48, 11; 10.1175/JPO-D-17-0264.1
For small Ri, the dominant instability types in the BBL are therefore likely to be of the centrifugal, symmetric, or mixed types, with the centrifugal mode becoming dominant for large S, where the topography begins to act as an effectively vertical boundary (Dewar et al. 2015). For 
7. Summary
In this article, we considered baroclinic instability over sloping topography, extending earlier QG results to the nongeostrophic regime appropriate for the ocean BBL. Importantly, weakly stratified BBLs can support a submesoscale baroclinic instability, a BBL counterpart to the surface baroclinic mixed layer instability (Boccaletti et al. 2007). In the BBL these submesoscale instabilities are relatively insensitive to topographic slope, suggesting they are likely a robust part of the dynamics of the ocean BBL, along with the symmetric (Allen and Newberger 1998) and centrifugal (Molemaker et al. 2015) modes.
Two 1D theories of the BBL structure over topography were also shown to result in solutions that are susceptible to growing baroclinic instability, with important implications for our understanding of the dynamics of both the BBL and the interior. In the case of a BBL generated by turbulent mixing, a thick outer layer supports a rapidly growing baroclinic mode that releases available potential energy and provides a restratifying tendency that is not accounted for in the 1D formulation. This suggests that deep submesoscale baroclinic instabilities could alter the stratification and across-slope flow along topographic features such as the mid-Atlantic ridge, with important implications for closure of the abyssal circulation (Callies 2018). Likewise, interior flow along a slope generates available potential energy in the BBL through an across-slope Ekman flow, leading to baroclinic growth rates that quickly exceed the rate of Ekman adjustment. These instabilities thus have the potential to affect both the BBL dynamics and the exchange between the boundary layer and interior, in both the coastal and deep ocean.
Extensive work on the surface boundary layer has demonstrated the leading-order role that submesoscale baroclinic instability plays in boundary layer restratification (Fox-Kemper et al. 2008; Johnson et al. 2016; Su et al. 2018), vertical exchange between the boundary layer and interior (Mahadevan and Tandon 2006), and the surface flux of potential vorticity (Wenegrat et al. 2018). The similarities between the BBL and surface mode (section 2), and the large-amplitude eddy fluxes found in the nonlinear simulation of section 5, suggest that many of the same results may hold in the BBL. However, there are also important differences between the surface and bottom boundary layer that require further consideration. For example, as noted in section 2, topography shapes parcel trajectories, modifying the instability energetics and potentially the associated eddy fluxes of buoyancy and tracers (Spall 2004; Isachsen 2011; Brink 2013). Further, variations in topography, not considered here, can impose an external length scale on instabilities (de Szoeke 1983), generate slope-normal flow and secondary circulations (Benthuysen et al. 2015), and modify instability growth rates (de Szoeke 1983; Solodoch et al. 2016), all of which may alter the role of baroclinic instability in BBL dynamics. Relaxing the assumption of across-slope uniformity in boundary layer height or structure may also lead to an arrest of the inverse cascade due to the topographic beta effect (Rhines 1975; Brink 2012). Full nonlinear simulations to explore the finite-amplitude behavior, and parameter dependence, of the submesoscale BBL baroclinic mode will therefore be the subject of a future paper.
Acknowledgments
The authors thank the contributors to the Dedalus Project (www.dedalus-project.org) and GOTM (www.gotm.net), both of which were extremely useful for the analyses in this paper. Authors JOW and LNT were supported during this work by NSF Grant OCE-1459677. Comments from three anonymous reviewers are also gratefully acknowledged.
APPENDIX A
Energetics
APPENDIX B
Configuration of the Nonlinear Model
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