1. Introduction

Frontal patterns in temperature and velocity have long been recognized as common occurrences, especially near the surface in the lower atmosphere or upper ocean. In the atmosphere fronts are central features in synoptic-scale wintertime storms, and the so-called Bergen School based an early conceptual model with an intimate relation between a cyclone and its fronts (Bjerknes 1919). Later it was understood that winter storms arise primarily from the baroclinic instability of the planetary-scale westerly winds, but they nevertheless typically develop subsynoptic-scale fronts through the frontogenetic effect of a synoptic-scale deformation flow [i.e., with a nonzero horizontal strain rate, Bergeron (1928)]. In the ocean, surface fronts are also common, both as quasi-permanent features of the general circulation (e.g., in the Antarctic Circumpolar Current) as well as more ephemeral but abundant, densely packed, submesoscale structures that are evident in satellite surface temperature images (Castelao et al. 2006; Ullman et al. 2007).

Surface-intensified, submesoscale frontogenesis caused by a mesoscale straining deformation field is an active process in simulations of several turbulent rotating, stratified flow regimes (Capet et al. 2008; Molemaker et al. 2009). The essential dynamics of surface frontogenesis in these turbulent simulations appears to be generally consistent with the model of Hoskins and Bretherton (1972) [also see the general theoretical summary in Hoskins (1982)]. Additionally, in these simulations three-dimensional (3D) fluctuations sometimes grow vigorously on the front, although the circumstances and mechanisms for their growth are presently unclear. Flament et al. (1985) gives an observational example of an oceanic, horizontally meandering, submesoscale, frontal instability.

If the deformation rate α were zero and the frontal flow steady (e.g., hydrostatic, geostrophic, and inviscid), then this situation could be analyzed as a familiar type of linear instability process, where baroclinic, barotropic, and—at larger Rossby number Ro—ageostrophic and centrifugal instability modes each might occur for various flow configurations. There is extensive literature on the linear instability of stationary frontal flows in the absence of deformation (e.g., Schär and Davies 1990; Barth 1994). However, when the deformation field is present and the frontal flow is intensifying, there is no general theoretical framework for analyzing frontal fluctuation growth. An analogy is sometimes suggested for a vorticity filament (i.e., a thin shear layer) in a barotropic fluid. Without deformation the flow is barotropically unstable. With a uniform deformation field oriented to narrow progressively the shear-layer width, the layer is stabilized (Dritschel et al. 1991). By this analogy one expects deformation to suppress the frontal fluctuation growth; hence, frontogenesis would be antithetical to frontal instability. Furthermore, in Eady’s flow [with uniform potential vorticity between two vertical boundaries; i.e., Eady (1949)], the growth of baroclinic instability is suppressed by active deformation (Bishop 1993), further suggesting that frontogenesis limits frontal instability even in a baroclinic fluid.

In this paper we show that this expectation is false for baroclinic frontogenesis by demonstrating that surface frontogenesis manifests an exponentially growing fluctuation mode even in the presence of active deformation. Furthermore, the fluctuation growth rate is strongly amplified by frontogenesis that causes rapidly increasing Ro and Froude number (Fr) values for the frontal flow. This growing fluctuation mode is a continuation of the dominant, geostrophic, normal-mode instability that occurs for a stationary frontal flow in the absence of deformation.

Spall (1997) numerically simulates a surface, baroclinic, mesoscale frontal flow in a deformation field that exhibits baroclinic instability and finite-amplitude equilibration as long as the deformation rate is weak enough and Ro and Fr are not too large. He interprets the latter limitation as consistent with the Bishop (1993) deformation–stabilization model. Our linear analysis supports both the importance of baroclinic instability and the frontogenetic equilibration tendency induced by the growing mode (although we do not examine the nonlinear equilibration process). However, we do not find evidence for frontal stabilization with large α, Ro, and Fr; in fact, we find that frontal instability is enhanced by increases in each of these parameters. This discrepancy is further addressed in section 9.

2. Governing equations and parameters

We first pose an idealized problem for two-dimensional (2D), baroclinic, upper-surface frontogenesis induced by a barotropic deformation flow u_d with a deformation rate α(t) (equal to half the rate of strain):

The configuration for the problem is sketched in Fig. 1, including the secondary circulation that arises in the frontal plane. This deformation flow has no vorticity, no horizontal divergence, and no associated buoyancy variation even in the presence of a background vertical stratification. The deformation flow is conducive to frontogenesis with a frontal axis aligned with ŷ. We take the view that u_d arises from flow dynamics controlled remotely away from the frontogenetic region and is externally specified in the problems we solve. For example, a time-varying vorticity field with compact support and finite circulation has a far-field velocity that is dominated by strain and can be locally Taylor expanded about the frontal location, (x, y) = 0, as a dynamically inconsequential uniform velocity plus the shear flow in (1).

We define the total flow as u + u_d. Substituting this flow into the incompressible Boussinesq equations and discarding terms independent of u (considered part of the deformation field dynamics) yields interaction terms in the advection operator ∝ u_d; hence, ∝ (x, y). These terms are computationally awkward to deal with in a large but finite horizontal domain centered around the front, so we make a coordinate transformation moving with the deformation flow (1) that eliminates these secular terms from the advection operator (and has the further benefit of preserving better spatial resolution during frontogenesis than do fixed coordinates):

with (cf. Hoskins and Bretherton 1972). To a first approximation (as shown in section 4), frontogenesis sharpens the x gradients in u while nearly preserving the structure in X. Often there is a similar sharpening of the vertical gradients; therefore, for computational accuracy we further define a transformed vertical coordinate:

with . In all the solutions analyzed in this paper, we choose α_z(t) = α(t). The important physical boundary is the top surface at z = Z = 0. Finally, the time coordinate is unchanged in the transformation, T = t. The solutions considered below have a specified Y structure and have vanishing flow speeds as |X|, |Z| → ∞.

The ideal 3D Boussinesq equations are nondimensionalized with the following scales:

Both V and Y are oriented along the frontal flow, and U and X are across it; Z and w are vertical components; φ is the pressure normalized by mean density ρ_o; and b = −gρ/ρ_o is the buoyancy field. Nondimensional parameters are

Here f is the Coriolis frequency assumed to be spatially uniform, and N is the buoyancy frequency assumed to approach a constant at depth. For an incipient oceanic front all of the parameters in (5) typically have small values for mesoscale currents, and in this paper we assign them nominal values¹ of Ro = Fr = ε = 0.1 and λ = 0.01. These imply that the frontal flow will be approximately geostrophic and hydrostatic, at least before the deformation flow causes significant frontogenesis.

In (X, Z, T) coordinates the nondimensional, inviscid, 2D frontal equations are

The full buoyancy field is the sum of the background stratification 〈b〉(z) and its frontal deviation:

The 2D substantial time derivative is

We further consider 3D fluctuations whose dynamics are linearized about the preceding 2D frontal flow. Another advantage of the coordinate transformation is that a separable solution form exists for the alongfront direction when α ≠ 0; for example, the fluctuation cross-front velocity in complex notation is represented as û(X, Z, T)e^ikY. The nondimensional fluctuation amplitudes are solutions of the following PDE system:

In this paper we do not include nonlinear fluctuation advection effects in integrating either (6) or (9), but we do analyze the emerging averaged fluctuation buoyancy flux divergence in section 6.

3. Computational method

The code uses staggered-grid, finite-difference discretization in space for (6)–(9), and a third-order Adams–Bashforth scheme is used to advance the variables in time. It uses a QUICK algorithm for the advection of velocities and scalars, and a multigrid Poisson solver for determination of φ and φ̂ within each time step. These methods have been successfully used before (Molemaker and Dijkstra 2000; Molemaker et al. 2009).

The model equations have anisotropic viscous and buoyancy diffusion terms, partly for discretization noise control. In the results presented below, they have negligible dynamical influence because of their small coefficients. The boundary conditions are zero-normal flow at x = ±L_x/2 and z = −L_z, 0. Viscous and diffusive terms at near-boundary grid points are evaluated by extrapolation from adjacent interior grid values² [e.g., near z = 0, a ghost point is defined by υ(−dz/2) = 3υ(dz/2) −3υ(3dz/2) + υ(5dz/2)]. This choice is made to avoid boundary layers and limit the boundary fluxes of stress, buoyancy, and potential vorticity. The latter in particular may be large during frontogenesis, even with zero stress and buoyancy flux; this can lead to symmetric frontal instability if the potential vorticity evolves to change its sign within the domain (Hoskins 1974). However, this regime is not examined in this paper; the solutions we present do not have large changes in their potential vorticity values, nor do they exhibit strong sensitivity to the particular representation for stress- and flux-free boundary conditions. The usual nondimensional domain size is L_x = 8 and L_z = 2, while the initial frontal scales are O(1) in each direction. We use the same boundary conditions in (X, Z). Except for z = Z = 0, these boundary conditions are artificial approximations to an unbounded domain as (x, X) → ±∞ and (z, Z) → −∞. We stop the time integrations before the shrinking domain boundary (due to the deformation-flow coordinate transformation; see section 2) encroaches too severely on the region with significant frontal and fluctuation amplitudes. The solutions below are obtained with an (X, Z) grid size of (128, 128), and we tested that similar solutions are found with larger grid size. Since frontogenesis has a superexponential rate (section 4), we expect an ever stiffer grid-resolution requirement at later times; however, the solutions here are not integrated into a regime where our resolution becomes inadequate.

In the solutions in sections 4 and 5, a particular initial condition is used for the 2D fields. It has an initial alongfront flow υ(x, z) in geostrophic, hydrostatic balance with the buoyancy field b(x, z), which has a uniform stratification away from the front:

Our standard parameter choices are A = 3.0, B = 2.0, and G = 1.2 (Fig. 3, top row). [Different initial conditions are examined in section 8, including some that have more of a surface mixed layer structure, at least in 〈b〉(z).]

The Ertel potential vorticity associated with (6)–(8) is

With (10), Π exhibits reversals in both its vertical and horizontal gradients: the υ(x, z) profile implies cyclonic relative vorticity for x < 0 and anticyclonic for x > 0, and the weak, near-surface vertical density stratification in the far field as x → −∞ reduces the potential vorticity (i.e., is anticyclonic relative to the spatial mean value), while the opposite far field has stronger stratification (i.e., is cyclonic). These sign-reversing lateral potential-vorticity gradients satisfy the Rayleigh necessary condition for geostrophic barotropic and baroclinic instabilities of a stationary, parallel flow. In the geophysical fluid literature this is often referred to as the Charney–Stern condition, which explicitly implicates the interior potential vorticity gradient ∂_xΠ and the boundary buoyancy gradient ∂_xb′(x, 0) (Eliassen 1984). We show in section 5 that such 3D instabilities do occur with (10), both with and without active deformation α, and in section 8 we show that the character of the instability can shift from baroclinic to barotropic as the relative contributions of ∂_zυ and ∂_xυ change. In contrast, the special case frontogenetic problem with uniform potential vorticity, often adopted in the frontogenesis literature (e.g., Hoskins and Bretherton 1972), does not satisfy the Rayleigh necessary condition for linear instability in a semi-infinite domain (i.e., with one boundary, unlike Eady’s flow with two boundaries), and thus this type of frontogenesis cannot manifest the essentially geostrophic instabilities that are the focus of this paper. The absence of linear frontal instability holds in particular for the stationary fronts in the small-Ro asymptotic model, surface quasigeostrophy (Held et al. 1995). Many previous instability studies have been made for variants of Eady’s flow (Bishop 1993; Song and Nakamura 2000; Heifetz et al. 2004), but we view this two-boundary model as dubiously relevant to a single-boundary front, at least when its solution behavior is qualitatively different.

We consider this υ(x, z) profile as a representative balanced, surface, frontal flow. It is susceptible to both barotropic and baroclinic normal-mode instabilities for small Ro and Fr due to its combined horizontal and vertical shears. Abundant experience has shown that such flows are quantitatively sensitive to the details of their shear profiles. Our goal in this paper is to demonstrate what we found to be generic frontogenesis and fluctuation growth behavior under active deformation without exhaustively surveying the profile sensitivities.

The deformation field α(t) is initially zero but ramps up to an asymptotic dimensional value of α₀ over a nondimensional time period of Δt ≈ 0.25. Consequently, the cross-front flows are also initially zero: u = w = 0. The reason for a ramp up is to diminish early time generation of 2D inertia–gravity waves caused by deformation shock (Snyder et al. 1993). Our analyses in sections 4–8 are restricted to t ≥ Δt whenever α ≠ 0.

The 3D fluctuation equations are solved with spatially white, random noise as an initial condition³ in (û, υ̂, ŵ, b̂)(x, z) and a specified value for the alongfront wavenumber k at time t₀ ≥ Δt. The combined 2D–3D system is integrated forward in time until time t₁. At that point, the integration is restarted at t = t₀ ≤ t₁ with the 2D solution reinstated at t₀, but the 3D fields at t₁ are reinterpreted as initial conditions at t₀. This is repeated as an iterative “breeding” procedure that allows the most rapidly growing fluctuation to emerge from the initial noise (cf. Toth and Kalnay 1997). It is important that the integration interval, t₁ − t₀, not be too large so that the fluctuations not be too disrupted by restarting at an earlier phase of the 2D flow and associated deforming grid. (A symptom of an excessively large interval is spurious excitation of inertia–gravity waves in the 3D fields by the restart.) In the results below, the interval is usually chosen as t₁ − t₀ = 0.006 25, and the number of breeding cycles is 800 with a total fluctuation breeding time interval of 5. This breeding procedure is nearly the same as freezing the 2D frontogenesis fields at t₀ while continuing to integrate the 3D fluctuations forward with nonzero α, which is justifiable a posteriori if the amplifying fluctuation exponential growth rate σ is much larger than the frontogenetic deformation rate α. A frozen-2D fluctuation formulation could be analyzed as a generalized eigenvalue problem; however, we can obtain essentially the same answer by doing breeding time integrations with a short t₁ − t₀, so we did not feel it was necessary to separately develop the eigenvalue method. If the resulting σ were smaller than α₀, then, insofar as the latter is a rough characterization of the 2D frontal rate of change (section 4), we should not expect the 3D fluctuation growth to be very accurate with either breeding or frozen-2D calculations since the frontal changes would alter the background state before the fluctuations changed. Solutions are presented with a consistent, simultaneous 2D and 3D evolution without this breeding approximation in section 7.

4. Two-dimensional frontogenesis

The essential phenomenology and dynamical understanding for 2D frontogenesis are well known, at least within the context of the assumptions of semigeostrophic balance and uniform potential vorticity in the interior (Hoskins 1982)—neither of which are assumed here, although this does not change the essential 2D results. A surface-intensified lateral buoyancy gradient and its associated geostrophic alongfront flow in the presence of a favorably aligned deformation flow (Fig. 1) has its cross-front lateral gradients sharpen with time and develops a closed, secondary circulation with an upward flow on the buoyant side and a surface horizontal flow across the front toward the dense side that further accelerates the frontal sharpening. With semigeostrophic balance and uniform potential vorticity, the front becomes infinitely thin in finite time (Hoskins and Bretherton 1972), but this singular behavior is a priori implausible even though proof of finite-time regularity is still a mathematically open question for fluid dynamical equations such as (6). Two-dimensional simulations have shown an increasingly unbalanced evolution as frontogenesis proceeds (Snyder et al. 1993). A heuristic rationalization of the rate of development of the secondary circulation comes from the x momentum equation in (6), assuming exact geostrophic balance for υ and ∂_xφ and neglecting advective nonlinearity:

that is, exponential amplification in time. Two-dimensional continuity requires that w amplify along with u (neglecting any evolving differences in their X and Z scales). A similar rationalization of frontal sharpening comes from assuming passive advection of the frontal fields by the deformation flow (1), which implies x gradients also grow ∝ exp(β). The solutions below show that these heuristic estimates are qualitatively apt but quantitatively imprecise.

The expected behaviors are confirmed in our solutions. Here, we consider a standard case with the parameter values in section 2 and initial conditions in (10), together with a nondimensional deformation rate of α₀ = 1.0. As time-evolving measures for the magnitude of the alongfront shears, we define local Rossby and Froude numbers by and , respectively. Figure 2 shows that the frontal shears monotonically increase with time, indicating the occurrence of frontogenesis with decreasing cross-frontal and vertical scales. The functional form of the increase is approximately ∝ exp(β). At times later than t ≈ 1, their growth curves are convincingly superexponential, but accurate computations are increasingly difficult in a collapsing front, and in this paper we do not pursue the late time regime with large . (Other initial profiles sometimes show quantitatively different growth curves; e.g., see section 8.) Nevertheless, even over this limited period of frontogenesis up to t = 1.5, the Rossby and Froude numbers grow to O(1) values.

The 2D frontal fields are illustrated in Figs. 3 –5. Both υ(x, z) and b(x, z) remain approximately in thermal wind balance as their horizontal and vertical scales shrink. The frontal strength is measured by the square of the horizontal buoyancy gradient, (∂_xb)²/2, and steadily increases in magnitude with its maximum located at the surface and moving to the left with advection by u(x, 0, t). The secondary circulation streamfunction ψ (with u = ∂_zψ, w = −∂_xψ) spins up from zero to a counterclockwise flow in the (x, z) plane: upward on the light side and downward on the dense side, toward the dense side near the top surface and toward the light side at depth. The center of the front moves toward the dense side due to advection by this secondary circulation. The direction of this secondary circulation is to convert frontal potential energy into frontal kinetic energy since wb′ > 0 on average. Similarly, this correlation implies that the secondary circulation acts to increase the stratification in the vicinity of the front; this is sometimes called restratification since it acts opposite to destratification by vertical buoyancy mixing.

The maximum in υ shows only a moderate increase with time, while its gradients are growing approximately exponentially (Fig. 2). The magnitude of ∂_xb—whose square is often taken as a pointwise measure of frontal strength (section 6)—also grows approximately exponentially, and its spatial pattern shows horizontal and vertical scale contractions toward the moving frontal center at the surface (Fig. 4). The secondary circulation is expressed in the cross-front streamfunction field ψ (Fig. 5). It is initially zero because α(0) = 0 (section 3), but it amplifies in magnitude early in frontogenesis as u and w increase; however, because the rate of scale contraction is ultimately stronger than the rate at which (u, w) increase, the amplitude of ψ peaks at t ≈ 0.3 and thereafter declines during frontogenesis.

In the energy balance for the 2D frontal flow (cf. the appendix for the 3D fluctuation flow), there is an energy loss to the background deformation flow [i.e., α(ε²u² − υ²) < 0 for small ε] and a conversion from frontal potential energy to kinetic energy (i.e., wb′ > 0 for the secondary circulation in Fig. 5). In a diagnostic test of cross-front geostrophic, hydrostatic balance for the alongfront flow (i.e., ∂_zυ − ∂_xb, normalized by the magnitude of either term alone), the deviations were found to grow during frontogenesis but never to exceed a few percent for the regime examined here, as long as the deformation field α(t) is slowly varying and does not excite inertia–gravity waves. The alongfront momentum balance, however, is ageostrophic to leading order; frontogenesis is an unbalanced process with respect to the secondary circulation it generates.

5. Three-dimensional fluctuation growth

Now consider the evolution of separable 3D fluctuations in (9). We first solve the stationary-front problem without deformation. This is done by choosing t₀ = 0.0. In this case the bred fluctuation is equivalent to the most unstable normal mode for each k value. For the initial frontal profile (10), the maximum 3D fluctuation growth occurs for k_max = 2.0. An exponential fit to the time history of fluctuation magnitudes (Fig. 6) is very accurate after an initial period of adjustment, indicating both the success of our breeding technique and the unimportance of 2D frontal evolution during the brief breeding interval. It shows a nondimensional growth rate of σ_max = 2.00. Thus, even at t₀ = 0.0 in our standard case of 2D frontogenesis (section 4), σ_max > α₀, which is consistent with the necessary conditions for the fluctuation breeding approximation (section 3). We will see that the accuracy of this approximation further improves with increasing t₀ since σ_max increases. The relation σ_max > α₀ implies that 3D fluctuations grow faster than 2D gradients in frontogenesis (Fig. 2). This implies that perturbed fronts will usually manifest a 3D instability that becomes more evident as the 2D frontogenesis proceeds.

Since the growing 3D fields have complex coefficients multiplying e^ikY (section 2), we first show only the spatial patterns of their absolute values in Fig. 7. The larger modal magnitudes are in a horizontal and vertical region concentrated near the front except for |ŵ|, which reaches deeper into the interior. Notice that the magnitudes in υ̂ and b̂ are comparable to each other (consistent with approximately geostrophic, hydrostatic balance in the cross-front momentum equation), while |û| is larger and |ŵ| is smaller. A larger |û| indicates that the 2D frontal scaling with ε ≪ 1 does not apply to the 3D fluctuations. Its emerging magnitude, with nondimensional |û| ∼ kε⁻¹, is consistent with an approximately geostrophic, hydrostatic balance in the alongfront momentum equation. The smaller |ŵ| indicates that the fluctuation structure is dominated by meanders in the horizontal plane (Fig. 8), even though ŵ ≠ 0 is essential to fluctuation growth by baroclinic energy conversion from the 2D frontal flow (see the appendix and Fig. 10). The largest |υ̂| values are displaced from the frontal center at the cross-front edges of the meanders.

Because of the initially rather small values of Ro and Fr, we interpret this mode as approximately a quasigeostrophic instability. The energy balance analysis later in this section shows that the instability is a mixed type, with a baroclinic energy generation rate due to vertical shear ∂_zυ that is more than twice as large as the barotropic generation rate due to ∂_xυ. In this regime the most unstable wavenumber (k = 2.0) is approximately the same as both the inverse baroclinic deformation radius for the background stratification and the vertical scale of the frontal flow (FrA/GRo = 2.5 here, characteristic of baroclinic instability) and the inverse frontal width B [=2 here, characteristic of barotropic instability; e.g., McWilliams (2006)].

The experimental protocol for investigating the consequences of frontogenesis is to examine a succession of increasing t₀ values and at each one to make a survey in k values to identify the range in k where significant 3D fluctuation growth occurs and to identify its maximum growth rate, σ_max at k_max. Because the amplifying 3D fluctuations are bred over a given short time interval (section 3), this is essentially equivalent to freezing the 2D frontal fields in time and thus solving a normal-mode instability problem that exhibits temporal exponential growth with growth rate σ > 0. This is illustrated in Fig. 6 at t₀ = 1.0 for the fastest-growing mode with k = k_max(t₀) = 9.5 and in Fig. 9 for the unstable k range for several t₀. With our breeding procedure we are unable to identify modes with weak or no growth and, for any particular k, cannot find the unstable modes, if any, with weaker growth rates than the fastest-growing mode.

The emergent spatial patterns in (û, υ̂, ŵ, b̂) associated with the dominant mode at t₀ = 1.0 are shown in Fig. 7 (bottom). The 3D frontogenetic mode is somewhat similar in shape to the stationary-front instability mode except for its greatly reduced spatial scales in the cross-front plane due to 2D frontogenetic scale contraction. In addition, there are some more subtle differences in the mode shape, such as the shift of its larger amplitude regions toward the dense side of the front approximately following the motion of the frontal center itself (Fig. 3). Figure 8 shows the horizontal phase structure of this mode in a horizontal plane near the top surface. The horizontal flow [rescaled to have a common dimensional horizontal velocity scale; i.e., nondimensional (εu, υ)] has recirculations around the b′(x, y) extrema, consistent with geostrophic, hydrostatic balance. The vertical velocity w(x, y) is fractionally displaced from b′(x, y) in the upshear direction, consistent with baroclinic energy conversion from the 2D frontal flow to the 3D fluctuations (wb′, BF > 0; see the appendix).

We find that σ_max systematically increases with t₀, that is, with the frontal sharpening (Fig. 9 and Table 1). The associated alongfront wavenumber k_max as well as the bandwidth Δk for growing modes similarly increase. These results show that the horizontal scale of the instability is comparable to the frontal width with σ decreasing at both larger and smaller k values; this is familiar behavior from unstable stationary flows with both vertical and horizontal shear (Barth 1994). In a first approximation σ_max increases in pace with expβ (Table 1), similar to the rate of growth of the 2D shear magnitudes in the sharpening front (Fig. 2). The rates of increase in k_max and Δk are much faster than expβ. This can be rationalized as a combination of an increase following the true shrinking 2D frontal scale and an increase to counter the artificial stretching associated with the transformed alongfront coordinate (n.b., kY = ke^βy); however, k_max(t₀) in Table 1 does not increase quite as fast as e^2β(t₀).

To further interpret the 3D fluctuation growth, we perform an energy balance analysis as defined in the appendix. Several of the associated fields are shown in Fig. 10 for both a stationary front (t₀ = 0.0) and actively sharpening frontogenesis (t₀ = 1.5). Again, we see qualitative similarity between the initial [α(0) = 0] and frontogenetic [α(1.5) = α₀] stages. The fluctuation energy density K + A is concentrated in the frontal zone (as expected from the modal patterns in Fig. 7), shrinking as the front sharpens. The integral conversion rates include the energy exchange with the deformation flow DF, the conversion of fluctuation potential to kinetic energy PK, and the energy conversions from the 2D frontal flow by both Reynolds stress RF and buoyancy flux BF. Broadly over the domain all of these conversions are positive except for DF, which is generally small but is sometimes negative (Table 1; n.b., negative conversion with the background deformation flow also occurs for the 2D frontal flow; see section 4). In Fig. 7 the buoyancy-flux conversion generation is concentrated in the frontal zone with approximately the same distribution as the vertical shear ∂_zυ and the frontal strength (Fig. 4; notice the different spatial domains in the plots). The Reynolds stress conversion is relatively weaker and is concentrated on the edges of the frontal flow where |∂_xυ| is largest. On the cyclonic side RF is even locally negative near where ∂_xυ and |υ̂| have maxima (Figs. 3 and 7); this feature becomes more pronounced in wider fronts where the instability becomes more purely baroclinic (section 8). The integrated conversion rates (normalized by the total fluctuation energy) for several t₀ values are listed in Table 1.

Baroclinic generation from the unstable front, BF > 0, is the dominant source of fluctuation growth. Hence, we can identify this growing mode as a form of baroclinic instability (but also see section 8). Here RF is also positive for all t₀ values but not as a large as BF. Although positive except for late-time frontogenesis, DF is much smaller. Hence, PK > 0 to maintain the partition between K < A in the baroclinically unstable mode. With increasing frontogenetic development (i.e., increasing t₀), the dominance of baroclinic energy generation increases. The primary frontal instability mechanism in Capet et al. (2008) was by baroclinic conversion.

A diagnostic test of geostrophic, hydrostatic balance for the fluctuations shows unbalanced deviations that grow with t₀ (hence with and ). The deviations reach a relative level of several tenths at the largest t₀ examined here, in contrast to the apparent origin as a fully balanced instability mode as Ro, Fr → 0. Nevertheless, in spite of the great changes in spatial scale and growth rate induced during frontogenesis for both the 2D frontal flow and the 3D fluctuation fields (with more modest changes in their (X, Z) spatial patterns), there is a nearly geostrophic, baroclinic-instability-like, 3D fluctuation growth mode whose growth and shape are robustly persistent during frontogenesis. As an approximate summary, we can say that the changes of scale and growth rate in the 3D instability with increasing t₀ simply track the sharpening front and increasing 2D shear.

As stated above, the maximum 3D fluctuation growth rate, σ_max, approximately keeps pace with the quasi-exponentially increasing and values during frontogenesis (Fig. 2). This suggests that the frontal instability rate is primarily determined by the frontal shear magnitude, rather than the frontogenetic deformation rate itself. This is tested by a sequence of cases with different values of α₀. In each case, , , and σ_max grow quasi-exponentially in time, with exponents approximately proportional to β ∼ α₀t₀. Thus, with increasing α₀, σ_max(t) is a much steeper function of t since a given 2D frontal strength occurs at increasingly earlier t₀. To factor out this effect, we compare different α₀ cases by picking comparison times t₀ that have similar values of and (Table 2 is based on a survey with finite resolution in t₀ and k). The result is that (k_max, σ_max) are only weakly increasing functions of α₀, although their enhancement increases somewhat for larger α₀. [In practice we are limited by our experimental protocol (section 3) from looking at very large values of α₀ since the comparable t₀ value drops well below the spinup interval Δt for α(t).] A visual comparison of the 3D modal patterns for these different cases with those in Fig. 7 shows that they differ only moderately with α₀. This result, that even the normalized frontal instability rate increases with the deformation rate, is contrary to Bishop (1993). This discrepancy is discussed in section 9.

6. Fluctuation frontolysis and restratification

Section 5 shows that 3D fluctuations extract energy mostly from the 2D frontal flow, increasingly with t₀. This indicates that fluctuations play some role in retarding frontogenesis (often referred to as frontolysis) at least in an integral sense. Here we refine that characterization by examining the frontogenetic tendency balance (Giordani and Caniaux 2001) derived from the alongfront-averaged buoyancy conservation equation including the nonlinear eddy flux. For simplicity we write it in untransformed coordinates (also used in the appendix for the energy balance), even though in practice we evaluate its component terms from our solution in transformed coordinates:

The eddy-flux symbol ĉ : d̂ is defined as Re[ĉ] Re[d̂] + Im[ĉ] Im[d̂]. The final line collects the frontogenetic tendencies in the preceding right-side line into advection by the 2D circulation T_2D and eddy fluxes T_F.

In Fig. 11 the pattern of the advective effect of the secondary circulation T_2D is familiar from previous studies of frontogenesis. It has a dipole structure, mainly due to lateral advection, indicating displacement of the frontal center toward the dense side. Since the positive lobe is stronger than the trailing negative lobe, it implies a frontogenetic growth of frontal strength on the dense side. On the other hand, the eddy-flux effect T_F is overwhelmingly negative (i.e., frontolytic): T_F is nearly centered on the peak frontal strength (Fig. 4) and it somewhat trails the position of the peak frontogenesis in T_2D. This suggests that fluctuation frontolysis will retard frontogenesis but might not entirely stop it since the peaks of T_F and T_2D are not exactly coincident. Nevertheless, fluctuation frontolysis may play a significant role in reducing the 2D frontogenesis induced by the deformation flow, changing the shape of the evolving frontal flow at the least, but potentially providing a mechanism for frontal arrest at a finite scale. We leave further investigation of finite-amplitude fluctuation effects in modifying frontogenesis to the future.

Another diagnostic measure of fluctuation effects on the front is the eddy-induced circulation that provides a mean Lagrangian transport of buoyancy and other material tracers (Gent and McWilliams 1990). The eddy-induced overturning streamfunction in the cross-front plane is defined by

with associated “bolus” velocities u_* = ∂_zψ_* and w_* = −∂_xψ_* [Ferrari et al. (2008), but note the opposite sign convention here].

The overturning streamfunction ψ_* for our standard frontogenetic case (α₀ = t₀ = 1.0, k = 9.5) is shown in Fig. 12. It is one signed (positive) in ψ_*, implying a counterclockwise circulation centered on the instantaneous frontal center. Its sense of circulation therefore generally reinforces the 2D secondary circulation ψ in the lower part of the front (Fig. 5).⁴ The most significant structural difference is that ψ_* does not have an interior maximum whereas ψ does. In general, ψ_* defines an incompressible circulation that must satisfy the surface boundary condition of ψ_* = 0 at z = 0. However, in the problem we have posed—with an essentially inviscid flow and significant stratification extending up to the top surface (hence, neither (û : b̂) nor ∂_zb decrease toward the surface)—the leftward return-flow branch, where ψ_* decreases toward zero at the top boundary, is singularly compressed into the topmost grid cell (and not plotted in Fig. 12). In the ocean and in oceanic general circulation models that include ψ_* as a parameterization for eddy flux, the surface return flow primarily occurs within the turbulent boundary layer where ∂_zb ≈ 0, as sketched in Fig. 4 of Ferrari et al. (2008) for an unstable stationary frontal flow. The proper interpretation of ψ_* in this solution is that the 3D fluctuations contribute to buoyancy restratification and potential-to-kinetic energy conversion (as expected for a baroclinic instability), reinforcing these same effects in 2D frontogenesis, but unlike ψ they do not advectively contribute to frontogenesis within the interior. Further consideration of the implied return-flow branch in ψ_* is deferred to cases with a weakly stratified surface layer (section 8) where it is explicitly manifest.

7. Transient fluctuation growth

In the breeding approximation (section 3), we determine the fastest-growing 3D modes for each wavenumber k while effectively freezing the 2D frontogenesis at a particular time t₀. This is justified by the approximation σ ≫ α that is confirmed a posteriori (section 5). Now, however, we relax the approximation and examine the simultaneous 2D and 3D evolution, using the fastest-growing mode and its associated k value as the 3D initial condition at t = t₀. In Fig. 13, we show the results of three different t₀ values, each integrated forward to the final time of t = 1.5. In each case the 3D amplitude grows exponentially at the expected growth rate σ (Table 1) for an early interval, but later the growth curve falls below a simple exponential form, although significant growth continues to occur. For the t₀ = 1.0 case, the exponential growth does not appreciably abate over the time of integration. The evolving 3D fluctuation patterns (not shown) continue to show the familiar scale contractions in (x, z) (e.g., as in Fig. 7) following along with the 2D frontogenesis. This continuing transient growth occurs in spite of substantial changes in the 2D frontal shape and the k value and associated 3D fluctuation pattern with maximum growth rate at t₀. It is also consistent with the growing k bandwidth of unstable modes as t₀ increases (Fig. 9); for example, the transient growth initiated at t₀ = 0.5 with k = 4.0 exhibits growth at t = 1 with a rate very close to the bred mode at t₀ = 1 with k = 4.0. Thus, the reliability of the breeding approximation is further confirmed. More importantly, the transient evolutions show widespread and efficient fluctuation growth during active frontogenesis, even among nonexponentially growing 3D fluctuation patterns. In combination with the rather broad σ(k) curves with overlap in their k ranges at different t₀ (Fig. 9), we can expect broadband fluctuation growth as frontogenesis proceeds, where the emergent 3D fluctuation patterns will be influenced by the extant perturbations in particular realizations.

8. Profile sensitivities

As remarked in section 3, geostrophic shear instability is notoriously sensitive to details of the mean-flow profile. We will not make a comprehensive survey for the present problem. Nevertheless, we briefly describe the results for some alternative initial frontal profiles to give a sense of the typicality of the behavior presented in sections 4–6.

9. Discussion

Near-surface 2D frontogenesis induced by a deformation flow enhances the growth of 3D fluctuations that occur on an ever smaller scale and with an ever larger growth rate as the 2D front progressively sharpens. This behavior is consistent with previous instability studies of stationary flows in its dependences on the scale and shear magnitude (or Ro and Fr value) of the background flow. The 3D fluctuation growth further increases with a larger deformation rate. The fluctuations grow by a combination of baroclinic and barotropic energy conversions from the frontal flow, with the former dominating for most of the situations that we have examined. The alongfront-averaged buoyancy fluxes of the growing fluctuations resist the 2D frontogenesis through a frontolytic tendency in (∂_xb)² and they also augment the buoyancy restratification and potential-to-kinetic energy conversion tendencies of the 2D frontogenesis itself. The eddy-induced cross-frontal overturning streamfunction ψ_* is also indicative of isopycnal flattening and restratification by the 3D fluctuations. These behaviors are remarkably robust across a range of initial horizontal and vertical shear and stratification profiles. In the ranges surveyed here, all of the 3D instability examples are essentially of a single type, morphing from a primarily baroclinic energy source to a barotropic one as the ratio of 2D shear to horizontal shear decreases, consistent with previous instability studies of stationary frontal flows. Our primary conclusion, therefore, is that active deformation in frontogenesis does not greatly alter the instability mechanism.

This conclusion is starkly in contrast to traditional frontogenesis models with uniform potential vorticity where either such 3D frontal-shear instabilities do not occur [with only one vertical boundary (Held et al. 1995)] or where deformation acts to suppress the instability [with two vertical boundaries (Bishop (1993)]. The conclusion also differs from the fully barotropic regime where a deformation flow can stabilize a sharpening, otherwise unstable, horizontal shear layer. These differences imply that the nature of the instability in a deformation flow for each of these three regimes (i.e., mixed baroclinic–barotropic instability with interior potential vorticity gradients, baroclinic instability due to surface buoyancy gradients in the Eady model, and barotropic instability of a vorticity filament) is importantly different in its behavior. However, the same kinds of instability differences also occur for stationary flows in these three regimes in a way characterized by their different contributions to the Rayleigh/Charney–Stern conditions.

Our results are mostly consistent with those of Spall (1997), whose flow configuration is qualitatively similar to ours. Here we have explored 3D fluctuation growth with larger values of Ro, Fr, and α, albeit with a linearized dynamical approximation. However, since we find an increasing growth rate with all these parameters, we disagree with his interpretation that a larger deformation rate stabilizes the flow by the mechanism presented in Bishop (1993); rather, it seems more likely that large α, narrow-front instabilities are suppressed by his choices for too large an eddy viscosity value and/or too coarse a grid resolution to resolve the small instability scales. (This speculation could only be confirmed by extending his study.) Thus, strong frontal instability is most likely to occur in the submesoscale regime in the ocean.

Frontogenesis and frontal instability, and the associated material transport and turbulent cascade of variance toward small-scale mixing, are evidently important dynamical processes in the submesoscale regime in the upper ocean (and likely the lower atmosphere as well). The present study demonstrates a significant linear instability for Rossby and Froude numbers ranging from small to O(1) values. This result naturally points toward further studies of finite-amplitude 3D frontal dynamics, later-stage frontogenetic instabilities for larger Ro and Fr values, and nonconservative influences by surface fluxes and boundary layer turbulence. We have also seen, but largely avoided in this paper by the problem selection, that spontaneous inertia–gravity wave emission during frontogenesis is occasionally quite efficient in both 2D and 3D, for example, in association with rapidly changing deformation flows. It remains an open question as to what type of instability or turbulence ultimately limits frontogenesis and frontal “collapse” under conditions of persistent deformation (Hoskins 2003), but the instability mode presented here, with its attendant frontolytic effects, is one candidate.