2.1 Balanced secondary circulation

To set-up the diagnostic balanced model for $\boldsymbol{u}_{a3}=(\boldsymbol{u}_{a},w)$ given $(b,\boldsymbol{u}_{g},\unicode[STIX]{x1D749},{\mathcal{B}}^{\prime },\unicode[STIX]{x1D708}_{v},\unicode[STIX]{x1D705}_{v})$ , the PE system (2.6) is approximated by neglecting ageostrophic acceleration $\unicode[STIX]{x2202}_{t}\boldsymbol{u}_{a}$ in the momentum equations. Many previous proposals for momentum-balanced models have been made (e.g. McWilliams Reference McWilliams, Velasco Fuentes, Sheinbaum and Ochoa2003) that suppress inertia–gravity wave behaviour and that have useful higher-order accuracy in $Ro$ compared to the leading-order asymptotic model, quasigeostrophy. Often the alternatives do not give greatly different solutions. In particular, different approximations have been made about which ageostrophic advective terms to include beyond the leading-order geostrophic ones. The approach taken here is the maximalistic approximation of including complete advection in buoyancy and momentum within the hydrostatic constraint. Another distinction is whether the neglected acceleration is ageostrophic ( $\unicode[STIX]{x2202}_{t}\boldsymbol{u}_{a}$ ), as here, or horizontally divergent ( $\unicode[STIX]{x2202}_{t}\boldsymbol{u}_{\unicode[STIX]{x1D712}}$ , where $\boldsymbol{u}_{\unicode[STIX]{x1D712}}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D712}$ ); because vertical momentum mixing mixes rotational and divergent components of $\boldsymbol{u}$ , the former choice is what allows the SCFT model here to be wholly diagnostic. A further simplifying approximation, suitable for submesoscale structures, is an $f$ -plane approximation; i.e.  $\unicode[STIX]{x1D6FD}=0$ . This simplification is not essential for the resulting structure of the diagnostic model.

With the replacement of the pressure gradient force by integration of the hydrostatic balance for $b^{\prime }$ , the SCFT balanced model is

The geostrophic substantial derivative is $\text{D}_{g}=\unicode[STIX]{x2202}_{t}+J[\unicode[STIX]{x1D713}_{g},~]$ , with $J$ the horizontal Jacobian operator, $J[p,q]=\unicode[STIX]{x2202}_{x}p\unicode[STIX]{x2202}_{y}q-\unicode[STIX]{x2202}_{y}p\unicode[STIX]{x2202}_{x}q$ . The left-hand-side terms in (2.11) are those retained in a quasigeostrophic plus Ekman-layer approximation (with vanishing $\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}^{\prime }$ at depth), and the right-hand side ${\mathcal{R}}$ terms collect the residuals of the terms in (2.6) not otherwise written explicitly here; they are defined by

with the ageostrophic advective nonlinear terms ${\mathcal{N}}$ defined by

As with (2.6) this system is to be understood as written in deformation coordinates (2.2) but evaluated at $t=0$ where the metric factors $\exp [\pm \tilde{\unicode[STIX]{x1D6FD}}]$ in the spatial derivatives are equal to one; this has consequences for further derived relations below.

This balanced dynamical system can, in a familiar way, be separated into an independent prognostic equation for the evolution of its geostrophic streamfunction $\unicode[STIX]{x1D713}_{g}$ – a generalization of the quasigeostrophic potential vorticity equation with additional right-hand side forcing terms, designated as the balanced potential vorticity equation (BVPE) – and an independent diagnostic equation for its ageostrophic velocity or secondary circulation – the Omega equation $\unicode[STIX]{x1D6FA}\text{E}$ . These two equations, respectively, are defined by the operations $\text{BPVE}=\text{curl}^{z}$ [horizontal momentum equations]  $+$   $f_{0}\unicode[STIX]{x2202}_{z}N^{-2}$ [buoyancy equation] and $\unicode[STIX]{x1D6FA}\text{E}=f_{0}\unicode[STIX]{x2202}_{z}\,\text{curl}^{z}$ [horizontal momentum equations] $-\unicode[STIX]{x1D6FB}^{2}$ [buoyancy equation]. The formulas for BPVE and $\unicode[STIX]{x1D6FA}\text{E}$ are given in appendix A for completeness. The $\unicode[STIX]{x1D6FA}\text{E}$ operations lead to the cancellation in the $\unicode[STIX]{x2202}_{t}\boldsymbol{u}_{g}$ and $\unicode[STIX]{x2202}_{t}b^{\prime }$ terms from (2.11), showing that $\unicode[STIX]{x1D6FA}\text{E}$ is the consistency condition that assures the equivalence of the $\unicode[STIX]{x1D713}_{g}$ and $b^{\prime }$ tendencies in the balanced model. It also may be used as a substitute for the buoyancy equation in (2.11) in comprising a complete dynamical system for determining $\boldsymbol{u}_{3}$ from $b$ .

The $\unicode[STIX]{x1D6FA}\text{E}$ is central to the SCFT diagnostic approach taken here. However, because of the vertical mixing and ageostrophic advection, this leads here to a generalized 3-D nonlinear Sawyer–Eliassen equation (Eliassen Reference Eliassen1962) for $(\boldsymbol{u}_{a},w)$ that is fourth order in $\unicode[STIX]{x2202}_{z}$ and second order in $\unicode[STIX]{x1D735}$ with spatially varying coefficients; this can be a difficult operator to invert. A diagnostic Sawyer–Eliassen equation for the secondary circulation in a front or parallel flow or vortex has a long history, in particular in situations of quasigeostrophic or geostrophic momentum balance approximations with specified right-hand side forcings from the geostrophic flow, diabatic heating or eddy momentum flux (e.g. Hoskins & Bretherton Reference Hoskins and Bretherton1972).

The present generalization to fully 3-D flows with vertical mixing and nonlinear functional dependency on the ageostrophic flow leads to the alternative SCFT formulation solved here. Because the generalized $\unicode[STIX]{x1D6FA}\text{E}$ is difficult to solve directly, an iterative approach is taken. Furthermore the balanced system (2.11)–(2.13) is split into two subsystems that are each readily solvable at each iteration step. The associated ageostrophic velocity decomposition is designated by

with the subscript $T$ denoting the TTW component and $O$ the $\unicode[STIX]{x1D6FA}\text{E}$ component.

The balanced dynamical system is a generalization of conservative geostrophic, hydrostatic dynamics with higher-order validity in $Ro$ and parameterized turbulent vertical mixing. An iterative approach to solving its equations, starting with a geostrophic first evaluation, is likely to succeed if the converged answer is similar in some sense to this first approximation even when $Ro$ is not small (except where balanced solutions do not exist, as can be expected for large $Ro$ ; see the discussion at the end of § 2). For upper-ocean submesoscale flow structures that partly occupy the turbulent surface boundary layer, the starting dynamical approximation in the iteration needs to be generalized to include vertical momentum mixing (i.e. TTW). This is the rationale for organizing the iterative solution procedure for (2.11)–(2.13) into the two subsystems defined below by utilizing the flow decomposition (2.14). This partitioned subsystems approach to solving (2.11)–(2.13) is not a unique choice, and other approaches may also be efficacious.

The first subsystem is designated TTW. It is based on the TTW approximation to the left-hand side of the horizontal momentum equation in (2.11). After subrtacting out geostrophic balance, the TTW subsystem solves the following equations for $(\boldsymbol{u}_{T},w_{T})$ :

This is solved in an iterative sequence, $n=1,2,\ldots ,$ for $(\boldsymbol{u}_{T},w_{T})^{n+1}$ , where the right-hand side terms are evaluated with total $(\boldsymbol{u},w)^{n}$ and starting fields of $(\boldsymbol{u},w)^{0}=(\boldsymbol{u}_{g},0)$ ; their definitions are

The field $\unicode[STIX]{x1D707}_{g}$ is the velocity potential for the divergent component of the geostrophic momentum advection in (2.11); it and the ageostrophic advection term ${\mathcal{N}}_{T}^{\boldsymbol{u}}$ are defined in (2.18) and (2.19).

The TTW subsystem (2.15)–(2.16) does not contain all the terms in the horizontal momentum equations in (2.11); in particular, it lacks the geostrophic tendency terms $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}$ , and it may be incomplete in its advection terms; this incompleteness will be corrected in the $\unicode[STIX]{x1D6FA}\text{E}$ subsystem (2.20)–(2.21) below.

With ${\mathcal{R}}_{T}^{\boldsymbol{u}}={\mathcal{Q}}_{T}=0$ , (2.15) is exactly the simple turbulent thermal wind balance discussed in § 1, and the vertically interiorward values of $w_{T}$ can be identified with Ekman pumping. The technical advantage of this TTW subsystem is that it fully contains all the vertical momentum mixing terms and thus allows the associated $\unicode[STIX]{x1D6FA}\text{E}$ subsystem (2.20) to be posed as a second-order 3-D elliptic partial differential equation (PDE) problem. The TTW subsystem can be solved as a 1-D linear problem for $\boldsymbol{u}_{T}$ independently at each $(x,y)$ , analogous to solving the Ekman-layer problem with variable $\unicode[STIX]{x1D708}_{v}(z)$ and specified right-hand side forces. Then $w_{T}$ is solved from the resulting 3-D $\boldsymbol{u}_{T}$ field by a simple vertical integration downward from the boundary condition at $z=0$ .

For the system (2.11), there is a constraint on the partition of various terms in the horizontal momentum equations between the TTW and $\unicode[STIX]{x1D6FA}\text{E}$ subsystems. It arises from the fact that TTW solves the horizontal momentum equations, but $\unicode[STIX]{x1D6FA}\text{E}$ only includes information about their curl, hence not about their divergence. In the particular partition defined here this constraint is only relevant for the advection terms. Consider a Helmholtz decomposition of the horizontal vector ${\mathcal{N}}^{\boldsymbol{u}}$ into its rotational and divergent components,

All of $\unicode[STIX]{x1D707}$ must stay in the right-hand side of TTW as $\unicode[STIX]{x1D707}_{T}$ with $\unicode[STIX]{x1D707}_{O}=0$ , while $\unicode[STIX]{x1D706}$ can either partly stay as $\unicode[STIX]{x1D706}_{T}$ or be assigned to $\unicode[STIX]{x1D6FA}\text{E}$ as $\unicode[STIX]{x1D706}_{O}$ , as long as $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{T}+\unicode[STIX]{x1D706}_{O}$ . (Because $\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D706}_{g}$ enters explicitly into $\unicode[STIX]{x1D6FA}\text{E}$ , $\unicode[STIX]{x1D706}_{g}$ itself does not need to be solved for.) For the SCFT applications in this paper, the choice is made separately (i) for the geostrophic advection,

which are assigned to TTW and $\unicode[STIX]{x1D6FA}\text{E}$ , respectively, so that they occupy their usual positions in quasigeostrophic theory, and (ii) for the ageostrophic advection, which is wholly assigned to $\unicode[STIX]{x1D707}_{T}$ and $\unicode[STIX]{x1D706}_{T}$ ; i.e.  $\unicode[STIX]{x1D707}_{O}=\unicode[STIX]{x1D706}_{O}=0$ , and

where the right-hand side is specified in (2.13). For SCFT problems with additional dynamical effects represented in ${\mathcal{R}}^{\boldsymbol{u}}$ , a similar constraint must be respected. For 2-D fronts and filaments, the geostrophic momentum advection is zero, $\unicode[STIX]{x1D707}_{g}=\unicode[STIX]{x1D706}_{g}={\mathcal{N}}_{O}^{\boldsymbol{u}}=0$ (§ 3).

The $\unicode[STIX]{x1D6FA}\text{E}$ subsystem is derived by the previously indicated operation on (2.11), whose result is in appendix A, and by subtracting the terms already contained within the TTW subsystem. It solves the following equations for $(\boldsymbol{u}_{O},w_{O})^{n+1}$ after the solution of the TTW subsystem at $n+1$ :

where

The right-hand side terms are evaluated using $(\boldsymbol{u}_{T},w_{T})^{n+1}$ and $(\boldsymbol{u}_{O},w_{O})^{n}$ . The geostrophic acceleration from momentum has been cancelled by the buoyancy tendency, the geostrophic momentum advection has been combined with the geostrophic buoyancy advection to form the first right-hand side term and the term $f_{0}^{2}\unicode[STIX]{x2202}_{z}^{2}w_{T}$ has been cancelled as part of the subtracted TTW subsystem balance. The derivation of the right-hand side $\unicode[STIX]{x1D6FC}$ term is in appendix B. This subsystem is a 3-D linear, second-order elliptic problem for $w_{O}$ with independent 2-D Poisson equation problems for $\boldsymbol{u}_{O}=\unicode[STIX]{x1D735}\unicode[STIX]{x1D712}_{O}$ and $\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D712}_{O}=-\unicode[STIX]{x2202}_{z}w_{O}$ at each $z$ . The left-hand side operator inversion will be well behaved as long as $N^{2}>0$ . With ${\mathcal{N}}^{\unicode[STIX]{x1D6FA}}=0$ , it is equivalent to the quasigeostrophic $\unicode[STIX]{x1D6FA}\text{E}$ ; however, notice that even in this simple quasigeostrophic case $w_{O}$ imposes a correction on $w_{T}$ that expresses the suppressive effect on vertical motion of the stable stratification $N^{2}$ in the oceanic interior. By comparison with (A 3)–(A 4), the composition of (2.20)–(2.21) can be recognized after the subtraction of the TTW subsystem balances.

A flow chart of the solution procedure is thus the following: given $b$ , ( $\unicode[STIX]{x1D708}_{v},\unicode[STIX]{x1D705}_{v}$ ), and ( $\unicode[STIX]{x1D749},~{\mathcal{B}}$ ), $\boldsymbol{u}_{g}$ is obtained from (2.4), then in an iterative sequence, $(\boldsymbol{u}_{a},w)$ are obtained from the TTW subsystem for $(\boldsymbol{u}_{T},w_{T})$ and the $\unicode[STIX]{x1D6FA}\text{E}$ subsystem for $(\boldsymbol{u}_{O},w_{O})$ . With a convergence of the iteration sequences, this yields a complete solution of the balanced system (2.11). The experience in obtaining the SCFT solutions in this paper is that convergence is readily achieved in most situations, barring excessively large $Ro_{g}$ values; near the convergence limit small values of $\unicode[STIX]{x1D6FE}$ in (2.24) can be helpful.

The general ${\mathcal{R}}$ formulations in (2.12)–(2.13), (2.16) and (2.21) are seemingly complicated, but the explanatory utility of a SCFT analysis will be greatest in particular idealized situations where the ${\mathcal{R}}$ forms are simpler (e.g. for 2-D $b(x,z)$ in § 3). As with most balanced models, there is no guarantee of solution existence for finite $Ro$ ; in particular, when the geostrophic advective tendencies are zero for a coherent structure, finite $Ro$ is a measure of the magnitude of ageostrophic advection (§ 6).

In more general evolutionary dynamics, e.g. the full PE (2.6), a failure of balance would manifest itself with an outbreak of faster oscillations and/or smaller spatial scales. Non-existence of a balanced solution would be manifested here as non-convergence of the $\text{TTW}+\unicode[STIX]{x1D6FA}\text{E}$ iteration cycle, although non-convergence does not necessarily imply non-existence. In this paper the limits of convergence will be approached but not probed assiduously. In some approximate balanced models, the solvability conditions can be identified analytically (e.g. McWilliams et al. Reference McWilliams, Yavneh, Cullen and Gent1998), but that is not readily accomplished in the present situation with vertical mixing. Furthermore, experience with circulation models (e.g. Gula et al. Reference Gula, Molemaker and McWilliams2014; Sullivan & McWilliams Reference Sullivan and McWilliams2016; Suzuki et al. Reference Suzuki, Fox-Kemper, Hamlington and Van Roekel2016) indicates that often, even at quite large $Ro$ , the simulated secondary circulations and frontogenesis in fronts and filaments are qualitatively similar to the solutions shown below, suggesting that their dynamics can remain close to, even if not exactly, balanced. The present paper provides an analytic basis for the balanced SCFT when $Ro\sim 1$ , but as yet there is no such basis for the phenomenologically similar behaviours at $Ro\gg 1$ .

Finally, it should be understood that these diagnostic solutions for $\boldsymbol{u}_{3}$ , given $b$ , are not steady in time. Rather, they have nonzero time derivatives that have been minimized by the selection of appropriate ‘coherent-structure’ spatial configurations for $b$ and by the balance approximations in the TTW $+$ $\unicode[STIX]{x1D6FA}\text{E}$ system that both preclude inertia–gravity wave dynamics and incorporate much of the advective tendency in cancellation with other tendency terms. (In a weather forecasting context this would be called an initialization procedure.) Given the SCFT solution, the tendencies for $\boldsymbol{u}_{g}$ and $b$ can be evaluated in (A 1) and (2.11), as can the frontogenetic tendencies in § 2.2.

2.2 Frontogenetic tendency

A complementary diagnostic perspective comes from evaluating the buoyancy gradient tendency associated with the diagnosed secondary circulation. The frontogenetic tendency equation is formed by taking the horizontal derivative of the buoyancy equation in (2.6) and then taking its dot product with $\unicode[STIX]{x1D735}b^{\prime }$ . Again, the derivation is made in the deformation-following transformed coordinates, and the result is expressed in Cartesian coordinates (appendix B):

These tendency terms, respectively, are the geostrophic self-straining ${\mathcal{T}}_{g}^{b}$ , two ageostrophic horizontal strain terms associated with buoyancy advection combined in ${\mathcal{T}}_{adv}^{b}$ , external straining deformation ${\mathcal{T}}_{\unicode[STIX]{x1D6FC}}^{b}$ , and boundary-layer vertical buoyancy mixing ${\mathcal{T}}_{\unicode[STIX]{x1D705}_{v}}^{b}$ . The final line indicates the notation for each of these right-hand side terms. All of them are known from the geostrophic flow and the secondary-circulation solution of the $\text{TTW}+\unicode[STIX]{x1D6FA}\text{E}$ system in the preceding section. The interpretation is that the right-hand side terms indicate the rate of change in $(\unicode[STIX]{x1D735}b)^{2}/2$ in a Lagrangian reference frame. They have no time derivatives; i.e. their evaluation is diagnostic. Positive right-hand side terms in (2.22) indicate a frontogenetic tendency.

In McWilliams et al. (Reference McWilliams, Molemaker and Olafsdottir2009b ), the frontogenetic tendency effects were diagnostically evaluated in Cartesian coordinates without explicitly implicating the external strain $\unicode[STIX]{x1D6FC}$ frontogenesis effect. The results there are not wrong, but this neglect of showing the dominant frontogenetic tendency effect explicitly is regrettable in retrospect. The derivation of this $\unicode[STIX]{x1D6FC}$ term is in appendix B.

An analogous diagnostic equation can be derived for the tendency in horizontal shear variance, $(\unicode[STIX]{x1D735}\boldsymbol{u})^{2}$ . In some cases (e.g. a dense filament; McWilliams et al. Reference McWilliams, Gula, Molemaker, Renault and Shchepetkin2015), this perspective is more informative about the subsequent frontogenetic evolution than is the buoyancy tendency (2.22). By operating on the horizontal momentum equation $j$ in (2.6) with $(\unicode[STIX]{x2202}_{i}u^{\,j})\unicode[STIX]{x2202}_{i}$ , the following frontogenetic tendency relation is obtained:

A horizontal index notation ( $i,j,k=1,2$ ) is used with repetition implying summation. The notation $\unicode[STIX]{x2202}_{i}$ is a shorthand for the partial derivative with respect to $x^{i}$ , and $u^{k}$ is the $k$ th component of $\boldsymbol{u}$ . The tendency terms are due to, respectively, pressure gradient, horizontal and vertical advection, external deformation, and boundary-layer vertical momentum mixing. The derivation of ${\mathcal{T}}_{\unicode[STIX]{x1D6FC}}^{\boldsymbol{u}}$ is in appendix B. No approximation to (2.6) is made here other than $\unicode[STIX]{x1D6FD}=0$ ; i.e. the left-hand side of (2.23) formally includes the total acceleration. Even in 3-D flows, there is no contribution to ${\mathcal{T}}_{adv}^{\boldsymbol{u}}$ and ${\mathcal{T}}_{\unicode[STIX]{x1D719}}^{\boldsymbol{u}}$ from wholly $\boldsymbol{u}_{g}$ terms.

2.3 Numerical methods

A numerical solution method is developed to solve the $\text{TTW}+\unicode[STIX]{x1D6FA}\text{E}$ subsystems iteratively (§ 2.1) and then to evaluate the diagnostic frontogenetic tendencies (§ 2.2) and buoyancy fluxes (§ 3). It is based on a uniform spatial grid in $x$ and $y$ and a stretched, staggered grid in $z$ , with $(u,v,b)$ at cell centres and $(w,N^{2})$ at cell edges and finer resolution near the surface. The differential operators are discretized with second-order accuracy. The elliptic operators are solved by matrix inversion. Iteration convergence is accelerated by relaxation, e.g.

for $\unicode[STIX]{x1D6FE}\leqslant 1$ , after each solution cycle in the TTW and $\unicode[STIX]{x1D6FA}\text{E}$ subsystems.

The fields $b(\boldsymbol{x}_{3})$ , $\unicode[STIX]{x1D708}_{v}(\boldsymbol{x}_{3})$ and $\unicode[STIX]{x1D705}_{v}(\boldsymbol{x}_{3})$ are specified analytically and particular numerical parameters are listed in § 3.2.

3.1 Two-dimensional SCFT equations

For this problem the $\text{TTW}+\unicode[STIX]{x1D6FA}\text{E}$ system (2.15)–(2.21) in § 2.1 is greatly simplified. The along-axis derivative is zero, $\unicode[STIX]{x2202}_{y}=0$ . The only geostrophic flow is $v_{g}=\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D713}_{g}$ and the vertical vorticity is $\unicode[STIX]{x1D701}=\unicode[STIX]{x2202}_{x}v_{T}$ . There is no geostrophic advection; thus, $\unicode[STIX]{x1D707}_{g}=\unicode[STIX]{x1D706}_{g}=J[\unicode[STIX]{x1D713}_{g},b^{\prime }]=0$ . Ageostrophic advection is only in the cross-axis plane: $D=\unicode[STIX]{x2202}_{t}+u\unicode[STIX]{x2202}_{x}+w\unicode[STIX]{x2202}_{z}$ . With these simplifications the 2-D forms of the TTW, $\unicode[STIX]{x1D6FA}\text{E}$ , ${\mathcal{T}}^{b}$ and ${\mathcal{T}}^{\boldsymbol{u}}$ equations are listed in appendix C.

Because the frontal spatial scale is small, the wind stress $\unicode[STIX]{x1D749}$ is assumed to be spatially uniform. The focus is on the interplay among wind stress, the boundary-layer vertical mixing, and the submesoscale structure in $b(x,z)$ . Thus, any consideration of surface buoyancy flux forcing effects is deferred: ${\mathcal{B}}^{\prime }$ is set to zero, and $\unicode[STIX]{x1D705}_{v}(x,z)\approx \unicode[STIX]{x1D708}_{v}(x,z)$ , as in the KPP scheme. However, oceanic circulation models (Gula et al. Reference Gula, Molemaker and McWilliams2014) and large eddy simulations with resolved boundary-layer turbulence (Sullivan & McWilliams Reference Sullivan and McWilliams2016) show similar SCFT behaviours with convective ${\mathcal{B}}^{\prime }$ for the essential reason that the turbulence, no matter how it is generated, causes vertical momentum mixing which in turn causes a frontogenetic secondary circulation for a front or filament. Nagai et al. (Reference Nagai, Tandon and Rudnick2006) analyses frontal secondary circulations when $\unicode[STIX]{x1D705}_{v}\neq \unicode[STIX]{x1D708}_{v}$ .

The bottom of the domain is at $z=-H$ , which is much deeper than the boundary-layer depth at $z=-h(x)$ .

3.2 Analytic formulas for $b^{\prime }$ , $\unicode[STIX]{x1D708}_{v}$ and $\unicode[STIX]{x1D705}_{v}$

The buoyancy field for a straight (2-D), upper-ocean front or filament is idealized as

The square of the buoyancy frequency for the interior stratification is $N_{0}^{2}$ , and $N_{0}^{2}$ is a weak background minimum stratification ( $0<N_{b}^{2}\ll N_{0}^{2}$ ) that assures that $\unicode[STIX]{x2202}_{z}b$ is positive everywhere. The fractional reduction of the surface boundary-layer stratification (relative to the interior $N_{0}^{2}$ ) is $\unicode[STIX]{x1D6E4}$ (i.e.  $0\leqslant \unicode[STIX]{x1D6E4}\ll 1$ ). The vertical scale of the transition between these two regimes $\unicode[STIX]{x1D706}^{-1}$ occurs around the base of the boundary layer at $z=-h(x)$ . The upper free surface is at $z=\unicode[STIX]{x1D702}(x)$ , with $\unicode[STIX]{x1D702}\ll h$ . The surface elevation is evaluated by vertical integration of hydrostatic balance, $\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D719}=b$ , upward from $z=-H$ where $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D719}^{\prime }=\unicode[STIX]{x2202}_{x}b^{\prime }=0$ to reach the surface boundary condition of a constant (atmospheric) value of $\unicode[STIX]{x1D719}$ at $z=\unicode[STIX]{x1D702}(x)$ ; equivalently, $\unicode[STIX]{x1D719}^{\prime }(x,0)\approx g\unicode[STIX]{x1D702}(x)$ by linearizing this surface pressure condition about $z=0$ .

The distinction between a front or filament is determined by the choice of $h(x)$ , i.e. a step in the former case and a localized extremum in the latter. For a front the choice is

with $x=0$ the centre of the front, $\ell$ the half-width, $h_{0}$ the central depth and $\unicode[STIX]{x1D6FF}h$ the depth change across the front. For $\unicode[STIX]{x1D6FF}h>0$ , the deep side of the frontal mixed layer is $x<0$ (west), which therefore is also the dense side in (3.1). The relation between the surface buoyancy horizontal change across the front $\unicode[STIX]{x1D6FF}b_{s}$ and $\unicode[STIX]{x1D6FF}h$ from (3.1) is

In the far fields at depth and on either side, $h_{fr}$ goes to $h_{0}\pm (\unicode[STIX]{x1D6FF}h)/2$ and $b(z)$ goes to two different horizontally uniform stratification profiles. With $f_{0}>0$ the associated geostrophic velocity $v_{g}$ is a northward surface jet around the front centre, mostly confined to the weakly stratified upper layer, with equal cyclonic and anti-cyclonic vorticity extrema on the west (dense) and east (light) sides, respectively. An analogous dense surface filament has

Its surface buoyancy anomaly $\unicode[STIX]{x1D6FF}b_{s}$ is again given by (3.3), and its $v_{g}(x,z)$ is a double jet, with $v_{g}>0$ on the east side and a dominant cyclonic $\unicode[STIX]{x1D701}_{g}=\unicode[STIX]{x2202}_{x}v_{g}>0$ in the centre with weaker negative lobes on the sides. A light surface filament would occur for (3.4) with $\unicode[STIX]{x1D6FF}h<0$ ; it has a weaker strain-induced frontogenesis rate (McWilliams et al. Reference McWilliams, Colas and Molemaker2009) and is thought to be rarer and/or weaker in nature and simulations; so the focus will be on dense filaments.

In association with the particular $h(x)$ shape functions in (3.2) and (3.4), the maximum values at the surface for the geostrophic along-axis velocity and vorticity are

with geometric shape coefficients $C_{v}=0.5$ and 1.0 and $C_{\unicode[STIX]{x1D701}}=0.5$ and 2.4, respectively, for the analytic formulas (3.2) and (3.4) for the front and dense filament. From these relations and (3.3), the strength of a submesoscale flow and its geostrophic Rossby number, $Ro_{g}=\unicode[STIX]{x1D701}_{g}/f_{0}$ , increase with the following influences: a stronger surface density anomaly $\unicode[STIX]{x1D6FF}b_{s}$ (which in turn increases with stronger interior stratification $N_{0}$ and boundary-layer depth anomaly $\unicode[STIX]{x1D6FF}h$ ); lower latitude $f_{0}$ ; deeper boundary-layer depth $h_{0}$ ; and smaller horizontal width $\ell$ (i.e. deeper within the submesoscale range). In particular, if such flow structures are formed by frontogenetic processes that decrease $\ell$ while holding these other quantities approximately constant, then $v_{g}$ and $Ro_{g}$ will increase as the frontogenesis proceeds.

The variation in $h(x)$ implies a variation in the vertical mixing if $h$ is interpreted as the boundary-layer depth. A KPP prescription (Large et al. Reference Large, McWilliams and Doney1994) for the vertical eddy viscosity in a wind-driven boundary layer is

The von Kármán constant is $k=0.4$ ; $u_{\ast }=\sqrt{|\unicode[STIX]{x1D70F}|/\unicode[STIX]{x1D70C}_{0}}$ is the oceanic wind-friction velocity; $\unicode[STIX]{x1D70E}$ is a normalized depth coordinate, and $\unicode[STIX]{x1D70E}_{0}=0.005$ is a velocity logarithmic singularity regularization constant as $\unicode[STIX]{x1D70E}\rightarrow 0$ ; and $\unicode[STIX]{x1D708}_{v0}$ is a small interior background eddy viscosity. In the absence of surface buoyancy flux ( ${\mathcal{B}}^{\prime }=0$ ), $\unicode[STIX]{x1D705}_{v}\approx \unicode[STIX]{x1D708}_{v}$ in the boundary layer, but usually $\unicode[STIX]{x1D705}_{v0}\leqslant \unicode[STIX]{x1D708}_{v0}$ in the interior. This is a reduced version of KPP by including only its dependencies on $b$ and the surface wind stress. A more complete version would depend also on $\boldsymbol{u}$ , but that would complicate the analysis here; furthermore, as yet not much is known about how vertical mixing might change inside a submesoscale flow structure.

For purposes of illustration a standard set of parameter values is used:

Most of these parameters are defined in the formulas (3.1), (3.2), (3.4) and (3.6). This $u_{\ast }$ value corresponds to an atmospheric surface wind speed of $U_{atm}=8.2$ m s $^{-1}$ under neutral stratification conditions in (2.10); its direction is $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ , with the convention that it increases counter-clockwise from an origin in the east direction. To have roughly comparable $\unicode[STIX]{x1D6FF}b_{s}$ , maximum $v_{g}$ , and maximum $\unicode[STIX]{x1D701}_{g}$ values, widths of $\ell =1.5$ km for the front and $\ell =3$ km for the dense filament are chosen. The corresponding buoyancy, geostrophic velocity and eddy viscosity fields are shown in figure 1. These parameter values represent a mid-sized submesoscale surface front or filament, rather than, e.g. a mesoscale flow structure with a larger $\ell$ and a deeper reach below the surface layer.

Figure 1.

Idealized shapes for a 2-D front (a,c,e) and dense filament (b,d,f) with (3.1), (3.2), (3.4) and (3.7):  $b(x,z)$ (a,b), $v_{g}(x,z)$ (c,d) and $\unicode[STIX]{x1D708}_{v}(x,z)$ (e,f). The boundary-layer depth $z=-h(x)$ is indicated by the thick black line. Only a portion of the full domain is plotted. The $\unicode[STIX]{x1D6FF}b_{s}$ values here correspond to a surface temperature difference of $0.25\,^{\circ }\text{C}$ with a thermal expansion coefficient of $\unicode[STIX]{x1D6FC}^{\ast }\approx 2\times 10^{-4}~\text{K}^{-1}$ .

For the solutions reported in this paper, the discrete calculations are made with a horizontal grid size of $\text{d}x=0.24$  km and a stretched vertical grid with $\text{d}z=2$  m near the surface. This seems adequate to resolve the solution structure except perhaps for strong nonlinearity near the limit of $\text{TTW}+\unicode[STIX]{x1D6FA}\text{E}$ system convergence when some flow structures become horizontally narrow (§ 6).

3.3 Buoyancy flux

Another diagnostic for the solutions below is the integrated buoyancy flux. For a 2-D $b(x,z)$ , vertical and horizontal fluxes are defined by

where the prime denotes the deviation from the average value in the $x$ far field. Because $\overline{w}=0$ in almost all the SCFT solutions below, $w^{\prime }$ is usually the total vertical velocity. (An exception is for a front when $U_{atm}$ is very large, discussed at the end of § 4.2; $BF_{v}$ is not analysed in this case.) The total $u$ is retained in $BF_{h}$ because $\overline{u}$ is typically non-zero when $u_{\ast }\neq 0$ , and the far-field velocity in $x$ is the Ekman current. The horizontal domain width $L_{x}$ is chosen to be much larger than $\ell$ .

4.1 Far-field velocity

The TTW subsystem solution in the far field where $\unicode[STIX]{x2202}_{x}b$ vanishes is simply an Ekman-layer velocity profile $\boldsymbol{u}_{\infty }(z)$ (e.g. figure 2), and $\unicode[STIX]{x1D6FA}\text{E}$ adds no correction. Because it is independent of $x$ , it has no associated $w$ or $BF_{v}$ , and it is uninfluenced by advection even when ${\mathcal{N}}^{\boldsymbol{u}},{\mathcal{N}}^{b}\neq 0$ . It has the expected attributes known from constant- $\unicode[STIX]{x1D708}$ solutions: the surface current is rotated anti-cyclonically (clockwise for $f>0$ ) from the surface stress, and with depth it rotates further in this direction while decreasing in amplitude. Its depth-integrated ageostrophic horizontal transport is

With the variable eddy viscosity $\unicode[STIX]{x1D708}_{v}$ profile in (3.6), these same attributes still occur, but the profile shape is altered with increased vertical shear at the top and bottom of the $[-h,0]$ layer where $\unicode[STIX]{x1D708}_{v}$ becomes small. The anti-cyclonic rotation angle at the surface is smaller, and $\boldsymbol{u}\approx 0$ for $z\leqslant -h$ . A scaling estimate for the Ekman current based on (4.1) is

which is representative of the mid-layer speed, with the surface value approximately twice larger. With changing wind direction $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ , the $\boldsymbol{u}_{\infty }(z)$ profile is unchanged except for a rotation with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ .

4.2 Secondary circulation

The velocity decomposition (2.14) represents the primary TTW solution $(\boldsymbol{u}_{T},w_{T})$ due to the wind stress, buoyancy gradient and vertical momentum mixing. The $\unicode[STIX]{x1D6FA}\text{E}$ correction $(\boldsymbol{u}_{O},w_{O})$ is due to the stratification and vertical buoyancy mixing, with an iteration between them for dynamical consistency. Figure 3 shows that overall $w$ is close to $w_{T}$ in this linear dense filament solution, but the $w_{O}$ correction is not trivial, especially near the bottom of the surface layer where the stratification begins to increase. At depth, the cancellation between $w_{T}$ and $w_{O}$ is complete, and $w$ is zero below. (In figure 3 this cancellation is only slightly evident because all $w$ components are vanishing near the bottom of the surface layer, but in § 6 ageostrophic advection causes $w_{T}=-w_{O}\neq 0$ to extend to the bottom of the domain.)

A vertical integral of the momentum equations in (2.15) with ${\mathcal{N}}_{T}^{\boldsymbol{u}}=0$ shows that the ageostrophic transport relation (2.15) for $\boldsymbol{T}_{T}$ is unaltered by the presence of $b(x,z)$ and $\unicode[STIX]{x1D708}_{v}(x,z)$ , so there can be no Ekman pumping even in the near field of the submesoscale structure, i.e.  $w_{ek}\approx \unicode[STIX]{x2202}_{x}T_{T}^{x}=0$ (neglecting a small contribution from $u(x,-h)\unicode[STIX]{x2202}_{x}h$ because $u(x,-h)$ is small). This restricts the vertical velocity to the surface layer in this linear TTW case.

The associated ageostrophic secondary circulation for a dense filament is shown in figure 4. The horizontal components are defined by

where the prime again denotes a departure from the far-field average. The circulation in the $(x,z)$ plane consists of two counter-rotating cells. Because the velocity is 2-D incompressible, this circulation can be expressed with an overturning streamfunction $\unicode[STIX]{x1D6F7}(x,z)$ defined by

The total circulation is $(u_{a}^{\prime },w)(x,z)$ plus the far field $u_{\infty }(z)$ (figure 2). The central downwelling jet is stronger than the peripheral upwelling flows. In the along-front direction, $v_{a}^{\prime }(x,z)$ mirrors $u_{a}^{\prime }(x,z)$ in its spatial pattern; relative to $v_{g}(x,z)$ (figure 1 d), it reduces the geostrophic shear by weakening $v$ in the upper part of the boundary layer and strengthening it in the lower part. It also reduces the vertical vorticity in the filament; the peak value for $\unicode[STIX]{x1D701}_{g}/f$ is 1.33 and the reduced peak for $\unicode[STIX]{x1D701}/f$ is 0.99 (not shown). Note that these normalized values comprise a $Ro$ , and the values are not small here, as for many submesoscale flows.

This orientation relationship between $\boldsymbol{u}_{a}^{\prime }(z)$ and $\boldsymbol{u}_{g}(z)$ holds at each horizontal location in the TTW subsystem with only modest adjustments from the $\unicode[STIX]{x1D6FA}\text{E}$ subsystem, and it is nearly independent of $\unicode[STIX]{x1D749}$ when $\unicode[STIX]{x1D735}b$ is as relatively large as it is here (but see figures 6 and 7 below). Figure 5 shows these velocity profiles in the centre of the northward along-front jet. The surface ageostrophic flow is rotated cyclonically by an obtuse angle relative to the surface geostrophic shear and it reverses in the lower part of the boundary layer.

In the $\text{TTW}+\unicode[STIX]{x1D6FA}\text{E}$ system there are two external drivers to the circulation, $\unicode[STIX]{x1D749}$ and $\unicode[STIX]{x1D735}b$ . To assess their relative importance in figures 4 and 8, artificial alternative solutions are obtained by setting $\unicode[STIX]{x1D749}=0$ , while retaining $\unicode[STIX]{x1D708}_{v},\unicode[STIX]{x1D705}_{v}$ as before. The results (not shown) are TTW circulations that are visually nearly indistinguishable from ones with $\unicode[STIX]{x1D749}\neq 0$ , except that now $\boldsymbol{u}_{\infty }=0$ . This is consistent with the fact that the secondary circulation is nearly independent of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ for the parameter values (3.7).

Figure 4.

Linear TTW secondary circulation for a dense filament with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ : (a,b) $u_{a}^{\prime }$ and $v_{a}^{\prime }$ after subtracting the far field $\boldsymbol{u}_{\infty }(z)$ in figure 2 and (c,d) $w$ and $\unicode[STIX]{x1D6F7}$ . The accompanying $b$ , $v_{g}$ , $\unicode[STIX]{x1D708}_{v}$ fields are in figure 1(b,d). Again and throughout the paper, $z=-h(x)$ is indicated by the thick black line.

Figure 5.

Vertical profiles of $\boldsymbol{u}$ in the centre of the northward jet at $x=2$ km for a dense filament with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ (as in figure 1 b,d,f): $v_{g}(z)$ is the solid line, $u_{a}^{\prime }$ is dashed and $v_{a}^{\prime }$ is dotted. The TTW surface $\boldsymbol{u}_{a}^{\prime }(0)$ is rotated cyclonically relative to $\boldsymbol{u}_{g}(0)$ by a little more than $3\unicode[STIX]{x03C0}/4$ , reverses sign with depth and vanishes near $z=-h$ .

Figure 6.

Ratio of the maximum absolute value of the linear TTW secondary-circulation streamfunction $\unicode[STIX]{x1D6F7}$ for a dense filament to the scaling estimate $\unicode[STIX]{x1D6F7}_{ttw}$ as a function of the wind speed $U_{atm}$ . The filament axis is parallel to $\hat{\boldsymbol{y}}$ . The black line is for cases with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ (eastward wind), and the red points are for selected wind speed values for cases with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x03C0}/2$ (northward wind).

A scaling estimate for the TTW secondary circulation comes from a primary balance in the $y$ -momentum equation in (2.15),

where the far-field values are used for $\unicode[STIX]{x1D708}_{v}$ and $h$ and the maximum value of $\unicode[STIX]{x2202}_{x}b$ is used for the buoyancy gradient. This leads to a scaling estimate,

where, from (2.10) and (3.7),

(The subscript $ttw$ here is meant to distinguish a particular dynamical balance that is close to simple turbulent thermal wind from the velocity decomposition (2.14), where T denotes the solution of the TTW subsystem (2.15) that can be valid for a much wider range of dynamical balances.) Thus, using (3.5), the TTW scaling estimates are

The perturbation ageostrophic velocities $u_{a}^{\prime }$ and $v_{a}^{\prime }$ both scale with $V_{ttw}$ because of their similarity in figure 5. These quantities are a combination of the buoyancy gradient (or $v_{g}$ ) and the vertical mixing strength induced by the wind stress.

A comparison can be made with an analogous scaling estimate for a field of mixed-layer eddies generated by the surface-layer baroclinic instability of a mean horizontal buoyancy gradient in Fox-Kemper, Ferrari & Hallberg (Reference Fox-Kemper, Ferrari and Hallberg2008), Bachman & Taylor (Reference Bachman and Taylor2016), viz.  $\unicode[STIX]{x1D6F7}_{mle}\sim 0.1h_{0}^{2}\unicode[STIX]{x2202}_{x}\overline{b}/f=0.1h_{0}\overline{V}_{g}$ , which is accompanied by a vertical restratification buoyancy flux estimate of $\overline{w^{\prime }b^{\prime }}\sim \unicode[STIX]{x1D6F7}_{mle}\unicode[STIX]{x2202}_{x}\overline{b}=0.1f\overline{V}_{g}^{2}$ (cf.  $BF_{v}/\ell$ in (4.19)).

Notice by comparing figures 2 and 5 that the magnitudes of $u_{a}^{\prime }$ and $u_{\infty }$ are similar. The ratio of their scaling estimates in (4.8) and (4.2) is

which is approximately one for the parameters in (3.7).

In figure 6 the aptness of this scaling estimate is demonstrated over nearly three decades in $|\boldsymbol{U}_{atm}|$ . For low and intermediate winds, there is no directional dependence in $\unicode[STIX]{x1D6F7}$ , as discussed above and as presumed in (4.6), while for high winds there is a directional dependence that is evident in both figures 6 and 7; nevertheless over this whole wind speed and direction range, $\unicode[STIX]{x1D6F7}_{ttw}$ is an $O(1)$ accurate estimate for $\max [|\unicode[STIX]{x1D6F7}|]$ .

Figure 7.

Linear TTW secondary-circulation streamfunction $\unicode[STIX]{x1D6F7}(x,z)$ for a dense filament with $|\boldsymbol{U}_{atm}|=32.8~\text{m}~\text{s}^{-1}$ : $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ (a) and $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0.5\unicode[STIX]{x03C0}$ (b). The $b(x,z)$ , $h(x)$ and $v_{g}(x,z)$ fields are the same as in figure 1(b,d,f), and the $\unicode[STIX]{x1D708}_{v}(x,z)$ field has the same pattern but an amplitude proportional to the different $u_{\ast }$ value (i.e. four times larger).

At large $U_{atm}$ , the presence of $v_{g}$ is less important in the secondary circulation within the central filament region than is the variation in $\unicode[STIX]{x1D708}_{v}\propto h(x)$ in the KPP parameterization (3.6); i.e. this regime is one with a spatially varying Ekman layer across the submesoscale flow structure. (The model here assumes that the filament structure in $b(x,z)$ survives the intense boundary-layer turbulence under a very high wind, which it might not do.) A scaling estimate is $u_{a}^{\prime }\sim u_{\infty }(\unicode[STIX]{x1D6FF}h/h_{0})$ because $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}_{v}/\unicode[STIX]{x1D708}_{v}=\unicode[STIX]{x1D6FF}h/h_{0}$ in (3.6). Using (3.3), (3.5), and (4.2), this implies

Because of the abundance of scaling estimates in this paper, in this and most of the further estimates below an $O(1)$ non-dimensional coefficient, fitted to the actual solutions, is left out for brevity; in any event its magnitude will depend on the submesoscale functional shapes, as noted for $C_{v}$ and $C_{\unicode[STIX]{x1D701}}$ after (3.5).

As $u_{\ast }$ increases for fixed $V_{g}$ , the ratio of this secondary-circulation streamfunction to the TTW one in (4.8) is

i.e. the product of a Rossby number based on $u_{\ast }$ and an inverse Burger number for the surface layer $h$ and the interior stratification $N_{0}$ . This reaches a value around one for $U_{atm}\approx 40~\text{m}~\text{s}^{-1}$ , consistent with the $O(1)$ directional difference in figure 7 at that $U_{atm}$ value. The reason for the directional difference is that $u_{a}(z)$ is more surface trapped in an Ekman layer for $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ , where the Ekman transport, $T_{T}^{y}<0$ , is along the filament, than it is for the case of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x03C0}/2$ , where $T_{T}^{x}>0$ is across the filament. The difference of two analytic Ekman-layer solutions with different constant $\unicode[STIX]{x1D708}_{v}$ values (i.e. a larger one in the filament centre), shows an analogous sign reversal and magnitude difference in the associated $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}(z)$ between the two $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ values in figure 7.

Figure 8.

Linear TTW secondary circulation for a front with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ : (a) $u_{z}^{\prime }(x,z)$ and $v_{a}^{\prime }(x,z)$ (b) after subtracting the far-field averaged $\overline{\boldsymbol{u}}(z)$ , and $w(x,z)$ (c) and $\unicode[STIX]{x1D6F7}(x,z)$ (d). Note that the far-field values are small but non-zero because of the modestly different Ekman profiles $\boldsymbol{u}_{\infty }(z)$ on the two sides. The accompanying $b$ , $v_{g}$ , $\unicode[STIX]{x1D708}_{v}$ fields are in figure 1(a,c,e).

For a front the linear TTW solution is very similar to that from a dense filament except for the change of shape for $b(x,z)$ and $(\unicode[STIX]{x1D708}_{v},\unicode[STIX]{x1D705}_{v})(x,z)$ (figure 1 a,c,e). The far-field velocity profile is now different on both sides because $h$ is different: the two far-field $\boldsymbol{u}(z)$ shapes are similar to figure 2, but with the vertical scale compressed or expanded with the different far-field $h$ values, and the associated $\boldsymbol{u}$ magnitude is increased or decreased accordingly to conform to the transport integral constraint (4.1). The secondary circulation is again confined to the vicinity of the front and to within the surface layer (figure 8). Its shape is now a single cell in $\unicode[STIX]{x1D6F7}$ , with upwelling on the light side and surface flow toward the dense side. The $v_{a}^{\prime }(x,z)$ again acts to reduce the geostrophic shear. The maximum value of $Ro_{g}=\unicode[STIX]{x1D701}_{g}/f_{0}$ is 1.04, and again the vorticity is reduced by TTW to a value of $Ro=\unicode[STIX]{x1D701}/f_{0}=0.70$ . The peak value of $\unicode[STIX]{x1D6F7}$ is reduced by approximately 10 % for the front compared to the filament, but, because of shape differences in their $b(x,z)$ , the peak downwelling value of $w$ is reduced by about 30 % for the front.

For the parameters (3.7), the secondary circulation for a front is again independent of wind direction $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ . For large $U_{atm}$ the vertical velocity is one-signed at the front, with a monotonic $x$ variation across the front in $\unicode[STIX]{x1D6F7}(x,z)$ from (4.4) (not shown). Again the sign and magnitude of $w$ differ with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ : $w$ is larger and positive for a north wind when $U_{atm}$ is large, compared to smaller and negative with an east wind. The scaling estimates (4.2), (4.8), and (4.10) remain valid for fronts as well.

4.3 Frontogenetic tendency and buoyancy flux

In the presence of the secondary circulations in the preceding section, there are implications for further evolution of the submesoscale structure expressed as frontogenetic tendencies in velocity and buoyancy (§ 2.2) and as vertical and horizontal buoyancy fluxes (3.8) across the structure.

For the velocity gradient the frontogenetic tendencies are shown in figure 9. The sum of the advective and pressure force tendencies, ${\mathcal{T}}_{adv\,\unicode[STIX]{x1D719}}^{\boldsymbol{u}}={\mathcal{T}}_{adv}^{\boldsymbol{u}}+{\mathcal{T}}_{\unicode[STIX]{x1D719}}^{\boldsymbol{u}}$ , is clearly positive in the upper centre of the filament, as well as on both sides of the front, especially on the dense side. These are partly balanced by negative tendencies associated with the vertical mixing, ${\mathcal{T}}_{\unicode[STIX]{x1D708}_{v}}^{\boldsymbol{u}}$ . The sum of these, ${\mathcal{T}}^{\boldsymbol{u}}$ , is also positive in the upper centre of the dense filament and the upper dense side of the front, but negative on the upper light side. Thus, there is an important positive frontogenetic tendency near the surface for both types of submesoscale flow structure, with a net frontolytic tendency on the upper light side of the front. (It is known from time integration that in fact the TTW secondary circulation drives a dense filament toward a frontogenetic singularity in $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x2202}_{x}u$ (McWilliams et al. Reference McWilliams, Gula, Molemaker, Renault and Shchepetkin2015).) The maxima of ${\mathcal{T}}^{\boldsymbol{u}}$ are 0.8 and $1.8\times 10^{-13}~\text{s}^{-3}$ for the front and filament, respectively. In all the contributing tendencies for ${\mathcal{T}}^{\boldsymbol{u}}$ , the magnitudes are about 50 % smaller for the front compared to the filament.

Figure 9.

Frontogenetic velocity-gradient tendencies from (2.23). The sum of ${\mathcal{T}}_{adv\,\unicode[STIX]{x1D719}}^{u}={\mathcal{T}}_{\unicode[STIX]{x1D719}}^{\boldsymbol{u}}+{\mathcal{T}}_{adv}^{\boldsymbol{u}}$ is in (a,b), ${\mathcal{T}}_{\unicode[STIX]{x1D708}_{v}}^{\boldsymbol{u}}$ is in (c,d) and the total ${\mathcal{T}}^{\boldsymbol{u}}$ is in (e,f). The front and dense filament solutions are in (a,c,e) and (b,d,f), respectively. These are for the standard parameters (3.7) with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ .

The patterns in figure 9 can be understood by identifying the dominant contributions to ${\mathcal{T}}^{\boldsymbol{u}}$ . They are associated with the horizontal strain component $\unicode[STIX]{x2202}_{x}u$ and the along-axis velocity $v$ that is primarily geostrophic $v_{g}$ ; i.e.

Because $f_{0}$ and $\unicode[STIX]{x1D701}_{g}=\unicode[STIX]{x2202}_{x}v_{g}$ are of the same order in these solutions, both terms in ${\mathcal{T}}_{adv}^{\boldsymbol{u}}$ here contribute comparably. The relevant patterns of $u_{a}^{\prime }$ and $v_{g}$ for the dense filament and front are evident in figures 1, 4 and 8.

This leads to a scaling estimate for ${\mathcal{T}}_{adv}^{\boldsymbol{u}}$ based on the first right-hand side term in (4.12) and $\unicode[STIX]{x1D6F7}_{ttw}$ in (4.8), viz.

With the parameter values in (3.7), this estimate is within an $O(1)$ value of the solutions for both the front and filament. It is evidently a highly nonlinear quantity, so the actual results have numerical values that are sensitive to the specific shape choices, though the patterns are robust. For solutions with smaller $Ro_{g}$ values, the estimate should be based on the second right-hand side term in (4.12) that will be the larger one; i.e.

whose ratio to (4.13) is $Ro_{g}^{-1}$ . With the same scaling for $\unicode[STIX]{x1D708}_{v}$ following (4.6), ${\mathcal{T}}_{\unicode[STIX]{x1D708}_{v}}^{\boldsymbol{u}}$ has the same scaling as ${\mathcal{T}}_{\unicode[STIX]{x1D719}}^{\boldsymbol{u}}$ . Thus, it is a parametrically robust result that the total ${\mathcal{T}}^{\boldsymbol{u}}$ has a frontogenetic shape similar to figure 9, bottom, because of the similar scaling among its component tendencies when $Ro_{g}\sim 1$ .

A similar evaluation is made for the buoyancy-gradient advective tendency ${\mathcal{T}}_{adv}^{b}$ in figure 10. For these solutions the diffusive tendency is negligible because $\unicode[STIX]{x2202}_{z}b$ is small where $\unicode[STIX]{x1D705}_{v}$ is not small. The dominant contribution is again associated with the horizontal strain,

Thus, the patterns of $u_{a}^{\prime }$ and $b^{\prime }$ explain the four-centred structure for the front, with moderate frontogenesis in the upper dense side where there is surface horizontal convergence and somewhat stronger frontolysis in the lower dense side where there is divergence in the lower part of the boundary layer. For the dense filament, this pattern is simply doubled, as expected from the remark that a filament is a kind of double front in $u_{a}^{\prime }$ and $b$ . (This perspective is not as useful for ${\mathcal{T}}^{\boldsymbol{u}}$ , where the central cyclonic shear in a dense filament is the single dominant influence compared to the peripheral shears.) By the same rationale as for (4.13), the scaling estimate here is

and it is quantitatively apt for the solutions here.

Figure 10.

Advective frontogenetic buoyancy-gradient tendency ${\mathcal{T}}_{adv}^{b}$ from (2.22) for the front (a) and dense filament (b). Because ${\mathcal{T}}_{\unicode[STIX]{x1D705}_{v}}^{b}$ is small, these are approximately the total tendency ${\mathcal{T}}^{b}$ . These TTW solutions are for the standard parameters (3.7) with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ .

If ${\mathcal{T}}_{adv\,ttw}^{\boldsymbol{u}}$ and ${\mathcal{T}}_{adv\,ttw}^{b}$ are normalized by the velocity- and buoyancy-gradient variances, respectively, a frontogenetic rate is estimated as

which is the same for both frontogenetic tendencies. The same normalization for ${\mathcal{T}}_{\unicode[STIX]{x1D719}\,ttw}^{\boldsymbol{u}}$ yields an estimate of

This latter estimate can be interpreted as the inverse of a multiple of the turbulent eddy-turnover time in a wind-driven boundary layer, where the turbulent kinetic energy scales as $u_{\ast }^{2}$ . Notice that $r$ does not depend on the submesoscale structural parameters $V_{g}$ and $\ell$ except through $Ro_{g}$ .

Another comparison for $r_{ttw}$ is with the geostrophic $t_{eddy}^{-1}$ discussed in § 1. Their scaling ratio is 0.1 $u_{\ast }/(f_{0}h_{0})$ , which for present parameters has a value of approximately 0.2. Thus, even in coherent structures like 2-D fronts and filaments, with no conservative evolutionary tendencies (i.e.  $t_{cs}^{-1}$ is zero), the ageostrophic advective frontogenetic rate associated with the TTW circulation is not that much smaller than the turbulent eddy-turnover time for geostrophic advection in an incoherently configured eddy field.

As with the secondary circulation (§ 4.2), there is little dependence of ${\mathcal{T}}^{\boldsymbol{u}}$ and ${\mathcal{T}}^{b}$ on the wind direction $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ as long as $\unicode[STIX]{x1D6F7}_{ttw}$ in (4.8) is not small compared to $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FF}h}$ in (4.10).

Figure 11.

Integrated buoyancy fluxes (3.8) for the front (dashed lines) and dense filament (solid lines) with the parameters in (3.7). Panels (a,c) and (b,d) are $BF_{v}(z)$ and $BF_{h}(x)$ , respectively. Panels (a,b) and (c,d) are for $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ and $\unicode[STIX]{x03C0}/2$ , respectively. Note the change of scale for $BF_{h}$ between the two wind directions.

The buoyancy fluxes for both the front and filament are plotted in figure 11. The vertical flux $BF_{v}$ is everywhere positive and peaks in the middle of the surface layer. This is a restratification flux and represents a conversion from available potential energy to kinetic energy. Its vertical shape is controlled mainly by $w(z)$ because $b^{\prime }$ varies little with $z$ in the surface layer. It is somewhat larger for the filament than the front, consistent with its larger $w$ . Because it is controlled by the secondary circulation (i.e. by $w$ ), it also does not depend on $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ for the standard parameters (3.7). A scaling estimate based on $\unicode[STIX]{x1D6F7}_{ttw}$ is

whose magnitude is close to the peak value in figure 11(b,d), and whose approximate vertical buoyancy flux per unit area is $\overline{w^{\prime }b^{\prime }}=BF_{v~ttw}/\ell$ . This restratification flux can be interpreted by comparison with an equivalent surface heat flux per unit area, $Q=(\unicode[STIX]{x1D70C}_{0}c_{p}/\unicode[STIX]{x1D6FC}^{\ast }g)\max _{z}[BF_{v}(z)]/\ell \approx 200~\text{W}~\text{m}^{-2}$ (where $c_{p}\approx 4\times 10^{3}~\text{m}^{2}~\text{s}^{-2}~\text{K}^{-1}$ is the heat capacity of seawater and $\unicode[STIX]{x1D6FC}^{\ast }\approx 2\times 10^{-4}~\text{K}^{-1}$ is the thermal expansion coefficient); i.e.  $BF_{v}$ is not small.

For $BF_{h}$ , however, there is an important dependency on wind direction, as demonstrated by the much larger values for $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x03C0}/2$ than for $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ . The reason is the influence of $u_{\infty }(z)$ . For all wind directions, $u_{a}^{\prime }(z)$ reverses with depth (figure 4 a) and so has partly cancelling contributions to the integrand in $BF_{h}$ . The same is true for $u_{\infty }(z)$ for an eastward wind whose $T_{T}^{x}$ in (4.1) is zero. However, for a northward wind, $u_{\infty }(z)$ is mostly eastward at all depths and $T_{T}^{x}=u_{\ast }^{2}/f_{0}>0$ , so the horizontal buoyancy flux is much larger. A scaling estimate for the TTW secondary-circulation buoyancy flux is

which is the same as in (4.19). An estimate for the horizontal Ekman buoyancy flux, when $\boldsymbol{T}_{T}$ in (4.1) is directed across the front or filament, is

Their ratio is formally the same as (4.9), i.e.  ${\approx}1$ for the parameters in (3.7), but the actual solutions in figure 11 indicate that the estimator $BF_{h~ttw}$ is too large by approximately a factor of 20 because of the sign reversal in $u_{a}^{\prime }$ with depth, which causes opposing fluxes across the buoyancy gradient.

The shape of $BF_{h}$ is controlled by the shape of $u(x,z)$ times the surface $b^{\prime }(x,0)$ , which is approximately the shape of $b^{\prime }(x,z)$ throughout the surface layer. For the secondary circulation component $u_{a}^{\prime }$ , the sense of the horizontal flux is from higher buoyancy toward lower. For the dense filament this implies a horizontal buoyancy convergence toward the centre, while for the front it is a flux across the front toward the dense side. For the Ekman component, because $u_{\infty }(z)$ is independent of $x$ , $BF_{h}(x)$ has the shape of $b^{\prime }(x,0)$ , and thus is across the buoyancy gradient; in fact, replacing $u$ by $u_{\infty }$ in the $BF_{h}$ integrand in (3.8), so that $BF_{h}(x)\propto b^{\prime }(x,0)$ , is quite an accurate approximation for $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x03C0}/2$ . The simplest evolutionary interpretation in this case is that the Ekman buoyancy flux implies a bulk horizontal movement of the submesoscale $b^{\prime }$ structure in the direction of $T_{T}^{x}$ , which would therefore have no reflection in $w$ , ${\mathcal{T}}^{\boldsymbol{u}}$ or ${\mathcal{T}}^{b}$ , all of which are nearly independent of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ . Alternatively, there may be vertically differential advection of $b^{\prime }(x,z)$ by $(u,w)$ , which would be expressed in other evolutionary changes in the structure’s shape than those evident in the $BF$ profiles. In some circumstances with down-front wind stress, especially with weak interior stratification $N_{0}$ , the horizontal buoyancy flux can advect dense fluid over lighter fluid and induce gravitational instability; furthermore, down-front winds can deplete the potential vorticity of the surface layer and induce centrifugal (symmetric) instability (Thomas Reference Thomas2005; Taylor & Ferrari Reference Taylor and Ferrari2009).

As the wind direction varies away from $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=0$ , there will be a transition from $BF_{h~ttw}$ , which is essentially independent of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ , to $BF_{h~ek}$ with a weighting factor of $\sin [\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}]$ .

Figure 12.

Linear strain-induced, secondary-circulation streamfunction $\unicode[STIX]{x1D6F7}(x,z)$ for a front (a) and dense filament (b). These cases have $\unicode[STIX]{x1D6FC}=10^{-5}~\text{s}^{-1}$ but no wind stress or vertical mixing. These can be compared to the topologically similar TTW $\unicode[STIX]{x1D6F7}$ patterns in figure 8(b,d), and figure 4(d), respectively.

6.1 Secondary circulation

One effect of nonlinearity is to alter the Ekman transport relation (4.1), which generalizes to

when $\unicode[STIX]{x1D707}_{g}=\unicode[STIX]{x1D6FC}=0$ ; i.e. for a 2-D front or filament, cross-axis momentum advection causes Ekman pumping at the base of the surface layer, with $w_{ek}\approx \unicode[STIX]{x2202}_{x}T_{T}^{x}\neq 0$ . On the other hand, as in § 5, the induced pycnocline circulation will only penetrate over the rather small depth scale of $h_{p}$ for submesoscale flow structures. As in figure 3, $w_{T}\gg w_{O}$ in the surface layer, but with the advective contribution to (6.1), both components extend to the bottom of the domain, with $w_{O}=-w_{T}$ and $w=0$ at depth.

Figure 15.

Spatial minima of $w(x,z)$ for nonlinear TTW solutions for a front (a) and a dense filament (b) with parameters (3.7) as a function of wind direction $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ . The dashed lines correspond to the minima in the linear TTW solutions in figures 4(c) and 8(c), respectively.

Unlike the linear TTW solutions in § 4 (when $V_{\unicode[STIX]{x1D6FF}h}\ll V_{ttw}$ from (4.10)), advection causes a significant dependence on the wind direction $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ . The reason for this is that the dominant advection in the 2-D flows examined here is the cross-axis component, $u_{a}\unicode[STIX]{x2202}_{x}\cdot$ , and $u_{a}$ has an important contribution from the far-field (Ekman) velocity, $u_{a\infty }(z)$ (e.g. figure 2). Figure 15 shows how the peak downwelling velocity varies with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ for both fronts and dense filaments. The strongest $w$ minima occur when $u_{a}^{\prime }$ constructively reinforces $u_{a\infty }$ in the upper part of the surface layer. In a dense filament with its approximate anti-symmetry with two signs of $u_{a}^{\prime }$ (figure 4 a), this occurs for two wind directions, approximately NE and SW, where the near-surface Ekman velocity is most nearly across axis. In a front with only a single sign for $u_{a}^{\prime }$ (figure 8 a), the SW direction supports this constructive reinforcement. There is also a weak enhancement of $w<0$ for a front with N to NE winds; in this case the upper-ocean TTW and Ekman $u$ are opposing, but still there is a modest enhancement of $\unicode[STIX]{x2202}_{x}u<0$ .

This rationalization of the $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ dependency is based on the nonlinear secondary circulation not being too different in its spatial patterns from the linear one. This is clearly so in the far field, where the linear and nonlinear solutions are exactly the same. It is only approximately so in the central region. It is clear from figure 15 (cf. the dashed lines) that the nonlinear circulation can be substantially stronger (by a factor up to 3-4 in $w$ ). The associated spatial patterns for the peak $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ values are shown in figures 16 and 17, and they are indeed generally similar in shape to their linear counterparts (i.e. figures 8 c, and 4 c, respectively). In particular, in all cases there is a downwelling $w$ on the dense side of a front and near the centre of a dense filament. Besides the larger magnitudes for favourable $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ , the differences due to advection are mainly ones of sharper gradient, smaller width and greater $x$ asymmetry.

Figure 16.

Nonlinear TTW secondary circulation for a front with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=-3\unicode[STIX]{x03C0}/4$ and $Ro_{g}=1.04$ : (a,b) $u_{a}^{\prime }$ and $v_{a}^{\prime }$ after subtracting the far field $\boldsymbol{u}_{\infty }(z)$ in figure 2 and (c,d) $w$ and $\unicode[STIX]{x1D6F7}$ . This angle choice corresponds to the largest $\min [w]$ value in figure 15(a). The accompanying $b$ , $v_{g}$ , $\unicode[STIX]{x1D708}_{v}$ fields are in figure 1(a,c,e), and the corresponding linear solution for the ageostrophic circulation is in figure 8.

Figure 17.

Nonlinear TTW secondary circulation for a dense filament with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=3\unicode[STIX]{x03C0}/8$ and $Ro_{g}=1.32$ : (a,b) $u_{a}^{\prime }$ and $v_{a}^{\prime }$ after subtracting the far field $\boldsymbol{u}_{\infty }(z)$ in figure 2 and (c,d) $w$ and $\unicode[STIX]{x1D6F7}$ . This angle choice corresponds to the largest $\min [w]$ value in figure 15(b). The accompanying $b$ , $v_{g}$ , $\unicode[STIX]{x1D708}_{v}$ fields are in figure 1(b,d,f), and the corresponding linear solution for the ageostrophic circulation is in figure 4.

For completeness, $T_{T}^{x\prime }(x)$ from (6.1) is plotted for these nonlinear solutions in figure 18. It shows a tripole pattern for the front and a dipole pattern for the dense filament. The implied $w_{ek}(x)$ for the front is a tripole with a downwelling centre slightly west of the middle of the front, echoing the $w$ pattern in figure 16 but different from the basic linear TTW dipole pattern in figure 8. For the filament, the implied $w_{ek}$ is a dipole with downwelling west of the centre, as in figure 17 but again different from the basic linear TTW tripole in figure 1. Were $h_{p}$ larger (i.e. with smaller $N_{0}$ , bigger $\ell$ , or larger $f_{0}$ ), then these $w_{ek}(x)$ patterns would be more evident in the pycnocline, while they are rather faint in the present solutions. The strongest nonlinear $w(x,z)$ values occur in the surface layer, and they preserve the linear TTW shapes of a dipole for the front and an almost central downwelling for the filament. The secondary peak in $\min [w](\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}})$ for the dense filament with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=-5\unicode[STIX]{x03C0}/8$ has a  $w(x,z)$ pattern with a similar narrow, downwelling branch as in figure 17 but mirror reflected to be displaced to the east of the filament centre.

Figure 18.

Ageostrophic transport anomaly $T_{T}^{x\prime }(x)$ from (6.1) across a front (a) and a dense filament (b) for $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=-3\unicode[STIX]{x03C0}/4$ and $3\unicode[STIX]{x03C0}/8$ , respectively.

A more formulaic way to understand the influence of nonlinearity is to focus on the $y$ -momentum equation in (2.15) and estimate the incremental cross-axis velocity $u_{TNL}$ by a balance with the Coriolis force; viz. with successive approximations,

These several approximations are rough ones (especially the last step with $u\approx u_{\infty }$ ), but they preserve much of the spatial structure and order of magnitude of the more general dynamical balances. This then is connected to the incremental effect on $w$ by the continuity equation in the upper part of the surface layer,

hence, the advectively generated vertical velocity in the centre of the surface layer is

From the simple shapes for the front (i.e.  $u_{\infty }<0$ with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=-3\unicode[STIX]{x03C0}/4$ , $\unicode[STIX]{x1D701}_{g}$ a dipole with a positive lobe to the west, and $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{g}$ with a central negative extremum) and the dense filament (i.e.  $u_{\infty }>0$ for $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=3\unicode[STIX]{x03C0}/8$ , $\unicode[STIX]{x1D701}_{g}$ a positive monopole, and $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{g}$ a central dipole with its negative lobe to the west side), the resulting shapes are thus explained for $w_{TNL}(x)$ . It has a central downwelling for the front that adds to the dipole pattern for the linear TTW $w$ to make an unequal dipole with strong downwelling in the west (cf. figure 16), and it has a dipole for the filament with its downwelling lobe to the west that adds to the central downwelling for linear TTW central $w$ to make a dominant downwelling centre in the west (cf. figure 17).

The same formulas as in (6.2) can be used to give a scaling estimate for the incremental secondary circulation due to nonlinearity. Here separate estimates for $u$ are made based on either the Ekman velocity $V_{ek}$ in (4.2) or the TTW circulation anomaly $V_{ttw}$ in (4.8), whose inverse ratio is (4.9). The result is

where ${\mathcal{F}}=\max [0.1V_{g},u_{\ast }]$ . As expected their ratio with their linear counterparts in § 4 is $Ro_{g}$ . For the solutions with parameters (3.7), the values of the arguments in ${\mathcal{F}}$ are approximately the same, which indicates that it is indeed the $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ values for constructive reinforcement between $u_{a}^{\prime }$ and $u_{\infty }$ that generates the large $w$ extrema in figures 16 and 17. Furthermore, although the previous argument emphasizes the role of ${\mathcal{N}}^{v}$ , the contributions from ${\mathcal{N}}^{u}$ and ${\mathcal{N}}^{b}$ are also quantitatively important here where $Ro_{g}\sim 1$ . For weaker submesoscale flows with $Ro_{g}\ll 1$ , $\unicode[STIX]{x1D6F7}_{NL}\propto u_{\ast }^{2}V_{g}$ , which is linear in $V_{g}$ . As $Ro_{g}$ further increases, $\unicode[STIX]{x1D6F7}_{NL}\propto u_{\ast }V_{g}^{2}$ ; i.e. it becomes quadratic in $V_{g}$ .

The other diagnostic quantities of interest also vary with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ with curves at least broadly similar to the shape of $\text{min}[w](\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}})$ in figure 15, but the magnitudes of their $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ dependencies differ; e.g. in figures 16 and 17, the nonlinear amplification of the extrema for $\boldsymbol{u}_{a}^{\prime }$ and $\unicode[STIX]{x1D6F7}$ is not as large as for $w$ , by about 50 % and 75 %, respectively, rather than the 200 %–300 % for $w$ . The amplification factors for each of the fields are similar between the front and filament. Furthermore, sharper horizontal gradients are evident in the nonlinear solutions.

As a function of $Ro_{g}$ , the dependencies in (6.5) have begun to make their transition from the linear to the steeper quadratic variation with $V_{g}$ by these $Ro_{g}\sim 1$ solutions. Furthermore, both of the extreme $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ cases shown here are close to the limit of iteration convergence for the $\text{TTW}+\unicode[STIX]{x1D6FA}\text{E}$ system; i.e. very small values of $\unicode[STIX]{x1D6FE}$ in (2.24) and large iteration numbers are required, and convergence fails for further increases in $\unicode[STIX]{x1D6FF}h_{0}$ (i.e. in $Ro_{g}$ ). As discussed at the end of § 2.1, this is suggestive of a breakdown of balanced solutions. As $Ro_{g}$ approaches the convergence limit, the diagnostic SCFT solutions do develop sharper spatial structures and rapidly increasing magnitudes, and, not surprisingly, a dependency develops on the discrete grid resolution (§ 3), which is not pursued here. The magnitude increases with $Ro_{g}$ are possibly steeper than the quadratic dependency in the scaling estimate (6.5); the range of successful large $Ro_{g}$ solutions is too small to determine this here. One can at least contemplate the possibility that balanced dynamics is approaching a finite-time singularity (n.b. Hoskins & Bretherton (Reference Hoskins and Bretherton1972) and McWilliams et al. (Reference McWilliams, Gula, Molemaker, Renault and Shchepetkin2015) for balanced strain-induced and TTW evolutionary frontogenesis, respectively).

Figure 19.

Frontogenetic velocity gradient tendencies ${\mathcal{T}}^{\boldsymbol{u}}$ from (2.23) in nonlinear TTW solutions for a front with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=-3\unicode[STIX]{x03C0}/4$ (a) and dense filament with $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=3\unicode[STIX]{x03C0}/8$ (b). These angle choices correspond to the largest $\min [w]$ values in figure 15. They can be compared with their linear counterparts in figure 9(e,f).

6.2 Frontogenetic tendency and buoyancy flux

The frontogenetic velocity gradient tendencies ${\mathcal{T}}^{\boldsymbol{u}}$ remain similar in pattern but amplified in magnitude and shrunk in scale due to nonlinearity for both the front and dense filament (figure 19). The peak positive values are on the west side of the centres. For a front, this is already the favoured location for ${\mathcal{T}}^{\boldsymbol{u}}>0$ in the linear TTW solution (figure 9 e,f), and for both front and filament nonlinear TTW solutions it is also the location of peak surface convergence above the peak downwelling flow (figures 16 and 17). The nonlinear amplification factor for the peak values of ${\mathcal{T}}^{\boldsymbol{u}}$ is similar to that for $w$ . For these TTW solutions there is some but smaller nonlinear amplification in ${\mathcal{T}}^{b}$ (not shown).

The nonlinear TTW scaling estimate for ${\mathcal{T}}^{\boldsymbol{u}}$ is amplified compared to the linear one (4.13) by the factor $Ro_{g}$ , as it is for $w$ in (6.5) as well. This implies a highly nonlinear dependency, ${\mathcal{T}}^{\boldsymbol{u}}~\propto ~(V_{g}/\ell )^{4}$ . However, because of the substantial shape changes between linear and nonlinear $w$ and ${\mathcal{T}}^{\boldsymbol{u}}$ , it is implausible that a simple magnitude scaling estimate like these will be very accurate when $Ro_{g}$ is large.

The buoyancy flux profiles, $BF_{v}(z)$ and $BF_{h}(x)$ in (3.8), are relatively little affected by ageostrophic advection. The profile for $BF_{v}(z)$ always has a convex shape as in figure 11(b,d). Its peak amplitude increases somewhat with nonlinearity especially for $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ values close to the ones with the largest - min $[w]$ values, but never by more than approximately 30 %. Thus, in spite of the large amplification in the peak value of $w$ , the horizontally integrated buoyancy flux is not greatly changed. The dominant influence on $BF_{h}(x)$ remains the Ekman transport dependency on $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}$ , with peak positive and negative fluxes for $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70F}}=\pm \unicode[STIX]{x03C0}/2$ where the cross-front transport is largest.

As with the linear solutions, in combined nonlinear TTW and $\unicode[STIX]{x1D6FC}$ solutions, the SCFT behaviours are largely additive.

6.3 Inaccuracy of the ‘nonlinear Ekman-layer’ model

A widely used approximate model for the wind-driven surface boundary layer is the so-called nonlinear Ekman layer (NLE; Stern Reference Stern1965). With the assumptions that (i) the horizontal scale of the wind stress is larger than that of the surface geostrophic current, (ii) the vertical scale of the geostrophic flow is larger than the depth of the boundary layer and (iii) the relevant Rossby numbers are small but finite, then the ageostrophic horizontal transport $\boldsymbol{T}_{T}$ and the vertical velocity at the base of the boundary layer, the Ekman pumping $w_{ek}$ , are given by

i.e. simple generalizations of the usual Ekman relations replacing $f$ by $f+\unicode[STIX]{x1D701}_{g}$ to include some advective nonlinearity. These formulas are convenient because they do not require solution of a differential equation in $z$ , are independent of the vertical mixing profile $\unicode[STIX]{x1D708}_{v}(\boldsymbol{x},z)$ , and have a clear implication of a secondary circulation generated by wind over mesoscale currents and eddies. A partial motivation is the vector identity for horizontal momentum advection,

where at least part of the first left-hand side term (the vortex force) is retained in the NLE model. Niiler (Reference Niiler1969) made a similar Ekman pumping analysis for the Gulf Stream with $v_{g}(x,y)$ by including an advective term in the $y$ -momentum equation, $u_{a}\unicode[STIX]{x2202}_{x}v_{g}$ , but not its vortex force companion in the $x$ -momentum equation; as explained in the rest of this section, this alternative avoids much of the criticism of Stern’s proposal.

In the $\text{TTW}+\unicode[STIX]{x1D6FA}\text{E}$ system, without explicitly invoking Stern’s assumptions behind (6.6), the NLE approximation can be represented by replacing ${\mathcal{N}}_{a}^{\boldsymbol{u}}$ in (2.13) by

and otherwise neglecting ageostrophic advection. No assumption is made a priori that $\unicode[STIX]{x1D701}_{g}$ is independent of $z$ within the boundary layer but obviously such a situation is contained within (6.8). When SCFT solutions obtained with (6.8) are compared to the more complete solutions in the preceding part of this section, they are found to be consistently inaccurate in representing nonlinear advective effects even in the limit $Ro_{g}\rightarrow 0$ .

There are several important deficiencies in (6.8), hence in (6.6). One has to do with the neglect of ${\mathcal{N}}^{b}$ in (2.13), but the NLE solution accuracy is not much better with or without including this term in $\unicode[STIX]{x1D6FA}\text{E}$ . Another has to do with neglecting the suppressive effect of stratification, as expressed in $w_{O}$ , but again this is not the primary cause of inaccuracy here (and will not be important when $h_{p}$ is large enough). Rather it is the false inclusion of the term ${\mathcal{N}}_{a\,NLE}^{u}=v_{a}\unicode[STIX]{x1D701}_{g}$ , which is absent in ${\mathcal{N}}_{a}^{u}$ and has a leading-order magnitude in a $Ro_{g}$ expansion, larger than any of the other terms in ${\mathcal{N}}_{a}^{u}$ . (In contrast, ${\mathcal{N}}_{a\,NLE}^{v}=-u_{a}\unicode[STIX]{x1D701}_{g}$ is a correct leading-order term in the $y$ -equation.) The fallacy is to keep this vortex force term in the $x$ -momentum equation while neglecting its cancellation in the Bernoulli head, i.e. the second left-hand side term in (6.7). Thus, there often will be no convenient alternative for representing the effect of momentum advection in the boundary layer other than by solving a more complete system like the $\text{TTW}+\unicode[STIX]{x1D6FA}\text{E}$ one used here.