3.1. Overview

In this section, we provide an overview of the life cycles of the simulated front and dense filament. The evolution of the front or the filament is analysed by calculating the maximum vertical vorticity $\zeta _{z,max}$ following Sullivan & McWilliams (Reference Sullivan and McWilliams2018), defined as

(3.1) \begin{equation} \zeta _{z,max}(t) = {\max }_{(x)}\left (\langle \overline \zeta _z\rangle _y\Big \vert _{z=0}\right )\!. \end{equation}

The term $\langle \overline \zeta _z\rangle _y$ represents the along-front averaged vertical component of vorticity $\boldsymbol{\overline \zeta } = (\overline \zeta _x,\overline \zeta _y, \overline \zeta _z)$ , and the maximum is taken in the cross-front direction. The time series of $\zeta _{z,max}$ are compared between the filament and front simulations (figures 2 $a$ and 2 $d$ ). Here $\zeta _{z,max}$ is normalized by the Coriolis frequency $f$ , which effectively represents a local Rossby number. We note that frontal strength can be quantified either by cross-front density gradients (e.g. $M^2/f^2$ as defined in (2.4)) or by cross-front velocity gradients (e.g. $\zeta _{z,max }$ ) (Sullivan & McWilliams Reference Sullivan and McWilliams2018). In this study, unless otherwise specified, we define frontal strength in terms of velocity gradients and use the time series of $\zeta _{z,max }$ to characterize frontal evolution.

Figure 2. $(a)$ Maximum along-front averaged vertical vorticity $\zeta _{z,max}/f$ as a function of time for the filament simulation. $(b)$ Maximum along-front averaged TKE $_{max}/w_*^2$ as a function of time. $(c)$ Along-front averaged surface temperature $\langle \overline \theta \rangle _y$ as a function of cross-front distance $x$ . The red and blue lines represent the initial time $t=0$ and peak frontal strength $t=t_p$ , respectively, corresponding to the times indicated by the vertical lines in $(a)$ . $(d{-}f)$ Same as $(a){-}(c)$ , but for the single-sided front simulation. Note that the vertical axis ranges in $(d)$ and $(e)$ are different from those in $(a)$ and $(b)$ .

The evolution of submesoscale fronts and filaments typically involves the growth of frontal instability and turbulence. We calculate the maximum along-front averaged TKE $_{max}$ (Sullivan & McWilliams Reference Sullivan and McWilliams2024) to analyse the development of turbulence (figures 2 $b$ and 2 $e$ ),

(3.2) \begin{equation} \textrm {TKE}_{max}(t) = {\max }_{(x,z)}\left (\langle \textrm {TKE}\rangle _y\right )\!, \end{equation}

where the maximum is taken in the cross-sectional $x{-}z$ plane. The TKE is normalized by $w_*^2$ , with $w_*$ representing the Deardorff convective velocity scale. In our simulations $w_*=[\alpha g Qh_0/(\rho c_p)]^{1/3}=0.014$ m s⁻¹, where $Q$ is the surface heat flux, $h_0$ is the initial mixed layer depth and $c_p=4186$ J kg $^{-1}$ K $^{-1}$ is the ocean specific heat.

Both the filament and front exhibit a similar life cycle, including frontogenesis, frontal arrest and decay. During frontogenesis, the frontal (or filament) strength increases rapidly and peaks at around $t=2$ to 3 hr (figures 2 $a$ and 2 $d$ ). Snapshots taken at the time of peak frontal strength $t=t_p$ , compared with the initial state, illustrate the sharpening and narrowing of the frontal zone (figures 2 $c$ and 2 $f$ ).

Frontogenesis can trigger various instability mechanisms and promote the growth of turbulence (McWilliams Reference McWilliams2016, Reference McWilliams2021). As a result, a rapid increase in TKE is observed in both cases (figures 2 $b$ and 2 $e$ ), which lags behind the increase in frontal strength by around 1 hr. Visualizations of frontal instability and turbulence structures are shown in the map views of figure 3. Notably, turbulence is not confined to the surface frontal zone, but also extends into deeper regions (e.g. $z=-40$ m, figures 3 $c$ and 3 $d$ ). While the maximum TKE is typically found near the frontal zone, it is not always located at the sea surface; instead, the maximum can occur at depths of several tens of metres. The momentum fluxes associated with frontal turbulence generally oppose the frontogenetic tendency, leading to the halting (or arrest) of frontogenesis (Sullivan & McWilliams Reference Sullivan and McWilliams2018).

Figure 3. $(a,b)$ Map views of surface temperature for the filament and front simulations, respectively, around the time of peak frontal strength. $(c,d)$ Turbulence structures revealed by isosurfaces of Q-criterion for the filament and front simulations, with $Q=0.0001$ s $^{-2}$ . The Q-criterion is defined as $Q = ({1}/{2}) (||\boldsymbol{\varOmega }||^2 - ||\boldsymbol{S}||^2)$ , where $\boldsymbol{S}$ and $\boldsymbol{\varOmega }$ are the symmetric and antisymmetric components of the velocity gradient tensor, respectively (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988). Colours indicate the depth of the isosurfaces. Note that the isosurfaces extend from the sea surface to depths of around 40 m.

Overall, the filament reaches a greater peak strength and higher peak TKE than the front, although both are initialized with identical frontal strength. This difference is expected due to the symmetric configuration of the filament, which can lead to stronger surface convergence and a faster frontogenetic rate (McWilliams et al. Reference McWilliams, Colas and Molemaker2009). Another difference between the two simulations is the propagation of the surface front towards the colder (denser) side, moving approximately 1 km over several hours, while the filament remains stationary at the domain centre. Frontal propagation is evident in the Hovmöller diagram of vertical vorticity (figure 4). In addition, stripe patterns are observed in the Hovmöller diagrams of both cases. Their formation likely relates to weak symmetric instability triggered during the initialization, which can result in slantwise convection cells. The stripes tend to converge towards the frontal zone over time, presumably due to horizontal advection by the secondary circulation.

Following the peak strength, the front and filament undergo a decay phase (figures 2 $a$ and 2 $d$ ) and eventually reach a relatively steady state after 10 hr, with a Rossby number of around 5. In this study we primarily examine the frontogenesis and arrest processes; the frontal decay stage is not the focus of our analyses and will be discussed in a later section.

In summary, both the front and filament follow a similar life cycle, characterized by frontogenesis (sharpening and narrowing), arrest (halting) and decay. However, differences arise in the stronger peak strength of the filament and the propagation of the front. These distinct behaviours may lead to variations in the frontal momentum balance, instability and turbulence generation. Detailed comparisons and analyses of the front and filament will be presented in the subsequent sections.

Figure 4. Hovmöller diagrams of along-front averaged surface vertical vorticity $\langle \overline \zeta _z\rangle _y/f$ (as a function of time and cross-front distance): $(a)$ filament; $(b)$ front. The end of the relaxation period is $t=0$ .

3.2. Frontogenesis

The TTW is a classical mechanism driving frontogenesis (e.g. McWilliams Reference McWilliams2016). The TTW represents an approximate momentum balance between the pressure gradient, Coriolis force, and the divergence of vertical turbulent momentum flux in (2.7). In particular, turbulence (vertical turbulent momentum flux) plays a key role in TTW, which disrupts the classical thermal wind balance and leads to the initial development of secondary circulation.

Figure 5. Filament simulation: side views of temperature and velocity at $t=t_p-0.5$ hr. $(a)$ Along-front averaged temperature $\langle \overline \theta \rangle _y$ . $(b)$ Along-front averaged vertical velocity $\langle \overline w\rangle _y$ . $(c)$ Along-front averaged cross-front velocity $\langle \overline u\rangle _y$ . $(d)$ Along-front averaged along-front velocity $\langle \overline v\rangle _y$ . White or black lines show contours of temperature.

In the filament case, TTW results in a pair of counter-rotating secondary circulation cells centred around the filament in the cross-front section, which converge near the surface (figure 5 $c$ ). The surface convergence associated with the secondary circulation enhances frontogenesis, sharpening the gradients of momentum and temperature at $x=0$ km (e.g. when comparing figure 5 with figure 1 $a$ ). In the meantime, strong downwelling occurs at the filament due to mass conservation and horizontal convergence (figure 5 $b$ ). Figure 5 shows a representative snapshot at $t=t_p-0.5$ hr, and the pattern of secondary circulation remains consistent throughout frontogenesis and arrest. Although the presence of winds and waves could lead to asymmetry in the cross-front direction (Hamlington et al. Reference Hamlington, Van, Luke, Fox-Kemper, Julien and Chini2014; Sullivan & McWilliams Reference Sullivan and McWilliams2018), the current simulations, which only include convective turbulence, show a symmetric filament structure (Sullivan & McWilliams Reference Sullivan and McWilliams2024).

Unlike the filament, where density gradients exist on both sides, the front case features a single density step, leading to the occurrence of only one secondary circulation cell. The front qualitatively resembles one half of a filament, displaying similar structures in both the temperature and velocity fields (figure 6). However, while the filament is stationary at $x=0$ , the surface front exhibits noticeable movement, with the convergence and downwelling zone reaching around $x = 1$ km at peak frontal strength.

Figure 6. Front simulation: side views of temperature and velocity at $t=t_p-0.5$ hr. Note that the range of the horizontal axis differs from that in figure 5. $(a)$ Along-front averaged temperature $\langle \overline \theta \rangle _y$ . $(b)$ Along-front averaged vertical velocity $\langle \overline w\rangle _y$ . $(c)$ Along-front averaged cross-front velocity $\langle \overline u\rangle _y$ . $(d)$ Along-front averaged along-front velocity $\langle \overline v\rangle _y$ . White or black lines show contours of temperature.

Additionally, secondary circulation causes frontal slumping and restratification in the surface boundary layer (figures 5 $a$ and 6 $a$ ). The mixed layer depth, defined as the depth where temperature first departs from its surface value, is decreased compared with its initial depth of $h_0=55$ m on both sides of the filament. In the front case, the shallowing of the boundary layer occurs on the warmer side of the surface frontal zone, i.e. the region left behind the propagating front. Restratification also limits the vertical extent and strength of convection cells, as seen in the vertical velocity field (figures 5 $b$ and 6 $b$ ). The initialized temperature field may trigger weak symmetric instability away from the frontal zone, leading to the development of slantwise convection cells (Taylor & Ferrari Reference Taylor and Ferrari2009). In addition, geostrophic jets develop in the along-front direction, giving rise to organized convective rolls roughly aligned with the front (Salesky, Chamecki & Bou-Zeid Reference Salesky, Chamecki and Bou-Zeid2017). Both processes may result in non-zero along-front averaged vertical velocities away the frontal zone, as is found in figures 5 $b$ and 6 $b$ .

The frontogenetic rate typically depends on the strength of surface convergence, as demonstrated in the frontogenetic equations for temperature and velocity gradients (e.g. Hoskins & Bretherton Reference Hoskins and Bretherton1972; McWilliams et al. Reference McWilliams, Gula, Molemaker, Renault and Shchepetkin2015; Barkan et al. Reference Barkan, Molemaker, Srinivasan, McWilliams and D’Asaro2019). Frontogenesis, particularly in the case of TTW frontogenesis, is expected to tend towards a finite-time singularity after a time inversely proportional to the initial convergence rate, assuming that the influences of turbulent mixing are negligible (Barkan et al. Reference Barkan, Molemaker, Srinivasan, McWilliams and D’Asaro2019; McWilliams Reference McWilliams2021). In our filament and front simulations, the initial horizontal convergence is approximately $0.0005-0.001$ s $^{-1}$ , which is consistent with the observed frontogenesis time scale of around 1 hr. However, the growth of instabilities and turbulent processes will eventually invalidate the assumption of negligible mixing, which leads to the arrest of frontogenesis and prevents the occurrence of a finite-time singularity.

3.3. Frontal arrest

A common explanation for frontal arrest is that it occurs when the growth rate of frontal instability exceeds the frontogenetic rate, resulting in opposing turbulent fluxes that halt further frontal sharpening and narrowing (McWilliams Reference McWilliams2016; Sullivan & McWilliams Reference Sullivan and McWilliams2018). Submesoscale frontal arrest can arise from various instability mechanisms, and it remains unknown what determines the minimum horizontal scale at which arrest occurs (e.g. McWilliams Reference McWilliams2021; Sullivan & McWilliams Reference Sullivan and McWilliams2024). In this section, we examine the momentum balance and investigate the frontal arrest mechanism. More detailed analyses of instability and turbulence, both of which play a critical role in causing frontal arrest, will be presented in the subsequent sections.

Figure 7. Filament simulation: side views of the dominant terms in the along-front averaged momentum equation at $t=t_p$ . $(a)$ Horizontal advection $\langle \overline u (\partial \overline v/\partial x)\rangle _{y}$ . $(b)$ Horizontal turbulent flux divergence $\partial \langle \overline {u'v'}\rangle _{y}/\partial x$ .

For the dense filament, previous studies suggest that horizontal (cross-front) shear instability plays a key role in the arrest process, i.e. a barotropic instability or inflection point instability (Gula et al. Reference Gula, Molemaker and McWilliams2014; Sullivan & McWilliams Reference Sullivan and McWilliams2018, Reference Sullivan and McWilliams2024). This arrest mechanism is reflected by the cross-front gradients of momentum fluxes in the along-front averaged (2.7b ). The horizontal advection term $\langle \overline u (\partial \overline v/\partial x)\rangle _{y}$ (associated with secondary circulation) tends to enhance the filament jets and increase the cross-front gradient, while the turbulent flux divergence term $\partial \langle \overline {u'v'}\rangle _{y}/\partial x$ acts to weaken the filament. Frontal arrest is achieved when the horizontal turbulent flux increases sufficiently to counteract the advection by secondary circulation. This results in an approximate balance of $\langle \overline u (\partial \overline v/\partial x)\rangle _{y}\approx \partial \langle \overline {u'v'}\rangle _{y}/\partial x$ (figure 7), consistent with Sullivan & McWilliams (Reference Sullivan and McWilliams2024). The time evolution of these terms further supports this mechanism: the advection term dominates during the early stage, contributing to frontogenesis, while the turbulent flux divergence term rapidly intensifies and balances the advection term at around $t = t_p$ .

Figure 8. Front simulation: side views of the dominant terms in the along-front averaged momentum equation at $t=t_p$ . $(a)$ Advection $\langle \overline u (\partial \overline v/\partial x)\rangle _{y}$ . $(b)$ Horizontal turbulent flux divergence $\partial \langle \overline {u'v'}\rangle _{y}/\partial x$ . $(c)$ Unsteady term $\partial \langle \overline v \rangle _{y}/\partial t$ . $(d)$ Vertical turbulent flux divergence $\partial \langle \overline {v'w'}\rangle _{y}/\partial x$ . Note that $(a)$ and $(c)$ , which are indicative of frontal propagation, have a larger colormap range than $(b)$ and $(d)$ . The SGS stress and Coriolis force are negligible compared with the terms shown in this figure. Here the dominance of the unsteady and advection terms primarily reflects the propagation of the front; the momentum budget with the propagation effect removed is shown in figure 9.

For the single-sided front, the dominant terms in the $v$ -momentum (2.7b ) are the unsteady and advection terms (figures 8 $a$ and 8 $c$ ). The horizontal and vertical turbulent flux divergence terms are smaller and play a secondary role (figures 8 $b$ and 8 $d$ ). It is important to note that the dominance of the unsteady and advection terms primarily reflects the movement of the front. However, the contributor to frontogenesis is not the overall movement of the front but rather local disturbances in advection. The separated effects of frontal movement and frontogenesis can be demonstrated by taking the $x$ -derivative of the advection term $\langle \overline u(\partial \overline v/\partial x)\rangle _{y}$ , which results in two components: $\langle \overline u (\partial ^2\overline v/\partial x^2)\rangle _{y}$ , associated with the movement of the front, and $\langle (\partial \overline u/\partial x) (\partial \overline v/\partial x)\rangle _{y}$ , associated with frontogenesis, which is primarily driven by horizontal convergence (spatial variability in the cross-front velocity).

Figure 9. Front simulation: near-surface $v$ -momentum budget as a function of cross-front distance $x$ at $t=t_p$ . Solid black line, modified advection $\langle (\overline u-u_{\textit{front}}) (\partial \overline v/\partial x)\rangle _{y}$ ; red line, horizontal turbulent flux divergence $\partial \langle \overline {u'v'}\rangle _{y}/\partial x$ ; blue line, vertical turbulent flux divergence $\partial \langle \overline {v'w'}\rangle _{y}/\partial x$ ; dashed black line, sum of the blue and red lines, but with the opposite sign. Note that the subtracted term $u_{\textit{front}} (\partial \langle \overline v \rangle _{y}/\partial x)$ is nearly an order of magnitude larger than the terms shown here. The subtracted term represents frontal propagation, which balances with the unsteady term (figure 8).

To isolate the processes dominating frontal arrest, we eliminate the influence of frontal propagation in the momentum (2.7b ) by subtracting the front movement speed $u_{\textit{front}}(t)$ . Here $u_{\textit{front}}(t)$ is estimated from the Hovmöller diagram shown in figure 4 $(b)$ . This results in the modified advection term $\langle (\overline u-u_{\textit{front}}) (\partial \overline v/\partial x)\rangle _{y}$ , which represents advection in a moving reference frame and allows us to separate the contribution of advection to frontogenesis from the effects of frontal propagation. The modified advection term tends to strengthen the frontal jet (figure 9). In contrast, both the horizontal momentum flux divergence $\partial \langle \overline {u'v'}\rangle _{y}/\partial x$ and vertical momentum flux divergence $\partial \langle \overline {v'w'}\rangle _{y}/\partial x$ act to weaken the frontal jet. The combined effect of these two terms generally balances the modified advection term, leading to the arrest of frontogenesis (figure 9).

It is noteworthy that, unlike the filament case where the horizontal turbulent flux divergence dominates over the vertical turbulent flux divergence (Figure 10 a), the vertical flux term is comparable to the horizontal flux term in the front case and both terms are important for frontal arrest. In the time series of $v$ -momentum budget terms (figure 10 $b$ ), the modified advection term contributes to frontogenesis in the early stage, and the sum of the horizontal and vertical turbulent flux divergence terms grows rapidly, and exceeds the advection term after $t = t_p$ . The Coriolis term is over an order of magnitude smaller than the turbulent flux terms in both the filament and front cases, making its influence on frontal arrest negligible. This is expected given the high Rossby number, e.g. greater than 20, at around $t=t_p$ .

Figure 10. Maximum along-front averaged terms in the $v$ -momentum budget as a function of time. $(a)$ The filament simulation. Lines show absolute values of horizontal advection (associated with secondary circulation), horizontal and vertical turbulent flux divergences, and their combined sum. See legends for details. $(b)$ The front simulation. Lines show absolute values of modified advection, horizontal and vertical turbulent flux divergences, and their combined sum. Vertical black lines mark the time of peak frontal strength $t = t_p$ . Note that the vertical axis ranges in the two panels are different.

Next, we provide an explanation for the propagation of the single-sided front. In the filament case, the frontal zone remains stationary at the centre of the domain, because the cross-front velocity is always zero at $x=0$ due to the symmetric structure of the filament. In contrast, for the front case, secondary circulation introduces a non-zero cross-front velocity $\langle u \rangle _y$ at the surface frontal zone, which therefore drives the self-propagation of the front. The front movement speed $u_{\textit{front}}$ ranges between $1-3$ cm s⁻¹ during the frontogenesis and arrest phases. Generally, the region with maximum convergence $-\partial \langle \overline u \rangle _{y}/\partial x$ is located ahead of (i.e. on the colder side of) the region with the maximum $\partial \langle \overline v \rangle _{y}/\partial x$ . The frontogenetic tendency driven by surface convergence acts to locally increase $\partial \langle \overline v \rangle _{y}/\partial x$ , which consequently causes the forward propagation of the maximum $\partial \langle \overline v \rangle _{y}/\partial x$ . The distance $\Delta x$ between the regions of maximum $-\partial \langle \overline u \rangle _{y}/\partial x$ and maximum $\partial \langle \overline v \rangle _{y}/\partial x$ is approximately $10$ m, with the maximum convergence rate $-\partial \langle \overline u \rangle _{y}/\partial x$ being of the order of $0.001$ s $^{-1}$ . These yield an estimated velocity scale for frontal propagation, $\Delta x(-\partial \langle \overline u \rangle _{y}/\partial x) \approx 1$ cm s⁻¹, which is of the same order of magnitude as the actual front movement speed $u_{\textit{front}}$ .

The frontal propagation gradually slows down during the decay phase, e.g. after $t=6$ hr (figure 4 $b$ ). Similar frontal propagation during frontogenesis, followed by a gradual decrease in movement speed in the later stage, has been reported by Dauhajre et al. (Reference Dauhajre, Srinivasan, Molemaker, Gula, Hypolite, McWilliams, Barkan and Young2025). We note that in a previous LES study by Pham & Sarkar (Reference Pham and Sarkar2018), submesoscale fronts led to the formation of gravity currents, with frontogenesis continuing unabated and gravity currents propagating for over 10 hr until the simulations ended. However, in our simulation, frontogenesis and propagation are arrested after several hours. We attribute these contrasting behaviours to differences in frontal strength: our front has an $M^2/f^2$ ratio of 50, whereas theirs is 500 – an order of magnitude larger – leading to stronger gravity current propagation.

4.1. The TKE budget

Turbulence plays a crucial role in frontogenesis and arrest processes in the ocean surface boundary layer, as demonstrated in the previous sections. On one hand, turbulence initiates secondary circulation in the cross-front direction via TTW, which results in frontogenesis (Sullivan & McWilliams Reference Sullivan and McWilliams2024; Dauhajre et al. Reference Dauhajre, Srinivasan, Molemaker, Gula, Hypolite, McWilliams, Barkan and Young2025). On the other hand, turbulence drives horizontal and vertical momentum fluxes that counteract the frontogenetic tendency, consequently leading to frontal arrest. In this section, we examine the characteristics of turbulence and further investigate their generation mechanisms in the front and filament cases.

Figure 11. $(a,\!b)$ Side views of TKE for the filament and front, respectively, at the peak frontal strength $t=t_p$ . $(c,\!d)$ Side views of the ratio of $(1/2)\langle \overline {w'^2}\rangle _y/\langle \textit{TKE}\rangle _y$ for the filament and front. White lines show contours of temperature.

The normalized TKE is less than unity in the far field, where turbulence is primarily generated by convection (figures 11 $a$ and 11 $b$ ). Convective turbulence is typically characterized by greater variance in the vertical component compared with the horizontal components (e.g. Chor et al. Reference Chor, Yang, Meneveau and Chamecki2018), and the ratio of $(1/2)\langle \overline {w'^2} \rangle _y/ \langle \textit{TKE} \rangle _y$ exceeds 0.5 (figures 11 $c$ and 11 $d$ ).

In contrast, near both the filament and front, TKE reaches values up to an order of magnitude higher than the convective turbulence in the far field. The TKE level at the filament is higher than that at the front, consistent with higher peak strength of the filament. The ratio of $(1/2)\langle \overline {w'^2}\rangle _y/\langle \textit{TKE}\rangle _y$ is less than 0.2 in the frontal zones for both cases, suggesting that more energy is contained in the horizontal dimensions and TKE generation mechanisms differ from those in boundary layer convection. Note that the vertical variance $\overline {w'^2}$ per se remains higher in the frontal zone than in the far field, consistent with the distribution of total TKE. The ratio of $(1/2)\langle \overline {w'^2} \rangle _y / \langle \textit{TKE} \rangle _y$ only indicates the relative partitioning of energy among the TKE components. Figure 11 presents the turbulence distribution at the time of peak frontal strength. Although not depicted in the figure, it is worthwhile to mention that high TKE near the front persists for over 10 hr, i.e. the frontal zones maintain significantly higher TKE levels than the far field even in the relatively steady frontal decay stage.

Figure 12. Side views of terms in the TKE (2.10) at $t=t_p$ . All terms are normalized by $w_*^3/h_0$ . Here $(a{-}f)$ filament; $(g{-}l)$ front. Black lines show contours of temperature.

Analysis of the TKE budget (2.10) reveals that horizontal shear production $P_{hor}$ is the dominant source of turbulence near the filament (figure 12 $a$ ). This is consistent with findings in Sullivan & McWilliams (Reference Sullivan and McWilliams2024), indicating the key role of horizontal shear instability in generating turbulence and arresting frontogenesis. Vertical shear production $P_{vert}$ is significantly smaller than $P_{hor}$ , with negative values near the surface and positive values below $z=-20$ m (figure 12 $b$ ). Similar negative $P_{vert}$ has been reported by Sullivan & McWilliams (Reference Sullivan and McWilliams2024), and it is presumably due to $\partial \langle \overline w\rangle /\partial z\gt 0$ near the surface. The generated TKE is largely dissipated into internal energy as shown by the $\epsilon$ term in figure 12 $(d)$ . Analysis in Sullivan & McWilliams (Reference Sullivan and McWilliams2024) suggests that local TKE dissipation cannot fully balance the horizontal production, resulting in TKE transport from the centre of the dense filament towards the far field. Similar TKE transport is also found in our case. While the transport pattern is less distinct in the snapshot shown in figure 12 $(f)$ , it becomes clearer when averaged over time.

For the single-sided front, horizontal shear production $P_{hor}$ plays a major role in the TKE budget (figure 12 $g$ ), similar to the filament case. However, vertical shear production $P_{vert}$ has a comparable magnitude to $P_{hor}$ , and mostly occurs in the restratified region beneath the surface frontal zone (figure 12 $h$ ). These results suggest that both horizontal and vertical shear instabilities can be responsible for turbulence generation and the subsequent arrest of the single-sided front. The unsteady and transport terms mainly capture the horizontal transport of TKE associated with the movement of the front (figures 12 $k$ and 12 $l$ ).

The buoyancy production $B$ is predominantly negative near the filament (figure 12 $c$ ), while small positive values of $B$ are found near the sea surface associated with convective heat flux. The negative $B$ suggests that turbulence primarily draws energy from the sheared along-filament currents and secondary circulation, rather than from potential energy, leading to mixing that erodes stratification. A consistent result was reported in the filament simulation by Sullivan & McWilliams (Reference Sullivan and McWilliams2024). In our comparative study, buoyancy production $B$ is predominantly negative at the front, similar to the filament, but with a substantially smaller magnitude (figure 12 $i$ ). This weaker negative $B$ reflects reduced turbulence intensity and less vigorous mixing at the front compared with the filament. The presence of negative buoyancy production, alongside positive vertical shear production, indicates vertical shear instability (Kelvin–Helmholtz instability) in both cases, which will be examined in more detail in a subsequent section.

It is also worthwhile to mention that the volume integral of the unsteady term is positive in both the filament and front cases at $t=t_p$ . This is consistent with the time series presented in figure 2, where the growth of turbulence is delayed relative to the increase in frontal strength in both cases. As a result, TKE continues to increase while the frontal strength reaches its maximum at $t=t_p$ .

Figure 13. Terms in the TKE budget as a function of cross-front distance $x$ . Here $(a)$ filament; $(b)$ front. These results are averaged between $z=-h_0=-55$ m and $z=0$ m and from $t=0$ to $t=6$ hr, and are normalized by $w_*^3/h_0$ . See legends for details.

We calculate the time-averaged TKE budget to compare the relative contributions of different production terms over the life cycles of the front and filament (figure 13). In the filament case, horizontal shear production $P_{hor}$ is the primary source of turbulence, whereas in the front case, vertical shear production $P_{vert}$ is more prominent. This increased contribution of $P_{vert}$ is likely due to the more tilted shape of the front, which leads to stronger vertical shear. Additionally, the magnitude of horizontal shear is smaller in the single-sided front compared with the filament, where a pair of opposing jets develops. It is noteworthy that, in the front case, $P_{vert}$ is comparable to $P_{hor}$ at the peak frontal strength (figures 12 $g$ and 12 $h$ ). However, $P_{vert}$ becomes increasingly dominant in the later stage, so that it exceeds $P_{hor}$ by a factor of three in the time-averaged results (figure 13 $b$ ). In regions adjacent to the frontal zone (the near field), TKE is transported away from the primary production zone towards the far field (not shown), similar to Sullivan & McWilliams (Reference Sullivan and McWilliams2024). In the far field under free convection, the TKE budget in both cases is governed by the balance between buoyancy production and dissipation, and the spatially and temporally averaged contributions of other terms are small.

4.2. Temporal evolution of turbulence

In this section, we further investigate the temporal evolution of turbulence to elucidate its role throughout frontogenesis and arrest. The wavenumber spectra for the turbulent components $u'$ , $v'$ and $w'$ are examined in the filament case (figure 14). Here we focus on the spectra in the frontal zone. Turbulence is more isotropic before frontal arrest, but a noticeable shift in energy distribution occurs at later stages, with a greater proportion of energy concentrated in the horizontal components as turbulence intensifies. In particular, a distinct dominant wavenumber appears in the spectra of $u'$ and $v'$ at and after frontal arrest, indicating the emergence of horizontal coherent structures associated with horizontal shear instability. This is consistent with the dominance of horizontal shear production, as revealed by the TKE budget analysis presented earlier.

Figure 14. Filament simulation: turbulent energy spectra and cross-component coherence versus $y$ -wavenumber. $(a)$ Spectra of streamwise velocity fluctuations, $E_{u'u'}$ , shown at three stages $t = t_p - 0.5$ hr, $t = t_p$ and $t = t_p + 1.5$ hr, illustrating spectral evolution over time. $(b)$ Spectra of cross-stream velocity fluctuations, $E_{v'v'}$ . $(c)$ Spectra of vertical velocity fluctuations, $E_{w'w'}$ . $(d)$ Coherence functions between streamwise and cross-stream velocity components, $\gamma ^2_{u'v'}$ .

Figure 15. Front simulation: turbulent energy spectra and cross-component coherence versus $y$ -wavenumber. $(a)$ Spectra of streamwise velocity fluctuations, $E_{u'u'}$ , shown at three stages $t = t_p - 0.5$ hr, $t = t_p$ and $t = t_p + 1.5$ hr. $(b)$ Spectra of cross-stream velocity fluctuations, $E_{v'v'}$ . $(c)$ Spectra of vertical velocity fluctuations, $E_{w'w'}$ . $(d)$ Coherence functions between streamwise and cross-stream velocity components, $\gamma ^2_{u'v'}$ .

The coherence function $\gamma ^2_{u'v'}$ is calculated to investigate the temporal evolution of horizontal turbulent flux, which is defined as

(4.1) \begin{equation} \gamma ^2_{u'v'}=\frac {|G_{u'v'}|^2}{E_{u'u'}E_{v'v'}}. \end{equation}

Here $E_{u'u'}$ and $E_{v'v'}$ are the autospectral density functions, and $G_{u'v'}$ is the cross-spectral density function (Jenkins & Watts Reference Jenkins and Watts1968; Bendat & Piersol Reference Bendat and Piersol2011). Similar to the energy spectra, a peak in the coherence function emerges at and after $t=t_p$ , signifying the growth of horizontal (cross-front) turbulent flux, which leads to frontal arrest. Moreover, the dominant wavenumber decreases after frontal arrest, suggesting the formation of larger horizontal coherent structures along with the widening of the filament.

As a comparison, energy spectra and coherence are examined in the single-sided front (figure 15). Both horizontal turbulence components and horizontal turbulent flux grow during frontogenesis and frontal arrest, same as the filament case. However, the growth of horizontal eddies is less pronounced after $t = t_p$ in the front case. This aligns with the TKE budget analysis in which vertical shear production becomes more dominant in the later stages, and horizontal turbulent flux and horizontal shear instability play a relatively less dominant role for the single-sided front.

Figure 16. Temporal evolution of spectral characteristics in the filament and front simulations. $(a)$ Centroid wavenumber ( $k_C$ ) and spectral bandwidth ( $k_W$ ) of the streamwise velocity fluctuation spectra $E_{u'u'}$ in the filament simulation. $(b)$ Same as $(a)$ , but for the front simulation. $(c)$ Integrated spectral energy centred around $k_C$ as a function of time, in the filament simulation. The integration is performed over a fixed narrow bandwidth $2\delta _k$ around $k_C$ , with $\delta _kL_y\approx$ 20. $(d)$ Same as $(c)$ , but for the front simulation. Vertical black lines indicate the time of peak frontal strength $t = t_p$ .

Subsequently, we calculate spectral characteristics such as centroid wavenumber and effective bandwidth, and investigate their temporal evolution. Here we present results from the autospectral density function $E_{u'u'}$ as an example, and similar trends are found in the $v'$ component and the $u'v'$ turbulent flux. The centroid wavenumber $k_C$ is calculated as

(4.2) \begin{equation} k_C = \frac {\int k_y E_{u'u'} {\textrm d}k_y}{\int E_{u'u'} {\textrm d}k_y}. \end{equation}

The effective bandwidth $k_W$ is given by

(4.3) \begin{equation} k_W = \sqrt {\frac {\int (k_y - k_C)^2 E_{u'u'} {\textrm d}k_y}{\int E_{u'u'} {\textrm d}k_y}}, \end{equation}

which is essentially the standard deviation of the energy distribution around the centroid wavenumber.

In the filament case, both $k_C$ and $k_W$ decrease with time (figure 16 $a$ ), indicating the dominance of lower wavenumbers and concentration of energy around the dominant wavenumber. The integrated energy near the centroid wavenumber increases over time (figure 16 $c$ ), reflecting the development of horizontal eddies driven by horizontal shear instability. For the front case, similar trends are observed during the early stages (figures 16 $b$ and 16 $d$ ); however, the growth of horizontal eddies is interrupted around 1 hr after $t = t_p$ , as vertical shear production becomes more important.

Figure 17. Near-surface vertical velocity field in the filament simulation. $(a)$ A map view of vertical velocity at $z=-5$ m, shown at $t=t_p+1.5$ hr, illustrating the modulation of turbulence characteristics induced by the filament across different regions. $(b-d)$ Enlarged views of selected subregions marked by black boxes in $(a)$ , representing the far field, near field and filament core, respectively.

The modification of turbulence characteristics is not confined to the frontal zone, but extends into regions adjacent to it. We present a map view of the filament to further illustrate influences of frontal evolution on turbulence (figure 17). In the frontal zone (filament core), horizontal eddies emerge driven by the opposing jets. Note that, at this later time, the dominant wavelength of these eddies becomes larger compared with the earlier stage in figure 3 $(a)$ . In addition, in regions slightly away from the frontal zone (the near field), the influence of filament jets and secondary circulation alters the structure of convection cells (figure 17 $c$ ), in contrast to the classical convection cells found in the far field (figure 17 $b$ ). In the front simulation (not shown), large horizontal eddies do not develop in the frontal zone at later times, as opposed to the filament case. Nevertheless, the structure of convection cells in the near field is also modified by the frontal dynamics, as seen in the filament. These results demonstrate that submesoscale frontal processes exert a broader influence on turbulence, reshaping structures well beyond the frontal zone.

4.3. Frontal instability

Various instability mechanisms can arise during the evolution of submesoscale fronts and filaments (e.g. McWilliams Reference McWilliams2016; Gula et al. Reference Gula, Taylor, Shcherbina and Mahadevan2022). Frontal instability often plays a crucial role in generating turbulence and driving cross-frontal turbulent fluxes, consequently leading to frontal arrest. In this section, we examine the criteria for different instability mechanisms and explore their potential impact on the dynamics of the front and filament. The submesoscale processes examined in this study are inherently nonlinear and involve interactions with strongly inhomogeneous turbulence. Consequently, classical linear instability analyses are insufficient to fully capture the complex dynamics of ocean fronts under these realistic conditions. Nonetheless, here we employ these theoretical criteria to diagnose the possibility of various instabilities, in line with common practice in the oceanographic community.

Previous studies suggest that horizontal shear instability is the dominant mechanism responsible for frontal arrest in dense filaments (e.g. Gula et al. Reference Gula, Molemaker and McWilliams2014). According to Rayleigh–Kuo’s inflection point criterion (Rayleigh Reference Rayleigh1880; Kuo Reference Kuo1949), a necessary condition for horizontal shear instability is that $\beta - ( {\partial {^2 v}}/{\partial {x^2}})$ must change sign within the boundaries of the current. Here the current $v$ is in $y$ -direction, and $x$ is perpendicular to the current. The term $\beta$ is the derivative of the planetary vorticity (Coriolis) $f$ with respect to $x$ . This criterion essentially indicates the existence of critical points where the absolute vorticity reaches an extreme value (Kuo Reference Kuo1949).

Figure 18. Side views of the filament simulation at $t=t_p$ . $(a)$ Vertical temperature gradient $\partial \langle \overline {\theta }\rangle _y/\partial z$ . $(b)$ Gradient Richardson number $Ri_g$ . Note that regions with $Ri_g\lt 0$ are whited out. $(c)$ Second-order velocity gradient $\partial ^2\langle \overline {v}\rangle _y/\partial x^2$ . $(d)$ Ertel’s potential vorticity (PV) $\langle q\rangle _y$ . All quantities are averaged in the along-front direction.

Figure 18 $(c)$ shows that the along-front averaged gradient $\partial ^2\langle \overline {v}\rangle _y/\partial x^2$ changes its sign at the filament’s centre, satisfying the condition for horizontal shear instability (with $\beta =0$ in our study). Additionally, map views of instantaneous temperature and velocity fields also reveal patterns of horizontal shear instability between the pair of filament jets (figures 3 $a$ and 3 $c$ ; figure 17 $a$ ).

Figure 19. Side views of the front simulation at $t=t_p$ . $(a)$ Vertical temperature gradient $\partial \langle \overline {\theta }\rangle _y/\partial z$ . $(b)$ Gradient Richardson number $Ri_g$ , with regions with $Ri_g\lt 0$ being whited out. $(c)$ Second-order velocity gradient $\partial ^2\langle \overline {v}\rangle _y/\partial x^2$ . $(d)$ Ertel’s PV $\langle q\rangle _y$ . All quantities are averaged in the along-front direction.

For the single-sided front, $\partial ^2\langle \overline {v}\rangle _y/ \partial x^2$ also changes sign in the frontal zone (figure 19 $c$ ). Horizontal shear instability patterns are evident at the time of peak frontal strength (figure 3 $b$ ). However, these patterns become less pronounced at a later time (not shown), when compared with the filament. This aligns with the TKE budget analysis of the front, where horizontal shear production is dominant at peak frontal strength, but it diminishes over time as vertical shear production increases (§ 4.1).

Next, we calculate the gradient Richardson number

(4.4) \begin{equation} Ri_g = \frac { \frac {\partial {\langle \overline b\rangle _y}}{\partial {z}}} {\left (\frac {\partial {\langle \overline u\rangle _y}}{\partial {z}}\right )^2 +\left (\frac {\partial {\langle \overline v\rangle _y}}{\partial {z}}\right )^2}, \end{equation}

where $b$ is the buoyancy variable. The gradient Richardson number compares the stabilizing effect of stratification with the destabilizing effect of velocity shear (e.g. Kundu, Cohen & Dowling Reference Kundu, Cohen and Dowling2015). The critical Richardson number $Ri_c$ is approximately 0.25 (Howard Reference Howard1961; Miles Reference Miles1961), and $Ri_g\lt Ri_c$ is a necessary condition for the growth of vertical shear instability, also known as Kelvin–Helmholtz instability. Note that reported values of $Ri_c$ can range between 0.2 and 1 (e.g. Galperin, Sukoriansky & Anderson Reference Galperin, Sukoriansky and Anderson2007), and here we use $Ri_c=0.25$ as a commonly used criterion in the ocean.

The gradient Richardson number is less than $Ri_c$ near the single-sided front (figure 19 b), indicating conditions favourable for vertical shear instability. Moreover, regions where $Ri_g\lt Ri_c$ is consistent with regions with enhanced vertical shear production (figure 12 $h$ ). Vertical shear instability can also be inferred in the map view of turbulence structures (figure 3 $d$ ), where isosurfaces of Q-criterion subduct to depths of $\sim\!40$ m. Regions with $Ri_g\lt Ri_c$ are also found on both sides of the filament (figure 18 $b$ ). The $Ri_g$ calculation suggests that both the front and filament are susceptible to vertical shear instability. However, as revealed by the TKE budget, vertical shear production is much weaker than horizontal shear production near the filament, whereas vertical shear production is more significant in the front case. This implies that vertical shear instability plays a more important role in the front, but has a limited impact on the dynamics of the filament.

Previous studies of submesoscale currents have focused on baroclinic instability, also known as mixed layer instability (Haine & Marshall Reference Haine and Marshall1998; Capet et al. Reference Capet, McWilliams, Molemaker and Shchepetkin2008; Fox-Kemper, Ferrari & Hallberg Reference Fox-Kemper, Ferrari and Hallberg2008). Baroclinic instability primarily draws energy from background potential energy. Given the negative buoyancy production discussed in § 4.1, baroclinic instability is unlikely to be a dominant mechanism for the arrest of either the filament or the front.

Symmetric instability can be regarded as a combination of inertial instability and gravitational (convective) instability (Haine & Marshall Reference Haine and Marshall1998), and it typically involves motion that is aligned with isopycnals, referred to as slantwise convection. Buoyancy perturbations associated with symmetric instability are usually small (Gula et al. Reference Gula, Taylor, Shcherbina and Mahadevan2022), and consequently, unlike baroclinic instability, symmetric instability primarily gains energy from the mean kinetic energy.

The criterion for symmetric instability can be written as (Hoskins Reference Hoskins1974; Haine & Marshall Reference Haine and Marshall1998; Gula et al. Reference Gula, Taylor, Shcherbina and Mahadevan2022)

(4.5) \begin{equation} fq=f[(\boldsymbol{\overline \zeta }+f\boldsymbol{e}_z)\boldsymbol{\cdot }\boldsymbol{\nabla }\overline b]\lt 0, \end{equation}

where $q$ is Ertel’s PV. In the filament case, $q$ is mostly positive throughout the frontogensis and arrest stages, suggesting that symmetric instability is generally suppressed (given $f\gt 0$ in our simulations). Localized regions of $q\lt 0$ appear in the filament core very close to the sea surface (e.g. above $z=-10$ m), where surface cooling reduces PV and creates conditions favourable for symmetric instability. In the front case, negative $q$ is found in the frontal zone near the sea surface, indicating the possibility of symmetric instability. Taylor & Ferrari (Reference Taylor and Ferrari2009) demonstrated that symmetric instability can trigger secondary Kelvin–Helmholtz (vertical shear) instability that is associated with the slantwise convection cells. Nevertheless, in our case, strong vertical shear production is found in the restratified region behind the propagating frontal zone, e.g. below $z=-20$ m (figure 12 $h$ ), and it does not coincide with regions of negative PV. This indicates a different mechanism from that revealed by Taylor & Ferrari (Reference Taylor and Ferrari2009): here the vertical shear instability is probably not associated with symmetric instability but instead directly results from frontal slumping. Our findings align with the prediction made by Hoskins & Bretherton (Reference Hoskins and Bretherton1972) over 50 years ago, which proposed that frontogenesis would ultimately be arrested by Kelvin–Helmholtz instability, and consistent results have been reported by Samelson & Skyllingstad (Reference Samelson and Skyllingstad2016).