3.1. Generation of vorticity by non-traditional effects

Neglecting terms of order $O(\delta ^2)$ from (2.5) and depth averaging (2.5a), (2.5b) and (2.5e) gives

(3.2a)$$\begin{gather} \frac{\partial \bar{u}}{\partial t}+\delta \bar{w} - \bar{v} ={-}\frac{\partial \bar{p}}{\partial x}, \end{gather}$$

(3.2b)$$\begin{gather}\frac{\partial \bar{v}}{\partial t}+ \bar{u} ={-}\frac{\partial \bar{p}}{\partial y}, \end{gather}$$

(3.2c)$$\begin{gather}\frac{\partial \bar{u}}{\partial x}+\frac{\partial \bar{v}}{\partial y} = 0, \end{gather}$$

which may be combined to give

(3.3)\begin{equation} \frac{\partial }{\partial t}\left(\frac{\partial \bar{v}}{\partial x}-\frac{\partial \bar{u}}{\partial y}\right) = \delta \frac{\partial \bar{w}}{\partial y}. \end{equation}

Equation (3.3) states that the non-traditional component of the Coriolis force acts to generate vorticity over long times, $t \sim O(1/\delta )$. This suggests the inclusion of a second time scale, $T = \delta t$, corresponding to this vorticity generation. Using a multiple scales approach the time derivative may be expanded as

(3.4)\begin{equation} \frac{\partial }{\partial t} \to \frac{\partial }{\partial t}+\delta\frac{\partial }{\partial T}, \end{equation}

where now the ${\partial }/{\partial t}$ term corresponds to transient inertial oscillations resulting from an unbalanced initial condition. From Crowe & Taylor (Reference Crowe and Taylor2018), a longer time scale on the order of $t \sim O(\delta ^4)$ is also expected to be important. This slow scale corresponds to shear dispersive spreading of the front and will be discussed in § 3.6.

From now on transient oscillations are neglected by setting the fast time derivative, ${\partial }/{\partial t}$, to zero. Therefore, the system is assumed to be balanced over the inertial time scale $t$ and only the slow evolution is considered.

3.2. The $O(1)$ solution

At leading order in $\delta$ (2.5) gives

(3.5a)$$\begin{gather} - v_0 ={-}\frac{\partial p_0}{\partial x}+E\frac{\partial ^2u_0}{\partial z^2}, \end{gather}$$

(3.5b)$$\begin{gather}u_0 ={-}\frac{\partial p_0}{\partial y}+E\frac{\partial ^2v_0}{\partial z^2}, \end{gather}$$

(3.5c)$$\begin{gather}0 =\frac{E}{Pr}\frac{\partial ^2b_0}{\partial z^2}, \end{gather}$$

(3.5d)$$\begin{gather}0 ={-}\frac{\partial p_0}{\partial z}+b_0, \end{gather}$$

(3.5e)$$\begin{gather}\frac{\partial u_0}{\partial x}+\frac{\partial v_0}{\partial y}+\frac{\partial w_0}{\partial z} = 0, \end{gather}$$

corresponding to the leading-order (in $Ro$) TTW system of Crowe & Taylor (Reference Crowe and Taylor2018). The leading-order buoyancy equation may now be solved for

(3.6)\begin{equation} b_0 = b_0(x,y,T); \end{equation}

hence, the layer is vertically well mixed to leading order in $\delta$. The leading-order pressure may now be solved as

(3.7)\begin{equation} p_0 = \bar{p}_0+z b_0, \end{equation}

where $\bar {p}_0$ balances the depth-averaged component of velocity through geostrophic balance. This depth-averaged flow may be represented as a streamfunction by

(3.8a,b)\begin{equation} \bar{u}_0 ={-}\frac{\partial \psi_0}{\partial y},\quad \bar{v}_0 = \frac{\partial \psi_0}{\partial x}, \end{equation}

where $\psi _0 = \bar {p}_0$ so the depth-averaged pressure acts as a streamfunction for this horizontal flow. The depth-dependent velocity fields, $(u_0',v_0',w_0)$, may be calculated (see Crowe & Taylor Reference Crowe and Taylor2018) by solving a fourth-order linear system to obtain solution

(3.9a)$$\begin{gather} u_0' ={-}\sqrt{E}\left[K''(\zeta)\frac{\partial b_0}{\partial x}-K(\zeta)\frac{\partial b_0}{\partial y}\right], \end{gather}$$

(3.9b)$$\begin{gather}v_0' ={-}\sqrt{E}\left[K(\zeta)\frac{\partial b_0}{\partial x}+K''(\zeta)\frac{\partial b_0}{\partial y}\right], \end{gather}$$

(3.9c)$$\begin{gather}w_0 =E\, K'(\zeta) \nabla_H^2 b_0, \end{gather}$$

where $\zeta = z/\sqrt {E}$ and $K(\zeta )$ is an $E$ dependent vertical structure function satisfying

(3.10)\begin{equation} \begin{cases} K^{(4)}(\zeta)+K(\zeta)+\zeta = 0 & \textrm{for}\ \zeta \in[-\zeta_0,\zeta_0],\\ K'(\zeta) = 0 & \textrm{at}\ \zeta ={\pm} \zeta_0,\\ K'''(\zeta) = 0 & \textrm{at}\ \zeta ={\pm} \zeta_0, \end{cases} \end{equation}

where $\zeta _0 = 1/(2\sqrt {E})$ is the value of $|\zeta |$ on the top and bottom surfaces. Note that primes ($'$) on $K$ are taken to mean derivatives with respect to $\zeta$ rather than deviations from a vertical average as used elsewhere. The full solution for $K(\zeta )$ is given by $K_0(\zeta )$ in Appendix A of Crowe & Taylor (Reference Crowe and Taylor2018). For $E \ll 1$, it can be shown that $K(\zeta )\sim -\zeta$ and, hence, thermal wind balance holds outside of thin boundary layers of width $O(\sqrt {E})$ near the top and bottom boundaries.

3.3. The $O(\delta )$ solution

At order $O(\delta )$ (2.5) gives

(3.11a)$$\begin{gather} \frac{\partial u_0}{\partial T}+w_0 - v_1 ={-}\frac{\partial p_1}{\partial x}+E\frac{\partial ^2u_1}{\partial z^2}, \end{gather}$$

(3.11b)$$\begin{gather}\frac{\partial v_0}{\partial T}+ u_1 ={-}\frac{\partial p_1}{\partial y}+E\frac{\partial ^2v_1}{\partial z^2}, \end{gather}$$

(3.11c)$$\begin{gather}\frac{\partial b_0}{\partial T}=\frac{E}{Pr}\frac{\partial ^2b_1}{\partial z^2}, \end{gather}$$

(3.11d)$$\begin{gather}- u_0 ={-}\frac{\partial p_1}{\partial z}+b_1, \end{gather}$$

(3.11e)$$\begin{gather}\frac{\partial u_1}{\partial x}+\frac{\partial v_1}{\partial y}+\frac{\partial w_1}{\partial z} = 0. \end{gather}$$

It can be shown that the only solutions satisfying (3.11c) along with no-flux boundary conditions are

(3.12a,b)\begin{equation} \frac{\partial b_0}{\partial T} = 0 \quad \textrm{and} \quad b_1 = b_1(x,y,T). \end{equation}

Therefore, $b_0$ does not change over the time scale $t = O(1/\delta )$ and the buoyancy is also depth independent to $O(\delta )$. The pressure may now be calculated using (3.9a) and (3.11d) as

(3.13)\begin{align} p_1 &= \bar{p}_1+(b_1 + \bar{u}_0) z -E \left[\left(K'(\zeta)-\frac{K(\zeta_0)}{\zeta_0}\right) \frac{\partial b_0}{\partial x}\right.\nonumber\\ &\quad \left.+\left(K'''(\zeta)-\frac{K''(\zeta_0)}{\zeta_0} +\frac{\zeta^2}{2}-\frac{\zeta_0^2}{6}\right)\frac{\partial b_0}{\partial y}\right], \end{align}

where the final term arises from the integral of $u_0'$ and has been set to be depth independent.

3.3.1. The depth-averaged system

From (3.11a), (3.11b), and (3.11e), the depth-averaged velocity and pressure satisfy

(3.14a)$$\begin{gather} \frac{\partial \bar{u}_0}{\partial T}+ \bar{w}_0 - \bar{v}_1 ={-}\frac{\partial \bar{p}_1}{\partial x}, \end{gather}$$

(3.14b)$$\begin{gather}\frac{\partial \bar{v}_0}{\partial T}+ \bar{u}_1 ={-}\frac{\partial \bar{p}_1}{\partial y}, \end{gather}$$

(3.14c)$$\begin{gather}\frac{\partial \bar{u}_1}{\partial x}+\frac{\partial \bar{v}_1}{\partial y} = 0, \end{gather}$$

which may be combined to give

(3.15)\begin{equation} \frac{\partial }{\partial T}\left(\frac{\partial \bar{v}_0}{\partial x}-\frac{\partial \bar{u}_0}{\partial y}\right) = \frac{\partial \bar{w}_0}{\partial y} \ \implies \ \frac{\partial }{\partial T}\nabla^2 \psi_0 = \frac{\partial \bar{w}_0}{\partial y}, \end{equation}

which describes the generation of depth-averaged vorticity. Substituting for $\bar {w}_0$ gives that

(3.16)\begin{equation} \frac{\partial \psi_0}{\partial T} = 2 \sqrt{E^{3}} K(\zeta_0) \frac{\partial b_0}{\partial y} \ \implies \ \psi_0 = \varPsi_0 + 2\sqrt{E^{3}} K(\zeta_0) \frac{\partial b_0}{\partial y} T, \end{equation}

where $\varPsi _0 = \varPsi _0(x,y)$ is the value of $\psi _0$ at $T = 0$. The depth-averaged geostrophic flow can now be determined from $\psi _0$. From (3.14c), the $O(\delta )$ depth-averaged flow may now be written as

(3.17a,b)\begin{equation} \bar{u}_1 ={-}\frac{\partial \psi_1}{\partial y},\quad \bar{v}_1 = \frac{\partial \psi_1}{\partial x}, \end{equation}

where, by (3.14a) and (3.14b), $\psi _1$ is related to $\bar {p}_1$ through

(3.18)\begin{equation} \bar{p}_1 = \psi_1 - 2\sqrt{E^{3}} K(\zeta_0) \frac{\partial b_0}{\partial x}. \end{equation}

To determine the evolution of $\psi _1$ it is necessary to consider the $O(\delta ^2)$ system.

3.3.2. The depth-dependent system

The depth-dependent quantities may now be considered by subtracting the depth-averaged horizontal momentum equations in (3.14) from (3.11a) and (3.11b) to obtain

(3.19a)$$\begin{gather} w_0' - v_1' ={-}\frac{\partial p_1'}{\partial x}+E\frac{\partial ^2u_1'}{\partial z^2}, \end{gather}$$

(3.19b)$$\begin{gather}u_1' ={-}\frac{\partial p_1'}{\partial y}+E\frac{\partial ^2v_1'}{\partial z^2}, \end{gather}$$

where the time derivative terms vanish as $(u_0',v_0')$ does not depend on $T$. Substituting for $w_0'$ using (3.9c) and $p_1'$ using (3.13), this system may be solved (see Appendix A) for solution

(3.20a)$$\begin{gather} u_1'={-}\sqrt{E}\left[K''(\zeta)\frac{\partial }{\partial x}-K(\zeta)\frac{\partial }{\partial y}\right]\left(b_1-\frac{\partial \psi_0}{\partial y}\right) +E\frac{\partial }{\partial y}\left[A(\zeta)\frac{\partial b_0}{\partial x}-B(\zeta)\frac{\partial b_0}{\partial y}\right], \end{gather}$$

(3.20b)$$\begin{gather}v_1'={-}\sqrt{E}\left[K(\zeta)\frac{\partial }{\partial x}+K''(\zeta)\frac{\partial }{\partial y}\right]\left(b_1-\frac{\partial \psi_0}{\partial y}\right) +E\frac{\partial }{\partial y}\left[B(\zeta)\frac{\partial b_0}{\partial x}+A(\zeta)\frac{\partial b_0}{\partial y}\right]. \end{gather}$$

Finally, $w_1$ may be calculated using (3.11e) as

(3.21)\begin{equation} w_1 = E K'(\zeta) \nabla_H^2 \left(b_1 - \frac{\partial \psi_0}{\partial y} \right) - \sqrt{E^3} C(\zeta) \nabla_H^2 \frac{\partial b_0}{\partial y}, \end{equation}

where $C(\zeta )$ is the integral of $A(\zeta )$. The functions $A$, $B$ and $C$ are complicated functions of $\zeta$, $K(\zeta )$ and $\zeta _0$ and are given in Appendix B.

3.4. The $O(\delta ^2)$ solution

In Crowe & Taylor (Reference Crowe and Taylor2018) it was shown that an $O(Ro)$ stratification is induced and maintained by an advection–diffusion balance in the buoyancy equation. Here this effect is expected to appear at orders $O(\delta ^2) = O(Ro)$ and $O(\delta ^3)$ and the $O(\delta ^2)$ system is considered first.

3.4.1. The buoyancy field

Since it has been assumed that $Ro = O(\delta ^2)$, it is convenient to define $Ro = \mathcal {R}\, \delta ^2$, where $\mathcal {R}$ is an $O(1)$ number. The $O(\delta ^2)$ buoyancy equation is

(3.22)\begin{equation} \frac{\partial b_1}{\partial T} + \mathcal{R} \left(u_0\frac{\partial b_0}{\partial x}+v_0\frac{\partial b_0}{\partial y}\right) = \frac{E}{Pr} \frac{\partial ^2 b_2}{\partial z^2}, \end{equation}

and noting that $b_1$ is depth independent, (3.22) may be depth averaged to obtain

(3.23)\begin{equation} \frac{\partial b_1}{\partial T} + \mathcal{R} J(\psi_0,b_0) = 0, \end{equation}

where $J(\phi ,\varphi ) = (\partial _x \phi )(\partial _y \varphi ) - (\partial _y \phi )(\partial _x \varphi )$ is the Jacobian derivative. Substituting for $\psi _0$ gives

(3.24)\begin{equation} b_1 ={-}\mathcal{R} \left[ J(\varPsi_0,b_0)T + \sqrt{E^3} K(\zeta_0) J\left(\frac{\partial b_0}{\partial y},b_0\right) T^2\right], \end{equation}

assuming that $b_1 = 0$ at $T=0$.

Subtracting (3.23) from (3.22) gives

(3.25)\begin{equation} \mathcal{R}\left(u_0'\frac{\partial b_0}{\partial x}+v_0'\frac{\partial b_0}{\partial y}\right) = \frac{E}{Pr} \frac{\partial ^2 b_2}{\partial z^2}. \end{equation}

This equation was considered in Crowe & Taylor (Reference Crowe and Taylor2018) and describes the restratification of the front by the TTW circulation. The solution is

(3.26)\begin{equation} b_2 = \bar{b}_2(x,y,T) - \mathcal{R}\,Pr\, \sqrt{E}\, K(\zeta) |\nabla_H b_0|^2. \end{equation}

3.4.2. The streamfunction for the depth-averaged flow

Depth-dependent velocity components of order higher than $O(\delta )$ are not required in the subsequent calculations. However, higher-order components of $\psi$ are required to determine the higher-order depth-averaged buoyancy terms and may be determined by considering the vertical vorticity.

The depth-averaged vertical vorticity equation may be derived by cross-differentiating (2.5a) and (2.5b) and depth averaging to obtain

(3.27)\begin{equation} \delta \frac{\partial \bar{\eta}}{\partial T} + Ro\, \nabla_H \boldsymbol{\cdot} \left[ \overline{\boldsymbol{u}_H \eta} - \overline{\boldsymbol\omega_H w} \right] = \delta \frac{\partial \bar{w}}{\partial y}. \end{equation}

Here $\boldsymbol {u}_H = (u,v,0)$ is the horizontal velocity, $\eta = {\partial v}/{\partial x}-{\partial u}/{\partial y}$ is the vertical vorticity and

(3.28)\begin{equation} \boldsymbol\omega_H =\begin{pmatrix} \displaystyle \frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}\\ \displaystyle \frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}\\ 0 \end{pmatrix} \end{equation}

is the horizontal vorticity. At $O(\delta ^2)$ (3.27) gives

(3.29)\begin{equation} \frac{\partial \nabla_H^2\psi_1}{\partial T}+\mathcal{R}\, J[\psi_0,\nabla_H^2 \psi_0] = \mathcal{R} \nabla_H \boldsymbol{\cdot} [ -\overline{\boldsymbol{u}_{H0}' \eta_0'} + \overline{\boldsymbol\omega_{H0} w_0}] + \frac{\partial \bar{w}_1}{\partial y}, \end{equation}

where the flux terms can be expressed in terms of $b_0$ to give

(3.30)\begin{equation} \frac{\partial \nabla_H^2\psi_1}{\partial T}+\mathcal{R} J[\psi_0,\nabla_H^2 \psi_0] = \mathcal{R} \nabla_H \boldsymbol{\cdot} [{\boldsymbol{\mathsf{P}}}\boldsymbol{\cdot} \nabla_H b_0 \nabla_H^2 b_0] + \frac{\partial \bar{w}_1}{\partial y} \end{equation}

for

(3.31)\begin{equation} {\boldsymbol{\mathsf{P}}} = E \begin{pmatrix} 2\overline{K'^2} & \overline{K^2}-\overline{K''^2} \\ \overline{K''^2}-\overline{K^2} & 2\overline{K'^2} \end{pmatrix}. \end{equation}

The flux term in (3.30) corresponds to both the generation of vorticity due to vortex stretching and the horizontal transport of vorticity due to a correlation between the vertically sheared profiles for the horizontal velocity and the vertical vorticity. Over time scales longer than $T$, these terms have been shown to generate along-front jets (Crowe & Taylor Reference Crowe and Taylor2019b) and play a role in baroclinic instability (Crowe & Taylor Reference Crowe and Taylor2019a). Vorticity is also generated by the non-traditional component of the Coriolis force through the $y$ variations in $\bar {w}_1$, as discussed in § 3.1.

Equation (3.30) may be solved for $\nabla _H^2\psi _1$ by a simple integration in $T$. However, solving for $\psi _1$ requires inverting the Laplacian operator so it is not possible to present a simple analytic solution. Solutions for (3.30) could be easily found numerically for given fields $b_0$, $\psi _0$ and $b_1$.

3.5. The $O(\delta ^3)$ solution

Now the order $O(\delta ^3)$ balance is considered to determine the stratification maintained by the $O(\delta )$ velocity component. The $O(\delta ^3)$ vorticity equation will not be examined though it may be derived from (3.27) similarly to (3.30). The buoyancy equation is

(3.32)\begin{equation} \frac{\partial b_2}{\partial T} + \mathcal{R} \left( u_1 \frac{\partial b_0}{\partial x}+v_1\frac{\partial b_0}{\partial y} + u_0\frac{\partial b_1}{\partial x}+v_0\frac{\partial b_1}{\partial y}\right) = \frac{E}{Pr} \frac{\partial ^2 b_3}{\partial z^2}, \end{equation}

which may be depth averaged to obtain

(3.33)\begin{equation} \frac{\partial \bar{b}_2}{\partial T} + \mathcal{R} [ J(\psi_1,b_0) + J(\psi_0,b_1) ] = 0. \end{equation}

This equation may be solved using the expression for $\psi _1$ if required. Since the depth-averaged buoyancy is known to the first two orders in $\delta$ and it is not possible to find a simple analytic expression for $\psi _1$, expressions for $\bar {b}$ are not calculated explicitly at $O(\delta ^2)$ or higher. Instead, the focus is on determining the vertical structure of $b$, denoted $b'$, to the lowest two orders. Since the lowest-order term in $b'$ is $b'_2$ (see (3.26)), the next order term, $b_3'$, must also be determined.

Subtracting (3.33) from (3.32) and noting that ${\partial b_2'}/{\partial T} = 0$ gives the equation for the depth-dependent buoyancy

(3.34)\begin{equation} \mathcal{R} \left( u_1' \frac{\partial b_0}{\partial x} + v_1' \frac{\partial b_0}{\partial y} + u_0' \frac{\partial b_1}{\partial x} + v_0' \frac{\partial b_1}{\partial y} \right) = \frac{E}{Pr}\frac{\partial ^2 b_3}{\partial z^2}, \end{equation}

with solution

(3.35)\begin{align} b_3 &= \bar{b}_3(x,y,T) + \mathcal{R}\,Pr \left[ E \left( \frac{D_1(\zeta)}{2} \frac{\partial }{\partial y} |\nabla_H b_0|^2 + D_2(\zeta) J\left[ \frac{\partial b_0}{\partial y},b_0\right] \right) \right. \nonumber\\ &\quad + \sqrt{E} \left( K(\zeta) \left( \nabla_H \frac{\partial \psi_0}{\partial y} \cdot \nabla_H b_0 - 2 \nabla_H b_0 \cdot \nabla_H b_1 \right)\right. \nonumber\\ &\quad \left.\left. -\left( K''(\zeta)+\frac{\zeta^3}{6}-\frac{\zeta\zeta_0^2}{2} \right) J\left[\frac{\partial \psi_0}{\partial y},b_0\right]\right)\right], \end{align}

where the vertical structure functions $D_1(\zeta )$ and $D_2(\zeta )$ are given in Appendix B. The term $\bar {b}_4$ can be determined by depth averaging the $O(\delta ^5)$ buoyancy equation, as noted above, this calculation is not done here.

3.6. Higher-order terms and shear dispersive spreading

The asymptotic approach may be continued as above to $O(\delta ^4)$ and higher. However, from Crowe & Taylor (Reference Crowe and Taylor2018, Reference Crowe and Taylor2019b), slow frontal spreading is expected due to a buoyancy flux resulting from the correlation between the leading-order velocity and the $O(\delta ^2)$ stratification, $\overline {\boldsymbol {u}_{H0}' b_2'}$. This spreading is due to shear dispersion and was found to appear in the equations at $O(Ro^2) = O(\delta ^4)$ and occur over a time scale of $t = O(1/Ro^2) = O(1/\delta ^4)$. Similarly, the flux terms $\overline {\boldsymbol {u}_{H1}' b_2'}$ and $\overline {\boldsymbol {u}_{H0}' b_3'}$ resulting from non-traditional effects might be expected to drive some buoyancy change at $O(\delta ^5)$. Therefore, new time scales are introduced to examine the effects of this shear dispersion.

The time scale $T = \delta t$ was shown in § 3.1 to be the time scale over which an $O(1)$ amount of depth-averaged vorticity is generated by non-traditional effects. Over time scales longer than $T$, many of the terms in $\bar {b}$ and $\psi$ demonstrate secular growth and as such it is necessary to introduce additional slow time scales corresponding to this slow frontal spreading. This is done by equating the size of the time derivative of the leading-order buoyancy, $b_0$, with the shear dispersion terms

(3.36a,b)\begin{equation} \frac{\partial b_0}{\partial t} \sim \delta^4 \mathcal{R} \nabla_H \boldsymbol{\cdot} [ \overline{\boldsymbol{u}_{H0}' b_2'} ] \quad \textrm{and} \quad \frac{\partial b_0}{\partial t} \sim \delta^5 \mathcal{R} \nabla_H \boldsymbol{\cdot} [ \overline{\boldsymbol{u}_{H1}' b_2'} + \overline{\boldsymbol{u}_{H0}' b_3'}], \end{equation}

to get two time scales, $T_4 = \delta ^4 t$ and $T_5 = \delta ^5 t$, and letting $b_0$ depend on $T_4$ and $T_5$. Here $T_4$ corresponds to the slow spreading time scale from Crowe & Taylor (Reference Crowe and Taylor2018) while $T_5$ corresponds to a longer time scale on which the evolution of depth-averaged buoyancy occurs due to non-traditional effects. Determining a closed system in full generality requires knowing how $\psi$ evolves over the slow scales $T_4$ and $T_5$ which requires examining high-order equations for the depth-averaged vorticity (Crowe & Taylor Reference Crowe and Taylor2019a,Reference Crowe and Taylorb). Instead, the simplifying assumption of a straight front is made. Under this assumption, the $\psi$ dependent terms vanish and equations purely in terms of $b_0$ are recovered as

(3.37)\begin{equation} \frac{\partial b_0}{\partial T_4} = \mathcal{R}^2\,Pr\, \nabla_H \cdot [ {\boldsymbol{\mathsf{Q}}} \boldsymbol{\cdot} \nabla_H b_0 |\nabla_H b_0|^2], \end{equation}

and

(3.38)\begin{equation} \frac{\partial b_0}{\partial T_5} = \mathcal{R}^2\,Pr\, \nabla_H \cdot \left[ {\boldsymbol{\mathsf{R}}}_1 \boldsymbol{\cdot} \frac{\partial \nabla_H b_0}{\partial y} |\nabla_H b_0|^2 + {\boldsymbol{\mathsf{R}}}_2 \boldsymbol{\cdot} \nabla_H b_0 \frac{\partial }{\partial y}|\nabla_H b_0|^2\right], \end{equation}

where

(3.39)\begin{equation} {\boldsymbol{\mathsf{Q}}}(E) = E \begin{pmatrix} \overline{K'^2} & \overline{K^2} \\ -\overline{K^2} & \overline{K'^2} \end{pmatrix}, \end{equation}

and

(3.40a,b)\begin{equation} {\boldsymbol{\mathsf{R}}}_1(E) = E\sqrt{E} \begin{pmatrix} \overline{AK} & -\overline{BK} \\ \overline{BK} & \overline{AK} \end{pmatrix}, \quad {\boldsymbol{\mathsf{R}}}_2(E) = \frac{E\sqrt{E}}{2} \begin{pmatrix} \overline{AK} & -\overline{D_1K} \\ \overline{D_1K} & \overline{AK} \end{pmatrix}. \end{equation}

Equation (3.37) is identical to the result derived in Crowe & Taylor (Reference Crowe and Taylor2018) and describes the spreading of a front due to a horizontal buoyancy flux resulting from the correlation between the induced stratification and the cross-front flow. Equation (3.38) similarly describes a horizontal buoyancy flux, with terms arising from the non-traditional corrections to the stratification and cross-front flow.

It is worth noting that over long time scales the generation of significant background vorticity is expected, both by non-traditional effects as discussed in § 3.1 and due to the correlation between along-front and cross-front velocity fields as shown in (3.30) and discussed in Crowe & Taylor (Reference Crowe and Taylor2019b). These correlation terms appear as a consequence of vertical mixing driving a cross-front flow and do not appear in the limit of $E \to 0$. The generated vorticity manifests as along-front jets and can become large enough to significantly modify the absolute vorticity of the system resulting in a modification of the TTW velocity solution and, hence, a modified stratification and frontal spreading. Additionally, frontal systems are susceptible to baroclinic instability (Stone Reference Stone1966; Crowe & Taylor Reference Crowe and Taylor2019a) which may lead to a breakdown of the straight front assumption.

4.1. The velocity fields

Correct to $O(\delta )$, the velocity fields are given by

(4.1)\begin{align} \boldsymbol{u}_H &={-}\boldsymbol{\nabla}\times[ (\psi_0+\delta\psi_1)\hat{\boldsymbol{z}}]-\sqrt{E} {\boldsymbol{\mathsf{K}}}\boldsymbol{\cdot} \nabla_H \left(b_0+\delta\,b_1 - \delta\frac{\partial \psi_0}{\partial y}\right)\nonumber\\ &\quad + \delta E {\boldsymbol{\mathsf{A}}}\boldsymbol{\cdot} \nabla_H\frac{\partial b_0}{\partial y} + O(\delta^2), \end{align}

and

(4.2)\begin{equation} w = E K'(\zeta) \nabla_H^2 \left(b_0+\delta\,b_1 - \delta\frac{\partial \psi_0}{\partial y}\right) - \delta \sqrt{E^3} C(\zeta) \nabla_H^2 \frac{\partial b_0}{\partial y} +O(\delta^2), \end{equation}

where

(4.3a,b)\begin{equation} {\boldsymbol{\mathsf{K}}}(\zeta) = \begin{pmatrix} K''(\zeta) & -K(\zeta) \\ K(\zeta) & K''(\zeta) \end{pmatrix} \quad \textrm{and} \quad {\boldsymbol{\mathsf{A}}}(\zeta) = \begin{pmatrix} A(\zeta) & -B(\zeta) \\ B(\zeta) & A(\zeta) \end{pmatrix}. \end{equation}

The depth-averaged velocity is described by a streamfunction where

(4.4)\begin{equation} \psi_0 = \varPsi_0 + 2\sqrt{E^{3}} \delta t K(\zeta_0) \frac{\partial b_0}{\partial y}, \end{equation}

for some initial streamfunction $\psi _0 = \varPsi _0$ at $t = 0$. It should be noted that $t = O(1/\delta )$ so all terms here are leading order. The $O(\delta )$ streamfunction component, $\psi _1$, satisfies (3.30).

The leading-order flow can be split into components in the cross-front direction (described by the diagonal terms in ${\boldsymbol{\mathsf{K}}}$) and along-front direction (described by the off-diagonal terms in ${\boldsymbol{\mathsf{K}}}$). However, the $O(\delta )$ terms are aligned relative to gradients of the north–south ($y$) derivatives of $b_0$ and $\psi _0$ which do not necessarily correspond to the direction of $\nabla _H b_0$.

Two special cases are $b_0 = b_0(x)$ and $b_0 = b_0(y)$. The case of $b_0 = b_0(x)$ describes a front with the along-front direction aligned north–south. In this case all $y$ derivatives can be neglected and the non-traditional terms have no effect on the front as discussed in § 2. Conversely, $b_0 = b_0(y)$ describes a front with the along-front direction aligned east–west. In this case non-traditional effects are maximised and the gradients of $b_0$ are aligned with the gradients of ${\partial b_0}/{\partial y}$ so the horizontal velocity terms driven by the non-traditional rotation can be easily split into cross-front and along-front components similarly to the leading-order flow.

4.2. The buoyancy field

The buoyancy field can be split into depth-averaged and depth-dependent components. Correct to the lowest two orders in $\delta$ the solutions are

(4.5)\begin{equation} \bar{b} = b_0+\delta b_1 +O(\delta^2) = b_0 -Ro \,t J\left[\varPsi_0 + \sqrt{E^3} \delta t K(\zeta_0)\frac{\partial b_0}{\partial y},b_0\right] + O(\delta^2), \end{equation}

where $Ro\,t = O(\delta )$. The depth-dependent buoyancy is given by

(4.6)\begin{align} b' &= Ro\,Pr\, \sqrt{E} \left[ \left(- K(\zeta) +\delta \sqrt{E} \frac{D_1(\zeta)}{2} \frac{\partial }{\partial y} \right) |\nabla_H b_0|^2 + \delta \left( \sqrt{E} D_2(\zeta) J\left[ \frac{\partial b_0}{\partial y},b_0 \right]\right.\right. \nonumber\\ &\quad + \left.\left. K(\zeta) \nabla_H\left( \frac{\partial \psi_0}{\partial y} - 2 b_1 \right)\cdot\nabla_H b_0 -\left( K''(\zeta)+\frac{\zeta^3}{6}-\frac{\zeta\zeta_0^2}{2} \right) J\left[\frac{\partial \psi_0}{\partial y},b_0\right]\right)\right] \nonumber\\ &\quad + O(\delta^4). \end{align}

Similarly to the velocity fields, if $b_0 = b_0(x)$ then the non-traditional rotation has no effect on the front and the solution reduces to the results of Crowe & Taylor (Reference Crowe and Taylor2018). From $b'$ the vertical buoyancy gradient, $N^2$, may be determined as

(4.7)\begin{align} N^2 &= \frac{\partial b'}{\partial z} = Ro\,Pr\left[ \left(- K'(\zeta) +\delta\,\sqrt{E} \frac{D_1'(\zeta)}{2} \frac{\partial }{\partial y} \right) |\nabla_H b_0|^2 + \delta \left( \sqrt{E} D_2'(\zeta)\, J\left[ \frac{\partial b_0}{\partial y},b_0\right] \right.\right. \nonumber\\ &\quad + \left.\left. K'(\zeta) \nabla_H\left( \frac{\partial \psi_0}{\partial y} - 2 b_1 \right)\cdot\nabla_H b_0 -\left( K'''(\zeta)+\frac{\zeta^2}{2}-\frac{\zeta_0^2}{2} \right) J\left[\frac{\partial \psi_0}{\partial y},b_0\right]\right)\right] \nonumber\\ &\quad + O(\delta^4). \end{align}

The horizontal buoyancy gradient may be similarly calculated using

(4.8)\begin{equation} M^2 = \nabla_H b = \nabla_H \bar{b} + \nabla_H b', \end{equation}

where the first term on the right-hand side is leading order and depth independent while the second term is order $O(Ro)$ and depth dependent.

4.3. Frontal spreading and shear dispersion

Over very long times the front is expected to evolve through shear dispersion. Equations (3.37) and (3.38) may be combined to give

(4.9)\begin{equation} \frac{\partial b_0}{\partial t} = Ro^2\,Pr\, \nabla_H \cdot \left[ \left({\boldsymbol{\mathsf{Q}}} \boldsymbol{\cdot} \nabla_H b_0+\delta\,{\boldsymbol{\mathsf{R}}}_1 \boldsymbol{\cdot} \nabla_H \frac{\partial b_0}{\partial y} + \delta\,{\boldsymbol{\mathsf{R}}}_2 \boldsymbol{\cdot} \nabla_H b_0\, \frac{\partial }{\partial y}\right)|\nabla_H b_0|^2\right], \end{equation}

which is valid for a straight front provided the vorticity generated by non-traditional effects and vertical mixing is less than the background vorticity. Expressions for ${\boldsymbol{\mathsf{Q}}}$, ${\boldsymbol{\mathsf{R}}}_1$ and ${\boldsymbol{\mathsf{R}}}_2$ are given in (3.39) and (3.40a,b). It should be noted that (4.9) reduces to the results of Crowe & Taylor (Reference Crowe and Taylor2018) for $b_0 = b_0(x)$, similarly to the results for velocity and buoyancy. If $b_0 = b_0(y)$ is an odd function of $y$, then solutions to (4.9) will remain odd in $y$ for all time for the case of $\delta = 0$. However, for $\delta \neq 0$, the addition of an extra $y$ derivative in the non-traditional correction terms leads to an asymmetry and, hence, different evolution on each side of the front.

5.1. Depth-independent jets

Taking the initial streamfunction of $\varPsi _0 = 0$, the depth-averaged velocity is given by

(5.2)\begin{equation} \psi_0 = 2\sqrt{E^{3}} \delta t K(\zeta_0)\textrm{sech}^2 y \ \implies \ (\bar{u}_0,\bar{v}_0) = 4\sqrt{E^{3}} \delta t K(\zeta_0)(\textrm{sech}^2 y \tanh y,0), \end{equation}

corresponding to two jets running in opposite directions along the edges of the front. As expected, motion is confined to the frontal region. Since the Jacobian terms vanish for $b_0 = b_0(y)$, (3.30) may be solved for $\psi _1$ as

(5.3)\begin{equation} \psi_1 = \mathcal{R}E \delta t \overline{K'^2}\left(\frac{\partial b_0}{\partial y}\right)^2 - 2\,E^3(\delta t)^2 [K(\zeta_0)]^2 \frac{\partial ^3b_0}{\partial y^3}. \end{equation}

The first term of $\psi _1$ in (5.3) describes the vorticity generated by the correlation between the cross-front and along-front TTW velocities (Crowe & Taylor Reference Crowe and Taylor2019b), while the second term describes the generation of vorticity through the action of the non-traditional Coriolis force on the $O(\delta )$ vertical velocity. The streamfunction and along-front velocity of the depth-independent jets are shown in figure 2 correct to $O(\delta )$ as a function of $y$ for $E = 0.1$, $\delta = 0.2$, $\delta \,t = 1$ and $\mathcal {R} = 1$. These jets grow with time and are expected to become large for $T\gg 1$.

Figure 2. The streamfunction (a) and velocity (b) of the along-front jets. Solutions are shown correct to $O(\delta )$ for $E = 0.1$, $\delta = 0.2$, $\delta \,t = 1$ and $\mathcal {R} = 1$.

5.2. Frontal circulation

For an $x$ independent front, the cross-front velocity ($v$) and vertical velocity ($w$) satisfy the mass conservation equation

(5.4)\begin{equation} \frac{\partial v'}{\partial y} + \frac{\partial w}{\partial z} = 0, \end{equation}

and, hence, the circulation around the front in the $y-z$ plane can be represented by a circulation streamfunction, $\phi$, defined using

(5.5a,b)\begin{equation} v' = \frac{\partial \phi}{\partial z}, \quad w ={-}\frac{\partial \phi}{\partial y}. \end{equation}

Note that there is no depth-independent flow in the $y$ direction as $\psi = \psi (y)$ so $v = v'$ here. The circulation components, $\phi _0$ and $\phi _1$, are given by

(5.6a,b)\begin{equation} \phi_0 ={-}E K'(\zeta) \frac{\partial b_0}{\partial y}, \quad \phi_1 = \sqrt{E^3} C(\zeta) \frac{\partial ^2b_0}{\partial y^2} + E K'(\zeta)\frac{\partial ^2\psi_0}{\partial y^2}. \end{equation}

The two terms of $\phi _1$ in (5.6a,b) each arise due to different components of the non-traditional Coriolis force. The horizontal component appears directly in the horizontal momentum balance resulting in the first term of (5.6a,b) while the vertical component drives the system out of hydrostatic balance, modifying the pressure field and giving the second term.

Figure 3 shows a comparison between the TTW solutions of Crowe & Taylor (Reference Crowe and Taylor2018) (corresponding to $\delta = 0$) and the modified TTW solutions presented here with $\delta = 0.4$. Solutions are given correct to $O(\delta )$ using $\varphi = \varphi _0+\delta \,\varphi _1$ for a given field $\varphi$ and shown for $E = 0.01$ and $T = 1$. The TTW solution consists of a flow from the high buoyancy side of the front to the low buoyancy side near the top surface and the opposite on the bottom surface. This results in upwelling on the high buoyancy side and downwelling on the low buoyancy side resulting in a anti-clockwise net circulation (shown by positive $\phi$). This behaviour was discussed in Crowe & Taylor (Reference Crowe and Taylor2018) and is consistent with previous results and observations (Eliassen Reference Eliassen1962; Orlanski & Ross Reference Orlanski and Ross1977; McWilliams Reference McWilliams2017). Non-traditional effects act to tilt the circulation cell and drive a flow in the centre of the layer. This flow may lead to a topological change in the structure of the circulation with a streamline in figure 3(c) seen to split into two separate cells.

Figure 3. Comparison between the TTW and modified TTW solutions for $E = 0.01$ and $T = 1$; (a) $\phi$ for $\delta = 0$, (b) $\phi$ for $\delta = 0.4$, (c) $v$ for $\delta = 0$, (d) $v$ for $\delta = 0.4$, (e) $w$ for $\delta = 0$ and (f) $w$ for $\delta = 0.4$. Solutions are shown correct to $O(\delta )$.

Figure 4 shows separately the two terms of $\phi _1$ from (5.6a,b) for $E = 0.1$ and $T = 1$. The associated cross-front velocities are also shown. The first term consists of four counter-rotating cells resulting in regions of convergence near the top and bottom boundaries and acting to tilt the leading-order circulation cell. The second term consists of three counter-rotating cells and results from the along-front jets modifying the vertical pressure gradient away from hydrostatic balance. As these jets grow, the second term of $\phi _1$ grows linearly with time for $T = O(1)$. Over very long time scales, it is predicted that these jets can become large enough to modify the absolute vertical vorticity in the frontal region. Therefore, while small in figure 3(b), this circulation component may become large at late times leading to further topological changes in the structure of the frontal circulation.

Figure 4. (a) The first term of the $O(\delta )$ circulation streamfunction component, $\phi _1$, from (5.6a,b). (b) The second term of $\phi _1$ from (5.6a,b). Panels (c) and (d) show the cross-front velocity ($v$) components associated with the streamfunction components shown in panels (a) and (b), respectively. Results are shown for $E = 0.1$ and $T = 1$.

Additionally, by depth integrating the vertical velocities corresponding to the two terms in (5.6a,b) the net vertical transport of fluid may be calculated. The first term depth averages to zero so does not correspond to any vertical transport, instead this term describes a tilting of the circulation cell as noted above. The second term does, however, have a non-zero depth average which suggests that the circulation cells in figure 4(b) may act to enhance the vertical exchange of tracers through the surface mixed layer.

The $O(\delta )$ cross-front velocities shown in figure 4 contain regions of surface convergence. This velocity convergence can lead to a sharpening of surface buoyancy gradients resulting in frontogenesis (Hoskins Reference Hoskins1982; Shakespeare & Taylor Reference Shakespeare and Taylor2013) and, hence, non-traditional effects may be frontogenetic. The asymptotic framework used here assumes $Ro \ll 1$ so this model is not strictly valid for studying frontogenesis where the Rossby number is typically order $1$. However, for $Ro = O(1)$, the frontal sharpening predicted here will be an $O(\delta )$ effect, therefore, away from the equator, non-traditional effects are unlikely to be a dominant frontogenetic mechanism when compared with other mechanisms such as external strain, spontaneous adjustment and the secondary circulation induced by finite Rossby number effects (Hoskins & Bretherton Reference Hoskins and Bretherton1972; Blumen Reference Blumen2000; Gula et al. Reference Gula, Molemaker and McWilliams2014; McWilliams Reference McWilliams2017). Non-traditional frontogenesis may be relevant in a small region around the equator where $\delta \geq O(1)$, though, since TTW is unlikely to be the dominant balance in this region, it is not possible to draw any conclusions from this analysis.

5.3. The along-front flow

The depth-dependent along-front velocity components are given by

(5.7a,b)\begin{equation} u_0' = \sqrt{E} K(\zeta) \frac{\partial b_0}{\partial y}, \quad u_1' ={-} E B(\zeta)\frac{\partial ^2b_0}{\partial y^2} -\sqrt{E} K(\zeta) \frac{\partial ^2\psi_0}{\partial y^2}, \end{equation}

where the two terms of $u_1'$ arise from the modified horizontal momentum balance and the modified hydrostatic balance similarly to the terms of $\phi _1$ in (5.6a,b).

Figure 5 shows the depth-dependent along-front velocity correct to $O(\delta )$ for the cases of $\delta = 0$ and $\delta = 0.4$ with $T = 1$. The along-front flow is dominated by a thermal wind shear modified by vertical mixing (Crowe & Taylor Reference Crowe and Taylor2018) and non-traditional effects are seen to be small. Therefore, the most significant effect of non-traditional rotation on the along-front flow is the development of the depth-independent jets shown in figure 2 though if the jets become large, significant modification of the depth-dependent flow may occur through the second term of (5.7a,b).

Figure 5. Comparison of the along-front velocities for the TTW and modified TTW systems showing (a) $u'$ for $\delta = 0$ and (b) $u'$ for $\delta = 0.4$. Parameters are $E = 0.01$ and $T = 1$ and solutions are given correct to $O(\delta)$.

5.4. Buoyancy and stratification

As noted above, the depth-averaged buoyancy remains equal to $b_0$ for the simple case of $b_0 = \tanh y$. However, the vertical structure of the buoyancy field and the associated stratification are determined by the frontal circulation so are still affected by non-traditional rotation. These terms appear at orders $O(\delta ^2)$ and $O(\delta ^3)$ and from (4.6) are given by

(5.8)\begin{equation} \left.\begin{gathered} b_2 ={-}\mathcal{R}\,Pr\sqrt{E} K(\zeta)\left(\frac{\partial b_0}{\partial y}\right)^2,\\ b_3 =\mathcal{R}\,Pr\left( E D_1(\zeta)\frac{\partial b_0}{\partial y}\frac{\partial ^2b_0}{\partial y^2} + \sqrt{E} K(\zeta) \frac{\partial b_0}{\partial y}\frac{\partial ^2\psi_0}{\partial y^2}\right). \end{gathered}\right\} \end{equation}

The lowest-order buoyancy term with vertical structure, $b_2$, describes the stratification maintained by an advection–diffusion balance between the advection of buoyancy by the leading-order circulation, $\phi _0$, and the vertical mixing of buoyancy (Crowe & Taylor Reference Crowe and Taylor2018). A similar balance occurs at $O(\delta ^3)$ so the first (second) term of $b_3$ in (5.8) describes the stratification maintained by the first (second) term of $\phi _1$.

Figure 6 shows the depth-dependent buoyancy, $b'$, and vertical stratification, $N^2 = {\partial b}/{\partial z}$, correct to $O(\delta ^3)$ for $E = 0.01$ and $T = 1$. Solutions are shown for $\delta = 0$ and $\delta = 0.4$. Since $b'$ and $N^2$ are linear in $Ro\,Pr$ through the factor of $\mathcal {R} \delta ^2\,Pr$, results are plotted for $b'/(Ro\,Pr)$ and $N^2/(Ro\,Pr)$ to remove this dependence. The advection–diffusion balance is seen to drive a stable restratification of the front and modification by non-traditional effects is small unless $\psi _0$ becomes large.

Figure 6. Comparison between the TTW and modified TTW solutions for $E = 0.01$ and $T = 1$; (a) $b'/(Ro\,Pr)$ for $\delta = 0$, (b) $b'/(Ro\,Pr)$ for $\delta = 0.4$, (c) $N^2/(Ro\,Pr)$ for $\delta = 0$ and (d) $N^2/(Ro\,Pr)$ for $\delta = 0.4$. Solutions are shown correct to $O(\delta )$.

From (5.8), the order $O(\delta ^3)$ horizontal buoyancy gradient may be calculated as

(5.9)\begin{equation} M^2_3 = \frac{\partial b_3}{\partial y} = \mathcal{R}\,Pr\left( E D_1(\zeta)\frac{\partial }{\partial y}\left[\frac{\partial b_0}{\partial y}\frac{\partial ^2b_0}{\partial y^2}\right] + \sqrt{E}\,K(\zeta) \frac{\partial }{\partial y}\left[\frac{\partial b_0}{\partial y}\frac{\partial ^2\psi_0}{\partial y^2}\right]\right). \end{equation}

The two terms of (5.9) are plotted in figure 7 for $E = 0.01$, $\mathcal {R}\,Pr = 1$ and $T = 1$. Regions of positive horizontal buoyancy gradient are observed for both terms in $M_3^2$, these regions correspond to frontal sharpening due to the cross-front velocity convergence seen in figure 4.

Figure 7. (a) The first term of $M_3^2$ from (5.9). (b) The second term of $M_3^2$ from (5.9). Results are shown for $E = 0.01$, $\mathcal {R}\,Pr = 1$ and $T = 1$.

5.5. Shear dispersive spreading

For $b_0 = b_0(y)$, (4.9) becomes

(5.10)\begin{equation} \frac{\partial b_0}{\partial t} = Ro^2\,Pr\, \frac{\partial }{\partial y}\left[c_3 \left(\frac{\partial b_0}{\partial y}\right)^3+\delta\,c_4 \left(\frac{\partial b_0}{\partial y}\right)^2\frac{\partial ^2b_0}{\partial y^2}\right] \end{equation}

for

(5.11a,b)\begin{equation} c_3(E) = E\,\overline{K'^2} \quad \textrm{and} \quad c_4(E) = 2 \sqrt{E^3}\,\overline{AK}. \end{equation}

This equation is derived using the $\psi _0$ independent terms from $b'_3$ and $\boldsymbol {u}'_{H1}$ and corresponds to the case of weak vorticity generation, $\psi _0 = 0$. Over the long time scale of shear dispersive spreading, $t=O(1/\delta ^4)$, significant vorticity generation is expected. However, the case of $\psi _0 = 0$ is considered here to isolate the effect of the $\psi$ independent terms.

As $\delta \to 0$, (5.10) reduces to the result of Crowe & Taylor (Reference Crowe and Taylor2018) where the front approaches a self-similar solution and spreads as $y \sim t^{1/4}$. This self-similar solution is odd in $y$, hence, the high buoyancy and low buoyancy sides evolve in the same way. However, the $c_4$ term breaks this $y$ symmetry due to an odd number of $y$ derivatives so both sides are expected to evolve differently for non-zero $\delta$. To test this prediction, (5.10) is solved numerically using the Dedalus framework (Burns et al. Reference Burns, Vasil, Oishi, Lecoanet and Brown2020). The units of time are rescaled such that $Ro^2Pr\,c_3 = 1$ leaving $r = \delta c_4/c_3$ as the only free parameter. Simulations are run for $r = 0$ and $r = 0.2$ and initialised using the profile $b_0(t = 0) = \tanh y$. Sixth-order hyperdiffusion with a hyperdiffusivity of $\nu _6 = 3\times 10^{-9}$ is included for numerical stability and simulations are run until $Ro^2Pr\,c_3 t = 10$ using a third-order implicit–explicit Runge–Kutta scheme and a domain of $y \in [-5,5]$ with $N_y = 256$ grid points.

Figure 8 shows the numerical solutions for $b_0$ for $r = 0$ and $r = 0.2$ at a range of times. The difference between these two solutions is plotted in panel (b) allowing the expected asymmetry due to non-traditional effects to be observed. The effect of the non-traditional term is found to be small and of greatest importance near the frontal edges where the curvature, ${\partial ^2b_0}/{\partial y^2}$, is large. Therefore, it is expected that non-traditional effects will not play a significant role in the shear dispersive spreading of a front without the inclusion of strong vorticity generation.

Figure 8. (a) The buoyancy field, $b_0$, from numerically solving (5.10). Solutions are calculated for $r = \delta c_4/c_3 = 0$ (red) and $r = 0.2$ (blue) and shown for $Ro^2Pr\,c_3 t \in [0, 2.5, 5, 7.5, 10]$ with steeper solutions corresponding to earlier times. (b) The difference in $b_0$ between the $r = 0$ results and the $r = 0.2$ results for the same time values as (a).

Over long times, a large amount of vorticity is expected to be generated by both vertical mixing (Crowe & Taylor Reference Crowe and Taylor2019b) and non-traditional effects. This vorticity manifests as depth-independent jets and can act to modify the total vorticity of the system. If this vorticity is sufficiently strong, local vorticity terms can appear in the leading-order TTW balance. These terms will modify the Coriolis force, resulting in a modified depth-dependent velocity and, hence, a modified circulation. Therefore, it is predicted that the generation of vorticity will play a more important role in the evolution of $b_0$ than the $\psi _0$ independent circulation considered here. The effects of large $\psi _0$ could be considered using the approach of Crowe & Taylor (Reference Crowe and Taylor2019b); however, such solutions are expected to be complicated and provide no new insight into the problem so will not be considered here.

6.1. Typical frontal parameters

The small parameter describing non-traditional effects is the ratio of the vertical and horizontal components of the rotation vector scaled by the aspect ratio and is given by

(6.1)\begin{equation} \delta = \frac{H}{L} \frac{1}{\tan \theta} \end{equation}

for latitude $\theta$, layer depth $H$ and typical frontal width $L$. The requirement that $Ro = O(\delta ^2)$ therefore implies that

(6.2)\begin{equation} B \sim H \tilde{f}^2, \end{equation}

where $B$ is the typical buoyancy difference across the front. Taking typical values of $\tilde {f} \approx 10^{-4} \ \textrm {s}^{-1}$ and $H = 100\ \textrm {m}$ gives a buoyancy difference which is much smaller than the typical values of $B\approx 10^{-4}\ \textrm {ms}^{-2}$. Therefore, for this asymptotic regime to hold, the frontal velocities and, hence, the Rossby number would have to be much smaller than would be expected physically.

To determine more physical values of the relevant parameters, note that non-traditional effects are most important in the tropical and subtropical regions where $\tan \theta < 1$. Here, fronts with small horizontal scales, $L \sim 1 \ \textrm {km}$, may have order $1$ values of $\delta$; however, the Rossby number would also be order $1$ for these small-scale fronts. This case of $Ro = O(1)$ and $\delta = O(0.1 - 1)$ is likely to be the most physically relevant regime. Previous studies (Crowe & Taylor Reference Crowe and Taylor2019b) have noted that TTW balance can remain valid even for the case of finite Rossby numbers so numerical simulations are now performed to test if the phenomenon described above are relevant to this regime.

6.2. Numerical simulations for $Ro = 1$

Here, (2.5) is solved subject to no-stress and no-flux boundary conditions using the Dedalus package (Burns et al. Reference Burns, Vasil, Oishi, Lecoanet and Brown2020). These numerical simulations are two dimensional; the cross-front direction is taken to align with the $y$ axis where non-traditional effects are maximised and ${\partial }/{\partial x}$ is set to zero. Fields are expanded in terms of a Fourier basis in the horizontal $(y)$ direction and a Chebyshev basis in the vertical $(z)$ direction and time stepped using a third-order implicit–explicit Runge–Kutta scheme. Horizontal mixing with a viscosity of $10^{-4}$ is included for numerical stability. Simulations are initialised using the $O(1)$ TTW solution for velocity and buoyancy given in § 4 for $b_0 = \tanh y$ and $\psi _0 = 0$.

Figure 9 shows numerical results for $Ro = 0.1$, $\delta = 1$ and $E = 0.1$. Figure 9(a) shows the development of the along-front jets through the growth of the streamfunction of the depth-averaged flow, $\psi$, with time. These profiles for $\psi$ are consistent with the analytical predictions shown in figure 2(a), differing by less than $3\,\%$ from the theory despite the use of an order $1$ value of $\delta$ in an asymptotic expression that is known only to $O(\delta )$. Figure 9(b) shows the $O(\delta )$ component of the streamfunction of the frontal circulation, $\phi _1$, at $t = 2$. Here $\phi _1$ is calculated as the difference between the total value of $\phi$ and the value of $\phi _0$ calculated using (5.6a,b). Again the results are well described by the theory as the structure of these circulation cells can be seen to be a sum of the components shown in figures 4(a) and 4(b). The accuracy of the theoretical predictions for order $1$ values of $\delta$ suggests that the solutions of § 4 are valid, even outside of the asymptotic regime considered.

Figure 9. Results from a numerical simulation with $(Ro,\delta ,E) = (0.1,1,0.1)$ showing (a) the streamfunction, $\psi$, describing the depth-averaged along-front flow and (b) the circulation component $\phi _1$ at $t = 2$.

Figure 10 shows numerical results for $Ro = 1$, $\delta = 1$ and $E = 0.1$. Here, the effects of nonlinearity become significant and it is necessary to separate the effects of finite $Ro$ from the effects of finite $\delta$. This may be done by running another simulation with $(Ro,\delta ,E) = (1,0,0.2)$ and calculating the difference

(6.3)\begin{equation} \Delta \varphi = \varphi|_{\delta = 1} - \varphi|_{\delta = 0} \end{equation}

for some field $\varphi$. Figure 10(a) shows the value of $\Delta \psi$ for a range of values of $t$. Similarly to the case of $Ro = 0.1$, the along-front flow is found to be well described by the theoretical predictions with a difference of around $4\,\%$ between $\Delta \psi$ and the prediction for $\psi _0+\delta \psi _1$. The difference in frontal circulation, $\Delta \phi$, is shown in figure 10(b). While qualitatively similar to theoretical predictions and the case of $Ro = 0.1$, the nonlinearity and non-traditional components appear to interact nonlinearly resulting in some deviation from the predictions. In particular, the centre of the middle circulation cell in figure 10(b) is seen to split in two. Nonetheless, even for cases far outside the asymptotic regime considered analytically, the effects of including non-traditional rotation appear to be qualitatively the same as discussed above, with the generation of along-front jets and a modification of the frontal circulation.

Figure 10. Results from a numerical simulation with $(Ro,\delta ,E) = (1,1,0.1)$ showing (a) the difference in streamfunction, $\Delta \psi$, between the results for $\delta = 1$ and $\delta = 0$, and (b) the difference in frontal circulation, $\Delta \phi ,$ at $t = 2$.