5.1. Self-similar near-field vertical stratification and cross-shore exchange flow

The mean Lagrangian (Stokes + Eulerian) cross-shore velocity $\langle \bar {u}_{L}\rangle$ contributes to exchange between the near-field ($x\gtrapprox -500$ m) and ambient inner-shelf farther offshore, or far-field $x\lessapprox -500$ m. The near field is characterized by reduced vertical stratification, indicated by shoreward broadening isotherms (figure 3d–f), and an overturning streamfunction $\langle \bar {\varPsi }\rangle$ with two counter-circulating cells (figure 3g–i). Over the $24\text {--}48$ h averaging period, the near-field stratification and exchange flow are approximately constant. Here, we evaluate whether this dynamic equilibrium state is self-similar with respect to varying $\mathrm {d}{T_0}/\mathrm {d}{z}$. To account for variations in the cross-shore distribution of $\langle \bar {\varPsi }\rangle$ and $\langle \bar {u}_{L}\rangle$ with $\mathrm {d}{T_0}/\mathrm {d}{z}$, the vertical length scale $d_{s}$ is used to re-define the boundary between near field and far field as $x=-L_{mix}$, where the depth is three times $d_{s}^{48\ \mathrm {h}}$,

(5.1)\begin{equation} h({-}L_{mix}) = 3d_{s}^{48\ \mathrm{h}},\end{equation}

which is equivalent to $L_{mix}=3d_{s}^{48\ \mathrm {h}}/\beta _{SZ}$ for planar beach slope $\beta _{SZ}$. The factor of $3$ reflects that $d_{s}$ is the depth where the $x\ge -L_{mix}$ area average of $T_0(z)$ is equivalent to $T_0(-d_{s})$, see (A1). The cross-shore location $x=-L_{mix}$ roughly tracks the varying cross-shore location of inner-shelf $\langle \bar {\varPsi }\rangle$ maxima for $\mathrm {d}{T_0}/\mathrm {d}{z}\ge 0.05\,^\circ \text {C}\ \text {m}^{-1}$ (triangle, figure 3h,i). As the associated $d_{s}^{48\ \mathrm {h}}$ decreases from $4$ to $2.8$ m (figure 9b), the resulting maximum $h=12$ m ($L_{mix}\approx 6L_{SZ}$) for weakest $N_0$ decreases to $h=8$ m ($L_{mix}\approx 3L_{SZ}$) for highest $N_0$. Thus, the near-field $(x,z)$-area shrinks as stratification increases.

The vertical structures of $\langle \bar {N}\rangle$ and $\langle \bar {u}_{L}\rangle$ are scaled at $x=-L_{mix}$. The mean stratification $\langle \bar {N}\rangle (z)$ is scaled by initial stratification $N_0$ and the non-dimensional depth $z/d_{s}^{48\ \mathrm {h}}$ is used to account for the shifting cross-shore location $L_{mix}$ (figure 11a). The near-field $\langle \bar {u}_{L}\rangle (z)$ flow is related to $\mathrm {d}{T_0}/\mathrm {d}{z}$ using a baroclinic velocity scale (e.g. Manins Reference Manins1976; Maxworthy & Monismith Reference Maxworthy and Monismith1988),

(5.2)\begin{equation} \mathcal{U} = {C}_0 N_0 d_{s}, \end{equation}

with ${C}_0$ analogous to an intrusive gravity current (IGC) Froude number (e.g. Cheong, Kuenen & Linden Reference Cheong, Kuenen and Linden2006; Bolster et al. Reference Bolster, Hang and Linden2008). Using $d_{s}^{48\ \mathrm {h}}$ in (5.2) with constant $C_0$ estimated by least-squares fit to the vertical minimum of $\langle \bar {u}_{L}\rangle (z)$ at $x=-L_{mix}$ (varying in the range $3\text {--}7\ \text {mm s}^{-1}$) yields ${C_0} \approx 0.07$. The $x=-L_{mix}$ mean cross-shore velocity $\langle \bar {u}_{L}\rangle (z)$ is scaled by $\mathcal {U}$ and shown vs non-dimensional depth $z/d_{s}^{48\ \mathrm {h}}$ in figure 11(b).

Figure 11. Non-dimensional mean (a) stratification $\langle \bar {N}\rangle /N_0$ and (b) cross-shore Lagrangian velocity $\langle \bar {u}_{L}\rangle /\mathcal {U}$ vs non-dimensional depth $z/d_{s}^{48\ \mathrm {h}}$ for $\mathrm {d}{T_0}/\mathrm {d}{z} \ge 0.05\,^\circ \text {C}\ \text {m}^{-1}$ at $x=-L_{mix}$ (5.1), which varies from $h=12$ m ($L_{mix}{\approx }6L_{SZ}$) to $h=8$ m ($L_{mix}{\approx }3L_{SZ}$) for the weakest to strongest $N_0$. In (b) the unstratified $\mathrm {d}{T_0}/\mathrm {d}{z}=0$ simulation profile of $\langle u _L \rangle /U$ (dashed black line) is shown for $h(-6L_{SZ})\approx 12$ m with $U = 8 \times 10^{-4}$. (c) Non-dimensional inner-shelf cross-shore exchange velocity $\bar {u}_{EX}/\mathcal {U}$ (5.3) and (5.2) vs shifted non-dimensional cross-shore coordinate $\mathcal {X}_{u}/L_{mix}$ (5.4). In (c), the non-dimensional cross-shore locations of the far-field along-shore velocity maximum $x=-L_{max}$ are shown on top as coloured down triangles.

Non-dimensional mean stratification profiles are self-similar below $z/d_{s}^{48\ \mathrm {h}}\approx -0.35$, as the curves collapse (figure 11a). Sub-surface stratification is reduced for $-1 < z/d_{s}^{48\ \mathrm {h}} < -0.5$, and slightly enhanced for $z/d_{s}^{48\ \mathrm {h}}<-1.5$. The minimum $\langle \bar {N}\rangle /N_0 \approx 0.5$ occurs slightly above $z/d_{s}^{48\ \mathrm {h}}=-1$, consistent with the minimum $\partial _z \Delta T_{s}^{48\ {\rm h}}$ occurring shallower than $d_{s}^{48\ \mathrm {h}}$ (figure 9), potentially due to aliasing of $d_{s}$ time dependence. As $d_{s}$ growth is large until $t\approx 12$ h, fluid exported offshore earlier than $t=24$ h would occupy a depth shallower than $z/d_{s}^{48\ \mathrm {h}}=-1$. Near-surface stratification is also reduced for small $\mathrm {d}{T_0}/\mathrm {d}{z}$, owing to the proximity to where the well-mixed fluid intersects the surface and due to larger near-surface $\langle \bar {K}_{V}\rangle$ for weak $N_0$ (e.g. figure 3a). Below $z/d_{s}^{48\ \mathrm {h}}=-1.5$, stratification is elevated, reaching $\langle \bar {N}\rangle /N_0 \approx 1.25$ at $z/d_{s}^{48\ \mathrm {h}}\approx -2$, consistent with the compressed $\langle \bar {T}\rangle$ isotherms and elevated $\mathrm {r.m.s.}\{T\}$ at depth (figure 4a).

The non-dimensional cross-shore velocity $\langle \bar {u}_{L}\rangle /\mathcal {U}$ is also self-similar for $\mathrm {d}{T_0}/\mathrm {d}{z}\ge 0.05\,^\circ \text {C}\ \text {m}^{-1}$, and has a $3$-layer exchange flow structure with near-surface and bottom onshore flow and offshore flow in between (figure 11b). Near the surface ($z/d_{s}^{48\ \mathrm {h}}>-0.5$), onshore flow is strong, with a maximum $\langle \bar {u}_{L}\rangle /\mathcal {U}\approx 2$. The maximum offshore flow occurs at $z/d_{s}^{48\ \mathrm {h}}\approx -1$, with $\langle \bar {u}_{L}\rangle /\mathcal {U}\approx -1$. Occurrence of the maximum offshore directed $\langle \bar {u}_{L}\rangle /\mathcal {U}$ flow where vertical stratification $\langle \bar {N}\rangle /N_0$ is minimum indicates sub-surface export of relatively low stratification (mixed) fluid, consistent with previous TRC exchange modelling (Kumar & Feddersen Reference Kumar and Feddersen2017c; Grimes et al. Reference Grimes, Feddersen and Kumar2020b) and differential mixing experiment and theory (e.g. Maxworthy & Monismith Reference Maxworthy and Monismith1988). Below $z/d_{s}^{48\ \mathrm {h}}=-1.5$, flow becomes onshore again with a weak maximum $\langle \bar {u}_{L}\rangle /\mathcal {U}\approx 0.25$ at $z/d_{s}^{48\ \mathrm {h}}=-2$. The relative strength of near-surface onshore transport ($z/d_{s}^{48\ \mathrm {h}}>-0.5$) compared to sub-surface onshore transport ($z/d_{s}^{48\ \mathrm {h}}<-1.5$) implies stratified near-surface fluid is the dominant source to the TRC region. The $3$-layer exchange flow for stratified cases is contrasted by the unstratified $2$-layer $\langle \bar {u}_{L}\rangle /\mathcal {U}$ profile at $x=-6L_{SZ}$ (black dashed, figure 11b), which is scaled by $\mathcal {U}=8\times 10^{-4}~\text {m s}^{-1}$ (roughly a factor of 3 smaller than weakest $N_0$) to give an offshore minimum $\langle \bar {u}_{L}\rangle /\mathcal {U}\approx -1$. The unstratified $2$-layer structure is characteristic of Stokes drift and a quasi-uniform undertow velocity profile (e.g. Lentz et al. Reference Lentz, Fewings, Howd, Fredericks and Hathaway2008; Hally-Rosendahl, Feddersen & Guza Reference Hally-Rosendahl, Feddersen and Guza2014). The difference between $2$-layer unstratified and $3$-layer stratified $\langle \bar {u}_{L}\rangle$ demonstrates the different dynamics of the inner-shelf response.

The collapse of stratified equilibrium near-field $\langle \bar {N}\rangle /N_0$ and $\langle \bar {u}_{L}\rangle /\mathcal {U}$ profiles supports a dynamical similarity to IGC. Noting that the intrusion has $\langle \bar {N}\rangle /N_0\approx 1/2$ in (5.2), that is, $\mathcal {U}={C_0}d_{s} N_0/2$, increases the Froude number to ${C_0}\approx 0.13$ which is comparable to energy-conserving IGC from lock-release theory and experiments with ${C_0}\approx 0.25$. Based on dimensional analysis, the near-field velocity scaling $d_{s} N_0$ applies for both inertia and turbulent stress dominated exit flow, with the latter having ${C_0} \ll 1$, as observed in a laboratory experiment (Maxworthy & Monismith Reference Maxworthy and Monismith1988). The estimated advective speed $\mathcal {U} \approx 0.07 (N_0 d_{s})$ is small relative to the background mode-1 internal wave phase speed $\approx N_0 d_{s}$ (e.g. White & Helfrich Reference White and Helfrich2008), suggesting radiated internal waves may contribute to the gravitational adjustment of the TRC-mixed fluid. Also, since the time averages span $t=24\text {--}48$ h, after an inertial period ($\,f^{-1}$), the cross-shore speed may be reduced owing to the potential vorticity dynamics, as suggested by mean along-shore velocity $\langle \bar {v}\rangle$ (e.g. Thomas & Linden Reference Thomas and Linden2007). The inner-shelf eddy activity in the TRC region $(x>-4L_{SZ})$ suggests viscous effects are non-negligible.

With the non-dimensional vertical structure at $x=-L_{mix}$ well scaled, the cross-shore variation in baroclinic circulation $\langle \bar {\varPsi }\rangle$ with initial stratification is evaluated using a cross-shore exchange velocity based on the mean cross-shore velocity $\langle \bar {u}_{L}\rangle$ (e.g. Grimes et al. Reference Grimes, Feddersen and Kumar2020b),

(5.3)\begin{equation} {u}_{EX}(x) = \dfrac{-2}{h+\langle\bar{\eta}\rangle}\int_{{-}h}^{\langle\bar{\eta}\rangle} \langle\bar{u}_{L}\rangle^{-}(x,z) \,\mathrm{d}z,\end{equation}

where the vertical integral is restricted to offshore directed flow, indicated with superscript $(^{-})$. For a bimodal vertical $\langle \bar {\varPsi }\rangle (z)$, the offshore transport quantified with the vertical integral in (5.3) is equivalent to the difference between $\langle \bar {\varPsi }\rangle$ vertical extrema, ${max (\langle \bar {\varPsi }\rangle )-{min}(\langle \bar {\varPsi }\rangle )}$. Cross-shore exchange velocity $u_{EX}$ is a commonly used metric to evaluate cross-shore exchange potential (Suanda & Feddersen Reference Suanda and Feddersen2015; Kumar & Feddersen Reference Kumar and Feddersen2017a,Reference Kumar and Feddersenb). For $\mathrm {d}{T_0}/\mathrm {d}{z}\gtrapprox 0.05\,^\circ \text {C}\ \text {m}^{-1}$, $u_{EX}(x)$ has a local inner-shelf maximum near $x=-L_{mix}$. A shifted non-dimensional near-field cross-shore coordinate is introduced,

(5.4)\begin{equation} \mathcal{X}_{u}/L_{mix} = (x+L_{mix})/L_{mix},\end{equation}

such that $\mathcal {X}_{u}>0$ is the near field and $\mathcal {X}_{u}<0$ is the far field (figure 11c).

The non-dimensional $u_{EX}/\mathcal {U}$ is self-similar with non-dimensional $\mathcal {X}_{u}/L_{mix}$ (figure 11c), indicating that $d_{s}$ is the fundamental vertical and horizontal (through $L_{mix}$) length scale describing the near-field distribution of $\langle \bar {\varPsi }\rangle$. Maxima of $u_{EX}/\mathcal {U}$ generally occur in the range $0<\mathcal {X}_{u}/L_{mix}< 0.2$ with magnitude $\approx 1/2$. There is a local near-field exchange velocity minimum at $\mathcal {X}_{u}/L_{mix}\approx 0.5$, owing to closed inner-shelf and surf-zone $\langle \bar {\varPsi }\rangle$ streamlines, partially due to the strong barotropic dominance of TRCs in this region (Kumar & Feddersen Reference Kumar and Feddersen2017b). Offshore of the inner-shelf maximum, in the far-field $(\mathcal {X}_{u}/L_{mix}<0)$, the horizontal decay length scale weakly increases relative to $L_{mix}$ with increasing stratification, suggesting an additional far-field horizontal length scale that increases with $\mathrm {d}{T_0}/\mathrm {d}{z}$. Far-field circulation is evaluated in the next section.

Cross-shore exchange flows are common in many oceanographic settings, and either maintain or modify cross-shore tracer distributions (e.g. buoyancy, or pollutants). An alternative means of quantifying exchange flow strength is with a flushing time scale $(\mathcal {T}_{f})$, a bulk measure of the transport time needed to fill a control volume (e.g. Lemagie & Lerczak Reference Lemagie and Lerczak2015). A near-field flushing time scale is estimated as the ratio of the cross-sectional near-field area, roughly $(9/2) d_{s}^2/\beta _{SZ}$ with bottom slope $\beta _{SZ}=0.025$, to the offshore directed transport, approximately $(d_{s} \mathcal {U})$, giving,

(5.5)\begin{equation} \mathcal{T}_{f} \sim \frac{9}{2} \frac{d_{s}^2/\beta_{SZ}}{d_{s}({C_0}{N_0}d_{s})} = \frac{9}{2} (\beta_{SZ} {C_0} N_0)^{{-}1}. \end{equation}

Thus, increasing $\mathrm {d}{T_0}/\mathrm {d}{z}$ (larger $N_0$) decreases the flushing time $\mathcal {T}_{f}$ due to enhanced $\mathcal {U}$ and decreased near-field area (smaller $d_{s}$). Using modelled $d_{s}^{48\ \mathrm {h}}$, $\mathcal {T}_f$ ranges from $5~\text {d}$ (smallest $N_0$) to $0.93~\text {d}$ (largest $N_0$). For reference, the unstratified $2$-layer exchange flow requires $\approx 8~\text {d}$ to flush the near field.

5.2. Self-similar far-field cross-shore stratification and along-shore thermal wind

The strength of along-shore current $\langle \bar {v}\rangle$ increases with $\mathrm {d}{T_0}/\mathrm {d}{z}$ (figure 3d–f), forming a far-field ($x<-L_{mix}$) sub-surface positive along-shore current roughly centred between the broadening $\langle \bar {T}\rangle$ isotherms (figure 3d–f). Kumar & Feddersen (Reference Kumar and Feddersen2017b) found that, offshore of $\approx 4.5L_{SZ}$, along-shore flow reached an approximate thermal wind balance,

(5.6)\begin{equation} \dfrac{\partial{\langle\bar{v}_{g}\rangle}}{\partial{z}} = \dfrac{g\alpha}{f}\dfrac{\partial{\langle\bar{T}\rangle}}{\partial{x}}\equiv \dfrac{\langle\bar{M}^2\rangle}{f}, \end{equation}

where $\langle \bar {v}_g\rangle$ is the geostrophic component of along-shore velocity and $\langle \bar {M}^2\rangle$ the mean horizontal stratification. As the far-field intrusion has mean isotherms that slope upward (downward) above (below) $z=-d_{s}^{48\ \mathrm {h}}$ (e.g. figure 10a), thermal wind relation (5.6) also implies a sub-surface $\langle \bar {v}_g\rangle$ maximum, consistent with figure 3(j–l). A geostrophic horizontal length scale, based on the Rossby deformation radius ($L_{R}$) and zero-crossing depth ($d_{s}$),

(5.7)\begin{equation} L_{R} = d_{s} N_0/f, \end{equation}

is used to scale the far-field response. In contrast to $L_{mix}$, which decreases weakly with increasing $\mathrm {d}{T_0}/\mathrm {d}{z}$, the magnitude of $L_{R}$ (5.7) using $d_{s}^{48\ \mathrm {h}}$ increases with $\mathrm {d}{T_0}/\mathrm {d}{z}$, ranging from $\approx 260$ m to $\approx 1300$ m. The far-field location of maximum mean along-shore velocity, denoted $x=-L_{max}$, is well represented by,

(5.8)\begin{equation} L_{max} = {C_{mix}}L_{mix} + {C_{R}}L_{R}, \end{equation}

with constant ${C_{mix}} = 1.23$ and ${C_{R}} = 0.3$, and varies between $L_{max}=800$ and $900$ m (not shown). Vertical profiles of $\langle \bar {M}^2\rangle (z)$ and $\langle \bar {v}\rangle (z)$ at $x=-L_{max}$ are used to evaluate the structure and scale of the far-field response.

An along-shore velocity $\langle \bar {v}\rangle$ scaling based on the thermal wind relation (5.6) is formed by relating the far-field horizontal temperature gradient to the initial stratification using the geostrophically balanced isotherm slope scaling (Lentz & Chapman Reference Lentz and Chapman2004),

(5.9)\begin{equation} \mathrm{d}z/\mathrm{d} x=\langle\bar{M}^2\rangle/\langle\bar{N}^2\rangle\sim f/N_0, \end{equation}

and assuming self-similar far-field vertical stratification $\langle \bar {N}\rangle \propto N_0$, such that,

(5.10)\begin{equation} \langle\bar{M}^2\rangle = {C_1} f N_0. \end{equation}

The isotherm slope scaling (5.9) implies a local far-field intrusion aspect ratio proportional to $d_{s}/L_{R}$. Thus, $L_{R}$ scales the horizontal distance over which far-field isotherms broaden. A geostrophic velocity scale $\mathcal {V}$ is formed by substituting (5.10) in (5.6) and assuming $d_{s}$ parameterizes the vertical scale over which $\langle \bar {v}\rangle$ increases to $\mathcal {V}$ giving,

(5.11)\begin{equation} \mathcal{V} \approx {C}_1 d_{s} N_0, \end{equation}

with constant ${C_1}=0.26$ determined using $d_{s}^{48\ \mathrm {h}}$ in (5.11) through least-squares fit to the $z$-maximum $\langle \bar {v}\rangle$ at $x=-L_{max}$.

Non-dimensional mean horizontal stratifications $\langle \bar {M}^2\rangle /({C_1}f N_0)$ at $x=-L_{max}$ generally collapse vs non-dimensional depth $z/d_{s}^{48\ \mathrm {h}}$ (figure 12a). Near the surface, $\langle \bar {M}^2\rangle /({C_1}f N_0)$ is negative and has an ${O}(1)$ local minimum at $z/d_{s}^{48\ \mathrm {h}}\approx -0.3$. Over $-2 < z/d_{s}^{48\ \mathrm {h}} < -0.75$, $\langle \bar {M}^2\rangle /({C_1}f N_0)$ is positive, with an ${O}(1)$ local maximum at $z/d_{s}^{48\ \mathrm {h}}\approx -0.8$. Below $z/d_{s}^{48\ \mathrm {h}}=-2$, $\langle \bar {M}^2\rangle /({C_1}f N_0)$ becomes weakly negative again. Using the near-field minimum stratification $\langle \bar {N}\rangle /N_0\approx 1/2$ in (5.9) and (5.10) gives a theoretical estimate for ${C_1}=1/4$ with extrema of $\langle \bar {M}^2\rangle /({C_1}f N_0)\approx \pm 1$, consistent with the fit ${C_1}=0.26$ and the magnitude of non-dimensional extrema (figure 12a). The collapse is poorer for the weakest stratification runs with $\mathrm {d}{T_0}/\mathrm {d}{z}\le 0.1\,^\circ \text {C}\ \text {m}^{-1}$. Non-dimensional mean along-shore velocity $\langle \bar {v}\rangle /\mathcal {V}$ at $x=-L_{max}$ also collapses (figure 12b), with vertical slope generally of the same sign as $\langle \bar {M}^2\rangle$, consistent with the thermal wind relation (5.6). There is a weak near-surface $\langle \bar {v}\rangle /\mathcal {V}$ minimum at $z/d_{s}^{48\ \mathrm {h}}\approx -0.25$. Below $z/d_{s}^{48\ \mathrm {h}}\approx -0.75$, $\langle \bar {v}\rangle /\mathcal {V}$ becomes positive and has a sub-surface ${O}(1)$ maximum at roughly $z/d_{s}^{48\ \mathrm {h}}=-1$ and then decreases again as $\langle \bar {M}^2\rangle$ becomes positive. For $z/d_{s}^{48\ \mathrm {h}}<-1.75$, $\langle \bar {v}\rangle /\mathcal {V}$ is again weakly negative and tends toward zero as $z/d_{s}^{48\ \mathrm {h}}$ decreases below $-3$. Consistency of $\langle \bar {M}^2\rangle /({C_1}f N_0)$ and $\langle \bar {v}\rangle /\mathcal {V}$ indicates the far-field mean along-shore velocity has reached an approximate geostrophic balance with the mean cross-shore horizontal stratification, with $x=-L_{max}$ isotherm slopes well parameterized by $\mathrm {d}z/\mathrm {d} x\sim f/N_0$ (5.9).

Figure 12. (a) Non-dimensional mean horizontal stratification $\langle \bar {M}^2\rangle /({C_1} f N_0)$ (5.10) and (b) non-dimensional mean along-shore velocity $\langle \bar {v}\rangle /\mathcal {V}$ (5.11) vs non-dimensional depth $z/d_{s}^{48\ \mathrm {h}}$ for $\mathrm {d}{T_0}/\mathrm {d}{z}{\ge }0.05\,^\circ \text {C}\ \text {m}^{-1}$ at $x=-L_{max}$ (5.8) which varies from $x=-800$ to $-900$ m. (b) Non-dimensional inner-shelf zero-crossing depth mean along-shore velocity $v_{ZC}/\mathcal {V}$ (5.12) and (5.11) vs shifted non-dimensional cross-shore coordinate $\mathcal {X}_{v}/L_{R}$ (5.13). In (c), the non-dimensional cross-shore location of the near-field/far-field division $x=-L_{mix}$ is shown on top as coloured up-triangles.

The far-field cross-shore structure of $\langle \bar {v}\rangle$ is evaluated using a cross-shore transect at the zero-crossing depth, denoted,

(5.12)\begin{equation} v_{ZC}(x) = \langle\bar{v}\rangle(x,-d_{s}^{48\ \mathrm{h}}). \end{equation}

Based on the isotherm slope scaling (5.9) at $x=-L_{max}$, a far-field shifted non-dimensional cross-shore coordinate is introduced,

(5.13)\begin{equation} \mathcal{X}_{v}/L_{R} = (x+L_{max})/L_{R}, \end{equation}

where $\mathcal {X}_{v}=0$ is the maximum along-shore current location. Non-dimensional $v_{ZC}/\mathcal {V}$ generally collapses using $\mathcal {X}_{v}/L_{R}$ (figure 12c), particularly for $\mathcal {X}_{v}/L_{R}<0$. The non-dimensional zero-crossing depth velocity $v_{ZC}/\mathcal {V}$ decays to near zero by $\mathcal {X}_{v}/L_{R}\approx -2$, indicating $L_{R}$ accurately scales the offshore decay of the geostrophic current. Moving onshore from $\mathcal {X}_{v}/L_{R}=0$, toward the near field at $\mathcal {X}_{u}=0$ (up-triangles, top of figure 12c), the curves separate with lower stratification having larger shoreward decay scale, consistent with transition to a near-field horizontal scale characterized by $L_{mix}$. Thus, inner-shelf response to TRC mixing is determined by two length scales, $L_{mix}\sim d_{s}/\beta _{SZ}$ in the near field, and Rossby deformation radius $L_{R}\sim d _{s} N_0/f$ in the far field, both based on the fundamental vertical scale $d_{s}$. The $L_{R}$ dependence suggests that the offshore extent of the far-field intrusion is larger at lower latitudes (smaller $f$). As the intrusion growth is constrained by the near-field exchange flow (e.g. Maxworthy & Monismith Reference Maxworthy and Monismith1988), the time scale for the far-field intrusion to reach length $L_{R}$ roughly scales as $L_{R}/\mathcal {U}\sim f^{-1}$, indicating that, at lower latitudes, geostrophic balance may not occur over the $24\text {--}48$ h time scale considered here.

5.3. Scaling the domain-integrated potential energy growth rate

As the surf-zone to inner-shelf kinetic energy flux ${F}_{k'}^{(sz)}$ is independent of $N_0$, the near-field mean exchange flow and stratification scalings are used to relate the continuous TRC-region mixing to the domain-integrated potential energy growth rate $\langle \partial {}_t{E}_{p}^{(x,z)}\rangle$, or equivalently $\gamma _{E}$ (4.11). The scaling analysis assumes that the near-field exchange flow $\mathcal {U}$, zero-crossing depth $d_{s}$ and stratification $N$ are constant, implying that $\langle \partial {}_t {E}_{p}^{(x,z)} \rangle$ is representative of the increase in far-field $(\mathcal {X}_{u}<0)$-integrated ${E}_{p}^{(x,z)}$. With this assumption, and noting that far-field vertical mixing is relatively weak (figure 3a–c), $\langle \partial {}_t {E}_{p}^{(x,z)} \rangle$ can be approximated with a cross-shore advective buoyancy flux between the near-field and far-field regions, or,

(5.14)\begin{equation} \langle \partial{}_t {E}_{p}^{(x,z)} \rangle \sim \varrho \int_{{-}3d_{s}}^{0} z' \mathcal{B}(z') \,\mathrm{d}z', \end{equation}

where $\mathcal {B}(z)=-g\varrho ^{-1}\langle \overline {u \rho }\rangle$ is the vertically varying mean cross-shore buoyancy flux at $\mathcal {X}_{u}=0$. Leveraging the self-similar vertical stratification and cross-shore exchange flow, the buoyancy flux ($\mathcal {B}$) is approximated as constant,

(5.15)\begin{equation} \mathcal{B} \sim \mathcal{U} d_{s} N_{0}^2, \end{equation}

where $(d_{s} N_{0}^2)$ parameterizes the buoyancy anomaly between onshore and offshore directed transports. Lastly, scaling the vertical integral of $(z' \mathrm {d}z')$ in (5.14) with $d_{s}^2$ gives,

(5.16)\begin{align} \langle \partial{}_t {E}_{p}^{(x,z)} \rangle & \sim \varrho \mathcal{U} d_{s}^3 N_{0}^2, \nonumber\\ &\propto \varrho d_{s}^4 N_{0}^3. \end{align}

Using modelled $d_{s}^{48\ \mathrm {h}}$ and dimensionless proportionality constant $0.05$ in (5.16), scaling estimates of $\gamma _{E}$ generally reproduce the modelled $\mathrm {d}{T_0}/\mathrm {d}{z}$ dependence (crosses, figure 8), particularly for $\mathrm {d}{T_0}/\mathrm {d}{z}\ge 0.1\,^\circ \text {C}\ \text {m}^{-1}$. The consistency of scaling (5.16) supports the assumption of steady near-field $\mathcal {U}$ and $d_{s}$, and dynamic equilibrium between the cross-shore buoyancy flux at $x=-L_{mix}$ (5.15) and the near-field mixing. Conceptually, the exchange flow, scaled by $\mathcal {U}$, continuously supplies the TRC region with stratified fluid from offshore, such that, when mixed, the cross-shore pressure gradient and near-field available potential energy (figure 10) are maintained. Thus, the surf-zone and TRC mixing-driven circulation is largely self-sustaining, provided sufficient supply of stratified fluid to maintain it. Consequently, the approximately geostrophic far-field intrusion slowly grows throughout the averaging period. At very long time scales, it is not clear how the far-field dynamics impacts the near-field exchange flow.