A Scale-Aware Parameterization of Restratification Effect of Turbulent Thermal Wind Balance

Peiran Yang; Zhao Jing; Haiyuan Yang; Lixin Wu

doi:10.1175/JPO-D-23-0169.1

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Numerical experiments
3. A scale-aware parameterization of the VBF induced by the TTW balance
4. Discussion
5. Conclusions
Acknowledgments.
Data availability statement.
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Fig. 1.

Snapshots of the normalized relative vertical vorticity ς/f in the (a) NP27, (b) KE9, (c) KE3, and (d) KE1. The model domain of the KE9, KE3, and KE1 is enclosed by black lines in (a), and the region used for validating the parameterization is enclosed by black lines in (d).

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Fig. 2.

Vertical profiles of model-resolved (a) turbulent vertical viscosity $\bar{K_{m}}$ and (b) squared horizontal density gradient ${(\bar{\nabla_{H} b})}^{2}$ averaged over 150°–166°E, 33°–42°N from 1 Dec 2003 to 31 Mar 2004 in the NP27 (red), KE9 (blue), KE3 (orange), and KE1 (black).

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Fig. 3.

The squared horizontal density gradient averaged within the mixed layer ${(\bar{\nabla_{H} b^{z}})}^{2}$ over 150°–166°E, 33°–42°N from 1 Dec 2003 to 31 Mar 2004 as a function of Δs⁻¹. The blue line denotes the linear regression line derived from samples in the KE3, KE9, and NP27.

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Fig. 4.

(a) Probability density function (PDF) of the mixed layer deformation radius $\bar{L_{f}}$ and (b) bin-averaged $\bar{K_{m}}$ at its peaking depth as a function of ${(\bar{\nabla_{H} b^{z}})}^{2}$ conditional on $\bar{L_{f}} < 1 km$ (dashed) and $\bar{L_{f}} \geq 1 km$ (solid) in the KE1.

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Fig. 5.

Vertical profiles of the (a) model-resolved VBF under the TTW balance $\bar{B_{f}^{TTW}}$ averaged over 150°–166°E, 33°–42°N from 1 Dec 2003 to 31 Mar 2004 in the KE1, KE3, KE9, and NP27. (b)–(d) As in (a), but for $\bar{B_{f}^{TTW}}$ , the parameterized VBF $B_{f}^{{TTW}^{'}}$ , and their sum $B_{f}^{TTW} = \bar{B_{f}^{TTW}} + B_{f}^{{TTW}^{'}}$ in the KE1-P, KE3-P, KE9-P, and NP27-P, respectively. The dashed lines in (c) show $B_{f}^{{TTW}^{'}}$ computed offline based on the model output in the KE1, KE3, KE9, and NP27.

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Fig. 6.

(a) Temporal variations of the surface buoyancy loss (black) and wind stress (blue) averaged over 150°–166°E, 33°–42°N. Temporal and vertical variations of (b) horizontally averaged squared buoyancy frequency $\bar{N^{2}}$ in the KE1 and its ratio to that in the (c) NP27 and (d) NP27-P, respectively.

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Fig. 7.

Vertical profiles of $\bar{K_{m}} {(\bar{\nabla_{H} b})}^{2}$ averaged over 150°–166°E, 33°–42°N from 1 Dec 2003 to 31 Mar 2004 in the KE1-P (black), KE3-P (orange), KE9-P (blue), and NP27-P (red). The dashed lines show their counterparts in the KE1, KE3, KE9, and NP27.

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Fig. 8.

Vertical profiles of (a) $⟨ \bar{K_{m}} {(\nabla_{H} b)}^{2} ⟩$ , (b) $⟨ \bar{K_{m}} ⟩ ⟨ {(\nabla_{H} b)}^{2} ⟩$ , (c) $⟨ \bar{K_{m}} ⟩$ , (d) 〈(∇_Hb)²〉, (e) the covariance, (f) the correlation coefficient r between (∇_Hb)² and $\bar{K_{m}}$ , and (g),(h) the standard deviation of (∇_Hb)² and $\bar{K_{m}}$ in the KE1-P (black), KE3-P (orange), KE9-P (blue), and NP27-P (red). The angle brackets denote the average over 150°–166°E, 33°–42°N from 1 Dec 2003 to 31 Mar 2004. The (∇_Hb)² is derived from the scaled ${(\bar{\nabla_{H} b})}^{2}$ based on Eq. (8).

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Fig. 9.

(a) Vertical profiles of $⟨ B_{f}^{TTW} ⟩$ in the KE1-MY25-P (black) and $⟨ \bar{B_{f}^{TTW}} ⟩$ (dashed blue) and $⟨ B_{f}^{TTW} ⟩$ (solid blue) in the KE9-MY25-P. (b) Vertical profiles of $⟨ \bar{K_{m}} ⟩$ in the KE1-MY25-P (black) and KE9-MY25-P (blue). (c),(d) As in (a) and (b), but for the simulations using the KPP scheme.

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Editorial Type:: Article

Article Type:: Research Article

A Scale-Aware Parameterization of Restratification Effect of Turbulent Thermal Wind Balance

Peiran Yang

Peiran Yang ^aLaoshan Laboratory, Qingdao, China
^bFrontiers Science Center for Deep Ocean Multispheres and Earth System and Key Laboratory of Physical Oceanography, Ocean University of China, Qingdao, China

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Zhao Jing

Zhao Jing ^aLaoshan Laboratory, Qingdao, China
^bFrontiers Science Center for Deep Ocean Multispheres and Earth System and Key Laboratory of Physical Oceanography, Ocean University of China, Qingdao, China

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Haiyuan Yang

Haiyuan Yang ^aLaoshan Laboratory, Qingdao, China
^bFrontiers Science Center for Deep Ocean Multispheres and Earth System and Key Laboratory of Physical Oceanography, Ocean University of China, Qingdao, China

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Lixin Wu

Lixin Wu ^aLaoshan Laboratory, Qingdao, China
^bFrontiers Science Center for Deep Ocean Multispheres and Earth System and Key Laboratory of Physical Oceanography, Ocean University of China, Qingdao, China

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Online Publication:: 21 May 2024

Print Publication:: 01 May 2024

DOI:: https://doi.org/10.1175/JPO-D-23-0169.1

Page(s):: 1169–1181

Received:: 23 Aug 2023

Final Form:: 26 Feb 2024

Accepted:: 04 Mar 2024

Published Online:: 21 May 2024

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

The vertical buoyancy flux B_f under the turbulent thermal wind (TTW) balance $B_{f}^{TTW}$ plays an important role in restratifying the surface mixed layer in winter. So far, most of the global ocean models are too coarse to resolve this process. In this paper, a scale-aware parameterization is proposed for $B_{f}^{TTW}$ and implemented in a hierarchy of regional ocean simulations over the winter Kuroshio Extension with horizontal resolutions ranging from 27 to 1 km. The parameterization depends on the Coriolis parameter, model-simulated turbulent vertical viscosity, horizontal density gradient, and a scaling relationship to adjust for the effects of model horizontal resolution on the simulated horizontal density gradient. It shows good skills in reconciling the difference between $B_{f}^{TTW}$ in the coarse-resolution simulations (27, 9, and 3 km) and in the 1-km simulation where $B_{f}^{TTW}$ is well resolved. Furthermore, implementation of the parameterization improves the simulated stratification in the surface mixed layer in coarse-resolution simulations.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Zhao Jing, jingzhao198763@sina.com

Abstract

Corresponding author: Zhao Jing, jingzhao198763@sina.com

Keywords: Ocean models; Parameterization; Small scale processes; Oceanic mixed layer

1. Introduction

The current generation of climate models shows significant biases in simulating the sea surface temperature and mixed layer depth that are key to the air–sea interactions (Fox-Kemper et al. 2021; Huang et al. 2014). One possible cause of such biases is the incapability of resolving oceanic small-scale processes at O(0.1–100) km due to the coarse model resolution. By generating a prominent restratifying vertical buoyancy flux (VBF), these small-scale processes act to warm the sea surface and shallow the surface mixed layer (Boccaletti et al. 2007; Callies and Ferrari 2018; Fox-Kemper et al. 2011; Jing et al. 2020; Su et al. 2018). A reliable parameterization of their restratification effect is thus crucial to improve the fidelity of climate simulations without increasing the model resolution and computational burden.

The mixed layer instability (MLI), frontogenesis, symmetric instability (SI), and turbulent thermal wind (TTW) balance are considered the major mechanisms for the VBF of small-scale processes in the surface mixed layer (Mahadevan and Tandon 2006; McWilliams 2016; Yang et al. 2021a). The MLI is a type of baroclinic instability occurring in the surface mixed layer (Boccaletti et al. 2007; Fox-Kemper et al. 2008), converting the available potential energy stored in the mixed layer fronts to kinetic energy. The frontogenesis occurs at an intensifying front caused by background confluent flows, inducing an ageostrophic secondary circulation (ASC) with upwelling and downwelling on the lighter and denser side of the front, respectively (Barkan et al. 2019; Hoskins 1982; McWilliams 2021). The SI is triggered if the Ertel potential vorticity (PV) has an opposite sign to the planetary vorticity and tends to occur in the surface frontal regions when the PV is extracted from the ocean due to the surface buoyancy loss (D’Asaro et al. 2011; Jing et al. 2021; Thomas 2005; Thomas et al. 2013). In readjusting to a symmetrically neutral state, the SI restratifies the surface mixed layer with the magnitude of its VBF related to the surface buoyancy loss (Bachman et al. 2017; Taylor and Ferrari 2010). The TTW balance is a linear momentum balance between Coriolis force, horizontal pressure gradient, and turbulent vertical mixing (Gula et al. 2014; Wenegrat and McPhaden 2016; Yang et al. 2021a). The viscous effect destroys the vertical shear of the front, inducing a restoring ASC that has a similar pattern to that of the frontogenesis (Gula et al. 2014; McWilliams 2016; McWilliams et al. 2015).

Parameterizations of the VBF induced by the MLI, frontogenesis, and SI have been proposed (Bachman et al. 2017; Fox-Kemper et al. 2011; Zhang et al. 2023). Among them, the parameterization of the MLI (Bodner et al. 2023; Fox-Kemper et al. 2011) is relatively more mature and has been widely applied in climate simulations (Calvert et al. 2020; Chang et al. 2020; Fox-Kemper et al. 2011; Griffies et al. 2015; G. Xu et al. 2022). This parameterization is scale aware and formulated as an overturning streamfunction related to the horizontal density gradient in the surface mixed layer scaled by model resolution, the mixed layer depth, and the Coriolis frequency (Bodner et al. 2023; Fox-Kemper et al. 2011, 2008; Fox-Kemper and Ferrari 2008). However, the parameterization of the VBF of the TTW balance is still lacking. Yet, a numerical study in the winter Kuroshio Extension (Yang et al. 2021a) suggests that the VBF of the TTW balance is comparable to the sum of those induced by the MLI and frontogenesis. In this study, we aim to propose a scale-aware parameterization for the VBF of the TTW balance and test its performance in a set of ocean simulations in the winter Kuroshio Extension with horizontal resolutions varying from 1 to 27 km. The VBF of the TTW balance is well resolved in the 1-km simulation, where the 27 km is close to the highest resolution affordable for long-term global climate simulations nowadays (Haarsma et al. 2016; Hewitt et al. 2020). The manuscript is organized as follows. The configuration of numerical models is described in section 2. Section 3 proposes the parameterization and evaluates its performance. Discussion on the parameterization is given in section 4 followed by conclusions in section 5.

2. Numerical experiments

To evaluate the performance of the parameterization for the VBF of the TTW balance, the Regional Ocean Modeling System (ROMS) (Haidvogel et al. 2000; Shchepetkin and McWilliams 2005) is used to configure a series of offline-nested simulations over the North Pacific focused on the Kuroshio Extension. The lowest-resolution simulation, named NP27, has a modeling domain over the North Pacific (99°–270°E, 3.6°–66°N; Fig. 1a), with a horizontal resolution of ∼27 km and 50 terrain-following vertical levels. The initial state and boundary conditions for the NP27 are obtained from the Simple Ocean Data Assimilation (SODA; Carton et al. 2018). The simulation is initialized on 1 January 1998 and integrated to 31 March 2004, outputting 3-hourly snapshots of temperature, salinity, three-dimensional velocity, and turbulent vertical viscosity as well as daily averaged diagnostic terms in momentum and tracer equations.

Fig. 1.

Citation: Journal of Physical Oceanography 54, 5; 10.1175/JPO-D-23-0169.1

The finer simulations in the hierarchy at ∼9-, ∼3-, and ∼1-km resolution are configured over the Kuroshio Extension (named KE9, KE3, and KE1 hereinafter). The domains cover 142.5°–173°E, 27°–46°N for the KE9, 147°–170°E, 29°–44.5°N for the KE3, and 148°–168°E, 31°–43.5°N for the KE1, shrinking with the increasing resolution (Figs. 1b–d). The vertical levels of the KE9 are the same as those of the NP27, while the KE3 and KE1 have 65 vertical levels with ∼5-m grid size in the upper 100 m. The initial states and the one-way nested open boundary conditions for each KE simulation are obtained from the 3-hourly snapshots of its immediate parent simulation. After the spinup of their immediate parent simulations, the KE9, KE3, and KE1 are initialized on 1 January, 1 July, and 1 October 2003, respectively. All these three simulations are run until 31 March 2004.

In all the simulations, the horizontal and vertical advections for momentum are discretized using a third-order upwind scheme and fourth-order central difference scheme, respectively. The multidimensional positive definite advection transport algorithm (MPDATA; Margolin and Smolarkiewicz 1998) is used for the horizontal and vertical advection of tracers. We use the K-profile parameterization (KPP) turbulent mixing closure scheme for the vertical mixing of momentum and tracers (Large et al. 1994) and a biharmonic horizontal Smagorinsky-like mixing scheme for momentum (Griffies and Hallberg 2000; Smagorinsky 1963). A biharmonic horizontal mixing scheme of tracers is used in the NP27 with diffusivity set as −1 × 10¹⁰ m⁴ s⁻¹, whereas no horizontal mixing is used for tracers in the rest simulations. In all the simulations, the atmospheric forcing is calculated from a bulk formula based on the 6-hourly Climate Forecast System Reanalysis (CFSR; Saha et al. 2010), and the sea surface salinity is restored to that in the SODA with a nudging time scale of 10 days. In addition to the 3-hourly snapshots, all the KE simulations output the 3-hourly averaged diagnostic terms in momentum and tracer equations. The above model configurations are summarized in Table 1.

Table 1.

A summary of numerical configurations of the NP27, KE9, KE3, and KE1.

Zhang et al. (2023) reported that the VBF of the TTW balance has strong seasonality with its value in summer one order of magnitude smaller than that in winter. Therefore, we focus on the performance of the parameterization in winter, based on the model output from 1 December 2003 to 31 March 2004 in an inner domain of the KE1, i.e., 150°–166°E, 33°–42°N (Fig. 1d).

3. A scale-aware parameterization of the VBF induced by the TTW balance

a. VBF induced by the TTW balance

The governing equation of the TTW balance (Gula et al. 2014)^¹ is

v_{a} \times k = - \frac{\partial}{\partial z} (\frac{K_{m}}{f} \frac{\partial v_{a} + v_{g}}{\partial z}),

(1a)

with the boundary conditions

ρ_{0} K_{m} {(\frac{\partial v_{g}}{\partial z} + \frac{\partial v_{a}}{\partial z}) |}_{z = 0} = τ_{w},

(1b)

ρ_{0} K_{m} {(\frac{\partial v_{g}}{\partial z} + \frac{\partial v_{a}}{\partial z}) |}_{z = - \infty} = 0,

(1c)

where k is the unit vector directed upward, v_g = (u_g, υ_g) is the geostrophic velocity and v_a = (u_a, υ_a) is the ageostrophic horizontal velocity under the TTW balance, f is the Coriolis frequency, ρ₀ is the reference density set as 1025 kg m⁻³, K_m is the turbulent vertical viscosity parameterized by the KPP scheme (Large et al. 1994), and τ_w = (τ_w_,_x, τ_w_,_y) is the surface wind stress. The lower boundary condition Eq. (1c) is posed as a no-stress condition following Cronin and Kessler (2009) and Wenegrat and McPhaden (2016).

Equation (1) models the ASCs driven by both τ_w and the geostrophic stress τ_g = K_m∂u_g/∂z (Cronin and Kessler 2009; F. Xu et al. 2022; Yang et al. 2021b). The former can generate a positive VBF via the current feedback (Dawe and Thompson 2006; Duhaut and Straub 2006), but its magnitude is generally much smaller than that induced by τ_g at least in winter in the Kuroshio Extension region (F. Xu et al. 2022; Yang et al. 2021b). In the following parameterization, we focus only on the VBF induced by τ_g by setting τ_w = 0.

The vertical velocity is derived based on the continuity equation under the rigid-lid and f-plane approximations as

w = - \nabla_{H} \times (\frac{K_{m}}{f} \frac{\partial v_{a} + v_{g}}{\partial z}) \cdot k,

(2)

with ∇_H = (∂/∂x, ∂/∂y). The three-dimensional ageostrophic velocity u_a = (v_a, w) can be written in terms of a vector streamfunction Ψ_TTW as u_a = ∇ × Ψ_TTW with ∇ = (∇_H, ∂/∂z). One of the streamfunctions satisfying Eq. (1) is in the form of

Ψ_{TTW} = - \frac{K_{m}}{f} \frac{\partial v_{a} + v_{g}}{\partial z} .

(3)

Accordingly, Eq. (1) can be re-expressed to form an equation of Ψ_TTW as

\frac{K_{m}}{f} \frac{\partial^{2} Ψ_{TTW}}{\partial z^{2}} + Ψ_{TTW} \times k = - \frac{K_{m}}{f^{2}} \nabla_{H} b,

(4a)

with the boundary conditions

{Ψ_{TTW} |}_{z = 0} = 0,

(4b)

{Ψ_{TTW} |}_{z = - \infty} = 0 .

(4c)

The VBF of the TTW balance,

B_{f}^{TTW} = w b

, can be expressed in the form of the vertical component of a skew flux plus a residue term:

B_{f}^{TTW} = (Ψ_{TTW} \times \nabla_{H} b) \cdot k + \nabla_{H} \times (Ψ_{TTW} b) \cdot k,

(5)

where b = −g(ρ − ρ₀)/ρ₀ is the buoyancy with g as the gravitational acceleration. On the one hand, the first term on the right-hand side of Eq. (5) is always positive and peaks within the surface mixed layer, capturing the restratifying effect (F. Xu et al. 2022). On the other hand, the second term is negligible in terms of its horizontal average over a region (Yang et al. 2021b). In the following analysis, we will focus on the first term only and parameterize its effect in coarse-resolution ocean models.

Decompose each variable into the component resolved (simulated) by a numerical model (denoted by overbars) and unresolved component (denoted by primes) whose effects need to be parameterized. The VBF induced by the unresolved anomalies under the TTW balance, i.e.,

B_{f}^{{TTW}^{'}} = w^{'} b^{'}

, is

B_{f}^{{TTW}^{'}} = B_{f}^{TTW} - \bar{B_{f}^{TTW}} = Ψ_{TTW} \times \nabla_{H} b \cdot k - \bar{Ψ_{TTW}} \times \bar{\nabla_{H} b} \cdot k .

(6)

Parameterizing

B_{f}^{{TTW}^{'}}

in a coarse-resolution ocean model is to estimate

B_{f}^{TTW}

based on the model-resolved quantities. As indicated by Eq. (4a), Ψ_TTW depends linearly on the nonhomogeneous term f⁻²K_m∇_Hb. Therefore,

B_{f}^{TTW}

is proportional to

B_{f}^{TTW} \propto f^{- 2} K_{m} {(\nabla_{H} b)}^{2} .

(7)

Equation (7), consistent with the scaling of

B_{f}^{TTW}

proposed by McWilliams (2017), suggests that a scale-aware parameterization should take into consideration the potential dependence of K_m and (∇_Hb)² on the horizontal model grid size Δs. Numerical simulations in the Kuroshio Extension reveal that the magnitude of

\bar{K_{m}}

varies a little for Δs ranging from 1 to 27 km (Fig. 2a) so that K_m can be approximated as

\bar{K_{m}}

. The insensitivity of

\bar{K_{m}}

to Δs is expected because

\bar{K_{m}}

within the surface boundary layer parameterized by the KPP scheme is largely determined by the atmospheric forcing (Large et al. 1994) that is the same across all the simulations.

Fig. 2.

Citation: Journal of Physical Oceanography 54, 5; 10.1175/JPO-D-23-0169.1

The magnitude of ${(\bar{\nabla_{H} b})}^{2}$ shows a substantial decrease in magnitude as Δs becomes coarser (Fig. 2b). Such a decrease reflects that the simulated width of mixed layer fronts is limited by Δs rather than physical processes in coarse-resolution simulations. A scaling of ${(\bar{\nabla_{H} b})}^{2}$ is thus necessary for coarse-resolution simulations and will be detailed in the following subsection.

b. Scaling of the horizontal density gradient

The above analysis suggests that a scale-aware parameterization of $B_{f}^{{TTW}^{'}}$ has to adjust for the effect of Δs on ${(\bar{\nabla_{H} b})}^{2}$ . As b is almost uniform vertically within the surface mixed layer, we use the horizontal gradient of the vertical mean b over the surface mixed layer ${(\bar{\nabla_{H} b^{z}})}^{2}$ as a measurement of the front intensity, where the superscript z means the vertical average within the surface mixed layer. In this study, the surface mixed layer depth H is defined following Large et al. (1997) as the shallowest depth where the local interpolated stratification N² matches the maximum vertically averaged N² between the surface and any discrete depth within that water column. Figure 3 compares ${(\bar{\nabla_{H} b^{z}})}^{2}$ averaged over 150°–166°E, 33°–42°N from 1 December 2003 to 31 March 2004, denoted by $⟨ {(\bar{\nabla_{H} b^{z}})}^{2} ⟩$ hereinafter, in the different simulations. The value of $⟨ {(\bar{\nabla_{H} b^{z}})}^{2} ⟩$ changes approximately linearly with Δs⁻¹ in the NP27, KE9, and KE3 (Fig. 3), whereas $⟨ {(\bar{\nabla_{H} b^{z}})}^{2} ⟩$ in the KE1 is only 63% of that extrapolated from the linear regression line, implying that the mixed layer fronts in the KE1 should be largely resolved and not strongly limited by Δs.

Fig. 3.

Citation: Journal of Physical Oceanography 54, 5; 10.1175/JPO-D-23-0169.1

Fox-Kemper et al. (2011) suggest that the width of mixed layer fronts could be estimated as the mixed layer deformation radius L_f = N^zH/f with N^z as the mean buoyancy frequency over the surface mixed layer. We use the scaling of

{(\bar{\nabla_{H} b})}^{2}

proposed by Fox-Kemper et al. (2011) to relate (∇_Hb)² to

{(\bar{\nabla_{H} b})}^{2}

in the surface mixed layer, i.e.,

{(\nabla_{H} b)}^{2} = \max (\frac{Δ s}{\bar{L_{f}}}, 1) {(\bar{\nabla_{H} b})}^{2} .

(8)

It should be noted that even the KE1 is still too coarse to resolve the SI and its associated restratification effect. This may make

\bar{N^{z}}

bias low under strong surface cooling and further cause overly small

\bar{L_{f}}

. To remedy this shortage, a mixed layer deformation radius after the SI adjustment to an O(1) Richardson number state (Tandon and Garrett 1994) is imposed as a lower bound of

\bar{L_{f}}

, i.e.,

\bar{L_{f}} \geq | \nabla_{H} b^{z} | H / f^{2}

, where H is approximated as

\bar{H}

and |∇_Hb^z| is related to

| \bar{\nabla_{H} b^{z}} |

through Eq. (8). This leads to the lower bound expressed as

\bar{L_{f}} \geq {(| \bar{\nabla_{H} b^{z}} | \bar{H} f^{- 2})}^{2 / 3} Δ s^{1 / 3}

In the winter Kuroshio Extension, $\bar{L_{f}}$ is generally around or larger than 1 km (Fig. 4a), consistent with the observed values (Dong et al. 2020). Accordingly, 〈(∇_Hb^z)²〉 in the KE1 is only 16% larger than $⟨ {(\bar{\nabla_{H} b^{z}})}^{2} ⟩$ (Table 2a), lending supports that the mixed layer fronts are largely resolved in the KE1. The slight difference between 〈(∇_Hb^z)²〉 and $⟨ {(\bar{\nabla_{H} b^{z}})}^{2} ⟩$ in the KE1 is primarily due to the small $\bar{L_{f}}$ (<1 km) in the highly stratified frontal regions associated with small $\bar{K_{m}}$ (Fig. 4b). As $B_{f}^{TTW} \propto f^{- 2} K_{m} {(\nabla_{H} b)}^{2}$ , the difference between (∇_Hb^z)² and ${(\bar{\nabla_{H} b^{z}})}^{2}$ in these regions has negligible impacts on $⟨ B_{f}^{TTW} ⟩$ , providing us confidence that $⟨ B_{f}^{TTW} ⟩$ should be well resolved in the KE1. To further verify this point, we conduct another simulation with ∼0.3-km resolution (denoted as KE0.3; see Text S1 in the online supplemental material for configuration details). The mean values of $⟨ {(\bar{\nabla_{H} b^{z}})}^{2} ⟩$ do not show a significant difference between KE0.3 and KE1, nor do the mean values of $\bar{B_{f}^{TTW}}$ (see Fig. S1).

Fig. 4.

Citation: Journal of Physical Oceanography 54, 5; 10.1175/JPO-D-23-0169.1

Table 2.

A list of variables averaged over 150°–166°E, 33°–42°N from 1 Dec 2003 to 31 Mar 2004 in the (a) NP27, KE9, KE3, and KE1 and (b) NP27-P, KE9-P, KE3-P, and KE1-P. The peaking value of $\bar{K_{m}}$ , $\bar{B_{f}^{TTW}}$ , and $B_{f}^{TTW}$ is shown with their peaking depth in parentheses.

The scaling Eq. (8) works reasonably well in reconciling the differences of

{(\bar{\nabla_{H} b})}^{2}

across the simulations with different Δs. Specifically, 〈(∇_Hb^z)²〉 in the NP27, KE9, and KE3 are close to that in the KE1 (Table 2a), differing by 24%, 17%, and 15%, respectively. Based on Eqs. (6) and (8), we propose a scale-aware parameterization of

B_{f}^{{TTW}^{'}}

B_{f}^{{TTW}^{'}} = \max (\frac{Δ s}{\bar{L_{f}}} - 1, 0) \bar{Ψ_{TTW}} \times \bar{\nabla_{H} b} \cdot k,

(9)

where

\bar{Ψ_{TTW}}

can be solved from Eq. (4) by replacing ∇_Hb and K_m with

\bar{\nabla_{H} b}

and

\bar{K_{m}}

. The presence of the maximum function on the right-hand side of Eq. (9) reflects the scale-aware nature of the parameterization. In particular, the parameterization is activated only at locations with

Δ s > \bar{L_{f}}

c. Implementation of the parameterization in a numerical model

To implement the parameterization equation [Eq. (9)] in a numerical model like ROMS, we derive $\bar{\nabla_{H} b}$ from the model-simulated horizontal pressure gradient under the hydrostatic approximation, i.e., $\bar{\nabla_{H} b} = ρ_{0}^{- 1} \partial \bar{\nabla_{H} p} / \partial z$ . The lower boundary for the computation of Eq. (4c) is moved from infinity to 5 times the surface boundary layer depth diagnosed from the KPP scheme or the sea floor (h), whichever is shallower. Such a lower boundary is chosen because K_m is negligible at this level compared to its peaking value within the surface boundary layer (Fig. S2). As the lower boundary condition may become problematic in the coastal region, the application of the parameterization is confined to the open ocean. For this reason, the value of $\bar{Ψ_{TTW}}$ is set as 0 for h smaller than 1000 m and multiplied by a factor of (h − 1000)/500 for h ranging from 1000 to 1500 m. However, this nudging has no influence on $B_{f}^{{TTW}^{'}}$ in our analysis domain (150°–166°E, 33°–42°N) where h is larger than 4000 m.

In ROMS and other ocean general circulation models, b is not a prognostic variable but diagnosed from potential temperature θ and salinity S based on the equation of state. It can be easily shown that parameterizations of the vertical flux of θ and S by unresolved anomalies under the TTW balance are

w^{'} θ^{'} = \max (\frac{Δ s}{\bar{L_{f}}} - 1, 0) \bar{Ψ_{TTW}} \times \bar{\nabla_{H} θ} \cdot k,

(10)

w^{'} S^{'} = \max (\frac{Δ s}{\bar{L_{f}}} - 1, 0) \bar{Ψ_{TTW}} \times \bar{\nabla_{H} S} \cdot k,

(11)

where

Δ s = \sqrt{(Δ x^{2} + Δ y^{2}) / 2}

with Δx (Δy) as the zonal (meridional) model grid size. Once w′θ′ and w′S′ are available,

B_{f}^{{TTW}^{'}}

can be derived as

B_{f}^{{TTW}^{'}} = g (\bar{α} w^{'} θ^{'} - \bar{β} w^{'} S^{'})

by assuming b′ is a linear function of θ′ and S′, i.e.,

b^{'} = g \bar{α} θ^{'} - g \bar{β} S^{'}

, where

\bar{α}

and

\bar{β}

are the thermal expansion and haline contraction coefficients, respectively.

d. Performance of the parameterization in numerical simulations with different resolutions

1) Parameterized VBF under the TTW balance

Equations (10) and (11) are implemented online in the KE1, KE3, KE9, and NP27, costing about 2.7% of the total computational time with the potential for further optimizations. To distinguish numerical simulations with and without the parameterizations, we add a suffix “P” to the former (e.g., NP27-P vs NP27). In the numerical simulations without the parameterization, the difference of $\bar{B_{f}^{TTW}}$ is substantial. Its peaking value increases and peaking depth shoals as Δs becomes finer (Fig. 5a and Table 2a). The $⟨ \bar{B_{f}^{TTW}} ⟩$ peaks around 41 m with a value of 2.62 × 10⁻⁸ m² s⁻³ in the KE1, whereas it peaks around 64 m with a value of 0.67 × 10⁻⁸ m² s⁻³ in the NP27 (Table 2a).

Fig. 5.

Citation: Journal of Physical Oceanography 54, 5; 10.1175/JPO-D-23-0169.1

The $⟨ B_{f}^{{TTW}^{'}} ⟩$ is negligible compared to $⟨ \bar{B_{f}^{TTW}} ⟩$ in the KE1-P due to the resolving of the mixed layer fronts but becomes more dominant as Δs becomes coarser (Figs. 5b,c). The sum of $⟨ B_{f}^{{TTW}^{'}} ⟩$ and $⟨ \bar{B_{f}^{TTW}} ⟩$ , i.e., $⟨ B_{f}^{TTW} ⟩$ agrees well with each other in the KE1-P, KE3-P, and KE9-P (Fig. 5d). The values of $⟨ B_{f}^{TTW} ⟩$ differ by less than 5% at their peaking depths (Table 2b). The parameterization leads to a moderate overestimation of $⟨ B_{f}^{TTW} ⟩$ in the NP27-P, with its peaking value about 32% larger than that in the KE1-P (Fig. 5d and Table 2b). The reasons for this overestimation will be discussed in section 4a. Nevertheless, the relative difference of $⟨ B_{f}^{TTW} ⟩$ between the KE1-P and NP27-P is much smaller compared to that of $⟨ \bar{B_{f}^{TTW}} ⟩$ between the KE1 and NP27, suggesting that implementing the parameterization improves the representation of the VBF under the TTW balance.

2) Restratification effect of the parameterization

The primary role of $B_{f}^{TTW}$ is to restratify the mixed layer. To test the restratification effect of the parameterization, we compare $\bar{N^{2}}$ in the NP27 and NP27-P with that in the KE1 ( $\bar{N^{2}}$ in the KE1 and KE1-P are almost the same). In the KE1, there is a downward extension of weak $\bar{N^{2}}$ of O(10⁻⁶) s⁻² from the sea surface to the ocean interior from December to February (Fig. 6b), as a result of surface buoyancy loss and wind stirring (Fig. 6a). This destratification effect is much more evident in the NP27 than in the KE1 due to the lack of restratified VBF induced by the small-scale processes in the surface mixed layer. The value of horizontal mean $\bar{N^{2}}$ in the surface mixed layer in the KE1 can reach 4–12 times of that in the NP27 (Fig. 6c). With the parameterization implemented, the NP27-P shows an improved simulation of $\bar{N^{2}}$ in the surface mixed layer. The horizontal mean $\bar{N^{2}}$ in the surface mixed layer in the KE1 and NP27-P generally agree within a factor of 4 (Fig. 6d). Nevertheless, the horizontal mean $\bar{N^{2}}$ in the surface mixed layer in the NP27-P still biases low. This may be partially due to the missing restratification effects by the MLI, frontogenesis, and SI.

Fig. 6.

Citation: Journal of Physical Oceanography 54, 5; 10.1175/JPO-D-23-0169.1

3) Interaction between resolved and parameterized VBFs under the TTW balance

There is a significant interaction between $B_{f}^{{TTW}^{'}}$ and $\bar{B_{f}^{TTW}}$ . On the one hand, $⟨ \bar{B_{f}^{TTW}} ⟩$ in the KE3, KE9, and NP27 are systematically larger than their counterparts in the KE3-P, KE9-P and NP27-P (Figs. 5a,b; Table 2). On the other hand, $⟨ B_{f}^{{TTW}^{'}} ⟩$ computed offline in the KE3, KE9, and NP27 are also larger than those in the KE3-P, KE9-P, and NP27-P (Fig. 5c; Table 2). Such differences become more evident as Δs becomes coarser. Therefore, there is negative feedback of the implementation of the parameterization on both $⟨ B_{f}^{{TTW}^{'}} ⟩$ and $⟨ \bar{B_{f}^{TTW}} ⟩$ . This negative feedback can be understood based on the restratification effect of the parameterization which acts to reduce ${(\bar{\nabla_{H} b})}^{2}$ and $\bar{K_{m}}$ (Table 2). Accordingly, implementing the parameterization leads to a reduction of $⟨ \bar{K_{m}} {(\bar{\nabla_{H} b})}^{2} ⟩$ (Fig. 7) that further weakens $⟨ B_{f}^{{TTW}^{'}} ⟩$ and $⟨ \bar{B_{f}^{TTW}} ⟩$ . The above findings suggest the necessity for online tests of the performance of the parameterization.

Fig. 7.

Citation: Journal of Physical Oceanography 54, 5; 10.1175/JPO-D-23-0169.1

4. Discussion

a. The overestimation of the parameterization in coarse-resolution simulation

In the NP27-P, the parameterization overestimates the VBF induced by the unresolved small-scale processes under the TTW balance. As

⟨ B_{f}^{TTW} ⟩ \propto ⟨ K_{m} {(\nabla_{H} b)}^{2} ⟩

, the larger

⟨ B_{f}^{TTW} ⟩

in the NP27-P than in the KE1-P should be due to the larger

⟨ \bar{K_{m}} ⟩

, 〈(∇_Hb)²〉 computed from

⟨ {(\bar{\nabla_{H} b})}^{2} ⟩

through Eq. (8), or their covariance, i.e.,

⟨ B_{f}^{TTW} ⟩ \propto ⟨ \bar{K_{m}} ⟩ ⟨ {(\nabla_{H} b)}^{2} ⟩ + r std (\bar{K_{m}}) std [{(\nabla_{H} b)}^{2}]

(12)

where std represents the operator of standard deviation and r represents the correlation coefficient between

\bar{K_{m}}

and (∇_Hb)².

The first term on the right-hand side of Eq. (12) has similar values between the NP27-P and KE1-P (Fig. 8b) because both 〈(∇_Hb)²〉 and $⟨ \bar{K_{m}} ⟩$ in the KE1-P are close to their counterparts in the NP27-P (Figs. 8c,d). Therefore, the scaling of ${(\bar{\nabla_{H} b})}^{2}$ works reasonably well in representing the spatiotemporal mean value of (∇_Hb)² and cannot account for the overestimation of $⟨ \bar{K_{m}} {(\nabla_{H} b)}^{2} ⟩$ and $⟨ B_{f}^{TTW} ⟩$ in the NP27-P (Figs. 5d and 8a). In contrast, the second term on the right-hand side of Eq. (12) differs evidently between the NP27-P and KE1-P and is responsible for the overestimated $⟨ B_{f}^{TTW} ⟩$ in the NP27-P (Fig. 8e). Its value is negative and partially offsets the contribution to $⟨ B_{f}^{TTW} ⟩$ by the first term. The negative correlation between $\bar{K_{m}}$ and (∇_Hb)² is due to the strong restratifying VBF in the frontal regions (Yang et al. 2021b) and makes $⟨ B_{f}^{TTW} ⟩$ not only depend on 〈(∇_Hb)²〉 but also on std[(∇_Hb)²] and $std (\bar{K_{m}})$ . Although the values of $std (\bar{K_{m}})$ between the KE1-P and NP27-P are close to each other (Fig. 8g), the value of std[(∇_Hb)²] in the NP27-P in the mixed layer is only 33% of that in the KE1-P (Fig. 8h). This overly small std[(∇_Hb)²] reduces the offset effect of the second term in Eq. (12) and is responsible for the overestimation of $⟨ B_{f}^{TTW} ⟩$ in the NP27-P. The effect of std[(∇_Hb)²] bias is partly canceled by the larger negative value of r in the NP27-P than in the KE1-P (Fig. 8f), so that the relative difference of the second term in Eq. (12) between the NP27-P and KE1-P is smaller than that inferred from the difference of std[(∇_Hb)²] alone. It should be noted that similar biases in std[(∇_Hb)²] and r also exist in the KE9-P and KE3-P but become less evident with the decreasing Δs.

Fig. 8.

Citation: Journal of Physical Oceanography 54, 5; 10.1175/JPO-D-23-0169.1

The above analysis indicates that the scaling of ${(\bar{\nabla_{H} b})}^{2}$ used in this study fails to capture the variance and possibly higher-order statistics of (∇_Hb)², causing biases in $⟨ B_{f}^{TTW} ⟩$ . We remark that the deficiency of the scaling in representing the variance of (∇_Hb)² is likely to bias the parameterized VBF induced by the MLI $B_{f}^{MLI}$ (Fox-Kemper et al. 2011) in a similar way. The $B_{f}^{MLI}$ depends on the product of H² and (∇_Hb)². As H² is negatively correlated to (∇_Hb)² due to the restratifying VBF in the frontal region, the overly small std[(∇_Hb)²] in coarse-resolution simulations should lead to the overestimation of $⟨ B_{f}^{MLI} ⟩$ . This is confirmed by the offline computation of $⟨ B_{f}^{MLI} ⟩$ that increases as Δs becomes coarser (Fig. S3) and is consistent with the findings in the previous studies (Calvert et al. 2020; Griffies et al. 2015).

b. Interactions between the parameterized VBF induced by the TTW balance and MLI instability

By decomposing the VBF induced by different submesoscale dynamical processes (including the MLI and TTW balance), Yang et al. (2021a) find that the MLI- and TTW-induced VBFs coexist and have comparable magnitudes in the winter Kuroshio Extension. However, the MLI- and TTW-induced VBFs are not independent and are likely to interact with each other and so are their parameterizations. These two parameterizations are likely to exert negative feedback on each other, as both the parameterizations act to restratify the surface mixed layer. On the one hand, including the parameterization of TTW-induced VBF shoals the surface mixed layer, diminishing the parameterized MLI-induced VBF as it is proportional to the square of surface mixed layer depth (Fox-Kemper et al. 2008, 2011). On the other hand, including the parameterization of MLI-induced VBF enhances the stratification, reducing the turbulent vertical viscosity and parameterized TTW-induced VBF.

c. Effect of the turbulent vertical mixing schemes on the parameterization

In this study, the parameterization is implemented in simulations with the turbulent vertical mixing parameterized by the KPP scheme. Technically, it can also be implemented in simulations with a generic length scale (GLS) closure scheme, as long as the Reynolds stress can be expressed in the form of K_m ∂u/∂z, where K_m varies smoothly with depth.

For example, the Mellor–Yamada level 2.5 (MY25) scheme is a widely used GLS scheme in ocean and climate models (Mellor and Yamada 1974). We conduct another 1- and 9-km resolution simulation using the MY25 scheme with the TTW parameterization implemented (KE1-MY25-P and KE9-MY25-P). The KE1-MY25-P and KE9-MY25-P share the same configurations with the KE1-P and KE9-P, respectively, except for the turbulent vertical mixing scheme. As shown in Figs. 9a and 9c, our parameterization works qualitatively well for both the KPP and MY25 schemes.

Fig. 9.

Citation: Journal of Physical Oceanography 54, 5; 10.1175/JPO-D-23-0169.1

Quantitatively, the performance of our parameterization depends on the turbulent vertical mixing scheme. The magnitude of $⟨ B_{f}^{TTW} ⟩$ in the KE9-MY25-P is about 17% larger than that in the KE1-MY25-P at the peaking depth, whereas the difference is less than 2% when the KPP scheme is used (Fig. 9c). The quantitative difference of $⟨ B_{f}^{TTW} ⟩$ between the KE9-MY25-P and KE1-MY25-P is primarily due to different K_m between these two simulations (Figs. 9a,b). In our parameterization, K_m is assumed to be invariant with respect to the model grid size, so that no scaling is applied to K_m to reconcile the difference of K_m among simulations with different grid sizes. This assumption is approximately valid for the KPP scheme (Fig. 9d) but less so for the MY25 scheme (Fig. 9b). This is because K_m from the KPP scheme is largely controlled by sea surface forcing, whereas K_m from the MY25 scheme depends mainly on the vertical shear and stratification, both becoming stronger in the KE1-MY25-P than in the KE9-MY25-P.

d. Width of mixed layer fronts

In this study, the width of mixed layer fronts is estimated as the mixed layer deformation radius following Fox-Kemper et al. (2011). Recently, Bodner et al. (2023) propose another formula to estimate the width of mixed layer fronts, by taking into consideration the arresting effect of turbulent flux on the front width under the TTW balance. Application of Bodner et al.’s (2023) formula to our simulations suggests that the mixed layer fronts can be as sharp as O(100) m, an order of magnitude smaller than the mixed layer deformation radius in the winter Kuroshio Extension. Bodner et al.’s (2023) formula implies that $⟨ {(\bar{\nabla_{H} b^{z}})}^{2} ⟩$ would not converge unless the model resolution increases to O(100) m or finer, which contradicts the fact that the values of $⟨ {(\bar{\nabla_{H} b^{z}})}^{2} ⟩$ between the KE1 and KE0.3 are close to each other (see Fig. S1). However, this contradiction may result from several aspects and should not be treated as conclusive evidence of the superiority of mixed layer deformation over Bodner et al.’s (2023) formula as an estimate of the mixed layer front width in the real ocean. On the one hand, the arrested front width proposed by Bodner et al. (2023) is an estimate of the minimum width a mixed layer front can reach under the TTW balance. As the mixed layer fronts are constantly generated and damped, it is likely that many of the mixed layer fronts do not reach the limiting stage of their width. On the other hand, the convergence between $⟨ {(\bar{\nabla_{H} b^{z}})}^{2} ⟩$ in the KE1 and KE0.3 could be a numerical effect of model configurations, such as the hydrostatic approximation (Mahadevan 2006) and diffusive nature of MPDATA scheme (Margolin and Smolarkiewicz 1998). In a word, although the mixed layer deformation radius is likely to be a more reasonable estimate of the mixed layer front width than Bodner et al.’s (2023) formula in simulations of ocean general circulation models like ROMS, it is not necessarily so in the real ocean.

5. Conclusions

In this study, we propose a scale-aware parameterization for the VBF induced by the TTW balance $B_{f}^{TTW}$ . The scheme is implemented in a hierarchy of ocean simulations with a horizontal resolution of 27, 9, 3, and 1 km in the winter Kuroshio Extension where $B_{f}^{TTW}$ is resolved in the 1-km simulation. The major conclusions are summarized as follows:

The parameterization depends on the Coriolis parameter, model-resolved turbulent vertical viscosity, and horizontal density gradient. It employs a scaling relationship as a function of horizontal model resolution and mixed layer deformation radius to adjust for the effect of horizontal model resolution on the simulated width of mixed layer fronts. In particular, the parameterization is automatically inactivated if the mixed layer fronts are resolved in the models.
The parameterization shows good skills in reconciling the difference of $B_{f}^{TTW}$ across models with different horizontal resolutions. Their peaking values of $⟨ B_{f}^{TTW} ⟩$ differ by less than 32% for horizontal resolutions ranging from 27 to 1 km. Implementing the parameterization leads to a better representation of the stratification within the surface mixed layer in coarse-resolution simulations.

There is a moderate but noticeable overestimation of $⟨ B_{f}^{TTW} ⟩$ in the 27-km simulation by the parameterization. This overestimation is primarily due to the deficiency in the scaling of the front intensity. Although the scaling used in this study performs well in representing the spatiotemporal mean value of the front intensity, 〈(∇_Hb)²〉, it fails to capture its variance and higher-order statistics. Yet, the true $⟨ B_{f}^{TTW} ⟩$ does not only depend on 〈(∇_Hb)²〉 but also on the standard deviation of front intensity, std[(∇_Hb)²]. A new scaling of the front intensity able to represent both 〈(∇_Hb)²〉 and std[(∇_Hb)²] is thus crucial to improve the performance of the parameterization and will be addressed in a future study.

Acknowledgments.

This research is supported by the National Key Research and Development Program of China (2022YFC3104801), National Natural Science Foundation of China (42206024 and 42176218), Science and Technology Innovation Project of Laoshan Laboratory (LSKJ202202501), and computational resources are supported by Laoshan Laboratory (LSKJ202300302). Supports are also obtained from Taishan Scholar Funds (tsqn201909052). The model simulation and many of the computations were executed at the High Performance Computing Center of Laoshan Laboratory.

Data availability statement.

The simulation data used in this work can be reproduced using the ROMS code available at https://www.myroms.org/projects/src/wiki.

Please note that Eq. (1a) does not appear in the same form as Eq. (12) of Gula et al. (2014). The latter can be derived by taking the vertical derivative of Eq. (1a).

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Numerical experiments
3. A scale-aware parameterization of the VBF induced by the TTW balance
4. Discussion
5. Conclusions
Acknowledgments.
Data availability statement.
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A Scale-Aware Parameterization of Restratification Effect of Turbulent Thermal Wind Balance

Abstract

Abstract

1. Introduction

2. Numerical experiments

3. A scale-aware parameterization of the VBF induced by the TTW balance

a. VBF induced by the TTW balance

b. Scaling of the horizontal density gradient

c. Implementation of the parameterization in a numerical model

d. Performance of the parameterization in numerical simulations with different resolutions

1) Parameterized VBF under the TTW balance

2) Restratification effect of the parameterization

3) Interaction between resolved and parameterized VBFs under the TTW balance

4. Discussion

a. The overestimation of the parameterization in coarse-resolution simulation

b. Interactions between the parameterized VBF induced by the TTW balance and MLI instability

c. Effect of the turbulent vertical mixing schemes on the parameterization

d. Width of mixed layer fronts

5. Conclusions

Acknowledgments.

Data availability statement.

REFERENCES

Supplementary Materials

Share Link

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A Scale-Aware Parameterization of Restratification Effect of Turbulent Thermal Wind Balance

Abstract

Abstract

1. Introduction

2. Numerical experiments

3. A scale-aware parameterization of the VBF induced by the TTW balance

a. VBF induced by the TTW balance

b. Scaling of the horizontal density gradient

c. Implementation of the parameterization in a numerical model

d. Performance of the parameterization in numerical simulations with different resolutions

1) Parameterized VBF under the TTW balance

2) Restratification effect of the parameterization

3) Interaction between resolved and parameterized VBFs under the TTW balance

4. Discussion

a. The overestimation of the parameterization in coarse-resolution simulation

b. Interactions between the parameterized VBF induced by the TTW balance and MLI instability

c. Effect of the turbulent vertical mixing schemes on the parameterization

d. Width of mixed layer fronts

5. Conclusions

Acknowledgments.

Data availability statement.

REFERENCES

Supplementary Materials

Share Link

Headings

References

Figures

Cited By

Metrics

Related Content

AMS Publications

Get Involved with AMS

Affiliate Sites

Follow Us

Contact Us