
The budget equation for the ocean heat content (Liu et al., 2010) can be expressed in a general form as
$$ \dfrac{\partial \left({C}_{p}\rho T\right)}{\partial t}=\nabla \left({C}_{p}\rho {\boldsymbol{v}}T\right)+{F}_{\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{c}}+R , $$ (1) where
$\boldsymbol{v}$ is the threedimensional velocity vector of the ocean current,$ \rho $ is the seawater density,${C}_{p}$ is the specific heat of seawater in the upper ocean [4100 J (kg °C)^{−1}], and the water temperature T is in degrees Celsius. Here,$ {F}_{\mathrm{Forc}} $ is the external forcing, which is the downward heat flux from the atmosphere, and the residual term R includes heat diffusion and subgridscale processes.For the upperocean in the Niño3.4 region, the volume average of Eq. (1) can be expanded as
$$ \begin{aligned}[b] & \dfrac{1}{V}\iiint \dfrac{\partial \left({C}_{p}\rho T\right)}{\partial t}{\rm{d}}x{\rm{d}}y{\rm{d}}z=\dfrac{1}{V}\iiint \{{C}_{p}[\dfrac{\partial \left(\rho uT\right)}{\partial x}\\ & \quad\quad + \dfrac{\partial \left(\rho vT\right)}{\partial y}+\dfrac{\partial \left(\rho wT\right)}{\partial z}]+{F}_{\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{c}}+R\}{\rm{d}}x{\rm{d}}y{\rm{d}}z , \end{aligned}$$ (2) where V =
$\iiint {\rm{d}}x{\rm{d}}y{\rm{d}}z$ , and the lefthand side (LHS) of Eq. (2) represents the change in ocean heat over time per unit volume (W m^{−3}). The first term on the righthand side (RHS) of Eq. (2) represents the volumeaveraged total net heat transport through all surfaces of the closed volume.The ocean density in the equations mentioned above varies both over time and over the grid, but the magnitude of this variation is miniscule. Comparing the heat transport terms calculated by using a varying density and the regionally averaged density
$ {\rho }_{0} $ , as shown in Fig. S1 in the supplementary material, reveals that the difference in the values of the terms does not exceed 1%. Then, Eq. (2) can be approximately expressed as$$\begin{aligned}[b] \dfrac{1}{V}{C}_{p}{\rho }_{0} & \iiint \dfrac{\partial T}{\partial t}{\rm{d}}x{\rm{d}}y{\rm{d}}z\approx \dfrac{1}{V}\iiint \{{C}_{p}{\rho }_{0}[\dfrac{\partial \left(uT\right)}{\partial x}\\ & + \dfrac{\partial \left(vT\right)}{\partial y}+\dfrac{\partial \left(wT\right)}{\partial z}]+{F}_{\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{c}}+R\}{\rm{d}}x{\rm{d}}y{\rm{d}}z . \end{aligned}$$ (3) Positive values for u, v, and w are assumed to represent eastward, northward, and upward, respectively. Then the first and second terms on the RHS of Eq. (3) can be expressed as
$$\begin{aligned}[b] {F}_{\mathrm{E}\mathrm{W}} & \equiv \dfrac{1}{V}\iiint [{C}_{p}{\rho }_{0}\dfrac{\partial \left(uT\right)}{\partial x}]{\rm{d}}x{\rm{d}}y{\rm{d}}z\\ & = {C}_{p}{\rho }_{0}\dfrac{\iint {\left[\right(uT)}_{{\rm{E}}}{\left(uT\right)}_{{\rm{W}}}]{\rm{d}}y{\rm{d}}z}{V} , \end{aligned} $$ (4) $$ \begin{aligned}[b] {F}_{\mathrm{N}\mathrm{S}} & \equiv \dfrac{1}{V}\iiint [{C}_{p}{\rho }_{0}\dfrac{\partial \left(vT\right)}{\partial y}]{\rm{d}}x{\rm{d}}y{\rm{d}}z\\ & = {C}_{p}{\rho }_{0}\dfrac{\iint {\left[\right(vT)}_{{\rm{N}}}{\left(vT\right)}_{{\rm{S}}}]{\rm{d}}x{\rm{d}}z}{V} , \end{aligned}$$ (5) $$ \begin{aligned}[b] {F}_{\mathrm{T}\mathrm{B}} &\equiv \dfrac{1}{V}\iiint [{C}_{p}{\rho }_{0}\dfrac{\partial \left(wT\right)}{\partial z}]{\rm{d}}x{\rm{d}}y{\rm{d}}z\\ & ={C}_{p}{\rho }_{0}\dfrac{\iint {\left[\right(wT)}_{{\rm{T}}}{\left(wT\right)}_{{\rm{B}}}]{\rm{d}}x{\rm{d}}y}{V} . \end{aligned}$$ (6) $$ {F}_{Q}\equiv \dfrac{1}{V}\iiint {F}_{\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{c}}{\rm{d}}x{\rm{d}}y{\rm{d}}z=\dfrac{{Q}_{\mathrm{n}\mathrm{e}\mathrm{t}}{S}_{{\rm{w}}}}{h} . $$ (7) Here,
$ {F}_{\mathrm{E}\mathrm{W}} $ ,$ {F}_{\mathrm{N}\mathrm{S}} $ , and$ {F}_{\mathrm{T}\mathrm{B}} $ denote the net heat transport in the zonal, meridional, and vertical directions, respectively. The variables${\left(uT\right)}_{{\rm{E}}}$ ,${\left(uT\right)}_{{\rm{W}}}$ ,${\left(vT\right)}_{{\rm{N}}}$ ,${\left(vT\right)}_{{\rm{S}}}$ ,${\left(wT\right)}_{{\rm{T}}}$ , and${\left(wT\right)}_{{\rm{B}}}$ show the values of$ u\cdot T $ at the east and west sides, the values of$ v\cdot T $ at the north and south sides, and the values of$ w\cdot T $ at top and bottom boundaries, respectively. The variables$ {Q}_{\mathrm{n}\mathrm{e}\mathrm{t}} $ and${S}_{{\rm{w}}}$ are the downward net sea surface heat flux and the penetrating shortwave radiation at the bottom of the upper ocean layer, respectively; h is the depth of the upperocean. According to the widely used shortwave radiation transfer scheme (Paulson and Simpson, 1977),$$ {S}_{{\rm{w}}}=I\left(z\right)={I}_{0}({A}_{1}{\rm{e}}^{\dfrac{h}{{B}_{1}}}+{A}_{2}{\rm{e}}^{\dfrac{h}{{B}_{2}}}) , $$ (8) where
$ {I}_{0} $ is the net shortwave radiation flux, A_{1} and A_{2} are scale factors, and B_{1} and B_{2} are the depths of penetration, we follow the classification of seawater turbidity by Jerlov (1968) and set the optical properties of seawater as class I ($ {A}_{1} $ = 0.58,$ {A}_{2} $ = 1 −$ {A}_{1} $ = 0.42,$ {B}_{1} $ = 0.35 m, and$ {B}_{2} $ = 23.0 m). For the upper ocean in our study, h = 40 m, and then${S}_{{\rm{w}}}\approx 0.07\times{{Q}}_{\mathrm{n}\mathrm{e}\mathrm{t}}$ . Therefore,${F}_{Q}=\dfrac{{0.93\times{Q}}_{\mathrm{n}\mathrm{e}\mathrm{t}}}{h}$ .Hereafter, we use the bracket
$ \left\langle{ \quad }\right\rangle $ to denote the threedimensional average over the upperocean layer in the Niño3.4 region. Then, Eq. (3) can be simplified as$$ {C}_{p}{\rho }_{0}\dfrac{\partial \left\langle{T}\right\rangle}{\partial {t}}={F}_{\mathrm{E}\mathrm{W}}+{F}_{\mathrm{N}\mathrm{S}} + {F}_{\mathrm{T}\mathrm{B}}+{F}_{Q}+\left\langle{R}\right\rangle , $$ (9) where the residual term
$ \left\langle{R}\right\rangle $ , which includes turbulent transport, heat conduction, and other subgrid processes, has a very small contribution to the variation in$ \left\langle{T}\right\rangle $ and can be neglected. The average heat tendency is mainly determined by the net heat transports in the zonal, meridional, and vertical directions and the net surface heat flux. Equation (9) can be rewritten as$$ {C}_{p}{\rho }_{0}\dfrac{\partial \left\langle{T}\right\rangle}{\partial t}\approx {F}_{{\rm{E}}}+{F}_{{\rm{W}}}+{F}_{{\rm{N}}}+{F}_{{\rm{S}}}+{F}_{{\rm{B}}}+{F}_{Q} , $$ (10) where the vertical velocity w = 0 at the sea surface.
${F}_{{\rm{E}}}={C}_{p}{\rho }_{0}\dfrac{\iint {\left(uT\right)}_{{\rm{E}}}{\rm{d}}y{\rm{d}}z}{V}$ ,${F}_{{\rm{W}}}={C}_{p}{\rho }_{0}\dfrac{\iint {\left(uT\right)}_{{\rm{W}}}{\rm{d}}y{\rm{d}}z}{V}$ ,${F}_{{\rm{N}}}=$ ${C}_{p}{\rho }_{0}\dfrac{\iint {\left(vT\right)}_{{\rm{N}}}{\rm{d}}x{\rm{d}}z}{V}$ ,${F}_{{\rm{S}}}={C}_{p}{\rho }_{0}\dfrac{\iint {\left(vT\right)}_{{\rm{S}}}{\rm{d}}x{\rm{d}}z}{V}$ , and${F}_{{\rm{B}}}={C}_{p}{\rho }_{0}\dfrac{\iint {\left(wT\right)}_{{\rm{B}}}{\rm{d}}x{\rm{d}}y}{V}$ represent the lateral heat transports at the east, west, north, south, and bottom boundaries, respectively. Their positive values denote heat transport toward the interior. In other words, the average heat tendency of a given volume is mainly determined by the heat transports across the six boundaries of the volume according to Eq. (10).Equation (10) differs from Eq. (11), which has generally been used in previous studies (e.g., Huang et al., 2012; Graham et al., 2014; Ballester et al., 2016),
$$ \begin{aligned}[b] & \dfrac{1}{V}{C}_{p}{\rho }_{0}\iiint \dfrac{\partial T}{\partial t}{\rm{d}}V\approx \dfrac{1}{V}\iiint [{C}_{p}{\rho }_{0}(u\dfrac{\partial T}{\partial x}\\ & \quad + v\dfrac{\partial T}{\partial y}+w\dfrac{\partial T}{\partial z})+{F}_{\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{c}}+R]{\rm{d}}x{\rm{d}}y{\rm{d}}z . \end{aligned}$$ (11) Equation (11) can be deduced from Eq. (3) which is decomposed as,
$$ \dfrac{1}{V}{{C}}_{{p}}{\mathrm{\rho }}_{0}\iiint \dfrac{\partial T}{\partial t}{\rm{d}}x{\rm{d}}y{\rm{d}}z\approx \dfrac{1}{V}\iiint [{\underbrace{  {C_p}{\rho _0}\left( {u\dfrac{{\partial T}}{{\partial x}} + v\dfrac{{\partial T}}{{\partial y}} + w\dfrac{{\partial T}}{{\partial z}}} \right)}_{{\text{①}}}}\underbrace {  {C_p}{\rho _0}T \left( {\dfrac{{\partial u}}{{\partial x}} + \dfrac{{\partial v}}{{\partial y}} + \dfrac{{\partial w}}{{\partial z}}} \right)}_{{\text{②}}}+{{{F}}_{{\rm{Forc}}}} + R]{\rm{d}}x{\rm{d}}y{\rm{d}}z ,$$ (12) assuming that ocean water is an incompressible liquid [
$\nabla \left({\rho \boldsymbol{v}}\right)=0$ ].Equation (10) can be used to analyze the effect of largescale currents outside the region on the mean temperature change, as the heat transports from different directions are independent of its inside temperature. However, Eq. (11) ignores the effect of the divergence term and makes all advection terms dependent on not only the temperature and ocean current inside the region but also those outside the region. Lee et al. (2004) pointed out that the spatial integration of local temperature advection in Eq. (11) can only reflect the internal redistribution of heat and cannot explain external processes that control the heat content of a domain.
The primary data used in this study are monthly temperature, zonal and meridional currents, net surface heat flux, and salinity from 1979 to 2018 with a horizontal resolution of 0.25° and 75 levels in the vertical direction. The data are produced by the Ocean Reanalysis System 5 (ORAS5; Zuo et al., 2019) operated at ECMWF with the widelyused Nucleus for European Modelling of the Ocean (NEMO). The vertical discretization is approximately 1 m near the surface, and 24 levels are in the upper 100 m of the ocean. Carton et al. (2019) showed that the simulated ocean temperature and salinity in ORAS5 are nearly free from largescale errors when compared with observations.
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Zhao, H., T. W. Wu, L. Z. X. Li, et al., 2022: Upperocean lateral heat transports in the Niño3.4 region and their connection with ENSO. J. Meteor. Res., 36(2), 360–373, doi: 10.1007/s1335102211756 
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