## 1. Introduction

The surface circulation of the Arabian Sea is characterized by its strong seasonal variability. In the open ocean, its dynamics are predominantly driven by the seasonally reversing monsoon winds, which result in large-scale cyclonic flow during the northeast monsoon (boreal winter) and anticyclonic flow during the southwest monsoon (boreal summer) (Luther and O’Brien 1985; Schott et al. 2009). The strong seasonality of both wind forcing and the surface circulation indicate rapid adjustment of the Sverdrup balance by Rossby waves, which, as suggested in previous studies of other low-latitude ocean basins of similar size (e.g., the South China Sea, Liu et al. 2001; Xie et al. 2007), can propagate across the basin in a few months. The latitudinal dependence of Rossby wave phase speed also results in variation of arrival time along the western boundary.

In addition, downwelling and upwelling long Rossby waves of annual frequency propagate across the Arabian Sea and are particularly noticeable during the intermonsoon periods (Brandt et al. 2002; Beal et al. 2013). Rather than being locally excited by the surface winds, the annual Rossby waves are generated at the eastern boundary, and are possibly linked to coastal Kelvin waves from the Bay of Bengal (McCreary et al. 1993; Rao et al. 2010). The annual cross-basin planetary waves play a key role in the reversal of the western boundary current (McCreary et al. 1993; Beal et al. 2013; Vic et al. 2014; Wang et al. 2018).

The western boundary current of the Arabian Sea, the Somali Current, is unique among its counterparts (e.g., the north-poleward-flowing Kuroshio and Gulf Stream). During each monsoon phase, the Somali Current can be described by classical Stommel and Munk solutions for narrow western boundary currents, in which the viscosity terms are of leading-order importance in the vorticity balance. During the intermonsoons, its reversal is influenced by both local and remote forcings. Locally, alongshore southwesterly (northeasterly) winds can lead to geostrophic northward (southward) flow through the cross-shore pressure gradient established by Ekman upwelling (downwelling). Remotely, the early boreal spring reversal of the current prior to the local winds switching directions is associated with the arrival of the annual downwelling Rossby wave (Beal et al. 2013; Wang et al. 2018).

The Great Whirl (Swallow and Fieux 1982; Wirth et al. 2002; Beal and Donohue 2013), a seasonal anticyclonic system, is a prominent circulation feature of the Somali Current system. Even though the Great Whirl appears primarily during the southwest monsoon, it is found to persist into the northeast monsoon (e.g., Melzer et al. 2019). The Great Whirl makes a significant contribution to the upwelling of cold water (Schott et al. 2002), which has major impacts on local and remote air–sea heat exchange (Schott 1983; Hitchcock et al. 2000; Wirth et al. 2002).

In our previous study (Wang et al. 2018), we constructed a momentum budget for the surface Somali Current, using output from the global ocean general circulation model (OGCM) used here, whose horizontal resolution resolves mesoscale eddies in the Arabian Sea. We identified three regimes in the Somali Current based on differences in flow directions and underlying dynamics during the annual cycle (Fig. 1). Specifically, in the *Great Whirl regime* (north of 5°N), the Somali Current lies on the shoreward side of the nonlinear anticyclonic system, thus its flow is northward for most of the year. In the *equatorial regime* (between the equator and 2°N), equatorial dynamics and the northward inertial overshoot of the East Africa Coastal Current (EACC) control the dynamics of the Somali Current. In the *transition regime* (2°–5°N), the western boundary current is largely driven by the local winds.

While the momentum budget analysis can reveal the dynamical differences of the western boundary current during the seasonal cycle, it has limitations when representing recirculation systems like the Great Whirl and the Southern Gyre, and when illustrating open ocean processes including the annual Rossby waves. Therefore, in this study, we use a vorticity budget to establish a more comprehensive dynamical picture of the Arabian Sea. We aim to answer questions regarding seasonally varying open ocean dynamical balances and nonlinear processes involved in the seasonal evolution of the western boundary current and the Great Whirl.

Vertically integrated vorticity budgets have been used broadly to aid in understanding ocean dynamics (Hughes and De Cuevas 2001; Murray et al. 2001; Lu and Stammer 2004; Yeager 2015; Azevedo Correia de Souza et al. 2015; Schoonover et al. 2016). The approach chosen to vertically integrate the vorticity equation (i.e., whether a vertical integral or average is calculated before or after taking the curl of the momentum equations) will result in appearance of different terms [e.g., bottom pressure torque, bottom vortex stretching or the joint effect of baroclinicity and relief (JEBAR) term] that represent different physical processes or the same process in different forms (Mertz and Wright 1992; Bell 1999). In addition, vorticity analysis relies heavily on output from OGCMs, as they can provide the four-dimensional fields which are required to construct the budget. However, there are differences between the discretized and continuous equations and often further steps and assumptions need to be made to reproduce the continuous vorticity equations from numerical output (Bell 1999; Foreman and Bennett 1989).

Most of these earlier vorticity budget analyses focused on large-scale dynamical balances and largely did not pursue detailed analysis of the nonlinear term. In many cases, the nonlinear and diffusion terms, rather than being calculated directly, were treated as the residual of the leading-order balances, as they are found only to be important near the boundaries as part of non-quasigeostrophic balances. As well, these terms are not included as standard model output. Further, a complete and accurate decomposition of the nonlinear term (which includes advection of relative vorticity, tilting of horizontal vorticity, and stretching of vertical vorticity) should be calculated during the model run, but most numerical models do not include the option or algorithm for that computation.

In this study, we utilize nonstandard and high-frequency model output from an OGCM that is mesoscale eddy-resolving in the study region to perform a full vorticity analysis of the Arabian Sea. A new aspect of this study is to examine the role of eddy forcing on the seasonal cycle of the western boundary current, which is found to be a key process for the annual reversal of the Somali Current and the life cycle of the Great Whirl. In addition, we provide a comprehensive dynamical picture of the seasonal cycle of the basin, which provides further support for many earlier findings, including the annual cross-basin Rossby waves.

The rest of the paper is organized as follows: the configuration of the model is described in section 2. The details of the vorticity budget analysis, its vertical integration, and treatment of the nonlinear terms are presented in section 3. We show the analysis results in section 4, which is followed by discussion and conclusions in section 5. A full description of the algorithm of the vorticity budget is provided in the appendix.

## 2. Model description

We calculate and analyze a full vorticity budget using output from a forced global OGCM, coupled to a sea ice model, and run in the Community Earth System Model (CESM; Hurrell et al. 2013) framework (McClean et al. 2018). The model output is the same product used in Palóczy et al. (2018), Wang et al. (2018), Palóczy et al. (2020), and Castillo-Trujillo et al. (2021). The ocean component is the Los Alamos National Laboratory (LANL) Parallel Ocean Model 2 (POP 2). The horizontal mesh is on an Arakawa B-grid (Arakawa and Lamb 1977) with a nominal resolution of 0.1°; in our study region, the Arabian Sea, mesoscale eddies are resolved as the grid size is approximately 11 km. The vertical grid consists of 42 geopotential levels with shaved bottom cells (Pacanowski and Gnanadesikan 1998). The *K*-profile parameterization (KPP; Large et al. 1994) is used for vertical mixing, and the momentum and tracer horizontal mixing is represented by biharmonic operators [see Wang et al. (2018) for further details]. The simulation uses the Coordinated Ocean Reference Experiment-II corrected interannual forcing (CORE-II CIAF; Large and Yeager 2009) to provide surface fluxes for the ocean/sea ice model. The interannually varying atmospheric forcing is applied from 1948 to 2009.

Standard model output including the velocity and hydrographic fields was archived as monthly averages. For the last five years of the simulation (2005–09), we archived daily averaged variables. The daily averaged fields consist of both the standard model output as was archived in the monthly averaged fields and nonstandard terms that required POP code modifications in order to be archived; these latter terms include nonlinear advection, the baroclinic pressure gradient, and the horizontal and vertical diffusion terms. The daily averaged velocity fields and additional momentum equation terms allow us to decompose mean and eddy flows and calculate closed momentum and vorticity budgets offline.

## 3. Methods

Here, *ζ* is the vertical relative vorticity (*ζ* = ∂_{
x
}
*υ* − ∂_{
y
}
*u*); **v** and **u** are the three-dimensional and two-dimensional velocity [**v** = (**u**, *w*) = (*u*, *υ, w*)], respectively; *f* is the Coriolis parameter and *β* is its meridional gradient; *ϕ* is the geopotential (*ϕ* = *p*/*ρ*
_{0}, where *p* is pressure and *ρ*
_{0} is the reference density); **F**
_{
υ
} and **F**
_{
h
} represent horizontal and vertical diffusion, respectively; ∇ is the three-dimensional gradient operator [∇ = (∂_{
x
}, ∂_{
y
}, ∂_{
z
})] and ∇_{
h
} is its horizontal component.

From left to right in Eq. (1), the terms composing the vorticity budget are time tendency, the nonlinear term, stretching of planetary vorticity, advection of planetary vorticity, curl of the pressure (geopotential) gradient, curl of the vertical diffusion, and curl of the horizontal diffusion. For the remainder of the paper, we will use monthly averages of these terms for analysis unless otherwise specified.

The nonlinear, pressure gradient, horizontal diffusion, and vertical diffusion terms in the momentum equation were archived during the model run. We can then close the horizontal momentum equation offline [see more details in the appendix, as well as the methods section in Wang et al. (2018)], which ensures the closure of the vertical vorticity balance after applying the linear curl operator (∇_{
h
} ×) to each term.

### a. Bridging the continuous and discretized forms of the vorticity equations

A major difference between the discretized and continuous vorticity equations lies in the pressure term. In its continuous form, the curl of the pressure (geopotential) gradient term [the first term on the right-hand side of Eq. (1)] is zero. For the discretized form, this is generally the case for the interior ocean,^{
1
} but not for the grid cells adjacent to the boundary (Bell 1999).

This is due to the model’s discretized nature manifested by the discontinuity at the boundary. To satisfy no-slip and no-normal-flow boundary conditions, (i.e., zero velocities at the boundary), the momentum terms, including the horizontal pressure gradient term ∇_{
h
}
*p*, which are assigned at the corner of the tracer cells, are set to be zero as well. As a result, the curl of the pressure gradient is not zero and acts as a sidewall torque in the tracer cells next to the boundary (see, e.g., appendix C.3 in Yeager 2013).

The discretized equation also differs from its continuous form in the vertical integrations. For the continuous equation, there are three different forms (vertical integral of the vertical vorticity equation, curl of the vertically integrated momentum equations, and curl of the vertically averaged momentum equations), and their utilities have been discussed in previous studies (Mertz and Wright 1992; Cane et al. 1998; Mellor 1999; Bell 1999). It is, however, less straightforward to translate these forms of depth integration into discretized equations (Mertz and Wright 1992; Bell 1999).

In finite-volume models with piecewise constant configuration, the values of the vorticity terms in each grid box can be seen as an integral or average. Provided the operations are over the same set of grid points, the orders of the model’s curl and vertical integration are interchangeable. This means that the model does not distinguish between the curl of the vertically integrated momentum equations and the vertically integrated vorticity equation (the curl of the momentum equations) as in the continuous form. Thus, when Eq. (1) is integrated to the bottom of the ocean, it results in a form more similar to the curl of the vertically integrated momentum equations rather than the vertically integrated vorticity equation: the pressure (sidewall torque) term is nonzero and can be interpreted as the equivalent of the bottom pressure torque term in the curl of the vertically integrated momentum equations; the bottom vortex stretching term *fw*
_{
b
} (*w*
_{
b
} is the vertical velocity at the bottom of the ocean) vanishes with *w*
_{
b
} being zero at the bottom of each column, which is required by the no-normal-flow boundary condition.

**A**is shorthand for the nonlinear term;

*i*stands for the interior cells and

*b*for boundary cells; and

In this study, we focus on both near surface and barotropic vorticity budgets of the Arabian Sea. Therefore, we adopt the interior-boundary method following Bell (1999) to present vertically integrated vorticity balances, which unlike other forms, are not limited to full-depth integration. Vertical integration of the curl of the vertical diffusion becomes ∇ × (*τ*
_{
t
} − *τ*
_{
b
}) (Yeager 2013), in which *τ*
_{
t
} and *τ*
_{
b
} are the interfacial stresses at the surface and bottom of the ocean, respectively. Like the momentum equation, the surface wind stress dominates the vertical diffusion term and the curl of the vertical diffusion term essentially measures the surface wind stress curl. In addition to the vertically integrated interior cell balances [Eq. (2a)], we show vertical profiles of the vorticity terms based on Eq. (1). We will also discuss the balances in the boundary cells [Eq. (2b)] and the different meaning and role of the pressure term in different parts of the ocean.

### b. Decomposition of the nonlinear term

We perform two types of decomposition of the nonlinear term [the second term on the left-hand side of Eq. (1)]. First, we separate the contributions from the mean flows and the eddies. For the mean part, we first calculate the momentum advection term offline,^{
2
} strictly following the model’s momentum advection scheme (Smith et al. 2010), using the mean velocity field

By applying the curl operator to the mean momentum advection term, the nonlinear term due to the mean flow is obtained. The difference between the curl of the total momentum advection term (online) and the curl of the mean momentum advection term (offline) is then the eddy contribution. The eddy contribution therefore includes variability that is not captured by the mean, including both above annual and below monthly time scales.

Equation (3b) is the full flux form of Eq. (3a), where the three-dimensional relative vorticity *ω* = [(∂*ω*/∂*y*) − (∂*υ*/∂*z*), (∂*u*/∂*z*) − (∂*ω*/∂*x*), *ζ*]. To minimize the residual from the decomposition, which is primarily due to a nonzero divergence from POP’s momentum advection scheme, we use the flux form of the decomposition. Readers are referred to the appendix for the details of the algorithm of the decomposition in Eq. (3). Note that in this form, vertical advection [the second term on the right-hand side of Eq. (3b)] and vertical stretching of relative vorticity [the fourth term on the right-hand side of Eq. (3b)] are in fact two identical terms and cancel each other.

Ideally, this decomposition should be applied at each model time step so that the covariance between velocity components is conserved. Practically, global simulations save, average, and store output only at a monthly frequency. Offline calculation of the decomposition in Eq. (3) using monthly averaged velocity fields will result in large errors, which are caused by the missing covariance at time scales shorter than a month. Here, we archived the physical fields as daily averages. As we will show in the next section, calculating the decomposition offline using the daily averaged output leaves only a small residual contributed by the diurnal variability that the daily average does not resolve. Similarly to the mean-eddy decomposition of the total nonlinear term, we can decompose each component of the nonlinear advection term in Eq. (3) into mean and eddy parts.

## 4. Results

In this section, we build on results from the Somali Current momentum budget analysis in Wang et al. (2018) by using vorticity budgets to overview the dynamics of the Arabian Sea, which includes open ocean dynamics, their connection to the western boundary current, and the role of the nonlinear eddy term in the Great Whirl.

### a. Seasonal cycle of the surface circulation

We first review the seasonal surface circulation in the Arabian Sea simulated by the model (Fig. 2), as described in detail in Wang et al. (2018). The figure also shows the seasonal cycle of relative vorticity, whose governing dynamics are described below in section 4b.

Seasonally averaged velocity (arrows) and relative vorticity (color) in the surface (top 100 m) Arabian Sea, averaged from 2005 to 2009. (a) Boreal winter (December–February). (b) Boreal spring (March–May). (c) Boreal summer (June–August). (d) Boreal fall (September–November).

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Seasonally averaged velocity (arrows) and relative vorticity (color) in the surface (top 100 m) Arabian Sea, averaged from 2005 to 2009. (a) Boreal winter (December–February). (b) Boreal spring (March–May). (c) Boreal summer (June–August). (d) Boreal fall (September–November).

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Seasonally averaged velocity (arrows) and relative vorticity (color) in the surface (top 100 m) Arabian Sea, averaged from 2005 to 2009. (a) Boreal winter (December–February). (b) Boreal spring (March–May). (c) Boreal summer (June–August). (d) Boreal fall (September–November).

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

During the northeast monsoon, defined here as December–February (DJF, Fig. 2a), the Somali Current is southward north of 2°N (region of positive relative vorticity eastward of a strip of negative vorticity near the western boundary). The southward flow is strongest between 2° and 5°N, i.e., in the Somali Current’s *transition regime*. North of 5°N, in the *Great Whirl regime*, the surface flow is southward but very weak. In the *equatorial regime* south of 2°N, the DJF Somali Current is northward and eventually turns offshore at 2°N. It can be viewed as an extension of the northward EACC that occurs south of the equator. The EACC also feeds another offshore branch south of the equator, which is connected to the South Equatorial Countercurrent (SECC).

The surface circulation, especially the western boundary current, begins to transform during the boreal spring intermonsoon period in March–May (MAM, Fig. 2b). In the Great Whirl regime of the Somali Current, northward flow starts to appear, primarily due the arrival of an annual downwelling Rossby wave (Brandt et al. 2002; Beal et al. 2013; Wang et al. 2018). In the transition regime, the southward current is still noticeable but has significantly weakened with the decline of the northeasterly winds, relative to the DJF period. Meanwhile, clockwise circulation structures begin to form offshore in both the Great Whirl regime and south of the equator, which are the signatures of the initial stages of the Great Whirl and Southern Gyre, respectively. The split of the EACC at the equator is still apparent. The northern branch has become a steady zonal flow while the SECC is now a meandering zonal flow with a wave pattern.

During the southwest monsoon in June–August (JJA, Fig. 2c), the northward surface current is continuous along the northeast coast of Africa. The fully developed Great Whirl is prominent north of 5°N, and has merged with the Southern Gyre that migrated northward in August (Wang et al. 2018). The coherent northward flow collapses in the boreal fall intermonsoon during September–November (SON, Fig. 2d); at the same time, the Great Whirl starts to break down and the Southern Gyre retreats. The flow turns southward in the transition regime as the local winds diminish, while the northward flow north of 5°N lasts until December, maintained by a remnant anticyclonic structure from the Great Whirl.

### b. Vorticity balance of the surface circulation

We further examine the seasonal cycle of the surface circulation (top 100 m) through all of the terms in the vorticity equation (Fig. 3). Table 1 explains each of the terms.

Seasonal vorticity terms (see Table 1) in the surface Arabian Sea (averaged over top 100 m and years 2005–09). A three-point Gaussian filter has been applied to smooth fields for the purpose of presentation.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Seasonal vorticity terms (see Table 1) in the surface Arabian Sea (averaged over top 100 m and years 2005–09). A three-point Gaussian filter has been applied to smooth fields for the purpose of presentation.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Seasonal vorticity terms (see Table 1) in the surface Arabian Sea (averaged over top 100 m and years 2005–09). A three-point Gaussian filter has been applied to smooth fields for the purpose of presentation.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

List of vorticity budget terms. The names of the vorticity terms are shorthand versions of each term in either the interior balance described in Eq. (2a) (used in Fig. 3 and Fig. 4) or the original vorticity equation in Eq. (1) (used in Figs. 6–8).

In the interior basin, wind forced planetary geostrophic dynamics dominates the vorticity balance, which is between planetary vorticity advection [the beta effect, Fig. 3, panel (iii)], planetary vorticity stretching [Fig. 3, panel (iv)], and surface wind stress curl [represented by curl of the vertical diffusion, Fig. 3, panel (vi).].

During the peaks of both monsoon seasons (DJF and JJA), the Ekman balance dominates the dynamics north of the maximum wind speed axis (zero vertical diffusion curl) in the northern Arabian Sea: negative (positive) wind stress curl is balanced by positive (negative) *f*∂_{
z
}
*w*, i.e., downwelling (upwelling) in the subsurface during the northeast (southwest) monsoon. Sverdrup dynamics primarily controls the regions to the south of the maximum wind speed axis, where wind stress curl is primarily balanced by −*βυ*.

In addition, during DJF, a narrow band of negative −*βυ* between the equator and 10°N is balanced by positive *f*∂_{
z
}
*w* that extends from the tip of India to the western boundary of the African coast around 5°N [Figs. 3a(iii) and 3a(iv)]; this balance represents the annual downwelling Rossby wave propagating westward identified in earlier studies (McCreary et al. 1993; Brandt et al. 2002; Beal et al. 2013), which roughly follows the potential vorticity contours in Fig. 1. During boreal fall, there is also evidence of an upwelling Rossby wave with the leading order vorticity balance between positive −*βυ* and negative *f*∂_{
z
}
*w*. Unlike the downwelling Rossby wave, the upwelling wave arrives at the western boundary after the local wind switches direction (not shown). It has been suggested in previous studies (e.g., McCreary et al. 1993; Vic et al. 2014) that the upwelling Rossby wave can contribute to the destruction of the Great Whirl through perturbations of the local pressure field. The horizontal distribution of the two vorticity terms suggests that the upwelling Rossby wave follows an even more slanted pathway [Fig. 3c(iii)] compared with its downwelling counterpart [Fig. 3a(iii)], which can contribute to a slower phase speed and hence a later arrival.

The western boundary balance consists of diffusion terms, nonlinear terms, and planetary vorticity stretching (*f*∂_{
z
}
*w*) and advection terms (−*βυ*). The horizontal diffusion acts primarily as a friction term and is always of the opposite sign to −*βυ*. Its contribution is limited to a narrow band close to the coast. The planetary vorticity stretching term (*f*∂_{
z
}
*w*) is a key term in the western boundary vorticity balance north of 5°N, especially during the life cycle of the Great Whirl. During boreal summer and fall, the shape of the Great Whirl is evident in the dipole pattern of *f*∂_{
z
}
*w* [Figs. 3c(iv) and 3d(iv)]. Upwelling near the shore is associated with northward and offshore flow, while downwelling further offshore marks the center of the anticyclonic gyre. Both upwelling and downwelling are balanced by the wind stress curl (vertical diffusion). The planetary vorticity stretching (*f*∂_{
z
}
*w*) and advection terms (−*βυ*), which have the same sign during the peaks of the monsoons, oppose the local vorticity input from the surface wind stress.

The nonlinear terms [Fig. 3, panels (i) and (ii)] are of leading-order magnitude along the boundaries. Together, the mean and eddy nonlinear terms balance −*βυ* in the band 2°–5° offshore depending on the latitude and season.

During DJF, the nonlinear mean term is most important south of 5°N, where the southward boundary current in the transition regime and the northward inertial flow in the equatorial regime are strong. On the other hand, the nonlinear eddy term is the major contributor to the balance north of 5°N in the Great Whirl regime. The eddy term remains significant during MAM [Fig. 3b(ii) ] in the Great Whirl regime, where the Great Whirl starts to form, and the equatorial regime, where the northward migration of the Southern Gyre has begun. Another prominent feature during boreal spring is the meandering SECC; it can be identified by the repeating offshore dipole pattern in the −*βυ* and mean nonlinear terms.

During JJA, around the location of the Great Whirl, the pattern of the nonlinear mean term resembles that of *f*∂_{
z
}
*w* and is of leading-order importance in addition to −*βυ*, *f*∂_{
z
}
*w* and wind stress curl. The sign indicates that it contributes to balancing −*βυ*. In other words, the mean flow advection acts as the centrifugal force to balance the Coriolis force in the inertial balance. South of the equator, −*βυ* is mostly balanced by the nonlinear mean term, suggesting the cross-equatorial inertial flow is a steady northward current. The eddy term is most prominent in the transition and Great Whirl regimes. In the transition regime, the contribution from the eddy term represents the northward migration of the Southern Gyre on time scales shorter than one month. In the Great Whirl regime, the eddy contribution likely indicates the highly energetic behavior of the Great Whirl.

During boreal fall (SON, Fig. 3d), the Great Whirl persists. The eddy term makes the greater contribution of the two nonlinear terms in the leading-order balance at this location, suggesting its importance in maintaining the Great Whirl and the northward western boundary current north of 5°N.

In summary, analysis of the seasonal cycle of the vorticity balances leads to a better understanding of the processes driving the evolution of the surface circulation in the Arabian Sea. In particular, by separating the relative contributions of the nonlinear eddy and mean terms, we have highlighted the important role of nonlinear eddies in the western boundary system, especially during the life cycle of the two recirculation systems. The analysis also lends further support to earlier studies, including the relationship between the seasonal cycle of the Somali Current, the surface wind, and planetary waves in the interior Arabian Sea.

### c. Regional vorticity budgets in the Arabian Sea

In this section, we provide detailed analyses of regional vorticity budgets in the Arabian Sea including those in the open ocean and the western boundary current. In addition to the surface and barotropic balances, we also compare balances at different depths.

#### 1) Open ocean

As illustrated in Fig. 3, all terms in the vorticity equation in the open ocean are at least an order of magnitude smaller than in the western boundary current system. Nonetheless, the seasonal cycle of the open ocean vorticity balances reveals the interaction of different forcings at various time scales, which can in turn influence the western boundary current and its seasonal cycle. To represent the open ocean, we use a box in the central northern Arabian Sea (Fig. 1) that is located along the pathway of the annual Rossby waves, and examine the barotropic vorticity balances within it.

The leading-order terms are the time tendency ∂_{
t
}
*ζ*, −*βυ*, *f*∂_{
z
}
*w*, and vertical diffusion (Fig. 4a). Two processes dominate. The first is the large-scale Sverdrup balance driven by the surface wind stress curl (represented in the vertical diffusion term), which results in column stretching (squeezing) (*f*∂_{
z
}
*w*) and southward (northward) transport (−*βυ*) during the northeast (southwest) monsoon. The second is westward propagating barotropic Rossby waves. Here, the vorticity balance is dominated by ∂_{
t
}
*ζ*, −*βυ*, and *f*∂_{
z
}
*w*.

Time series of full-depth integrated vertical vorticity terms (cm s^{−2}; see Table 1) averaged in several boxes in the Arabian Sea (Fig. 1). (a) Open ocean (64°–66°E, 10°–12°N). (b) Great Whirl recirculation region (53°–55°E, 6.5°–8.5°N). (c) Somali Current Great Whirl regime (50°–52°E, 7°–9°N). (d) Somali Current transition regime (47.5°–49.5°E, 2.5°–4.5°N). (e) Somali Current equatorial regime (42°–45°E, 1°S–1°N). For each subplot, the upper panel shows vorticity terms integrated over the interior cells while the lower panel shows terms integrated over the boundary cells.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Time series of full-depth integrated vertical vorticity terms (cm s^{−2}; see Table 1) averaged in several boxes in the Arabian Sea (Fig. 1). (a) Open ocean (64°–66°E, 10°–12°N). (b) Great Whirl recirculation region (53°–55°E, 6.5°–8.5°N). (c) Somali Current Great Whirl regime (50°–52°E, 7°–9°N). (d) Somali Current transition regime (47.5°–49.5°E, 2.5°–4.5°N). (e) Somali Current equatorial regime (42°–45°E, 1°S–1°N). For each subplot, the upper panel shows vorticity terms integrated over the interior cells while the lower panel shows terms integrated over the boundary cells.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Time series of full-depth integrated vertical vorticity terms (cm s^{−2}; see Table 1) averaged in several boxes in the Arabian Sea (Fig. 1). (a) Open ocean (64°–66°E, 10°–12°N). (b) Great Whirl recirculation region (53°–55°E, 6.5°–8.5°N). (c) Somali Current Great Whirl regime (50°–52°E, 7°–9°N). (d) Somali Current transition regime (47.5°–49.5°E, 2.5°–4.5°N). (e) Somali Current equatorial regime (42°–45°E, 1°S–1°N). For each subplot, the upper panel shows vorticity terms integrated over the interior cells while the lower panel shows terms integrated over the boundary cells.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

To better illustrate the different balances at different time scales, we show the power spectral density of the vorticity terms for the full depth (Fig. 5a). The terms ∂_{
t
}
*ζ*, −*βυ*, and *f*∂_{
z
}
*w* are found at time scales ranging from several weeks to a couple of months, in which the barotropic Rossby wave balance (∂_{
t
}
*ζ* ~ −*βυ*) is of leading-order importance; the waves of mixed length scales are most likely excited by local wind stress curl changes. The corresponding low-mode baroclinic Rossby waves, can be implied from the the near surface vorticity spectra in Fig. 5b. Here, the Rossby wave balance is still evident at time scales of a couple of months. The relatively rapid phase speed (about 13 m s^{−1} for the barotropic mode, not shown) of the Rossby waves at these time scales enables a relatively fast Sverdrup adjustment compared with higher latitudes. On the other hand, the balance *βυ* ~ *f*∂_{
z
}
*w*, which represents the long Rossby waves (barotropic mode in Fig. 5a and baroclinic mode indicated in Fig. 5b), is evident at annual frequencies in addition to the Sverdrup balance (between the planetary vorticity advection term −*βυ* and the vertical diffusion term). This provides further support to the nature of the westward propagating sea surface height signal found in previous studies (Brandt et al. 2002; Beal et al. 2013).

Power spectral density of the vorticity terms integrated over the (left) full depth and (right) top 300 m in the (a),(b) open ocean box and (c),(d) Great Whirl recirculation box (Fig. 1). The power spectra are calculated using daily averaged model output, therefore the nonlinear term is not decomposed into its eddy and mean components. The dotted vertical lines denote annual, semiannual, and monthly frequency.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Power spectral density of the vorticity terms integrated over the (left) full depth and (right) top 300 m in the (a),(b) open ocean box and (c),(d) Great Whirl recirculation box (Fig. 1). The power spectra are calculated using daily averaged model output, therefore the nonlinear term is not decomposed into its eddy and mean components. The dotted vertical lines denote annual, semiannual, and monthly frequency.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Power spectral density of the vorticity terms integrated over the (left) full depth and (right) top 300 m in the (a),(b) open ocean box and (c),(d) Great Whirl recirculation box (Fig. 1). The power spectra are calculated using daily averaged model output, therefore the nonlinear term is not decomposed into its eddy and mean components. The dotted vertical lines denote annual, semiannual, and monthly frequency.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

The annual long Rossby wave can be distinguished from the annually reversing Sverdrup balance by their phase differences (Fig. 4a). Barotropic −*βυ* has two maxima, one during boreal spring as a part of the downwelling Rossby wave vorticity balance (balanced by *f*∂_{
z
}
*w*) and the other during boreal summer as a part of the Sverdrup balance (balanced by vertical diffusion). Therefore, the annual long Rossby wave balance happens about a quarter cycle earlier than the local Sverdrup balance. This is especially noticeable in 2006, 2008, and 2009. The boreal spring downwelling long Rossby wave that occurs before the onset of the southwest monsoon is a key process in the formation of the nascent Great Whirl (Beal and Donohue 2013) and the appearance of northward flow in the Great Whirl regime of the Somali Current (Beal et al. 2013; Wang et al. 2018).

The vorticity balance is further shown as a function of depth in Fig. 6, based on Eq. (1). The vertical structure of a low-mode baroclinic Rossby wave is evident in −*βυ* and *f*∂_{
z
}
*w*. The direct contribution from the vertical diffusion term is limited to near the surface, where the surface-intensified structure is primarily attributed to the surface wind stress curl; the depth of the surface Ekman layer can be identified by the abruptly decreasing vertical diffusion term at a depth of about 100 m (*f*∂_{
z
}
*w* and vertical diffusion). The seasonal deepening of the Ekman layer is represented by the depth range over which the vertical diffusion term changes. The Sverdrup balance is apparent in the balance between −*βυ* and *f*∂_{
z
}
*w* below the direct effect of the wind forcing in the Ekman layer.

Time series of monthly vertical profiles of vorticity equation terms [in Eq. (1), see Table 1] averaged in a box in the open ocean (63°–67°E, 9°–13°N).

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Time series of monthly vertical profiles of vorticity equation terms [in Eq. (1), see Table 1] averaged in a box in the open ocean (63°–67°E, 9°–13°N).

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Time series of monthly vertical profiles of vorticity equation terms [in Eq. (1), see Table 1] averaged in a box in the open ocean (63°–67°E, 9°–13°N).

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

#### 2) Western boundary and the Great Whirl

Vorticity balances in the western boundary region are shown in Figs. 4b–e. Here we focus on the three regimes of the Somali Current as well as a location in the return flow of the Great Whirl (Fig. 1), whose strength and shape are expected to be highly variable on both seasonal and interannual time scales. The leading-order vorticity terms in the western boundary are at least an order of magnitude larger than those in the open ocean regions.

In the Great Whirl recirculation box (Fig. 4c), the vorticity terms appear to be more variable. The time tendency, nonlinear eddy, and planetary vorticity advection terms constitute the leading-order balance. The time tendency of relative vorticity is primarily driven by nonlinear terms on time scales smaller than a couple of months (Figs. 5c and 5d). The short barotropic Rossby wave balance (*ζ*
_{
t
} ~ −*βυ*) can be seen throughout the five years of simulation. This vorticity signature of the barotropic Rossby wave is found at time scales of several months (Figs. 5c and 5d), which echoes that in the open ocean (Figs. 5a and 5b). On the seasonal time scale, the leading-order balance is between the nonlinear eddy term and −*βυ*, which indicates the key role eddies play in the Great Whirl. The Great Whirl forms and intensifies in early summer, indicated by the vorticity tendency becoming negative after June (especially during 2005, 2006, 2008, and 2009 in Fig. 4c). The change in vorticity at this time is shown to be caused by the nonlinear eddy term (recall that in Table 1, all terms except for the tendency term are on the right-hand side of the equation). With the intensification of the Great Whirl, the southward return flow and thus −*βυ* increases, and during the peak of the southwest monsoon, the leading-order balance is largely between −*βυ* and the nonlinear eddy term.

Further west and closer to the coast, the impact of eddies is still evident, although the magnitude of the vorticity terms is much smaller than in the box further to the east (Fig. 4b). Similarly to the Great Whirl recirculation box, the nonlinear eddy term acts to balance the planetary advection term during the southwest monsoon. The influence of the Rossby wave is still present (e.g., during the winter of 2008/09) but less evident. A key difference between this box and the Great Whirl recirculation box is the importance of the horizontal diffusion term. The boundary friction is of leading-order importance during most of the year. Together with the nonlinear eddy term, the smaller-scale horizontal diffusion contributes to the leading-order balance of −*βυ*.

The horizontal diffusion appears to be of leading-order importance only during the first half of the year, i.e., during the early reversal of the northward flow in the Great Whirl regime (northern part of the Somali Current; see Wang et al. 2018). The leading-order balance is again between the nonlinear eddy term and −*βυ* after June. This is a key feature that differentiates the Great Whirl regime from the transition regime (Fig. 4d). While in the Great Whirl regime the northward flow can last until early winter, presumably due to the existence of the nonlinear Great Whirl, the northward flow in the transition regime only appears until the peak of the southwest monsoon in August. Figure 4b suggests that the nonlinear eddy term in the Great Whirl regime remains important during the declining phase of the southwest monsoon, while it is not present in the transition regime at this time.

The vertical profiles of the vorticity budgets in the two boxes around the Great Whirl are shown in Fig. 7. Near the surface, the vertical diffusion term is confined within the top 100 m, where the stretching of planetary vorticity (*f*∂_{
z
}
*w*) and the nonlinear terms also dominate. The Ekman pumping/suction relationship between the surface wind stress curl (vertical diffusion term) and *f*∂_{
z
}
*w* is similar to the open ocean (Fig. 6) and suggests the maximum upwelling occurs at a depth of 100 m. A key difference between the two boxes is the pressure term. Along the boundary and below the Ekman layer, the pressure term, which is the numerical equivalent of the bottom pressure torque, is a leading-order term in the balance; it acts as the primary source of drag to the northward flow (in −*βυ*). Away from the boundary, −*βυ* is instead balanced by the nonlinear eddy term and *f*∂_{
z
}
*w*, in the absence of the pressure term.

As in Fig. 6, but for the two boxes near the Great Whirl (Fig. 1). (top) Great Whirl recirculation region (53°–55°E, 6.5°–8.5°N). (bottom) Somali Current Great Whirl regime (50°–52°E, 7°–9°N).

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

As in Fig. 6, but for the two boxes near the Great Whirl (Fig. 1). (top) Great Whirl recirculation region (53°–55°E, 6.5°–8.5°N). (bottom) Somali Current Great Whirl regime (50°–52°E, 7°–9°N).

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

As in Fig. 6, but for the two boxes near the Great Whirl (Fig. 1). (top) Great Whirl recirculation region (53°–55°E, 6.5°–8.5°N). (bottom) Somali Current Great Whirl regime (50°–52°E, 7°–9°N).

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Finally, the vorticity balance in the *equatorial regime* of the Somali Current is shown in Fig. 4e. The nonlinear eddy term is relatively less important than the rest of the western boundary current boxes and the leading-order balance in the interior cells is between −*βυ* and the horizontal diffusion.

#### 3) Boundary cell balances

In the previous sections, the barotropic vorticity balances in various parts of the Arabian Sea were analyzed primarily focusing on the interior cells, leveraging the direct association with their continuous vertically integrated vorticity equation counterparts. In this section, we will discuss in more detail the balances in the boundary cells and the role of the bottom pressure torque term.

In studies focusing on the budget of the curl of the vertically integrated momentum equation (Hughes and De Cuevas 2001; Yeager 2015), it has been suggested that the bottom pressure torque is central to the leading-order dynamics in many parts of the ocean. Notably, for the vorticity balances in western boundary currents (Hughes and De Cuevas 2001), the bottom pressure torque, rather than the viscosity term, is found to be the largest term.

Figure 8 shows the budget from the vertically integrated vorticity equation over the entire water column (both interior and boundary cells are included), which is equivalent to the curl of the vertically integrated momentum equation in the model. Here time tendency and stretching of planetary vorticity terms are ignored as they are several orders of magnitude smaller than the other terms.

Terms from the vertically integrated vorticity equation, averaged from 2005 to 2009. The vertical integration is over the full depth, which includes both interior and boundary cells. Time tendency and planetary vorticity stretching terms are ignored. A three-point two-dimensional Gaussian filter is applied as a horizontal filter.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Terms from the vertically integrated vorticity equation, averaged from 2005 to 2009. The vertical integration is over the full depth, which includes both interior and boundary cells. Time tendency and planetary vorticity stretching terms are ignored. A three-point two-dimensional Gaussian filter is applied as a horizontal filter.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Terms from the vertically integrated vorticity equation, averaged from 2005 to 2009. The vertical integration is over the full depth, which includes both interior and boundary cells. Time tendency and planetary vorticity stretching terms are ignored. A three-point two-dimensional Gaussian filter is applied as a horizontal filter.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

The horizontal diffusion term is significantly smaller than the other terms even near the boundaries, which is consistent with previous analyses based on model results (Hughes and De Cuevas 2001; Yeager 2015). Our analysis from the previous sections suggests that breaking down the model’s barotropic vorticity into the interior and boundary cells may shed some light on the lack of significance of the full-depth integrated horizontal diffusion term.

As seen in Figs. 4b, 4d, and 4e, in the western boundary, the full-depth integrated horizontal diffusion term is cancelled by its integrals in the interior and boundary cells that have opposite signs. From the point of view of focusing on the bottom pressure torque, this may suggest that the effect of the bottom pressure torque is represented by the horizontal diffusion in the interior cells. From the point of view of horizontal diffusion, it may be argued that the horizontal diffusion term needs to change sign in the boundary cells in order to balance the bottom pressure torque term (whose presence in the model is partly the result of the no-slip and no-normal-flow boundary conditions).

A discussion of the roles of the pressure term would require further analysis and sensitivity experiments and is beyond the scope of this paper. However, it is worth addressing the usefulness of separating interior and boundary cells in the barotropic vorticity analysis. For instance, although the bottom pressure torque is balanced by horizontal diffusion near the western boundary at the equator (Fig. 4e), the nonlinear eddy term joins the horizontal diffusion in opposing the bottom pressure torque in the boundary layer at the latitudes further north (Figs. 4b,d). Therefore, it can be speculated that some of the frictional effects of the parameterized subgrid scale horizontal diffusion are shared by the resolved eddy processes.

Furthermore, for the interior ocean, we find that the bottom pressure torque is largely balanced by the planetary vortex stretching term *f*∂_{
z
}
*w* (lower panel of Fig. 4a). In other words, even though the no-normal-flow boundary condition prohibits a nonzero bottom velocity in the model, the bottom pressure torque term can act as an agent for the missing process through the balance ∇*h*
_{
b
} × ∇*ϕ* ~ −*fw*
_{
b
}, where *w*
_{
b
} is the ocean’s bottom vertical velocity in the continuous equations. After all, the model’s bottom pressure torque is due to the pressure torque on the sidewalls of the step-shape topography. One way to think of this is that the bottom velocity associated with the changing ocean depth is represented by the bottom pressure torque on the walls between adjacent grid cells.

Therefore, we conclude that the curl of the pressure gradient term, i.e., the bottom pressure torque when integrated vertically over the full depth, is important for the vorticity balance in the model. The separation of interior and boundary cells helps us understand the interpretation of this term. Different boundary cell balances in the open ocean and near the boundary shown in Fig. 4 suggest that the bottom pressure torque term may represent different physical processes.

### d. Decomposing the nonlinear term

In this section, we further decompose and evaluate the components of the nonlinear term. Here, we focus on the western boundary region during the southwest monsoon, when the two recirculation systems, the Great Whirl and Southern Gyre, are well developed and the nonlinear terms are most important. Figure 9 shows components of the nonlinear terms from July to September averaged over the 5-yr period of daily output (2005–09).

Components of the nonlinear term integrated over the top 100 m for (a) July, (b) August, and (c) September, averaged from 2005 to 2009: (i) horizontal advection (mean); (ii) horizontal advection (eddy); (iii) tilting of the horizontal vorticity (mean); (iv) tilting of the horizontal vorticity (eddy); and (v) remaining terms (diurnal variability ignored using the daily velocity fields and numerical errors from the cancellation of the two vertical terms). Topography is shown as contours with intervals of 1000 m. A three-point Gaussian filter is applied to each map.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Components of the nonlinear term integrated over the top 100 m for (a) July, (b) August, and (c) September, averaged from 2005 to 2009: (i) horizontal advection (mean); (ii) horizontal advection (eddy); (iii) tilting of the horizontal vorticity (mean); (iv) tilting of the horizontal vorticity (eddy); and (v) remaining terms (diurnal variability ignored using the daily velocity fields and numerical errors from the cancellation of the two vertical terms). Topography is shown as contours with intervals of 1000 m. A three-point Gaussian filter is applied to each map.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Components of the nonlinear term integrated over the top 100 m for (a) July, (b) August, and (c) September, averaged from 2005 to 2009: (i) horizontal advection (mean); (ii) horizontal advection (eddy); (iii) tilting of the horizontal vorticity (mean); (iv) tilting of the horizontal vorticity (eddy); and (v) remaining terms (diurnal variability ignored using the daily velocity fields and numerical errors from the cancellation of the two vertical terms). Topography is shown as contours with intervals of 1000 m. A three-point Gaussian filter is applied to each map.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

For both mean and eddy parts, horizontal advection (left two panels of Fig. 9) is the largest of the nonlinear terms; variability of the mean and eddy nonlinear terms discussed in the previous sections can be mostly attributed to that of the horizontal advection term. The magnitude of the horizontal tilting term (third and fourth panels) is about 20% that of the horizontal advection term and occurs very close to the boundary. The residual term (rightmost panel), which contains only the diurnal variability that is not resolved by daily averaged output, is almost negligible. This lends support to the accuracy of our decomposition algorithm and proves that the use of daily averages for model output is sufficient for offline nonlinear term decomposition in this region.

In July (Fig. 9a), the flow pattern can be largely inferred from the mean horizontal advection. The positive band of relative vorticity caused by mean horizontal advection contributes to the northward flow acceleration along the coast. The offshore tendency slightly south of 5°N is related to the Southern Gyre that migrated from the south. The eddy horizontal advection is of opposite sign to the mean and therefore shows a downgradient vorticity flux. The shape of the Great Whirl can be discerned by the core of negative mean horizontal advection ringed by positive values in August (Fig. 9b), especially on the western side near the coast. The northeast corner of the Great Whirl, on the other hand, is more manifested in the eddy horizontal advection. When the Great Whirl is weakening in September (Fig. 9c), the eddy term primarily maintains the vorticity balance north of 5°N, where the magnitude of the mean term reduces significantly.

The contribution from the horizontal vorticity tilting term mostly comes from the mean, which plays a nonnegligible role in the life cycle of the recirculation systems, noticeably in the offshore branch of the Southern Gyre [Fig. 9a(iii)] and the northeast corner of the Great Whirl [Fig. 9b(iii)]. It generally has the same sign as the eddy advection term, which could help maintain the Great Whirl in late boreal summer. Further analysis (not shown) shows that this is related to the horizontal gradient of vertical velocity near the recirculation systems.

## 5. Discussion and conclusions

In this study, we have analyzed the dynamics of the Arabian Sea using a full vorticity budget in a mesoscale eddy-resolving model. In the open ocean of the northern Arabian Sea, we have shown that the seasonal Sverdrup balance and Rossby waves dominate the large-scale dynamics. Fast barotropic and baroclinic Rossby waves in the low-latitude basin allow rapid Sverdrup adjustment. In addition, westward propagating annual Rossby waves, downwelling during boreal spring and upwelling during boreal fall, are shown to dominate leading-order balances during the intermonsoons. The annual Rossby waves are prominent features in many low-latitude ocean basins under the influence of monsoon forcing (Liu et al. 2001; Masumoto and Meyers 1998); it remains an open question as to whether they are forced locally or are free waves from remote locations. Arabian Sea annual Rossby waves are of particular interest due to their impacts on the flow direction of the northern sector of the Somali Current.

We then focused on the role of the nonlinear term at the western boundary, which had not been fully decomposed and discussed in detail in previous studies. The nonlinear eddy term is of leading-order importance in the Great Whirl, especially in its eastern return flow sector. This helps sustain the Great Whirl after the weakening of the southwest monsoon. Decomposition of the nonlinear term reveals the largest contribution is due to horizontal advection of relative vorticity. The twisting terms play nonnegligible roles close to the boundary and near the Great Whirl and the Southern Gyre.

In previous studies, the nonlinear term from the vorticity equation is usually treated as a single term. This is arguably a reasonable choice when the model does not necessarily resolve small-scale processes represented by the twisting terms. The effect of the nonlinear twisting terms is studied mostly in the context of submesoscale dynamics including frontogenesis (e.g., Barkan et al. 2019). However, with the resolution of large-scale models increasing and more smaller-scale processes being explicitly resolved, it will be valuable to study the relative roles of the individual components of the nonlinear term to better understand fine structures like fronts, which are important in air–sea interaction. We provide here an example of how to fully decompose the nonlinear term with the commonly used B-grid model.

We also analyzed the interpretation of the vorticity terms in the model’s vertically integrated vorticity equation and the utility of interior and boundary cells (Bell 1999). In the discretized model, the vertically integrated vorticity equation and the curl of the vertically integrated momentum equation are identical. The model’s bottom pressure torque term takes the place of various terms in different parts of the Arabian Sea.

In the open ocean, bottom pressure torque acts as the otherwise missing planetary vorticity stretching term, which is a function of bottom velocity. It is important to stress the effect of the partial bottom cells (Adcroft et al. 1997; Pacanowski and Gnanadesikan 1998). Partial bottom cells result in variable bottom layer thickness, which guarantees that there is not an absolutely “flat” ocean bottom across the cells. As the curl operator (details in the appendix) is weighted by cell thickness, the bottom pressure torque is always nonzero even in a region with relatively flat topography, where the vertical indices in adjacent cells are the same. The open ocean balance is due in large part to the existence of partial bottom cells.

Near the coast, the bottom pressure torque term is seemingly behaving as a drag in the full-depth vorticity balance, when the horizontal diffusion term is almost negligible (Fig. 8). This is consistent with previous studies (e.g., Hughes and De Cuevas 2001). We find the main reason for its behavior is that there is a cancellation of the horizontal diffusion term in the interior and boundary cells. In the full-depth balance, the role of the bottom pressure torque terms is equivalent to that of the horizontal diffusion and nonlinear terms in the interior cell balance. In the boundary cells where the bottom pressure torque term is confined, it is balanced by horizontal diffusion and nonlinear terms, especially the eddy terms. This leads to the question of the kinematical importance of the horizontal diffusion term, when the bottom pressure torque, which is independent of the specifics of the subgrid-scale parameterization, acts as de facto friction in the model. A potential future study of the sensitivity of the horizontal diffusion parameterization, especially at the western boundaries, would be helpful to better understand this question.

## Acknowledgments

We thank Dr. Michael J. Bell (U.K. Met Office) and two other anonymous reviewers for their comments on the paper. We are grateful to Dr. Stephen Yeager (NCAR) for providing the code for saving the nonstandard momentum fields in POP and the NCL code for the basic vorticity budget, which served as a prototype for the fully decomposed vorticity budget in this study. The work was funded by Office of Naval Research Grants N00014-15-1-2189 (J.L.M.) and N00014-15-1-2566 (L.D.T.). J.L.M. was also supported by DOE Office of Science Grants DE-SC0012778 and DE-SC0020073. We would like to acknowledge high-performance computing support from Yellowstone (ark:/85065/d7wd3xhc) provided by NCAR’s Climate Simulation Laboratory, sponsored by the National Science Foundation. Funding for the POP simulation was provided by a DOE BER grant entitled “Ultra High Resolution Global Climate Simulation” via a Los Alamos National Laboratory subcontract (J.L.M.); Caroline Papadopoulos (SIO/UCSD) carried out the 60-year POP simulation, and Elena Yulaeva (UCSD) performed the repeat POP simulation for years 2005–2009. Some POP analyses were carried out using Rhea in the Oak Ridge Leadership Computing Facility (OLCF) at Oak Ridge National Laboratory.

## Data availability statement

Output from the POP simulation for the region of the Arabian Sea is available through the UC San Diego Library Digital Collections (https://doi.org/10.6075/J0KW5D84). Code for constructing the vorticity budget can be found at https://github.com/herrwang0/vortbud.

## APPENDIX

### Derive the Fully Decomposed Vorticity Equation

#### a. Revisit the (continuous) vertical vorticity equation

Here, the three-dimensional velocity **v** = (**u**, *w*) = (*u*, *υ*, *w*); **Ω** is Earth’s rotation frequency, and 2**Ω** = (0, 0, *f*) under Cartesian approximation; *ϕ* = *p*/*ρ*
_{0}, in which *p* is the pressure and *ρ*
_{0} is the background density; and **F** is the general form for other forcing terms including (horizontal and vertical) viscous forcings, body forcings (gravity and tides), and the metric terms.

_{ y }, ∂

_{ x }), with which the relative vorticity

*ζ*= ∂

_{ x }

*υ*− ∂

_{ y }

*u*. By taking the vertical curl of (A1), we can derive the vertical vorticity equation:

There are several nonlinear terms in (A2), which come from the decomposition of the curl of the nonlinear advection term. The second term on the left-hand side of (A2) is the advection of the vertical relative vorticity. The fifth term is the stretching of the vertical relative vorticity, and the sixth term represents the tilting of the horizontal vorticity. The fifth and the sixth terms are often referred to as the twisting terms.

#### b. Momentum and vorticity budget closure in POP

##### 1) Momentum equation

*M*

_{ x },

*M*

_{ y }) is the metric term;

_{ x }

*u +*∂

_{ y }

*υ*+ ∂

_{ z }

*w*= 0.

During the last five years of the simulation, we archived the online calculation of nonlinear terms (including both advection and metric terms), baroclinic pressure gradient, and horizontal and vertical diffusion terms as daily averages. In addition, the barotropic pressure gradient terms can be calculated offline from sea surface height. The Coriolis term can be recovered through velocity.^{
A1
} Therefore, all terms in (A4) except for the time tendency term can be obtained as daily averages. The residual of all the other terms can be treated as the time tendency plus numerical truncation errors.

##### 2) Vorticity equation

*δ*(…) means the difference along the direction of its superscript. The Δ

_{ x }, Δ

_{ y }, and Δ

_{ z }are grid widths in their corresponding directions. As the curl operator is linear, the closure of the vorticity budget is guaranteed by that of the momentum budget. Taking the curl of (A4), the vorticity equation in POP can be written as

As discussed in section 3, the curl of the pressure gradient is nonzero in the discretized equation, unlike in the continuous form. The curl of the Coriolis term is further broken down into two terms, and the detailed descriptions of the decomposition is documented in section c(2) below.

*D*= ∇ ⋅

**v**. With that, the decomposition of the nonlinear advection becomes:

We can further combine the first two terms on the right-hand side of the equation into a flux form advection term for the vertical vorticity *ζ*. The last two terms would be the two additional terms in the discretized form and result in a potential residual when decomposing the nonlinear term.

#### c. Discretized nonlinear terms

##### 1) Momentum advection schemes in POP

Illustration of the B grid in the POP model.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Illustration of the B grid in the POP model.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

Illustration of the B grid in the POP model.

Citation: Journal of Physical Oceanography 51, 12; 10.1175/JPO-D-20-0223.1

*e*”), north wall (subscript “

*n*”) and top (subscript “

*t*”) of the U cells. The horizontal volume fluxes are calculated with a two-step average and the vertical volume flux is calculated through the continuity equation:

*x*direction advection as an example,

In its continuous form, the sum of the first three terms should equal zero. However, in the finite difference form here, even though

##### 2) Decompose the derivative of products

The terms *A*
_{0} and *B*
_{0} are often chosen as the two-dimensional arithmetic means of *A* and *B* (*A*
_{0} and *B*
_{0} can be taken out of the averages in the first two terms in (A14), and the third term is nonzero. We use the decomposition of the Coriolis term as an example to further illustrate this.

*f*

_{0}=

*f*|

_{ t }, i.e., the Coriolis parameter evaluated at the center of T cell, and apply the second-order approximation of the Coriolis parameter

*f*as in Yeager [2013, (C.13)–(C.16)], (A16) becomes

It can be shown that the third numerical term is at least two orders of magnitude smaller than the two leading physical terms. We will therefore ignore it. The major convenience of this form is that the first part of (A17) naturally becomes *f*∂_{
z
}
*w* in the model.

##### 3) Decompose the nonlinear term

To minimize the potential residuals from the decomposition of the nonlinear term, we need to carefully disassemble, rearrange, and assemble the discretized nonlinear momentum terms after applying the curl operator. The third term on the right-hand side of (A14) would have become a large residual term if we follow the decomposition similar to the Coriolis term, since the momentum components (*u*, *υ*, *w*) are highly correlated to each other.

*A*

_{0}and

*B*

_{0}in (A14) so that the third residual term is zero. The first part of the nonlinear term is

*uue*,

*υun*, and

*wuk*are volume fluxes through the interface as defined in (A9)–(A11). In the first step (A18a), the curl operator is applied to the momentum terms. Here a subscript is added to

*δ*operator to denote whether it is from the curl (

*c*) or advection (

*a*) operation. In the second step (A18b), the order of the difference operation from curl and from advection is switched. To avoid introducing additional terms, we move the denominator Δ

_{ x }inside of the inner difference operator

*δ*

^{ x }. We add an asterisk to this Δ

_{ x }to show that it is not located at the same point of

*υ*

_{ e }and

*u*

_{ e }, but rather at the U cell to the west of it. Finally, we break down the terms in the third step (A18c). The advection of the vertical vorticity in the

*x*direction is given by the first and third term in (A18c).

*y*and

*z*axes are derived similarly:

The twisting terms (including the stretching and tilting of the relative vorticity) are represented by the second and fourth terms in (A21)–(A23).

This term results from the nonzero divergence in the momentum equation, as we discussed in section b(1) above. Note that at this stage, this is the only nonphysical term introduced from the decomposition.

##### 4) Flux form of the twisting terms

Therefore, we can essentially *hide* the nonzero divergence term with the twisting terms with the help of the flux form. This allows us to cleanly decompose the nonlinear term without introducing any numerical terms.

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^{1}

Technically, nonuniform grid spacing can produce a small value of the curl of the pressure gradient term in the interior ocean, which is generally negligible when the size differences of the adjacent cells are small. For the 0.1° resolution model used here, this term is several orders of magnitude smaller than the other terms in the vorticity equation.

^{2}

In addition to advection, the total nonlinear term in the momentum equations also includes the metric term, which is a function of Earth’s radius. Although the metric terms are included in our offline calculation of the mean nonlinear term, it is worth noting that the term is generally two orders of magnitude smaller than the advection. Therefore, we will refer to the nonlinear term from the momentum equation simply as advection hereafter.

^{A1}

However, not exactly, as a semi-implicit time discretization scheme is used for the Coriolis term to damp the computational mode.