## 1. Introduction

The concern over rising sea level has led to numerous modeling studies and assessments of observational evidence from satellite altimetry, tide gauges, and other related datasets [see, e.g., Stocker et al. (2014), Church et al. (2011), and Milne et al. (2009) for a review]. However, because variability in sea level represents an integration of many aspects of climate change, to formulate projections and to understand contemporary changes in sea level involves consideration of changes in the hydrosphere and cryosphere, as well as the solid Earth, and the complexity of the problem remains challenging (e.g., Stammer et al. 2013). Ocean dynamics and associated redistributions of heat and freshwater are one of the major contributors to variability in sea level on a variety of time and spatial scales (Landerer et al. 2007; Yin et al. 2010; Stammer et al. 2013). Changes in ocean circulation can also affect the distribution of mass over the globe. The variable oceanic mass field in turn loads Earth and changes its gravity field through the processes of self-gravitation and crustal deformation. The ocean responds to such gravity field perturbations by adjusting its mass (and sea level) fields. Such adjustments are commonly referred to as self-attraction and loading (SAL) effects and can have a measurable impact on sea level (Mitrovica et al. 2001; Tamisiea et al. 2010). Their importance for tidal studies has long been recognized, as discussed by Ray (1998) and Arbic et al. (2004).

The theory of SAL, which describes the effects of self-gravitation and associated changes in geopotential and deformation of the ocean floor, has been formulated in the early works of Farrell and Clark (1976) and applied, for example, to tide gauge records to estimate implied twentieth-century land ice melting rates (Mitrovica et al. 2001) and understand the seasonal cycle in sea level (Tamisiea et al. 2010). Other studies have focused on the impact of SAL effects on variations in ocean mass on monthly to decadal time scales using GRACE (Vinogradova et al. 2011; Riva et al. 2010) and in situ bottom pressure data (Vinogradova et al. 2010). The effects of SAL on longer time scales, up to centennial, were discussed by Kopp et al. (2010). In all those studies, SAL effects are inferred by solving a sea level equation under the assumption that the ocean’s response to SAL is static and in equilibrium with the forcing. In such cases, the ocean is assumed to shift mass around rapidly and maintain negligible horizontal pressure gradients. While the equilibrium assumption is expected to hold at sufficiently low frequencies, the exact dependence on time scale was never properly addressed before and is one motivation of the present study.

One way to account for possible nonequilibrium sea level signals is to implement SAL physics in ocean general circulation models, but such instances are rare in part because of the high computational costs that can be involved. Among the first attempts of such implementation are studies by Stepanov and Hughes (2004) and Kuhlmann et al. (2011), who considered SAL within barotropic and baroclinic ocean models, respectively. In particular, Stepanov and Hughes (2004) show that integrating a model with the calculation of SAL effects, using a “prohibitively expensive” global convolution integral at each grid point and time step, can “occupy more than 90%” of the computing time. Kuhlmann et al. (2011) used a different computational approach based on spherical harmonics decomposition to incorporate SAL physics in their model. Their experiments were more successful in terms of computational efficiency, increasing computing time by only ~16%.

Here, we expand on these studies by implementing the physics of SAL in a baroclinic ocean model and focusing, in particular, on the possibility of having sea level dynamic signals induced by the SAL effects. Such deviations from equilibrium can depend on several factors, including coastal geometry, bottom topography, and so on. Drawing parallels with the ocean response to surface loading related to atmospheric pressure (Ponte 1993) and freshwater fluxes (Ponte 2006), such nonequilibrium signals are expected to be more significant for short time scales and in shallow and enclosed coastal regions.

In this initial SAL study, the only mass variations producing SAL effects are those induced by the ocean circulation. For simplicity, effects from other mass loadings such as those associated with land water, ice, and atmospheric pressure and discussed, for example, by Tamisiea et al. (2010) and Vinogradova et al. (2010, 2011) are not addressed here. In what follows, we describe the physics of SAL and its implementation in the ocean model in section 2, examine the results focusing on sea level variability induced by the SAL effects in section 3, and discuss the nonequilibrium response as a function of time scale and location in section 4. Summary and conclusions are presented in section 5.

## 2. Approach

### a. Basic equations

*α*can be written in the form of a Green’s function as

^{6}m) and mass (5.9736 × 10

^{24}kg), respectively. The constant of proportionality equal to

*θ*is latitude and

*λ*is longitude, taken to represent mass anomalies vertically integrated over the full ocean depth. For the purposes of this paper, we can treat

*α*is the angular distance between

*n*and

*m*are the degree and order of the spherical harmonic decomposition, and

^{−3}]. The computational cost then becomes mostly associated with transforming between grid space and spherical harmonics, but as discussed below, one can make use of efficient and readily available transform algorithms for significant computational gains.

### b. Implementing SAL physics in an ocean model

Similar to atmospheric pressure forcing in a baroclinic ocean (Ponte and Vinogradov 2007), the effects of

The above approach is implemented within the Massachusetts Institute of Technology (MIT) general circulation model [MITgcm; an evolved form of the model described in Marshall et al. (1997), Adcroft et al. (2004), and Campin et al. (2008)], which has an option to include surface loading such as atmospheric pressure in its forcing fields. The model setup used here provides an estimate of the ocean state on a 1° horizontal grid (but refined to about ⅓° near the equator in the meridional direction) and includes a dynamic sea ice component (Losch et al. 2010; Heimbach et al. 2010). Subgrid-scale parameterization of vertical mixing is achieved via the nonlocal *K*-profile parameterization (KPP) scheme of Large et al. (1994) and parameterization of geostrophic eddies are by Redi (1982) and Gent and McWilliams (1990). The model setup is that created by Gael Forget (MIT) as part of the latest version of the ECCO state estimates discussed in more detail by Wunsch and Heimbach (2013) and Speer and Forget (2013).

To implement SAL physics in the MITgcm, we have created a suite of codes that include Eq. (3), spherical harmonic decomposition, regridding routines, and so on and organized them using the basic MITgcm packaging structure so that one can easily enable and disable SAL physics if needed. The package is computationally efficient, and for the experiments considered here the inclusion of SAL physics leads to an increase of the computation time by less than 6% (timing tests were based on runs on 8 and 16 processors). The efficiency is in part because of a fast and accurate method of forward and inverse transform from a spatial grid to spherical harmonics and back. Here, we used the spherical harmonics software package SHTOOLS, developed by M. Wieczorek, which is freely available and includes routines that use the Driscoll and Healy (1994) sampling theorem to transform an equally spaced grid into spherical harmonics and the inverse transform. This sampling theorem, together with the use of a fast Fourier transform algorithm in longitude, requires *O*(*N*^{2} log*N*^{2}) operations to transform a *N* × *N* grid, versus *O*(*N*^{4}) operations using the basic formulas (Blais and Provins 2002). For our purposes, *N* was set to 360, which was chosen as a trade-off between computational cost and minimization of interpolation errors and which defines the maximum degree in spherical harmonic decomposition *l* = *N*/2 − 1 = 179. The spherical harmonic truncation to degree 179 is equivalent to 1° horizontal grid resolution. We use this maximum truncation degree because it is consistent with the ocean model resolution. Also, the computation of SAL is an inherent smoother that tends to minimize interpolation and truncation errors. The timing test for transforming the 1° grid to spherical harmonics and the inverse transform back to a grid takes ~0.01 s.

### c. Experiment design

To compute the sea level response to

To examine the ocean response to SAL effects on a wide range of time scales,

## 3. SAL effects associated with the variable ocean circulation

To better understand SAL-induced changes in sea level, let us first examine the magnitude of the expected mass variations, in our case those related to the ocean circulation. For most of the oceans, the standard deviation in

Figure 2 shows the standard deviation of SAL forcing

One of the common approaches to account for SAL effects is to use a simple parameterization by multiplying the mass field by a constant factor, a practice adopted for tidal models since Accad and Pekeris (1978) (see discussion in Ray 1998). To show possible limitations of such an approximation, Fig. 3 displays the ratio of standard deviations in SAL forcing

The standard deviation of

## 4. Dynamic and equilibrium response

^{1}As a crude test of our procedures, we compared monthly averaged values of

From the standard deviation of

For a quantitative assessment of the importance of the dynamic signals, one can look at the ratio of the standard deviation of

To assess the dependence of the results on time scale, Fig. 6b shows the same ratio but based on time series of ^{2} However, despite the general decrease, the ratio in Fig. 6b is still >0.2 in many places, indicating that the departures from equilibrium are not totally negligible at time scales longer than a week and that studies of

*ω*. To best resolve high-frequency fluctuations, the analysis here is done using hourly series of

Values of *r* ~ 1 indicate that power in *r* > 1 at periods <1 day, but that threshold period can be up to 2 days in the open-ocean regions and up to a week in the East Siberian Sea and Bay of Bengal. In general, the tendency for nonstatic response does increase with frequency. Values of *r* in Fig. 7 increase by more than a factor of 10 from the longest to the shortest periods. Similar conclusions on the IB approximation were reached by Ponte (1993), who reports global nonequilibrium behavior at periods shorter than ~2 days.

At some locations, for example, Bay of Bengal, the frequency structure of the

Besides the expected tendency for nonequilibrium response to occur at sufficiently short time scales, for which adjustment of the mass field may not occur fast enough (e.g., Ponte 1993), other factors that can lead to deviations from equilibrium are related to coastal geometry and ocean depth. For example, the potential for nonequilibrium response increases where resonance can occur, such as in semienclosed regions like Hudson Bay associated with Helmholtz-type resonant response (e.g., Mullarney et al. 2008). Shallow bathymetry, as found for example in the coastal Arctic and Patagonian shelf, is another factor that can induce a stronger dynamic response (Fig. 7) because of factors such as a much slower gravity wave propagation and consequent longer adjustment time scales.

Apart from their primary high-frequency nature,

## 5. Conclusions

Our main goals here were to include the full physics of SAL into an ocean model in a computationally efficient way and to assess the potential for dynamic behavior (i.e., departures from equilibrium) in the oceanic response to SAL effects as a function of location and time scale. In terms of computing cost, the implemented SAL package is very efficient and leads to an increase of the computation time by less than 6%, similar to (but smaller than) the values reported by Kuhlmann et al. (2011), who use similar method of Love number theory to compute SAL effects. The results confirm that implementation of SAL physics based on spherical harmonics becomes computationally feasible and is no longer prohibitive as found by Stepanov and Hughes (2004).

The effects of SAL associated with the variable ocean circulation, which we have examined, can result in measurable sea level signals, approaching 1 cm in places. Amplitudes of SAL-induced fluctuations are dependent on the size of the mass loads associated with the ocean circulation, and the omission of such effects amounts to approximately a 10% error in sea level values on average, which might be comparable to other model errors. Our results also indicate that simple parameterizations of SAL effects using constant scaling factors can induce further errors, in agreement with Stepanov and Hughes (2004).

An important innovation of this study and those of Stepanov and Hughes (2004) and Kuhlmann et al. (2011) is the ability to calculate potential dynamic signals associated with SAL effects. This dynamic response cannot be resolved, for example, by solving the sea level equation (Tamisiea et al. 2010), which is the more traditional approach. Our results show that the nature of the sea level response to SAL strongly depends on location and time scale, with most energetic deviations from equilibrium occurring at time scales ranging from daily up to a week and reaching 1 cm in amplitude. The nonequilibrium response produced by SAL also tends to have a large-scale spatial structure. Shallow depths and constricted coastal geometries also seem to enhance the propensity for dynamic behavior. These features are very similar to those found in the oceanic response to other surface loads (Ponte 1993, 2006).

Given the presence of high-frequency dynamic fluctuations in sea level produced by the SAL effects and their importance compared to the total variability, studies dealing with changes in sea level on subweekly time scales might benefit from including SAL physics implicitly into ocean models. Examples include modeling of short-period tides (Ray 1998) and modeling of high-frequency signals to dealias satellite altimetry and gravity missions (Quinn and Ponte 2011). In studies of low-frequency variability, the dynamic component is not an issue as long as one is averaging sea level records over relatively long periods of time, that is, monthly and longer.

Finally, we recall that the only mass variations considered here are those produced by the variable ocean dynamics. In the future, it would be useful to examine the ocean response to SAL effects produced by other mass loadings such as high-frequency land hydrology and atmospheric pressure changes. One could also try joint simulations of the tides and the ocean circulation to examine how the full implementation of the physics of SAL used here can affect high-frequency tidal dynamics as well. These topics are left for future study.

## Acknowledgments

This work was supported by the NASA Interdisciplinary Science Program through Grant NNX11AC14G and NSF Grant OCE-0961507, and partly by NASA Sea Level Change Team (N-SLCT) Project through Grant NNX14AP33G. We thank Gael Forget and Patrick Heimbach (MIT) for all their efforts on creating the model setup used in our experiments and for useful discussions on modeling issues.

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^{1}

The spatial mean of

^{2}

The reasons for these substantially higher values remain unclear and may be related to topography or other physical features, but numerical issues are also possible, and thus those results are to be treated with caution.