The Ability of Barotropic Models to Simulate Historical Mean Sea Level Changes from Coastal Tide Gauge Data

Abstract

The nature of mean sea level variation over the global coastal ocean is considered based on 219 historical tide gauge records and three barotropic ocean circulation models forced by reanalysis surface air pressure and wind stress. The consistency of the models and their ability to reproduce the data are considered on nonseasonal timescales (seasonal cycles and linear trends removed) from bimonthly to multidecadal over 1900–2010. Models consistently simulate stronger sea level variability at higher latitude, higher frequency, between winters, and over broad shallow shelves and semi-enclosed marginal seas; standard deviations in modeled monthly sea level grow from 1–2 cm on average at low latitude (0°–30°) to 5–10 cm at high latitude (60°–90°), with larger values simulated over some shelf areas (e.g., North Sea). Models are more consistent over narrow shelf regions adjacent to deep basins and less consistent along the broad shallow continental shelf. On monthly timescales, discrepancies between models arise mostly from differences in model configuration (e.g., fine vs. coarse horizontal resolution), whereas model configuration and surface forcing (i.e., choice of atmospheric reanalysis) contribute comparably to model differences on annual timescales. Model solutions become more uncertain at earlier times (e.g., prior to 1950). The models show more skill explaining variance in tide gauge data at higher latitude, higher frequency, between winters, and over broad shallow shelves and within semi-enclosed marginal seas; at middle and high latitudes (poleward of 45°), model sea level solutions on average explain 30–50% of the monthly variance and 35–70% of the variance from one winter to the next in the tide gauge data records. Statistically significant relationships between the model solutions and observational data persist on long decadal periods. The relative skill of individual models is sensitive to region and timescale, such that no one model considered here consistently performs better than the others in all cases. Results suggest that barotropic models are useful for reducing noise in tide gauge records for studies of sea level rise and motivate additional model comparison studies in the context of sea level extremes.

Appendix: Scaling of the Relative Influences of Barotropic and Baroclinic Effects on \(\zeta\) Variability

Following past works (Gill and Niiler 1973; Willebrand et al. 1980; Frankignoul et al. 1997), we perform a scaling analysis to establish the basic controls on the relative contributions of barotropic and baroclinic processes on $\zeta$ variability. The words “barotropic” and “baroclinic” have various usages and meanings in physical oceanography and geophysical fluid dynamics (Holton 1992; Pedlosky 1987). Here, we use these words in a similar sense to the separation of variables and normal vertical modes in a continuously stratified fluid (Gill 1982; Kundu and Cohen 2004). We define the barotropic component of a depth-dependent variable as the depth-averaged value, and the baroclinic component at a given depth as the deviation from the depth average. In the case of $\zeta$, which has no explicit depth dependence, we define the barotropic component as the scaled depth-averaged hydrostatic pressure p (cf. Gill 1982),

$$\begin{aligned} \left\langle \zeta \right\rangle \,{\mathop{=}\limits ^{.}}\, \frac{1}{\rho _0 g} \frac{1}{\zeta +H} \int _{-H}^\zeta p(z) \ \mathrm{d}z, \end{aligned}$$

(11)

where H is ocean depth, $\rho _0$ a constant reference density, and g gravitational acceleration. The baroclinic $\zeta$ component is defined as the difference between total and barotropic components,

$$\zeta ^{\prime\prime} \,\,{\mathop{=}\limits ^{.}}\,\, \zeta - \langle \zeta \rangle .$$

(12)

Usage of angled brackets and double primes to denote the barotropic and baroclinic components follows Peixoto and Oort (1992). It can be shown that, by applying the hydrostatic equation,

$$\begin{aligned} \frac{\text{d}p}{\text{d}z} = - \rho g, \end{aligned}$$

(13)

$\zeta ^{\prime\prime}$ can be rewritten in terms of a double integral of ocean density $\rho$,

$$\begin{aligned} \zeta ^{\prime\prime} = -\frac{1}{\rho _0} \frac{1}{\zeta +H} \int _{-H}^\zeta \int _z^0 \rho (z') \ \mathrm{d}z' \ \mathrm{d}z. \end{aligned}$$

(14)

See Piecuch et al. (2015) for more details on how (14) can be derived from (13) and (12).

To compare the magnitudes of $\left\langle \zeta \right\rangle$ and $\zeta ^{\prime\prime}$ variations, we consider basic ocean dynamics. Given the timescales of interest to this study (monthly to decadal), we generally expect an equilibrium (time-independent) barotropic response to surface atmospheric forcing, while we anticipate a transient (time-dependent) baroclinic adjustment. For simplicity, we suppose that the only surface forcing is by a variable wind stress $\varvec{\tau }$. (Indeed, the above expressions for $\left\langle \zeta \right\rangle$ and $\zeta ^{\prime\prime}$ neglect any surface loading by barometric pressure.) Thus, to scale the p integrand in Eq. (11), we assume that the equilibrium barotropic adjustment takes the form of a Sverdrup balance,

$$\begin{aligned} {\mathbf J} \left( p, \frac{H}{f} \right) = - \nabla \times \left( \frac{\varvec{\tau }}{f} \right) , \end{aligned}$$

(15)

where ${\mathbf J}$ is the Jacobian operator, $\nabla \times$ is the vertical component of the curl operator, and f is the Coriolis parameter. Eq. (15) represents a balance between vertical stretching of the water column by the curl of $\varvec{\tau }$ and advection of the water column across contours of ambient potential vorticity by geostrophic flows (Pedlosky 1987). Combining Eqs. (11) and (15), we find that $\left\langle \zeta \right\rangle$ is of order,

$$\begin{aligned} \mathcal {O} \left( \left\langle \zeta \right\rangle \right) \sim \frac{\varvec{\tau } f}{\rho _0 g H \beta }, \end{aligned}$$

(16)

where $\beta$ is the meridional derivative of f.

To scale the $\rho$ integrand in Eq. (14), we assume the transient baroclinic response is such that density changes arise solely due to heaving of the mean stratification by an anomalous velocity w,

$$\begin{aligned} \frac{\partial \rho }{\partial t} = \rho _0 \frac{N^2}{g} w, \end{aligned}$$

(17)

where $N^2$ is the square of the buoyancy frequency. For simplicity, we assume a vertical structure for w such that it is equal to the Ekman pumping velocity,

$$\begin{aligned} w_{\text{ek}} = \nabla \times \left( \frac{\varvec{\tau }}{\rho _0 f} \right) , \end{aligned}$$

(18)

within some shallow upper layer of vertical thickness $\delta$, and zero everywhere below that depth. Combining Eqs. (14), (17), and (18), we find that $\zeta ^{\prime\prime}$ scales as

$$\begin{aligned} \mathcal {O} \left( \zeta ^{\prime\prime} \right) \sim \frac{N^2 \delta \varvec{\tau }}{\rho _0 \omega g L f}, \end{aligned}$$

(19)

where $\omega$ is the angular frequency of the density variation and L is a representative length scale of the surface forcing. Thus, the ratio of the barotropic to the baroclinic $\zeta$ term is on the order of

$$\begin{aligned} \frac{\mathcal {O} \left( \left\langle \zeta \right\rangle \right) }{\mathcal {O} \left( \zeta ^{\prime\prime} \right) } \sim \frac{\omega f^2 L}{N^2 \delta H \beta }. \end{aligned}$$

(20)

That is, barotropic response becomes increasingly important relative to baroclinic adjustment for progressively higher latitude ($f^2/\beta$ increases in magnitude with latitude), higher frequency, larger spatial scale, weaker stratification, and shallower water column thickness. These simple arguments are borne out and corroborated by more realistic ocean general circulation model experiments (Vinogradova et al. 2007; Bingham and Hughes 2008).