1. Introduction
Sea level variations impact coastal communities, making them a subject of great societal importance. Climate change induces long-term trends in global and regional sea level (e.g., Nicholls and Cazenave 2010; Kemp et al. 2011). Meanwhile, air–sea fluxes of buoyancy and momentum account for much of the nontidal variability on shorter time scales, ranging from hours to months. Understanding the ocean sensitivity to atmospheric forcing is useful for illuminating local dynamics and thus predicting the regional ocean response to weather events and climate variability. This study highlights the physical pathways that force high-frequency variability of coastal sea surface height (SSH) in a state estimate of the California Current System.
The California Current System is a region of high biological productivity sustained by wind-driven coastal upwelling (Huyer 1983), which brings nutrient-rich deep waters to the sunlit surface layers. Seasonal and interannual variability in upwelling due to shifting wind patterns influences plankton and fish populations of the California Current Ecosystem (Rykaczewski and Checkley 2008; Chenillat et al. 2013). Understanding the variability in coastal SSH, which is intrinsically linked to coastal upwelling, may bring new insight on the processes leading to observed fluctuations in nutrient supply and biological productivity. Surface wind stress and buoyancy fluxes also influence the local ecosystem by setting up the stratification and horizontal circulations (Checkley and Barth 2009). As surface geostrophic transport is determined by the difference between near- and offshore SSH, understanding the sensitivity of coastal SSH is a first step in understanding what controls the transport of the California Current and undercurrent.
While local winds affect coastal SSH through Ekman transport and pumping, coastally trapped waves can carry the effect of nonlocal winds (Brink 1991). In the California Current System, coastally trapped waves are found to propagate sea level anomalies over hundreds of kilometers (Battisti and Hickey 1984; Chapman 1987; Pringle and Riser 2003). Using a numerical simulation, Pringle and Dever (2009) showed that coastal currents in the Southern California Bight are influenced by remote winds both within and outside the California coast region. Ryan and Noble (2006) examined the relation between sea level on the California coast and alongshore winds through a regression analysis between tide gauge and wind buoy data, including local (within ~100 km) and remote [O(1000 km)] winds. They found that nonlocal winds contribute significantly, their effect lasting more than 3 months at some stations. They also considered responses at longer periods, including ENSO, but did not exclude ENSO effects on sea level that are unrelated to winds.
An adjoint model separately evaluates the sensitivity to each forcing mechanism, thus avoiding statistical noise and correlations between forcing mechanisms that can affect regression analyses. Veneziani et al. (2009) used an adjoint model to analyze the sensitivity of several aspects of the California Current System, including coastal SSH. Consistent with the aforementioned studies, they find that coastal SSH is sensitive to both local and remote forcing, that the sensitivity persists for at least the 2-week duration of their adjoint run, and that the sensitivity depends on the forward model ocean state. Identifying and quantifying the mechanisms and pathways of sensitivity propagation remains, however, an open question, as is the propagation of sensitivity southward past Point Conception.
This work uses the adjoint of an ocean general circulation model to diagnose the response of SSH near Port San Luis, off the coast of central California, to atmospheric forcing. Linearized physical pathways propagating the effects of surface wind stress, heat fluxes, and freshwater fluxes are identified through the adjoint sensitivity analyses. The goal is to analyze the evolution of the sensitivity over the model domain for times ranging from hours to months, thus addressing the question of what forces SSH variability in both time and space.
Our analysis extends the work of Veneziani et al. (2009), who performed a 14-day adjoint model run to compute the sensitivity of squared SSH over the 14-day period and over an approximately 50-km strip adjacent to the coast between 35° and 40°N. In contrast, by using higher viscosity to damp small-scale structures, which tend to be nonlinear, the experiments presented here use a 100-day adjoint model run and compute the sensitivity of SSH (i.e., a linear objective function) at a single point in space and time. The lack of averaging in the objective function makes it easier to follow the back propagation of different wave types and modes from a single target point.
The adjoint-derived sensitivities are SSH response functions that can be Fourier transformed into transfer functions for comparison with those estimated from observations by Ryan and Noble (2006). We examine both the response (section 4a) and the transfer function (section 4c) with respect to momentum input from the wind, the main driver of SSH variability. The SSH response to local and remote winds clearly shows the barotropic storm surge response during the first few hours as well as the slower propagation of coastally trapped waves and inertia–gravity waves. Integrations in summer and winter reveal differences between the wind-driven responses due to the influence of seasonally varying stratification (as discussed in section 5).
Because the adjoint model solves a set of equations that are linearized around the background ocean state, adjoint-derived sensitivities are a linear approximation to the nonlinear SSH response. Linear analysis provides an insight into the physics, and is commonly employed to compute transfer functions (e.g., Ryan and Noble 2006) and wave propagation (e.g., Pringle and Riser 2003). Nonlinear effects may be significant, especially at shorter length scales, and adjoint model results should be interpreted cautiously (Powell et al. 2008; Moore et al. 2009; Zhang et al. 2011, 2012). However, for the problem of interest, the linearity assumption is shown to hold well for the 100-day period of integration. This is demonstrated by the nonlinearity check in perturbation experiments (section 3); the high skill from the hindcast experiment, which is linear in nature (section 4b); and the success of the multiyear state estimation process, which employs adjoint-derived sensitivities to minimize the model–observations misfit (as briefly described in section 2).
2. Methods: An adjoint model for state estimation and sensitivity analysis
A state estimate of the California Current System has been produced for the period from 1 January 2007 to 31 July 2009 using the machinery developed by the Estimating the Circulation & Climate of the Ocean (ECCO) consortium (Wunsch and Heimbach 2007). This involves optimizing inputs to the Massachusetts Institute of Technology (MIT) General Circulation Model [MITgcm; evolved from Marshall et al. (1997)] to bring the model solution into consistency with observations. Optimization is accomplished iteratively via the adjoint method [described, e.g., in Stammer et al. (2003) or Wunsch (2006)], using observations available for the time window of the simulation to adjust “controls” consisting of atmospheric state, open boundary conditions, and initial conditions for the model. The state estimate is a free-forward simulation using the optimized controls. The long-term assimilation window (942 days) means that the observations are combined in a continuous, dynamically consistent hindcast (without unphysical nudging terms).
The model domain, shown in Fig. 1, extends from 27.2° to 40°N and from 130°W up to the coast. It encompasses most of the California coastline and ends just south of Punta Eugenia in Baja California. The model has a 
Model domain. Shaded contours show the bathymetry with a 1000-m interval. White contours split the domain into three regions: the “local” area within 1° of Port San Luis tide gauge (Port San Luis), the “coastal” area with a depth <1500 m, and the “offshore” area. Letters mark specific locations of interest for the analysis.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
Parameter values used for forward (FWD) and adjoint (ADJ) model runs.
The model and solution have been previously described in Todd et al. (2011, 2012), who analyzed the 64th iteration of the assimilation. These studies found the state estimate to capture the structure and temporal variability of ocean currents on scales larger than about 30 km, although simulated current speeds tended to be smaller than observed. The present study uses the 108th iteration (described in section 2a), which is largely consistent with observations. Having a realistic forward model state and dynamics gives confidence that the response of the state estimate is representative of processes in the real ocean, and that it is useful for analyzing the response of SSH to wind perturbations.
a. Estimated ocean state
The state estimate is analyzed primarily in the central California coast area, around Port San Luis. This region is sufficiently far from the model’s open boundaries, which can sometimes introduce artifacts in the solution. The Port San Luis tide gauge, located at 35.18°N, 120.76°W, provides a continuous record of SSH during the time period considered. The simulated time series of SSH variations near the coast at 35.1°N (close to the actual location of Port San Luis) is compared to the tide gauge data. As can be seen in Fig. 2, the state estimate captures the amplitude of observed interannual variability, but not the timing, leading to a poor correlation of the two time series. At a low frequency, the power spectrum of SSH from the state estimate is similar to that of the observations (Fig. 2b); however, the model energy is low at higher frequencies, in the same way that the power spectrum of the local NAM winds is also low compared to wind buoy data (Fig. 2c). Farther offshore, the simulation is in good agreement with altimetric data. In Fig. 2d, the simulated SSH near Monterey Bay (36.5°N, 122.3°W) is compared to the optimally interpolated Archiving, Validation, and Interpretation of Satellite Oceanographic data (AVISO) product (http://www.aviso.oceanobs.com/). The model is constrained to along-track altimetric observations, among other data sources, but not to this gridded AVISO product. A Jason altimetry line falls close to that location, which is coastal but off of the shelf (the local ocean depth is 1300 m).
(a) Time series of SSH anomalies at Port San Luis: observed (gray) and modeled (black), smoothed with a 10-day running mean. (b) Power spectral density of observed and modeled SSH anomalies at Port San Luis, estimated via Welch’s method [data from the National Oceanographic and Atmospheric Administration (NOAA) Center for Operational Oceanographic Products and Services; http://tidesandcurrents.noaa.gov/]. (c) Power spectral density of meridional wind anomalies at the Santa Maria buoy, observed (gray; from National Data Buoy Center; http://www.ndbc.noaa.gov/) and NAM (black) winds. (d) Time series of SSH anomalies near Monterey Bay (36.5°N, 122.3°W): AVISO (dotted gray curve), Jason (crosses), and modeled (black curve).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
Stratification, which strongly influences the effect of winds on SSH (see sections 4a and 4c), is consistent with observations (e.g., the Spray gliders; Todd et al. 2011). Observed and simulated mixed layer depths (MLD) around Port San Luis are shown in Fig. 3. MLD is calculated, following de Boyer Montégut et al. (2004), as the depth where density exceeds the surface density by Δρ = 0.5 kg m−3 (other density thresholds give qualitatively similar results). The MLD near Port San Luis exhibits a strong seasonal cycle and there is good agreement with MLD calculated from densities inferred from glider data, with the exception of the amplitude of the deep mixed layers observed during the winters of 2007 and 2008.
MLD in a 1° × 1° box near Port San Luis. Running mean (30 day) of the simulated (solid curve) and monthly-averaged MLD calculated from observed temperature and salinity from Spray gliders (gray crosses; from Todd et al. 2012).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
Discrepancies between the California Current System state estimate SSH and the tide-gauge observations at Port San Luis (Fig. 2a) likely involve strong small-scale gradients not resolved in the model and, to some extent, the model not resolving finescale bathymetry and coastal structure. Comparing a single model grid point with a single tide gauge is a hard test for any model. At wider spatial scales, the state estimate agrees well with the observations, capturing the forward model ocean state and its statistics while obeying the physics as represented by an ocean general circulation model.
b. Adjoint sensitivities
The same adjoint model used for optimization of the state estimate can be employed to perform a sensitivity analysis. The adjoint model calculates the partial derivatives of all state variables around a time-dependent “forward” model trajectory. Adjoint sensitivities show where and when perturbations in the model forcing would lead to a specific response. (In contrast, a forward perturbation experiment shows where and when the system responds to a specific perturbation.) Here the response considered is the change in SSH at Port San Luis. In essence, we perturb the SSH and iterate the ocean physics backward to determine what model inputs are capable of making such a change. To determine the sensitivity of SSH to an earlier model state using the forward model would require many simulations with a suite of model input perturbations, and thus be computationally prohibitive. The adjoint method has been widely used to quantify the relative importance of different forcing mechanisms, whether it be the type or scale of forcing; recent examples include Moore et al. (2009), Veneziani et al. (2009), Zhang et al. (2011), Tulloch et al. (2011), and Todd et al. (2012). Losch and Heimbach (2007) provide additional information on the use of adjoint models to probe the sensitivity of ocean general circulation models. The equivalence between adjoint and regression analyses is explained in Zhang et al. (2011). A brief description is given below for readers who may not be familiar with the adjoint method.
Objective functions J can be arbitrarily complex functions of the response y. Quadratic functions are a common choice, mainly because of the similarity to the least squares objective function used in adjoint optimization. For example, Veneziani et al. (2009) considered the function J = η2, integrated along the coastline and over a 2-week period. For a quadratic objective function, the sensitivity is given by 
Three adjoint runs have been analyzed for this work. In the first, y is defined as the SSH at x = (35.1°N, 120.8°W) and t = 0000 UTC 1 February 2008. In the second, t = 0000 UTC 1 August 2008 and in the third, t = 1200 UTC 1 August 2008 (with x unchanged).
The adjoint is run backward in time for 100 days from the time of J (i.e., 1 February 2008 or 1 August 2008), such that sensitivities are calculated for time lags between forcing and response ranging from 0 to 100 days.
A caveat of the method is that the adjoint model is linearized around the forward model state. Thus, the adjoint-derived sensitivities are only accurate for small perturbations around the forward model state. The validity of the linearity assumption in MITgcm simulations can be verified through gradient checks and finite perturbation experiments (e.g., Losch and Heimbach 2007; Mazloff 2012). The fact that adjoint optimization can be employed to produce a multiyear state estimate (i.e., the cost function is reduced) implies that the linear method is useful over long periods of time. A perturbation experiment in the forward model, presented in the next section, confirms that nonlinearities in the response to finite wind stress perturbations remain small over the 100-day integration.
3. Coastal SSH response to a finite wind perturbation
The SSH response to wind stress fluctuations is probed by wind perturbation experiments in forward model runs. Quantifying the response to atmospheric forcing forward in time is more familiar and complements and aids interpretation of the adjoint sensitivity analysis presented in section 4. The experiment involves perturbing the alongshore wind stress near San Diego on 1 July 2007. The perturbation is Gaussian shaped with an amplitude 
The spatiotemporal evolution of the SSH perturbation is shown in Fig. 4. Daily averages are plotted, thus emphasizing the baroclinic response, as the fast barotropic response is smoothed by the averaging. It illustrates the regional dynamics taking place during the response to a positive perturbation (weakening of the poleward-flowing alongshore winds). Initially, the anomalous winds push water toward the coast, resulting in an anomalous tilt in the sea surface. The anomalous SSH excites coastally trapped waves that propagate northward, and Rossby waves that propagate westward and slowly dissipate.
Snapshots of the SSH response to a wind perturbation applied on 1 Aug 2008, in the forward model. The response is shown for t = 1, 3, 10, 30, 60, and 100 days after the end of the perturbation.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
Comparing forward model perturbation experiments forced by identical positive and negative wind stress perturbations tests the validity of the adjoint linearity assumption. We carry out a second experiment where the perturbation amplitude is negative (i.e., 
The evolution of δH1 and δH2, expressed by taking their root-mean-square (RMS) over an area surrounding the wind perturbation (32°–34°N, 117°–119°W), is shown in Fig. 5. Throughout the 100-day simulation, the RMS of the nonlinear response is small compared to the linear response. Thus the modeled response to the 0.1 N m−2 wind stress anomaly analyzed, which is an anomaly of typical magnitude for the region, is largely linear.
Linear (black) and nonlinear (gray) SSH response to a finite wind perturbation in the forward model. The alongshore wind stress is perturbed by ±0.1 N m−2 over 12 h and then brought back to the control value over the next 12 h; this perturbation is applied over a 1° × 1° area near San Diego. The linear response is approximated by the difference between the SSH anomalies resulting from the positive and negative perturbations, and the nonlinear response by their sum (see text for more details).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
The relatively small magnitude of the SSH response is due to the short temporal duration (1 day) and small spatial extent (~7 km) of the perturbation. For the wind perturbation of 0.1 N m−2, the largest SSH anomaly occurs near the perturbation and reaches 0.002 m (Fig. 4). Increasing the magnitude of the perturbation causes a larger SSH anomaly, and a more nonlinear response. For a (huge) wind stress anomaly of 1 N m−2 applied over the same area and the same period of time, the local SSH anomaly reaches 0.015 m. In that case, the magnitude of δH2 becomes comparable to the magnitude of δH1 after about 20 days (not shown). Increasing the duration of the perturbation 20-fold has a similar effect. Thus, the (linear) adjoint-derived sensitivities presented in the next section are good approximations to the true sensitivities over long time scales (100 days) as long as the perturbation is small, such as a typical wind anomaly lasting 1 day. However, for large perturbations, the sensitivities should be interpreted cautiously for time scales longer than a few weeks, as nonlinear effects become significant.
4. Adjoint sensitivity of coastal SSH to atmospheric forcing
In this section, the results of adjoint sensitivity experiments are presented. First, the spatial and temporal patterns of SSH sensitivity to atmospheric forcing are described. The SSH response is then reconstructed from the sensitivities and the forcing; this hindcast offers additional support of the validity of the adjoint method. Finally, spectral characteristics of the sensitivities are presented.
a. Spatial and temporal structure of the sensitivities
Insight into the physics is gained by examining the spatial patterns of sensitivities and their evolution in time. Figure 6 shows selected snapshots of the weighted sensitivity to wind stress 
Snapshots of the sensitivity of SSH on 1 Aug 2008 to (a) along- and (b) cross-shore wind stress over a 36-h period. Sensitivities are weighed by the spatially varying std dev of the wind stress. The time lag between wind perturbation and SSH response is indicated in each box. These maps show the forcing that can drive the coastal SSH response. A positive (negative) sensitivity implies that a positive wind perturbation at that location will cause a positive (negative) change in SSH at Port San Luis. A logarithmic color scale is used; only sensitivities exceeding the threshold (2.5 × 10−9 m) are colored.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
After 24 h, wavelike sensitivity patterns start to develop offshore. The period of oscillation varies between 0.6 and 0.9 day, which is similar to the inertial period near Port San Luis (0.87 day). When surface winds drive near-inertial motions in the mixed layer, horizontal convergence can occur, which leads to the generation of internal waves (e.g., D’Asaro et al. 1995; MacKinnon and Gregg 2005). Time-varying winds can efficiently inject kinetic energy into the near-inertial band of the internal wave spectrum (Alford 2003). Near-inertial waves can propagate onto the continental shelf (e.g., Shearman 2005; MacKinnon and Gregg 2005; Sobarzo et al. 2007), carrying the effects of offshore winds to the coast (Fig. 7).
As in Fig. 6, but for a longer time scale (2–30 days).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
Sensitivities along the coast south of Port San Luis are indicative of coastally trapped waves (Figs. 6 and 7). The sensitivities extend farther southward as the time lag increases, indicating a physical mechanism propagating the effects of the forcing northward. From Fig. 7, it is inferred that the signal propagates at a speed of up to 6 m s−1 and extends 50–100 km offshore. These values are consistent with the length scale and phase speed of coastally trapped waves, which are a mixture of Kelvin and shelf waves (Gill 1982).
While Fig. 6 shows the August sensitivities, the patterns are nearly identical for the February sensitivities. For the longer evolution of sensitivities (Fig. 7), the patterns are qualitatively very similar for August and February sensitivities, although the amplitude is different. The February sensitivities decay faster, especially near the tide gauge and offshore of Port San Luis. The difference is due to seasonal variations in the forward model ocean state, especially the stratification, as discussed in section 5b. Even in the August case, sensitivities have decayed significantly by day 30 (Fig. 7), and remain weak over the 100 days analyzed (not shown). Forward perturbation experiments (not shown) verified these seasonal differences.
These two metrics provide complementary information on the total sensitivity. Here, dJmean quantifies the effect of spatially uniform (basin scale) forcing perturbations, while dJvar is a metric of the magnitude of spatially varying sensitivities. For example, the sensitivity to alongshore winds is large (leading to a large dJvar), but to realize the potential response would require winds with small-scale structure (either concentrated in areas with sensitivities of similar sign, or with a spatial pattern that matches that of sensitivities); a spatially uniform wind stress perturbation will produce a small response (as quantified by the small dJmean). In practice, of course, atmospheric forcing is neither exactly uniform nor does it exhibit such a small-scale structure as seen in the sensitivities (Fig. 7), and care must be taken in interpreting the metrics.
The relative contribution of the different components of forcing to dJmean and dJvar is summarized in Table 2. Although the two metrics give different values, both indicate that alongshore wind is the dominant component (86% of the sensitivity to a uniform forcing and 63%–69% of the sensitivity to spatially varying forcing, depending on season). Cross-shore winds account for a small fraction of the sensitivity to a uniform forcing (1%–10%), but a larger fraction of the sensitivity to spatially varying forcing (21%–23%). The remaining sensitivity comes from heat fluxes and freshwater fluxes.
Attribution of the total sensitivity (measured as dJmean or dJvar) to the four components of atmospheric forcing.
The sensitivity can also be partitioned into local and remote contributions (Fig. 8). The short- and long-term responses are emphasized by looking at the cumulative sum of dJmean and dJvar over the time of adjoint integration. Only the sensitivities to wind stress are included here, because steric effects propagate and dissipate via different dynamics and persist on much longer scales in time and space. The domain is split into three regions for analysis: local, coastal (nonlocal), and offshore (see Fig. 1). The local region is defined as the area within 100 km of Port San Luis. The coastal region is defined as the area where ocean depth is less than 1500 m (excluding the local area). The remaining area is considered offshore. The relative contribution from each region is scaled by the corresponding area.
Cumulative sum of the total sensitivity, as defined by the metrics (a),(b) dJvar and (c),(d) dJmean for February and August. Only along- and cross-shore wind stress are included in the calculation (steric effects are excluded). Contributions from local and nonlocal sources are shown as different shades of gray. The local, coastal, and offshore regions are shown in Fig. 1.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
Local winds dominate the short-term response, whereas the effect of nonlocal winds accumulates more slowly (Fig. 8). The sensitivity to uniform forcing is largely local, with the nonlocal contribution being split almost equally between coastal and offshore winds. In February nonlocal effects exceed the local contribution after about 3 weeks, but in August local effects remain dominant. The sensitivity to spatially varying forcing is dominated by nonlocal winds after only 1 day, and the offshore contribution is approximately twice as large as the coastal contribution. This is explained by the presence of small-scale structures in the sensitivity, especially offshore (due to inertia–gravity waves). The potential for optimal excitation of these wave patterns makes dJvar exceed dJmean by an order of magnitude. Winds are not likely to exhibit such an optimal small-scale structure and this optimum sensitivity regime is not expected to be fully realized; indeed, the hindcast reveals that offshore winds contribute less to SSH variability than coastal winds (see section 3b). Over the 100-day period, the fraction of dJvar that is attributed to local winds is 13% in February and 19% in August. Thus, in both seasons, the cumulative effect of nonlocal winds accounts for more than 80% of the total sensitivity emphasizing connectivity and the potential for nonlocal effects to propagate via excitation of wave dynamics. The relative contribution of coastal winds is slightly larger in February (21%) than in August (18%), and the same is true for offshore winds (66% in February and 62% in August). Overall, the total sensitivity in summer is greater than in winter, because local and offshore sensitivities decay more rapidly in February, whereas they persist in August, because of the lower winter stratification (see section 5b).
To emphasize the seasonal differences, sensitivities are normalized by the cumulative value at day 100 and compared (Fig. 9). When considering the entire domain, both the dJvar and dJmean sensitivity metrics approach the final total faster in February. The evolution of sensitivities at the four specific locations marked on Fig. 1 (a point near Port San Luis, a point offshore, a point on the coast near San Diego, and a point in Sebastián Vizcaíno Bay at the southern end of the domain) clearly show that in February the effect of local and offshore winds becomes small after ~20 days, whereas in August it remains nonnegligible throughout the 100-day simulation. In contrast, winds blowing over the bay and along the coast (equatorward of Port San Luis) have similar effects in February and August.
Normalized cumulative sum of the total sensitivity, as defined by the metrics dJvar for February (gray) and August (black) at four locations. Only along- and cross-shore wind stresses are included in the calculation (steric effects are excluded). All curves are normalized by the value at t = 100 days. Sensitivities are averaged over a 0.5° square area around each location: (a) near Port San Luis, (b) offshore ~250 km southwest of Port San Luis, (c) on the coast to the south of Port San Luis, and (d) in the bay at the southern end of the domain (in Fig. 1 these locations are marked A, B, C, D, respectively).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
Figure 9d highlights the effects of barotropic and baroclinic coastally trapped waves. Rapidly propagating barotropic waves carry the effects of wind perturbations over the bay in less than a day. Farther north, the contribution from barotropic coastally trapped waves appears to be negligible (Fig. 9c). This suggests that barotropic waves are amplified in the bay through some resonance mechanism. Contributions from slower baroclinic coastally trapped waves persist over several months. Their effect builds up more slowly in February, which is consistent with a slower phase speed for Kelvin waves in a less stratified ocean.
b. Hindcasting with adjoint-derived sensitivities
Given that the model output is discrete in time and in space, with a spatial resolution corresponding to the model grid and a 6-h temporal resolution, the integration in Eq. (10) must be discretized as well. If the model was linear and sampled at high enough temporal resolution, the hindcast would correspond exactly to the value of J. Thus, the hindcast also serves as a test of the adjoint and the linearity assumption.
Assuming that sensitivities depend only on time lag and not on the actual time of the objective function calculation allows one to hindcast SSH variations given knowledge of the forcings. The time integral in Eq. (10) then becomes a convolution. Using the model forcings and the sensitivity of SSH determined in August does indeed produce a hindcast that captures most of the variability in the simulated SSH time series (Fig. 10a). Daily averaging the 6-hourly hindcast and comparing to the daily-averaged simulated SSH yields a correlation coefficient of 0.81. Discrepancies in the hindcast and simulated SSH are explained partly by the seasonal variations in the sensitivities, which are not taken into account in this hindcast (only August sensitivities are used). Other sources of differences include nonlinearity, coarse time sampling aliasing the fast response with 6-h sampling of the sensitivities, and numerical noise. Barotropic inflow originating outside the model domain sets the rate of domain-mean sea level change, and thus induces variability in coastal SSH. This implies sensitivity to open boundary conditions, which is not included in the hindcast. The effects of the barotropic inflow were removed by subtracting the domain-averaged SSH from the local SSH at every time step.
Hindcast of SSH at Port San Luis calculated from the sensitivities to atmospheric forcing on 1 Aug 2008. (a) Hindcast (black), modeled SSH (dashed gray), and SSH without barotropic inflow (solid gray; see text for explanation). A 10-day running mean is applied to all time series. The vertical line indicates 1 Aug 2008. (b) Cumulative contribution from each of the forcing components (τ‖, τ⊥, Q, and FW) to the hindcast for 1 Aug 2008.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
Truncating the convolution in time (i.e., integrating from δt to t, where δt varies from 0 to 100 days) shows the time scale of the contributions to the hindcast. This calculation reveals how the forcing components contribute to the SSH signal over in time. It is shown for the hindcast of SSH on 1 August 2008 in Fig. 10b. Alongshore winds account for the largest fraction of the SSH response, with significant contributions from cross-shore winds and heat fluxes. Freshwater fluxes have a negligible contribution to the SSH signal. The timing of the response varies between forcing mechanisms. Most of the response to alongshore winds occurs within 10 days. However, the response to cross-shore winds takes about 100 days to accumulate. The response to steric effects continues to build up even after 100 days. The differing behavior is expected because the kinetic energy input by the wind is eventually dissipated by viscosity, but steric changes are not subject to the same decay mechanisms, and can persist for much longer durations.
When the calculation is repeated using the sensitivities for February, the hindcast similarly captures the timing of SSH variations, but the magnitude is underestimated (not shown). This is to be expected because the short-term sensitivities are the same, and the long-term sensitivities are qualitatively similar, only weaker in February. This suggests that the SSH variability is largely controlled by the short-term response, which in turn is dominated by the response to alongshore wind stress.
The hindcast was decomposed into parts driven by local, coastal, and offshore winds, and the contributions from these components to the full hindcast variability are summarized in Table 3. This result confirms that both local and nonlocal winds are important drivers of coastal SSH variability. The contribution from offshore winds is weaker than anticipated based on the measures dJmean and dJvar. This suggests that, in addition to the small-scale spatial variability in offshore sensitivity, there is also high-frequency temporal variability in the sensitivity to forcing. Indeed the sensitivities offshore oscillate in time, and the effect of a long-lasting wind event will partially cancel by generating inertia–gravity waves that act on SSH with different phases.
Statistics of the hindcast realized with different winds: std dev σ , correlation with the modeled SSH ρSSH, and correlation with the total hindcast ρΔη. Each column is for a different hindcast that includes only the contribution from winds that are either local Δηlocal, coastal Δηcoastal, offshore Δηoffshore, or the sum of all winds Δη (=Δηlocal + Δηcoastal + Δηoffshore). February and August sensitivities are used separately in the hindcast calculation. Heat and freshwater flux forcing are not included.
Interestingly, the hindcasts realized with only local, coastal, or offshore winds are all highly correlated with modeled SSH (Table 3). This is explained by a high spatial autocorrelation of the surface wind stress in the California region. Local and remote winds tend to enhance each other’s effect on coastal SSH, such that each component of the hindcast is often in phase with the total hindcast. While the coastal winds hindcast has the highest correlation with modeled SSH, coastal winds themselves cannot account for the magnitude of SSH variability. Local and offshore winds are responsible for at least half of the variability in the total hindcast.
c. Spectral characteristics of the sensitivities
There is growing appreciation of how transfer functions can be used as a model diagnostic (e.g., Kim et al. 2009; MacMynowski and Tziperman 2010). These frequency-dependent linear relationships between in- and output were calculated here using Fourier transforms of the hourly time series of adjoint-derived sensitivities. The amplitude of transfer functions for local winds is shown in Fig. 11. Each gray curve corresponds to a single grid point in a 0.5° × 0.5° area near the tide gauge. The thick black curve is the average over that area. Peaks at a period of 0.87 days, the inertial period, indicate the effect of near-inertial waves. In August, there is an additional peak at a period of 1 day. This peak it not well separated from the inertial peak, and is most apparent close to the tide gauge location. It is caused by diurnal variations in mixed layer thickness, which modulates the sensitivities, with SSH being less sensitive to winds blowing at night (over a deeper mixed layer). This effect is more noticeable in summer, when the diurnal variation is a significant fraction of the shallow MLD. Shifting the objective function by 12 h confirms that the phase of oscillations in the sensitivity is determined by the time of day, not by the time lag relative to the objective function.
Amplitude of the transfer function for along- and cross-shore wind stress in (a),(b) February and (c),(d) August. Each gray curve is for one grid point in a 0.5° square area near Port San Luis; black curves show the average spectra. Transfer functions are obtained by taking the Fourier transform of sensitivity time series. There is a peak at the inertial period, T = 0.87 days, in both seasons. There is also a peak at T = 1 day in summer due to the diurnal cycle in the mixed layer.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
The transfer function amplitude is small at periods of a few hours meaning this high-frequency forcing is unable to drive a strong response. (Forcing at a higher frequency than a response cancels itself netting a vanishing effect.) The transfer functions flatten at periods between 10 and 100 days indicating that the time scale of the forcing exceeds that of the response. At these low frequencies, the forcing is essentially constant or unidirectional. (The period at which the transfer function flattens out indicates the maximum time scale of the response, such that forcing at periods longer than that time scale will yield the same response.) Peaks in the transfer function, here noticeable at the inertial period, indicate resonance.
Figure 12 shows transfer functions for winds from four locations (marked on Fig. 1, the same as in Fig. 9). For local and offshore winds, the February and August transfer functions are similar at short periods but diverge for longer periods. This indicates that the short-term response (1–2 days, see Fig. 6) is the same for the two seasons (i.e., it is unaffected by stratification). The response to low-frequency winds has a lower amplitude in February, when the mixed layer is deep, than in August, when the mixed layer is shallow (consistent with the results from section 4a).
Amplitude of the transfer function for (a)–(d) along- and (e)–(h) cross-shore wind stress at four locations (local, offshore, coastal, and in a bay) in February (gray) and August (black). Transfer functions are averaged over a 0.5° square area around each location marked A, B, C, and D in Fig. 1.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-018.1
In contrast, the coastal transfer functions at the San Diego and Sebastián Vizcaíno Bay locations do not exhibit notable seasonal differences. These transfer functions are still increasing at a 100-day period, suggesting that coastal winds have very long-lasting effects (that are carried by baroclinic coastally trapped waves). There is a peak at a period of 4 h in the response to winds in the Sebastián Vizcaíno Bay, reflecting the fast barotropic resonance seen in Fig. 6.
5. Discussion
a. Propagation of sensitivities
The adjoint calculations shown here provide a detailed look at the physical pathways linking the SSH response at a point to the atmospheric forcing elsewhere. In particular, significant influence is seen to pass through the Southern California Bight and around Point Conception. The short-time response shows characteristics of storm surge, and some interesting resonances in a bay in Baja California, while the long-term response shows the baroclinic dynamics. This extends the work of Veneziani et al. (2009) to both shorter and longer time scales, and uses a more local cost function and more time sampling for less blurring of the propagation features.
The response to wind stress reflects the transfer of momentum directly to the mixed layer during the first few hours. At time scales beyond a few hours the effects of rotation are seen, such as Ekman dynamics including inertia–gravity waves and propagation of barotropic Rossby and coastally trapped waves, which are insensitive to the stratification. On still longer time scales, the response shows propagation of baroclinic Rossby and coastally trapped waves and other wind-induced motions in the mixed layer at the coast. Coastally trapped waves are easily identified by the poleward propagation of the effect of coastal winds (Gill 1982).
Inertia–gravity waves are identified from the oscillating sensitivities offshore of Port San Luis. Such oscillations indicate that time-varying winds can efficiently inject kinetic energy into the mixed layer in the near-inertial-frequency band and that these motions can produce SSH changes at the coast. The local wind-driven response likely involves the generation of near-inertial currents and vertical propagation of internal waves, as described in MacKinnon and Gregg (2005). Most of these processes are, in different ways, affected by stratification, leading to large seasonal variations in the sensitivities.
b. Seasonal dependence of sensitivities
After the initial response, which is the same in summer and winter, sensitivities evolve differently depending on the season. The long-term response, examined here over a period of 100 days, decays more rapidly in winter due to seasonal variations in the forward model ocean state. While the spatial patterns are similar, the contributions from local and offshore winds persist for longer times in August than in February, leading to overall higher sensitivity of SSH in August.
Changes in stratification affect the decay of wind-induced currents in the upper ocean. The mixed layer around Port San Luis is 2–3 times deeper in winter (~70 m) than in summer (~30 m). These deeper turbulent mixed layers distribute the wind input of momentum over greater depths. As a result there is a seasonal cycle in near-inertial kinetic energy, and an inverse relation between near-inertial surface currents and mixed layer depth (Shearman 2005). Internal waves provide a primary pathway for turbulent dissipation of this kinetic energy (MacKinnon and Gregg 2005). On the continental shelf, the enhanced connectivity to topography in winter increases the rate of momentum dissipation, resulting in a faster decay rate of the sensitivity to local and offshore winds. In contrast, the sensitivity to remote coastal winds does not appear to vary seasonally (Fig. 9), suggesting that the MLD does not significantly increase the damping of coastally trapped waves. However the waves appear to propagate more slowly in winter (Fig. 9) as a consequence of the lower stratification.
Diurnal changes in stratification are also found to impact the sensitivities. In summer, diurnal variations are significant compared to the mean MLD, and the magnitude of the sensitivity to local winds reflects this periodic change in stratification. The sensitivity of winds near the gauge location is modulated by oscillations with a period of 1 day (see Figs. 11c,d). The sensitivity to nighttime winds is smaller than the sensitivity to daytime winds; this is confirmed by experiments changing the time of the objective function by 12 h. Thus, the modulation of the response by the diurnal cycle produces oscillations in the sensitivity of coastal SSH, the sensitivity being larger when the water column is more stratified. As with the seasonal dependence, this diurnal variation does not apply to the short-term response, which is insensitive to stratification.
Two measures of the domain-wide sensitivity are used to quantify the evolution of the sensitivities over time. The first, dJmean, gives the response to uniform forcing; the second, dJvar, is indicative of the response to spatially varying forcing. Both metrics show that the cumulative sensitivity after 100 days is twice as large in summer as it is in winter. Transfer functions indicate that the response to coastal winds exhibits little seasonal dependence. This implies that the generation of coastally trapped waves is weakly affected by stratification, as one expects because these waves originate as surface elevation changes in response to winds at the coast (this mechanism is bound to the surface region and does not involve ocean interior). In contrast, the long-term response to local and offshore winds varies strongly depending on the season, indicating that the mechanism carrying the effect of the wind stress is more complex and involves interactions with the ocean interior (not confined to the surface).
c. Dominance of alongshore winds
The effects of different components of forcing are quantified by their expected contributions to dJmean and dJvar. Both measures attribute the largest fraction of the total sensitivity to alongshore winds, with cross-shore winds, heat fluxes, and freshwater fluxes playing lesser roles. This is consistent with the results of Veneziani et al. (2009) who also find the sensitivity to alongshore wind stress to be the dominant component. The hindcast of SSH on 1 August 2008 gives an alternative way of evaluating the relative contributions (section 4b). The growing responses over time reveal that the short-term response is almost entirely attributed to alongshore winds. The longer-term response is partitioned between alongshore winds, cross-shore winds, and heat fluxes.
d. Local versus remote winds
The largest sensitivities to wind stress occur near the location of the tide gauge. However, the effect of these local winds tends to be relatively fast with most of the sensitivity occurring within a week in summer, and within just a few days in winter (Figs. 6 and 7). The sensitivity to nonlocal winds is weaker, but longer lasting and more widely distributed over the domain. Thus, the remote forcing accounts for a large component of the total sensitivity (integrated over space and time). Remote forcing can be attributed to coastal winds, this effect propagating up the coast via coastally trapped waves, and to offshore winds, this effect propagating via inertia–gravity waves.
The sensitivity to offshore winds is large, especially in summer, indicating the potential to drive significant coastal SSH variability. It is noteworthy, however, that much of the sensitivity has small-scale spatial structure, and the effects will cancel out under large-scale wind patterns. In the hindcast driven by NAM winds, offshore winds have a small effect on coastal SSH variability. This indicates that although the effect of wind-generated mixed layer motions offshore can reach the coast on time scales as long as 100 days or more, the contribution from winds in different locations and at different times tend to cancel each other.
Using coastal observations, Ryan and Noble (2006) infer that local winds dominate at short periods, and nonlocal (regional) winds dominate for periods longer than 10 days. They estimate the transfer function (“frequency response function” in their terminology) for SSH on the California coast from observed coherences, and find that transfer functions are growing at long time scales of 100 days, although they suggest that this may be due to correlations between forcing mechanisms. Our model results are consistent with these observations. While Ryan and Noble (2006) considered the remote effects of coastal winds, our adjoint sensitivity study also highlights the potential influence of offshore winds on coastal SSH via propagation of inertia–gravity waves, although the spatial scales of the forcing variability may be too large to give this mechanism much importance.
A correlation analysis found that remote coastal winds are highly correlated with SSH, yet they are not the sole driver of SSH variability. The hindcast shows that forcing by local winds is at least as important as remote wind forcing to get the right magnitude of SSH variations. This result emphasizes that correlation does not signify a causal effect, and that is especially true when surface winds tend to have large spatial scales, as they do near the California coast.
e. Limitations
The state estimate provides a useful platform for analyzing the dynamics of the California Current System. With a 
It is worth reiterating that the sensitivities to atmospheric forcing quantify the potential effects on SSH, but these effects may never be realized. For instance, the large sensitivities to offshore winds may not have much influence in SSH variability if the winds do not have the spatial and temporal variability needed to produce constructive interference at the target location. The results presented here show that a spatially uniform wind stress perturbation has a reduced effect on coastal SSH, and temporally persistent winds would show a similar result. The adjoint results do identify, however, that inertia–gravity waves are a potentially important mechanism for carrying the effect of offshore winds toward the coast.
The linear sensitivity analysis provides an insight into the physics, though it has obvious limitations, as the true sensitivities are expected to have some degree of nonlinearity. This should not be a substantial source of error for the particular problem examined in this study, because the SSH response to wind perturbations of typical magnitude is mostly linear over the 100-day period analyzed. The larger the perturbation, the sooner we expect the linearity of the response to break down. The linearity of the SSH response in our model results from a combination of the horizontal viscosity of 100 m2 s−1 and the model resolution, weakly unstable flows in the California Current System, and the particular wind perturbations that were considered. There may exist perturbations that would excite a more nonlinear response; we have not probed the entire domain with our forward perturbations. Overall, however, we expect the sensitivities presented here to capture the leading-order effects of atmospheric forcing on coastal SSH.
6. Conclusions
This paper addresses what controls sea surface height variability at one location on the coast. This is a first step in understanding what sustains pressure gradients, and thus determines the geostrophic circulation in the California Current System. To investigate cross-shore pressure gradients one would either use the gradient as a cost function or use the SSH at a point offshore of Port San Luis and take the difference of the sensitivities. The latter separates the two responses for clarity, and, because of linearity, the difference of the sensitivities is the same as the sensitivity of the differences. Addressing alongshore pressure gradients would require examination of SSH sensitivity at several locations along the coast. For both investigations, differencing the responses would shed light on pressure gradient sensitivities and reveal controls on the transport.
The results presented here emphasize the role of stratification in modulating the sensitivity of SSH to air–sea momentum fluxes. Higher stratification in summer increases the sensitivity of coastal SSH to wind stress, relative to lower stratification in winter. In a warming climate, the ocean may become more stratified. Barring other changes in ocean state that would affect the sensitivity of coastal SSH to atmospheric forcing, our analysis suggests that it could imply a higher sensitivity to wind stress and longer-lasting effects of the winds on coastal sea level.
Acknowledgments
The authors thank Jen MacKinnon and Dan Rudnick for insightful discussions, and the editor and two reviewers for their valuable comments and suggestions. This work was supported by NOAA Grant NA17RJ1231 to B.D.C., NASA Grant NNX12AQ27G to A.V. and M.R.M., and by funding from the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), Ministry of Trade, Industry and Energy (20114030200040) and the Basic Science Research Program through the National Research Foundation (NRF), Ministry of Education (2013R1A1A2057849) to S.Y.K.
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