Prediction of Sea Surface Elevation and Currents in the Gulf of California: Scales From

Prediction of sea surface elevation and currents in the Gulf of California: scales from tides to seasonal

S. G. Marinone*, I. González, and M. J. Figueroa

Department of Physical Oceanography

CICESE

* Corresponding author

Abstract

The GC etc

Keywords: Prediction, sea level, currents, Gulf of California, tides, HAMSOM.

Software availability:

1. Introduction

The objective of this article is to present a web page developed to predict sea level and currents in the Gulf of California (GC hereafter, see Fig. 1 for location).

The variability and evolution of ocean currents is a very important research topic in oceanography as to the moment we still don’t know in detail how they evolve as the different forcings, shape of the basins, nonlinear interactions, etc. all are complicated. However, at some particular scales, our capacities to reproduce and predict the sea level and currents have increased considerable. With long time series (which are difficult to obtain) and numerical models, seasonal and/or climatologies are known for many places. In particular, tides and tidal currents are even more predictable given the specific frequencies that they have.

Numerical models have been implemented in several places for different purposes (e.g., navigation in England, tourism in chinconcua1, etc.) which mainly use the predictability of the tides. In some cases, there are continuous measurements of currents which are assimilated by numerical models which produce predictions for operational purposes as well (e.g. chiconcua2). In the Gulf of California there is not a product at the moment which gives and idea of the currents at certain place and time. The actual knowledge of the gulf’s circulation is still at basic research level by mean of some measurements and numerical models. In the GC, as in any many places, there is a pressure to know the currents for both, scientiphycally and practical reasons. Fisheries studies require information of the currents to explain some aspects of the distribution of larvae, eggs, etc. For pollution problems it is indispensable to know the currents as well. In this article we explain how to use a web page that predicts sea level and currents at any location for any time from the results of a numerical model. The time scales that are solved are tidal and seasonal.

The rest of the paper is organized as follows: section 2 presents the model from which time series of sea level and currents were obtained and the analysis done to make the predictions, section 3 show some comparisons with observations, and section 4 explains how to use the web page developed. Finally, section 5 presents the conclusions.

A comment/warning: The current predictions are qualitatively only, it should not to be used for navigation or as a strict piece of information for decisions. The accuracy is not precisely known. We welcome your evaluations, opinions and suggestions. Please send comments or questions to . o .

2. Analysis and model

The numerical model used in this study is the layerwise vertically integrated Hamburg Shelf Ocean Model (HAMSOM). The model is forced by: (a) the tides, (b) the Pacific Ocean, (c) the winds, and (d) the heat and fresh water fluxes. The governing equations of the model are a set of vertically averaged equations for each layer. These are for momentum,

t u = -h-1·(v uh) – uH-1(w) - f v - 0-1x P + H-1(x) + 2H u,

and

t v = -h-1·(v vh) – vH-1(w) + f u -0-1y P + H-1(y) + 2Hv,

for continuity,

wzdx (uH)+ y (vH) wzu ,

for temperature and salinity,

t (T,S) = -·[v (T,S)] – h-1[(T,S) w] + Kh2H(T,S) + z Kv(T,S),

the hydrostatic equation,

z P = - g,

and the overall continuity equation,

t  = -·(V),

where v = (u,v) is the horizontal velocity, w is the vertical velocity, V = (U,V) are the transport for each layer, f is the local Coriolis parameter,  = (S,T,P) is the density field, (S, T, P) are salinity, temperature and pressure, g is the acceleration due to gravity, P = (-z) 0g + p(x,t) is the total pressure, p is the baroclinic pressure, h is the layer thickness which is equal to H, the nominal thickness, except in the first and last layers where it accommodates the surface elevation, , and the topography, respectively. The operator (…) is the difference of (…) taken between the upper (zu) and lower (zd) surfaces of the layer.

The vertical stresses are  = Av (z v) where Av is the vertical eddy coefficient defined, following Kochergin (1987), as

Av =  | z v | / (1 +  Ri),

where Ri is the Richardson number,  = 10 m2 and  = 10. At the top and bottom, the surface and bottom stress boundary conditions are

s = Cda Vw (Uw2+Vw2)½andb = Cdb v (u2+v2)½,

respectively, where Vw = (Uw,Vw) is the wind velocity vector, Cda and Cdb are drag coefficients for the air/water interface and for the sea bottom, respectively. Finally, Kh and Kv are the horizontal and vertical eddy diffusion coefficient for scalar quantities, which in the model are equal to and Av the horizontal and vertical eddy viscosities.

The model domain has a mesh size of 2.5' x 2.5' (3.9 km x 4.6 km). The number of layers depends on the local depth, but twelve nominal layers are used in the vertical, with lower levels at 10, 20, 30, 60, 100, 150, 200, 250, 350, 600, 1000 y 4000 m.

The tidal forcing includes the seven most important semidiurnal and diurnal tidal constituents, namely M2, S2, N2, K2, K1, O1, and P1, plus the semiannual and annual components, Ssa and Sa, respectively. The harmonic constants were estimated from several years of observations in Mazatlán, in the mainland side of the Gulf entrance, and in Cabo San Lucas, at the tip of the peninsula (see Fig. 1a). From these points, the sea surface elevation was obtained at each time step and then interpolated along the open boundary. The wind at the sea surface and Pacific hydrography at the mouth of the gulf forcing were restricted to vary seasonally only.

Once we produce the output of the model, we perform harmonic analysis to the 14 harmonics listed in Table 1 for every grid point of the model. These constituents explain more than 95% and 99% of the variance of the model currents and sea level, respectively. For the predictions, we reconstruct the currents and sea level as

 =  i A i cos (i t-i),

where  represents the sea surface elevation (), and the eastward and northward velocity components (u, v), (A, ) the amplitude and phase of each constituent for each variable, and the frequencies of the i component (see Table 1).

3. Comparison with observations

Figure 2 shows time series of the sea level for the places indicated in Figure 1. Two lines are shown in each plot, one for the predictions from the observations and the other from the model prediction. As can be appreciated, the difference is not significant. The sea level is predicted with a high degree of accuracy.

The associated tidal currents, both barotropic and baroclinic, produced by the model have been compared with observations by Marinone and Lavín (2005), who find good overall agreement for the different tidal constituents: the best agreement is for the semidiurnal components, while the diurnal components are a little underestimated. Instead of showing the harmonics, here we show directly the velocity field from current observations superimposed to those predicted by the model in Figure 3; the agreement is acceptable, but of course that there are areas where the agreement is poor, however, in order to have an idea of how the currents are in certain time and place, we feel that the actual level of prediction is fair. On seasonal scales, the gyres observed by Lavín et al (1997) and the overall heat balance are also reasonably well reproduced as reported by Marinone (2003).

4. The web site

The web site predicts the surface elevation and/or the currents. For clarity purposes of the web site output, only one every x data points are shown; also in order to speed up the calculations and transmission of the files. It was developed using JAVA, MATLAB, and APACHE technologies and targeted to work with the most common web browsers.

The main or homepage welcomes the user and give three (links) options: 1) “The Model” where some links and articles related to the model are available, 2) “The interactive interface” that sends the user to the form page where the user makes the selections, and 3) “The paper” which links to this article. Figure 3 shows both pages.

4.1 The interactive interface

This is the working/predictive part of the page (Fig. 3b) where the user selects from the following:

• Initial and Final date of the prediction.
• The selection of harmonics to make the prediction (which gives as options the groups of semidiurnal, diurnal, semidiurnal and diurnal, low frequency, and all of them).
• The choice of variables: surface elevation, currents or both. Here the user can choose the sampling rate (from minutes to monthly) and the depth for the currents (see model layers above).
• The output where the user selects from a single geographical point or the northern, central, southern of all gulf areas. When “A point in the map” is selected, the user must drag the square with the “mouse” to the desired location. Here also different options for the output, depending on the case, namely, data or graphics can be selected. For a “Single point selection” the user can retrieve data or a time series plot. When areas are chosen the user has different options: if heights are selected the options are animated (movie) time series in a two- or three-dimensional projection. If currents or both are selected the animation is only in two-dimensional mode. When movies are obtained, the new window that appears shows the last frame and the user must “click” on it with the mouse to retrieve the file which will be opened with the users default movie program.
• Once the decision is made for the above selections the user “submits” the information to the server and waits for the output.

At the end of the page a link to a help document is available where some restrictive “Rules” are explained. Also, there is a warning window that shows what rule is being violated.

Here are few examples:

a)“Single point in the map”. Figure 4 shows a time series of obtained with the following selection:

1. Initial Date: 20/10/2006, Final Date: 21/10/2006
2. All harmonics
3. Both, 30 min, 45 (means surface elevation and currents at 25 m)
4. A point in the map (in the default initial position), Graphic

b)“Heights” in a 3D view. Figure 5 shows the last frame of the three-dimensional projections obtained with the following selection:

1. Initial Date: 20/10/2006, Final Date: 21/10/2006
2. All harmonics
3. Heights, 30 min, 0
4. Full gulf, 3D view

c)“Heights and currents”. Figure 6 shows the last frame of the animation obtained with the following selection:

1. Initial Date: 20/10/2006, Final Date: 20/10/2006
2. All harmonics
3. Both, 1 hr, 25
4. Full gulf, 2D view

After clicking with the mouse in the web page shown in Figure 5, the system will send you the animation which will be started with the user program, as for example shown in Figure 6.

5. Conclusion

A web interface that predicts sea surface elevations and currents for the Gulf of California was developed using the output of a three-dimensional baroclinic model (HAMSOM) which was forced with tides, and climatological winds and hydrography at the mouth of the gulf. With the output of the model, harmonic analyses were performed and with the harmonic constants obtained the prediction is made. The results of these predictions should be taken with care, even though the model has been prove to work qualitatively well, especially for the surface elevations, and should not be taken in substitution of observations or more realistic modeling. However, with the latter in mind, the predictions of the site would assist the user in exploring what the currents can be or were in some area.

Acknowledgements. CONACyT, CICESE, PANGAS y revisores.

References

Kochergin, V. P. Three-dimensional prognostic models, in Three-Dimensional Coastal Ocean Models, edited by N. S. Heaps, AGU, Washington, D.C., 1987.

Lavín, M. F., R. Durazo, E. Palacios, M. L. Argote, and L. Carrillo. Lagrangian observations of the circulation in the northern Gulf of California. J. Phys. Ocean., 27, 2298-2305, 1997.

Marinone, S. G., A three-dimensional model of the mean and seasonal circulation of the Gulf of California. J. Geophys. Res., 108(C10), 3325, doi:10.1029/2002JC001720, 2003.

Marinone, S. G. and M. F. Lavín, Tidal Current Ellipses in a 3D Baroclinic Numerical Model of the Gulf of California, Estuarine, Coastal, and Shelf Sciences, 2005, 64: 519-530.

Table 1. Constituents used for harmonic analysis.

Name / Frequency  (cycles/hour) / Period (days)
M2 / 0.0805114 / 0.52
S2 / 0.0833333 / 0.50
K1 / 0.0417807 / 1.00
O1 / 0.0387306 / 1.08
N2 / 0.0789992 / 0.53
P1 / 0.0415526 / 1.00
K2 / 0.0835615 / 0.50
N2S2 / 0.0043341 / 9.61
Mf / 0.0030500 / 13.66
Msf / 0.0028219 / 14.77
Mm- / 0.0015122 / 27.55
Msm / 0.0013099 / 31.81
Ssa / 0.0002281 / 182.67
Sa / 0.0001141 / 365.18

Figure 1. The Gulf of California. Contour lines are the bathymetry in meters. SF, BLA, GUY, CSL, and MZT stand for San Felipe, Bahía de los Ángeles, Guaymas, Cabo San Lucas, and Mazatlán, respectively.

Figure 2. Surface elevation in the indicated places. Red and black lines are the elevations predicted from the observations and the model, respectively.

Figure 2. (a) u and (b) v components of velocity in the northern Gulf of California from the model (red) and the observations (blue) from 1990.

(a)

(b)

Figure 3. The Gulf of California web page. (a) Welcome page and (b) The interactive interface.

Figure 4. Example of a time series obtained for sea surface elevation and currents at 45 m depth.

Figure 5. Last frame obtained for Heights for a 3D animation. On this page the user clicks the mouse to obtain the animation file.

Figure 6. Last frame for the “Both” selection (sea level and currents and 25 m depth). After clicking with the mouse on this page the animation is retrieved.