Interview Questions for Tidal Predictions

Tidal prediction uses two primary approaches: harmonic analysis and numerical modeling. Harmonic analysis is a mathematical technique that decomposes the complex tidal signal into a sum of simpler, periodic waves (constituents) based on their frequencies. It’s like separating the notes of a chord in music. We identify these individual constituents and their amplitudes and phases, then recombine them to predict future tides. This method works well for locations with relatively stable bathymetry and predictable astronomical forcing.

Numerical modeling, on the other hand, simulates the actual movement of water in a given coastal area using hydrodynamic equations solved on a computer. It accounts for complex factors like coastline shape, bathymetry (seafloor topography), and even river discharge. This approach is more computationally intensive but can handle highly variable and complex scenarios where harmonic analysis might fall short. Imagine trying to predict the ripples in a pool after dropping a pebble versus calculating the tide in a complex bay with islands.

In essence, harmonic analysis is faster and simpler but less detailed, while numerical modeling provides more accuracy and detail but requires substantial computational resources and expertise.

The equilibrium tide theory provides a simplified model of tides by considering the gravitational effects of the sun and moon on a uniformly deep, frictionless ocean covering the entire Earth. It assumes that the ocean responds instantaneously to the changing gravitational forces. This theory predicts two tidal bulges: one facing the moon (due to the moon’s direct gravitational pull) and another on the opposite side of the Earth (due to the centrifugal force resulting from the Earth-Moon system’s rotation). The equilibrium theory is a fundamental starting point for understanding tides, but it significantly simplifies real-world conditions.

Its simplicity makes it a great introductory concept, but it lacks the complexity of real tides because it ignores crucial factors like friction, coastline geometry, and water depth variations, which significantly affect the actual tidal range and timing.

A tidal signal is composed of many different tidal constituents, each representing a periodic wave with a specific frequency related to astronomical cycles. The main constituents are identified by their names, such as M2 (principal lunar semidiurnal), S2 (principal solar semidiurnal), K1 (lunisolar diurnal), O1 (principal lunar diurnal), etc. These names reflect the celestial bodies influencing them (lunar or solar) and their periodicity (diurnal or semidiurnal).

Constituents are identified through harmonic analysis of long-term tidal observations. Software packages are employed to decompose the measured water levels into these individual components by fitting sine waves of known astronomical frequencies to the data. The strength (amplitude) and timing (phase) of each constituent are determined, allowing us to build a comprehensive description of the tidal signal.

For example, the M2 constituent represents the largest tidal component, characterized by two high tides and two low tides per lunar day. S2 is similarly a semidiurnal constituent, but driven by the sun. Understanding these constituents is crucial for accurate tidal prediction.

Astronomical factors significantly influence tidal predictions. The primary factors are:

• The Moon’s position: The moon’s gravitational pull is the dominant force driving tides. Its distance from the Earth (varying slightly along its orbit) and its position relative to the Earth (affecting the angle of the pull) influence the tidal range.
• The Sun’s position: The sun also exerts a gravitational pull on the Earth, although weaker than the moon’s. Its position relative to the moon and Earth influences the magnitude of tides, resulting in higher ‘spring’ tides when the sun and moon align (new and full moon) and lower ‘neap’ tides when they are at right angles (first and third quarter moon).
• The Earth’s rotation: Earth’s rotation affects the timing of tides, as it determines the time it takes for a given location to pass under the tidal bulges. The Earth’s slightly elliptical orbit around the sun causes minor variations in the Earth’s rotational speed throughout the year.

These factors interact in complex ways, and sophisticated prediction methods require accounting for all of them. Astronomical algorithms are integral parts of tidal prediction software, accurately calculating the positions and distances of the celestial bodies for any given time.

Tidal charts are graphical representations of predicted tidal heights and times for specific locations. They are essential navigational tools, particularly for shallow-water navigation. These charts typically show tidal ranges (the difference between high and low tides) and times of high and low water for several days. They might also include tidal streams (currents) information.

Navigators use tidal charts to determine the safe water depth at a given time and location. Knowing the predicted water level allows ships to avoid grounding on shoals or reefs, especially during low tide. They are particularly critical for planning the entry and exit of harbors, navigating channels, and planning offshore operations. Tidal charts provide a crucial layer of safety and efficiency for maritime navigation.

Simple tidal prediction methods, like using a simple harmonic constituent or a tabular approach based on past observations, have limitations. These methods:

• Ignore complex interactions: They often neglect the numerous interactions between different tidal constituents and the effects of non-linear processes (e.g., friction, shallow-water effects).
• Lack spatial resolution: They might be accurate for a single point but provide less reliable predictions for an entire area.
• Limited predictive capacity: Simpler methods might only accurately predict tides for a short period. Their accuracy declines when predicting tides further in the future.
• Insensitive to weather effects: These methods don’t typically account for meteorological effects, such as strong winds and atmospheric pressure changes, which can significantly alter tides.

For applications requiring high accuracy and reliable predictions over large areas and longer periods, more sophisticated methods like numerical modeling are necessary. Simple methods are useful for quick estimations or when more complex models are unavailable, but their limitations should be well understood.

Bathymetry (the depth and shape of the seafloor) and coastline shape have a profound impact on tidal patterns. The shape of the coastline funnels or focuses tidal energy, amplifying or reducing the tidal range in specific areas. Imagine a funnel; water gets squeezed and accelerated. Similarly, bays and estuaries can experience significantly larger tidal ranges than the open ocean due to the resonating effect of the incoming tidal wave.

Bathymetry also plays a crucial role. Shallow water areas cause friction and dissipate tidal energy, leading to reduced tidal ranges and altered timing. The depth profile of channels and inlets significantly impacts the speed and propagation of the tidal wave, creating complex tidal currents and patterns. These influences often result in variations in tidal range across relatively small distances.

Numerical models account for these effects by using detailed bathymetric and coastal data. High-resolution models are necessary to capture the influence of subtle changes in coastline geometry and underwater topography on the spatial distribution of tides.

Meteorological factors, primarily wind and atmospheric pressure, significantly influence tides. While the gravitational forces of the sun and moon are the primary drivers of tides, wind can generate what are known as ‘wind surges’ or ‘wind set-up,’ causing temporary increases in sea level. Strong winds blowing onshore can pile up water against the coast, leading to higher than predicted high tides. Conversely, offshore winds can lower sea levels.

Atmospheric pressure also plays a role. High atmospheric pressure depresses the sea surface, leading to slightly lower tides, while low atmospheric pressure has the opposite effect, causing slightly higher tides. This inverse barometric effect is relatively small compared to the effect of wind, but it’s still considered in accurate tidal predictions, especially in shallow coastal regions where the water column is more susceptible to pressure changes. Think of it like pressing down on a balloon – the higher the pressure, the less the balloon expands, representing a slight lowering of the sea level.

For example, during a strong storm with high winds and low pressure, the combined effects of wind surge and the inverse barometric effect can lead to significantly higher tides than predicted based solely on astronomical influences. This can have devastating consequences, particularly in low-lying coastal areas, resulting in coastal flooding.

Tidal resonance occurs when the natural period of oscillation of a body of water (like a bay, estuary, or even a portion of the open ocean) coincides with the period of the tide-generating forces. Imagine pushing a child on a swing – if you push at the right time (matching the swing’s natural frequency), the swing’s amplitude (height) will increase dramatically. Similarly, if the forcing period of the tide aligns with the natural frequency of a bay, the tidal range (difference between high and low tide) can be greatly amplified.

The shape and depth of the basin play a critical role. A long, narrow, and shallow bay is more susceptible to resonance than a wider, deeper one. The resonance effect is what leads to some of the world’s most extreme tidal ranges. The Bay of Fundy in Canada, for instance, experiences some of the highest tides globally due to its unique geometry and a natural resonant frequency that matches the tidal cycle. This means a relatively small astronomical tide can be amplified into a spectacularly high tidal range.

Handling missing or incomplete tidal data is a common challenge in tidal prediction. Several strategies are employed to address this issue. The most common approach involves using interpolation techniques (discussed in the next question) to estimate missing values based on the available data. However, the method chosen depends on the nature and extent of the missing data.

If only a few data points are missing, simple linear or spline interpolation might suffice. However, for significant data gaps, more sophisticated techniques, often involving harmonic analysis of existing data, are necessary. We can reconstruct the missing tide levels by fitting a harmonic model to the existing data and then using this model to predict the missing values. This approach leverages the known cyclical nature of tides. Finally, data from neighboring stations with similar characteristics may also be used, carefully accounting for differences in location and local effects. The accuracy of these estimations is crucial and requires careful evaluation and validation.

Several methods exist for interpolating tidal data, each with its strengths and weaknesses. The choice depends on the characteristics of the data and the desired accuracy.

• Linear Interpolation: The simplest method, connecting two known data points with a straight line. It’s quick and easy but can be inaccurate if the underlying tidal curve is not linear.
• Spline Interpolation: This method uses piecewise polynomials to fit the data more smoothly than linear interpolation, resulting in a more accurate representation of the tidal curve, particularly when dealing with more complex tidal patterns. Cubic splines are commonly used.
• Harmonic Interpolation: This powerful technique uses harmonic analysis to decompose the tidal signal into its constituent harmonic components (e.g., M2, S2, K1). The missing data can then be reconstructed by recombining these components. It is particularly effective for filling longer gaps and offers better accuracy for representing tidal variations.

In practice, a combination of these methods may be employed. For example, one might use harmonic analysis to fill larger gaps, then spline interpolation to smooth out any remaining irregularities, or possibly employing a weighted average if we have similar data from nearby locations.

I’m familiar with a variety of software and tools used for tidal analysis and prediction. These range from specialized commercial packages to open-source tools. Some of the most widely used software includes:

• T-Tide: A powerful open-source package for tidal harmonic analysis.
• MATLAB with specialized toolboxes: MATLAB offers excellent capabilities for data analysis, including signal processing techniques useful in tidal analysis. Specific toolboxes might be needed for advanced functionalities.
• Commercial hydrodynamic modeling software: Packages like Delft3D, MIKE by DHI, or XBeach allow for detailed simulations of tides and their interaction with coastal morphology. They often incorporate sophisticated tidal prediction components.

The choice of software often depends on the specific application. For instance, T-Tide is ideal for harmonic analysis of a time series of tidal data, while hydrodynamic modeling software is more appropriate for simulating complex tidal dynamics in a specific geographic area.

Validating the accuracy of tidal predictions is crucial. We employ several methods to assess the reliability of our predictions.

• Comparison with historical data: We compare our predictions with a long time series of historical tidal observations at the location of interest. Statistical measures like the root mean square error (RMSE) and correlation coefficients are calculated to quantify the agreement.
• Comparison with other prediction models: We compare our results with predictions from other established tidal models or prediction services to identify potential discrepancies and ensure consistency.
• Real-time validation: Where possible, we compare predictions with real-time observations (e.g., from tide gauges) to continuously monitor the accuracy of our models and make necessary adjustments.

Identifying systematic biases or errors is essential. If discrepancies are consistently observed, we need to revisit the underlying assumptions and parameters of our model, refining it to improve accuracy. Accurate validation is an iterative process and is crucial for building confidence in tidal prediction capabilities.

Tidal datums are reference levels used to measure water levels and depths. They are essentially standardized vertical references for describing tidal elevations. Their importance lies in providing a consistent and universally understood benchmark for charting, coastal engineering, and navigation. Imagine trying to describe a building’s height without a defined ground level – it would be chaotic.

Several types of tidal datums exist, each defined based on statistical analysis of tidal observations. The most commonly used include:

• Mean High Water (MHW): The average height of the high tides over a specific period (e.g., 19 years).
• Mean Low Water (MLW): The average height of the low tides over a specific period.
• Mean Sea Level (MSL): The average height of the water surface over a specific period, considering all high and low tides.

The specific datum used depends on the application. For example, chart depths are often referenced to Mean Low Water because it represents a safe navigation level, providing sufficient water depth below the surface even at low tide. Consistent use of well-defined datums is essential for safety and accuracy in various coastal activities.

Predicting tides in complex coastal areas presents significant challenges due to the intricate interplay of various factors. Imagine trying to predict the waves in a bathtub with many oddly shaped obstacles – that’s similar to the complexity we face. The key challenges include:

• Irregular coastlines: Bays, inlets, estuaries, and islands drastically alter tidal propagation, creating complex patterns of tidal currents and water levels that are difficult to model accurately.
• Variable bathymetry: The shape and depth of the seabed significantly impact tidal flow. Shallow areas cause friction, slowing down tidal waves and creating significant distortions.
• River discharge: Freshwater inflow from rivers can dramatically influence salinity and water levels, affecting tidal predictions, especially near river mouths.
• Meteorological effects: Wind and atmospheric pressure can significantly modify tidal heights and currents. Strong winds can push water against the coast, causing higher-than-predicted tides, while low pressure can lead to higher sea levels.
• Data limitations: Accurate tidal prediction relies on high-quality data. In remote areas or developing countries, data availability might be scarce, hindering precise modelling.
• Nonlinear interactions: Tidal propagation is governed by nonlinear equations that are computationally intensive to solve. This difficulty is compounded by the complexities mentioned above.

Addressing these challenges often involves using sophisticated numerical models that incorporate high-resolution bathymetry data, meteorological forecasts, and river discharge information. Advanced techniques like finite element analysis and spectral methods are often employed for improved accuracy.

Shallow water significantly affects tidal propagation, primarily through friction. As a tidal wave moves into shallower water, its speed decreases and its amplitude (height) increases. Think of a river – when it narrows, the water flows faster. The opposite happens with tides in shallow areas: they slow down but become higher. This is because the energy of the tidal wave is conserved, and as the speed decreases, the energy is concentrated in a smaller volume of water, leading to a higher water level.

We account for the effects of shallow water through the use of hydrodynamic models. These models solve the shallow water equations, which are simplified versions of the Navier-Stokes equations that account for the effect of friction and other factors. These equations explicitly include terms that account for the bottom friction proportional to the water depth, thereby capturing the essential physics of the shallow water effects. The models require detailed bathymetric data (seabed topography) to provide accurate predictions. The friction is commonly represented using a Manning’s roughness coefficient, which quantifies the frictional resistance of the seafloor based on its roughness and sediment type. More complex models might even include sediment transport processes that modify the seafloor’s roughness and thereby the friction.

Spring tides and neap tides represent the extremes of tidal range (the difference between high and low tide) during a lunar month. They are caused by the combined gravitational forces of the sun and the moon.

• Spring tides occur when the sun, moon, and Earth are aligned (during new moon and full moon). The gravitational forces of the sun and moon add up, resulting in larger tidal ranges. Imagine two people pulling a rope in the same direction – their combined force is much greater. Spring tides have higher high tides and lower low tides.
• Neap tides happen when the sun and moon are at right angles to each other (during the first and third quarter moons). The gravitational forces partially cancel each other out, leading to smaller tidal ranges. Think of two people pulling a rope in opposite directions – their forces partially counteract each other. Neap tides have lower high tides and higher low tides.

The difference in tidal range between spring and neap tides can be substantial, especially in coastal areas with significant tidal amplification. Knowing the type of tide is crucial for various activities like navigation, fishing, and coastal engineering.

Tidal currents are the horizontal movement of water caused by the rise and fall of the tide. They are essentially horizontal components of the tidal wave. Imagine a wave approaching a beach: the water rushes towards the shore (flood current) and then retreats (ebb current). Tidal currents can be quite strong, especially in channels and constricted areas.

Predicting tidal currents involves similar techniques used for predicting tidal heights. Hydrodynamic models, often coupled with numerical models of the water column, are used. These models solve the equations of motion for water, accounting for the forces due to the tide-generating potential, friction, and Coriolis effects (due to Earth’s rotation). The output provides predictions of both speed and direction of the current at various locations and times. The accuracy of these predictions depends on the quality of input data (bathymetry, meteorological data), the sophistication of the model, and the complexity of the coastal area. Real-time data from current meters can be incorporated into the models to improve their accuracy.

Tidal predictions are indispensable in coastal engineering projects. Accurate predictions are essential for the design and construction of various structures, considering factors like:

• Port and harbor design: Determining the required water depth for navigation and ensuring sufficient clearance for vessels at all tide levels.
• Coastal protection structures: Designing seawalls, breakwaters, and other structures to withstand the forces of tides and waves, especially during high tide and storm surges.
• Bridge design: Ensuring sufficient vertical clearance under bridges to accommodate vessels at all tide levels.
• Offshore structures: Assessing the forces acting on offshore platforms and wind turbines due to tidal currents.
• Sediment transport modeling: Understanding tidal influence on sediment transport patterns to evaluate coastal erosion and accretion.

For example, when designing a port, engineers need to ensure that the dredging depth is sufficient to accommodate ships at the lowest possible tide. Misjudging the tidal range could result in a port that is unusable for certain vessels during low tides.

Tidal predictions are critical for safe and efficient marine navigation. Accurate tidal information ensures that vessels can safely navigate shallow waters, avoid grounding, and optimize their routes. This information is typically provided in the form of tidal charts and tide tables.

• Determining safe water depths: Mariners use tidal predictions to calculate the available water depth over a seabed at any given time to avoid running aground.
• Timing of port entry and exit: Ships need sufficient water depth to enter and leave ports safely, therefore, tidal predictions allow for optimal timing of these maneuvers.
• Planning voyages: Navigators use tidal current predictions to adjust their routes and speeds to take advantage of favourable currents or avoid strong opposing currents.
• Search and rescue operations: Accurate tidal information is vital for search and rescue operations, as it helps to predict the movement of debris and people in the water.

Failure to account for tides can lead to accidents, delays, and even loss of life. For instance, a ship entering a port without considering the low tide could become stranded on a sandbank.

Tidal predictions play a crucial role in environmental monitoring, particularly in:

• Ecosystem studies: Understanding tidal influences on coastal ecosystems, like estuaries and salt marshes, which are highly sensitive to changes in salinity and water levels. Tidal patterns influence the distribution of organisms and nutrient cycling.
• Water quality monitoring: Tides affect the transport and dispersion of pollutants in coastal waters, and accurate tidal predictions are important for interpreting water quality data and modeling pollutant transport.
• Coastal erosion and accretion studies: Tides are a primary driver of coastal morphological changes, and understanding tidal patterns is crucial for monitoring erosion and sedimentation processes, informing strategies for coastal management.
• Habitat restoration projects: Restoration projects often rely on accurate tidal data to create habitats suitable for specific species, and to understand the connectivity between different parts of the coastal zone.
• Climate change impact assessment: Predicting sea level rise under climate change requires accurate modelling of tides, since sea level rise will interact with existing tidal patterns, impacting coastal flooding and erosion.

For instance, studying the impact of a proposed dredging project on a sensitive estuary requires the use of tidal models to predict the changes in water flow and salinity, which are crucial factors in evaluating potential environmental impacts.

Tidal predictions, while remarkably accurate, are not perfect. Several sources of error can affect their precision. These errors can broadly be categorized into model limitations, data limitations, and unforeseen events.

• Model Limitations: Tidal models are simplified representations of complex physical processes. They make assumptions about the shape of the ocean floor, water density, and friction, which can lead to discrepancies between predicted and observed tides. For instance, a model might not perfectly account for the complex interactions between tides and currents in a narrow channel.
• Data Limitations: The accuracy of predictions hinges on the quality of input data. Inaccurate or insufficient measurements of bathymetry (ocean depth), astronomical data, or historical tide gauge readings will propagate errors into the predictions. For example, if a crucial tide gauge in a region is malfunctioning, the predictions for that area would be less reliable.
• Unforeseen Events: Meteorological effects like strong winds or atmospheric pressure changes can significantly influence sea levels, causing deviations from predictions. Similarly, seismic activity can temporarily alter sea level and impact the accuracy of predictions. These events are difficult to incorporate into predictive models.

Understanding these error sources is crucial for interpreting tidal predictions and using them responsibly. It’s important to remember that predictions are probabilistic, not deterministic; there will always be some level of uncertainty.

Improving the accuracy of tidal predictions involves a multi-pronged approach that addresses the sources of error discussed previously.

• Refined Models: Employing more sophisticated hydrodynamic models that incorporate higher-resolution bathymetry data and advanced numerical techniques can reduce errors due to model limitations. Advances in computational power allow for more realistic simulations of complex coastal geometries and interactions.
• Enhanced Data Acquisition: Increasing the density and quality of observational data is paramount. This includes utilizing high-precision GPS technology for more accurate positioning of tide gauges, employing advanced satellite altimetry techniques, and implementing quality control measures for existing data sets. Regular maintenance and calibration of tide gauges are essential.
• Data Assimilation Techniques: Integrating real-time observations (e.g., from radar, satellite altimetry, and coastal sensors) into the predictive models using data assimilation techniques significantly improves accuracy. These techniques essentially ‘correct’ the model predictions by incorporating real-world measurements.
• Improved Meteorological Forecasting: Integrating accurate weather forecasts into tidal models helps account for the influence of wind and atmospheric pressure on sea level. Better forecasting of these meteorological factors leads to more precise predictions, particularly during storm surges.

By continuously refining models, enhancing data acquisition and processing, and adopting advanced data assimilation strategies, we can significantly improve the accuracy and reliability of tidal predictions.

A tidal gauge is a device used to measure the height of the tide relative to a known datum (reference point). Think of it as a sophisticated ruler for the ocean.

Operation typically involves a float resting on the water surface within a stilling well – a protected chamber that minimizes wave action. As the tide rises and falls, the float moves vertically, and this movement is translated into an electrical signal using various mechanisms (e.g., potentiometers, acoustic sensors).

This signal is then recorded, either locally or transmitted to a remote location for processing and analysis. The data collected provides a continuous record of sea level fluctuations at the gauge location, which is vital for understanding tidal patterns and validating tidal models. Modern gauges often incorporate GPS technology for precise positioning and data logging, ensuring high accuracy and reliability.

For example, a coastal city might use multiple tidal gauges to monitor sea levels and generate accurate predictions essential for flood warning systems.

Satellite altimetry plays a crucial role in modern tidal prediction. Altimeters on satellites measure the distance between the satellite and the ocean surface using radar pulses. By repeatedly orbiting the Earth, satellites can measure the sea surface height over vast areas of the ocean, providing valuable data for tidal analysis.

This data is particularly valuable for monitoring tides in remote or inaccessible locations, where deploying and maintaining traditional tide gauges is challenging. Satellite altimetry also provides a synoptic view of tidal patterns across large ocean basins, improving our understanding of global tidal dynamics. This global perspective is essential for calibrating and validating numerical tidal models.

However, satellite altimetry data has limitations. The spatial resolution is typically coarser than tide gauge measurements, and the data can be affected by atmospheric and oceanic conditions. Despite these limitations, it’s a powerful tool for enhancing our understanding and prediction of tides worldwide.

Incorporating real-time data into tidal predictions significantly improves accuracy, particularly for short-term forecasts. This is achieved through data assimilation techniques, which combine model predictions with real-time observations from various sources.

For example, data from coastal radar stations, tide gauges, and satellite altimetry is fed into the tidal model. These real-time measurements are used to correct or adjust the model’s predictions, essentially bringing the model into better agreement with the observed conditions. This process is iterative, with the model constantly being updated as new data arrives.

This approach is particularly effective in handling unpredictable events such as storm surges, where real-time data is critical for issuing timely and accurate warnings. The improved short-term forecast accuracy is essential for maritime navigation, coastal management, and disaster preparedness.

A tidal model is a mathematical representation of the physical processes that govern tides. It’s essentially a computer simulation of the ocean’s response to gravitational forces from the sun and moon.

These models use numerical techniques to solve complex equations that describe the motion of water in response to these forces, taking into account factors like bathymetry (ocean depth), coastline shape, and friction. The output of the model is a prediction of the tide height and current at specific locations and times.

Tidal models have widespread applications, including predicting tides for navigation, planning coastal engineering projects, assessing coastal flood risks, and understanding the dynamics of estuaries and coastal seas. They are crucial tools for coastal zone management and maritime safety.

My experience encompasses both hydrodynamic and harmonic tidal models. Each approach has its strengths and limitations.

• Harmonic Models: These models represent the tide as a sum of sinusoidal waves with different frequencies and amplitudes. They are relatively simple to implement and require less computational power. However, they are less accurate in complex coastal regions with strong currents and non-linear interactions.
• Hydrodynamic Models: These models solve the fundamental equations of fluid motion (Navier-Stokes equations) to simulate the flow of water. They are computationally intensive but are capable of representing complex coastal geometries, currents, and interactions with the atmosphere. They provide higher accuracy, especially in areas with complex topography and significant non-linear effects.

In practice, I often use a combination of these approaches. Harmonic analysis can be used to identify the dominant tidal constituents in a region, while hydrodynamic models can be used to simulate the detailed flow and water level variations. The choice of model depends on the specific application, the desired accuracy, and the available computational resources. For instance, for large-scale predictions across ocean basins, a global hydrodynamic model might be used, while a smaller-scale, localized study might utilize a harmonic model or a simplified hydrodynamic model.