Eastern boundary upwelling systems (EBUSs) are physically and biologically active regions of the ocean with substantial impacts on ocean biogeochemistry, ecology, and global fish catch. Previous studies have used models of varying complexity to study EBUS dynamics, ranging from minimal two-dimensional (2-D) models to comprehensive regional and global models. An advantage of 2-D models is that they are more computationally efficient and easier to interpret than comprehensive regional models, but their key drawback is the lack of explicit representations of important three-dimensional processes that control biology in upwelling systems. These processes include eddy quenching of nutrients and meridional transport of nutrients and heat. The authors present the Meridionally Averaged Model of Eastern Boundary Upwelling Systems (MAMEBUS) that aims at combining the benefits of 2-D and 3-D approaches to modeling EBUSs by parameterizing the key 3-D processes in a 2-D framework. MAMEBUS couples the primitive equations for the physical state of the ocean with a nutrient–phytoplankton–zooplankton–detritus model of the ecosystem, solved in terrain-following coordinates. This article defines the equations that describe the tracer, momentum, and biological evolution, along with physical parameterizations of eddy advection, isopycnal mixing, and boundary layer mixing. It describes the details of the numerical schemes and their implementation in the model code, and provides a reference solution validated against observations from the California Current. The goal of MAMEBUS is to facilitate future studies to efficiently explore the wide space of physical and biogeochemical parameters that control the zonal variations in EBUSs.

Eastern boundary upwelling systems (EBUSs) are among of the most biologically productive regions in the ocean, supporting diverse ecosystems and contributing to a significant portion of the global fish catch

The upwelling-favorable winds also drive baroclinic, equatorward geostrophic current, which sheds mesoscale eddies

While these qualitative patterns of productivity are common to upwelling systems, previous studies have shown that productivity varies substantially between EBUSs, but the causes of these inter-EBUS variations are not well understood.
Possible physical drivers of these inter-EBUS variations include the shape and strength of the wind-stress curl, which set the upwelling strength and source depth

Our understanding of these drivers is hindered in part by the observational limitations and in part by the computational expense of regional models that can resolve the processes mentioned above.
A range of models of varying complexity have been used to study EBUSs, from minimal two-dimensional (2-D) models

Here, we aim to close the current gap in understanding by developing an idealized, quasi-2-D model of the physics and biogeochemistry of EBUSs. The model includes parameterizations of the key three-dimensional processes, while retaining the computational efficiency of a 2-D model. The model is cast in a residual-mean framework

The rest of the paper is organized as follows. In Sect. 2, we describe the equations and physical parameterizations implemented in MAMEBUS, including general formulation of tracer advection and diffusion, the time-dependent turbulent thermal wind approximation of the momentum equations (T3W), eddy and boundary layer parameterizations, and our ecosystem formulation. In Sect. 3, we detail the algorithms and discretizations, including mesh specification, vertical coordinate transformation, and time integration. In Sect. 4, we describe the implementation of MAMEBUS including the various options available to the user, parameter choices, initialization, and output. In Sect. 5, we describe reference solutions for MAMEBUS, discussing model sensitivities to changes in bathymetry, wind forcing, and surface heat fluxes. Finally, in Sect. 6, we discuss further model development and future work.

A schematic of the essential components of the Meridionally Averaged Model of Eastern Boundary Upwelling Systems (MAMEBUS). This schematic highlights some components that the user is able to control including the offshore restoring conditions, the eddy mixing along isopycnals, the wind forcing, the surface mixed layer and bottom boundary layer parameterizations, and grid spacing.

MAMEBUS is comprised of a series of components that are necessary to capture physical–biogeochemical dynamics in EBUSs: (1) explicit momentum conservation in form of geostrophic, hydrostatic, and Ekman balances implemented as part of the T3W formulation; (2) eddies and their effect on material transport; (3) surface and bottom boundary layers; (4) nutrient and plankton cycles in form of a size-structured “NPZD”-type model

With the exception of the velocity field, all tracers in MAMEBUS evolve according to the following conservation equation:

We first formulate an evolution equation for the meridionally averaged concentration of an arbitrary tracer

To evolve a meridionally averaged tracer

As in Sect.

On the other hand, we have retained the time-evolution terms (leftmost terms in Eq.

In Eq. (

Additional boundary conditions are required to solve
Eq. (

In this section, we describe the parameterization of unresolved microscale mixing in the tracer evolution (Eq.

We formulate the diapycnal mixing coefficient

The diapycnal diffusivity in the surface mixed layer,

The diapycnal diffusivity in the bottom boundary layer,

At any point in space and time at which the water column is statically unstable, i.e., when

We now discuss the formulation of the eddy advection and isopycnal mixing terms in Eq. (

The eddy-induced velocity

The isopycnal mixing operator serves to mix tracers down their mean gradients, in a direction that is parallel to mean isopycnal surfaces in the ocean interior, following

We now discuss the formulation of

The simplest form for

The scheme described above for the SML relies on the fact that the ocean surface is approximately flat, which allows the same effective slopes

Analogous to the SML, we define the effective slope

The current biogeochemical model implemented in MAMEBUS is an NPZD (nutrient–phytoplankton–zooplankton–detritus) model. This NPZD model is modeled after the size-structured AstroCAT

The biogeochemical equations in MAMEBUS are formulated similarly to previous NPZD models but cast in terms of the meridionally averaged nutrient, phytoplankton, zooplankton, and detritus concentrations. We neglect additional terms that would be introduced by first formulating the equations and then taking the meridional average, e.g., covariances of the type

We drop the bar notation indicating a meridional average for this section, with the understanding that all variables denote meridionally averaged quantities. In the following, we include size-dependent uptake and grazing, along with variable sinking speeds for detritus, to retain essential size-dependent biogeochemical interactions and export fluxes. This will facilitate a future introduction of multiple size classes in the model. All variables and coefficients are given in Table

Parameters and values used in the ecosystem model implemented in MAMEBUS. Coefficients without explicit references are chosen by the user.

Common controls on phytoplankton population are bottom-up limitation (i.e., nutrient control), and top-down grazing by zooplankton

Parameters and values used in the ecosystem model implemented in MAMEBUS. Coefficients without explicit references are chosen by the user.

Top-down processes are represented by zooplankton grazing on phytoplankton.

Mortality closure terms often set important internal dynamics in ecosystem models

Sinking particles are an essential component of the vertical transport of nutrients from the surface to the deep ocean

Particles sink at a constant average speed in the water column, following Eq. (

In this section, we describe the treatment of all non-conservative terms in the tracer evolution equation. MAMEBUS allows arbitrary restoring of all tracers, which may be used, for example, to impose offshore boundary conditions or to impose restoring at the sea surface. Fixed fluxes of all tracers may also be imposed through the surface. More precisely, we formulate the non-conservative tracer tendency as

The restoring of a tracer is represented as an exponential decay to a prescribed, spatially varying tracer field,

Surface fluxes are represented as a tendency in the tracer concentration in the surface grid boxes. For an arbitrary tracer

In this section, we discuss the numerical solution of the model equations presented in Sect.

We solve the model equations presented in Sect.

We now write the physical tracer evolution (Eq.

Illustration of the numerical grid used to compute solutions to the model equations.

We solve Eq. (

To compute the advective tendency,

Finally, the meridional advection is discretized via a straightforward upwind advection scheme:

MAMEBUS evolves the model equations forward in time using Adams–Bashforth (AB) methods

Our time-integration scheme uses a family of time-step-variable Adams–Bashforth integrative methods.
This specific formulation of the AB methods allows for the model time step to be adjusted dynamically following the CFL conditions described in Sect.

MAMEBUS selects each model time step adaptively to ensure that time stepping is numerically stable. The time step is chosen to ensure that the CFL conditions for each of MAMEBUS's various advective and diffusive operators, described in preceding subsections, are satisfied.

The time step for advection of tracers is limited by the timescale associated with advective propagation across the width of a grid box (

We apply additional constraints on the time step to ensure that diffusive operators are stable. The standard numerical stability criterion for a Laplacian diffusion operator is

Note that although

In this section, we describe the discretization of the momentum equations presented in Sect.

The numerical time integration is calculated in a series of steps which include an explicit calculation of the non-diffusive time step, an implicit calculation of the vertical diffusion, and a barotropic corrector step in order to ensure that the flow is non-divergent. The calculation of the explicit time step is outlined in Sect.

Given the mean momentum at time step

We next compute the tendency due to vertical viscosity following Eq. (

Pressure-gradient calculations in

Stencil for the isopycnal slope and pressure-gradient scheme given by

In this section, we outline the implementation of the zonal baroclinic pressure-gradient calculation used in MAMEBUS following

First, we calculate all elementary differences in

We then calculate the hyperbolic differences of all variables. This step calculates an estimate of the derivatives following a cubic spline formalism outlined in more detail in

We the calculate the pressure field using the hydrostatic relationship. This is done via a vertical integration of the density reconstructed along the vertical lines in Fig.

The buoyancy gradient is calculated similarly to the pressure gradient. However, because we do not vertically integrate the buoyancy term, we opt to use the density Jacobian algorithm described in

First, we calculate the elementary differences, and the hyperbolic averages in

Finally, in order to calculate the horizontal buoyancy gradient, we divide by the area. Since the area of each cell is defined by the cell-centered

The along-shore pressure gradient in Eq. (

In this section, we outline the details for implementation in MAMEBUS. The model code is written in the C programming language. The model expects various user inputs that include initial conditions, along with user-defined model calculation details in Table

The software needed to run this model includes

a C compiler (e.g., GCC).

MAMEBUS has three active physical variables: the zonal and meridional momenta, and the temperature (buoyancy). The current implementation of the biogeochemical model has four active variables: nitrate (N), phytoplankton (P), zooplankton (Z), and detritus (D). A variable number of additional passive tracers may also be included.

Input parameters expected by the MAMEBUS model code. All parameters listed in this table are chosen by the user. The sample values listed in this table are those used in the reference experiments described in Sect.

MAMEBUS expects a list of parameters given in Table

For the solutions shown in Sect.

MAMEBUS numerical scheme options and descriptors.

A table outlining the initial profiles that MAMEBUS expects during initialization. To visualize the grid locations, see Fig.

The main function of the mamebus.c file has five major components and steps:

Calculate the time tendency of each tracer. The time step is calculated using the “tderiv” function detailed in Fig.

Add implicit vertical diffusion and remineralization (Eq.

Apply zonal barotropic pressure-gradient correction if the “momentumScheme” is MOMENTUM_TTW (Sect.

Enforce zero tendency where relaxation time is zero (Sect.

Write model state (Sect.

All of the model input and output are saved in binary files. Depending on the “monitorFreq” or the frequency of output, the model will interpolate the between time steps, calculate the correct model state if necessary, and write the data to file. The following list contains all files that are written to file during the time-integration step. For each model, there is an option to include an arbitrary number of passive tracers; however, this is the standard list of tracers that are included in the indicated modelTypes.

Residual stream function,

Mean stream function,

Eddy stream function,

Temperature field (all modelTypes)

Nitrate (NPZD model)

Phytoplankton (NPZD model)

Zooplankton (NPZD model)

Detritus (NPZD model)

In this section, we present reference solutions for MAMEBUS. Below, we discuss the choice of parameters, the non-conservative forcing, and profiles of restoring. We focus predominantly on the output of a single run and plan in the future to run parameter sweeps to better understand the response of the ecosystem dynamics to the physical forcing.

The model is configured to represent an idealized California Current System (CCS). While the model can be formulated to represent a general EBUS, we use the California Current System as a test case because this allows comparison of our results with measurements from

A list of input fields that MAMEBUS expects is given in Table

The topography for the reference solutions is

The call tree from the main function of mamebus.c.

Initial temperature profile with a profile of offshore restoring which is modeled as a sponge layer on the western side of the boundary, and at the surface, there is a surface restoring to an atmospheric profile, idealized to a profile of temperature from CalCOFI. The northward wind stress is shown at the top of the figure. The white lines in the temperature field are a few lines of constant initial temperature.

The initial conditions for the tracers in the model are the initial temperature profile, including timescales and inputs for restoring, and initial conditions for the NPZD model, which are tuned to give an approximate concentration of 30

The initial conditions for NPZD tracers include a constant concentration of nitrate,

The unresolved mesoscale and microscale mixing in the tracer evolution Eq. (

In our model reference configuration, the eddy and buoyancy diffusivities are functions of the baroclinic radius of deformation – the preferential length scale at which baroclinic instability occurs and closest to the fastest growing mode in the Eady model

The diapycnal diffusivities shown in the right panel of Fig.

We run the reference solutions of MAMEBUS for 25 model years, with initial conditions and physical forcing described in Sect.

Furthermore, we prescribed a continental shelf that is deeper than in nature in order to reduce the model's computation time. Further shallowing the continental shelf is possible, but the CFL constraint imposed by the finer vertical resolution on the shelf extends the computation time.

While the continental slope is tuned to have a similar slope as observations in central California near the shelf break, the mixed layers in this model run are set to a constant depth zonally and overlap on the shelf. This choice has been made for simplicity and could be refined via zonally varying mixed layer depths to improve agreement with specific EBUSs. The well-mixed area on the shelf is an analogue to the inner shelf, albeit somewhat deeper than those found in nature

Inputs of buoyancy diffusivity

The model temperature is generally in good agreement with observations for the upper ocean, reproducing sloping isotherms towards the coast, and realistic surface values. We observe a cold bias near the coast, which could be a result of the constant wind-stress curl forcing over the domain, inducing upwelling that is too strong in the model. A cold bias observed in the surface just outside the shelf, and a warm bias offshore, are likely caused by the prescription of a constant mixed layer depth, which may be too deep in the model for this particular section and time of the year.

As shown by the middle row of Fig.

The bottom row of Fig.

Model validation against in situ CalCOFI data taken along line 80 (point conception) during July 2015. The column on the left shows output from the model under constant wind forcing and is averaged over the last 5 model years. The column on the right shows values taken from CalCOFI and interpolated onto a

In order to compare physical solutions, we also include solutions which show the residual stream function, including the mean and eddy components in Fig.

In this section, we describe the changes in solutions due to model resolution. We chose four different resolutions and explored the results. Figure

Increasing the resolution leads to an overall shoaling of nutrients toward the surface. The largest overall change in near-slope nutrient concentration occurs when the resolution doubles from 32 to 64 horizontal points and vertical levels. Increasing the resolution beyond a

Stream functions calculated by MAMEBUS. This figure shows the residual stream function

A table outlining model run times of varying resolution between a computational cluster comprised of Intel Xeon E5-2650 v3 CPUs and a 2015 Mac laptop running macOS Catalina (version 10.15.7) for 20 model years, for both computing systems, the model is run on a single core. The highest resolution simulation (

This figure shows the model output of temperature, nitrate, and chl with varying resolution. The model was run for 30 years, and solutions shown are averaged over the final 30 years of the model run.

In this paper, we described the formulation, implementation, and main features of MAMEBUS, an idealized, meridionally averaged model of eastern boundary upwelling systems. The solutions are determined by a general evolution equations for materially conserved tracers (Sect.

MAMEBUS represents a simple, physically consistent tool in which to test and tune a variety of physical parameterizations and ecosystem model formulations. The ultimate goals of this research include exploration of physical–biogeochemical interactions in EBUS, mechanistic understanding of the factors that control cross-shore gradients in biogeochemical and ecological properties, and investigation of the processes that drive differences between distinct EBUSs.

Because of the 2-D framework, we acknowledge shortcomings to the model formulation, including physical aspects like intensification of upwelling around topographic features, for example, resulting from variations in the wind-stress curl

In future studies, we plan to use MAMEBUS to explore the effect of physical drivers such as wind stress, bathymetry, stratification, and eddies in controlling the zonal distribution of phytoplankton and food web processes, as informed by a size-structured ecosystem model. Furthermore, we plan to expand upon the physical framework in this paper by expanding eddy parameterizations to include the effect of submesoscale eddies on the shelf, where the mesoscale eddy activity is inhibited. An aspect of MAMEBUS that requires further investigation is the effect of meridional pressure gradients, which we neglected in our reference solutions in Sect.

With its limited computational cost, MAMEBUS can be used to investigate a wide parameter space in EBUSs and determine their sensitivity to a range of perturbations in major physical forcings, from changes in wind stress to increasing buoyancy forcing associated with climate change

In this Appendix, we discuss the partitioning of the mesoscale eddy tracer flux into components due to advection and isopycnal diffusion, used in Sect.

The effect of mesoscale eddies on the averaged tracer concentrations is given by the convergence of the eddy tracer flux (Eq.

While the above derivation is general, for application in MAMEBUS, we must make assumptions about the eddy tracer fluxes. Specifically, we assume (i) that approximately identical eddy stream functions

For a given tracer defined with an associated time tendency equation of the form

For higher-order AB methods, we consider a

Our representation of eddy advection and isopycnal stirring in the surface mixed layer (SML) and bottom boundary layer (BBL) is adapted from

As discussed in the “surface mixed layer” subsection, our SML scheme leads to the same eddy stream function as that of

Another difference between our formulation and that of

Finally, we compare the horizontal component of the eddy buoyancy flux in the SML:

Further to this comparison with the formulation of

The DOI for the MAMEBUS code is

This package includes the mamebus.c code along with example setup and processing functions that are used in MATLAB.

No data sets were used in this article.

ALS conceived and coordinated the development of MAMEBUSv1.0, and advised JEM in further model development. JEM developed and implemented the ecosystem models, updated the calculation of the momentum equation, and fine-tuned the pressure-gradient calculations. DB coordinated the development of the ecosystem model. JCM coordinated the development of the eddy parameterizations in MAMEBUS and advised JEM in further model development. JEM prepared the manuscript with contributions from all co-authors.

The authors declare that they have no conflict of interest.

This material is based in part upon work supported by the National Science Foundation with grant nos. OCE-1538702, OCE-1751386, and OCE-1635632, and by the National Aeronautics and Space Administration ROSES Physical Oceanography program under grant no. 80NSSC19K1192. Daniele Bianchi gratefully acknowledges funding from the Alfred P. Sloan Foundation. This work used the Extreme Science and Engineering Discovery Environment (XSEDE;

This research has been supported by the National Science Foundation (grant nos. OCE-1538702, OCE-1751386, OCE-1635632, and ACI-1548562) and the National Aeronautics and Space Administration (grant no. 80NSSC19K1192).

This paper was edited by Qiang Wang and reviewed by two anonymous referees.