Baroclinic Terms

ADCIRC can be operated in baroclinic mode, allowing it to simulate density-driven flows resulting from temperature and salinity variations. This section describes the theoretical foundation for baroclinic processes in ADCIRC.

Density Variation

In baroclinic mode, water density varies spatially and temporally due to variations in temperature and salinity. The density is computed using an equation of state:

\[\rho = \rho(T, S, p)\]

where:

  • \(\rho\) = water density

  • \(T\) = temperature

  • \(S\) = salinity

  • \(p\) = pressure

ADCIRC typically employs a simplified form of the UNESCO equation of state or the linear equation of state:

\[\rho = \rho_0 [1 - \alpha_T (T - T_0) + \beta_S (S - S_0)]\]

where \(\alpha_T\) is the thermal expansion coefficient and \(\beta_S\) is the saline contraction coefficient.

Baroclinic Pressure Gradient

The baroclinic pressure gradient is the primary driving force for density-driven flows. In the momentum equations, this appears as:

\[\frac{1}{\rho_0} \frac{\partial}{\partial \lambda} \int_{-h}^{\zeta} \int_{z'}^{\zeta} \frac{\partial \rho}{\partial \lambda'} g dz dz'\]
\[\frac{1}{\rho_0} \frac{\partial}{\partial \phi} \int_{-h}^{\zeta} \int_{z'}^{\zeta} \frac{\partial \rho}{\partial \phi'} g dz dz'\]

These terms represent the horizontal pressure gradient due to horizontal density variations. A mathematically equivalent but numerically advantageous form is:

\[\frac{g}{\rho_0} \int_{-h}^{\zeta} \left( z - \zeta \right) \frac{\partial \rho}{\partial \lambda} dz\]
\[\frac{g}{\rho_0} \int_{-h}^{\zeta} \left( z - \zeta \right) \frac{\partial \rho}{\partial \phi} dz\]

Transport Equations

In baroclinic mode, ADCIRC solves transport equations for temperature and salinity:

\[\frac{\partial (HT)}{\partial t} + \frac{\partial (UHT)}{\partial x} + \frac{\partial (VHT)}{\partial y} = \frac{\partial}{\partial x} \left( H K_h \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( H K_h \frac{\partial T}{\partial y} \right) + H Q_T\]
\[\frac{\partial (HS)}{\partial t} + \frac{\partial (UHS)}{\partial x} + \frac{\partial (VHS)}{\partial y} = \frac{\partial}{\partial x} \left( H K_h \frac{\partial S}{\partial x} \right) + \frac{\partial}{\partial y} \left( H K_h \frac{\partial S}{\partial y} \right) + H Q_S\]

where:

  • \(K_h\) = horizontal diffusion coefficient

  • \(Q_T\) = temperature source/sink term (e.g., heating, cooling)

  • \(Q_S\) = salinity source/sink term (e.g., evaporation, precipitation, river inflow)

Three-Dimensional Formulation

For fully three-dimensional baroclinic simulations, ADCIRC uses a sigma-coordinate system with the vertical domain divided into layers. The 3D momentum equations include additional terms for vertical advection and diffusion:

\[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} - fv + \ldots + \frac{\partial}{\partial z} \left( K_v \frac{\partial u}{\partial z} \right) = 0\]
\[\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} + fu + \ldots + \frac{\partial}{\partial z} \left( K_v \frac{\partial v}{\partial z} \right) = 0\]

where:

  • \(u, v, w\) = velocity components in 3D

  • \(K_v\) = vertical eddy viscosity

The vertical velocity (\(w\)) is diagnosed from the continuity equation.

The 3D transport equations for temperature and salinity are:

\[\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} = \frac{\partial}{\partial z} \left( K_T \frac{\partial T}{\partial z} \right) + \ldots\]
\[\frac{\partial S}{\partial t} + u \frac{\partial S}{\partial x} + v \frac{\partial S}{\partial y} + w \frac{\partial S}{\partial z} = \frac{\partial}{\partial z} \left( K_S \frac{\partial S}{\partial z} \right) + \ldots\]

where \(K_T\) and \(K_S\) are vertical diffusion coefficients for temperature and salinity, respectively.

Surface Heat Flux

The heat flux at the air-sea interface is parametrized by:

\[Q_{net} = Q_{sw} - Q_{lw} - Q_{sen} - Q_{lat}\]

where:

  • \(Q_{sw}\) = short-wave (solar) radiation

  • \(Q_{lw}\) = long-wave (thermal) radiation

  • \(Q_{sen}\) = sensible heat flux

  • \(Q_{lat}\) = latent heat flux (evaporation)

These fluxes are computed based on meteorological data such as air temperature, humidity, wind speed, and cloud cover.

Numerical Considerations for Baroclinic Simulations

Baroclinic simulations introduce several numerical challenges:

  1. Stable Stratification: In stably stratified regions, vertical mixing is suppressed, requiring adequate vertical resolution.

  2. Internal Waves: Baroclinic modes include internal waves, which have higher frequencies than external modes, potentially requiring smaller time steps.

  3. Mode Splitting: ADCIRC can use a mode-splitting approach, where external (barotropic) and internal (baroclinic) modes are solved with different time steps to enhance computational efficiency.

  4. Pressure Gradient Errors: Numerical errors in computing the baroclinic pressure gradient, particularly in regions of steep bathymetry, are mitigated through specialized algorithms.