MeridionalAdvectionDiffusion

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General solver of the 1D meridional advection-diffusion equation on the sphere:

\[\begin{split}\frac{\partial}{\partial t} \psi(\phi,t) &= -\frac{1}{a \cos\phi} \frac{\partial}{\partial \phi} \left[ \cos\phi ~ F(\phi,t) \right] \\ F &= U(\phi) \psi(\phi) -\frac{K(\phi)}{a} ~ \frac{\partial \psi}{\partial \phi}\end{split}\]

for a state variable \(\psi(\phi,t)\), arbitrary diffusivity \(K(\phi)\) in units of \(x^2 ~ t^{-1}\), and advecting velocity \(U(\phi)\). \(\phi\) is latitude and \(a\) is the Earth’s radius (in meters).

\(K\) and \(U\) can be scalars, or optionally vector specified at grid cell boundaries (so their lengths must be exactly 1 greater than the length of \(\phi\)).

\(K\) and \(U\) can be modified by the user at any time (e.g., after each timestep, if they depend on other state variables).

A fully implicit timestep is used for computational efficiency. Thus the computed tendency \(\frac{\partial \psi}{\partial t}\) will depend on the timestep.

In addition to the tendency over the implicit timestep, the solver also calculates several diagnostics from the updated state:

  • diffusive_flux given by \(-\frac{K(\phi)}{a} ~ \frac{\partial \psi}{\partial \phi}\) in units of \([\psi]~[x]\)/s

  • advective_flux given by \(U(\phi) \psi(\phi)\) (same units)

  • total_flux, the sum of advective, diffusive and prescribed fluxes

  • flux_convergence (or instantanous scalar tendency) given by the right hand side of the first equation above, in units of \([\psi]\)/s

Non-uniform grid spacing is supported.

The state variable \(\psi\) may be multi-dimensional, but the diffusion will operate along the latitude dimension only.

class climlab.dynamics.meridional_advection_diffusion.MeridionalAdvectionDiffusion(K=0.0, U=0.0, use_banded_solver=False, prescribed_flux=0.0, **kwargs)[source]

Bases: AdvectionDiffusion

A parent class for meridional advection-diffusion processes.

Attributes:
K
U
depth

Depth at grid centers (m)

depth_bounds

Depth at grid interfaces (m)

diagnostics

Dictionary access to all diagnostic variables

input

Dictionary access to all input variables

lat

Latitude of grid centers (degrees North)

lat_bounds

Latitude of grid interfaces (degrees North)

lev

Pressure levels at grid centers (hPa or mb)

lev_bounds

Pressure levels at grid interfaces (hPa or mb)

lon

Longitude of grid centers (degrees)

lon_bounds

Longitude of grid interfaces (degrees)

prescribed_flux
timestep

The amount of time over which step_forward() is integrating in unit seconds.

Methods

add_diagnostic(name[, value])

Create a new diagnostic variable called name for this process and initialize it with the given value.

add_input(name[, value])

Create a new input variable called name for this process and initialize it with the given value.

add_subprocess(name, proc)

Adds a single subprocess to this process.

add_subprocesses(procdict)

Adds a dictionary of subproceses to this process.

compute()

Computes the tendencies for all state variables given current state and specified input.

compute_diagnostics([num_iter])

Compute all tendencies and diagnostics, but don't update model state.

declare_diagnostics(diaglist)

Add the variable names in inputlist to the list of diagnostics.

declare_input(inputlist)

Add the variable names in inputlist to the list of necessary inputs.

integrate_converge([crit, verbose])

Integrates the model until model states are converging.

integrate_days([days, verbose])

Integrates the model forward for a specified number of days.

integrate_years([years, verbose])

Integrates the model by a given number of years.

remove_diagnostic(name)

Removes a diagnostic from the process.diagnostic dictionary and also delete the associated process attribute.

remove_subprocess(name[, verbose])

Removes a single subprocess from this process.

set_state(name, value)

Sets the variable name to a new state value.

set_timestep([timestep, num_steps_per_year])

Calculates the timestep in unit seconds and calls the setter function of timestep()

step_forward()

Updates state variables with computed tendencies.

to_xarray([diagnostics])

Convert process variables to xarray.Dataset format.
class climlab.dynamics.meridional_advection_diffusion.MeridionalDiffusion(K=0.0, use_banded_solver=False, prescribed_flux=0.0, **kwargs)[source]

Bases: MeridionalAdvectionDiffusion

A parent class for meridional diffusion-only processes, with advection set to zero.

Otherwise identical to the parent class.

Attributes:
K
U
depth

Depth at grid centers (m)

depth_bounds

Depth at grid interfaces (m)

diagnostics

Dictionary access to all diagnostic variables

input

Dictionary access to all input variables

lat

Latitude of grid centers (degrees North)

lat_bounds

Latitude of grid interfaces (degrees North)

lev

Pressure levels at grid centers (hPa or mb)

lev_bounds

Pressure levels at grid interfaces (hPa or mb)

lon

Longitude of grid centers (degrees)

lon_bounds

Longitude of grid interfaces (degrees)

prescribed_flux
timestep

The amount of time over which step_forward() is integrating in unit seconds.

Methods

add_diagnostic(name[, value])

Create a new diagnostic variable called name for this process and initialize it with the given value.

add_input(name[, value])

Create a new input variable called name for this process and initialize it with the given value.

add_subprocess(name, proc)

Adds a single subprocess to this process.

add_subprocesses(procdict)

Adds a dictionary of subproceses to this process.

compute()

Computes the tendencies for all state variables given current state and specified input.

compute_diagnostics([num_iter])

Compute all tendencies and diagnostics, but don't update model state.

declare_diagnostics(diaglist)

Add the variable names in inputlist to the list of diagnostics.

declare_input(inputlist)

Add the variable names in inputlist to the list of necessary inputs.

integrate_converge([crit, verbose])

Integrates the model until model states are converging.

integrate_days([days, verbose])

Integrates the model forward for a specified number of days.

integrate_years([years, verbose])

Integrates the model by a given number of years.

remove_diagnostic(name)

Removes a diagnostic from the process.diagnostic dictionary and also delete the associated process attribute.

remove_subprocess(name[, verbose])

Removes a single subprocess from this process.

set_state(name, value)

Sets the variable name to a new state value.

set_timestep([timestep, num_steps_per_year])

Calculates the timestep in unit seconds and calls the setter function of timestep()

step_forward()

Updates state variables with computed tendencies.

to_xarray([diagnostics])

Convert process variables to xarray.Dataset format.