39.10 meridional_tracer_budget

What are the dominant meridional balances (if any) in the model and how do they depend on time? Option meridional_tracer_budget contracts the tracer equation into a one dimensional equation in latitude by averaging over longitude and depth cells. Using operators described in Section 22.8.7, this yields

 

$$\frac{1}{Vol^T_{j}}\sum_{i=2}^{imt-1}\sum_{k=1}^{km}\Delta^T_{i,k,j} \cdot \Bigl[\delta_t(T_{i,k,j,n,\tau}) + ADV\_Tx_{i,k,j} + ADV\_Ty_{i,k,j} + ADV\_Tz_{i,k,j} =$$

$$DIFF\_Tx_{i,k,j} + DIFF\_Ty_{i,k,j} + DIFF\_Tz_{i,k,j} + source_{i,k,j}\Bigr]$$

where n is the tracer and the relation between j and jrow is as described in Section 14.2. The volume element and total volume as a function of latitude are given by

$$\Delta^T_{i,k,j} tmask_{i,k,j}\cdot\dxti\cdot\cstj\cdot\dytj\cdot dzt_k$$ = (39.158)

$$\sum_{k=1}^{km}\sum_{i=2}^{imt-1} \Delta^T_{i,k,j}$$ Vol^T_j = (39.159)

The quantity tmask_i,k,j is 1 for ocean cells and 0for land cells. Integrating and applying boundary conditions in depth and longitude to Equation (39.161) yields

 

$$\sum_{i,k}\delta_t(T_{i,k,j,n,\tau})\Delta^T_{i,k,j} + \sum_{i,k}(ADV\_Ty_{i,k,j})\Delta^T_{i,k,j} \sum_{i,k}(stf_{i,j,n})\Delta^T_{i,k,j} + \sum_{i,k}(DIFF\_Ty_{i,k,j})\Delta^T_{i,k,j}$$ =

$$\sum_{i,k}(source_{i,k,j})\Delta^T_{i,k,j}$$ + (39.160)

where $\sum_{i,k}$ is shorthand for summing over all cells in longitude and depth. Each term in Equation (39.164) is then averaged in time to produce stable estimates which can indicate the dominant meridional balances as a function of time. The meridional tracer equation becomes

 

$$\frac{1}{L}\sum_{i,k,\ell=1}^{L}\Bigl[ \!\!\!\!\!\!\!\!\!\!\!\delta_t(T_{i,k,j,n,\tau}) + ADV\_Ty_{i,k,j} = stf{i,j,n} + DIFF\_Ty_{i,k,j}$$

$$source_{i,k,j}\Bigr]$$ + (39.161)

where $\ell$ is the time step counter and L is the number of time steps in the averaging period described below. The i,k in the sum indicates a sum over all longitude and depth cells. The individual terms in Equation (39.165) are given as

$$\frac{1}{L}\sum_{\ell=1}^{L}\Bigl[\frac{1}{Vol^T_{j}}\sum_{k=1}^{... ...ac{t_{i,k,j,n,\tau+1} - t_{i,k,j,n,\tau-1}}{2\Delta\tau} \Delta^T_{i,k,j}\Bigr]$$ tstor_jrow,n = (39.162)

$$-\frac{1}{L}\sum_{\ell=1}^{L}\Bigl[\frac{1}{Vol^T_{j}}\sum_{k=1}^{km}\sum_{i=2}^{imt-1} ADV\_Ty_{i,k,j}\cdot \Delta^T_{i,k,j}\Bigr]$$ tdiv_jrow,n = (39.163)

$$-\frac{1}{L}\sum_{\ell=1}^{L}\Bigl[\frac{1}{Area^T_{j}}\sum_{i=2}^{imt-1} stf_{i,j,n}\cdot A^T_{i,j}\Bigr]$$ tflux_jrow,n = (39.164)

$$\frac{1}{L}\sum_{\ell=1}^{L}\Bigl[\frac{1}{Vol^T_{j}}\sum_{k=1}^{km}\sum_{i=2}^{imt-1} source_{i,k,j}\cdot \Delta^T_{i,k,j}\Bigr]$$ tsorc_jrow,n = (39.165)

$$-\frac{1}{L}\sum_{\ell=1}^{L}\Bigl[\frac{1}{Vol^T_{j}}\sum_{k=1}^{km}\sum_{i=2}^{imt-1} DIFF\_Ty_{i,k,j}\cdot \Delta^T_{i,k,j}\Bigr]$$ tdif_jrow,n = (39.166)

where the area element and total area of a latitude are given by

$$tmask_{i,1,j}\cdot \dxti\cdot\cstj\cdot\dytj$$ A^T_i,j = (39.167)

$$\sum_{i=2}^{imt-1} A^T_{i,j}$$ Area^T_j = (39.168)

The output from this diagnostic may be written as ascii to the model printout or as 32 bit IEEE unformatted data to file tracer_bud.yyyyyy.mm.dd.dta. If option netcdf or meridional_tracer_budget_netcdf is enabled, data is written in NetCDF format to file tracer_bud.yyyyyy.mm.dd.dta.nc rather than in unformatted IEEE. The ``yyyyyy.mm.dd'' is a place holder for year, month, and day and this naming convention is explained further in Section 38.2. The interval between output is specified by variable tmbint and the control is specified by variable iotmb. The averaging period tmbper is typically set equal to the interval but may be specified shorter. How about producing monthly averages when months vary in length? Enabling option monthly_averages will over-ride values in tmbper and tmbint and make it so. However, this option is global and applies to all time averaged diagnostics that are enabled. Refer to Section 39.11 for more details. Variables tmbper and tmbint are input through namelist. Refer to Section 14.4 for information on namelist variables. Note further that this dataset may be contracted in latitude and region space to yield a zero dimensional model which describes the total heat content of one tremendously gigantic cell as a function of time.

$$<tstor_{jrow,n}>^\phi \frac{1}{Area^T}\sum_{jrow=2}^{jmt-1}\Bigl[ tflux_{jrow,n} \Bigr] \Delta^T_{j}$$ = (39.169)

where the area element is

$$Area^T = \sum_{jrow=2}^{jmt-1}\Delta^T_{j}$$ (39.170)

However, this can only be done if the averaging period equals the interval for writing the output. But, when the averaging period equals the sampling interval this diagnostic is a significant time burner. For most cases, it may be adequate to specify a short averaging period at the end of the sampling interval (e.g. in the limit, a one time step average at the end of every sampling interval.). The terms in the above analysis may be further decomposed into regions based on mask  msktmb_i,jrow along the lines of Section 38.5. However,  msktmb_i,jrow =1 and it is left to the researcher to define other regions if desired. Note that the regional mask msktmb_i,jrow is also written to file tracer_bud.yyyyyy.mm.dd.dta when the initialization boolean itmb is true. It should be set true on the first run but false thereafter.