Column crossing and lateral transport in numerical weather models

Transport in numerical weather models is typically restricted to advection by the resolved wind field, and representation of other flow processes relies on forcing or coupling to other models. A discussion of non‐advective transport in numerical weather models is presented and illustrated using two examples. These are the sea breeze and surface water flow. Simple models of these phenomena are represented in an adapted version of the Met Office Unified Model (UM) by means of modifying the existing cold‐pool scheme in the UM. This approach of incorporating different transport processes in one framework could facilitate a better representation of the Earth system, by increasing interaction between meteorological, surface and subsurface processes.


| INTRODUCTION
Many phenomena relevant to environmental prediction involve flow or transport processes. For instance, atmospheric pollutants are advected by the prevailing wind, and the volume flux in rivers is an indicator of flood risk. Usually, a numerical weather prediction (NWP) model contains a dynamical core for predicting the wind field, but the majority of physical processes are column-based. That is, they take local or wind-advected properties as inputs to physical parametrizations of a (spatial) gridboxmean state, and rely on the wind to propagate the outputs.
For flows which are not modelled by NWP dynamics, such as volcanic 'umbrella' clouds or rivers, another model, forced by NWP output, is typically used (Dadson et al., 2011;Jones et al., 2007;Webster et al., 2020).
There are also local wind effects which may be poorly represented, due to lack of sufficient resolution or parametrization. These are often non-hydrostatic, driven by local buoyancy effects, and examples include convective cold pools, dust storms and sea breezes. (Buoyancydriven flows have in common with the movement on land of water, or ice, that they are driven by the gravitational acceleration.) They can produce significant local impacts, as well as feeding back into other processes such as atmospheric convection (Hirt et al., 2020;Meyer & Haerter, 2020;Rooney et al., 2022).
It is arguable that the present drive for 'digital twins' of the environment (e.g., Bauer et al., 2021) might be facilitated by a unified modelling infrastructure for any flows unresolved or unrepresented by the dynamical core. This may simplify the process of changing parametrizations and resolution, as well as allowing re-use of the same transport framework for different processes.
Lateral transport of cold pools was recently added as an optional process into the Met Office Unified Model (UM). Here, the extension of this method to other processes is discussed, providing an example of how the same basic infrastructure may be used for multiple flows.

| REPRESENTATION OF COLD POOLS IN THE UM
The UM is based on a rectangular lat-long grid. C-POOL, the cold-pool scheme recently added to the UM, is a 2-D representation of cold-pool generation, propagation and dissipation (see Rooney et al., 2022, for a full description). It stores information about cold-pool depth, buoyancy, velocity and residence time in a set of 2-D fields, which are separate from the 3-D fields characterizing the gridbox-mean atmospheric state. The convection-scheme downdraught parametrization forces the cold-pool scheme and, in turn, the cold-pool properties are drawn upon to modify the parametrized processes of convection.
C-POOL includes infrastructure which allows for cold-pool information to propagate laterally across gridboxes. If a cold pool is deemed to have a propagation lengthscale greater than the grid length, then the properties can be added into working arrays in neighbouring gridboxes, at the same time as being removed from the departure-point location. If more than one cold-pool front should arrive at the same location in the same timestep, the arrival point accommodates this using a set of rules whereby properties may be accumulated as sums or maxima, depending on the physical model. The working arrays generate the updated fields for use in the next timestep. Cold-pool propagation in the scheme does not yet incorporate additional flow acceleration or deceleration due to interaction with the background wind or surface topography, for instance. However, future model developments will hopefully include additional effects such as these.
Downdraught forcing initially produces a cold pool within a gridbox. With sufficient forcing or resolution, the cold pool will seek to propagate 'outward' on the grid in all horizontal directions. Once a front is in motion, each part of it acquires a direction of travel, and the scheme uses information about front velocity to allow this directed motion to continue (until fronts dissipate, or collide). Thus, at any timestep, the prognostic fields may include gridbox-based 'isotropic' features with no directional information, as well as propagating fronts with a specified velocity, and the scheme will evolve all these elements as required.

| Sea breeze
Sea breezes are similar to cold pools, being near-surface atmospheric flows driven by local buoyancy gradients. Each of these has the frontal structure of a gravity current (Simpson, 1987), with a sharp horizontal buoyancy gradient, and a significant along-front vorticity vector and hence vertical velocity. These features are difficult for NWP models to resolve accurately (Crosman & Horel, 2010). They may be expected to force convection in a similar way to cold pools, so that their representation would become an extension of the cold-pool scheme, primarily for the purpose of forcing parametrized convection (Bergemann et al., 2015). Crosman and Horel (2010) describe characteristic scales of sea breezes, namely depths of order 500 m, and a possible inland penetration of tens of km, although this scale of horizontal propagation may be conservative in some regions. Miller et al. (2003) give a broad possible depth range of 300-2500 m for the flow behind the seabreeze 'head'.
To adapt the cold-pool scheme to represent sea breezes, the important differences are in initiation. Sea breezes are generated by a coastal temperature contrast rather than a cold downdraught and, forming along coasts, have a preferred (inland) direction of travel.
In the UM, sea-breeze initiation is aided by 'coastal tiling', which is the calculation of both land-and seasurface energy balances in coastal gridboxes. These may be identified, for example, as points where the land fraction is between 0.01 and 0.99. Coastal gridboxes provide co-located (on the grid) land and sea surface temperatures, T L and T S respectively, at coastal points. From these, a representative breeze windspeed u b may be obtained by rearranging the threshold formula for breeze formation (Biggs & Graves, 1962), to obtain where C P is the specific heat capacity of the air and ϵ is a dimensionless constant of order 10 À3 . Conventionally, if u b is greater than the background wind, then a sea breeze may be expected. Various measures of background wind have been used previously, such as surface (10 m) wind from weather stations in data-based studies (Biggs & Graves, 1962;Laird et al., 2001), or the geostrophic wind in model studies (Arritt, 1993). Here, a comparison of u b with the windspeed on the lowest model level is made, and if u b is the greater then a sea-breeze component may be added to the cold-pool fields.
The necessary steps to add a sea-breeze component are to initiate the depth and buoyancy fields, h and g 0 respectively (which determine the breeze speed), and to set a direction of propagation. As in Rooney et al. (2022), h represents the depth of the cold pool or sea breeze i.e. the distance from the surface to the upper limit of the feature, and the buoyancy (or reduced gravity) is given by g 0 ¼ gΔT=T 0 , where g is the gravitational acceleration and ΔT is a representative temperature reduction from the background temperature T 0 .
C-POOL has upper limits of depth and buoyancy, h max and g 0 max , which may be used for initiation via the formulae These are empirical, but formulated in such a way that g 0 h ¼ u 2 b , that is, the Froude number is unity. Previous observational evidence for the buoyancy-inertia balance that this represents has been summarized by Miller et al. (2003). This formulation also implies that g 0 =h ¼ g 0 max =h max , that is, the proportion of these sea-breeze properties is aligned with that of parameters already in the model. Ancillary model information on orography gradients is used to provide direction, under the assumption that the inland and uphill directions will often be similar at coastal points. Zonal (West-East) and meridional (South-North) orography gradients are already provided in all model configurations for use by the radiation scheme, see Figure 1. It is trivial to generate velocity components to match the magnitude u b and such that the velocity vector is aligned with the uphill direction.
Having initialized a breeze-front in this manner, it will be evolved by C-POOL in the same way as a coldpool front which has acquired direction by propagating outward at some previous timestep. Thus, the model requires no modification beyond the initiation process.
As a demonstration, a climate-resolution configuration of the UM was run for a short period, with the code modified as described above. The horizontal resolution is classified as N96, that is, a gridbox size of around 130 km in midlatitudes. The model has 85 vertical levels (L85) with the lowest level at 20 m above the surface (Walters et al., 2019).
In this run, sea-breeze forcing at coastal points was enabled but the usual downdraught forcing of cold pools was disabled. Thus, all the cold-pool fields originate from sea breeze forcing. Otherwise, the model configuration was as described by Rooney et al. (2022), including the run start time, namely 0000 UTC on 1 September 2006. Figure 2 shows example results at 36 and 40 h into the run, that is, 1200 UTC and 1600 UTC on 2 September 2006, or local noon at 0 and 60 W, respectively. Panels 2(a) and (b) show the zonal component of the sea-breeze velocity, and (c) and (d) show the meridional component. Thus, panels (a) and (c) demonstrate sea-breeze activity in the afternoon around the Mediterranean and the Arabian peninsula, for instance. This has largely subsided in the later panels (b) and (d), but some activity has begun on the West coast of S.America.
There is some evidence of inland propagation during the local afternoon, for example on the Arabian peninsula and the West coast of Southern Africa. Enlargements of the Arabian area are shown in Figure 3 for illustration. Thus, the inland propagation in the simulation may not be unrealistic. It should be noted that the details of this experiment will depend on model specifics such as the height of the lowest model level. However, the surface wind scale is not expected to vary greatly across the 10-20 m elevation range, so limited sensitivity to increased vertical resolution is anticipated.

| Surface water
The sea-breeze representation described above combines three elements; a physical model of gravity currents, gridbox-crossing infrastructure, and information on orographic gradients. The first two of these come from the C-POOL scheme. For other gravity-driven processes, the last two may be retained, while substituting a different process model. This section presents an idealized demonstration of how surface water movement could be modelled with the same infrastructure. In this idealization, forcing is applied constantly and uniformly on the model grid, with a simple gridbox-based parametrization controlling its redistribution.
The orographic gradients in the zonal and meridional directions, denoted d x and d y respectively, are used to redistribute the load of accumulated forcing. Letting then s yields values between 0 and 1, indicating the steepness of terrain. If the loading of a gridbox at a particular timestep is L, then the amount f which may be offloaded by gravitydriven transport is modelled as and the flowspeed is modelled as proportional to f. The (downhill) horizontal direction of flow is given by tan À1 d y =d x À Á þ π. This very simple gridbox-based model then makes use of the same infrastructure as C-POOL for flow propagation. The values of L are reduced by the amount f, which is relocated (or not) depending on the speed and residence time in the gridbox. The working array, initially zero everywhere at the start of the timestep, is updated by adding values of LÀ f ð Þ and f at the appropriate locations, This automatically accommodates propagation into a gridbox from more than one source location, if necessary. At each timestep, uniform forcing is also added. The working array then becomes the updated field of L. The values from the updated field may then be used to force other parametrizations such as surface infiltration etc.
The values of L at different times during a trial integration are shown in Figure 4. In this trial, the UM in the same climate configuration as above was run for 30 days with the flow model incorporated (although not interacting with any other model processes), and the panels in Figure 4 show the gridbox loading after 1 h, 9.5 days, 19.5 days and 29.5 days.
The absolute values are not meaningful, nor are the absolute travel times, since the flow speed is taken as proportional to the offload. However, the emerging pattern shows how the model operates. There is no transport on sea points away from the coast, so the loading accumulates uniformly there. On land, steep terrain sheds the loading effectively, and it then accumulates near the base of mountain ranges. Even on less-steep terrain, further transport will still take place, at a slower rate. There is evidence of this from the highlighted edges of landmasses, where transport terminates in the neighbouring sea points, and hence loading is relatively increased there.
Further development would be required to produce a properly functional surface-runoff model. However creating the lateral-transport infrastructure is arguably the most difficult step, since other model refinement and the incorporation of processes such as surface infiltration, drag etc. can be applied locally in each gridbox, following the usual development approach for model physics. Orographic gradients would probably also need to be replaced with dedicated ancillaries to direct the flow (Yamazaki et al., 2009).
Having all of the hydrological cycle within one model could facilitate synthesis and closure of the energy, water and carbon budgets (Galy et al., 2015;Hathway & Sharples, 2012;Yang et al., 2021), and also the linking of lake models to the river network (Rooney & Bornemann, 2013;Williamson et al., 2009). With separate models for hydrology and meteorology, as is typical at present, an extensive model coupler could potentially provide the same visibility of hydrological variables to the meteorology and vice versa. Separate models also allow for differences in time and space resolution. However, the incorporation of all processes in a single model framework may promote further interactions between meteorology and hydrology, and encourage efforts to develop model resolution-independence.

| CONCLUDING DISCUSSION
The above two examples show how the transport infrastructure of the C-POOL scheme can be adapted to potentially represent other processes involving lateral movement on the model grid. Rather than needing to couple in a separate model for each process, having such an infrastructure in place allows for a multiplicity of processes to be incorporated in a single system. The transport routines are generic, and may be called by different processes, with different methods of determining propagation speed. Computation and memory overheads would possibly be less than those required to couple and run separate models. Thus the scope for interactions, such as heat or carbon exchange, increases.
C-POOL is essentially a physically based cellular automaton. A lattice-based approach to modelling flows of this sort was also followed by Böing (2016). It is arguable that, at very high resolution, an NWP model may be able to explicitly resolve flows such as cold pools. However, there are other processes, such as horizontal subsurface diffusion of heat or moisture, which would still be unrepresented, even though they may be more significant at higher resolution. It is possible that diffusive (or other) flows could also be modelled with some adaptation of the present model infrastructure (Chopard & Droz, 1991;Packard & Wolfram, 1985).
As described by Rooney et al. (2022), in the present formulation of C-POOL the prognostic fields are given a so-called 'small halo' in multi-processor decomposition of the model integration. This means that field edges, which are swapped across processor boundaries, are one gridbox wide. This limits propagation to nearest neighbours only during a single timestep. For the representation of faster processes, adaptation to fields with large haloes would allow transport beyond the nearest neighbours.
NWP at the Met Office is currently in transition between the UM, which has been in use for many years, and LFRic, a new model formulation designed to take advantage of highly parallel computer processing (Adams et al., 2019). LFRic is, comparatively, still at an early stage of development, so this may be a good point at which to promote discussion of these issues.
AUTHOR CONTRIBUTIONS Gabriel G. Rooney: Conceptualization; software; writing original draft.