Glacial fjords are host to a range of processes that serve to influence a fjord's circulation and properties e.g.. In this first iteration of FjordRPM, we focus on a small number of key processes that we deem to be of primary importance (Fig. ). Two of these are specific to glacial environments and are recognised to be of particular significance in glacial fjords. Firstly, the input of subglacial discharge at the glacier grounding line drives buoyant plumes that mix with and upwell ambient waters before reaching the fjord surface or finding neutral buoyancy at an intermediate depth . Secondly, the melting of icebergs calved into the fjord from tidewater glaciers acts as a heat sink and freshwater source while driving convective upwelling adjacent to icebergs . In addition to these processes, we include the exchange of water between the fjord and continental shelf, as driven by pressure gradients in response to spatial and temporal variation in water properties e.g., and vertical mixing and advection between model layers.

We split the fjord (Fig. ) into N vertically stacked layers, indexed by j from the surface (j=1) to the bottom (j=N). Each layer has a volume Vj, which is fixed in time. At present, the model assumes a cuboid-shaped fjord of length L and width W, such that Vj=WLHj, where Hj is the thickness of layer j. Each layer also has a static iceberg area in contact with the ocean Ij and a time-varying temperature Tj(t) and salinity Sj(t) that can be considered as an average over the volume of the layer. The number of layers and their volumes are user-defined; the layers do not have to have equal volume or thickness (e.g. see Fig. ), though the sum of the layer thicknesses has to equal the fjord depth. Fewer layers means less vertical resolution, whereas more layers limits the time step of the model. If the fjord has a sill, then the model will slightly adjust the thicknesses to ensure that the sill depth coincides with a layer boundary (e.g. Fig. ), simplifying the formulation of the fjord–shelf exchange.

The layers experience time-dependent exchanges (Fig. ) with the subglacial discharge-driven plumes (entrainment and intrusion), with the shelf (fjord–shelf exchange), with icebergs (melting and upwelling), and with adjacent layers (vertical mixing and vertical advection). Note that the model allows for multiple plumes because multiple marine-terminating glaciers can terminate in a single fjord. Other glacial fjord exchange processes and drivers of circulation, such as ambient melting of the glacier, surface fluxes, sea ice, winds, and tides, were considered lower priority and are not yet implemented, but this is not intended to imply that these processes are not important in some situations. Exchange fluxes are denoted by QjXy, where j denotes the layer where the exchange is happening, X denotes the flux quantity being exchanged (V for volume, T for heat, or S for salt) and y denotes the process driving exchange (p for plume, s for shelf, i for icebergs, k for vertical mixing, or v for vertical advection).

For fluxes that represent exchange between adjacent layers (iceberg-driven upwelling, vertical mixing, and vertical advection), a further line of explanation is required. These fluxes naturally take a general form Qj+1,jXy, where j+1 and j are the layers exchanging. These fluxes can, however, be written in the single-subscript format previously introduced by noting that the net effect on layer j is the difference between the exchange with layer j+1 and the exchange with layer j-1; that is, QjXy=Qj+1,jXy-Qj,j-1Xy. In this expression, there is no exchange between the bottom layer and the sea floor or between the surface layer and atmosphere (QN+1,NXy=Q1,0Xy=0).

For sign convention, a volume flux QjVy is defined as positive if it is adding volume to a layer. A volume flux between layers Qj+1,jVy is defined as positive if it removes volume from the deeper layer j+1 and adds it to the shallower layer j. To give some examples, Q2Vp is the volume exchange between the plume and layer 2, which is positive if the plume is adding volume to layer 2, and Q5Vs is the volume exchange between fjord layer 5 and the shelf, which is positive if it is directed from the shelf into the fjord.

Estimating many of the exchanges requires knowledge of the density differences between two water masses. To do this, we employ a linear equation of state ρ(T,S)=ρref1+βS(S-Sref)-βT(T-Tref). Here, ρref is the reference density of water with salinity Sref and temperature Tref, and βS and βT are the haline contraction and thermal expansion coefficients, respectively. The relative buoyancy of two water masses a and b is then defined by 1gab′=gρa-ρbρref=gβSSa-Sb-βTTa-Tb, in which g is the gravitational acceleration and the reference properties have cancelled out.

We now describe in turn how we parameterise exchange with the plume (Sect. ), with the shelf (Sect. ), with icebergs (Sect. ), due to vertical mixing (Sect. ), and due to vertical advection (Sect. ). For reference, a full list of model variables and parameters, including their units, is given in Tables  and .

If there is subglacial discharge emerging from beneath a marine-terminating glacier, this will drive an upwelling plume adjacent to the glacier that entrains water at depth and intrudes into the fjord at a level of neutral buoyancy . Within FjordRPM, this is represented by a flux leaving the deeper layers and a flux entering a shallower layer (Fig. ). To quantify these fluxes, we employ a now-standard approach that has been termed “plume-melt theory” by , consisting of the classic theory for buoyant plumes coupled to the standard three-equation parameterisation for submarine melt . If there are multiple discharge plumes emerging from a single glacier or multiple glaciers with discharge plumes draining into a single fjord, the following approach is applied separately for each discharge plume. Note that submarine melting and melt-driven convection outside of subglacial discharge plumes are not currently accounted for in FjordRPM.

We assume a line plume geometry with a fixed plume width in the across-fjord direction of Wp. As the plume rises, plume volume, momentum, heat, and salt flux per unit width evolve according to 2addzbpup=αpup+mp,2bddzbpup2=bpgp′-Cdup2,ddzbpupTp=αpupT+mpTb2c-Cd1/2ΓTupTp-Tb,ddzbpupSp=αpupS+mpSb2d-Cd1/2ΓSupSp-Sb, in which bp is the plume width in the along-fjord direction, up is the plume (vertical) velocity, Tp is the plume temperature, Sp is the plume salinity, αp is the plume entrainment coefficient, gp′=gβSS-Sp-βTT-Tp is the relative buoyancy of the fjord and plume, and T and S are the fjord temperature and salinity (see also Table ). All other variables and parameters relate to the submarine melt rate induced by the plume, mp, which is given by three equations that balance heat and salt flux through the ice–ocean boundary layer and maintain the boundary layer at the in situ freezing point e.g.: 3ampl+mpciTb-Ti=cwCd1/2ΓTupTp-Tb,3bmpSb=Cd1/2ΓSupSp-Sb,3cTb=λ1Sb+λ2+λ3|z|, where Tb and Sb are the temperature and salinity of the ice–ocean boundary layer, l is the latent heat of ice melt, ci and cw are the heat capacities of ice and seawater, Cd is the plume–ice drag coefficient, ΓT and ΓS are heat and salt transfer coefficients, and λi denotes the constants that control the linearised freezing point (see also Table ).

The boundary conditions for Eq. () are the fjord temperature T and salinity S, which are taken from the FjordRPM layers. The plume is initiated by a flux of subglacial discharge Qsg emerging from beneath the glacier at the grounding line depth Hgl, with initial salinity Ssg=0 and temperature Tsg=λ2+λ3Hgl (i.e. the in situ freezing point). The initial plume width and velocity are set by ensuring that the plume has a balance of buoyancy and momentum e.g. and an initial flux of Qsg, giving bp=αpQsg2/gp′Wp21/3 and up=Qsg/bpWp.

At each FjordRPM time step, Eqs. () and () are numerically integrated using an Euler scheme on the vertical grid defined by the interfaces between the FjordRPM model layers (Fig. ). To provide an example based on Eq. (a), if the plume width, velocity, and melt rate at the interface between layers j+1 and j are bpj+1,j, upj+1,j, and mpj+1,j, then the plume volume flux at the next interface up, between layers j and j-1, is estimated by bpj,j-1upj,j-1=bpj+1,jupj+1,j4+Hjαpupj+1,j+mpj+1,j. Beginning from the grounding line, this numerical integration continues to shallower layers until the plume finds neutral buoyancy and intrudes into the fjord. The layer of plume neutral buoyancy is defined as the layer where the plume–fjord relative buoyancy, gp′, switches from positive to negative, indicating that the plume is negatively buoyant in that layer and will stop rising and intrude horizontally. If the plume reaches the fjord surface without becoming negatively buoyant, then it intrudes into the surface layer. In this manner, we obtain the plume velocity up and submarine melt rate mp at each of the layer interfaces.

The volume flux per unit plume width entrained from the fjord into the plume is the first term on the right-hand side of Eq. (a), while the associated heat and salt fluxes are the first terms on the right-hand side of Eqs. (c) and (d). For layers where the plume is rising, the exchange fluxes between the model layers and the plume (Fig. ) are therefore given by 5aQjVp=-αpWpupj+1,jHj,5bQjTp=-αpWpupj+1,jHjTj,5cQjSp=-αpWpupj+1,jHjSj, where we know upj+1,j from the numerical integration. For any layers that are deeper than the grounding line or shallower than the layer where the plume intrudes into the fjord, all plume fluxes are 0.

In the layer where the plume intrudes into the fjord, say j0, the flux from the plume into the fjord consists of the subglacial discharge, the submarine meltwater, and all of the water that has been entrained into the plume. Thus, the volume flux from the plume into the layer of neutral buoyancy is 6Qj0Vp=Qsg+Wp∑jHjαpupj+1,j+mpj+1,j, where the sum runs over all layers j between the grounding line and the layer of neutral buoyancy. To obtain the heat flux exchange, we apply the same principle, but we first use Eq. (a) to rewrite Eq. (c) as 7ddzbpupTp=αpupT+mpTeff, where 8Teff=Tb-lcw-cicwTb-Ti is the effective meltwater temperature e.g.. Noting the relative size of the terms, we furthermore approximate Teff≈-l/cw. The heat flux from the plume into the layer of neutral buoyancy can then be calculated as 9Qj0Tp=QsgTsg+Wp∑jHjαpupj+1,jTj+mpj+1,jTeff. Lastly, the salt flux from the plume into the layer of neutral buoyancy can be calculated as 10Qj0Sp=Wp∑jHjαpupj+1,jSj, where we have used the fact that the salinity of subglacial discharge, Ssg, is 0 and the fact that the last two terms on the right-hand side of Eq. (d) cancel according to Eq. (b). The sums in Eqs. () and () again run over all layers j between the grounding line and the layer of neutral buoyancy.

By these definitions, the plume removes water from the deeper layers, mixes it with subglacial discharge and submarine meltwater, and puts the resulting mixture back into a layer closer to the surface, representing the upwelling and transformation of fjord waters by the plume and driving an overturning circulation. While the plume exchange fluxes in a given layer may be large, note that because all of the volume flux entrained from deeper layers is put back into a shallower layer, the net volume flux over all layers associated with the plume is simply the subglacial discharge and the submarine melt flux; that is, 11∑j=1NQjVp=Qsg+∑j=1NWpHjmpj+1,j≡Qsg+Qsm. Similarly, all of the heat and salt entrained from deeper layers is put back into a shallower layer, and the net heat and salt flux associated with the plume come only from the subglacial discharge and submarine melting.

We assume that exchange between the shelf and fjord is driven by pressure gradients. The pressure difference, Δpj, between the shelf and fjord in layer j may be quantified as 12Δpjρref=gΔη+∫gΔρ(z)ρrefdz, where Δη is the sea surface height difference between the shelf and fjord, Δρ(z) is the density difference between the shelf and fjord, and the integral runs over all depths from the surface down to the midpoint of layer j. The integrand in Eq. () is the relative buoyancy between the shelf and fjord, given by 13gsj′=gβSSjs-Sj-βTTjs-Tj, where Sjs and Tjs are the shelf temperature and salinity profiles mapped onto the FjordRPM grid (Fig. ). That is, 14Sjs(t)=1Hj∫jSs(z,t)dz, and the integral runs over the depth range covered by layer j. An equivalent expression applies for temperature. If we now define a quantity ϕ, given for each layer by 15ϕj=gsj′Hj/2+∑k=1j-1gsk′Hk, then Eq. () may be written as Δpj/ρref=gΔη+ϕj. That is, ϕj is the integral in Eq. () evaluated using the model layers. The shelf-to-fjord pressure gradient is then finally estimated by dividing by the fjord length, L. We assume that the sea surface height difference adjusts quickly to ensure that the net flux into the fjord is 0 and thus treat the barotropic pressure term separately below.

To estimate the shelf–fjord exchange, previous studies have considered cases where this pressure gradient is (i) balanced by friction and mixing ; (ii) in some form of geostrophic balance ; or (iii) under hydraulic control . The relevant balance for a given fjord will depend on the fjord and sill geometry, the stratification, and the circulation. We have derived our exchange parameterisation with regimes (i) and (ii) in mind.

To proceed, we consider the along-fjord momentum balance, neglecting the acceleration and advection terms. Under regime (i), the Coriolis term is additionally neglected, and the pressure gradient can be balanced by vertical stress divergence, obtaining an exchange velocity that scales as H2AϕjL, in which H is the fjord depth and A is a vertical eddy viscosity e.g.. Alternatively, the pressure gradient can be said to be balanced by more general sources of friction, obtaining an exchange velocity that scales as 1rϕjL, in which 1/r is a frictional timescale . Under regime (ii), argued that boundary currents should set up an along-fjord pressure head of similar magnitude to the across-fjord pressure head, which leads to a velocity scaling like 1fϕjL, where f is the Coriolis parameter. Generalising these cases, we take an exchange velocity of the form C0ϕjL and treat C0 as an empirical tunable parameter.

We then define the fjord–shelf volume exchange fluxes for above-sill layers as 16QjVs=WHjub+C0ϕjL, where WHj is the cross-sectional area of contact between layer j and the shelf and ub is a barotropic velocity that ensures overall conservation of the fjord volume (see below). C0 controls the strength of the exchange and accounts for the fjord–shelf balance that is being assumed. For layers that are below the sill (Fig. ), there is no direct exchange between that layer and the shelf.

Regarding the barotropic velocity ub, note that if the sum of all the volume fluxes into the fjord is not 0, there will be a change in the total fjord volume. This could be accounted for by dynamically tracking the sea surface height of the fjord relative to the shelf and its impact on fjord-to-shelf pressure gradients. However, the adjustment timescale for the sea surface height is very fast (on the order of minutes), and therefore tracking sea surface height requires a very small model time step that limits the rest of the model. In addition, the sea surface height rapidly approaches a steady state in which the fjord–shelf volume fluxes have adjusted to ensure there is no net volume flux into the fjord. On the timescales of the model time step (hours to days), it is therefore a very good approximation to choose ub in Eq. () to ensure there is no net volume flux into the fjord. This is equivalent to the “rigid lid” approach sometimes applied in general circulation models. Other than the shelf fluxes, the only contribution to net volume change is from the plume fluxes, where the net volume flux comprises the subglacial discharge and submarine meltwater and is given by Eq. (). Thus, if there are Nabove above-sill layers, conservation of overall fjord volume means 17Qsg+Qsm+∑j=1NaboveQjVs=0, from which, combining with Eq. (), we can obtain the barotropic velocity, ub, as ub=-1W∑j=1NaboveHj18×Qsg+Qsm+WC0L∑j=1NaboveHjϕj.

The associated heat and salt fluxes are obtained by noting that if, for layer j, the volume flux is directed out of the fjord, then the salinity associated with the volume flux should be that of the layer, while if the volume flux is directed into the fjord, then the salinity should be that of the shelf. Thus, the salt fluxes associated with exchange with the shelf are given by 19QjSs=QjVsSjif QjVs<0QjVsSjsif QjVs≥0, with similar expressions for heat, completing the definition of the shelf fluxes.

Iceberg melting freshens and cools fjords , and the input of the buoyant meltwater gives melt-driven convective plumes rising up the sides of icebergs . These plumes could be represented by plume-melt theory , as for the subglacial discharge plumes in Sect. , but we would require many such plumes and would have to decide, on the basis of iceberg characteristics, at what depth to initiate these plumes. We are not convinced that such complexity is merited within a reduced-physics model, and so we have aimed to take a simpler approach without losing the key physics.

Starting with iceberg melting, numerous parameterisations exist for iceberg melt rate (e.g. ). Ideally, we would use the three-equation melt rate parameterisation (Eq. ); however, it is not straightforward to estimate the relative velocity of the icebergs and ocean that would be required. We therefore instead take the simple approach of assuming that the melt rate of icebergs in layer j is proportional to the thermal forcing of that layer, with a constant of proportionality M0. Once scaled for the layer iceberg surface area, Ij, the iceberg melt flux is given by 20Qjmelt=M0Tj-TjfIj, where M0 is a constant and Tjf=λ1Sj+λ2+λ3zj is the linearised in situ freezing point, which depends on the layer salinity Sj and the mean depth of the layer, zj=Hj/2+∑n=1j-1Hn.

To estimate the resulting upwelling flux, we follow scalings from for melt-driven convection. Leaving the details to Appendix , the scale of the (vertical) convection velocity resulting from iceberg melting in layer j can be taken to be 21vj=Qjmeltgj,melt′HjαiIj1/3, where αi is the entrainment coefficient and gj,melt′ is the relative buoyancy of the layer and meltwater, given by 22gj,melt′=gβSSj-βTTj-Teff, in which Teff=-l/cw is the effective meltwater temperature as in Sect. .

By making an analogy with a line plume of width Wj=Ij/Hj, the entrainment flux into the rising melt-driven convection plume is then 23Qjent=αivjHjWj=αivjIj. In a stratified fjord, however, the melt-driven convection cell will reach neutral buoyancy some height after initiating. The length scale that controls this height can be estimated, again following scalings in and detailed in Appendix , by 24lj+1,jice=vj+12Hj+1Hj+Hj+12gj+1,j′. If lj+1,jice>Hj+1, then any melt-driven convection cell that initiates in layer j+1 will upwell to layer j. If, however, lj+1,jice=Hj+1/2 (say), then only melt-driven convection cells initiating in the upper half of layer j+1 would upwell over the layer boundary to layer j. Assuming there is an equal probability of melt-driven convection cells initiating at any point in the layer, the resulting upwelling flux would be half of that in Eq. (). The final iceberg upwelling volume flux from layer j+1 to j is thus given by Eq. () scaled by a fraction fice that accounts for how far iceberg meltwater is able to upwell: 25Qj+1,jVi=min1,lj+1,jiceHj+1Qj+1ent≡fj+1,jiceQj+1ent. Note that fN+1,Nice=f1,0ice=0 because there is no upwelling into the bottom layer or from the surface layer. The associated temperature and salt fluxes are given by multiplying by the temperature and salinity of the layer from which the volume is upwelling.

Having accounted for the upwelling driven by iceberg melting, we return to the melting itself. The impact of melting ice on ambient ocean waters is equivalent to adding freshwater of effective temperature Teff to the ambient . In addition, the volumes of iceberg meltwater are much smaller than the layer volumes, such that we may treat the meltwater as a virtual flux, meaning that we account for its impact on layer temperature and salinity but not on layer volume. Under these assumptions, and if all of the iceberg meltwater in layer j stays in layer j, then the layer heat flux associated with the iceberg melt flux Qjmelt is -Qjmeltl/cw, and the salt flux is -QjmeltSj.

Putting it all together, the net volume flux into layer j due to iceberg melting is 26QjVi=Qj+1,jVi-Qj,j-1Vi, where the terms on the right-hand side are given by Eq. (). The net heat flux into layer j is QjTi=Qj+1,jViTj+1-Qj,j-1ViTj-fj+1,jiceQj+1meltlcw27-1-fj,j-1iceQjmeltlcw, where the first term is the upwelling of ambient water from the layer below, the second term is the upwelling of ambient water to the layer above, the third term is the upwelling of meltwater from the layer below, and the fourth term is the meltwater generated in the present layer that stays in the present layer. Finally, the net salt flux into layer j is QjSi=Qj+1,jViSj+1-Qj,j-1ViSj-fj+1,jiceQj+1meltSj28-1-fj,j-1iceQjmeltSj.

Note that, by definition, ∑j=1NQjVi=0; hence, there is no fjord-scale volume flux associated with iceberg melting. In doing the same sum for heat and salt flux, the first two terms on the right-hand sides of Eqs. () and () cancel because upwelling vertically redistributes heat and salt but does not change the fjord-scale content. The second two terms on the right-hand sides of Eqs. () and () do not, however, cancel, giving a net cooling and freshening due to iceberg melting.

Exchange between layers occurs due to vertical mixing at the layer interfaces. We assume that this causes an exchange of heat and salt but no net exchange of volume and calculate the vertical turbulent tracer fluxes as the product of an eddy diffusivity, Kz, and the vertical gradient of the tracer. Once scaled for the area of contact between layers j+1 and j, this gives a salt exchange of 29Qj+1,jSk=WLKz∂S∂z=2WLKzSj+1-SjHj+1+Hj, and an equivalent expression applies for temperature (we make no distinction between the eddy diffusivity of heat and salt). This eddy diffusivity is estimated as a function of the local Richardson number, Ri, using the KPP scheme : 30Kz=Kb+K0Ri≤0K01-Ri/Ri023Ri0>Ri>00Ri≥Ri0, where Ri0=0.7 is the Richardson number above which shear-driven mixing is suppressed, K0=5×10-3 m2s-1 scales the shear-driven mixing, and Kb=1×10-5 m2s-1 is the background vertical mixing associated with internal waves (Table ). We adopt these default values from , but we note evidence from Greenland fjords that smaller values may be more appropriate . The Richardson number takes the usual definition Ri=N2/(∂u/∂z)2 and is estimated at the interface between layers j+1 and j as Ri=2gj+1,j′/(Hj+1+Hj)2uj+1-uj/Hj+1+Hj2,31=gj+1,j′Hj+1+Hj2uj+1-uj2, where gj+1,j′ is the relative buoyancy of the two layers according to Eq. (). Lastly, the horizontal velocity scale for each layer, uj, is estimated using the plume and shelf exchange fluxes as 32uj=QjVp-QjVs2WHj.

By these definitions, tracer vertical mixing is inhibited by stratification between layers but enhanced by velocity shear driven by the exchange between layers and the plume or shelf. Like for the iceberg fluxes, these layer-to-layer tracer fluxes can be converted into net fluxes for each layer using 33QjSk=Qj+1,jSk-Qj,j-1Sk, and an equivalent expression applies for net heat fluxes.

By the described parameterisations, there are volume fluxes into or out of the layers due to the plume, shelf, and icebergs. If, for a given layer, these fluxes do not sum to 0, then in a model with fixed layer volumes, there must be an additional volume flux that ensures conservation of layer volume. By analogy with models that calculate vertical velocities by imposing incompressibility, we take this additional volume flux to be vertical advection between layers. As an illustrative example, consider a fjord with a plume and a shallow sill that significantly restricts the inflow and outflow. The water that is upwelled by the plume is not able to be replaced directly (i.e. horizontally) by inflow from the shelf. There must therefore be downwelling (or reflux) within the fjord to replace the water entrained into the plume. This downwelling would be represented by the vertical flux now described.

Suppose that the net volume flux imbalance in layer j by the parameterisations described so far is Qjimb, given by 34Qjimb=QjVp+QjVs+QjVi. For the surface layer (j=1), and because there can be no volume flux to the atmosphere, we ensure conservation of volume by imposing a flux from the second layer into the surface layer given by Q2,1Vv=-Q1imb. To ensure conservation of volume for the second layer, we must then impose a flux from the third layer into the second layer given by Q3,2Vv=-Q2imb+Q2,1Vv. We can continue iterating to get the required vertical fluxes for all boxes with the general expression 35Qj+1,jVv=-∑n=1jQnimb. The associated heat and salt fluxes depend on the direction of the flux. For salt flux, this is written as 36Qj+1,jSv=Qj+1,jVvSj+1if Qj+1,jVv≥0Qj+1,jVvSjif Qj+1,jVv<0. That is, if the flux is directed from layer j+1 to j, then the relevant salinity is Sj+1, whereas if the flux is directed from layer j to j+1, the relevant salinity is Sj. As for icebergs and vertical mixing, these layer-to-layer fluxes can finally be converted to net fluxes for each layer using 37QjSv=Qj+1,jSv-Qj,j-1Sv, with equivalent expressions applying for volume and heat fluxes.

Having now defined all of the required fluxes (Fig. ), we turn to the equations giving the time evolution of the model. Because layer volumes are fixed in time, the only evolution equations are for the salt and heat content of each layer. For layer salt content, we have 38VjdSjdt=QjSp+QjSs+QjSi+QjSk+QjSv, where the terms are, respectively, the salt fluxes associated with the plume(s) (Eqs. c and ), shelf (Eq. ), icebergs (Eq. ), vertical mixing (Eq. ), and vertical advection (Eq. ). For layer heat content, we have the equivalent 39VjdTjdt=QjTp+QjTs+QjTi+QjTk+QjTv. These evolution equations, together with all of the flux definitions throughout Sect. –, form a closed set of equations that can be numerically integrated in time.

Before describing the numerics of FjordRPM, we note one last way in which layer temperature and salinity can evolve, which is that we enforce total mixing of two adjacent layers if the upper layer becomes denser than the lower layer. Physically, this represents convection. Specifically, at each time step and for each pair of adjacent layers j+1 and j, we check the buoyancy jump gj+1,j′ between the layers using Eq. (). For any pair of layers having gj+1,j′<0, we reset the salinity of the layers to 40Sj+1=Sj=Sj+1Vj+1+SjVjVj+1+Vj, with an equivalent equation for temperature.

FjordRPM is coded in MATLAB, though future plans include a translation to an open-source language such as Python and to Fortran for ease of coupling with earth system models. We make use of a number of structures to efficiently carry around the required variables: “p” contains all the model parameters, “f” contains the forcing boundary conditions, “a” contains the initial conditions, and “s” contains the solution. The time vector for the model is denoted “t”. Full lists of the variables in these structures are included for reference in Tables  and . The code is arranged in a modular fashion (Fig. ) and proceeds as follows. The top-level routine (run_model) takes as input the structures p, t, f, and a and outputs the solution s. Within this routine, the code first makes a number of checks on the inputs (check_inputs), such as ensuring they have the correct dimensions. The solution variables are initialised and the initial conditions read in (initialise_variables). The forcing boundary conditions may be provided on a depth and time grid that differs from the model layers and time stepping, so some spatial and temporal interpolation of the forcings may be required (bin_forcings).

The model then enters the main time-stepping loop (Fig. ). At each time step ti, with dynamic variables Tj(ti) and Sj(ti), the first action is to check for and resolve any density inversions (homogenise_unstable_layers; Sect. ). The code then uses the temperature and salinity of the layers at the current time step Tj(ti),Sj(ti), together with all of the parameters and variables that do not vary in time, to calculate the heat and salt fluxes described in Sect. – (compute_fluxes and functions therein). With these fluxes, the model makes a forward Euler time step of Eqs. () and () to obtain the temperature and salinity Tj(ti+1),Sj(ti+1) of the layers at the next time step (step_solution_forwards). The final stage of the time-stepping loop is to check that the sum of the volume flux exchange for each layer, which should be 0, is below a certain tolerance (check_solution). If it is not, which only happens if the model becomes unstable (see Sect. ), the simulation is aborted. Once the time-stepping loop has finished, a final function (get_final_output) processes the solution into a final form for saving as output.

A special note is required here for the plume fluxes. Variability in the plume dynamics is determined by the subglacial discharge and fjord water properties. Because subglacial discharge and fjord water properties will generally vary only on timescales of days, while the model time step may be several hours, it is not essential that we “renew” the plume dynamics every single time step. Indeed, integrating the plume-melt theory (Eqs.  and ) is one of the slowest parts of the model, and so updating the plume dynamics only every so often can substantially speed up the model. To implement this, note that the exchange fluxes between the plume and fjord (Eqs. , , , and ) depend on the plume velocity, the induced submarine melt rate, and the fjord properties. Thus, we can use the same plume velocity and induced submarine melt rate for a number of time steps while using the fjord properties from the current time step. Essentially, this decouples the dynamics of the plume from the fjord properties for a number of time steps but maintains conservation of heat and salt because the temperature and salinity that enter the plume fluxes are always from the current time step. The parameter run_plume_every defines the number of time steps between updates of the plume dynamics.

As inputs, the model first requires specification of all of the physical parameters contained in the structure “p”. Default values for all of these parameters are provided in a function default_parameters; some, like the heat capacity of seawater cw, are unlikely to need to be changed, while others, such as the shelf exchange parameter C0, are more likely to be varied by the user. The geometry of the problem must also be specified, including the length, width, and depth of the fjord, the presence and depth of a possible sill, and the grounding line depth of any glaciers with plumes.

The user must also specify the number and thickness of the model layers. The layers do not all have to have the same thickness, so the user could choose to have higher vertical resolution at depths of interest. A large number of layers gives the solution better vertical resolution but will require a shorter time step (see next section) and increase the computation time, though this is not normally an issue unless doing an ensemble of hundreds of simulations. We recommend roughly 20–80 layers, giving a vertical resolution of O(10 m) in Greenland's fjords. When there is a sill, the model may make a small adjustment to the layer thicknesses (in the initialise_variables routine) to ensure that the sill depth lies at a layer boundary, simplifying the calculation of the shelf fluxes.

The forcing boundary conditions must be provided in the structure “f”. These include the subglacial discharge for each plume as a function of time and the shelf temperature and salinity as a function of depth and time. Finally, the initial conditions must be specified in the structure “a”. These include the initial temperature and salinity of the layers and also the surface area of icebergs in each layer. The latter is, for now, classified as an initial condition rather than a forcing because it does not vary in time.

The last required input is the time vector, “t”, encompassing the duration and time step for the simulation. Like most models, a time step that is too large can lead to numerical instability. The complexity of the right-hand side of Eqs. () and (), however, makes it difficult to derive a general limit on the stable time step. If we take into account only the mixing fluxes, Eq. () becomes a 1D heat equation with a spatially and temporally variable diffusivity. With a forward Euler scheme, as employed here, stable time steps of the 1D heat equation must satisfy Δt≤(Δz)2/2Kz, where Δz would be our layer thickness and Kz is the eddy diffusivity of heat. With Kz at a reasonable maximum possible value (5×10-3 m2s-1, Sect. ) and Δz=10 m, the maximum stable time step is around 0.1 d. However, in FjordRPM, this condition guarantees neither stability nor instability, as Kz is a function of the Richardson number and only reaches such a high value in a few layers and/or at a few time steps and because vertical mixing is just one of the total five fluxes.

Another handle on stability is the fraction of a layer volume that is exchanged in a given time step. For example, if a vertical volume flux is Q (Sect. ), then the fraction of layer volume V that is exchanged in a time step Δt due to this flux is QΔt/V. Writing the volume flux as Q=WLv, with v the vertical velocity, and the volume as V=WLΔz, with Δz the layer thickness, the fraction of layer volume exchanged in a time step can be expressed as vΔt/Δz, which is an advective Courant–Friedrichs–Lewy condition. Investigations have shown that instability is often associated with this fraction exceeding ∼0.5; and with a typical vertical velocity scale of 10-4 ms-1 and a layer thickness Δz=10 m, this gives a maximum stable time step of around 0.5 d. But again, this condition is not always relevant, as instability can also arise from horizontal fluxes or the vertical mixing or from interactions between these fluxes. Furthermore, we do not have these fluxes in advance of running the simulation, so this does not provide an easy way of choosing the time step.

In practice, the model is sufficiently cheap to run such that the pragmatic way to choose the time step is to try a number of values and check the solution for instability. As a guide, we have found that a stable time step is on the order of 0.1–4 d for realistic forcings and layer thicknesses of 10 m. Such simulations take no more than a few seconds per model year on a laptop.

Before moving onto example results, we provide a few insights into the character of the model. Although there are many parameters and possible choices, the key dynamics boil down to just a handful of parameters – essentially one per exchange flux process.

The plume fluxes depend principally on the subglacial discharge Qsg and plume width Wp. For a given subglacial discharge, the main “adjustable” parameter is the plume width. A smaller value will decrease the flux of plume waters that are input to the layer at neutral buoyancy, and this neutral buoyancy layer will tend to be closer to the surface. Conversely, a larger plume width will increase the flux of plume waters, and the neutral buoyancy level will tend to be deeper.

In summary, for a given geometry and forcings, the main adjustable parameters controlling the dynamics are the plume width Wp, the shelf exchange parameter C0, the iceberg melt parameter M0, and the vertical mixing parameter K0.

The idealised MITgcm simulations (Fig. ) consider a fjord that is 60 km long and uniformly 6 km wide and 800 m deep. At the fjord head, there is a single glacier with grounding line depth 800 m, and at the fjord mouth, there is a sill of depth 400 m. The fjord opens out onto a shelf that is uniformly 600 m deep, 182 km wide (i.e. in the along-fjord direction), and 225 km long (i.e. in the across-fjord direction); note that only part of the shelf is shown in Fig. a. The horizontal resolution is 250 m within the fjord, telescoping to around 3 km at the far reaches of the shelf. The vertical resolution is 10 m near the surface, increasing to 50 m at the bottom. Temperature and salinity at the boundaries of the shelf (Fig. b and c) are, depending on the simulation, restored to either “summer” or “winter” conditions, with summer conditions coming from a conductivity–temperature–depth (CTD) profile on the continental shelf close to Sermilik Fjord, SE Greenland, and winter conditions coming from an expendable CTD in the mouth of the same fjord . The initial conditions throughout the domain are taken to be the same as these far-field boundary conditions, but the simulations are run for long enough that there should be no memory of the initial conditions. The intermediary circulation simulations involve a time-varying wind stress applied to the shelf (Fig. d; further details in Sect. ), and the iceberg melt-driven simulations require an iceberg concentration (Fig. e; further details in Sect. ). The simulations take place on an f-plane with f=1.35×10-4 s-1. In sum, these simulations are intended to be representative of large glacial fjords in Greenland.

The equivalent FjordRPM simulations use the same fjord geometry, with 60 layers of 13.3 m thickness and a time step of 0.1 d. The “shelf” profiles Ts(z,t) and Ss(z,t) used to force FjordRPM are extracted from the MITgcm simulations in a 6km×6km region on the shelf adjacent to the fjord (Fig. a). This is because shelf dynamics mean that conditions in this region can differ significantly from the MITgcm boundary conditions (Fig. b and c), particularly for the intermediary circulation simulations, and we wish FjordRPM to experience the same shelf forcing as the fjord in MITgcm does.

As a final test of FjordRPM, we compare to a nearly 3-year, realistic MITgcm simulation of Sermilik Fjord and the adjacent shelf in SE Greenland (Fig. ), conducted by . In this context, “realistic” means that their simulation used the actual bathymetry of the fjord and shelf and was forced at the glacier by subglacial discharge from a regional climate model RACMO; at the surface by an atmospheric reanalysis ERA5; and at the ocean boundary by an ocean reanalysis ASTE;. Together with observational validation from this well-studied fjord, the simulations in are as close to realistic as we can currently get (aside from the presence of icebergs, which they did not consider). Full details may be found in their paper.

We attempt to undertake the same simulation in FjordRPM, making the setup as close as possible given the much-simplified nature of the reduced-physics model. Whereas MITgcm used realistic bathymetry (Fig. ), FjordRPM has no sill and comprises a cuboid fjord of width 6 km, depth 900 m, and length 101 km (ensuring that the total volume of the fjord is the same in both MITgcm and FjordRPM). Whereas in MITgcm there are three glaciers with plumes in side-arms of the main fjord, FjordRPM has three plumes in the cuboid fjord with respective initiation depths of 650, 160, and 465 m, corresponding to the grounding line depths of the three glaciers. The plume width of 280 m in MITgcm is inherited from the MITgcm model resolution and is set to be the same in FjordRPM, and the subglacial discharge forcing is identical. Unlike MITgcm, FjordRPM has no surface forcing. For the shelf forcing in FjordRPM, we extract properties from the MITgcm simulation close to the fjord mouth (Fig. ). We configure FjordRPM with 60 layers of 15 m thickness, a time step of 0.1 d, and parameters as in Table . The FjordRPM simulation runs in a few seconds on a laptop.

We compare MITgcm and FjordRPM in terms of simulated fjord properties (Fig. ) and fjord–shelf exchange fluxes (Fig. ). The fjord properties extracted from MITgcm are averaged over all model points up-fjord of a flux gate at the fjord mouth (Fig. ), while from FjordRPM, they are simply the layer properties. The exchange fluxes from MITgcm are those passing through the flux gate at the fjord mouth, while from FjordRPM, they are the fjord–shelf exchange fluxes.

When compared to MITgcm, FjordRPM captures the evolution of fjord properties relatively well (Fig. ). It captures the timing and magnitude of the seasonal cycle in both temperature and salinity at all depths. It also recreates the high-frequency variability in properties that arises during winter storm systems (e.g. Fig. e). There are differences between MITgcm and FjordRPM – in general, FjordRPM appears warmer and saltier than MITgcm. There are also specific depths and times where the properties diverge, for example, during summer at 50–100 m depth (Fig. a), where FjordRPM becomes warmer than MITgcm, likely due to differences in plume neutral buoyancy. We have not dug further into these differences, as to do so properly would require more space than feels appropriate for the present paper, but possible reasons for the differences include the specifics of how the shelf forcing is sampled (Fig. ) and the lack of realistic hypsometry in FjordRPM.

FjordRPM also does well at capturing fjord-to-shelf exchange at approximately monthly or longer timescales when compared to MITgcm (Fig. , thicker lines). FjordRPM captures the seasonal cycle in fjord-to-shelf volume exchange (Fig. a), with increased exchange during winter and more quiescent periods during summer. Considering freshwater fluxes, FjordRPM captures the variable exchange of freshwater during winter and the emergence of a net freshwater flux of 1–5 mSv from the fjord to the shelf during summer (Fig. b). In fjord-to-shelf heat exchange, FjordRPM has much of the same variability as MITgcm, but in terms of absolute value, FjordRPM exceeds MITgcm during summer, and MITgcm exceeds FjordRPM during winter (Fig. c). We suspect that these differences may result from air–sea heat fluxes, which are included in MITgcm but not in FjordRPM – indeed, excluding the surface 30 m from the exchange flux changes the sign of the heat flux in MITgcm during summer, leading to better agreement between the models (Fig. d).

The greatest differences in exchange fluxes between FjordRPM and MITgcm are at weekly or shorter timescales (Fig. , thinner lines), where MITgcm shows a greater magnitude of variability, especially during summer. This is reflected in the mean absolute errors of FjordRPM compared to MITgcm, which are significantly larger when incorporating shorter timescales (Fig. ). We have forced FjordRPM with daily output from the MITgcm simulation – it is possible that FjordRPM would show greater variability at short timescales if we extracted, say, hourly output from MITgcm. However, this “sluggish” nature of FjordRPM appears similar to the idealised intermediary circulation simulations above and hence may represent something more fundamental about the implemented physics. We expand on this in the discussion and conclude at this point that FjordRPM shows good promise at capturing fjord properties and longer-term fjord–shelf exchange fluxes when applied to real fjords.