In an iceberg-choked fjord, meltwater can drive circulation. Down-fjord of the ice, buoyancy and rotation lead to an outflowing surface coastal current hugging one side of the fjord with an inflowing counter-current below. To predict the structure and evolution of these currents, we develop an analytical model – complemented by numerical simulations – that involves a rectangular fjord initially at rest. Specifically, we (i) start with the so-called Rossby adjustment problem; (ii) reconfigure it for a closed channel with stratification; and (iii) generalize the conventional “dam-break” scenario to a gradual-release one that mimics the continual, slow injection of meltwater. Implicit in this description is the result that circulation is mediated by internal Kelvin waves. The analytical model shows that if the total meltwater flux increases (e.g., a larger mélange, warmer water, or enhanced ice–ocean turbulence) then circulation strength increases as would be expected. For realistic parameters, a given meltwater flux induces an exchange flow that is

“Nowhere in the sea could a melting iceberg be expected to have a more pronounced effect on its environment than in the enclosure of a fjord”

The obvious consequences of iceberg melt are cooling and freshening of near-surface waters. Less obvious are the consequences for circulation. As

By comparison, there is an abundance of information on how glacial fjords respond to their other dominant source of freshwater, namely subglacial discharge. We know the influence of discharge strength

Perhaps the same level of understanding is possible for iceberg-melt-induced flows.

It is easy to intuit the idea that meltwater is buoyant and therefore rises up and out of the fjord at the surface and that a compensating inflow is needed at depth. But this does not help with more quantitative questions. Why are the currents the speeds they are? What controls the depths of the inflow and outflow? What would happen in wider or narrower channels? How would results change with different stratification or a different melt parameterization? These questions have yet to be addressed in detail.

The core contribution of this paper is an analytical model explaining the first-order dynamics of a fjord's response to hundreds of melting icebergs. To gain process-level insights and to make the problem tractable, we use a semi-realistic approach. For example, the fjord has a realistic width and depth but is idealized as a rectangle. And the icebergs have realistic sizes but are distributed such that there is a clear line separating mélange and open water. Further, we ignore other forcings like subglacial discharge and shelf-driven baroclinic flow, and we investigate the initial value problem of a fjord starting from rest. These simplifications let us best illustrate the role of waves in setting the circulation.

Before presenting the analytical model, we develop and run a high-resolution numerical model of the same semi-realistic scenario (Sect.

We simulate the dynamics of a rectangular fjord that is 600 m deep (

Numerical model setup.

The icebergs each extend over many grid cells horizontally and together act as upside-down topography

The icebergs follow a power-law size distribution: the number of icebergs of a given horizontal area scales as

The ambient water initially has a constant potential temperature and linear salinity profile and, hence, a constant buoyancy frequency:

The coordinate system has

Vertical mixing is parameterized with the

The Coriolis parameter is set for 70° N (

The simulated icebergs melt at 0.3–0.4 m d

Dynamics of the top half of the fjord after 1 week of simulation. A simple description is of surface outflow in the top

Most of the surface flow within the mélange is down-fjord, as expected. The up-fjord flow in the southwest corner (Fig.

Averaged over the week-long simulation, fluxes of freshwater and heat are 110 m

Starting from rest, the coastal current down-fjord of the mélange is the first major feature to form (Fig.

Development of fjord circulation from rest as depicted by vertical and horizontal slices at the southernmost side and surface, respectively (

Advection

We will build the core of our analytical model in nine steps (Sect.

The first three sections consider the Rossby adjustment problem in a channel (also called geostrophic adjustment). We start with a two-dimensional problem of flow generated by an abrupt release of a region of high sea surface height in an open channel (Sect.

The middle three sections consider a different two-dimensional problem: baroclinic adjustment in the

The last three sections consider the three-dimensional problem with rotation. For the third time, we start with open and closed settings with abrupt releases (Sect.

A sea surface height discontinuity in a fluid at rest is unbalanced. When released, the discontinuity generates waves and currents that restore equilibrium. Here, we consider this evolution toward equilibrium for a rotating, shallow-water system in a channel. The dominant signals will be Kelvin waves propagating along the boundaries and leaving behind steady boundary currents with

Consider a single discontinuity perpendicular to the channel axis (Fig.

Rossby adjustment in a channel for three different scenarios.

Evaluating Eq. (

If the channel is closed on the high surface end as in Fig.

In Fig.

We turn now to a depth-dependent scenario: a non-rotating, open channel in two dimensions (

Baroclinic adjustment in a stratified fluid of a density anomaly near the surface of a non-rotating, open channel.

The density field of the final, steady state is easy to predict. Assuming no mixing occurs, the final state must contain the same fluid parcels as the initial state but sorted vertically such that no available potential energy remains. Hence, the final density field is horizontally homogeneous (Fig.

Provided the initial density anomaly is small relative to the background stratification, it will

The rearranging of fluid parcels produces pressure perturbations that lead to internal waves with a range of mode numbers spreading out in both directions from the location of the initial density discontinuity (Fig.

The two mode-1 waves spread out rapidly from the initial discontinuity. On the right-hand side, there is a wave with negative vertical displacements (green shades in Fig.

In the long time limit, the velocity at any horizontal location is given by the generalization of Eq. (

The coefficients

Like

Examples of the link between

The proportionality constant

Baroclinic adjustment in a closed channel follows the same steps as for an open channel, but the boundary breaks the symmetry and means that no steady state arises. Consider the scenario in Fig.

Baroclinic adjustment of density anomalies near the closed end of a stratified channel.

The potential and kinetic energy in the closed case are similar to Eqs. (

With our assumption that the low-density region is small compared to the channel length, the final equilibrium state can be approximated as

The globally integrated kinetic energy expression involves

The velocity field at any given time, depth, and position (assuming

Consider now the same closed system but with the low-density anomaly gradually added (Fig.

If we want to know the velocity field at any time

With this gradual release in a stratified channel, we can start to see a resemblance between the analytical and numerical models (e.g., compare Fig.

Three-dimensional baroclinic Rossby adjustment in a channel combines concepts from the

Baroclinic Rossby adjustment in an open-ended, stratified channel predicted by analytical and numerical models. The values of velocity are unimportant here, but the initial densities and the color maps for panels

When the density discontinuity is released, it generates two counter-propagating families of Kelvin waves (Fig.

For regions away from the initial discontinuity, the velocity is parallel to the walls, and the expression for

A curious consequence of the superposition in Eq. (

To confirm that the sum of Kelvin waves is a good approximation of the true system, we test it against an MITgcm simulation with the same initial conditions (this simulation is separate to those described in Sects.

Not shown in Fig.

In a closed channel, the baroclinic Kelvin waves propagating toward the closed end start at

The final step before incorporating melt is anticlimactic because the hard work has been done. To adapt the closed, abrupt-release case from the previous section to the gradual-release case, we simply repeat the methodology described by Eqs. (

In Sect.

Iceberg (and glacier) melt in the numerical model.

In a glacier context, melt rates are often discussed in velocity units, with a convenient unit being meters per day (m d

Let

To use the analytical model, we need an expression for

In Sect.

At time

The analytical model should work in regions governed by linear wave dynamics; it is not expected to work where advection dominates (e.g., at

Consider first the cross-channel structure. At

Snapshots at

When compared carefully, the analytical model slightly overpredicts the velocities. This is best quantified by evaluating the total outflow

Given the number of approximations and assumptions that go into the analytical model, we deem it to be that it agrees reasonably well with the numerical model. One notable assumption not yet discussed is that the icebergs have no dynamical effect as obstacles. In the numerical model, the icebergs are stationary and induce drag on near-surface flows

The analytical model predicts the outflow well (Eq.

To further test the analytical model, we repeat the

Three comparisons are, of course, far from an exhaustive test of the plausible parameter space. Yet, given the agreement in all cases, there is no reason not to trust the analytical model.

There are several obvious ways that melt-induced fjord circulation could increase: warmer ambient water, a larger mélange, or enhanced turbulent transfer at the ice–ocean interface.

To examine these dependencies – and the less intuitive role of stratification – in detail, we undertake a parameter space study using the analytical model to predict

Predictions from the analytical model of how outflow (

If the fjord geometry is fixed but the ice–ocean interface conditions are changed then

All panels in Fig.

An ambitious goal for our analytical model would be to make large-scale predictions like those that exist for the role of subglacial discharge. These start with classic buoyant-plume theory

A more immediate goal is to help interpret observations or numerical models for specific settings. For example, as part of ongoing related work, we are analyzing multi-year simulations of Sermilik Fjord with and without icebergs (using a setup like

The analytical model is built on linear wave dynamics; nonlinear advection and instabilities are ignored. In parts of the domain, this can quickly become a limitation. In the numerical model, we saw advection dominating in the

Instead, it makes more sense to simply ask whether the analytical model would remain skillful after several months. Without running simulations to properly answer this, our best guess follows from

Other obvious uncertainties surround whether the analytical model still works in fjords that feature sills or have realistic coastlines, whether it remains useful in the presence of competing forcings such as shelf waves and subglacial discharge, and how it should be adapted for the case where there is not a convenient demarcation of mélange and open water but rather where iceberg concentration varies along the fjord.

The analytical model involved many steps. A summarizing example helps bring all of these steps together.

The continual input of meltwater generates a continual fjord response. Discretizing the problem in time makes this response easier to understand. In Fig.

Summary of the analytical model. To predict the total fjord circulation at a given time

For the response to the last 1 % released at

For the

Figure

Ultimately, the analytical model – and the scalings that follow from it (Sect.

In our simulations, icebergs produce meltwater through only subsurface melting; wave erosion and melting at the ice–air interfaces are ignored. Thermodynamics at all ice–ocean interfaces are treated with the three-equation formulation, and the same velocity-dependent turbulent-transfer coefficients are used for the vertical sides and the basal face. Specifically, we adapt the “icefront” package implementation from

Conceptually, the analytical model does not change if the reference density is nonlinear. The only difference is that mode shapes and internal wave speeds need to be calculated numerically with matrix methods.

Mode shapes for vertical velocity, denoted

The archive at

The author has declared that there are no competing interests.

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Thank you to the two anonymous reviewers that helped clarify and improve several aspects of this paper.

This work was funded by the Heising–Simons Foundation (grant no. 2019-1159) as part of the program “Eyes at the front: a megasite project at Helheim Glacier”. The numerical simulations were run on the Expanse system at the San Diego Supercomputer Center through allocation no. EES220032 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services and Support (ACCESS) program, which is supported by the National Science Foundation (grant nos. 2138259, 2138286, 2138307, 2137603, and 2138296).

This paper was edited by Christian Haas and reviewed by two anonymous referees.