Two approaches to ice-sheet modeling are available. Analytical modeling is
the traditional approach (Van der Veen, 2016). It solves the force
(momentum), mass, and energy balances to obtain three-dimensional solutions
over time, beginning with the Navier–Stokes equations for the force balance.
Geometrical modeling employs simple geometry to solve the force and mass
balance in one dimension along ice flow (Hughes, 2012a). It is useful
primarily to provide the first-order physical basis of ice-sheet modeling for
students with little background in mathematics. The geometric approach uses
changes in ice-bed coupling along flow to calculate changes in ice elevation
and thickness, using a floating fraction

Cornelis “Kees” Van der Veen's comparison of geometric and analytic
approaches to the force balance in glaciology in

My interest in the force balance for ice sheets spans four decades, beginning when I used glacial geology to reconstruct former ice sheets from the bottom up based on the strength of ice-bed coupling deduced from glacial geology, an approach that also produced the concave surface of ice streams for the first time (Denton and Hughes, 1981, chap. 5 and 6). I developed the geometric approach after observing the huge arcing transverse crevasses at the head of Byrd Glacier, and realized it was actually pulling ice out of the East Antarctic Ice Sheet (Hughes, 1992). Since then it has been a work in progress.

Referring to Hughes (2008), Van der Veen (2016) states on his page 1332 that
I believe lateral drag vanishes at the center of an ice stream. Lateral shear
stress

Van der Veen (2016) states that his Eq. (9) is similar to my Eq. (36) in Hughes (2003), but it is not the same. We cannot readily translate term by term the geometric balance in the conventional notation of the force balance. It is just the same equation that holds.

In the geometric force balance, the driving force is the area of a right
triangle and all the resisting forces are areas of triangles and a rectangle
(or parallelogram) that fit into the triangle so the driving and resisting
forces are identical. All signs are positive in my Eq. (36). His

The proof that my

Figure 4 from Hughes et al. (2016). Under an ice stream, basal ice
is grounded in the shaded areas and floating in the unshaded areas (top) as
seen in a transverse cross section (bottom) for incremental basal area

Resisting stresses linked to floating fraction

Referring to my Fig. 3 (left), Fig. 3 in Van der Veen (2016), line AF should
be parallel to line BE because they both show ice pressure increasing
linearly with depth. Line CE shows how water pressure increases linearly with
depth, as is obvious at the calving front. In my geometrical force balance,
the longitudinal gravitational driving force is area ADF of the big right
triangle. Fitted inside ADF are a resisting flotation force given by area BDE
for floating ice fraction

Figure 5 from Hughes et al. (2016). Top: stresses at

Figure 3 (left) and Fig. 4 (right) from Van der Veen (2016).

Van der Veen (2016) correctly states that his Eq. (16) represents my longitudinal gravitational driving force, but then he states it “does not represent the gravitational driving force” (p. 1335). It does. The analytic and geometric approaches to the force balance must be presented and understood each on their own terms. Attempts to mix the two, as Van der Veen (2016) did, leads only to confusion.

Van der Veen (2016) states on his page 1335 that a longitudinal force balance
along

Van der Veen (2016) states that his Fig. 4a, reproduced in my Fig. 3 (right panels), should represent my geometrical force balance because his area ADF equals his area APD. It would if he divided his area APD into my smaller areas of triangles and a rectangle shown in my Fig. 2, areas that resist gravitational forcing from his area ADF. He states that both areas ADF and APD are “lithostatic stresses”. They are not. Area ADF is my gravitational driving force and area APD is the sum of my resisting forces opposing the driving force, as he shows by his horizontal arrows in his Fig. 4a. There is no surface slope in his Fig. 4a. That condition applies to an unconfined linear ice shelf with constant thickness (Weertman, 1957; Robin, 1958), in which case only my areas 3 and 4 in my Fig. 2 (bottom) add to give his area APD, since there are no basal and side drag forces represented by my areas 1 and 2. Raymond (1983) analyzed deformation near interior ice divides where the surface slope is also zero.

In his Fig. 4b, shown in my Fig. 3b, Van der Veen (2016) correctly shows the
geometrical force balance in my Fig. 2 (bottom) for a sloping ice surface
above a horizontal bed. From these figures we can both obtain the geometric
longitudinal force balance over incremental length

Resistance from my

My geometrical force balance aims to teach the fundamentals of glaciology to students with an inadequate background in mathematics, usually students studying to be glacial geologists (Hughes, 2012a). My geometrical approach was designed to make maximum use of glacial geology in reconstructing former ice sheets from the bottom up (Hughes, 1998, chap. 9 and 10; Fastook and Hughes, 2013) and in demonstrating how basal thermal conditions produce glacial geology under the Antarctic Ice Sheet today (Hughes, 1998, chap. 3, Wilch and Hughes, 2000; Siegert, 2001). Previously I had spent more time teaching calculus than glaciology because the Navier–Stokes equations had to be integrated in the force balance. Everyone knows the area of a rectangle is base times height, and of a triangle is half that; yet knowing that delivers the same results as integrating the Navier–Stokes equations for linear sheet, stream, and shelf flow.

I developed the geometrical force balance over some decades, from
Hughes (1992) through to Hughes et al. (2016). My papers are a work in progress;
see pages 201–202 of Hughes et al. (2016) regarding

This response gives me an opportunity to correct three mistakes in
Hughes (2012a) that will be apparent to careful readers. The first line in
Eq. (12.9) should be

No data sets were used in this article.

The author declares that he has no conflict of interest.

I thank Cornelis van der Veen for giving me the opportunity to further
explain my geometric force balance in relation to the standard analytic force
balance. I thank the editor, Frank Pattyn, for allowing my explanation to appear in