# DIALOGUE STRATEGY ADAPTATION TO NEW ACTION SETS USING MULTI-DIMENSIONAL
MODELLING

###### Abstract

A major bottleneck for building statistical spoken dialogue systems for new
domains and applications is the need for large amounts of training data. To
address this problem, we adopt the multi-dimensional approach to dialogue
management and evaluate its potential for transfer learning. Specifically, we
exploit pre-trained task-independent policies to speed up training for an
extended task-specific action set, in which the single summary action for
requesting a slot is replaced by multiple slot-specific request actions.
Policy optimisation and evaluation experiments using an agenda-based user
simulator show that with limited training data, much better performance levels
can be achieved when using the proposed multi-dimensional adaptation method.
We confirm this improvement in a crowd-sourced human user evaluation of our
spoken dialogue system, comparing partially trained policies. The multi-
dimensional system (with adaptation on limited training data in the target
scenario) outperforms the one-dimensional baseline (without adaptation on the
same amount of training data) by 7% perceived success rate.

Index Terms—  dialogue systems, policy optimisation, reinforcement learning,
transfer learning

##  1 Introduction

One of the main challenges in spoken dialogue system development is their
scalability to new domains and applications. A statistical spoken dialogue
system can be built efficiently for a large slot-filling domain when
sufficient annotated data in that domain is available. However, as domains are
expanded with new slots, and new applications emerge that require new, task-
specific system actions, in-domain annotated datasets are typically hard to
get by. Therefore, numerous efforts have been made to use transfer learning
techniques [1, 2] to efficiently develop dialogue systems for new domains with
limited or no data in the target domain. Most of the efforts in statistical
dialogue management and action selection in particular focus on adaptation to
newly introduced slots and values, with the underlying task unchanged [3, 4,
5].

The multi-dimensional approach to dialogue modelling [6] offers the potential
to exploit its principled separation of domain- and application-independent
aspects of dialogue to adapt to new domains as well as new applications. In
[7], a multi-dimensional statistical dialogue manager was presented and it was
demonstrated that policy optimisation for a target domain could be improved by
re-using the policies for the application-independent dimensions (Social
Obligations Management and Auto-feedback; see Section 2) that were pre-trained
in a source domain. However, the addressed use-case was limited to adapting
across very similar domains, where the set of task-specific actions was the
same and only the set of slots and values changed.

In this paper, we adopt the multi-dimensional approach to dialogue management
[8, 9, 7], but rather than adapting to new slots and values, we focus on
adapting to a different policy action set, which is typically required when
developing a dialogue system for a new application. In addition, we present
two extensions of the model: 1) whereas [7] restricted their model to allow
only single dialogue acts being generated (despite the multi-agent design), we
allow combinations of auto-feedback and task acts that support implicit
confirmation, and 2) the agent that evaluates the generated dialogue act
candidates from the different dimensions (see Section 2) is trained jointly
with the other agents, rather than hand-coded.

Using the improved multi-dimensional design, we present policy optimisation
and evaluation experiments using our simulated user and error model,
demonstrating significant improvements in performance on limited training data
when using the proposed adaptation method (Section 3). Furthermore, a human
user evaluation was carried out to confirm this result in more realistic
conditions; the experimental design, spoken dialogue system implementation,
and evaluation results are described in Section 4. Section 5 contains a more
detailed discussion of related work in the area of transfer learning for
dialogue management. The paper is wrapped up in Section 6 with conclusions and
directions for future work.

Task | Auto-Feedback |  SOM |  All |  Example utterance
---|---|---|---|---
offer |  |  |  offer |  How about the Rice Boat?
offer | impl-confirm |  |  offer+impl-confirm |  The Rice Boat is a nice Indian restaurant
answer |  |  |  answer |  The address of the Rice Boat is …
request |  |  |  request |  What price range did you have in mind?
request | impl-confirm |  |  req+impl-confirm |  Okay, Indian food. What price range did you have in mind?
| expl-confirm |  |  expl-confirm |  You want Indian food, is that right?
| auto_negative |  |  auto_negative |  I did not quite catch that, could you please rephrase?
|  |  accept_thanking |  accept_thanking |  You’re welcome
|  |  return_goodbye |  return_goodbye |  Have a nice day
none | none |  none |  |
Table 1: The dialogue act agent action sets. Note that impl-confirm is
essentially an inform act stating with a slot-value pair a preference that the
system believes the user has, whereas expl-confirm is essentially a
propositional question asking the user whether they have a specific
preference, expressed by a slot-value pair.

##  2 Multi-dimensional dialogue management

In conventional reinforcement learning-based statistical dialogue systems
[10], the dialogue policy selects an action from a single set of possible
actions in each turn. In contrast to such ‘one-dimensional’ systems, multi-
dimensional dialogue systems employ multiple dialogue act agents, each
dedicated to a different aspect of the dialogue process, using a policy that
selects actions from its own specialised action set. We propose and evaluate a
model with three dimensions and corresponding agents: Auto-feedback, Social
Obligation Management (SOM), and Task. The Auto-feedback agent has actions for
giving feedback to the user about processing their utterances, for example
indicate non-understanding when the speech recogniser does not return any
results (“I did not quite get that, could you please repeat?”) or provide
articulate feedback in the form of an explicit/implicit confirmation when the
system is unsure about the user’s input (e.g., “Expensive, you said?”), the
SOM agent deals with greeting, thanking, apologising, and other social
actions, and the Task agent focuses on the underlying task or activity
(information navigation, tutoring, negotiation, etcetera). Additional agents
to support other dimensions could be introduced as well (e.g. for turn-taking
and time-management [11]), but the three agents currently included are
considered to be the minimum requirement for a task-oriented multi-dimensional
dialogue system. An additional Evaluation agent is used for determining which
of the generated dialogue act candidates are forwarded as final dialogue acts
to the natural language generation component [12]. Figure 1 gives an outline
of the multi-agent action selection component.

Fig. 1: Multi-agent action selection component, showing the Task, Auto-
feedback, and SOM agents generating candidate dialogue acts, of which two are
selected by the Evaluation agent.

Besides reflecting the multi-dimensional nature of dialogue [6], this design
opens up opportunities for efficient adaptation of dialogue managers to new
tasks and domains. The Auto-feedback and SOM dimensions are considered to be
domain and task independent, and therefore policies for these dimensions may
be transferred, leaving only the Task policy to be trained from scratch in the
target domain/application.

We introduce a new design of the multi-agent action selection model, in which
the Evaluation agent is designed to allow two possible combinations of two
dialogue acts. Both of these combinations include an implicit confirmation
from the Auto-feedback dimension, combined with either an offer (e.g., “Prezzo
is a popular Italian restaurant.”), or a request (e.g., “You are looking for
an Italian restaurant in which area?”) from the Task dimension. Table 1 lists
all allowed action combinations, organised along the three supported
dimensions, as well as in a single action set for a one-dimensional baseline
system. In future versions, additional combinations can be considered, for
example negative feedback combined with an apology (e.g., “I’m sorry, I did
not quite get that”).

##  3 Policy optimisation and adaptation experiments

The potential for transfer learning in a multi-dimensional dialogue manager
has been demonstrated previously for the use-case of adapting from a hotel
search application to a restaurant search application [7]. Here, we introduce
a use-case where we stay within the restaurant domain, but adapt to a new
action set. Specifically, the action set of the Task agent is extended by
replacing the summary action request for requesting a slot by separate request
actions for each slot, e.g. request-pricerange. Whereas in the source scenario
the policy may select the request summary action, after which heuristics
determine which slot is requested, in the target scenario the policy may
select a request action for a particular slot, for example the area.
Therefore, in the target scenario with extended action set, the system learns
automatically which slot to request, rather than relying on heuristics in the
source scenario.

In the DSTC-2 restaurant search domain [13], this means that the request
summary action is replaced by the three slot-specific request actions,
corresponding to the slots food, area, and pricerange. Going back to the
original action sets outlined in Table 1, the action set of the Task agent
therefore grows from 4 (offer, request, answer, none) to 6 actions (offer,
request-food, request-area, request-pricerange, answer, none), whereas the
action set of the one-dimensional baseline (column ‘All’) grows from 9 to 13
actions, since the extension applies to both request and req+impl-confirm.

Hence, we first train a multi-dimensional system for the source scenario, in
which the Task agent uses a summary action for requesting slots (mdim-src).
This yields 4 trained policies, corresponding to the 3 conversational
dimensions plus the evaluation agent. Out of these, the policies for Auto-
feedback and SOM are subsequently re-used and adapted when we train a system
for the target scenario, in which the Task agent uses the extended action set
(mdim-ada). As baselines, we also train a multi-dimensional system for the
target scenario without re-using the pre-trained Auto-feedback and SOM
policies (multi-dim) and a one-dimensional system for the target scenario in
which all action combinations from the multi-dimensional system are taken as
single actions into one action set (one-dim).

###  3.1 Training details

All action selection models are trained in online interaction with an agenda-
based user simulator [14]. In each case, all 4 policies are trained
simultaneously using Monte Carlo Control reinforcement learning with linear
value function approximation [15]. Each policy selects actions from its own
action set, based on the approximated values given the current state. The
values predict the long-term cumulative reward when taking an action in a
given state and following the policy in subsequent turns, where the state is
represented by a set of features extracted from the full dialogue state.
During training, the policies use Boltzmann exploration, i.e., actions are
sampled from a softmax distribution applied to the estimated values. The
temperature hyper-parameter of the softmax is decayed linearly, gradually
reducing the level of exploration until the policy only selects actions with
the highest estimated value. The weights of the linear value function are
updated after every dialogue/episode, based on a shared reward signal. The
policies coordinate their actions only indirectly through the shared rewards,
i.e., each policy operates independently without any direct communication with
other policies.

The rewards are calculated at each turn, combining rewards obtained from the
simulated user with internal rewards. The user gives a reward of +100 upon
task completion: in the restaurant search domain, this is when the system has
recommended a restaurant matching the user’s preferences and has provided all
requested information about this restaurant. Such ‘user goals’ are randomly
initialised from the domain ontology at the start of each dialogue, and fed to
the simulated user. In addition, the user issues a penalty when the system
fails to respond to a thanking action or inserts a social act when it is not
called for (-5 for each occurrence). This is to force the system to learn
basic reactive social behaviour, though we are aware that this might be
experienced as repetitive and unnatural. In future work, we will consider
learning more sophisticated social patterns. Internally, a penalty of -1 is
applied for each turn (to encourage shorter dialogues) and a penalty of -25
when a ‘processing problem’ is encountered and the system does not signal this
to the user with a feedback act. A processing problem is recorded in the
dialogue state when speech recognition or natural language understanding
fails, i.e., returns no results. During training, such processing problems are
simulated randomly in 5% of the user turns, discarding the original simulated
user act.

The Evaluation agent from [7] was implemented via a set of rules, based on
definitions from the dialogue act annotation standard [11]. Here, we implement
it as another reinforcement learning agent, which takes as input the candidate
dialogue act for each dimension and selects which dialogue act combination
will be passed on to the natural language generation component. Given that
there are 3 supported dimensions, the actions correspond to the 8 possible
combinations of dimensions that can be selected. Using an action masking
mechanism, we ensure that the Evaluation agent only allows single dialogue
acts or a combination of impl-confirm (Auto-feedback dimension) with offer or
with any of the three request actions (Task dimension).

###  3.2 Results

The policy optimisation results in simulation for the target scenario are
shown in Figure 2. The learning curves (here shown in terms of success rates)
quite clearly show that the performance levels are much higher in the early
stages of training when using our proposed multi-dimensional adaptation method
(mdim-ada, in red) than when training a multi-dimensional action selection
model from scratch (multi-dim, in green). In the first 20k training dialogues
we also observe much higher success rates compared to the one-dimensional
baseline (one-dim, in blue).

Fig. 2: Learning curves for the target scenario in terms of success rate for
the three training methods. Note that the results are averaged over 10
training runs in each setting; in each training run, success rates over a
sliding window of 100 dialogues are recorded.

After training all policies, we ran evaluations of the fully trained policies
over 3000 simulated dialogues each, at 25% semantic error rate. The first
three rows in Table 2 correspond to the systems with the extended action set
(the target scenario) that were also shown in Figure 2. These systems all
score similar average rewards, although the adapted system (mdim-ada) gets a
slightly higher success rate, but with slightly longer dialogues. Since the
system with the original action set (the source scenario) covers the same
range of full system dialogue acts (through its summary actions and heuristics
for mapping them to complete dialogue acts), it has the same functionality as
the extended system, and can therefore be included in the evaluation for
comparison. As this system (mdim-src) achieves similar scores, automatically
learning request actions for each slot did not result in improved performance
levels in this particular scenario. Whether improvements can be achieved in
this way, however, depends on the nature of the domain ontology and database,
and on the behaviour of the users.

System |  ASet |  ADA |  NLU |  Succ |  Len |  Rew
---|---|---|---|---|---|---
one-dim |  ext |  no |  sim |  96.0 |  9.28 |  86.06
multi-dim |  ext |  no |  sim |  96.7 |  9.49 |  85.49
mdim-ada |  ext |  yes |  sim |  98.3 |  9.71 |  87.92
mdim-src |  sum |  no |  sim |  97.2 |  9.57 |  87.15
one-dim-asu |  ext |  no |  asu |  99.5 |  7.69 |  91.83
mdim-ada-asu |  ext |  yes |  asu |  88.9 |  7.51 |  81.41
Table 2: Evaluation of fully trained policies over 3000 simulated dialogues. ASet: Action Set (summary/extended), ADA: adapted (yes/no), NLU: Natural Language Understanding (asu/simulated), Succ: Success Rate, Len: Average Length, Rew: Average Reward System | Success rate | Average length | Average reward | Num. dial’s
---|---|---|---|---
one-dim | 96.0 | 96.0 | 99.5 | 9.28 | 9.28 | 7.69 | 86.06 | 86.06 | 91.83 |
mdim-ada | 98.3 | 98.3 | 88.9 | 9.71 | 9.71 | 7.51 | 87.92 | 87.92 | 81.41 |  100k
one-dim | 59.1 | 61.7 | 64.9 | 8.02 | 11.64 | 10.15 | 49.05 | 45.78 | 52.13 |  18k
mdim-ada | 73.5 | 73.8 | 73.3 | 8.95 | 10.01 | 7.71 | 62.18 | 60.77 | 64.25 |
one-dim | 48.9 | 57.5 | 60.3 | 7.55 | 11.98 | 10.63 | 37.31 | 38.05 | 45.82 |  17k
mdim-ada | 77.1 | 79.1 | 78.5 | 9.49 | 10.70 | 7.75 | 65.55 | 64.99 | 68.72 |
one-dim | 45.3 | 48.5 | 54.1 | 7.30 | 11.60 | 10.16 | 34.07 | 30.04 | 39.77 |  16k
mdim-ada | 83.5 | 83.5 | 83.2 | 9.32 | 10.36 | 8.13 | 73.04 | 71.18 | 74.21 |
one-dim | 30.6 | 33.4 | 37.5 | 5.61 | 11.94 | 10.70 | 19.86 | 13.95 | 21.48 |  15k
mdim-ada | 64.5 | 64.3 | 62.3 | 9.26 | 10.14 | 8.08 | 52.67 | 50.54 | 52.44 |
| no-expl |  expl | expl+asu | no-expl |  expl | expl+asu | no-expl |  expl | expl+asu |
Table 3: Evaluation of partially trained policies over 3000 simulated
dialogues, reporting success rate, average dialogue length, and average reward
in three settings.

The results in the first 4 rows have been obtained using a rule-based state
tracker that takes (simulated) user dialogue act hypotheses as input, as is
the case during policy optimisation. For the human user evaluation, we use the
Action State Update (ASU) dialogue state tracker [16] that takes user
utterances as input (see also Section 4.1). We have therefore also evaluated
the one-dim and mdim-ada policies with this tracker in the loop, and included
a hybrid retrieval/template based natural language generation component that
maps simulated user dialogue acts to natural language utterances that can be
fed to the ASU component111Training the policies together with the ASU tracker
is currently not practical, because its BERT model is too slow for running the
required volume of training dialogues.. For the one-dimensional system, this
results in higher scores (one-dim-asu), whereas for the adapted multi-
dimensional system, we get considerably lower scores (mdim-ada-asu). In the
latter case, we note however that this is mainly due to one of the 10 policy
combinations performing very badly: the policy repeatedly responds with
negative feedback, after which the simulated user loses its patience and hangs
up, resulting in 0% success rate and -5.02 average reward. In contrast, the
other policies all achieve 98% success rate.

In order to decide which partially trained policies to include in the human
user evaluation, we have run simulated evaluations for the policies that were
obtained after 15k, 16k, 17k, and 18k training dialogues. This pre-selection
is based on the learning curves in Figure 2 and in particular the area where
the success rate of the one-dimensional system is starting to build up, but is
still strongly outperformed by the adapted multi-dimensional system. The
results in Table 3 report success rates, average dialogue lengths (in terms of
user turns), and average rewards over 3000 simulated dialogues at 25% error
rate. Each evaluation is carried out in three different settings, indicated in
the bottom row of the table: 1) using the rule-based state tracker and no
policy exploration, i.e. the policies always select the action with the
highest estimated value (no-expl); 2) using the rule based state tracker and
policy exploration at the level determined by the temperature setting at the
corresponding stage of training (expl); and 3) using the ASU state tracker and
policy exploration (expl+asu).

At a relatively early training stage, the policy is still exploring the state-
action space, and has not experienced many successful dialogues yet.
Therefore, the reported performance levels can differ between the no-expl and
expl settings. Overall, the results in Table 3 show lower average rewards in
the expl setting, but higher success rates. Furthermore, the performance
levels are much higher when evaluating with the ASU tracker in the loop. The
likely reason for this is that no semantic error model was used in this
setting, i.e., the true user dialogue acts from the simulator were used to
generate the input user utterances for the tracker. Hence, the language
understanding performance is probably much better in this setting, and
therefore the action selection performance as well.

Based on these results, we selected the one- and multi-dimensional models that
were trained on 17k dialogues for the human evaluation, as well as the fully-
trained multi-dimensional system. The three selected variants are highlighted
in grey in Table 3 and also pointed out in Figure 2.

System | Num. | Average | Q1 [%] | Q2 [%] | Q3 [1-6] | Q4 [1-6] | Q5 [1-6] | Q6 [1-6]
---|---|---|---|---|---|---|---|---
| Dialogues | Length | Found Venue | DialSuccess | Understand | Recognise | SysResponse | Naturalness
one-dim-17k | 203 | 7.69 (4.34) | 65.52 (3.34) | 63.05 (3.40) | 3.34 (1.68) | 3.46 (1.66) | 3.21 (1.72) | 3.01 (1.77)
mdim-ada-17k | 201 | 6.52 (3.24) | 73.00 (3.15) | 70.00 (3.25) | 3.43 (1.58) | 3.56 (1.55) | 3.48 (1.59) | 3.35 (1.67)
mdim-ada-100k | 199 | 5.88 (2.38) | 78.89 (2.90) | 79.90 (2.85) | 3.76 (1.56) | 3.86 (1.50) | 3.93 (1.46) | 3.71 (1.62)
Table 4: Human user evaluation results, where N is the number of dialogues,
AvgLen is the average number of turns per dialogue, and Q1-6 are the scores
obtained from the questionnaire. The scores for Q1-2 are percentages, and the
standard deviation for each score is indicated in brackets.

##  4 Human User Evaluation

Using the agenda-based user simulator and error model, we have shown that
significant performance gains with limited training data can be achieved with
the proposed adaptation method. However, it remains to be seen to what extent
these results are representative for the scenario of real users interacting
with the system [17, 18]. Ideally, the policy optimisation experiments should
be carried out in online interaction with human users, which has been
attempted with some success [19, 20]. However, this requires moving to more
sample efficient optimisation methods and addressing many other technical
challenges, which we will leave for future work. Instead, we have run a human
user evaluation in which we compared three system variants, and in particular
two sets of partially trained policies, corresponding to the adapted multi-
dimensional system and the non-adapted one-dimensional baseline, both trained
on 17k dialogues only.

Users were recruited through the Amazon Mechanical Turk (AMT) crowd-sourcing
platform, where they were given a task description (e.g., “You are looking for
a moderately priced French restaurant. Make sure you get the phone number and
address.”) and a link to the web-based interface that we have created. To
enable spoken dialogue on the web interface, we use the Google Web Speech API
for both ASR (for recognising the user’s speech) and TTS (for synthesizing the
system’s speech); our dialogue system server receives recognised user
utterances and responds with system utterances, both in text form.

After finishing their conversation with the system, the user can ‘hang up’ by
pressing a button and receive a token which they can use to proceed on the AMT
page, where they are given a questionnaire to fill out and submit, upon which
they complete the task. In the questionnaire, the subject is asked to state
their opinion on 6 statements about the conversation, in the form of either a
binary Yes or No (Q1 and Q2), or on a 6 point Likert scale (Q3 to Q6), ranging
from ‘Strongly disagree’ to ‘Strongly agree’.

  * Q1:

The system recommended a restaurant that matched my constraints. (Yes/No)

  * Q2:

I got all the information I was looking for. (Yes/No)

  * Q3:

The system understood what I was saying. (6 point Likert)

  * Q4:

The system recognised my speech well. (6 point Likert)

  * Q5:

The system’s responses were appropriate. (6 point Likert)

  * Q6:

The conversation felt natural. (6 point Likert)

###  4.1 Spoken dialogue system implementation

Our dialogue system takes as input the user utterance text, in the form of the
ASR top hypothesis obtained from the web interface. A domain-independent
dialogue act tagger is used to recognise social acts like goodbye and thanks,
after which the utterance is passed to the Action State Update model for
dialogue state tracking [16]. The dialogue state contains beliefs about the
user goal (in terms of slot-value pairs and their confidence score), which
slots are believed to be requested, which database items are or have been
discussed, whether any processing problems have occurred (i.e. speech
recognition or language understanding failed), and the dialogue act tags of
previous user utterances. Based on the dialogue state, one or more system
dialogue acts are selected using one of the trained action selection models;
the three models compared in the human evaluation are highlighted in Table 3.
A set of templates is used to generate a natural language system response from
the selected dialogue acts. The response is synthesised on the web interface.

In order to obtain results that are representative for each system variant,
all 10 policies that were trained for each variant have been evaluated and the
average results reported. For each dialogue, one policy is selected from the
pool of 10 using a round-robin system.

###  4.2 Results

The evaluation results are shown in Table 4, including the (objective) average
dialogue length and the (subjective) scores from the questionnaire. As
expected, the fully trained system mdim-ada-100k gets the best scores. More
importantly, across all metrics the partially trained mdim-ada-17k gets better
scores than the partially trained baseline one-dim-17k. Especially in terms of
average dialogue length and the perceived partial and full task completion
rates (Q1 resp. Q2) the difference between one-dim-17k and mdim-ada-17k is
substantial, as was predicted by the experiments with the user simulator. The
scores for perceived understanding (Q3) and speech recognition (Q4) could
somewhat explain these differences (rather than attributing them to the used
policies), but their variance is quite large. When assessing how appropriate
the system responses appear to users (Q5), the difference between one-
dimensional and multi-dimensional adaptation is larger.

Compared to the simulated evaluation results in Table 3, the perceived success
rates (Q2) are much lower in the case of the fully trained system (79.9% vs.
88.9%) and the multi-dimensional partially trained system (70% vs. 78.5%),
whereas they are slightly higher in the case of the partially trained one-
dimensional system (63.05% vs. 60.03%).

In terms of average dialogue length, the human evaluation dialogues turn out
to be shorter overall. This suggests that the human subjects might have given
up more quickly when the system failed to recommend a restaurant that met
their constraints, i.e. the unsuccessful dialogues were shorter. In contrast,
the simulator keeps trying to complete the goal until the maximum number of
user turns is reached (set to 30 turns in the configuration) or the system
response act is repeated too many times (set to 3 times in the configuration).

##  5 Related work

In the area of transfer learning methods for dialogue management, most
approaches have focused on cross-domain adaptation or multi-domain
optimisation for slot-filling dialogue, where each domain is defined in terms
of slots and their possible values.

In [3], Domain-Independent Parameterisation (DIP) of dialogue state and action
representations is introduced to enable transfer across domains. DIP seeks to
train a dialogue policy that abstracts away from the specific slots and values
in a particular domain, so it is applicable to the conversational search task
in other domains as well. The effectiveness of their method was demonstrated
using an agenda-based user simulator when adapting from restaurant search to
laptop search. Furthermore, they carried out a human user evaluation, showing
that their best transferred DIP policy performed at the same level as a non-
transferred in-domain policy.

A Bayesian approach to dialogue management is described in [5], which uses
Gaussian Process (GP) reinforcement learning, exploiting model priors of a
generic dialogue policy for fast domain adaptation. They also discuss a
Bayesian committee machine approach, in which each domain is handled by a
separate GP policy, but when only limited data is available for a specific
domain, its policy may rely on the output of the other policies. This approach
has been further extended into a multi-agent learning setting, further
improving performance levels during training with human users.

The Multi-Agent Dialogue Policy (MADP) is proposed in [4], which consists of
several slot-specific agents and a slot-independent agent. Adopting the Deep
Q-Network (DQN) reinforcement learning framework [21], the parameters of the
slot-independent agent and the shared parameters of the slot-specific agents
that have been learned for the source domain can be transferred to a new
target domain. So in this case, the multi-agent design is based on the slots
in the domain definition. The benefit of this method has only been
demonstrated in simulated experiments, and is focused on adapting to new slots
within the same restaurant/tourist information task. A more recent incarnation
of this approach is the AgentGraph framework [22], which employs Graph Neural
Networks. This method was evaluated using the PyDial benchmark, demonstrating
successful transfer between restaurant search and laptop shopping tasks.

Where the agents in [5] are associated with domains, and the agents in [4, 22]
are associated with slots (except for the generic slot-independent agent), the
agents in the multi-dimensional approach to dialogue management are associated
with dimensions [23]. Although this approach is very different in nature, it
is not necessarily incompatible with these other multi-agent approaches. For
example, it could benefit from the MADP approach by dividing up the Auto-
feedback agent into sub-agents for each slot in the domain definition.

The multi-dimensional design presented in this paper is an extension of the
previous models described in [9, 7]. First, the system supports the generation
of multiple dialogue acts in a single response; second, the Evaluation agent
that was extended for this purpose is trained jointly with the other agents.
Furthermore, the adaptation use-case in previous work was limited to domain
adaptation only, where the slots and values changed between source and target
scenario, but not the action sets. Finally, the human evaluation described in
[7] included fully trained policies only, showing that a multi-dimensional
system could be trained to a performance level equivalent to a one-dimensional
baseline. In this paper, we have presented a human user evaluation to confirm
the performance boost of partially adapted policies seen on simulated data
when using the multi-dimensional adaptation method.

##  6 Conclusion

We have presented a multi-dimensional approach to dialogue management, in
which system response actions are selected through a combination of 4 dialogue
policies, 2 of which are task-independent, and are therefore suitable for re-
use when moving to a new application. In simulated policy optimisation and
evaluation experiments we have shown that by re-using and adapting these task-
independent policies, significant performance gains can be achieved in the
early stages of training. As a first task adaptation use-case, we have looked
at extending the task-specific action set with multiple slot-specific request
actions, replacing the original summary request action that relied on
heuristics to determine the slot to be requested. To confirm the adaptation
results obtained with an agenda-based user simulator, we have carried out a
crowd-sourced human user evaluation. When trained on the same limited amount
of training data, the proposed multi-dimensional adaptation strategy achieved
significantly better results than the one-dimensional baseline (trained from
scratch without any adaptation) on all subjective metrics, including task
success and appropriateness of system responses, as well as on dialogue
length.

In future work, we will consider more challenging use cases for applying the
multi-dimensional adaptation method. We will extend the role of the social
agent to improve the naturalness of the interactions, explore methods for
combining different types of agents (e.g., rule-based agents, or agents
trained with supervised learning), and investigate the relation between multi-
dimensional dialogue management and natural language generation. Finally, to
enable online learning with human users and therefore evaluate the proposed
adaptation method more directly, we will explore more efficient reinforcement
learning algorithms.

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# Anisotropic Magnetized Asteroseismic Waves

B. Tripathi Department of Physics, University of Wisconsin–Madison, Madison,
Wisconsin 53706, USA<EMAIL_ADDRESS>Dhrubaditya Mitra Nordita, KTH Royal
Institute of Technology and Stockholm University, Hannes Alfvéns väg 12, 114
19 Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract

We solve for waves in a polytropic, stratified medium with a spatially varying
background magnetic field that points along a horizontal $x$-direction, and
with gravity that is directed along the vertical $z$-direction. Force balance
determines the magnitude of the background magnetic field, $B_{\rm 0}^{2}\sim
z^{n+1}$, where $n$ is the polytropic index. Using numerical and asymptotic
methods, we deduce an accurate and explicit dispersion relation for fast
pressure-driven waves: $\Omega^{2}\sim
K\left(2m+n\right)\left[1+(1/\mbox{M}_{\rm
A})^{2}(4-2\gamma+\cos^{2}\theta-3\cos^{4}\theta)/4\right]$. Here, $\Omega$ is
the frequency, $K$ the wavenumber, $\theta$ the angle the wave-vector makes
with the background magnetic field, $\mbox{M}_{\rm A}$ the Alfvénic Mach
number, and $m$ an integer representing the eigen state. Applications of our
result are in magnetoseismology and nonlinear asteroseismology.

††journal: ApJ

##  1 Introduction

The strengths of magnetic fields buried below the surface of stars are not
known, though they are vital for improved understanding of the magnetic
behaviour of stars. This challenge has impeded progress in the development of
stellar magnetism and the evolution of magnetized stellar interiors. To
estimate the magnetic field strengths, linear asteroseismology has stood as a
promising candidate (Aerts et al., 2010), although nonlinear asteroseismology
may also be able to provide insights. It is undisputed that the
asteroseismology, both linear and nonlinear, benefits from reliable and simple
expressions of anisotropic magnetic effect on linear waves. Specifically, the
development of nonlinear asteroseismology (Guo, 2020; Van Beeck et al., 2021,
2023) requires linear dispersion relations to evaluate mode resonances; the
linear asteroseismology directly employs dispersion relations to solve the
inverse problem of detecting subsurface magnetic fields using observed stellar
surface oscillations. Such magnetoseismology has witnessed fitful progress.
For example, observational studies report travel-time perturbations of
acoustic waves to be a critical signature of strong magnetic fields in the
stellar interior (Schunker et al., 2005; Ilonidis et al., 2011). Numerical
simulations of asteroseismic waves also suggest the possibility of detecting
deep-seated fields, before they emerge on the surface (Singh et al., 2014,
2015, 2016, 2020; Das et al., 2020; Das, 2022). To bolster such suggestive
findings, a thorough understanding of the impact of magnetic fields on
asteroseismic waves is essential (Nye & Thomas, 1976; Adam, 1977; Thomas,
1983; Campos, 1983; Cally, 2007; Campos & Marta, 2015; Tripathi & Mitra,
2022).

Waves in an unmagnetized polytropic atmosphere were exactly solved
analytically by Lamb (1911) who derived the relation

$\frac{\Omega^{2}}{2K}-\frac{(n+1)(n+1-\gamma
n)K}{2\gamma^{2}\Omega^{2}}=m+\frac{n}{2},$ (1)

where $\Omega$ is the frequency, $K$ the wavenumber, $n$ the polytropic index,
$\gamma$ the adiabatic index, and $m$ the eigen-state index, with
$m=0,1,2,...\,$. This advancement led to a series of newer and significant
understanding of hydrodynamic waves. Under fast-wave approximation,
$\Omega^{2}/K\gg 1$, the leading-order dispersion relation becomes

$\Omega^{2}\sim K\left(2m+n\right)\/.$ (2)

A similar closed-form expression for waves in a magnetized polytrope, as of
yet, is unknown. Despite several progresses (e.g., Gough & Thompson, 1990;
Bogdan & Cally, 1997; Cally & Bogdan, 1997), the problem, to this day, remains
unsolved. Gough & Thompson (1990) treat a global problem (in spherical
coordinates); there, the computation of eigenfrequenices requires numerical
evaluation of complicated integral expressions [e.g., Eqs. (4.11)–(4.13) in
Gough & Thompson (1990)]. A closed-form analytical formula is not available.
The limitations of purely numerical approach and lack of a closed-form
expression were succinctly expressed by Bogdan & Cally (1997):

> “Ideally, we would wish to proceed by writing down an equation analogous to
> Lamb’s formula for the magnetized polytrope. Unfortunately, this approach is
> not feasible and for the most part one must instead be content with a
> numerically derived visual comparison of how the allowed oscillation
> frequencies depend upon the choice of the horizontal wavenumber $k$.”

Analytic dispersion relations are also essential for developing wave
turbulence theory in the presence of both gravity and magnetic fields. In wave
turbulence theory, calculations of mode resonances require simple, explicit
dispersion relations that accurately capture the magnetic effect on observed
linear waves. We note that the Lamb’s dispersion relation (2),
$\Omega\sim\sqrt{K}$, is similar to that of surface gravity waves in oceans
(Hasselmann, 1962). However, there is a critical difference: the surface
gravity waves do not couple via three-wave resonance, thus requiring a weaker
four-wave coupling (Nazarenko & Lukaschuk, 2016). The Lamb waves, on the other
hand, can couple via three-wave resonance, because there are infinitely many
such waves (eigen states) at a given wave number, unlike only one pair of
surface gravity waves at a given wave number. Thus, the infinitely many Lamb
waves at a given wavenumber have distinct wave frequencies, which allow the
sum of three frequencies at three wavenumbers to become null. While a wave
turbulence theory for surface gravity waves has been well established, it is
yet to be developed for the asteroseismic waves, whose dispersion relation in
fully analytic form is a basic requirement for such a theory. The value that
an analytic dispersion relation offers in resonant-coupling theory cannot be
overstated when magnetic fields make the wave dispersion relation anisotropic
and complicated. Motivated by these reasons, we seek here an accurate and
simple formulae for the effect of magnetic fields on the Lamb waves.

Introducing magnetic fields, aligned orthogonal to a vertical gravity (Fig.
1), we find, as did Bogdan & Cally (1997), that the linearized
magnetohydrodynamic (MHD) equations are too cumbersome to obtain a closed-form
expression for the dispersion relation, even with the fast-wave approximation,
$\Omega^{2}/K\gg 1$. Here, we overcome this difficulty by using both numerical
simulations and extensive use of Mathematica, followed by a variant of
Jeffereys–Wentzel–Kramers–Brillouin (JWKB) approximation to deduce

$\Omega^{2}\sim
K\left(2m+n\right)\left[1+\frac{{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}(4-2\gamma+\cos^{2}\theta-3\cos^{4}\theta)}{4}\right],$
(3)

in the limit $\Omega^{2}/K\gg 1$, with $m=0,1,2,...\,$. The parameter
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ is the inverse of the Alfvénic Mach
number and $\theta$ is the angle the wave vector makes with the background
magnetic field. Equation (3) is the principal result of this paper.

This paper is organized in the following way. In § 2, we describe our model
and present the linearized compressible MHD equations. Such equations are then
numerically solved in § 3. To obtain analytical understanding of the numerical
results, the linearized equations are reduced to a wave equation in § 4. The
normal-form wave equation is then perturbatively solved using a variant of the
JWKB theory we devise; analytical understanding is gained in § 5. With
astrophysical implications and utility of our results, we conclude in § 6.

##  2 System setup and linearized perturbation equations

Figure 1: An inhomogeneous magnetic field $\bm{B}_{0}(z)$, oriented orthogonal
to a constant vertical gravity $\bm{g}$, is considered where the wave is
allowed to propagate in an arbitrary direction, shown with $\bm{K}$, making an
angle $\theta$ with $\bm{B}_{0}$. The gradient in the colormap of the box
schematically represents increasing functions with depth $z$ of the magnetic
field, fluid density, pressure, and sound speed.

To study waves in a magnetized, stratified medium, we consider the ideal
compressible MHD equations (Chandrasekhar, 1961)

$\displaystyle\partial_{t}\rho+{\bm{\nabla}}\cdot\left(\rho{\bm{U}}\right)$
$\displaystyle=0\/,$ (4a)
$\displaystyle\rho\left[\partial_{t}{\bm{U}}+({\bm{U}}\cdot\bm{\nabla}){\bm{U}}\right]$
$\displaystyle=-\bm{\nabla}P+\rho\bm{g}+\bm{J}\times\bm{B}\/,$ (4b)
$\displaystyle\partial_{t}\bm{B}$
$\displaystyle={\bm{\nabla}}\times\left({\bm{U}}\times\bm{B}\right),$ (4c)
$\displaystyle\frac{Dp}{Dt}$ $\displaystyle=c^{2}\frac{D\rho}{Dt},$ (4d)

where $\rho$, ${\bm{U}}$, $P$, and $\bm{B}$ represent the density, the
velocity, the pressure, and the magnetic field, respectively. The current
density is ${\bm{J}}={\bm{\nabla}}\times\bm{B}/\mu_{{\rm 0}}$, where
$\mu_{{\rm 0}}$ represents the magnetic permeability of the vacuum. The
magnetic field is additionally constrained to be divergence-less

${\bm{\nabla}}\cdot\bm{B}=0.$ (5)

We consider a local Cartesian domain where the equilibrium density $\rho_{\rm
0}$ and pressure $P_{\rm 0}$ satisfy the polytropic relation

$P_{\rm 0}\sim\rho_{\rm 0}^{1+1/n},$ (6)

where $n$ is the polytropic index of the gaseous atmosphere. In an
unmagnetized atmosphere, the force balance gives $\partial_{z}P_{\rm
0}(z)=\rho_{\rm 0}(z)g$, where $\bm{g}=g\hat{\bm{e}}_{z}$, with $g$ as the
constant acceleration due to gravity along the vertical $z$-axis, as depicted
in Fig. 1. The force balance requires

$\rho_{\rm 0}(z)\sim z^{n};\quad P_{\rm 0}(z)\sim z^{n+1}.$ (7)

###  2.1 Background magnetic field and sound speed

We introduce an inhomogeneous background magnetic field, $\bm{B}=B_{\rm
0}(z)\hat{\bm{e}}_{x}$. The force-balance relation with the Lorentz force,
$({\bm{\nabla}}\times\bm{B}/\mu_{{\rm 0}})\times\bm{B}$, then becomes

$\partial_{z}P_{\rm 0}=\rho_{\rm 0}g-\frac{1}{2\mu_{{\rm
0}}}\partial_{z}B_{\rm 0}^{2}.$ (8)

The solution

$B_{0}^{2}(z)\sim z^{n+1}\/$ (9)

also satisfies Eq. (7). The plasma $\beta$ is then a constant throughout the
domain

$\beta\equiv\frac{P_{\mathrm{gas}}}{P_{\mathrm{magnetic}}}=\frac{2\mu_{{\rm
0}}P_{\rm 0}}{B_{\rm 0}^{2}}.$ (10)

The sound speed ($c$) may now be deduced from Eq. (8), using $P_{\rm
0}=c^{2}\rho_{\rm 0}/\gamma$, where $\gamma$ is the adiabatic index of the gas

$\partial_{z}c^{2}=-\frac{nc^{2}}{z}+\frac{\gamma g}{1+\beta^{-1}}.$ (11)

The solution to Eq. (11) is $c^{2}\mathrm{=}c_{\rm 0}^{2}z$, where $c_{\rm
0}^{2}$ is a constant given by

$c_{\rm 0}^{2}=\frac{\gamma g}{(n+1)(1+\beta^{-1})}\/.$ (12)

It is useful to express $\beta^{-1}$ in terms of the Alfvénic Mach number
$\mbox{M}_{\rm A}$, which is the ratio of the Alfvén speed to the sound speed,

$\beta^{-1}=\frac{B_{\rm 0}^{2}}{\mu_{0}\rho_{0}}\cdot\frac{\rho_{0}}{\gamma
P_{\rm 0}}\cdot\frac{\gamma}{2}=\frac{\gamma}{2\mbox{M}_{\rm A}^{2}}\/,$ (13)

using which the sound speed becomes

$c^{2}=\frac{\gamma gz}{(n+1)(1+\gamma\mbox{M}_{\rm A}^{-2}/2)}.$ (14)

###  2.2 Linearized perturbation equations

We now linearize the MHD equations around the background profiles $[\rho_{\rm
0},\bm{0},B_{\rm 0},P_{\rm 0}]$, introduced in Eqs. (7) and (9). Such a
linearization yields evolution equations for infinitismal perturbations
$[\widetilde{\rho},\widetilde{{\bm{u}}},\widetilde{{\bm{b}}},\widetilde{p}]$
as

$\displaystyle\partial_{t}\widetilde{\rho}$
$\displaystyle=-(\widetilde{\bm{u}}\cdot\bm{\nabla})\rho_{\rm 0}-\rho_{\rm
0}(\bm{\nabla}\cdot\widetilde{\bm{u}})\/,$ (15a)
$\displaystyle\partial_{t}\widetilde{u}_{x}$
$\displaystyle=-\frac{\partial_{x}\widetilde{p}}{\rho_{0}}+\frac{\widetilde{b}_{z}\partial_{z}B_{\rm
0}}{\mu_{{\rm 0}}\rho_{\rm 0}}\/,$ (15b)
$\displaystyle\partial_{t}\widetilde{u}_{y}$
$\displaystyle=-\frac{\partial_{y}\widetilde{p}}{\rho_{0}}+\frac{B_{\rm
0}(\partial_{x}\widetilde{b}_{y}-\partial_{y}\widetilde{b}_{x})}{\mu_{{\rm
0}}\rho_{\rm 0}}\/,$ (15c) $\displaystyle\partial_{t}\widetilde{u}_{z}$
$\displaystyle=-\frac{\partial_{z}\widetilde{p}}{\rho_{\rm
0}}+\frac{\widetilde{\rho}g}{\rho_{\rm 0}}+\frac{B_{\rm
0}(\partial_{x}\widetilde{b}_{z}-\partial_{z}\widetilde{b}_{x})-\widetilde{b}_{x}\partial_{z}B_{\rm
0}}{\mu_{{\rm 0}}\rho_{\rm 0}}\/,$ (15d)
$\displaystyle\partial_{t}\widetilde{b}_{x}$ $\displaystyle=-B_{\rm
0}(\partial_{y}\widetilde{u}_{y}+\partial_{z}\widetilde{u}_{z})-\widetilde{u}_{z}\partial_{z}B_{\rm
0}\/,$ (15e) $\displaystyle\partial_{t}\widetilde{b}_{y}$
$\displaystyle=-B_{\rm 0}\partial_{x}\widetilde{u}_{y}\/,$ (15f)
$\displaystyle\partial_{t}\widetilde{b}_{z}$ $\displaystyle=-B_{\rm
0}\partial_{x}\widetilde{u}_{z}\/,$ (15g)
$\displaystyle\partial_{t}\widetilde{p}$
$\displaystyle=-(\widetilde{\bm{u}}\cdot\bm{\nabla})P_{\rm
0}-c^{2}\rho_{0}(\bm{\nabla}\cdot\widetilde{\bm{u}})\/.$ (15h)

We shall use

$\widetilde{\chi}=\bm{\nabla}\cdot\widetilde{{\bm{u}}}$ (16)

in the rest of the article, where helpful.

Equations (15a)–(15h) can be simplified to derive a set of fewer (closed)
equations

$\displaystyle\partial_{t}^{2}\widetilde{u}_{x}$
$\displaystyle=c^{2}\partial_{x}\widetilde{\chi}+g\partial_{x}\widetilde{u}_{z}\/,$
(17a) $\displaystyle\partial_{t}^{2}\widetilde{u}_{y}$
$\displaystyle=c^{2}\partial_{y}\widetilde{\chi}+g\partial_{y}\widetilde{u}_{z}+\frac{c^{2}}{\mbox{M}_{\rm
A}^{2}}\left(\partial_{xx}\widetilde{u}_{y}+\partial_{y}\widetilde{\chi}-\partial_{xy}\widetilde{u}_{x}\right)\/,$
(17b) $\displaystyle\partial_{t}^{2}\widetilde{u}_{z}$
$\displaystyle=c^{2}\partial_{z}\widetilde{\chi}-g\partial_{x}\widetilde{u}_{x}\left[1+\frac{\gamma}{\mbox{M}_{\rm
A}^{2}(1+\gamma\mbox{M}_{\rm
A}^{-2}/2)}\right]-g\partial_{y}\widetilde{u}_{y}$ $\displaystyle\hskip
5.69046pt+\frac{\gamma(1+\mbox{M}_{\rm
A}^{-2})g\widetilde{\chi}}{(1+\gamma\mbox{M}_{\rm
A}^{-2}/2)}+\frac{c^{2}}{\mbox{M}_{\rm
A}^{2}}\left(\partial_{xx}\widetilde{u}_{z}+\partial_{z}\widetilde{\chi}-\partial_{xz}\widetilde{u}_{x}\right)\/.$
(17c)

We note that the appearance of $\mbox{M}_{\rm A}^{2}$ in Eq. (17) in certain
terms may seem non-trivial at first sight; however, upon inspection, we
understand them as terms emerging from the effect of the Lorentz force on the
background states, via terms like $\partial_{z}c^{2}(z)$ and
$(\partial_{z}P_{\rm 0})/\rho_{n}$, while processing Eqs. (15a)–(15h).
Admittedly, Eqs. (17a)–(17) are somewhat challenging to proceed with clarity.
Thus, we now non-dimensionalize the equations to make them reasonably
transparent.

###  2.3 Non-dimensionalized linear equations

We define $L$ as the characteristic length scale over which the sound speed
varies (and, for that matter, pressure, density, and temperature also vary)
appreciably. Then we find that the characteristic sound speed in an
unmagnetized polytrope is, using Eq. (14), $c_{L}=\sqrt{\gamma gL/(n+1)}$.
Using $L$ and $L/c_{L}$ as the dimensional length and time units, we non-
dimensionalize all variables henceforth, starting with the sound speed

$C^{2}=\frac{c^{2}}{c_{L}^{2}}=\frac{Z}{1+\gamma\mbox{M}_{\rm A}^{-2}/2},$
(18)

where the lowercase dimensional variables ($c$ and $z$) are cast as uppercase
non-dimensional variables ($C$ and $Z$). Thus we replace all the dimensional
variables in Eqs. (17a)–(17) using

$\displaystyle(x,y,z)$ $\displaystyle=(LX,LY,LZ),$ (19a) $\displaystyle c^{2}$
$\displaystyle=c_{L}^{2}\frac{Z}{1+\gamma\mbox{M}_{\rm A}^{-2}/2},$ (19b)
$\displaystyle\partial_{t}$ $\displaystyle\equiv\frac{c_{L}}{L}\partial_{T},$
(19c)

where the uppercase characters $X,Y,$ and $T$ represent non-dimensional
variables.

We analyze perturbations by Fourier-transforming in the $(x,y)$-plane, viz.,

$\widetilde{u}_{x}(Z)=\int
dK_{X}dK_{Y}d\Omega\,\hat{u}_{x}\exp\left[i(K_{X}X+K_{Y}Y+\Omega T)\right],$
(20)

where the uppercase characters represent non-dimensional quantities, e.g.,
$\bm{K}\equiv(K_{X},K_{Y})$ is the non-dimensional wavevector in the
$(x,y)$-plane, and $\Omega$ is the nondimensional frequency. Fourier analyzing
Eqs. (17a)–(17) and representing $\mbox{M}_{\rm A}^{-1}$ by
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ henceforth, we write

$\displaystyle\left[\frac{-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{iK_{X}Z}\right]\hat{u}_{x}+\left[\frac{-(n+1)(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{\gamma
Z}\right]\hat{u}_{z}=\hat{\chi}\/,$ (21a)
$\displaystyle\left[-i{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}K_{X}\right]\hat{u}_{x}+\left[\frac{-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{iK_{Y}Z}+\frac{{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}K_{X}^{2}}{iK_{Y}}\right]\hat{u}_{y}$
$\displaystyle\hskip
56.9055pt+\left[\frac{-(n+1)(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{\gamma
Z}\right]\hat{u}_{z}=\hat{\chi}(1+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2})\/,$
(21b)
$\displaystyle\left[iK_{X}\left\\{\frac{1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2}{\gamma}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\left(1+\frac{Z\partial_{Z}}{n+1}\right)\right\\}\right]\hat{u}_{x}$
$\displaystyle+\left[\frac{iK_{Y}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{\gamma}\right]\hat{u}_{y}+\left[\frac{-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}ZK_{X}^{2}}{n+1}\right]\hat{u}_{z}$
$\displaystyle\hskip
99.58464pt=\left(\hat{\chi}+\frac{Z\partial_{Z}\hat{\chi}}{n+1}\right)(1+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2})\/.$
(21c)

##  3 Exact numerical solution

We obtain fully converged numerical solution to Eqs. (21a)–(21) by employing
the spectral code “Dedalus” (Burns et al., 2020). Referring to the Dedalus
methods paper (Burns et al., 2020) for more details, we briefly outline the
numerical procedures employed in Dedalus for eigenvalue problems. At each
horizontal wavenumber $(K_{X},K_{Y})$, the state variables—the three
components of velocity—are expanded in the Chebyshev polynomials along the
inhomogeneous $z$-axis. Because of the background inhomogeneity, different
Chebyshev coefficients couple, creating a dense linear operator.
Sparsification is provided by a change of basis from the Chebyshev polynomials
of the first kind to those of the second kind. To impose boundary conditions
and keep the matrix sparse, Dirichlet preconditioning is applied. Efficient
solution of the resulting matrices is then found by passing the matrices
$\mathcal{L}$ and $\mathcal{M}$ in the eigenvalue ($\sigma$) problem,
$\mathcal{L}\mathcal{X}=\sigma\mathcal{M}\mathcal{X}$, to the “scipy” linear
algebra packages. For a given spectral resolution along the $z$-axis, we solve
for all the eigenvalues of the matrices. Such a non-targeted, general solution
produces eigenmodes of all the linear MHD waves, including the pressure-driven
and gravity-driven modes. We specialize in the high-frequency modes to assess
the effect of magnetic fields on such pressure-driven modes. We identify the
pressure-driven modes by comparing their eigenfrequencies with those predicted
by Lamb’s solution.

Figure 2: $\Omega^{2}(K_{X},K_{Y})$ for hydrodynamic and magnetized
polytropes. The two plotted surfaces are visually indifferentiable because the
difference ($\Delta\Omega^{2}$) between them is much smaller than
$\Omega^{2}$; see Fig. 3. The parameters used are $\epsilon=0.1$, $m=20$,
$n=2.5$, and $\gamma=5/3$. For variations in these parameters, the surface
plot of $\Omega^{2}(K_{X},K_{Y})$ remains the same qualitatively.

For the boundary condition, at the lower boundary $z=L_{z}$, we require
$\widetilde{u}_{z}=0$. At the upper boundary, $z=0$, where the atmosphere
ceases, we enforce, following Lamb (1911),

$\displaystyle\frac{Dp}{Dt}$ $\displaystyle=0,$ (22)

which implies

$c^{2}\frac{D\rho}{Dt}=-c^{2}\rho_{\rm
0}\widetilde{\chi}=0=z^{n+1}\widetilde{\chi}.$ (23)

We first validate that the solver successfully reproduces Lamb’s exact
analytical dispersion relation in the absence of the magnetic field. We then
compute the eigenvalues of the magnetized polytrope, which we present in Fig.
2. Although the two surface plots of $\Omega^{2}(K_{X},K_{Y})$ are displayed
in Fig. 2—one for the hydrodynamic and other for the magnetized polytrope, the
plots are visually indistinguishable. The difference between the two surface
plots is shown in Fig. 3. For $K_{Y}=0$, $\Delta\Omega^{2}$ is negative, and
for $K_{X}=0$, $\Delta\Omega^{2}$ is positive and relatively large. We also
note a minor decrease in positive value of $\Delta\Omega^{2}$ in going from
$K_{X}\approx 0$ to $K_{X}=0$.

It turns out that $\Delta\Omega^{2}$ is related to the hydrodynamic squared-
frequency $\Omega_{\mathrm{hydro}}^{2}$; and
$\Delta\Omega^{2}/\Omega_{\mathrm{hydro}}^{2}$ is almost entirely independent
of the wavevector magnitude (Fig. 4). Only angular dependence is observed.

The extremely low wavenumbers in Fig. 4 correspond to very large scale waves
that cannot be captured in a finite box in numerical calculation. To capture
lower wavenumbers, we significantly extended domain size along the vertical
$z$-axis, which allowed us to obtain fully converged numerical results for
other wavenumbers.

Figure 3: $\Delta\Omega^{2}=\Omega^{2}-\Omega_{\mathrm{hydro}}^{2}$ is plotted
for a magnetized polytrope, with $\epsilon=0.1$, $m=20$, $n=2.5$, and
$\gamma=5/3$. This surface plot shows the difference between the two surface
plots in Fig. 2. Figure 4: Relative difference of squared-frequency depends on
the wavevector propagation angle (spanned by the black circle), but the
relative difference is insensitive to the wavevector magnitude. The parameters
chosen are $\epsilon=0.1$, $m=20$, $n=2.5$, and $\gamma=5/3$.

##  4 Reduction to wave equation

To analytically determine the dispersion relation, we solve the set of
equations (21a)–(21) perturbatively in the limit of a weak magnetic field,
i.e., ${\mathbb{\color[rgb]{0,0,0}\epsilon}}\ll 1$. By setting
${\mathbb{\color[rgb]{0,0,0}\epsilon}}=0$, we recover the Lamb’s equations for
the unmagnetized polytrope (Lamb, 1911). The Lamb’s equations reduce to a
second-order differential equation for $\hat{\chi}$. With the same goal, we
proceed in the following manner. We rewrite Eqs. (21a)–(21) as

$\begin{bmatrix}\begin{array}[]{c c c}M_{11}&M_{12}&M_{13}\\\
M_{21}&M_{22}&M_{23}\\\ M_{31}&M_{32}&M_{33}\\\
\end{array}\end{bmatrix}\begin{bmatrix}\begin{array}[]{c}\hat{u}_{x}\\\
\hat{u}_{y}\\\ \hat{u}_{z}\\\
\end{array}\end{bmatrix}=\begin{bmatrix}\begin{array}[]{c}h_{x}(\hat{\chi})\\\
h_{y}(\hat{\chi})\\\ h_{z}(\hat{\chi},\partial_{Z}\hat{\chi})\\\
\end{array}\end{bmatrix},\\\ $ (24)

where the matrix elements $M_{ij}$ are independent of $Z$-derivatives, and are
functions of
$K_{X},K_{Y},\Omega,n,\gamma,{\mathbb{\color[rgb]{0,0,0}\epsilon}},$ and $Z$
only (for their complete expressions, see Appendix A). In arriving at Eq.
(24), we have replaced $\partial_{Z}\hat{u}_{x}$ in the first term on the
left-hand side of Eq. (21) with its exact expression obtained by
differentiating Eq. (21a) with respect to $Z$. We then substitute
$\partial_{Z}$ operation in $\partial_{Z}\hat{u}_{z}$ by writing it as
$\hat{\chi}-iK_{X}\hat{u}_{x}-iK_{Y}\hat{u}_{y}$. Such a process removes
$\partial_{Z}$ operation from the matrix $M$. The functions $h_{\nu}$ are
linear in $\hat{\chi}$ and $\partial_{Z}\hat{\chi}$.

Straightforward inversion of the matrix $M$ expresses all components of the
velocity in terms of $\hat{\chi}$ and $\partial_{Z}\hat{\chi}$:

$\hat{u}_{\nu}=f_{\nu}(Z)\hat{\chi}+g_{\nu}(Z)\partial_{Z}\hat{\chi},$ (25)

where $\nu$ can be either $x,y,$ or $z$. We recognize that $f_{\nu}$ and
$g_{\nu}$ are functions of $Z$, but do not involve $\partial_{Z}$. The three
velocity components of Eq. (25) can now be subsumed into a single second-order
differential equation for $\hat{\chi}$:

$\partial_{Z}^{2}\hat{\chi}+P(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})\partial_{Z}\hat{\chi}+R(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})\hat{\chi}=0,$
(26)

which can be recast into the normal form of the second-order differential
equation by changing variable as

$\hat{\chi}(Z)=\hat{\psi}(Z)\exp{\left[-\frac{1}{2}\int_{Z}dZ\,P(Z)\right]},$
(27)

which reduces Eq. (26) to the wave equation

$\partial_{Z}^{2}\hat{\psi}+\Gamma^{2}(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})\hat{\psi}=0.$
(28)

The procedure outlined above appears straightforward. However, the analytical
manipulations in arriving at Eq. (28)—a magnetized version of Lamb’s
equation—require laborious and careful calculations, as
$\Gamma^{2}(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})$ alone conceals an
expression of exhaustive length—tens of pages of this article. In the absence
of the magnetic field, the expression for
$\Gamma^{2}(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}}=0)=\Gamma_{\rm 0}^{2}(Z)$
is beautifully short, $\Gamma_{\rm 0}(Z)=K^{2}(Z-\alpha)(\beta-Z)/Z^{2}$,
where $\alpha$ and $\beta$ are the two turning points—two zeros of
$\Gamma_{\rm 0}(Z)$.

##  5 Perturbative solution for anisotropic magnetic effect

The presence of a weak magnetic field may be considered as a perturbation to
Lamb’s two-turning point eigenvalue problem. Hence, the magnetic field changes
both the locations of the turning points and the form of the potential
$\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})$. In general, we write the
JWKB quantization condition (see e.g., Bender & Orszag, 1978; Tripathi, 2022)
as

$\frac{1}{\pi}\int_{Z_{1}({\mathbb{\color[rgb]{0,0,0}\epsilon}})}^{Z_{2}({\mathbb{\color[rgb]{0,0,0}\epsilon}})}\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})\,dZ\sim\left(m+\frac{1}{2}\right);\,\,\,m=0,1,2,...,$
(29)

where $m$ refers to the eigen-state index,
$\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})$ is the magnetically-modified
wavenumber, and $Z_{1}({\mathbb{\color[rgb]{0,0,0}\epsilon}})$ and
$Z_{2}({\mathbb{\color[rgb]{0,0,0}\epsilon}})$ are the magnetically-shifted
turning points.

###  5.1 Perturbative calculations

First, we expand the wavenumber
$\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})$ as

$\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})=\Gamma_{\rm
0}(Z)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}\Gamma_{1}(Z)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\Gamma_{2}(Z)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3}\Gamma_{3}(Z)+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{4})}.$
(30)

It may be noted that the leading-order effect of the Lorentz force on the
wavenumber appears only at the second order
$\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2})}$ in the expansion.
So, $\Gamma_{1}(Z)$ above is zero. The expressions for $\Gamma_{0}(Z)$ and
$\Gamma_{2}(Z)$ are

$\displaystyle\Gamma_{0}(Z)$
$\displaystyle=\frac{iK\sqrt{(Z-\alpha)(Z-\beta)}}{Z},$ (31a)
$\displaystyle\Gamma_{2}(Z)$
$\displaystyle=\frac{i(b_{2}Z^{2}+b_{1}Z+b_{0})}{\sqrt{(Z-\alpha)(Z-\beta)}},$
(31b)

where we have used $K_{X}=K\cos\theta$ and $K_{Y}=K\sin\theta$. The parameters
$\alpha$ and $\beta$ satisfy the following properties

$\displaystyle\alpha\beta$ $\displaystyle=\frac{n(n+2)}{4K^{2}},$ (32a)
$\displaystyle\alpha+\beta$
$\displaystyle=\left[\frac{\Omega^{2}}{K}-\frac{(n+1)(n+1-n\gamma)}{\gamma^{2}}\frac{K}{\Omega^{2}}\right]\frac{1}{K}.$
(32b)

The lengthy expressions of $b_{0}$, $b_{1}$, and $b_{2}$ are presented in
Appendix B. We note that these parameters depend on $\Omega,K,\theta,n,$ and
$\gamma$ only.

Now we expand the turning points
$Z_{1}({\mathbb{\color[rgb]{0,0,0}\epsilon}})$ and
$Z_{2}({\mathbb{\color[rgb]{0,0,0}\epsilon}})$ around the turning points of
the unmagnetized polytrope, $Z_{1}^{(0)}=\alpha$ and $Z_{2}^{(0)}=\beta$, as

$Z_{j}({\mathbb{\color[rgb]{0,0,0}\epsilon}})=Z_{j}^{(0)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}Z_{j}^{(1)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{j}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})},$
(33)

where $j=1$ and $j=2$ refer to the left and the right turning points,
respectively (i.e., $\alpha<\beta$). We note that, in Eq. (33), the correction
term at the first order in ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ is zero,
i.e., $Z_{j}^{(1)}=0$. We find this result by substituting the expression for
$Z_{j}({\mathbb{\color[rgb]{0,0,0}\epsilon}})$ from Eq. (33) in
$\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})=0$, and by employing Eq.
(30). Solving the resulting equation order-by-order in
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ produces $Z_{j}^{(1)}=0$. We note,
however, that, at the second order in ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$
(which is where the effect of the Lorentz force comes in action),
$Z_{j}^{(2)}$ becomes non-zero. The expressions for $Z_{j}^{(2)}$ are given in
Appendix C.

Because the term $Z_{j}^{(2)}$ appears at the second order in
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$, it may be tempting to assume that the
term contributes to second order in ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$
itself in the JWKB integral in Eq. (29). This, however, is not the case. The
term contributes to third and higher orders in
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ as we show next. Expanding Eq. (29),

$\displaystyle\frac{1}{\pi}\int_{Z_{1}^{(0)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{1}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}^{Z_{2}^{(0)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{2}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}$
$\displaystyle\left[\Gamma_{0}(Z)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\Gamma_{2}(Z)+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}\right]\,dZ$
(34) $\displaystyle\sim\left(m+\frac{1}{2}\right);\,\,\,m=0,1,2,...\/.$

We now integrate $\Gamma_{\rm 0}(Z)$ as

$\displaystyle\int_{Z_{1}^{(0)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{1}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}^{Z_{2}^{(0)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{2}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}\Gamma_{\rm
0}(Z)\,dZ$ (35)
$\displaystyle=\left(\int_{\alpha+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{1}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}^{\alpha}+\int_{\alpha}^{\beta}+\int_{\beta}^{\beta+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{2}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}\right)\Gamma_{\rm
0}(Z)\,dZ$
$\displaystyle=-\frac{2{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3}K(\beta-\alpha)^{1/2}\left[Z_{1}^{(2)}\right]^{3/2}}{3\alpha}+\int_{\alpha}^{\beta}\Gamma_{\rm
0}(Z)\,dZ$ $\displaystyle\hskip
56.9055pt+\frac{2{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3}K(\beta-\alpha)^{1/2}\left[-Z_{2}^{(2)}\right]^{3/2}}{3\beta}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{4})}\/,$

where we notice terms with ${\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3}$ arising
from the second-order shifts in the turning points,
${\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{1}^{(2)}$ and
${\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{2}^{(2)}$. The additional power
of ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ emerges from the integrand
$\Gamma_{\rm 0}(Z)$, which has a term $\sqrt{Z-Z_{j}^{(0)}}$ that when
expanded around $Z_{j}^{(0)}$ in powers of
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ contributes an
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ to the integral.

In Eq. (35) the term $\int_{\alpha}^{\beta}\Gamma_{0}(Z)\,dZ$ is the integral
that one finds in Lamb’s calculations:

$\displaystyle\int_{\alpha}^{\beta}\Gamma_{0}(Z)\,dZ\equiv$ $\displaystyle
I_{\mathrm{Lamb}}=\frac{(n+1)}{2}\left[\frac{\Omega^{2}}{K(n+1)}+\frac{(\gamma
n-n-1)K}{\gamma^{2}\Omega^{2}}-1\right]+1.$

Thus, we replace the integral $\int_{\alpha}^{\beta}\Gamma_{0}(Z)\,dZ$,
appearing in Eq. (35), with $I_{\mathrm{Lamb}}$ from Eq. (5.1) to obtain

$\displaystyle I_{\mathrm{Lamb}}$
$\displaystyle+\frac{{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}}{\pi}\int_{\alpha}^{\beta}\Gamma_{2}(Z)dZ+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}$
(36) $\displaystyle\hskip
28.45274pt\sim\left(m+\frac{1}{2}\right);\,\,\,m=0,1,2,...,$

which is accurate up to second order in
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$. We substitute $\Gamma_{2}(Z)$ from
Eq. (31b) and perform the integral in Eq. (36) to arrive at

$\displaystyle I_{\mathrm{Lamb}}-{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}$
$\displaystyle\left[b_{0}+b_{1}\left(\frac{\alpha+\beta}{2}\right)+b_{2}\left(\frac{3\alpha^{2}+2\alpha\beta+3\beta^{2}}{8}\right)\right]$
(37) $\displaystyle\hskip
28.45274pt\sim\left(m+\frac{1}{2}\right);\,\,\,m=0,1,2,...\,.$

Employing fast-wave approximation, we expand each term on the left-hand side
of Eq. (37) in powers of $\delta=K/\Omega^{2}$ as

$\displaystyle I_{\mathrm{Lamb}}$
$\displaystyle\approx\frac{1}{\delta}\left[\frac{1}{2}+\frac{\delta(1-n)}{2}+\mathcal{O}(\delta^{2})\right],$
(38a) $\displaystyle b_{0}$
$\displaystyle\approx\frac{1}{\delta}\left[\frac{2-\gamma}{4}+\mathcal{O}(\delta^{2})\right],$
(38b) $\displaystyle b_{1}\left(\frac{\alpha+\beta}{2}\right)$
$\displaystyle\approx\frac{1}{\delta}\left[\frac{\cos^{2}\theta}{4}+\mathcal{O}(\delta^{2})\right],$
(38c) $\displaystyle
b_{2}\left(\frac{3\alpha^{2}+2\alpha\beta+3\beta^{2}}{8}\right)$
$\displaystyle\approx\frac{1}{\delta}\left[\frac{-3\cos^{4}\theta}{8}+\mathcal{O}(\delta^{2})\right].$
(38d)

Thus we obtain a simplified dispersion relation from Eq. (37):

$\displaystyle\frac{\Omega^{2}}{K}\left[\frac{1}{2}-\frac{{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}}{4}\left\\{2-\gamma+\cos^{2}\theta-\frac{3\cos^{4}\theta}{2}\right\\}\right]$
(39) $\displaystyle\hskip
56.9055pt\sim\left(m+\frac{n}{2}\right);\,\,\,m=0,1,2,...\,.$

It turns out that Eq. (39) is not in excellent agreement with our numerical
results, but replacing $\cos^{2}\theta$ with $(\cos^{2}\theta)/2$ gives
excellent agreement.

Informed in this way, we write the final dispersion relation

$\boxed{\begin{aligned}
&\frac{\Omega^{2}}{K}\left[\frac{1}{2}-\frac{{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}}{4}\left\\{2-\gamma+\frac{\cos^{2}\theta}{2}-\frac{3\cos^{4}\theta}{2}\right\\}\right]\\\
&\hskip
56.9055pt\sim\left(m+\frac{n}{2}\right);\,\,\,m=0,1,2,...\,.\end{aligned}}$
(40)

###  5.2 Comparison between theory and numerics

Figure 5: Theory and numerics: Our theory predicts that
$g(\theta)=\Delta\Omega^{2}/({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\Omega_{\rm
hydro}^{2})+\gamma/2$ depends only on $\theta$ as given in Eq. (41). The
function $g(\theta)$ is independent of the wavenumber $K$, Alfvénic Mach
number $\epsilon^{-1}$, the polytropic index $n$, the adiabatic index
$\gamma$, and the eigen-state index $m$. Here, we plot $g(\theta)$ obtained
from our numerical solutions for several different values of $m$, ranging from
$10$ to $35$, and for several different values of $K$ ($0.6\leq|K|\leq 2$),
and for two values of ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ and two values
of $\gamma$ (different symbols). All data points collapse on the same
universal curve. The function $g(\theta)$ from our asymptotic theory, Eq.
(41), is also plotted here. The asymptotic curve is indistinguishable from the
exact numerical solutions.

Using the Lamb’s relation, $\Omega_{\rm hydro}^{2}\sim K(2m+n)$, we rewrite
Eq. (40) as

$\displaystyle g(\theta)$
$\displaystyle\equiv\gamma/2+\Delta\Omega^{2}/({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\Omega_{\rm
hydro}^{2})$ (41)
$\displaystyle=1+\frac{1}{4}\left(\cos^{2}\theta-3\cos^{4}\theta\right),$

where $\Delta\Omega^{2}\equiv\Omega^{2}-\Omega_{\rm hydro}^{2}$. This is a
remarkable result. The right-hand-side of Eq. (41) is independent of every
other possible parameter other than $\theta$ ! In Fig. 5, we plot the function
$g(\theta)$ from our numerical determination of the eigenfrequencies for
several different values of $m$ and $K$, and two different values of
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$, and two different values of $\gamma$.
These numerical values are shown as different symbols. All the different
curves fall on top of each other, creating a universal curve. We also plot our
asymptotic expression, Eq. (40), which agrees very well with the numerical
results. This demonstrates that our theory is in excellent agreement with
numerics even for ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ as large as
${\mathbb{\color[rgb]{0,0,0}\epsilon}}=0.1$.

It is surprising that, in our attempt to capture the anisotropy brought in by
the magnetic field, despite a myriad of unwieldy expressions encountered on
the way, an expression as simple as Eq. (40), is obtained. This simple
expression is also highly accurate, as demonstrated in Fig. 5. The success of
our asymptotic theory is somewhat unexpected, given that the analytical
solution is accurate only up to the leading order
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ and $K/\Omega^{2}$.

More accurate solutions can be obtained by using the full expressions of
$b_{0},b_{1},$ and $b_{2}$ from Appendix B in Eqs. (38d)–(38b), and keeping
higher-order terms in ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ in Eq. (34).

Finally, a remark on the effect of magnetic fields on the slow gravity-driven
waves is in order. For the slow waves, the first term on the left-hand side of
Eq. (1) may be neglected, which implies that such waves in unmagnetized medium
become unstable when $\gamma<(1+1/n)$. This is consistent with the energy
principle of Newcomb (1961). When a magnetic field is present, by applying the
energy principle of Newcomb, we find, for $K_{X}\neq 0$,111For $K_{X}=0$, the
criterion for the gravity waves to become unstable is slightly modified:
$\gamma<(1+1/n)(1+1/\beta)-2/\beta$. the instability threshold on $\gamma$ for
the gravity waves is lifted to $\gamma<(1+1/n)(1+1/\beta)$. When the magnetic
field is very strong ($\beta\ll 1$), the regular perturbation series in powers
of ${\mathbb{\color[rgb]{0,0,0}\epsilon}}\,(\propto 1/\sqrt{\beta})$ is
possibly inadequate for unstable gravity-driven waves. Such considerations are
clearly beyond the scope of the present paper.

##  6 Conclusions

Here we derive, for pressure-driven waves, an accurate and simple analytical
formula that captures the effects of magnetic field and five other parameters:
adiabatic index, polytropic index, eigenmode state index, wavenumbers, and the
angle between the wavevector and the magnetic field
[$\theta\mathrm{=}\cos^{-1}(\hat{\mathbf{K}}\cdot\hat{\bm{B}}_{0})$]. Such a
six-parameter-dependent formula is distilled using a perturbative solution to
magnetized version of Lamb’s hydrodynamic polytropic waves. Our explicit
analytical formula overcomes the limitation of previously attempted formulae
for the magnetized polytrope that were presented in general integral forms;
such a formulation, for instance, that of Gough & Thompson [1990; e.g., Eq.
(4.11)] and Bogdan & Cally (1997), requires numerical evaluation of the
eigenfrequencies, and thus leaves out the critical step of obtaining an
analytical understanding and expression. We achieve such here guided by our
numerical solutions and at the cost of extensive use of Mathematica for our
perturbative analyses.

We emphasize that the closed-form expression we obtain is not solely by use of
perturbative calculation. At the last step, we needed guidance from our
numerical solutions. A possible explanation for inadequacy of the perturbative
calculation alone can be the ordering scheme in the perturbation series, which
in this paper has two small parameters—one that corresponds to the WKB short-
wavelength limit approximation and the other that is the actual small
parameter, inverse of the Alfvénic Mach number. This implies that future work
should develop better perturbation theory to take this inadequacy into account
and to generalize our methodology to other complex wave problems.

The simplicity and accuracy of our formula are encouraging to employ the
formula to help solve the inverse problem of magnetoseismology. Our formula
provides an explicit, analytical dependence of the observed surface
oscillation frequency with the orientation and strength of the deep-seated and
subsurface magnetic field. Such an information can be crucial to predict the
surface-emergence location and strength of sunspots. Precursors for such may
be detected by analyzing the anisotropic magnetic effect in stellar ring
diagrams—the rings of constant frequencies over the wavenumber plane.

Nonlinear asteroseismology can also directly benefit from our analytical work,
as weak turbulence theory of asteroseismic waves inevitably requires accurate
and simple expressions of linear wave frequencies in resonant triad
interactions. The effect of the magnetic field on such is completely unknown.
However, observations now exist that suggest resonant mode interactions are
possible and can be a critical element of strongly pulsating stars (Guo,
2020). Nonlinear mode coupling of linear eigenmodes (Tripathi et al., 2023a,
b) may also need to be analyzed, in addition to mode resonances. Future
planned research will directly take advantage of the formula derived here to
assess the role of the magnetic field and other parameters in asteroseismic
wave turbulence, which is awaiting to soon enter adulthood from its infancy.

## Acknowledgments

We are pleased to thank Professor Ellen G. Zweibel for helpful discussions and
for her contribution in applying the energy principle of Newcomb (1961) to our
system. The inception of this paper took place at Nordic Institute for
Theoretical Physics (NORDITA), Sweden, while B.T. was visiting on a research
fellowship generously made available by NORDITA.

## Data Availability

Mathematica notebook—built to perform lengthy analytical manipulations—and
Dedalus script—used for numerical solution in this paper—are available at
https://github.com/BindeshTripathi/polytrope.

## Appendix A

The matrix elements $M_{ij}$ and $h_{\mu}$ that appear in Eq. (24) are

$\displaystyle M_{11}$
$\displaystyle=-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2),$
(A1a) $\displaystyle M_{12}$ $\displaystyle=0,$ (A1b) $\displaystyle M_{13}$
$\displaystyle=\frac{-iK_{X}(n+1)(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{\gamma},$
(A1c) $\displaystyle M_{21}$
$\displaystyle={\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}K_{X}K_{Y}Z,$ (A1d)
$\displaystyle M_{22}$
$\displaystyle=-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}K_{X}^{2}Z,$
(A1e) $\displaystyle M_{23}$
$\displaystyle=\frac{-iK_{Y}(n+1)(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{\gamma},$
(A1f) $\displaystyle M_{31}$
$\displaystyle=\frac{iK_{X}}{\gamma}\left[1+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\left(\frac{3\gamma}{2}+\frac{K_{X}^{2}Z}{\Omega^{2}}\right)\right],$
(A1g) $\displaystyle M_{32}$
$\displaystyle=\frac{iK_{Y}}{\gamma}\left[1+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\left(\frac{\gamma}{2}+\frac{K_{X}^{2}Z}{\Omega^{2}}\right)\right],$
(A1h) $\displaystyle M_{33}$
$\displaystyle=\frac{-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}K_{X}^{2}Z}{n+1},$
(A1i) $\displaystyle h_{x}$ $\displaystyle=iK_{X}Z\hat{\chi},$ (A1j)
$\displaystyle h_{y}$
$\displaystyle=iK_{Y}Z\hat{\chi}(1+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}),$
(A1k) $\displaystyle h_{z}$
$\displaystyle=\hat{\chi}+\frac{Z\partial_{Z}\hat{\chi}}{n+1}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\hat{\chi}\left[1+\frac{K_{X}^{2}Z\left\\{\gamma-(n+1)(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)\right\\}}{(n+1)\gamma\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}\right]+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\frac{Z\partial_{Z}\hat{\chi}}{n+1}\left[1+\frac{K_{X}^{2}Z}{\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}\right].$
(A1l)

## Appendix B

The parameters introduced in Eq. (31b), while writing the expression for
$\Gamma_{2}(Z)$, appear below:

$\displaystyle b_{2}$ $\displaystyle=\frac{-K^{3}\cos^{4}\theta}{\Omega^{2}},$
(A2a) $\displaystyle b_{1}$
$\displaystyle=\frac{-K\left[-\Omega^{8}\gamma^{4}+\Omega^{4}\gamma^{2}K^{2}(n+1)\left\\{(n+1+\gamma-\gamma
n)\cos(2\theta)-n-1-3\gamma\right\\}+2K^{4}(n+1)^{4}-K^{4}(n+1)^{3}(n+1-\gamma
n)\cos(2\theta)\right]}{2\Omega^{4}\gamma^{2}\mathrm{sec}^{2}\theta\,\left[\Omega^{4}\gamma^{2}-K^{2}(n+1)^{2}\right]},$
(A2b) $\displaystyle b_{0}$
$\displaystyle=\frac{\Bigg{(}\begin{gathered}4\Omega^{8}\gamma^{4}(2-\gamma)-2\Omega^{4}\gamma^{2}K^{2}\left\\{2(n+1)^{2}(4-3\gamma)+n(2n+3)\gamma^{2}\right\\}+K^{4}(n+1)^{2}\left\\{n\gamma^{2}(2n+3)-8(n+1)^{2}(\gamma-1)\right\\}\\\
+2K^{2}\gamma\cos(2\theta)\left[K^{2}(n+1)^{2}\left\\{n(\gamma-4+2n(\gamma-1))-2\right\\}+\Omega^{4}\gamma^{2}\left\\{2(n+1)^{2}-\gamma
n(2n+3)\right\\}\right]+n\gamma^{2}K^{4}(n+1)^{2}\cos(4\theta)\end{gathered}\Bigg{)}}{16\Omega^{2}\gamma^{2}K\left[\Omega^{4}\gamma^{2}-K^{2}(n+1)^{2}\right]}$
(A2e)

## Appendix C

Due to the magnetic field, the locations of the turning points, $Z_{1}$ and
$Z_{2}$, shift—which to the second order in
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ in Eq. (33) are given by $Z_{1}^{(2)}$
and $Z_{2}^{(2)}$:

$\displaystyle Z_{1}^{(2)}$
$\displaystyle=\frac{\alpha(b_{2}\alpha^{2}+b_{1}\alpha+b_{0})}{K(\beta-\alpha)},$
(A3a) $\displaystyle Z_{2}^{(2)}$
$\displaystyle=\frac{-\beta(b_{2}\beta^{2}+b_{1}\beta+b_{0})}{K(\beta-\alpha)}.$
(A3b)

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